The Spacetime Resonance Theorem and the Origin of Dark Mass
Abstract
This work applies a well-known idea from acoustics and optics—multiple scattering in a medium—to gravity itself. In linearised general relativity the gravitational field is determined by a retarded Green function, but in most applications its “causal tail’’ (the part inside the light cone) is discarded. When this tail is kept and combined with the linear damped-oscillator susceptibility that arises from the kinematic density–sloshing of baryonic matter (mathematically identical to the response used in acoustical and optical scattering theory), spacetime behaves as an effective transport medium for weak curvature waves. From this simple starting point we derive a single theorem—the Spacetime Resonance Theorem—showing how coherent, time-varying baryonic motion generates a small resonant curvature field. Multiple gravitational scattering causes part of this oscillatory field to accumulate as a slow, cycle-averaged component. This stored curvature energy behaves exactly like an effective dark-mass density. The resulting framework naturally reproduces the observed dark-mass profiles of spiral galaxies, ellipticals, clusters, and filaments; predicts “local silence’’ and “homogeneous silence’’ (matching the Bullet Cluster and the CMB); and remains fully consistent with standard GR. No new particles or modifications of gravity are introduced: dark-mass behaviour emerges from the resonant, multiple-scattering response of baryonic structure within ordinary weak-field GR.
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The Spacetime Resonance Theorem and the Origin of Dark Mass Bryan Hennelly ∗1,2,3 1Department of Electronic Engineering, Maynooth University, Maynooth, Co. Kildare, Ireland 2Department of Computer Science, Maynooth University, Maynooth, Co. Kildare, Ireland 3The Hamilton Institute, Maynooth University, Maynooth, Co. Kildare, Ireland 19 November 2025 Abstract In linearised general relativity the gravitational field is determined by a retarded Green function, but in most applications its “causal tail” is discarded. When this tail is kept and combined with the linear dampedoscillator susceptibility that arises from the kinematic density–sloshing of baryonic matter (mathematically identical to the response used in acoustical and optical scattering theory), this yields an effective–medium description of the weak–field gravitational response. Coherent, time–varying kinematic sloshing drives a collective curvature field whose delayed rescattering produces a small but persistent cycle–averaged component. This stored curvature energy appears as an effective dark–mass density and reproduces the observed dark–mass phenomenology. No new fields, particles, or modifications of GR are introduced: all dynamics arise from standard linearised gravity once the usual quasistatic truncation of the retarded kernel is avoided. The resulting causal–resonant framework reproduces the effective mass distributions of spiral galaxies, ellipticals, clusters, and filaments; yields natural “local” and “homogeneous” silence; and remains fully consistent with Solar–System tests and with the CMB. In this picture, dark–mass behaviour emerges as curvature energy stored by the collective resonant response of baryonic structure. ∗Email: [email protected] 1
Contents 1 Introduction 5 2 Spacetime Resonance Theorem 6 2.1 The Fundamental Oscillator–Field Principle . . . . . . . . . . . . 6 2.2 The Spacetime Resonance Theorem . . . . . . . . . . . . . . . . . 9 2.3 Corollary: Density–Controlled Restoring Frequency . . . . . . . . 12 2.3.1 Spectral origin of the density scaling . . . . . . . . . . . . 12 2.3.2 Agreement with known weak-field gravitational oscillators 13 2.4 Corollary: Density–Controlled Damping Rate . . . . . . . . . . . 13 2.5 Physical Interpretation and Preview . . . . . . . . . . . . . . . . . 14 3 Dark Mass 16 4 Resonant Field Dynamics 17 4.1 ChannelSeparation.......................... 17 4.2 Spacetime as an Effective Reversible Medium . . . . . . . . . . . 18 4.3 Collective Mode Dynamics and the Masson . . . . . . . . . . . . . 19 4.4 Linear Spatial Perturbations and the Propagation Law . . . . . . 21 5 Gravitational Scattering Theory 22 5.1 Single–Scatterer Response . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Multiple Scattering and the Foldy–Lax Hierarchy . . . . . . . . . 24 5.3 Effective–Medium and Ensemble Averaging . . . . . . . . . . . . . 26 5.4 Transport and Diffusion Limits . . . . . . . . . . . . . . . . . . . 27 5.5 Unified Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.6 Polarisation as Source–Projected Effective Susceptibility . . . . . 29 6 Spatiotemporal Feedback and Resonant Coherence 31 6.1 Constituent Entities & States of the Feedback System . . . . . . . 31 6.2 Feedback Channels and Interactions . . . . . . . . . . . . . . . . . 32 7 Temporal Coherence of the Resonant Channel 35 7.1 Phase Locking and Mutual Susceptibility . . . . . . . . . . . . . . 35 7.2 Global Frequency Selection in Inhomogeneous Baryonic Systems . 37 7.3 Polarisation of Temporal Coherence . . . . . . . . . . . . . . . . . 38 8 Spatial Coherence of the Resonant Channel 40 8.1 Kirchhoff–Sommerfeld Representation . . . . . . . . . . . . . . . . 42 8.2 Effective Potential and Energy Storage in the Resonant Medium . 44 8.3 Single Point Source in a Resonant Channel–1 Background . . . . 45 8.4 Interference of Two Point Sources . . . . . . . . . . . . . . . . . . 46 8.5 Temporal Memory and the Steady–State Reduction . . . . . . . . 51 2
9 Steady–State (Attractor) Solutions 52 9.1 Spiral Galaxies as Coherent m= 1 (Laguerre–Gaussian) Modes . . . . . . . . . . . . . . . . . . . . . 54 9.1.1 Observing the Steady–State Solution . . . . . . . . . . . . 54 9.1.2 Energy Source: Epicyclic Sloshing with Coherent Filtering 56 9.1.3 The First Moment . . . . . . . . . . . . . . . . . . . . . . 58 9.1.4 The Thin–Disk Model: Solving the 2D Helmholtz Equation RevealstheHalo ....................... 60 9.1.5 Extending the Thin–Disk to Three Dimensions . . . . . . . 62 9.1.6 Gain from Incoherent Foldy–Lax Scattering . . . . . . . . 63 9.1.7 Stored Energy, Dark–Mass Equivalence, and EnvironmentalGain ............................ 64 9.1.8 Numerical Results: 3D Thin–Disk Model . . . . . . . . . . 66 9.1.9 Physical Interpretation . . . . . . . . . . . . . . . . . . . . 69 9.2 Ellipticals as Random Resonant Media . . . . . . . . . . . . . . . 73 9.2.1 Observing the Statistical Steady State . . . . . . . . . . . 73 9.2.2 Complex Baryonic Forcing with Memory . . . . . . . . . . 75 9.2.3 Four–dimensional speckle geometry of the gravitational field 78 9.2.4 Energy Source: r–Mode Speckle Driving and Spectral Overlap 79 9.2.5 Ensemble reduction: masson–speckle summation . . . . . . 80 9.2.6 Transport Mean Free Path and Collective Damping . . . . 83 9.2.7 Core Regularization in the Non–Transport Regime . . . . 85 9.2.8 Gradient–Energy Boost and Final Transport–Regularized HaloLaw ........................... 86 9.2.9 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 88 9.2.10 Halo Formation from Coherent Energy Flux . . . . . . . . 88 9.2.11 Physical Interpretation . . . . . . . . . . . . . . . . . . . . 91 9.3 Clusters as Radiative Transport Extensions of Ellipticals . . . . . 95 9.3.1 Observing the Statistical Transport Equilibrium . . . . . . 95 9.3.2 Mathematical Formulation: Overlapping Elliptical TransportEnvelopes ........................ 98 9.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 103 9.3.4 Physical Interpretation . . . . . . . . . . . . . . . . . . . . 106 10 Local Silence and the Preservation of Classical Gravity 108 11 Prediction: Absence of Gain in Homogeneous Systems 110 11.1 Three Distinct Regimes for Homogeneous Clouds . . . . . . . . . 110 11.2 Theorem: Homogeneous Media Produce No Resonant Gain . . . . 111 11.3 Interpretation and Connection to Perturbation Theory . . . . . . 112 12 Emergence of Multiple Scattering from a Weakly Inhomogeneous Continuous Medium 113 12.1 A Brief Review of Garvitational Multiple Scatter Theory . . . . . 113 12.2 The Resonant Frequency in a Continuous Medium . . . . . . . . . 115 3
12.3 Weak Random Potential and the Ballistic Regime . . . . . . . . . 115 12.4 Nonlinear Growth Toward the Transport Threshold . . . . . . . . 116 12.5 Emergent Resonant Patches and Effective Discreteness . . . . . . 116 12.6 Linear Limit and Connection to the CMB Power Spectrum . . . . 117 13 Conclusion 118 14 Dedication and Acknowledgement 120 A Appendix: Bessel and Hankel Functions Used in the Main Text 121 A.1 Cylindrical Bessel and Hankel Functions . . . . . . . . . . . . . . 121 A.2 Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 122 A.3 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 122 B Appendix: Code for Simulating the 3D Thin–Disk Resonant Halo 123 C Appendix: Code for Simulating Elliptical Galaxies - Transport Theory 132 D Appendix: Code for Simulating Cluster Galaxies 142 4
1 Introduction The empirical success of Newtonian and Einsteinian gravity [ 1 , 2 ] on laboratory, Solar–System, and stellar scales is beyond dispute. Yet across galaxies, clusters, and cosmological environments, the observed gravitational field exceeds the response predicted from baryons alone. Rather than postulating a new particulate component, we revisit a well–known but rarely exploited aspect of general relativity: the full retarded Green–function structure of the linearised field. In the weak–field limit, curvature responds through the causal kernel Gret , whose support extends throughout the past light cone. These interior contributions, the “tails”, are genuine features of GR but are normally discarded when the field is approximated by its instantaneous Poisson limit. Restoring the full retarded response exposes a small but physically meaningful effect: curvature reacts to time–varying matter through a delayed, frequency–dependent, finite-memory response that is already present in the linearised theory. A central point is that we do not introduce an oscillator phenomenologically. It emerges automatically from four distinct ingredients. First and foremost, the Spacetime Resonance Theorem (SRT) (Section 2) predicts that any extended self–gravitating system possesses a collective susceptibility with a well-defined response frequency (the gal-mode). This arises directly from the causal retarded gravitational kernel of weak-field GR: once the instantaneous Poisson limit is lifted, the retarded Green function couples all masses through delayed curvature propagation, and an extended mass distribution automatically acquires a collective response frequency—even if the configuration were momentarily static (i.e. not driven by any energy source). Second, real astrophysical systems are not static. Their kinematic motions generate small but persistent density–sloshing (epicyclic drift in discs; irregular pseudo-random stirring in ellipticals), providing the broad driving spectra that excite the gal-mode selected by the SRT. Thus the SRT determines what collective mode the system can respond in, while the kinematic picture determines how real systems actually drive it. Third, once the gal-mode is driven, multiple-scattering effects can act as an amplification mechanism, enhancing the collective response. The precise form of this gain depends on the global geometry and dynamical state of the system, but it plays no role in defining the mode itself. Fourth, the oscillatory curvature field produced by these processes stores a slowly accumulated cycle-averaged energy density ρeff (the dark mass) through the long memory of the retarded kernel. This stored curvature forms the background gravitational field that shapes the large-scale attractor: it feeds back onto the very motions, gain pathways, and mode structure that generated it, yielding a self-organised resonant configuration over cosmic timescales. Within this multiple-scattering framework, coherent baryonic motion in rotating disks, orbiting cores, or clustered environments drives constructive curvature rescattering. Depending on spatial contrast and coherence length, this produces two distinct observational regimes: large-scale coherent modes in spirals, and 5
diffusive effective-medium halos in ellipticals and clusters. Homogeneous or weakly perturbed environments, lacking spatial contrast, exhibit no rescattering and remain resonantly silent—a behaviour consistent with the early Universe and the CMB. The present work develops this causal–resonant picture from first principles. Beginning with the retarded solution of linearised GR and the linear susceptibility of baryonic matter, we derive a self–consistent effective medium that accounts for spiral galaxies, ellipticals, clusters, and cosmological structure without introducing new fields or non–baryonic particles. The resulting Spacetime Resonance Theorem provides the microscopic foundation for the collective resonant mode and the stored curvature energy that behaves observationally as dark mass. 2 Spacetime Resonance Theorem The central result of this work rests upon a structural feature shared by all linear oscillator ensembles embedded in any causal medium described by a retarded Green function. Such behaviour is familiar across physics: acoustic scatterers in fluids, electromagnetic dipoles in dielectrics, plasma oscillators, elastic and seismological modes, collective excitations in condensed matter, and optical systems such as mode–locked lasers, where many cavity modes synchronise through retarded electromagnetic interactions. To make this universality explicit, we begin by formulating a general principle, independent of gravitational assumptions, that codifies the emergence of collective frequencies, phase coherence, renormalised propagation, and the universal damped–oscillator susceptibility produced by causal retardation and spatially distributed sources. Weak–field general relativity will then appear as a direct and natural specialisation of this structure. We emphasise that this Principle is not a theorem in the mathematical sense; it is a structural summary of universal behaviour in linear oscillator–field systems, intended as a conceptual framework for the rigorous results that follow. 2.1 The Fundamental Oscillator–Field Principle Principle (Fundamental Oscillator–Field Principle). Let {qi ( t ) } be an ensemble of linear oscillators (mechanical, electromagnetic, elastic, gravitational, or otherwise) that interact through a mediating field whose propagation is described by a causal retarded Green function Gret ( x, x′ ). Assume only linearity, causality, and that the oscillatory elements are spatially distributed, so that the field generated by one element is rescattered at distinct locations. Under these assumptions the coupled oscillator–field system exhibits the following universal structural features of linear response theory. 1. Collective behaviour and spectral emergence. Retarded field-mediated interactions couple the phases and amplitudes of the oscillators. In the frequency domain the interaction is encoded in the 6
coupling operator Kij(ω) = gi(ω)Gret(ω;xi,xj)gj(ω),(1) where gi ( ω ) denotes the linear coupling strength of oscillator i to the field. When the effective coupling bandwidth is sufficiently large compared to the natural detuning spread, the ensemble can exhibit collective long-time behaviour governed by the spectral structure of K(ω). 2. Damped–oscillator susceptibility. Each isolated oscillator responds linearly to a harmonic component of the mediating field. The susceptibility of an oscillator with natural frequency ω0and local damping coefficient γis the Lorentzian χ(ω) = 1 ω2 0−ω2−iγ ω,(2) the universal analytic structure enforced by linearity, causality, and the second–order local dynamics of the oscillator. This form appears identically across mechanical, electromagnetic, plasma, elastic, and weak-field gravitational systems. 3. Multiple scattering and self–energy renormalisation. Because the oscillators are spatially distributed, the field they source is rescattered from multiple locations. The net effect modifies the propagator through the Dyson relation G−1 eff (ω, k) = G−1 0(ω, k)−Σ(ω, k),(3) where G0 denotes the bare propagator of the mediating field and Σ( ω, k) is the self–energy encoding all multiple-scattering and feedback processes. In the long-wavelength, low-frequency regime, analyticity of the retarded response ensures an expansion Σ(ω, k)≃a0+a1(iω)+a2ω2+b2k2,(4) where a0 , a1 , a2 , and b2 are real coefficients determined by the statistics, spatial distribution, and coupling strengths of the oscillators. Substituting (4) into (3) yields a homogenised second–order effective evolution equation with collective parameters (Ω,Γ, ceff),(5) in which Ω is the emergent collective frequency, Γ the effective damping rate, and ceff an emergent propagation speed. The above features represent a broad and robust structural pattern of linear response theory. They arise whenever spatially distributed oscillatory elements 7
interact through a causal retarded field, independent of the specific physical medium. Spectral viewpoint. Because the field-mediated coupling (1) acts linearly on the vector of oscillator amplitudes, it may be regarded as an integral operator (or as a finite-dimensional matrix in discretised form). For an amplitude vector q= (q1, . . . , qN), the action of the operator is (K(ω)q)i= N X j=1 Kij(ω)qj,(6) or, in the continuous limit, ( Kq )( x ) = RK ( ω ; x, x′ ) q ( x′ ) d3x′ . The long-time dynamics of the coupled system are governed by the spectral properties of this operator. In particular, collective behaviour corresponds to the emergence of a normal mode associated with the dominant eigenvalue of K ( ω ), so that questions of uniqueness, stability, and coherence of the collective mode reduce to the spectral structure of the operator (6). Conditions for a Unique Dominant Collective Mode. For each fixed frequency ω , the coupling operator K ( ω ) acts linearly on the amplitude vector q and therefore possesses a (generally complex) spectrum defined by the eigenvalue equation K(ω)ψ=λ ψ, (7) with λ∈C and nonzero eigenfunction (or eigenvector) ψ . Because K ( ω ) is typically non-Hermitian, its eigenvalues need not be real. In linear response theory the long-time behaviour of the system is controlled by the eigenvalue whose real part ℜ ( λ ) is largest, since ℜ ( λ ) determines the temporal growth or decay rate of the associated collective mode. Thus ordering the spectrum by real part is the natural criterion for dynamical dominance and motivates the conditions below. To identify a unique collective normal mode and relate its frequency directly to the dominant spectral component of K ( ω ), one may further impose three conditions C1–C3: (C1) Irreducibility: no nontrivial partition of { 1 , . . . , N} blocks all cross-terms of Kij(ω). (C2) Dominant eigenvalue: K ( ω ) possesses a single eigenvalue whose real part strictly exceeds that of all others. (C3) Spectral gap: the dominant eigenvalue is separated from the remainder of the spectrum by a positive gap. When these conditions hold, the long-time behaviour is governed by a single collective mode whose emergent frequency is determined directly from this dominant eigenvalue. In the gravitational application that follows, these spectral 8
conditions are satisfied automatically by the positivity and compactness of the long-wavelength weak-field kernel. Principle 2.1 summarises a structural pattern so widespread that it is seldom stated explicitly: whenever spatially distributed oscillatory elements communicate through a causal retarded field, the system generically develops collective modes, a renormalised (effective) propagator, a well-defined self–energy encoding multiple scattering, and the universal Lorentzian susceptibility characteristic of linear second–order response. No reference to mechanical, electromagnetic, elastic, plasma, or gravitational specifics is required for the emergence of this general structure. 2.2 The Spacetime Resonance Theorem Before applying the Principle to gravity, we note that the weak-field Einstein equations satisfy its assumptions: the linearised curvature dynamics are governed by a causal retarded Green function, the equations are strictly linear in the perturbation, and baryonic matter provides spatially distributed oscillatory sources. Thus weak-field GR lies fully within the scope of Principle 2.1. When one does not impose the Newtonian instantaneous limit and instead retains the full retarded gravitational Green function, spacetime curvature plays the role of the causal medium; baryonic mass elements constitute the spatially distributed oscillatory sources; and gravitational rescattering generates the corresponding self–energy. The gravitational case therefore inherits the entire universal structure: collective behaviour, Lorentzian susceptibility, multiple scattering, Dyson renormalisation, and effective medium propagation. Before stating the theorem, we establish that the long-wavelength weak-field gravitational interaction automatically satisfies the spectral conditions (C1)–(C3) defined after Principle 2.1. Statement (Spectral Properties of the Weak–Field GR Kernel). In the long-wavelength, low-frequency regime, the linearised retarded gravitational kernel may be written schematically as the positive integral operator K(x, x′) = 4πG G0(x, x′)ρb(x′),(8) where G0 denotes the ω→ 0 (static, long-wavelength) limit of the retarded Green function of the linearised Einstein operator. In this limit the kernel reduces to the strictly positive Newtonian potential 1 / (4 π|x−x′| ), and it is in this static regime that positivity is invoked. We treat K as an integral operator acting on continuous functions over any bounded spatial domain. The restriction to a bounded spatial domain ensures that the Newtonian kernel 1 / (4 π|x−x′| ) defines a compact integral operator on this function space, as is standard in potential theory. In this regime the kernel K satisfies the abstract spectral conditions (C1)–(C3) introduced after Principle 2.1. Indeed: 9
The spatial coherence, scattering behaviour, and equilibrium structure of these attractors form the subject of the sections that follow. 3 Dark Mass The Spacetime Resonance Theorem establishes the exact decomposition Φ = ΦN+ Φres,(34) where Φ N is the static Newton–Poisson potential sourced by ρb,0 and Φ res is the coherent dynamical curvature field driven by the oscillatory baryonic component ρb,1 . Although Φ res averages to zero over a cycle, its quadratic energy does not. This section describes how that stored energy re-enters the static channel as an apparent dark mass. The effective–medium evolution equation (16) shows that Φ res carries a reversible quadratic energy density Eres =1 8πGh(∂tΦres)2+c2 eff(∇Φres)2+ Ω2Φ2 resi,(35) directly analogous to the conserved energy of a reversible acoustic or electromagnetic mode. In the weak–field Einstein equations, such quadratic terms appear in the stress–energy tensor at the same perturbative order as the Newtonian matter source. By the mass–energy equivalence, the resonant field therefore contributes an effective mass density ρeff =Eres c2=1 8πGc2h(∂tΦres)2+c2 eff(∇Φres)2+ Ω2Φ2 resi.(36) This effective mass enters the static sector in the usual Newton–Poisson way: ∇2ΦN= 4πGρb,0+ρeff.(37) Thus the observable gravitational potential is the quasi–static field Φ N sourced by the coarse–grained baryonic density ρb,0 together with the stored curvature energy ρeff . The oscillatory field Φ res never appears directly: only its cycle–averaged quadratic energy survives and contributes to the static Poisson channel. No modification of GR is introduced. The quantity ρeff is simply the gravitational imprint of the resonant curvature field generated by the baryonic ensemble, entering the Newtonian channel exactly as any other form of mass–energy. Its role as an apparent dark mass—and the slow feedback process through which it accumulates—is developed in detail in Section 6. 16
4 Resonant Field Dynamics 4.1 Channel Separation Retaining the full retarded Green–function structure of the weak–field Einstein equation yields a mathematically exact separation of the curvature response into its static and oscillatory frequency sectors, as established in the Spacetime Resonance Theorem. No approximation is involved: the split follows directly from the linearity of the retarded weak–field equations. The slowly varying component of the source generates the static Newtonian potential, while the time–dependent component excites the resonant curvature field, and the total solution is their exact superposition. Accordingly we decompose both the baryonic density and the curvature potential, ρb=ρb,0+ρb,1,Φ = ΦN+ Φres,(38) where ρb,0 is the slowly varying (quasi–static) density and ρb,1 is the genuinely time–dependent sloshing component. This exact frequency–sector split underlies the definitions of Channel 1 (the dynamic resonant response) and Channel 2 (the quasi–static Newtonian response) used throughout the remainder of this section. Channel 1: The Resonant (Dynamic) Channel. The time–dependent density component ρb,1 arises from ordinary kinematic sloshing of the baryons. In rotating discs this sloshing is dominated by the broad epicyclic spectrum (the “e–mode” spectrum), while in pressure–supported ellipticals it is generated by slow, pseudo–random gravitational stirring (the “r–mode” spectrum). These spectra supply a wide range of driving frequencies and inject the energy that activates Channel 1. The causal retarded kernel of weak–field GR does not respond equally to all frequencies. It filters the broadband kinematic drive and selects the unique collective response frequency predicted by the Spacetime Resonance Theorem: the gal–mode. Thus the physical sloshing provides the energy, while the SRT determines the narrow dynamical frequency in which the system actually responds. Once driven, each baryonic element acts as a weak phase–modulating scatterer of curvature, repeatedly exchanging delayed signals through the retarded propagator. This multiple–scattering process is encoded in the self–energy of the effective medium and yields the familiar resonant form ∂2 tΦres + 2Γ ∂tΦres + Ω2Φres −c2 eff∇2Φres = 4πG ρb,1,(39) with Ω, Γ, and ceff determined by the SRT and the scattering structure of the ensemble. Channel 2: The Newtonian Channel. If the source contains no time-varying oscillation, the frequency–dependent terms in Eq. (39) play no role. With ∂t Φ = 0 the retarded solution reduces to the usual instantaneous constraint ∇2ΦN= 4πG ρb,0,(40) 17
the standard Newtonian limit arising from the ω→ 0 behaviour of the retarded propagator. Thus Channel 2 is simply the static limit of the same underlying response. The two channels always coexist in general systems. Channel 1 carries the dynamic, oscillatory component of the curvature field, while Channel 2 sets the quasi–static geodesic background on which that field propagates. Their separation in Eqs. (38) – (40) is therefore ontological rather than merely conceptual: it follows directly from the linear retarded structure of weak–field GR, which uniquely decomposes the curvature response into a frequency–dependent (resonant) part and an instantaneous (Newton–Poisson) part. Their slow mutual interaction—dynamic energy accumulation modifying the quasi–static background—will be analysed later in the feedback architecture of Section 6. The remainder of this book is devoted entirely to Channel 1. Section 4.2 reformulates its dynamics in terms of a standard effective–medium wave equation; Section 5 develops the corresponding single– and multiple–scattering theory. Channel 2 requires no further development until the feedback structure is assembled in Section 6. 4.2 Spacetime as an Effective Reversible Medium Channel 1 requires a clear physical interpretation. The aim of this section is not to attribute material properties to spacetime, but to clarify that once the full retarded Green function of linearised GR is retained, the effective response generated by oscillatory baryonic forcing and multiple gravitational scattering behaves mathematically like a reversible, compressible medium with a single longitudinal mode. The “medium” in this sense is not spacetime itself but the collective gravitational response of the baryonic ensemble. This viewpoint mirrors the standard effective–medium interpretation used in multiple–scattering acoustics (e.g. the experiments of Derode, Tourin & Fink [ 4 , 5 ]), where microscopic scattering produces a macroscopic, reversible wave equation with well–defined resonance, linewidth, and effective propagation speed. Whenever the baryonic source contains an oscillatory component, the frequency dependent part of the effective propagator (Section 2) excites a collective mode governed by ∂2 tΦres + 2Γ ∂tΦres + Ω2Φres −c2 eff∇2Φres = 4πG ρb,1,(41) the fundamental evolution law of Channel 1. To expose its structure, compare Eq. (41) with the standard damped acoustic wave equation in a compressible, inviscid medium, ∂2 tu+ 2Γ ∂tu+ Ω2u−c2 s∇2u=fext ρ0 ,(42) where u is the velocity potential. This parallel is purely mathematical: both systems are governed by the same second–order operator with a restoring frequency Ω, a phase–lag coefficient Γ, and a wave speed ( ceff or cs ). The structural 18
correspondence is therefore Φres ↔u, ceff ↔cs,4πG ρb,1↔fext/ρ0.(43) No physical identification is implied: the effective curvature field is not a material compression, but its governing equation shares the same mathematical form as a reversible acoustic medium, exactly as in the effective–medium formulation of multiple–scattering acoustics [4, 5]. This equivalence extends to the instantaneous quadratic energy densities: Eu=1 2ρ0h(∂tu)2+c2 s(∇u)2+Ω2u2i,EΦ=1 8πGh(∂tΦres)2+c2 eff(∇Φres)2+Ω2Φ2 resi, (44) each conserved in the limit Γ → 0. Under harmonic driving at frequency ω , the resonant field attains the steady–state amplitude |Φres|RMS =4πG |ρb,1| p(Ω2−ω2)2+ (2Γω)2, Q =Ω 2Γ,(45) precisely analogous to a driven acoustic resonator. High– Q environments (Γ ≪ Ω) support long–lived, nearly lossless oscillations of the gravitational potential— the collective curvature mode of the baryonic ensemble, directly mirroring the coherent modes observed in multiply scattering acoustic media [4, 5]. Fundamental Equivalence (mathematical). Channel 1—the resonant gravitational response—is governed by the same linear operator that appears in effective reversible acoustic media. The equivalence is structural rather than material: pressure stress in a fluid ↔ curvature stress in the effective gravitational medium. This structural correspondence provides a robust dynamical interpretation of Channel 1 and motivates the use of standard wave–medium tools (normal modes, phase lag, quality factors, effective sound speeds). These concepts form the foundation for the scattering and coherence theory developed in the next section. 4.3 Collective Mode Dynamics and the Masson Channel 1 admits a natural variational formulation, reflecting its character as a genuine wave sector of the effective gravitational response of the baryonic ensemble. In the absence of damping (Γ=0), the resonant field Φ res is governed by the conservative Lagrangian density Lres =1 8πGh(∂tΦres)2−c2 eff(∇Φres)2−Ω2Φ2 resi,(46) whose Euler–Lagrange equation is the undamped form of Eq. (41) . The three terms correspond respectively to inertial response, spatial stiffness, and the collective restoring force determined by the baryonic self–energy. 19
The homogeneous solutions of Eq. (46) form a discrete set of coherent oscillatory wave packets—the normal modes of the effective gravitational medium. In condensed–matter language, a normal mode of a reversible, multiple–scattering medium is called a phonon. By strict structural analogy (established in Section 4.2), the oscillatory curvature mode of Channel 1 is the gravitational counterpart of such a phonon. For clarity, and to distinguish it from the familiar metric gravitational wave of vacuum GR, we refer to this collective excitation as a masson. The term is descriptive rather than ontological: a masson is not a new particle species, but the coherent curvature mode of the effective gravitational propagator generated by the baryonic ensemble itself. Each masson carries phase and reversible energy, propagates with effective speed ceff , and couples to baryonic structure through the coherent susceptibility χ⋆ (Ω) determined by the SRT, which is later driven by the system’s kinematic sloshing spectra. Once the kinematic driver is specified, this susceptibility acquires a well-defined polarisation structure, determining how the masson couples to disc-like flows (producing planar, anisotropic response) or to pressure-supported systems (producing an isotropically broadened response). The detailed decomposition of the masson’s susceptibility into its polarisation channels is developed in Section 5. When the gal-mode is driven, the associated reversible curvature energy is the quadratic energy density already defined in Eq. (35) . This energy then behaves gravitationally as an effective mass density and later feeds Channel 2 through the dark–mass mechanism developed in Section 6. In the driven system, the coefficient Γ controls linewidth and temporal phase coherence, with the corresponding quality factor Q=Ω 2Γ.(47) High– Q media (Γ ≪ Ω) support long–lived, spatially coherent curvature oscillations; macroscopic signatures of masson dynamics sustained by multiple gravitational scattering. The masson therefore provides the microscopic language for Channel 1: a collective curvature excitation of the effective gravitational medium. Its propagation, linewidth, and energy transport form the basis of the ensemble scattering and coherence theory developed in Section 5. Before constructing that scattering theory, we require the corresponding spatial propagation law for these modes. This follows from linear perturbation of Eq. (41) , expanding the resonant field about a quasi–static background and resolving it into spatial Fourier components. The resulting dispersion relation, group velocity, damping rate, and energy balance are summarised in the next subsection; these constitute the linear inputs for the multiple–scattering framework that follows. 20
4.4 Linear Spatial Perturbations and the Propagation Law In this section we summarise the spatial perturbation theory of the causal–resonant field. Nothing new is introduced here: this is the standard linearisation used in wave mechanics to extract dispersion, damping, and energy transport. Its purpose is simply to determine how the already–identified collective mode of Channel 1 propagates spatially as a weak gravitational wave within the effective medium established by multiple scattering. For completeness, the few algebraic steps that lead to the relations summarised below follow directly from inserting a small perturbation δ Φ into the effective-medium PDE of Channel 1 and performing the standard Fourier decomposition. Linearisation about a quasi–static background, applied to the already identified gal–mode of Channel 1, yields the scalar wave–type equation: ∇2δΦ + 1 c2 eff −∂2 t−2Γ ∂t+ Ω2δΦ = 4πG c2 eff δρb,(48) where δ Φ is the weak spatial perturbation of the resonant field and δρb its small baryonic source. This perturbative split is purely mathematical and should not be confused with the physical temporal split ρb = ρb,0 + ρb,1 used to define the two gravitational channels; here we examine only the spatial wave content of the established collective mode. A Fourier mode δΦ∝ei(k·x−ωt)satisfies the dispersion relation ω2+ 2iΓω−Ω2=−c2 effk2,(49) from which ω2 r= Ω2+c2 effk2,|δΦ|∝e−Γt,(50) and the phase/group velocities follow as vph =ωr k, vg=c2 effk ωr .(51) Thus Channel 1 supports weakly damped spatial waves with natural frequency Ω and linewidth Γ, exactly as in reversible acoustic or optical media—consistent with the effective–medium analogy established in [4, 5]. The corresponding energy balance is ∂tEΦ+∇·SΦ=1 c2 eff δρb∂tδΦ−2Γ(∂tδΦ)2,(52) where the Γ–term represents the reversible phase–lag attenuation associated with the frequency–dependent self–energy of the effective propagator. 21
Cycle–averaging gives the usual temporal coherence governed by the quality factor Q = Ω / (2Γ), already introduced in Section 4.3. Under harmonic driving at frequency ω , the spatial perturbation obeys the familiar Lorentzian response |δΦ|RMS =4πG |δρb| p(Ω2−ω2)2+ (2Γω)2,(53) peaking sharply at ω≃Ω with bandwidth 2Γ. For high driving frequencies ω≫Ω, |δΦ|RMS ∼4πG |δρb| ω2→0,(ω≫Ω),(54) demonstrating that the collective mode acts as a strict gravitational low–pass filter. Only frequencies near Ω can excite Channel 1; higher–frequency baryonic fluctuations are strongly suppressed. This mirrors the behaviour of ideal compressible media, where frequency response is controlled by the effective bulk modulus: high–frequency density perturbations cannot drive the medium. Together, Eqs. (48) – (53) show that the resonant field behaves as a weakly damped, energy–storing continuum supporting dispersive waves with finite bandwidth ∆ω= Ω/Q. The associated coherence time, τcoh ∼Q Ω,(55) is fixed entirely by the microscopic damping Γ that appears in the SRT. This same Γ determines the intrinsic memory time of the gal–mode, τmem = Γ −1 , and is independent of all environmental effects. Multiple scattering and transport processes do not modify these intrinsic timescales, but they do produce a much longer effective lifetime for the driven mode. In reverberant or diffusive environments, the repeated trapping of curvature signals generates an emergent effective damping rate Γ eff ≪ Γ and a correspondingly long trapping time τtrap = Γ −1 eff . This leads to an environmental amplification factor Genv which boosts the mode amplitude without altering the SRT–determined quantities Ω, Γ, τcoh, or τmem. These distinctions—microscopic damping from the SRT and emergent trapping from transport theory—form the basis for the multiple–scattering framework developed in Section 5. 5 Gravitational Scattering Theory Having established that the dynamical (Channel 1) sector of gravity is governed by the effective resonant wave equation defined by the SRT in Section 2, we now construct the corresponding gravitational scattering theory. Throughout this section we work exclusively with the oscillatory baryonic component ρb,1 , since 22
only temporally varying sources can excite and sustain the collective resonant field Φ res . The static component ρb,0 generates the background Newtonian field but does not participate in scattering. The goal in what follows is to develop the gravitational analogues of the Lippmann– Schwinger, Foldy–Lax, and Dyson equations using the propagator Gret associated with the effective–medium PDE. This provides the microscopic foundation for the collective behaviour analysed later—including coherent spiral modes and diffusive elliptical/cluster regimes (Section 9). In this framework each baryonic mass acts as a gravitational scatterer characterised by a linear susceptibility χ⋆ (Ω), which encodes its phase–lagged response at the gal–mode frequency selected by the SRT. The intrinsic susceptibility is scalar; however, the spatial structure of the driving field may introduce direction–dependent effects in the net response. A full account of this source–induced anisotropy is deferred to a dedicated subsection at the end of the present section, see Section 5.6. An incident potential Φ inc , generated by the oscillatory baryonic source elsewhere in the ensemble, is re–emitted with a phase delay described by the retarded Green function Gret . The resulting scattered field then propagates to all other masses. Repeated re–scattering among many inclusions generates a hierarchy of coupled responses, precisely analogous to classical multiple–scattering theory in acoustics and electromagnetism [6, 7, 8]. Thus baryonic matter plays a dual role: its oscillatory component ρb,1 provides the driving forcing spectrum, while the same masses act as gravitational scatterers that re–emit curvature perturbations through their susceptibility χ⋆ (Ω). The iterative network of scattering and re–scattering links the microscopic delayed response of individual baryons to the emergent, large–scale propagation properties of the collective resonant field. It therefore supplies the necessary bridge between the effective wave operator determined by the SRT and the macroscopic gravitational behaviour developed in the remainder of the book. 5.1 Single–Scatterer Response A single oscillatory baryonic perturbation—that is, a localised fluctuation δρb,1 of integrated mass m⋆ concentrated near x ⋆ —acts as a gravitational scatterer of the resonant field governed by the effective–medium wave equation (41) , derived from the Spacetime Resonance Theorem in Section 2. In the frequency domain it is convenient to denote by GΩ the frequency–Ωpropagator of this resonant operator, 1 which transports curvature perturbations at the collective frequency Ω from x′to x. A localised oscillatory element at x ⋆ with intrinsic (resonant–frequency) susceptibility χ⋆ (Ω) responds to an incident oscillatory field Ainc according to the 1 The formal definition of GΩ is given in Section 8 as the solution of the Helmholtz–type equation. Here we use only its role and notation as introduced for scattering theory. 23
standard Lippmann–Schwinger relation: Asc(x)=t⋆(Ω) GΩ(x−x⋆)Ainc(x⋆),(56) where the gravitational scattering amplitude is t⋆(Ω) = 4πG m⋆χ⋆(Ω).(57) Equations (56) – (57) constitute the basic microscopic scattering rule: an incident curvature perturbation drives a phase–lagged monopole response proportional to the oscillatory mass element m⋆ , which is then re–emitted through the propagator GΩ. Because the resonant sector is weakly damped (Γ ≪ Ω) and the underlying dynamics follow linearised GR, the exchange is fully reversible: no curvature energy is permanently absorbed. The perturbation stores curvature only transiently and re–emits it with the phase lag encoded by χ⋆ (Ω) and the causal propagation encoded by GΩ. The susceptibility χ⋆ (Ω) is scalar, as fixed by the SRT: the microscopic response of a pointlike oscillatory element is isotropic. Any tensorial or directional structure in the effective response arises solely from anisotropy in the incident field Ainc , whose geometry reflects the sloshing spectrum ρb,1 that drives the resonant channel. This source–projected anisotropy is analysed in Section 5.6. Thus Eqs. (56) – (57) supply the microscopic building block from which the Foldy–Lax hierarchy, ensemble self–energy, and effective–medium resonant propagation developed in subsequent sections are constructed. 5.2 Multiple Scattering and the Foldy–Lax Hierarchy In classical wave physics, the interaction of many scatterers with a coherent incident field is described by the Foldy–Lax hierarchy [ 6 , 7 , 8 ], which provides the standard self–consistent formulation of multiple scattering among discrete inclusions. Originally developed for acoustics and later generalised to electromagnetism, elasticity, and photonics, the same framework applies directly to the present gravitational problem once the effective propagator GΩ has been specified by the linearised dynamics of Section 4.4. Throughout this subsection the field amplitude A denotes the oscillatory (resonant) curvature perturbation Φres driven by the baryonic component ρb,1. Each baryonic inclusion re–emits the incident curvature fluctuation with a phase delay determined by its scalar susceptibility χ⋆ (Ω). These delayed re–emissions propagate to all other inclusions via the Green function GΩ , generating a network of mutual couplings that constitutes the gravitational analogue of standard multiple–scattering theory. Any directional structure present in Ainc reflects the source sloshing spectrum that drives the resonant field; such source–induced anisotropy affects the spatial organisation of coherence but does not modify the 24
scalar nature of χ⋆ (Ω). A full treatment of this source–projected anisotropy is deferred to the final subsection of this section. When the spacing between inclusions is much smaller than the resonant wavelength ( kΩd≪ 1), their phases remain locked to the incident field and the scattered amplitudes add coherently. In this fully coherent Foldy–Lax limit, Acoh ∝N|χ⋆(Ω)|Ainc,(58) so that the ensemble behaves as a single collective radiator with intensity Icoh ∝N2|χ⋆(Ω)|2|Ainc|2.(59) It is convenient to quantify this as an environmental gain Genv ≡|Atot|2 |Ainc|2,Genv =N2|χ⋆(Ω)|2(coherent limit).(60) For an ensemble of N scatterers, the total field satisfies the discrete Lippmann– Schwinger system A(x)=Ainc(x)+t(Ω) X n GΩ(x−xn)A(xn),(61) with t(Ω) = 4πG m⋆χ⋆(Ω). In operator notation this becomes A= (1 −tGΩ)−1Ainc,(62) whose Neumann expansion, A=Ainc +tGΩAinc + (tGΩ)2Ainc +··· ,(63) sums all orders of mutual gravitational re–scattering. In the limit Γ → 0 (purely real GΩ ), the response is strictly coherent and Eq. (62) reduces to the linear scaling of Eq. (58). If instead the relative phases of different inclusions become effectively random—because of geometric delays, finite memory (Γ > 0), or dynamical phase drift—then scattered amplitudes no longer combine coherently. They add in intensity only: Itot =N|χ⋆(Ω)|2|Ainc|2,(64) yielding the standard incoherent scaling Genv =N|χ⋆(Ω)|2,pGenv =√N|χ⋆(Ω)|.(65) The spiral–disk regime discussed in Section 9.1 lies in the long–wavelength, phase– ordered limit ( kΩd≪ 1), where both spatial phasing and delayed temporal response organise coherently into a global m = 1 pattern. As coherence decreases—due to finite Γ, increasing disorder, or large separations—the system transitions smoothly 25
curvature oscillations exchange energy and phase with the baryons and deposit curvature energy into the long–term state variable ρeff .Channel 2 provides the reactive, quasi–static response of the metric to the slowly varying mass distribution and the accumulated stored curvature energy. The interaction of the two channels forms a coupled spatiotemporal feedback system: Channel 1 continuously modifies the long–term metric state, while Channel 2 sets the background potential within which resonant oscillations occur. The architecture thus consists of two physical entities—baryonic matter and the spacetime response kernel—and two curvature states that evolve on fast (Channel 1) and slow (Channel 2) timescales, summarised in Table 1. 6.2 Feedback Channels and Interactions The curvature–matter system evolves through two coupled feedback channels, both arising from the same weak–field gravitational dynamics. Channel 1 is the resonant, wave–based response driven by the oscillatory baryonic density ρb,1 and governed by the causal, memory–bearing propagator derived in Section 8. Channel 2 is the Newtonian, effectively instantaneous response generated by the slowly varying baryonic density ρb,0 together with the accumulated effective (dark) mass ρeff . Their interaction forms the dual feedback architecture shown in Fig. 1. (1) Resonant Channel (Channel 1). The oscillatory component ρb,1 excites the resonant curvature field Φ res through the causal Green function GΩ of the damped resonant curvature equation. As developed in Section 8, GΩ encodes propagation, phase delay, and finite causal memory at the effective speed ceff . Because many baryons act as coherent emitters and scatterers, the resonant field is statistically amplified by the environmental gain Genv, giving Φres ∝pGenv GΩ∗ρb,1,(80) with ∗ denoting causal convolution. The interaction is reversible: resonant curvature perturbations exchange energy and phase with the baryons that generate them. Over the memory time τmem = Γ −1 , this oscillatory exchange accumulates into an effective (dark) mass density, ρeff ∝ ⟨|Φres|2⟩t,(81) which then enters the Newtonian response of Channel 2. (2) Newtonian Channel (Channel 2). The background baryonic density ρb,0 together with the stored effective mass ρeff source the Newtonian potential Φ N via the Poisson equation. This potential propagates effectively instantaneously at speed c within the weak–field regime and establishes the local geodesic structure. The resulting velocity field vredistributes ρb,0 and modulates the oscillatory component ρb,1 , thereby resetting the pattern of resonant sources that feed Channel 1. This constitutes the stress–advection loop: the accumulated effective 32
mass stabilizes the quasi–static potential, while baryonic motion continually reorganizes the drivers of the resonant field. Long–term coherence and equilibrium. Channel 1 continually deposits curvature energy into ρeff , while Channel 2 continually redistributes matter and reshapes the sources of resonant excitation. Together these processes drive the system toward a self–organized spatiotemporal equilibrium in which ρb,1 , Φ res , ρeff , and Φ N remain mutually consistent across their respective timescales. Figure 1 illustrates these causal links: red arrows denote the Newtonian stress–advection response; black and blue arrows denote resonant excitation and slow memory accumulation. Channel 1 Resonant channel feeding dark mass Channel 2 Newtonian channel feeding continuity ρb,1Φres ρeff GΩ rescat./prop. (∝ Genv) ρb,0, ρeff ΦNvρb,0, ρb,1 ∇2−∇ Continuity Advection long-term energy storage memory ∝(τmem) Figure 1: Two-channel gravitational feedback network. Channel 1 (top, black →blue): the oscillatory baryonic component ρb,1 drives the resonant gravitational field Φ res through the causal propagator GΩ . Multiple scattering produces an intensity gain Genv , and the resonant field deposits accumulated curvature into the slow, long-lived component ρeff over the memory time τmem = Γ −1 .Channel 2 (bottom, red): the slowly varying density ( ρb,0, ρeff ) sources the Newtonian potential Φ N , which induces bulk velocities and advects the baryonic distribution. Blue dashed arrows denote slow memory feedback; red arrows denote instantaneous Newtonian response and mass transport. Together the two channels form a closed spatiotemporal feedback loop linking baryonic motion, resonant gravitational response, and long-term curvature accumulation. 33
Table 2: Principal feedback links between baryons, effective curvature fields, and the spacetime response. From →To Physical Description Timescale ρb,1↔Φres Mutual excitation and re–scattering of resonant curvature perturbations. The field is driven through the causal propagator and statistically amplified by the ensemble gain factor √Genv . Feedback arises from propagation and rescattering of the metric perturbation. Fast Φ res → Spacetime response The resonant field contributes curvature energy to the local metric response described by the resonant operator, providing the instantaneous feedstock for long–term accumulation. Fast Spacetime response →ρeff Slow build–up of the time–averaged curvature energy density, producing the effective (dark) mass ρeff over the memory time τmem = Γ−1. Slow Channel 2: Static stress and advection ρb,0, ρeff →ΦN Static sourcing of the Newtonian potential that defines the background geodesics. The stored curvature energy acts gravitationally as effective mass and enters Poisson’s equation on equal footing with ρb,0. Instant aneous ΦN→Baryons The Newtonian potential sets the local acceleration field acting on the baryons, determining the background flow. Instant aneous v→ρb The induced velocity field advects the baryonic density, maintaining mass continuity and reorganizing the pattern of sources for the next resonant cycle. Fast ρb↬ρb Self–consistent continuity of baryonic redistribution, feeding back into the subsequent resonant excitation and Newtonian response. Same 34
Together these links form the dual–channel architecture shown in Fig. 1. The figure compresses the resonant feed and slow accumulation into a single arrow, whereas the table distinguishes their dynamical roles: a rapid deposition of curvature energy by Φ res , followed by slow integration over the memory kernel to form ρeff . Fast processes govern propagation and rescattering, while slow feedback through ρeff sustains the quasi–static halo supporting global stability. This coexistence of fast and slow regimes ensures that resonant dynamics, long–term memory, and the large–scale gravitational structure remain mutually consistent. The ensemble gain Genv clarifies that Φ res represents the coherent superposition of many baryonic emitters. Since | Φ res| ∝ √Genv , the stored component scales as ρeff ∝ Genv . This amplification is purely statistical—reflecting coherent phase alignment across many baryons—and does not violate causality; it follows from standard multiple–scattering theory applied to curvature perturbations within the weak–field limit. The causal relations illustrated schematically in Fig. 1 are expanded in Table 2. For compactness, the figure merges the resonant field Φ res and the formation of the stored effective mass ρeff into a single step. In the table this process is separated into two stages: (i) a fast feed, in which the oscillatory effective curvature field deposits energy into the spacetime response encoded by resonant propagation, and (ii) a slow accumulation, in which this energy is integrated over the memory interval τmem = Γ −1 to form the time–averaged effective mass density ρeff . This makes explicit that propagation and rescattering operate on the wave timescale, while the long–term stored component arises only after temporal coarse–graining over the resonant memory kernel. 7 Temporal Coherence of the Resonant Channel The behaviour described in this section is the concrete gravitational realisation of the collective phase-locking and emergent frequency predicted by the Spacetime Resonance Theorem (Section 2). 7.1 Phase Locking and Mutual Susceptibility The resonant channel linking baryonic oscillations ρb,1 to the effective curvature field Φ res behaves as a coupled oscillator system mediated entirely by causal propagation of curvature perturbations. Baryons drive the resonant field through the Green response GΩ of the effective resonant operator (Section 4), and the resulting Φ res feeds back on the baryons through the same phase–lagged gravitational interaction. No inverse operator acts directly on the mass distribution: feedback proceeds solely through delayed curvature forces generated by Φ res and their influence on baryonic motion. This wave–mediated, causal exchange constitutes Channel 1, the resonant feedback loop of the system. In an ensemble of baryonic oscillators (i.e. baryonic elements, each acting as a weak kinematic oscillator through local density–sloshing) the coupling is 35
collective. 3 Individual elements contribute coherently to the resonant curvature field, so that the resulting amplitude scales as Φres ∝pGenv GΩρb,1,(82) where Genv is the ensemble gain that measures statistical phase alignment across the environment. This factor modifies only the strength of collective reinforcement: it counts how many baryonic emitters share a common phase and does not alter causality or the underlying Green function. At the microscopic level this collective response may be described by an effective susceptibility χ (Ω) of the baryonic ensemble, so that the channel behaves as a weakly damped, polarizable medium whose overall gain is controlled by Genv|χ(Ω)|2. In the high– Q regime ( Q = Ω / 2Γ ≫ 1), the memory interval Γ −1 spans many oscillation periods, so baryons and curvature waves remain nearly phase–locked: their relative phase drifts only slowly compared with a single cycle. Repeated exchange between ρb,1 and Φ res then acts as a conservative mode–locking mechanism. Small phase offsets are continuously corrected by the delayed curvature forces, and the ensemble self–organises into a coherent global resonance at frequency Ω, statistically reinforced by Genv . This self–locking is directly analogous to mode locking in optical and acoustic resonators: reversible, phase–lagged exchange sustains coherence, while ensemble gain sets the effective field strength. The reversible exchange between ρb,1 and Φ res gradually loads energy into the resonant channel, which accumulates over long times as the effective stored component ρeff . Because ρeff evolves on the memory interval τmem ∼ Γ −1 , it acts as a slowly varying background contribution that modulates the effective resonant parameters B = ( ceff, Ω , Γ) entering the Channel 1 dynamics. Thus the coupled curvature–matter system continually re–tunes its own resonant environment as energy circulates among ρb,1↔Φres →ρeff,(83) with ρeff feeding back into the quasi–static Channel 2 potential Φ N via Poisson’s equation. Temporal coherence therefore emerges from the slow, collective evolution of mutual susceptibility within this coupled system. As long as the coherence time τcoh =Q Ω,(84) 3 The onset of coherence does not require a finely tuned coincidence of phases. As in all weakly damped oscillator ensembles, each baryonic element carries a small stochastic kinematic oscillatory component ( ρb,1 = 0 at arbitrarily low amplitude), and these fluctuations are continually exchanged through the causal Green response GΩ . Any neighbouring pair whose natural frequencies differ by less than the curvature–mediated locking bandwidth ( |ωi−ωj|≲ Γ) undergoes slow phase pulling and converges toward a common phase. Once a small coherence patch forms, its enhanced coherent emission strengthens the local forcing and recruits adjacent elements, leading to progressive domain coalescence. This nucleation-and-merging process is directly analogous to the formation of coherent modes in optical and acoustic resonators and requires no special initial alignment of phases. 36
exceeds the characteristic timescale of baryonic perturbations, the resonant channel remains phase–locked and globally coherent. Under these conditions the effective curvature field behaves as a self–sustaining resonator: its phase is stabilised collectively via √Genv and the effective susceptibility, while its spatial envelope evolves on the slower timescales associated with ρeff and the Newtonian background. 7.2 Global Frequency Selection in Inhomogeneous Baryonic Systems The Spacetime Resonance Theorem (Section 2) establishes that any finite coherence region must oscillate at the unique gal–mode frequency Ω, fixed entirely by the local coarse–grained baryonic density through the SRT scaling. The purpose of the present subsection is therefore not to derive the frequency—its value is already guaranteed by the SRT—but to elucidate how this unique global mode is dynamically selected in a realistically inhomogeneous medium. In particular, we show that causal curvature exchange (Channel 1), acting across neighbouring coherence patches with slightly different local natural frequencies, induces mutual frequency pulling and progressive domain coalescence. This Green-mediated communication forces the entire region to collapse onto the single gal–mode frequency required by the SRT, even when the underlying baryonic density varies smoothly in space. The analysis below is thus the dynamical realisation of the SRT in the inhomogeneous case: it demonstrates how the curvature–matter feedback mechanism enforces the SRT-selected frequency throughout the extended system. Consider an extended region in which the coarse–grained baryonic density varies smoothly with position. Each local coherence patch then possesses a slightly different natural frequency, inherited from the local mean density. Once curvature– mediated exchange is active, these patches no longer oscillate independently: the retarded curvature field sourced by each patch acts back on its neighbours with a finite phase lag, producing mutual frequency pulling. Two adjacent patches whose natural frequencies lie within one another’s locking range will synchronise to a common intermediate value, thereby forming a larger coherent enclosure. If the density variation is mild across the system, this merger process repeats iteratively. Curvature exchange continually transports phase–coherent perturbations between neighbouring regions, increasing the coherence volume and pulling the collective frequency toward the density–weighted mean of the expanding domain. Provided that the curvature–mediated coupling bandwidth ( ∼ Γ) exceeds the small natural detunings between adjacent patches, this inductive coalescence proceeds over the long memory time ( Q≫ 1), ultimately producing a single phase–locked resonant enclosure that spans the full inhomogeneous region and oscillates at one emergent global frequency Ω. This behaviour is the precise gravitational analogue of frequency pulling and domain coalescence in inhomogeneously broadened optical resonators (Siegman [ 12 ]). In such systems, local oscillators with slightly different natural frequencies do 37
not compete; delayed coupling through the common field drives them toward a shared frequency and enforces a common phase. In the present gravitational setting, curvature exchange plays the same role: it continually redistributes phase–coherent perturbations among neighbouring baryonic regions and selects a unique, globally coherent oscillation frequency for the entire inhomogeneous structure. 7.3 Polarisation of Temporal Coherence Section 5.6 established the central principle of gravitational “polarisation”: the microscopic susceptibility χ⋆ (Ω) is a scalar, and all tensorial features of the collective resonant response arise from its projection through the sloshing covariance of the baryonic driver, χij eff (Ω) = χ⋆ (Ω) Pij , with Pij defined by Eq. (76) . The previous subsection examined the spatial consequences of this projection (directional coherence, anisotropic gain). The purpose of the present subsection is to extend the same logic to the temporal organisation of the resonant channel: how the covariance Pij determines the temporal eigenmodes that the curvature–matter feedback loop can phase–lock and amplify. As explained in Section 5.6, only those components of ρi b,1 that carry spectral weight within the narrow gal–mode band selected by the SRT can act as efficient drivers of the resonant response. The temporal forcing of the resonant channel is therefore determined by the gal–mode–filtered portion of the sloshing covariance Pij , i.e. the components whose temporal power spectrum overlaps the SRT– selected gal–mode frequency window. Two physically distinct types of sloshing supply these components, and in the two limiting cases (spiral and elliptical galaxies) one of these spectra naturally dominates, simply because its frequency content has the strongest overlap with the gal–mode band appropriate to that environment: • epicyclic or orbital sloshing, with characteristic frequencies distributed around the local orbital rate; we refer to this as the e–mode spectrum. The detailed connection between the e–mode spectrum and the gal–mode excitations is developed later in Section 9.1.2. • rapid, pressure–driven or ‘speckle–curvature’ 4 sloshing, with characteristic timescales several orders of magnitude shorter than the e–mode cycles; we refer to this as the r–mode spectrum. Its origin and statistical properties are developed in Section 9.2. Although these sloshing mechanisms differ in physical origin and timescale, both possess components that overlap the gal–mode frequency band singled out by the SRT, and therefore both can act as efficient temporal drivers of the resonant channel. Crucially, this overlap reflects the deeper curvature–matter 4 The interpretation of the 4D gravitational speckle is developed later in Section 9.2 and connected to the Jeans-pressure description. 38
feedback structure: the large–scale geometry and kinematics encoded in Channel 2 set the dominant organisation of the sloshing spectrum (e–modes in rotation– supported systems), while in pressure–supported systems the high–frequency r–mode spectrum is further shaped by the small–scale curvature fluctuations generated within Channel 1 itself (the speckle dynamics developed in Section 9.2). Thus the same feedback framework that determines the global curvature field also determines which portions of the sloshing spectrum can effectively couple to, and drive, the gal–mode band. The temporal organisation of the resonant feedback loop follows directly from the same projected susceptibility that governs the spatial response. A curvature perturbation with incident amplitude Ajproduces a delayed re–emission Ai sc ∝χij eff(Ω) Aj=χ⋆(Ω) PijAj,(85) so the directions along which the feedback loop can maintain temporal phase coherence are determined entirely by the covariance Pij. The coherent temporal channels of the curvature–matter feedback loop are therefore obtained by diagonalising the covariance tensor: Pijvj (α)=λ(α)vi (α), χeff,(α)=χ⋆(Ω) λ(α).(86) Each eigenvector vi (α) identifies a direction in which the driver can sustain phase– coherent forcing, and the resonant channel amplifies the mode with the largest |λ(α)|2 . Temporal “polarisation” thus corresponds to preferential phase–locking onto the dominant eigenmode of Pij. To prepare for the physical interpretations developed later (spiral discs and elliptical galaxies), we now display the two canonical limiting forms of Pij that arise from their respective sloshing environments. These forms do not require any system–specific dynamics; they simply encode the geometry of the gal–mode– relevant forcing. Coherent planar forcing (e–mode dominated). When orbital or epicyclic sloshing dominates, the forcing is overwhelmingly planar. The covariance takes the approximate form Pij planar = 1 0 0 0 1 0 0 0 ϵ ,0< ϵ ≪1.(87) Diagonalising Eq. (86) yields a dominant planar mode with λ∥≃ 1 and a suppressed vertical mode with λ⊥≃ϵ . As shown later in the spiral–galaxy analysis, this planar temporal coherence underlies the emergence of global m = 1 structures. 39
Isotropic incoherent forcing (r–mode dominated). When rapid pressure– or speckle–driven fluctuations dominate, the forcing is statistically isotropic. The covariance becomes Pij iso = 1 0 0 0 1 0 0 0 1 ,(88) so the eigenvalues are equal. No preferred temporal direction exists, and the resonant channel forms three–dimensional coherence cells. The consequences for elliptical and cluster dynamics are developed in Section 9.2. In summary, temporal polarisation is the natural temporal extension of the source–projected spatial structure of Section 5.6. The sloshing spectrum defines a covariance Pij ; projection of the scalar susceptibility produces χij eff ; and the eigenstructure of Pij determines which temporal channels the feedback loop can lock onto and amplify. The contrasting temporal behaviours of rotation– dominated and pressure–supported systems therefore follow directly from the differing geometry of their sloshing covariances. 8 Spatial Coherence of the Resonant Channel The preceding sections developed the temporal structure of the resonant sector (Channel 1): the Spacetime Resonance Theorem selects a unique collective frequency Ω, and curvature–matter feedback synchronises all sloshing sources at this same frequency. With the temporal carrier fixed, the behaviour of the resonant channel is now determined entirely by its spatial organisation. Spatial coherence is governed by the frequency–domain Green function of the resonant operator. This object appeared notationally in Section 5 in the microscopic scattering formula (Eq. 56), but its formal definition is the solution of the Helmholtz problem Ω2−i2ΓΩ −c2 eff∇2GΩ(x,x′)=δ3(x−x′),(89) subject to retarded boundary conditions. Its explicit closed form is the standard damped Helmholtz/Yukawa kernel, GΩ(x,x′) = eikΩ|x−x′| |x−x′|, kΩ≡Ω ceff r1−i2Γ Ω,(90) which propagates a curvature perturbation of frequency Ω from x ′ to x. Equation (56) in the scattering section is now seen as the microscopic use of this same kernel. The spatially coherent resonant field is internally oscillatory and not directly observable; only its stored curvature component, ρeff ∝DGΩ∗ρb,1 2Eτmem ,(91) 40
accumulates over the memory time τmem = Γ −1 and sources the quasi-static Channel 2 potential. This point will soon be elucidated in more detail in Section 8.2. With the carrier frequency Ω fixed, the resonant potential admits the standard monochromatic ansatz Φres(x, t) = ReA(x)e−iΩt,(92) where A (x) is the complex spatial envelope. This envelope is governed by the stationary Helmholtz operator associated with Eqs. (89) – (90) , and constructive interference of phase–matched contributions generates extended, long–lived resonant envelopes—the gravitational analogue of standing–wave patterns in weakly damped media. Before analysing the envelope A (x) in detail, we summarise the coherence and polarisation structure that will organise all subsequent spatial calculations. • Temporal coherence. The SRT already fixes a unique carrier frequency Ω, and feedback over Channel 1 synchronises all emitters onto this single temporal mode. Thus every contribution to Φ res carries the common factor e−iΩt. • Polarisation (directional structure). As established in Sections 5.6 and 7.3, all anisotropy arises solely from the source covariance Pij . The microscopic susceptibility χ⋆ (Ω) is scalar; the only tensorial structure is the projection χij eff = χ⋆Pij . This projection determines both spatial and temporal polarisation (selection of coherent channels) but introduces no additional dynamics. Once the projection is absorbed into the definition of the coherent source term, the spatial analysis may be carried out for a scalar field A(x) without loss of generality. • Coherent versus incoherent limits. If the projected driver retains phase order across the relevant region (e.g. rotation–supported discs), then the field admits a coherent monochromatic envelope A (x) satisfying a Helmholtz equation. If instead the phases are effectively random (e.g. r–mode–dominated, pressure–supported systems), the same Green function GΩ governs propagation but the physically relevant quantity is the intensity ⟨A (x) A∗ (x ′ ) ⟩ . The incoherent limit is therefore obtained by replacing coherent superposition with statistical averaging and leads naturally to the effective–medium and transport equations developed later. With these coherence and polarisation considerations fixed, we may work with the standard monochromatic ansatz Φ res (x , t ) = Re [ A (x) e−iΩt ], with the understanding that A (x) represents the coherent envelope when phase order is present and that incoherent regimes are handled by replacing AA∗ with its ensemble average. All subsequent spatial structure follows from the Green function GΩ defined in Eq. (89) . The next subsection derives this envelope equation and its Kirchhoff–Sommerfeld representation, making explicit how the kernel GΩ sets the spatial coherence scale λres of the resonant gravitational medium. 41
Figure 2: Two–source interference (10 pc separation). Amplitude, phase, and effective density ρeff for two coherent point sources. The separation is much smaller than the resonant wavelength, so the two fields merge into a single envelope with gentle modulation. 48
Figure 3: Two–source interference (100 pc separation). Same setup as Fig. 2, but with the sources several wavelengths apart, producing strong constructive and destructive interference fringes. 49
In general the effective density contains both gradient and curvature contributions, as in Eq. (102) . For the present two–source configuration we again adopt the long–wavelength, slow–envelope approximation so that the gradient term provides only a small correction. Keeping only the dominant curvature contribution yields ρeff(x) = Ω2 4πGc2|A(x)|2,(113) showing directly that interference between oscillatory fields modulates the local effective gravitational density. The cross–term in Eq. (112) thus represents a real redistribution of stored curvature energy within the effective resonant medium; the total energy remains conserved in the absence of damping, but is shifted between regions of constructive and destructive curvature amplitude. The exponential envelope is again controlled by the same damping rate Γ that sets the temporal coherence time. This two–source configuration makes the phase structure of the Kirchhoff– Sommerfeld representation (Section 8.1) fully explicit. The Channel–1 driver takes the form δρb(x) = δρb(x)eiφd(x),(114) where φd (x) encodes the local phase delay of the baryonic motion relative to the global carrier e−iΩt . The propagation kernel GΩ (x , x ′ ) rephases each emitter before summation, so that A (x) is the phase–matched projection of the source onto the resonant field. For two isolated emitters this becomes transparent. Assign complex driver amplitudes δρ1eiϕ1, δρ2eiϕ2,(115) at x1and x2. Each generates a complex effective curvature field δΦp1(x)eiϕ1, δΦp2(x)eiϕ2,(116) so that the total Channel–1 envelope is A(x)=δΦp1(x)eiϕ1+δΦp2(x)eiϕ2.(117) The relative phase ∆ϕ=ϕ1−ϕ2(118) determines whether the superposition is constructive or destructive. If the sources maintain a fixed configuration or relative motion, ∆ ϕ remains constant in the global resonant frame, producing a stationary interference pattern and a time–independent composite envelope. In this discrete setting the role of the complex amplitudes is fully visible: δρ1,2eiϕ1,2 describes the phase–weighted baryonic drive, the kernel applies the additional propagation phase eikΩri , and Eq. (117) is the resulting phasor sum. This is precisely the two–element analogue of the coherent integral A(x) = G c2 eff Zd3x′GΩ(x,x′)δρb(x′) (119) 50
introduced in Eq. (94). If many emitters satisfy ∆ϕ(x)≈const,(120) their contributions add coherently and the envelope amplitude grows approximately with the number of participants. If instead their phases are random, ei∆ϕ= 0,(121) the cross–terms cancel statistically, leaving only self–intensity contributions and producing an incoherent, speckle–like background. This phase–locking behaviour is the discrete prototype of the coherent projection mechanism governing extended baryonic systems. It forms the conceptual bridge to the collective resonant modes developed in the following sections, where the interference of many emitters shapes the large–scale resonant envelope and the emergent effective density ρeff. 8.5 Temporal Memory and the Steady–State Reduction Although the steady–state resonant field is described by the stationary envelope A (x), its underlying dynamics remain explicitly oscillatory. The effective resonant medium possesses a finite causal memory τmem = Γ −1 : curvature perturbations do not vanish immediately after their source ceases, but decay exponentially and may interfere with later excitations. In the full time–dependent formulation this behaviour is encoded by the retarded Green function of the damped curvature wave operator, Gret(x, t −t′)∝e−Γ(t−t′) |x−x′|sinΩt−t′−|x−x′| ceff Θ(t−t′),(122) where the exponential encodes temporal fading and the oscillatory term expresses causal propagation at speed ceff . For a monochromatic, phase–locked driver at the global frequency Ω, the convolution of Gret with the source reduces, in the frequency domain, to the Helmholtz kernel with complex wavenumber kΩ=Ω+iΓ ceff , GΩ(x,x′) = eikΩ|x−x′| |x−x′|.(123) The imaginary part of kΩ reproduces the temporal decay e−Γ(t−t′) as the spatial attenuation e−r/Ldamp with Ldamp = ceff/ Γ. Thus temporal memory and spatial decay arise from the same damping parameter Γ; the resonant medium’s memory is expressed equally in time and space. Repeated re–excitation of the effective resonant medium over many cycles leads to the usual resonance enhancement. The curvature energy injected in each cycle accumulates with the quality factor Q=Ω 2Γ,(124) 51
so that in steady state the spatial envelope and its associated stored curvature energy acquire the multiplicative gains A−→ Q A, ρeff −→ Q2ρeff.(125) The squared dependence for ρeff follows from its scaling with the field intensity. Accordingly, the full causal dynamics of the resonant Channel 1 response may be represented in steady state by the spatial Helmholtz formulation supplemented by the single factor Q that encodes temporal memory. Temporal and spatial interference therefore represent two facets of the same phenomenon: the coherent superposition of curvature oscillations in an effective resonant medium with finite memory time τmem, unified by the damping rate Γ. 9 Steady–State (Attractor) Solutions The causal–resonant feedback framework developed above naturally gives rise to a hierarchy of long–lived, self–organised configurations that correspond to the observed families of galactic morphologies. From a dynamical–systems perspective, each configuration represents a steady–state attractor: a stable eigenstate of the coupled baryon–curvature feedback loop introduced in Section 6.2. The term “attractor” retains its mathematical meaning—the asymptotic state toward which the combined field–matter system converges under many cycles of curvature excitation, memory, and re–emission. In the steady regime, all Channel–1 curvature oscillations share the same carrier frequency Ω, and their spatial structure is encoded in the complex envelope A (x) introduced in Eq. (92) . In this limit, the governing field equation reduces to the Helmholtz envelope equation (93), whose spatial operator Lres−space[A]≡ ∇2A+k2 ΩA, k2 Ω=Ω2+i2ΓΩ c2 eff ,(126) is the Channel–1 resonant operator. In coherent systems (e.g. spiral discs) the wavenumber kΩ is set by the intrinsic resonant scale of the medium, while in disordered systems (e.g. ellipticals) it may be replaced by an effective value keff obtained through the scattering and transport formalisms of Sections 5.3 and 5.4. In all cases, Lres−space governs the spatial structure of the steady resonant envelope that sources the Channel–2 Newtonian potential through the stored energy density ρeff. A stationary attractor is obtained when the spatial curvature envelope A (x) becomes a fixed point of the feedback loop. In this limit the field satisfies a renormalised Helmholtz relation of the form Lres-space +G1/2 env IA= 4πG ρb,(127) where G1/2 env represents the scalar spatial gain inherited from Foldy–Lax rescattering, speckle–mediated phonon diffusion, and geometric reverberation within 52
the resonant medium, and I is the identity operator. This factor encodes how the environment modifies the effective susceptibility of the curvature field: in the absence of spatial structure one has G1/2 env = 0, whereas in a strongly structured system it provides the multiplicative correction that lifts the bare Helmholtz operator to its effective, feedback–supported form. Temporal memory—controlled by the quality factor Q = Ω / (2Γ) and discussed in Section 8.5—acts as an independent multiplicative gain and is not included explicitly in Eq. (127) . In practice we often absorb the temporal amplification into the definition of the renormalised envelope, A→Q A , leaving Eq. (127) to represent purely the spatial amplification encoded by G1/2 env . The resulting normalised eigenfunction A (x) describes the stable Channel–1 curvature mode selected by the local scattering environment, geometric boundary conditions, and damping state. Physically, these steady–state attractors are self–maintaining resonant states. Curvature energy dissipated by damping is continually replenished through coherent or statistical re–excitation by the baryonic ensemble. In the coherent limit—most clearly realised in spiral disks—the system forms a globally ordered, m = 1 standing–wave–like mode analogous to the Foldy–Lax multiple–scattering regime (Section 5.2), where phase–locked rescattering produces spatially coherent gain. For elliptical galaxies, the kinematic disorder increases, and the global phase order is lost and replaced by a statistical balance of intensity, described by the effective–medium and transport equations of Sections 5.3 and 5.4. In such regimes the effective curvature field no longer forms a standing wave but a dynamically sustained intensity distribution that is steady in its ensemble average. The continually fluctuating speckle field redistributes curvature energy stochastically, maintaining the velocity–dispersion pattern that sustains the mode. Despite these differences, all steady configurations satisfy the same causal energy balance: (i) curvature energy injected through organized baryonic motion must equal that removed by damping, and (ii) the local resonant state B = ( ceff, Ω , Γ) relaxes to a stationary form. Each attractor is therefore simultaneously an eigenmode of Eq. (127) and a fixed point of the causal feedback cycle. Once established, it perpetually re–excites itself through the spacetime memory channel described in Section 8.5, while remaining bounded by its own damping and phase statistics. The sections that follow identify three characteristic attractor classes emerging from this unified feedback framework: (i) the spiral–disc configuration, a globally coherent m = 1 resonant mode sustained by ordered rotational streaming; (ii) the elliptical configuration, a statistical or “random–resonant” state governed by diffusive curvature transport and 4D speckle statistics; and (iii) the cluster–core configuration, a nonlinear extension of the elliptical regime in which overlapping diffusive tails undergo collective curvature amplification, yielding a concentrated yet statistically stable resonant core. 53
9.1 Spiral Galaxies as Coherent m= 1 (Laguerre–Gaussian) Modes 9.1.1 Observing the Steady–State Solution The flat rotation curves of spiral galaxies provide a natural setting for a long–lived, single–frequency resonance of the Channel 1 effective curvature field. Empirically the rotational profile satisfies v(r)≃v0, ω(r) = v0 r,(128) so that all radii share nearly the same orbital speed while retaining their usual differential angular motion. In the present framework this kinematic structure does not impose any global azimuthal synchrony on the baryons; instead, it provides an ordered, predictable background flow in which the effective curvature field can select and sustain its own coherent spatial mode. Throughout this section we adopt the causal decomposition developed in Sections 6.2–8 with three clearly separate concepts. The resonance itself—the existence of a single collective frequency Ω and its scalar response kernel χ⋆ (Ω)— belongs to the SRT sector (Concept 1). The baryonic disc contributes only through the kinematic source term and its polarisation tensor Pij (Concept 2), while all environmental reinforcement arises through multiple scattering (Concept 3). Spiral galaxies lie squarely in the coherent regime, so the driver of the m = 1 mode is the complex, phase–weighted first moment of the baryonic density, whereas the gain is supplied by the full stellar population. The only synchrony required for the resonance is temporal. Because v0≪c , the relativistic factor γ(r) = 1 p1−v2 0/c2,(129) is essentially constant across the disk, giving a uniform proper–time relation dτ =dt γ0 .(130) This near–uniformity equalises the oscillation rate of all emitters in proper time, allowing every baryonic element to drive the medium at a common temporal frequency Ω. It synchronises the drive, not the azimuthal motion. The physical origin of this temporal driving is the familiar epicyclic sloshing of stars in a shearing disk. Each star oscillates radially at the local epicyclic frequency κ ( r ), but only the slow, shear–filtered component of this sloshing remains phase coherent across the flat–velocity annulus. It is this coherent, low–drift component—not the full epicyclic motion itself—that selects the effective driving frequency Ω ∼ω ( r0 ). A detailed discussion of this filtering mechanism is given in the next subsection. Once all emitters oscillate at a common temporal frequency Ω, the curvature medium can support a rotating eigenpattern. The large–scale spiral inherits an 54
m=1 azimuthal phase and rotates at the same frequency: Ωpattern = Ω,(m= 1).(131) This is a property of the curvature mode, not of the baryonic disk: the pattern rotates at Ω because the medium responds at this frequency, while the baryons continue to orbit with their natural rate ω(r) = v0/r. For a flat rotation curve, and for any chosen pattern frequency Ω pattern lying between the inner and outer angular velocities of the disk, there exists a corotation radius r0for which ω(r0) = Ωpattern.(132) The shear in the neighbourhood of r0 is relatively weak on dynamical timescales, so baryons there drift only slowly through the pattern and maintain a strong instantaneous projection onto the m = 1 Fourier harmonic. A broad annulus around r0 therefore contributes coherently to the global mode, while more distant radii contribute with smoothly decreasing weight. Thus a flat rotation curve enhances and stabilises the m = 1 curvature eigenpattern: the ordered, low–shear environment sustains a large region with high m= 1 coherence. Spatial organisation of the mode is maintained by rescattering of the Channel 1 field within the baryonic ensemble (Section 5). Each oscillating mass element emits curvature waves that propagate, scatter, and partially reinforce one another, producing a field whose dominant structure is the single–frequency, m = 1 curvature eigenpattern. The detailed gain mechanism (Foldy–Lax multiple scattering) is discussed later; it suffices here that this rescattering network supplies the spatial reinforcement needed for a persistent, phase–coherent mode. The crucial point is that the curvature mode is selected by the scalar SRT susceptibility χ⋆ (Ω), not by disk geometry. Geometry enters only through the source polarisation Pij , which fixes how the coherent sloshing component projects onto the scalar resonant kernel. Once Pij isolates the effective m = 1 driver, the spatial structure of the mode is governed entirely by the scalar Green function GΩand the ensuing multiple–scattering gain. Thus the global spiral morphology is the observable imprint of a spacetime– synchronous eigenstate: a coherent m = 1 mode of the Channel 1 operator Lres−space , temporally aligned by the single proper–time frequency Ω and spatially shaped by rescattering in the baryonic disk. The following theorem establishes the existence and stability of this coherent m= 1 spacetime mode. Theorem (Existence and Stability of a Coherent m = 1 Spacetime Mode). Consider a differentially rotating, finite–thickness disk with an approximately flat rotation curve v ( r ) ≃v0 . Let the curvature response of the surrounding medium be described by the driven, damped Helmholtz envelope equation ∇2+k2 ΩA=4πG c2 eff ρb,(133) 55
where A is the complex Channel–1 curvature–potential amplitude, kΩ = (Ω + iΓ)/ceff with Γ>0, and ceff is the curvature–wave speed. Suppose the baryonic emitters drive the medium at a single temporal frequency Ω. Then Eq. (133) admits nontrivial steady–state solutions whose azimuthal dependence consists of a single Fourier harmonic, A(r, θ, z, t)=ℜne−iΩteiθ ˜ A(r, z)o,(134) for some bounded spatial profile ˜ A ( r, z ). These solutions represent coherent, rotating m= 1 curvature patterns. A self–consistent mode arises when the driving frequency matches the orbital frequency at some radius r0, Ω = ω(r0) = v0 r0 ,(135) so that baryons near r0 experience minimal phase drift relative to the pattern. For flat rotation curves a unique such radius exists for any choice of Ω ∈ (0 , ωmax ), and the weak shear around r0 ensures that a broad annulus contributes coherently to the m= 1 harmonic. Because Γ > 0, the homogeneous part of Eq. (133) is exponentially damped. Thus the driven configuration (134) is a linearly attracting steady state: perturbations decay and the system converges toward the coherent m = 1 pattern. The resulting configuration defines a self–consistent steady state in which the effective dark–mass distribution and the baryonic rotation field remain mutually compatible. Corollary (Resonant Coherence Condition). The corotation relation (135) should be interpreted as a steady–state attractor: for the frequency Ω satisfying Ω = ω ( r0 ), the effective curvature field at each point remains phase coherent with its past, and the driven solution is stable under damping. Consequently the m= 1 Fourier sector dominates the long–time response of the disk. Remark on Vertical structure. For numerical work it is convenient to approximate the vertical dependence of ˜ A ( r, z ) by the lowest vertical eigenfunction Z0 ( z ) of the finite–thickness disk and to assume approximate separability, ˜ A ( r, z ) ≈M1 ( r ) Z0 ( z ). This yields an effective 2D in–plane problem for the radial envelope M1 ( r ); the approximation is useful but not required for the existence of the global, phase–locked m= 1 solution. 9.1.2 Energy Source: Epicyclic Sloshing with Coherent Filtering In Section 9.1.1 above, for clarity, we emphasised the breakdown of three concept levels. The following analysis concerns only the driver (Concept 2); the scalar susceptibility and resonant frequency have already been fixed by the SRT mechanism (Concept 1), and the gain mechanism is treated separately in Section 9.1.6 (Concept 3). 56
The temporal driving of Channel 1 ultimately originates in the familiar epicyclic motion of stars in a differentially rotating disk. A star on a nearly circular orbit executes small radial excursions, r(t)=rg+Acosκt +ϕ,(136) where rg is the guiding centre, A the oscillation amplitude, ϕ a phase, and κ ( r ) the epicyclic frequency. For a flat rotation curve the standard kinematic identity gives κ(r) = √2ω(r), ω(r) = v0 r,(137) so the sloshing of the radial coordinate—and therefore of the local surface density—occurs at a frequency higher than the orbital rate by a factor √2. Taken over the full disk, this epicyclic motion produces a broad continuum of sloshing frequencies κ ( r ). However, not all of this power is available to the coherent m = 1 mode. The curvature field is highly selective: only those sloshing components whose phases remain slowly drifting relative to the global pattern can contribute constructively. Over the flat–velocity annulus the orbital frequency ω ( r ) varies only weakly with radius, and shear therefore acts as a sharp temporal filter. The only stars whose radial sloshing remains phase coherent over many dynamical times are those whose epicyclic frequency lies within the linewidth of the resonant response, |κ(r)−Ω|≲Γ,(138) where Γ is the damping rate appearing in the Helmholtz operator. Over the coherence time τcoh ∼ (2Γ) −1 the relative phase drift |κ− Ω |τcoh then remains of order unity or smaller, allowing these stars to contribute to a long–lived coherent drive. This condition selects a narrow sloshing band centred on the pattern frequency Ω ≃ω ( r0 ), where r0 is the corotation radius defined by Eq. (132) . Accordingly, the effective driving frequency is not the full kinematic epicycle κ = √2ω , but the shear–filtered component of the sloshing that survives temporal dephasing. Only this small, shear–protected subset of the stellar population contributes constructively to the coherent first azimuthal moment developed in Section 9.1.3. Stars whose sloshing phases drift rapidly relative to the pattern do not add to the coherent drive, even though they continue to oscillate energetically. Crucially, however, all stars participate in rescattering of curvature waves. Every mass element contributes to the Foldy–Lax environmental amplification described in Section 9.1.6, so the gain scales with the full number of scatterers N⋆ , not merely with the coherent set. In this sense coherence determines the effective source, while the entire baryonic population determines the gain. Epicyclic sloshing therefore supplies the raw oscillatory energy; shear filtering isolates a single temporally coherent frequency Ω; and multiple scattering provides the large–scale spatial reinforcement. Together these mechanisms produce the temporally locked, spatially coherent m = 1 resonant mode that underlies the observed spiral halo. 57
rather than Q|χ⋆|N⋆ , reflecting the randomisation of temporal phases along the scattering network. Here Q = Ω / (2Γ) is the temporal memory factor, and ϵ is the coherent first–moment fraction extracted in Section 9.1.3. Equation (172) is the spiral–galaxy specialisation of the general environmental gain structure encoded in Eq. (79) : the root– N⋆ growth is the scalar gain Genv , while the factor ϵenforces the planar, m= 1 source projection. A crucial point is that this temporal incoherence does not destroy the m = 1 spatial symmetry of the mode. The rotational symmetry of the Helmholtz Green function ensures that the azimuthal harmonics form an orthogonal eigenbasis: GΩ(r, θ;r′, θ′) = GΩ(r, r′, θ −θ′).(173) Thus the propagation operator commutes with ∂/∂θ . Starting from a field of the form A(0)(r, θ) = A(0) r(r)eiθ,(174) a single Foldy–Lax scattering step gives A(1)(r, θ) = ZGΩ(r, r′, θ −θ′)A(0) r(r′)eiθ′r′dr′dθ′,(175) and performing the angular integral isolates only the m= 1 component: A(1)(r, θ) = eiθ A(1) r(r).(176) Since the kernel contains no m = 1 harmonics, no azimuthal mixing can occur unless the medium itself breaks axisymmetry. Iterating the Foldy–Lax hierarchy therefore preserves the m = 1 structure at all orders, even though temporal phases have been statistically erased. The resulting physical picture is therefore hierarchical and fully consistent with the three conceptual layers outlined earlier: 1. SRT and scalar susceptibility (Channel 1 definition): The scalar response χ⋆ (Ω) and the unique frequency Ω define the curvature mode that exists; 2. Source kinematics (polarisation): The sloshing covariance Pij projects the scalar susceptibility into the planar, m = 1 temporal–spatial driving channel that launches the Laguerre–Gaussian mode; and 3. Environmental gain (multiple scattering): Incoherent Foldy–Lax scattering produces the Genv = √N⋆ gain, with geometry fixed by Pij through Eq. (79) and amplitude scaling given by Eq. (172). Thus the spiral pattern is a global resonant state whose geometry is selected coherently by baryonic forcing and symmetry, while its amplitude is determined by the statistical accumulation of many incoherent scattering events within the same planar gain channel enforced by Pij and Gij env. 9.1.7 Stored Energy, Dark–Mass Equivalence, and Environmental Gain The observable (quasi–static) curvature bias arises from the stored oscillatory energy of the resonant field. From Section 8.2, the local field energy density and 64
its mass equivalent are EΦ(x) = 1 8πGhc2 eff |∇A|2+ Ω2|A|2i, ρeff(x) = EΦ(x) c2.(177) Substituting the normalized field A ( r, θ, z ) of Eq. (171) and separating variables gives |∇A|2=∂rAr 2|Z0|2+|Ar|2 r2|Z0|2+|Ar|2∂zZ0 2.(178) For the exponential vertical profile Z0 ( z ; r ) = e−|z|/Lz(r)/p2Lz(r) , we have ∂zZ0 2≃ |Z0|2/L2 z. The energy density therefore becomes ρeff(r, θ, z) = |A(r, θ, z)|2 8πGc2hΩ2+c2 eff∂rln Ar 2+1 r2+1 L2 zi.(179) In the adiabatic WKB limit, the local dispersion relation k2 Ω = k2 r + m2/r2 + k2 z (with the vertical term replaced by its evanescent magnitude 1 /L2 z ) renders the bracket approximately constant: Ω2+c2 effk2 r+m2 r2+1 L2 z≃Ω2+c2 effk2 Ω≈2 Ω2,(180) since kΩ= Ω/ceff and Γ≪Ω. Thus, to good approximation, ρeff(r, θ, z)≃2 Ω2 8πGc2|A(r, θ, z)|2.(181) To include the cumulative effect of multiple scattering in a statistically homogeneous ensemble, the field amplitude is scaled by the environmental reinforcement factor, A(r, θ, z)−→ Aeff (r, θ, z) = QpGenv A(r, θ, z),(182) where Q is the temporal quality factor of the curvature resonance, and Genv represents statistical amplification of field amplitude through repeated scattering and re–emission. For a long–wavelength field ( λΩ≫ mean inter–scatterer spacing), individual scatterers respond additively in amplitude, yielding Genv =N⋆|X|2.(183) where N⋆ is the number of baryonic scatterers within the effective coherence volume and |X| ≤ 1 quantifies their mean coupling efficiency to the global oscillation frequency Ω. The factor |X| thus measures how strongly each scatterer participates in the collective resonance. Substituting Eq. (182) into Eq. (181) gives the final expression for the effective energy density: ρeff(r, θ, z) = 2 Ω2 8πGc2Genv Q2|A(r, θ, z)|2,Genv =N⋆|X|2.(184) The result (184) therefore provides the complete description of the statistically amplified dark–mass density in terms of the intrinsic mode amplitude, resonance quality factor, and the effective number of coupled baryonic scatterers. 65
9.1.8 Numerical Results: 3D Thin–Disk Model The complete numerical results obtained from the three–dimensional thin–disk model of Section 9.1.5 are shown in Figures 4–6. All quantities and algorithms are implemented exactly as specified in the simulation code given in Appendix B. The rotation frequency is fixed by the flat circular speed v0 = 230 km s−1 at a reference radius r0 = 8 . 5 kpc , giving Ω = 8 . 77 × 10 −16 s−1 and a resonant wavelength λres ≃ 46 . 44 kpc . An effective propagation speed ceff = 2 × 10 5m s−1 is adopted, while the quality factor Q = 12 . 6 is chosen such that the resulting damping length Ldamp = ceff/ Γ ≃ 192 . 17 kpc (where Γ = Ω / (2 Q )) reproduces the observed halo extent and the asymptotic flattening of typical spiral–galaxy rotation curves. In this formulation the total resonant amplitude arises directly from the combination of the temporal coherence Q , coupling efficiency X , and geometric projection factor ϵ , giving an overall scaling proportional to Q|X|√N⋆ϵ . Unlike earlier parameterizations in which external gain was inserted by hand, these parameters now enter self–consistently through the integral Helmholtz solution. The resulting field strengths, density contrasts, and rotation curves remain identical in form to those of the original model, confirming the internal normalization of the updated implementation. Figure 4 presents the one–dimensional radial solutions derived from the coherent m = 1 integral formulation. Part (a) shows the field amplitude |A ( r ) | , which rises from zero at the origin to a well–defined maximum near r≃ 6 kpc , before declining exponentially on scales comparable to the damping length. Part (b) displays the logarithmic mass densities: the effective curvature density ρeff (blue) and the baryonic midplane density ρb (orange dashed). The effective component decreases almost linearly in log–space and approaches the background level near r≃ 650 kpc . Part (c) compares the baryonic, resonant, and total rotation curves. The total velocity remains approximately flat beyond r∼ 10 kpc , consistent with observed spiral–galaxy kinematics. Part (d) shows the enclosed–mass ratio Meff/Mb , which increases from ∼ 10 at 7 kpc to ∼ 14 at 100 kpc , ∼ 17 at 150 kpc , and approaches its asymptotic value of ≃ 20 by r≃ 200–250 kpc . These results confirm that the thin–disk model recovers both the outer flattening of the rotation curve and the canonical dark–to–baryon mass ratio inferred for Milky–Way–like galaxies. Figure 5 illustrates the two–dimensional midplane structure of the same m = 1 resonant mode over a 100 × 100 kpc domain. Part (a) shows the normalized amplitude |A| , revealing the expected toroidal morphology peaking at r≃ 6 kpc . Part (b) displays the normalized azimuthal phase arg ( A ), which forms a continuous one–armed spiral with a single 2 π winding, indicating a coherent global m = 1 mode. Part (c) overlays the baryonic (red) and resonant (green) densities, showing their partial spatial displacement within the disk plane. Part (d) shows the total (baryonic + resonant) density on a linear scale; only the bright baryonic core and nearby resonant envelope are visible, as the extended halo falls below 66
Figure 4: Radial profiles and rotation curves. (a) Field amplitude |A ( r ) | ; (b) effective and baryonic mass densities (logarithmic); (c) rotation curves for baryonic, resonant, and total components; (d) enclosed–mass ratio Meff/Mb . Results from the 3D thin–disk Helmholtz model (Section 9.1.5); numerical parameters as defined in Appendix B. 67
the display contrast. Figure 5: Midplane amplitude and density maps. (a) Normalized amplitude |A| showing the toroidal resonant structure; (b) normalized phase arg ( A ) exhibiting the spiral m = 1 winding; (c) combined baryonic (red) and resonant (green) densities; (d) total normalized density on a linear scale. Spatial domain: 100 ×100 kpc. Figure 6 shows the corresponding three–dimensional effective–density distribution. Part (a) presents the horizontal (XY) midplane slice, and part (b) the vertical (XZ) cross–section, each over a 200 × 200 kpc domain. The XY projection again displays the toroidal halo structure, whereas the XZ projection reveals a vertically stratified envelope extending to heights of order |z|∼Lz≃ 57 . 65 kpc , comparable to the resonant wavelength λres ≃ 46 . 44 kpc . This scale follows from the evanescent vertical branch of the disk dispersion relation and indicates that the effective curvature field remains confined within a few wavelengths 68
of the midplane rather than filling the entire damping region. The resulting morphology is a vertically stratified, wave–supported halo consistent with the expected resonant geometry. Figure 6: Three–dimensional effective density distribution. Normalized slices of ρeff for the (a) horizontal (XY) midplane and (b) vertical (XZ) cross–section, each across a 200 × 200 kpc domain. The vertical structure extends to |z| ∼ Lz≃57.65 kpc, consistent with the resonant wavelength scale λres. Although the separable thin–disk ansatz captures the essential resonant behaviour and global halo morphology, it does not constitute a full numerical solution of the three–dimensional Helmholtz equation. More accurate results could be obtained by solving the complete 3D problem without enforcing separability and using the exact baryonic density distribution as a source term. Such an approach would refine the vertical structure, mode coupling, and amplitude normalization within the stellar disk. 9.1.9 Physical Interpretation The global m =1 spiral mode emerges as the dominant steady–state solution of the driven Helmholtz equation when the baryonic disk is written as a complex, phase–weighted source. Its first azimuthal Fourier component ρ1 ( r ) isolates the shear–filtered, temporally coherent part of the stellar sloshing—the only portion of the broad epicyclic spectrum whose phase drifts slowly enough to drive the global curvature mode at the carrier frequency Ω 9 . This m =1 projection captures the largest shear–protected fraction of the organised in–plane motion and therefore excites a curvature field with the same azimuthal symmetry. The spiral 9 “Coherent” refers strictly to temporal coherence at frequency Ω; the stellar orbits themselves are not spatially phase–locked. 69
pattern is thus not an imposed disturbance but the natural, symmetry–selected eigenmode of a rotating, phase–biased baryonic source. A key physical ingredient is the planar geometry of the sloshing covariance Pij . As developed in Section 5.6, the microscopic susceptibility of each baryon is the scalar χ⋆ (Ω); all directionality in the collective resonant response arises from the source–projected covariance, χij eff =χ⋆(Ω) Pij,Gij env =Genv Pij.(185) In a rotating disk the sloshing field is almost entirely in–plane, so Pij possesses two large lateral eigenvalues and a strongly suppressed vertical one. The curvature channel can therefore accumulate energy only within this planar subspace: the resonant mode inherits the ( x, y ) geometry of Pij and cannot amplify motion or field structure in directions that the sloshing statistics do not supply. This explains why the global spiral mode is intrinsically planar from the outset. This planar geometry is not imposed externally; it is co–evolved through Channel 2. The quasi–static Newtonian curvature generated by the stored energy shapes the stellar sloshing field, while the same sloshing covariance projects the scalar susceptibility into the planar gain channel. The driver and the gain therefore reinforce each other: the disk provides the planar Pij that selects the m =1 mode, and the resulting mode maintains the very sloshing geometry that sustains it. Once this planar polarisation is established, the first azimuthal Fourier component ρ1 ( r ) of the phase-weighted baryonic density provides the dominant coherent driver. Because ρ1 ( r ) captures the largest organised part of the stellar motion, it naturally excites the m =1 curvature pattern with the same polarisation and the same azimuthal phase. This mode is not chosen arbitrarily: it is the symmetryselected eigen-solution of the anisotropically polarised resonant channel when driven by a rotating, phase-biased, planar mass distribution. One might ask why the steady–state attractor does not drive the stellar disk toward a perfectly synchronous m = 1 azimuthal motion, as would occur in a rigidly rotating system. The reason is dynamical: a rising rotation curve would eliminate the shear needed to propagate non–axisymmetric structure, while a falling (Keplerian) curve introduces excessive shear that destroys any pattern on a timescale much shorter than the oscillation period. Only a flat rotation curve—the characteristic kinematic feature of spiral galaxies—creates an extended near–corotation domain: a radial band over which the orbital frequency varies slowly enough that the epicyclic sloshing of stars remains close to the global driving frequency Ω. Within this band the sloshing phases drift slowly relative to the pattern, surviving the shear filtering described in Section 9.1.2. This broad near–corotation region is the essential dynamical ingredient that allows a low–order azimuthal mode, particularly m =1, to persist over tens of kiloparsecs without being sheared apart. The modest magnitude of the azimuthal projection factor, ϵ≃ 0 . 1–0 . 3, is not an adjustable parameter but a direct consequence of this shear–filtered coherence. 70
Only stars whose epicyclic sloshing frequencies lie within the resonant linewidth, i.e. those satisfying |κ ( r ) − Ω |≲ Γ, maintain temporal phase coherence over an oscillation cycle and therefore contribute to the coherent first moment. Material significantly inside or outside the near–corotation annulus experiences too rapid a phase drift (from differential rotation and the κ/ω gradient) to contribute coherently. As a result, only a fraction of the disk’s organised orbital energy projects into the m = 1 eigenmode, while the remainder contributes only incoherently. Fractions of order 10–30% are typical for global modes in shearing disks, and because the resonant curvature channel has a large quality factor ( Q≫ 1), even this partial coherent projection provides ample power to maintain the global m=1 mode and its associated halo amplitude. Although the baryonic disk selects the m =1 mode coherently through its first azimuthal moment, the subsequent build–up of the curvature field is governed by incoherent Foldy–Lax multiple scattering (see Section 5.2). The crucial point is that the three–dimensional Helmholtz Green function GΩ (x , x ′ ) depends only on the difference θ−θ′ when expressed in cylindrical coordinates. This azimuthal invariance ensures that the scattering operator commutes with ∂/∂θ: Seimθf(r, z)=eimθ S[f(r, z)],(186) for any integer m . Thus the m quantum number is preserved by all scattering events. The global m = 1 pattern is therefore robust: it is selected coherently at the source level (temporally at Ω) but its amplitude is amplified statistically through random rescattering without ever mixing into other azimuthal modes. The geometry is coherent; the gain is incoherent; the polarisation is protected by symmetry. The curvature envelope A ( r, θ, t ) carries a well-defined azimuthal phase eiθ and therefore possesses an associated orbital angular momentum (OAM), in complete analogy with optical vortex modes and OAM-carrying fields used in quantum communications and holographic optical tweezers [ 14 ]. Because the underlying Channel–1 field arises from a Lagrangian density, one expects a corresponding dynamical law to be obtainable by varying the action with respect to the baryonic degrees of freedom. Such a law would describe the torque exerted by the m = 1 curvature pattern on the stellar disk and quantify the transfer of angular momentum between the field and the orbiting baryons, thereby reducing azimuthal phase drift near corotation. Although we do not derive this torque equation here, its existence is physically well motivated: a rotating, phase-coherent field generically entrains material motion toward its own pattern speed. In the present context this implies that the m = 1 curvature mode can dynamically stabilise itself by gradually synchronising the baryons (in the temporal, frequency–locking sense), thereby increasing the effective coherence factor ε ( r ) over secular timescales and promoting long-term persistence of the global spiral resonance. Only four sets of physical parameters enter the global mode: (i) the resonant frequency Ω, (ii) the effective propagation speed ceff , (iii) the single–scatterer coupling parameters ( X, ϵ ), and (iv) the quality factor Q . There are no further 71
normalisation freedoms. The reference radius r0must lie somewhere within the observed flat portion of the rotation curve, but its precise location is not uniquely fixed by theory. In practice we selected a value near the inner edge of the flat profile, r0≃ 8 . 5 kpc , which yields v0≃ 230 km s−1 and therefore fixes the global frequency through Ω = v0/r0 . Although this choice is not a priori unique, it produces a broad near–corotation region and leads to excellent agreement with the observed halo structure. The effective propagation speed ceff is not introduced as a fit parameter but follows directly from the requirement that the resonant wavelength λres = 2 πceff/ Ω match the observed radial extent of spiral–galaxy halos. Given the extremely small global frequency Ω = 8 . 77 × 10 −16 s−1 fixed by the flat rotation curve, a natural wavelength of order λres ∼ 40–60 kpc can be obtained only if ceff ∼ 10 5m s−1 . Larger values would shift the resonant scale to unphysical, megaparsec lengths, while smaller values would confine it to only a few kiloparsecs. Thus the adopted value ceff = 2 × 10 5m s−1 is fixed by the requirement that the global curvature mode possess a wavelength comparable to the observed halo span of Milky– Way–like spirals, rather than by empirical fine-tuning. The geometric projection factor ϵ quantifies the m =1 component of the baryonic disk and is bounded by |ϵ| ≤ 1; a conservative value ϵ≃ 0 . 3 is adopted. Similarly, X≃ 1 corresponds to near-maximal single–scatterer coupling to the global mode. With these quantities fixed, the only remaining adjustable parameter is the quality factor Q = Ω / (2Γ), which controls the amplitude of the resonant mode and the radial extent of the halo through the damping length Ldamp = ceff/ Γ. A value Q≃ 12 . 6 simultaneously reproduces: (i) the observed halo extent ( Ldamp ≃ 192 kpc ), and (ii) the overall resonant amplitude required to obtain the empirical dark–to–baryonic mass ratio of ∼ 20. By contrast, the resonant wavelength ( λres ≃ 46 kpc ) and the vertical confinement scale ( Lz≃ 57 . 6 kpc ) are fixed by the chosen propagation speed ceff and temporal frequency Ω, and therefore do not depend on Q . Given that the correct amplitude and spatial extent emerge from a single tunable parameter, this appears to provide strong internal consistency for the resonant interpretation. It is enlightening to emphasise the extreme slowness of the resonant frequency itself. For the adopted parameters Ω = 8 . 77 × 10 −16 s−1 , the corresponding oscillation period TΩ≃2.3×108yr is a substantial fraction of a gigayear. Over the ∼ 1 . 4 × 10 10 yr age of the Universe the global mode executes only O (60) such oscillations. In this regime the quality factor Q≃ 12 . 6 has a transparent interpretation: it measures how many of these ultra–slow cycles contribute coherently to the stored curvature energy, corresponding to a memory interval of roughly one fifth of the cosmic age. Even if the entire stellar disk were to disappear instantaneously, the stored effective density ρeff would decay only at the intrinsic rate 2Γ, with a half-life τhalf ≃ ( ln 2) / (2Γ) ≈ 10 9yr . Thus the dark–mass analogue would remain substantially present long after the removal of its baryonic drivers. Finally, the resonant curvature mode behaves as a damped oscillator with natural 72
frequency Ω and damping rate Γ, so that microscopic dissipation removes effective gravitating density at the rate Ploss = 2Γ ρeff . In the steady state this loss is exactly replenished by coherent orbital driving from the baryons, and the resulting mode amplitude is fixed by the combined action of four physical factors: the quality factor Q = Ω / (2Γ), the single–scatterer coupling strength |X| , the geometric m =1 projection factor ϵ , and the statistical Foldy–Lax gain √N⋆ from the N⋆ baryonic emitters within a coherence volume. Together these give the scaling Aeff ∝Q|X|pN⋆ϵ, (187) expressing that the halo is a self–maintaining resonant structure whose long-term amplitude is determined by the balance between orbital energy injection and dissipation at rate Γ. 9.2 Ellipticals as Random Resonant Media 9.2.1 Observing the Statistical Steady State Unlike spirals, where shear filtering isolates a narrow coherent driving frequency from the sloshing spectrum, ellipticals possess no such frequency selection. Their epicyclic and orbital motions remain fully phase-mixed, so the curvature field is driven stochastically rather than coherently. Spirals maintain a coherent m =1 curvature mode through organized rotation and phase–locked rescattering; ellipticals, by contrast, lack any global azimuthal order. The random orbital motion of stars continually scrambles relative phases, so that coherence survives only locally while the ensemble as a whole behaves as a diffusive resonant medium. Curvature energy is still stored and re–emitted through the same causal mechanism, but now in a statistically fluctuating rather than phase–locked form. This transition from coherent to diffusive resonance is the same one described in radiative–transport theory for acoustic and elastic wave fields [ 15 , 16 , 9 , 4 , 5 ], where random rescattering replaces global phase locking with ensemble–averaged diffusion. The stochastic driver of this diffusive response is the broad r–mode sloshing spectrum of a pressure–supported stellar system. Unlike the e–mode sloshing in spirals, which is shear–filtered into a narrow m =1 band, the r–mode spectrum spans a wide range of frequencies set by the 3D orbital distribution. Although most of its power resides at high frequencies, it retains finite spectral weight down to the collective resonance frequency Ω, ensuring that random stellar motion can continue to excite the gal–mode even in the absence of any global ordering. This broadband, four–dimensional fluctuation field establishes the statistical geometry developed in Section 9.2.3, where the curvature response is shown to form a dynamically evolving 4D speckle pattern rather than a single coherent standing wave. It is the temporal power spectral-density of this speckle field at gal-mode frequency Ω that provides the stochastic driving required to sustain the resonant mode. 73
Because the curvature response possesses a Lorentzian window of width 2Γ around the carrier, any fluctuation satisfying |ω−Ω|≲2Γ (205) retains phase coherence across the interval τcoh and contributes to the driving. 13 Since the speckle spectrum itself is generated by stochastic rescattering with characteristic frequency ∼Ω, it necessarily overlaps this window. Because the speckle field continually decorrelates on the microscopic timescale τcoh = (2Γ) −1 (Section 9.2.3), the source term entering the forced–damped Helmholtz equation contains temporal fluctuations with characteristic frequencies of order 2Γ. After filtering through the Lorentzian susceptibility χ ( ω ) introduced earlier, only those components satisfying |ω− Ω |≲ Γ drive a significant response of the curvature amplitude A (x , t ). Since the effective gravitating density depends directly on the instantaneous curvature field through ρeff ∝ Ω 2|A|2 + c2 eff|∇A|2 (Section 9.2.8), these filtered r–mode fluctuations appear immediately as fluctuations in ρeff ( t ). Thus the r–mode agitation modifies the effective density on the microscopic coherence timescale set by Γ, while the long–time, orbit-averaged envelope of ⟨ρeff⟩ is governed separately by the transport–renormalised damping rate Γeff. Thus the very randomness of a pressure–supported stellar system supplies a self–consistent energy source for its own resonant curvature mode. Ellipticals sustain their dark–mass analogue not through orbital order (as spirals do) but through the high–frequency fluctuations inherent in their 3D speckle dynamics. 9.2.5 Ensemble reduction: masson–speckle summation We now rewrite the forcing in Eq. (194) in terms of the statistically independent four–dimensional coherence cells derived in Section 9.2.3. Each spatial coherence volume Vcell = α ℓ2 ⊥ℓ∥ defines an elementary unit of the ensemble environment, within which the stellar phasors remain phase–locked and the local effective curvature field evolves coherently over the memory time τcoh ≃ 1 / (2Γ). Cells separated by more than ℓ⊥in space or ∆tcell in time are fully decorrelated. Stars within a common cell therefore act as a single coherent source of effective mass Mcell , while the full galaxy forms a random, time–varying mosaic of such locally coherent “masson” excitations. The effective curvature field can thus be expressed as a superposition of statistically independent phasors: A(x, t) = 4πG c2 eff N3D X c=1 NT X j=1 M(c,j) cell GΩ x−xc,(206) 13 Spirals require a phase–locked m = 1 driver, so the relevant bandwidth is the phase–coherence window |κ− Ω |≲ Γ. Ellipticals have no global phase order; their stochastic 3D speckle field injects energy incoherently, so the appropriate condition is the full Lorentzian power width |ω− Ω |≲ 2Γ. Thus spirals use the coherence width, ellipticals the energy width. 80
where c indexes spatial cells and j indexes independent temporal realizations within the field’s memory window. Temporal multiplicity: curvature memory. Each spatial cell stores a finite number of statistically distinct phasors within its coherence time. Using the definitions of τcoh and the oscillation period 2 π/ Ω, the number of independent temporal realizations per cell is NT=τcoh ∆tcell = 1 2Γ 2π Ω =Ω 4πΓ=Q 2π,(207) so that each cell contributes NT independent temporal samples of its masson excitation. The density–controlled damping corollary (Section 2.4) established that Γ = κ Ω with κ = O (1). Consequently the ratio Ω / Γ entering NT is itself order unity, so that each coherence cell contributes only a modest number of independent temporal realisations per resonant period. This ensures that the temporal granularity of the r–mode driver is naturally matched to the resonance bandwidth, consistent with the speckle–driven forcing mechanism established in Section 9.2.4. Intensity expansion. Taking the square modulus of Eq. (206) gives |A(x, t)|2=4πG c2 eff 2 X c,j X c′,j′ M(c,j) cell M(c′,j′) cell ∗GΩx−xcG∗ Ωx−xc′,(208) from which ensemble averages may be computed. Statistical independence of distinct cells and temporal windows gives DM(c,j) cell M(c′,j′) cell ∗E=(|Mcell|2, c =c′, j =j′, 0,otherwise, (209) so all cross terms vanish. The mean field intensity is therefore D|A(x, t)|2E=4πG c2 eff 2 NT N3D X c=1 |Mcell|2GΩ(x−xc) 2.(210) For a Poisson stellar distribution, |Mcell|2=N2 cellm2 ⋆+NcellVar(m⋆),(211) and for Ncell ≫1 this simplifies to |Mcell|2≃M2 cell =N2 cellm2 ⋆.(212) Using N3D =Vgal/Vcell and Ncell =N⋆Vcell/Vgal,we obtain N3D M2 cell =Vgal Vcell N⋆Vcell Vgal 2 m2 ⋆=N2 ⋆m2 ⋆ Vgal Vcell.(213) 81
It is now convenient to collect the ensemble coupling terms into a single dimensionless environmental gain factor Genv . In Section 5.1 this quantity was defined microscopically as the total intensity gain relative to the incident field reducing in the fully coherent Foldy–Lax limit to Genv = N2|χ⋆|2 [Eq. (60) ]. In the effective–medium picture of Section 5.3, the same quantity was shown to scale inversely with the coherence volume Vcell through Genv ∝ 1 /Vcell reflecting the fact that larger coherence cells produce stronger local phase alignment. In the present, fully explicit cell–counting picture we include both the local coupling per cell and the number of cells filling the galaxy. To keep track of the mean stellar coupling efficiency introduced earlier, we write the ensemble environmental gain as Genv := |X|2N3D |Mcell|2 m2 ⋆≃ |X|2N3D N2 cell =|X|2N2 ⋆Vcell Vgal =|X|2N2 ⋆ Vgal α ℓ2 ⊥ℓ∥ =|X|2N2 ⋆ Vgal αλres 2π2v 2Γ,(214) where Vcell = α ℓ2 ⊥ℓ∥ is the spatial coherence volume from Eq. (198) , ℓ⊥ = λres/ (2 π ) is the transverse coherence length (Eq. (196) ), and ℓ∥ = v/ (2Γ) is the longitudinal coherence length (Eq. (197) ). The dimensionless factor |X|2≤ 1 represents the mean stellar coupling efficiency already introduced in the scattering theory: in the coherent Foldy–Lax limit it multiplies the intensity as Genv ∝|X|2N2 , while here it weights the total cell–ensemble gain in exactly the same way. These successive equalities make explicit how the ensemble environmental gain retains the same conceptual structure as in the scattering theory. The first line shows the statistical definition, where the coherent mass content of each cell, |Mcell|2 , is weighted by the mean coupling efficiency |X|2 and multiplied by the number of spatial and temporal realizations. The second line reformulates this in geometric terms, introducing the total stellar population N⋆ and the ratio Vcell/Vgal , which connects the microscopic cell statistics to the macroscopic galactic volume. The third line then expresses the same quantity in explicit physical scales, replacing the coherence lengths ( ℓ⊥, ℓ∥ ) with the resonant wavelength λres and damping time 1 / Γ, thereby revealing how Genv depends on the underlying causal and geometric parameters of the medium. Thus the earlier local relation Genv ∝ |X|2/Vcell for a single coherent patch is naturally extended here to the global ensemble gain: the per–cell response scales as |X|2/Vcell , while the total number of coherently contributing sources scales as N2 ⋆Vcell/Vgal . Their product yields the dimensionless Genv relevant for ellipticals. The replacements Γ → Γ eff and Ldamp →Ldamp,eff will be introduced in the following subsection when transport effects are included. 82
Substituting Eqs. (207), (213), and (214) into Eq. (210) finally gives D|A(x, t)|2E=4πG c2 eff 2Q 2πGenv GΩ(x−xc) 2,(215) which makes explicit how the environmental gain Genv , the temporal memory Q/ (2 π ), and the Green–function geometry combine to determine the observable curvature–field intensity in a statistically mixed stellar system. This result forms the starting point for the transport–enhanced description developed in the next subsection. 9.2.6 Transport Mean Free Path and Collective Damping The ensemble formulation developed above treats the effective curvature field as a superposition of kinematically driven stellar sources, each of which contributes phase-modulated curvature phasors within its local coherence window. In realistic galactic environments, however, each star acts simultaneously as (i) a kinematic source of curvature phonons generated by its orbital motion, and (ii) a compact elastic scatterer governed by the same forced–damped Helmholtz operator that propagates and damps the field. Through continual emission, scattering, and reabsorption, these mass elements generate a diffusive interference network whose statistics mirror those of resonant ultrasound and diffuse field correlations in complex materials [ 15 , 9 , 4 , 5 , 17 , 18 , 19 ]. The governing dynamics are elastic, obeying the same damped Helmholtz equation and transport identities as acoustic phonons in a diffuse medium. Following the formalism of multiple–scattering theory [ 8 , 20 ], the coherent field propagation is not determined by the microscopic damping rate Γ but by an effective transport wavenumber that incorporates the ensemble self–energy Σ(Ω): k2 eff =k2 0+ Σ(Ω), k0=Ω+iΓ ceff .(216) The self–energy Σ(Ω) encodes the cumulative phase delay and dwell time produced by the resonant multiple-scattering network, not by the emitters themselves. Its imaginary part defines the transport damping rate Γeff =ceff Im keff,(217) and the associated effective damping length Ldamp,eff =ceff Γeff .(218) In the diffusive regime, Γeff ≪Γ, Ldamp,eff ≫Ldamp =ceff Γ,(219) expressing the key transport result: resonant multiple scattering shortens spatial coherence but greatly extends the effective damping length. Curvature phonons 83
undergo repeated dwell–time interactions before decorrelating, continuously recycling part of their energy back into the coherent mode. In this regime the extended dwell time implied by Γ eff ≪ Γ does not alter the intrinsic temporal resonance window, which remains set by the microscopic linewidth 2Γ of the forced–damped curvature oscillator. The overlap condition for stochastic driving, |ω− Ω |≲ 2Γ, therefore continues to be governed entirely by Γ, not by Γ eff . The transport damping rate Γ eff controls only the long–time retention and recycling of curvature energy within the ensemble, enhancing the quality factor and the damping length, while leaving the driver–resonance frequency matching unchanged. This recycling produces an effective quality factor Qeff =Ω 2 Γeff ≫Q=Ω 2 Γ,(220) so that the observed large Q values in elliptical halos are not microscopic properties of single emitters but collective transport effects of the stellar ensemble. In this regime, all ensemble relations of Section 9.2.5 retain their formal structure under the replacements Γ→Γeff, Q →Qeff, Ldamp →Ldamp,eff,(221) and the environmental gain itself becomes transport–renormalized through the effective longitudinal coherence length ℓ∥,eff =v/(2Γeff ), yielding Genv,eff =|X|2N2 ⋆ Vgal αλres 2π2v 2Γeff .(222) This form preserves the same microscopic structure as Eq. (214) , but with Γ replaced everywhere by its transport value Γ eff to account for the collective damping reduction of the ensemble medium. The angularly averaged Green function of the multiply scattered curvature phonon field defines the causal transport kernel Keff(r) = exp−2r/Ldamp,eff (4π)2r2,(223) which represents the spherically symmetric energy envelope of the coherent curvature mode. It provides the physical basis for the closed halo law derived in the next subsection: a long–range, slowly decaying effective curvature field whose extended elliptical morphology arises naturally from the diffusive recycling of curvature energy within the transport medium. Substituting Eqs. (222) and (221) into the ensemble intensity (215) , and replacing the microscopic propagator |GΩ|2 by the transport kernel Keff ( r ), yields the compact transport–enhanced expression D|A(r)|2E=4πG c2 eff 2Qeff 2πGenv,eff Keff (r).(224) 84
This form makes the transport renormalization of both temporal and spatial coherence explicit: Genv,eff now embodies the reduced damping rate Γ eff , while Qeff and Keff ( r ) encode the extended memory time and spatial transport length. It provides the physical closure for the ensemble halo law, from which the core regularization and gradient–energy correction will follow. 9.2.7 Core Regularization in the Non–Transport Regime Before introducing the full energy conversion in Section 9.2.8, we first establish the spatial morphology implied by the transport solution for the field intensity ⟨|A ( r ) |2⟩ . At this stage, only the relative scaling of the envelope is needed: the absolute normalization and physical prefactors from the curvature–energy law will be applied in the next subsection. Accordingly, the discussion below treats ρeff ( r ) as a quantity proportional to ⟨|A ( r ) |2⟩ , with the understanding that the proportionality constant will be specified later. The transport theory described in the previous section predicts a far–field profile proportional to exp ( − 2 r/Ldamp,eff ) /r2 . This expression is exact for the ensemble–averaged coherent mode of a multiply scattered curvature–phonon field in the diffusive regime; that is, at radii beyond several coherence lengths where angular isotropy and statistical decorrelation hold [ 20 , 8 ]. However, the same form cannot apply arbitrarily close to the centre. At small radii: (i) stellar scatterers are closely packed, (ii) direct (ballistic) propagation dominates over diffusion, and (iii) the local speckle field becomes dominated by the coherent mean–field curvature rather than by ensemble statistics. In this inner region, the diffusion approximation implicit in the ensemble–transport kernel Keff ( r ) ∝ 1 /r2 breaks down. As r→ 0, Eq. (224) would therefore diverge unphysically, signalling the transition from the statistical (Dyson) regime back to the coherent Foldy–Lax limit.14 Curvature flux must remain finite at the centre of any bound system. For a spherically symmetric, stationary envelope, the net curvature energy flux crossing a sphere of radius rmust vanish smoothly as r→0; in particular, there can be no singular point source or sink at the origin. Writing the flux schematically as F(r)∝4πr2ρeff(r),(225) regularity requires that F(r) decrease at least as fast as r3, F(r)∝r3,(r→0),(226) which implies the small–rscaling ρeff(r)∝r, (r≪ℓ∥),(227) 14 Only the spatial transport approximation breaks down as r→ 0. The underlying speckle field—and therefore the temporal coherence scale τcoh = (2Γ)−1that sets the r–mode driving spectrum—remains unchanged. What fails is the diffusive propagation model, not the temporal forcing or the Lorentzian response window of the gal–mode. 85
where ℓ∥ = v/ (2Γ eff ) is the longitudinal coherence length of the ensemble–averaged transport field (not the microscopic coherence length v/ (2Γ) associated with the driver spectrum). This linear–core behaviour is therefore a natural, nonsingular choice: it regularizes the 1 /r2 divergence of the diffusive kernel while ensuring finite central flux and a smooth approach to the coherent (inner) regime. We choose a matching radius Rmatch of order the outer stellar scale radius, where the field has undergone many scatterings and the transport description becomes valid: Rmatch ∼ O(few Re)∼5 kpc for typical ellipticals.(228) The regularized envelope then takes the generic form ρeff(r)∝ r, r < Rmatch, exp[−2r/Ldamp,eff ]/r2, r ≥Rmatch. (229) Both the density and its effective flux 4 πr2ρeff are continuous at r = Rmatch by construction. The inner linear law is thus not an ad hoc empirical patch but a regularized limit of the transport solution: it removes the unphysical central divergence of the diffusive kernel, keeps the central flux finite, and matches smoothly onto the exponential–over– r2 transport tail. It predicts: cored density profiles in ellipticals and clusters, a flat circular–velocity contribution at small radii, and a smooth transition to the exponential–over–r2transport tail. The core radius is fixed by the coherence scale ℓ∥ = v/ (2Γ eff ), while the outer halo is controlled by the transport damping length Ldamp,eff . Together, these two scales define a two–zone structure—a coherent core joined continuously to a diffusive halo—that remains fully consistent with the transport hierarchy. In what follows, we introduce the curvature–energy conversion and gradient–energy correction that complete the physical normalization of this envelope, yielding the final quantitative halo law. 9.2.8 Gradient–Energy Boost and Final Transport–Regularized Halo Law Having established the spatial morphology of the coherent curvature envelope, we now complete its physical normalization by restoring the curvature–energy relation and including the gradient–energy contribution. This step connects the ensemble field intensity ⟨|A ( r ) |2⟩ to the observable effective density ρeff ( r ) and yields the final closed halo law. The local curvature–energy density is ρeff(x) = 1 8πGc2Ω2|A(x)|2+c2 eff |∇A(x)|2,(230) where the first term represents the potential energy of the global curvature oscillation and the second its elastic curvature (gradient) energy. 86
Defining the correlation length ℓcby ℓ−2 c:= |∇A|2 |A|2,(231) and substituting this into the averaged curvature–energy expression (230) gives ρeff(r)=Ω2 8πGc2D|A(r)|2E1 + c2 eff Ω2ℓ2 c≡Ξ∇ Ω2 8πGc2D|A(r)|2E,(232) where the dimensionless gradient–energy boost is Ξ∇= 1 + 2π Ldamp,eff λres 2 ,(233) obtained by setting ℓc≃Ldamp,eff and Ω = 2 πceff/λres . This gradient–energy boost enhances the total curvature energy stored in the coherent mode without changing the spatial shape of the halo. In the ballistic limit, Ξ ∇∼O (1); in the transport limit, it can become very large when Ldamp,eff ≫λres. Multiplying Eq. (224) by the gradient–energy boost Ξ ∇ and applying the small– r regularization from Eq. (229) yields the final, fully normalized transport halo law: ρeff(r)=4πG c2 eff 2Qeff 2πΞ∇Genv,eff r, r < Rmatch, e−2r/Ldamp,eff (4π)2r2, r ≥Rmatch. (234) For convenience, we collect the transport–renormalized parameters: Genv,eff =|X|2N2 ⋆ Vgal αλres 2π2v 2Γeff , Qeff =Ω 2Γeff ,Ξ∇= 1 +2πLdamp,eff λres 2 . (235) Together, these factors determine the amplitude of the regularized curvature phonon halo: Genv,eff encodes the coherent mass content of the stellar ensemble, Qeff sets its temporal memory, and Ξ ∇ provides the gradient–energy boost associated with transport–extended correlations. The resulting law defines a smooth, finite–core, exponentially damped envelope that links microscopic scattering dynamics to the macroscopic curvature structure of elliptical halos. 87
9.2.9 Numerical Results The boosted transport–halo law (Eq. 234) predicts a self-consistent, two–scale halo structure for elliptical galaxies: a small-radius linear core (Section 9.2.7) and a large-radius transport tail ∝exp ( − 2 r/Ldamp,eff ) /r2 , with the amplitude controlled solely by the resonant wavelength λres . The numerical results presented here are obtained using the reference model defined in Appendix C, with Ldamp,eff = 636 kpc tuned to match the observed halo extent of massive ellipticals, and λres = 27 pc chosen to reproduce a realistic dark–to–baryon mass ratio. Figure 7 shows radial diagnostics for the resulting ensemble–averaged halo. Panel (a) reveals that the effective dark density becomes comparable to the stellar density at ∼ 5 Re ( ∼ 25 kpc), rises above it, and gradually dominates throughout the transport tail. Panel (b) quantifies this with the radial density ratio ρeff/ρb , which reaches a maximum near r∼ 700 kpc. Panel (c) shows that the dark–induced rotation curve rises rapidly from the core, develops a modest ∼ 200 km,s −1 shoulder extending over several tens of kiloparsecs, and then declines slowly. This behavior is consistent with that observed in luminous ellipticals [ 21 ]. Panel (d) displays the enclosed mass ratio Meff/Mb : unity near 27 kpc, ∼ 3 by 100 kpc, ∼ 7 by 300 kpc, and asymptoting to ∼ 11 at ∼ 1 . 3 Mpc. These values fall squarely within the observed dark-to-luminous mass fractions for giant ellipticals in deep X-ray and weak-lensing surveys. Figure 8 visualizes the spatial structure of the halo in the central ± 50 kpc: panel (a) overlays the stellar and effective components using a log-scaled composite (red: baryons; green: curvature–transport dark mass), highlighting the strong baryon dominance near the center and the smoother, more extended spatial footprint of the resonant component. Panel (b) shows their combined log-normalized density distribution. Finally, Fig. 9 shows XY and XZ slices extending to ± 1 Mpc, demonstrating the near-spherical envelope of the coherent transport tail. The horizontal and vertical cuts are visually indistinguishable, confirming the expected isotropy of the far-field ensemble solution. The two free quantities in the model have clear and distinct roles: Ldamp,eff ⇒halo radial extent, λres ⇒total dark mass fraction. Tuning λres and Ldamp,eff therefore provides a transparent calibration of ellipticals along their observed halo size–mass sequence without any hidden or fine-tuned parameters. The results above constitute a first demonstration that resonant curvature–phonon transport alone can reproduce the canonical dark-mass phenomenology of pressure-supported galaxies. 9.2.10 Halo Formation from Coherent Energy Flux The boosted transport halo of Eq. (234) can equivalently be derived from energy–flux conservation in the source–free exterior region. In this picture, the 88
Figure 7: Radial halo diagnostics for the boosted transport–law model. (a) Effective vs baryonic mass densities. (b) Density ratio. (c) Circular velocities showing a modest ∼ 200 km s −1 shoulder. (d) Enclosed mass ratio rising to ∼ 11 at ∼1.3 Mpc. 89
with rs and ρs as characteristic scale radius and density, respectively. Recent extensions include the introduction of a “splashback” radius Rsp , beyond which ρ(r) rapidly steepens [24]. The theoretical basis for the cluster-scale treatment presented in this section, lies in the long-standing radiative transport framework developed for diffuse acoustic and optical propagation. Weaver [ 15 , 16 ] established that acoustic energy in heterogeneous solids obeys a radiative transfer equation for the intensity field, with distinct ballistic, coherent, and diffusive regimes. These works provided the first quantitative formulation of acoustic diffusion and localization, defining the statistical kernel that our own model generalises to astrophysical scales. The subsequent experimental validation was achieved by Derode, Tourin, and Fink [ 9 , 4 , 5 ], who investigated ultrasonic propagation through random ensembles of resonant scatterers. Their experiments confirmed the radiative transport regime predicted by Weaver, demonstrating the separation of coherent, ballistic, and diffuse components, as well as the emergence of reverberation gain in multiply scattered fields. These resonant-rod systems represent direct analogues of our elliptical galaxies: individual mesoscopic resonators embedded in a diffusive host that collectively generate extended energy-density tails. Later developments by Weaver and Lobkis [ 17 , 18 ] established that diffuse fields from independent random ensembles can exhibit weak cross-coherence, recoverable through cross-correlation measurements. This phenomenon—now widely used in seismology and acoustics—formalised the concept of overlapping diffuse transport fields, providing a rigorous physical basis for the small nonlinear coupling parameter ηused in our cluster model. Finally, Larose et al. [ 19 ] unified these concepts across acoustics, optics, and geophysics, showing that the superposition of diffuse fields from independent sources can reproduce effective Green’s functions through reverberant correlations. This provides the closest experimental and theoretical precedent for our treatment of overlapping elliptical transport envelopes within clusters: random, extended resonators (galaxies) embedded in a diffusive medium (intracluster baryons) that generate partially coherent, overlapping transport tails and a measurable ensemble reverberation gain. The ensemble behaviour demonstrated in the acoustic and optical transport experiments of Weaver and Derode provides the direct methodological foundation for the present cluster model. Here we apply the same radiative–transport formalism—previously shown to describe overlapping diffuse fields in random media—to the collective curvature–transport regime of spacetime itself. Each elliptical galaxy sustains its own diffusive envelope, generated by the random stellar motions that define its internal transport equilibrium, and these envelopes coexist within a shared effective curvature field (the common transport continuum of spacetime) that supports their mutual overlap. The resulting ensemble of overlapping transport intensities forms a statistically smooth superposition whose 96
weak cross–coherence produces the modest reverberation gain Grev =1+Ngalη, (251) which, when applied at the cluster scale, reproduces the NFW–like curvature of the effective density profile without invoking any additional physical assumptions. In the sections that follow, we adopt identical methodologies to those used in the acoustic and optical transport studies, applying them directly to the astrophysical transport regime to construct and normalize the cluster-scale effective density field. Each elliptical or lenticular acts as amesoscale driver of the effective curvature field, ρeff,i(r)∝exp(−r/Lmean),(252) where Lmean =ceffτtr (253) is the mean transport distance determined by the effective transport time τtr . Within a cluster, these envelopes overlap and interfere weakly. Their mutual correlation defines a small but finite reverberation parameter η, ⟨AiA∗ j⟩=ηq⟨|Ai|2⟩⟨|Aj|2⟩,(i=j),(254) yielding the statistical amplification Grev =1+Ngalη, (255) where Ngal is the number of contributing resonators. This weak collective gain reproduces the macroscopic curvature excess associated with the observed cluster halo. Theorem (Statistical Transport Theorem for Galaxy Clusters). Let a cluster contain Ngal elliptical or lenticular galaxies, each producing a stationary diffusive envelope ρeff,i ( r ) ∝exp ( −r/Lmean )within a shared effective transport field of speed ceff and damping rate Γ tr . If the ensemble satisfies the statistical balance condition dρeff dt =G −L ≃ 0,(256) then the ensemble–averaged effective density obeys ρeff(r)∝Grev exp[−2r/Lmean] r2,(257) with Lmean =ceff Γtr , Grev =1+Ngalη. (258) The convolution of these overlapping exponential envelopes approaches the observed NFW form ρ(r)∝r−1(1+r/rs)−2for Lmean ∼rs. 97
Interpretation. Equation (275) defines the steady transport equilibrium: the effective curvature energy density, governed by transport, is continually replenished by embedded elliptical and lenticular galaxies. The relevant length scale Lmean depends on the transport properties of the medium, not on the baryonic density or resonant frequency. Ellipticals act as isotropic diffusive emitters; lenticulars, with their oblate morphology, generate aspherical transport tails preferentially elongated toward the cluster centre—a behaviour consistent with X–ray and weak–lensing observations indicating that S0s are radially aligned and concentrated toward cluster cores. Spiral galaxies can contribute negligibly to the cluster effective curvature field because they have no transport tails and are correspondingly observed to populate only the periphery. The same transport equations extend naturally to filaments, which represent the cylindrical continuation of the cluster ensemble. In this geometry, overlapping diffusive envelopes align along a filamentary axis, and the mean gain is reduced by geometric dilution: Gfil ≃Grev Lmean Rfil ,(259) where Rfil is the filament radius. Although weaker in amplitude, these linear transport ensembles preserve the same physical mechanism—curvature energy diffusion and reverberation coupling among extended resonators—and thus naturally explain the emergence and persistence of cosmic filaments. Finally, the effective curvature field that sustains the cluster halo is self–maintaining in the same sense as in smaller systems. The apparent dark mass produced by the collective amplification Grev is also the mechanism that enhances the coupling among member galaxies: the increased curvature energy density focuses transport back toward dense regions, strengthening overlap and maintaining nonlinear equilibrium. The cluster is therefore a self–regulated, radiative–transport system—a statistically sustained effective curvature field whose ensemble dynamics reproduce both the amplitude and morphology of observed dark–matter halos. 9.3.2 Mathematical Formulation: Overlapping Elliptical Transport Envelopes Each elliptical galaxy acts as an effective source of the curvature–phonon field amplitude Ai (r), whose ensemble–averaged intensity ⟨|Ai|2⟩ is given by the single–galaxy transport solution derived in Eq. (224) . At the cluster scale, the total field amplitude is the coherent sum Atot(r) = Ngal X i=1 Ai(r),(260) so that the ensemble–averaged intensity becomes |Atot(r)|2=X i⟨|Ai|2⟩+X i=j⟨AiA∗ j⟩.(261) 98
The cross–correlation term ⟨AiA∗ j⟩ represents the weak mutual coherence between overlapping transport tails. Following the standard radiative–transport treatment of diffuse field superposition [15, 9, 4, 17, 19], we write ⟨AiA∗ j⟩=ηijq⟨|Ai|2⟩⟨|Aj|2⟩,(262) where 0 ≤ηij ≪ 1 measures the fractional coherence of the pair. For statistically independent galaxies, ηij fluctuates randomly about a small mean value η . The total intensity therefore reads |Atot|2≃X i⟨|Ai|2⟩(1+Ngalη),(263) identifying the dimensionless reverberation gain Grev =1+Ngalη. (264) This amplitude–level formulation maintains the same causal hierarchy as the elliptical case: the field intensity arises from the quadratic combination of statistically independent amplitude realizations, and the weak coherence factor η provides a small multiplicative enhancement due to overlapping transport regions. Analogous ensemble treatments appear in radiative–acoustic transport and time–reversal acoustics [ 15 , 9 , 5 , 18 , 19 ], where the reverberant energy of independent diffuse sources produces a measurable intensity gain. The corresponding effective energy–density field for the cluster follows directly from the ensemble intensity law of Eq. (224) . Each elliptical contributes a transport–renormalized amplitude envelope ⟨|Ai|2⟩ ∝ Keff ( | r − r i| ), including the gradient–energy boost factor Ξ ∇ defined in Eq. (233) . The cluster-scale superposition is therefore ρeff(r) = Grev Ξ∇ Ngal X i=1 Keff(|r−ri|),(265) where Grev accounts for the weak reverberant coupling between overlapping transport tails. This discrete form will subsequently be replaced by its continuous analogue through convolution with the galaxy number–density profile ngal ( r ) in the following subsection. This formulation directly parallels the intensity–based radiative transport models developed for multiply scattering acoustic and elastic media. Weaver [ 15 , 16 ] first expressed the diffuse energy field as an ensemble average over incoherent amplitude realizations, while Derode, Tourin, and Fink demonstrated experimentally that weak mutual coherence among independent resonant clusters produces a measurable reverberation gain [ 9 , 4 , 5 ]. Subsequent correlation analyses by Weaver and Lobkis [ 17 , 18 ] and by Larose [ 19 ] showed that such diffuse fields retain small but finite cross–coherence, recoverable through cross–correlation of 99
independent realizations. Our treatment of the cluster intensity follows this same logic: the total field is the incoherent sum of independent elliptical transport envelopes, augmented by a small global coherence factor η representing the ensemble–averaged reverberant coupling between them. In principle, one might attempt to model the total cluster field directly from the ensemble of overlapping galaxy amplitudes, fully accounting for all phase correlations. In practice, however, this approach is both computationally prohibitive and conceptually fragile, as the individual phases are rapidly randomized by multiple scattering and transport effects. We therefore proceed by employing the standard practice of radiative transport theory to model phenomenon of this type: by constructing the effective intensity (or energy–density) field instead, retaining the correct ensemble statistics while averaging over the random phases. This yields a tractable and physically transparent formulation, which naturally leads to the stepwise development outlined below. Continuous Galaxy Distribution and NFW Analogy Observational studies indicate that the radial number–density profiles of cluster galaxies are well described by an NFW–like form, albeit with systematically lower concentrations than the corresponding dark–matter halos [ 25 ]. For analytical convenience and physical realism, we therefore model the spatial distribution of member galaxies as an NFW–type profile smoothly truncated at large radii by a “splashback” [24] term: ngal(r)=n0 Wsp(r) x(1+x)2, Wsp(r) = 1 1 + exp[(r−Rsp)/∆sp],(266) where x = r/rs , and Rsp and ∆ sp represent the splashback radius and its characteristic transition width, respectively. This continuous parameterisation replaces the ∼ 100–500 discrete galaxies of a typical rich cluster with a smooth number–density field appropriate for convolution with the single–galaxy transport kernel. A key interpretative point is that these galaxies are not static particles. Their long diffusive decay times ensure that, over many dynamical periods, the transport envelopes of individual members overlap throughout the cluster volume, forming an effective continuum of resonant sources. The Necessity of 2.5D Convolution A direct three–dimensional convolution of the form ρeff(r) = ZK(|r−r′|)ngal(r′)d3r′(267) does not yield physically consistent results because K ( r ) describes an angle–integrated intensity rather than a Green’s function. This distinction is fundamental: a Green’s function preserves full phase and directional information, whereas an intensity kernel embodies an ensemble average over those angles. The 100
issue is well known in transport theory and has been extensively discussed in the contexts of optical diffusion, ultrasound propagation, and acoustic multiple scattering [8, 4, 5, 15, 20]. The appropriate formalism—long established in both radiative and acoustic transport—is to employ an angularly averaged, reduced–dimensional representation: ρeff(r)=2πZ∞ 0 ngal(r′)r′⟨K(s)⟩µdr′, s2=r2+r′2−2rr′µ, (268) where ⟨K ( s ) ⟩µ denotes the Legendre–weighted average over µ = cos θ . This “2.5D” formulation removes redundant angular degrees of freedom and guarantees mathematical stability while preserving the correct physical form of the transport field. It omits only a single geometric length factor, recoverable through explicit normalization. Such dimensional reductions are standard practice in radiative transfer [ 8 ], ultrasound diffusion [ 4 , 5 , 15 ], and optical multiple scattering [ 20 ]. In all of these contexts, the reduced formalism omits a single geometric length factor, producing an apparent scale ambiguity that is conventionally resolved by introducing a macroscopic normalization parameter, Λ geom , determined through global energy or mass conservation. In our case, because the convolution represents a direct superposition of intensities rather than fields, the appropriate normalization follows naturally from the total enclosed mass of the constituent galaxies. Writing ρeff(r)=Λgeom ρshape(r), we impose Z∞ 0 4πr2ρeff(r)dr =Ngal M(elliptical) DM ,(269) which yields the explicit expression Λgeom =Ngal M(elliptical) DM Z∞ 0 4πr2ρshape(r)dr .(270) This ensures that the integrated effective transport density recovers the combined dark–mass contribution of all elliptical members, anchoring the 2.5D convolution to a physically measurable mass scale without introducing additional free parameters. Reverberation Gain and Partial Coherence The parameter η quantifies weak statistical correlations among overlapping transport tails, representing the residual coherence that remains after ensemble averaging. Such partial coherence is a well–established feature of radiative and acoustic transport systems containing multiple scattering and long–lived resonances. In ultrasonic and optical analogues [ 4 , 5 , 15 , 20 , 18 ], the effect arises from recurrent scattering loops and time–delayed “reverberant” returns that couple neighbouring intensity fields. These processes lead to a small, positive excess in the ensemble–averaged intensity relative to the purely incoherent sum. 101
We model this effect phenomenologically through a nonlinear amplification, ρeff,rev(r)=Grev Λgeom ρshape(r), Grev =1+Ngalη, (271) where ρshape ( r ) is the normalized 2.5D convolution result and Λ geom enforces the global mass normalization. The reverberation gain Grev thus captures the small but finite cross–coherence among the overlapping transport fields of neighbouring galaxies. Laboratory measurements in acoustics and optics typically find η∼ 10 −3 –10 −2 , corresponding to total amplification factors Grev ≃3–10 for systems with many independent scatterers. This range coincides remarkably well with the amplification required for clusters: the ratio between the summed dark–mass contribution of all ellipticals and the total mass inferred from NFW fits is of precisely the same order. In this sense, the “reverberation gain” formalism provides a natural, physically motivated bridge between the mesoscopic transport behaviour of individual galaxies and the macroscopic gravitational potential of the cluster as a whole. Lognormal Distribution of Transport Lengths In many studies of radiative, optical, and acoustic transport through disordered or heterogeneous media, it is common practice to represent variability in local scattering or damping lengths using a lognormal ensemble rather than a single mean value. This reflects the multiplicative nature of scattering paths and the intrinsic heterogeneity of local environments, which naturally produce a lognormal distribution of effective transport lengths. Such formulations are well established in wave-diffusion and multiple-scattering analyses across acoustics [ 4 , 15], optics [20, 11], and mesoscopic transport theory [8]. Following this convention, we extend the kernel to an ensemble-averaged form, Kens(r) = ZK(r;L)P(L;Lmean, σln L)dL, (272) where P ( L ) is a lognormal probability density with logarithmic dispersion σln L . This procedure reproduces the empirically observed heavy tails in intensity transport and regularizes the large-scale overlap behaviour without introducing any ad hoc smoothing. Typical acoustic or optical diffusion studies employ 0 . 3 ≲σln L≲ 1 . 0 for moderately heterogeneous media, and occasionally σln L∼ 2 in strongly scattering or porous systems where transport paths vary over several orders of magnitude [ 20 , 8 ]. In the present context, a value of σln L = 2 . 5 yields a physically consistent and observationally well-matched profile: it preserves the correct NFW-like slope at large radii while capturing the required convex curvature near the core. The long-tail outliers in this distribution carry direct physical meaning—they represent rare instances in which a neighbouring galaxy’s transport tail penetrates a baryonic region, re-initiating a secondary radiative-transport process with an extended effective damping length. 102
While laboratory studies typically infer 0 . 5 ≲σln L≲ 2 . 0 for single-phase diffusive media [ 4 , 15 ], strongly resonant or dual-channel systems exhibit effective spreads approaching σln L∼ 2–3 [ 8 ], fully consistent with the broad heterogeneity expected in the cluster-scale double-transport regime considered here. Effective truncation of the ensemble. In practice, radiative-transport models rarely integrate the lognormal distribution to infinity. Rather, they approximate the ensemble average by sampling over a finite range of L/Lmean , typically within a few standard deviations of the logarithmic mean. This suppresses the statistical weight of extreme outliers without altering the intrinsic shape of K(r;L). Such effective truncation is standard practice in acoustic and optical diffusion analyses, where the physical coherence length of the medium imposes a natural limit on valid transport paths [ 4 , 15 , 8 ]. Our five-point sampling across ± 2 σln L follows this convention: it retains the full lognormal curvature while strongly suppressing contributions from exceedingly long paths ( L≫Lmean ). This preserves the physical broadness of the ensemble and the fidelity of the kernel’s form, exactly as in established radiative and ultrasonic transport treatments. 9.3.3 Numerical Results The numerical implementation follows directly from the formalism developed above. All calculations employ the reduced 2.5D convolution to evaluate the transport–overlap density ρeff ( r ), followed by application of the geometric normalization Λ geom to enforce total mass conservation, and the reverberation gain Grev = 1 + Ngalη to incorporate partial coherence between overlapping tails. The complete algorithm and parameter definitions are given in Appendix D, together with the Python code used to generate the figures. For a fiducial 10 15 M⊙ cluster, we adopt Ngal = 300 ellipticals, Lmean = 636 kpc , σln L = 2 . 5, and η = 8 . 3 × 10 −3 , corresponding to Grev ≃ 3 . 5. The resulting one-dimensional transport profiles are presented in Figure 10, which compares the model’s effective density ρeff ( r ) and enclosed mass Meff ( r ) with a standard NFW benchmark. All quantities were computed on a logarithmic grid spanning 0 . 2 kpc ≤r≤ 3 Mpc , using the 2.5D convolution for ρshape ( r ) and normalized by Λ geom as defined in Eq. (270) to preserve total enclosed mass. The reference NFW halo corresponds to M200 = 1015 M⊙and c200 = 4. Panels (a) and (b) of Figure 10 show the effective density in linear and logarithmic scales, respectively. The transport–overlap profile reproduces the NFW slope and amplitude across three orders of magnitude in radius, maintaining close agreement from ∼ 30 kpc to 3 Mpc . The slightly increased concavity of ρeff ( r ) at small radii ( r≲ 50 kpc ) possibly arises from the omission of the central brightest cluster galaxy (BCG), whose extended stellar envelope would naturally flatten the inner slope. The shallow dip near r→ 0 originates from the discrete angular integration and is purely numerical. Panels (c) and (d) display the corresponding enclosed-mass profiles. The total mass derived from the transport model follows the NFW prediction almost 103
Figure 10: Cluster density and enclosed mass. (a,b) Effective density in linear and log–log scales. (c,d) Corresponding enclosed-mass profiles. The transport–overlap model (solid) reproduces the NFW benchmark (dashed) over three decades in radius, with minor deviations at small r attributable to the missing central BCG component. 104
Figure 11: Circular velocity profiles. The transport–overlap prediction (solid) agrees closely with the NFW velocity curve (dashed) across the cluster, confirming that the cumulative mesoscopic transport reproduces the macroscopic gravitational potential. Figure 12: Two-dimensional distributions of effective density and velocity. (a) Normalized log10 ρeff intensity map. (b) Linear-scale circular-velocity field vc . Both panels exhibit smooth radial symmetry and NFW-like scaling. 105
and the stored curvature energy remains at the single–pass level: ρ(hom) eff ∼1 Nρ(inhom) eff ,(291) for systems with the same total mass, where N counts the discrete substructures available in the inhomogeneous case. Therefore a perfectly homogeneous baryonic medium cannot generate measurable curvature amplification in Channel 1 under any physical configuration. 11.3 Interpretation and Connection to Perturbation Theory The theorem shows that spatial contrast is essential for Channel 1. Without it, no Foldy–Lax loops form, no self–energy accumulates, and the collective resonant mode does not arise. This explains why the perturbative framework of Section 4.4 cannot be applied to a homogeneous system in isolation: the parameters (Ω , Γ , ceff ) exist only after rescattering has generated an effective medium. In the idealised circumstance of regime (3), where a homogeneous cloud is externally forced to oscillate at the global resonance within an already-existing effective medium, the response reduces to the driven Helmholtz convolution δΦ(x) = ZGΩ(|x−x′|)δρb(x′)d3x′,(292) but its amplitude remains faint, being limited to the single-pass level because no spatial contrast exists to sustain coherent feedback. The causal–resonant model therefore predicts that apparent dark structure correlates with baryonic contrast and temporal coherence, not with total baryonic mass: ρeff ∝(baryonic contrast) ×(coherence time).(293) In cluster collisions such as the Bullet Cluster (1E 0657 − 56), where dense, collisionless galaxies retain their spatial contrast while the diffuse plasma is stripped away, the curvature resonance must remain anchored to the galaxy component rather than to the displaced gas. This behaviour precisely matches observations without invoking non–baryonic matter, and yields a decisive test: if lensing peaks consistently coincide with regions of greatest baryonic contrast, the causal– resonant interpretation gains strong empirical support. 112
12 Emergence of Multiple Scattering from a Weakly Inhomogeneous Continuous Medium 12.1 A Brief Review of Garvitational Multiple Scatter Theory Before attempting to analyse the early Universe, it is useful to summarise the assumptions that underlie the effective–medium framework developed thus far. Throughout previous sections we have considered a baryonic mass distribution ρb(x) = N X j=1 mjδ(x−xj),(294) consisting of spatially separated stellar or galactic constituents. Within a coherent domain these constituents oscillate at a common curvature frequency Ω2∝G¯ρb,(295) where ¯ρbis the root–mean baryonic density over the coherent volume. Each baryonic element responds to the resonant curvature field as a damped oscillator, ¨ Xj+ 2Γ ˙ Xj+ Ω2Xj=αΦres(xj),(296) which reinjects curvature waves back into the medium. Substituting the induced oscillators into the Green–function representation yields the self–consistent Foldy– Lax equation, Φres(x)=Φinc(x) + ZGΩ(|x−x′|)σ(x′) Φres(x′)d3x′,(297) where the contrast–defined scattering strength is σ(x)∝δρb(x).(298) Applying the Foldy–Lax operator to both sides of Eq. (297) produces the renormalised effective–medium wave equation, ∇2Φres +1 c2 eff −∂2 t−2Γ ∂t+ Ω2Φres =4πG c2 eff ρb,1,(299) where ρb,1 denotes the slowly varying component of the baryonic distribution that drives the coherent mode. All renormalised parameters (Ω , Γ , ceff ) arise self– consistently from repeated rescattering encoded in Eq. (297) . The cycle–averaged stored curvature energy defines the effective density, ρeff(x)∝ ⟨|Φres(x)|2⟩.(300) 113
Crucially, nothing in Eqs. (295) – (300) requires ρb (x) to be discrete. Any baryonic distribution with sufficient spatial contrast—even a continuous density with internal structure— produces a nonzero σ (x) and therefore participates in the multiple–scattering mechanism. This observation is essential for cosmology: a continuous medium can become a resonant, scattering effective medium once it develops spatial inhomogeneity. However, the early Universe does not begin in that regime. The primordial baryon–photon plasma is extremely smooth, with δρb ρb∼10−5,(301) and therefore realises Regime 1 of Section 11.2, namely an isolated, homogeneous system with σ(x)≡0, Grev = 1,Φres = 0.(302) In this state no Foldy–Lax feedback operates, no effective self–energy can form, and the perturbation theory of Section 4.4 does not apply. There is therefore ρeff(x, t) = 0 (early homogeneous era).(303) Only when gravitational instability amplifies density fluctuations to δ∼ O (1) does σ (x) become nonzero and the Foldy–Lax transport mechanism turn on. The remainder of this section traces this transition: how a continuous, nearly homogeneous plasma dynamically evolves into the resonant, multiply–scattering environment required for the effective dark density described in the rest of this work. Although σ = 0 eliminates the multiple–scattering channel and therefore suppresses any renormalised feedback, it does not eliminate the underlying homogeneous curvature resonance. Even in a perfectly uniform baryon–photon plasma the bare oscillation ¨ Φres + Ω 2 0 ( t ) Φ res = 4 πG ¯ρb ( t ) δb remains active as a coherent, spatially homogeneous mode, exactly like the Langmuir oscillation in a uniform electron plasma. In this regime the cycle–averaged energy of the mode need not vanish, despite the absence of a scattering medium. The central result of the Spacetime Resoannce Theorem in Section 2 is that baryonic matter generically supports gravitational oscillation modes with natural frequency Ω 2∝G¯ρ , mirroring the familiar classical g–mode relation in stellar interiors. At late times such modes are driven by internal heat and pressure imbalances within stars, while in the early Universe they are excited by thermal and pressure fluctuations in the tightly coupled baryon–photon plasma. Thus, although the physical driver varies between epochs, the same gravitational restoring resonance is present throughout: baryonic matter at any density supports an oscillatory gravitational response, and any process capable of producing coherent compressions—whether stellar heat at late times or photon–baryon pressure at early times—can excite the same underlying curvature mode. 114
12.2 The Resonant Frequency in a Continuous Medium Before an effective medium forms, the local (bare) curvature resonance is determined solely by the coarse–grained baryon density: Ω2(x, t)∝G¯ρb(x, t),(304) In the early plasma, the coarse–grained density takes the form ¯ρb(x, t) = ¯ρb(t) [1 + δ(x, t)],|δ| ≪ 1,(305) so that Ω(x, t) = Ω0(t)1 + δ(x, t) 2+O(δ2).(306) Even tiny density perturbations therefore imprint a weak spatial modulation on the local curvature resonance. However, as the homogeneity theorem states, such fluctuations do not produce a nonzero scattering strength: σ(x)∝δρb(x)⇒σ≡0 (δ∼10−5).(307) Thus the early medium lies strictly in the non-scattering regime of Regime 1. 12.3 Weak Random Potential and the Ballistic Regime Formally inserting (306) into the resonant operator yields a Helmholtz equation of the form ∇2+k2 0+V(x, t)Φres =S, k0=Ω0(t) ceff ,(308) with weak random potential V(x, t)=2k0 δΩ(x, t) ceff +O(δ2)∝δ(x, t).(309) For a continuum with RMS amplitude σV and correlation length ℓc , random– medium theory gives the single–scattering mean free path ℓ−1 s(t)∝k4 0(t)σ2 V(t)ℓ3 c(t).(310) In the primordial plasma one has σV∼δ∼10−5, giving k0ℓs≫1,(311) so curvature propagation is entirely ballistic. There is no rescattering and hence no possibility of forming an effective medium. This resolves the apparent tension with Section 4.4: since no effective medium exists, the perturbative theory there does not apply at early times. 115
12.4 Nonlinear Growth Toward the Transport Threshold As gravitational instability amplifies density fluctuations, δ→ O(1), σV→ O(1), ℓc→galactic scales,(312) the combination in Eq. (310) grows by many orders of magnitude. There exists a cosmological epoch t∗defined implicitly by k0(t∗)ℓs(t∗)∼1,(313) which is the analogue of the Ioffe–Regel criterion for waves in random media. At t∗ curvature–phonon propagation transitions from the ballistic to the multiple– scattering regime. Beyond this point, the density field is no longer a weak perturbation, and Channel 1 becomes dynamically activated. 12.5 Emergent Resonant Patches and Effective Discreteness When δ∼ 1, the medium fragments into nonlinear overdensities. Within each patch, Ω(x)≃Ωpatch ∝√ρpatch,(314) so curvature energy becomes locally trapped. The cycle–averaged effective density grows as ρeff(x)∝|Φres|2,(315) leading to positive feedback: curvature trapping enhances mass inflow via Channel 2, sharpening contrast until each region behaves as a discrete, mesoscale resonant scatterer. These “resonant patches” constitute the progenitors of ellipticals and the basic units required for the Foldy–Lax and transport formalisms of Sections 5–9.3.4. This mechanism closely parallels modulational instability in plasmas, where spatial variations in ωp ( x ) ∝pne(x) produce a refractive index landscape that traps Langmuir waves. Here, fluctuations in Ω(x) trap curvature–phonons, giving rise to long–lived resonant envelopes that function as dark–matter analogues within the present theory. The predicted cosmological progression is therefore homogeneous plasma (Regime 1) −→ weak random medium (ballistic) −→ nonlinear resonant patches −→ multiple–scattering resonators (elliptical progenitors) −→ coherent disks (spirals) −→ transport ensembles (clusters, filaments). The onset of multiple scattering is not assumed but rather emerges dynamically when the mean free path falls to the resonant scale, enabling the same effective– medium physics that governs the present-day Universe. 116
12.6 Linear Limit and Connection to the CMB Power Spectrum The developments of Sections 5–9.3.4 demonstrate that multiple scattering, renormalised propagation, and ρeff arise only once the contrast σ (x) becomes nonzero. In the primordial plasma, however, Eqs. (301) – (303) imply that the Universe lies strictly in Regime 1, so that σ(x) = 0,Φres = 0,(Ω,Γ, ceff) undefined,(316) and no effective medium exists. Despite the absence of a scattering medium, the underlying bare oscillator— the curvature–phonon degree of freedom with frequency Ω 0 ( t ) from Eq. (304) — remains a well-defined linear response field. In Regime 1 all renormalised parameters revert to their bare values: ceff →c, Γ→0,Ω→Ω0(t),(317) and the effective–medium PDE (299) consistently reduces to the ballistic Helmholtz equation ∇2+1 c2−∂2 t+ Ω2 0(t)Φres(x, t) = 4πG c2¯ρb(t)δb(x, t),(318) which is simply the linear curvature response to the baryon overdensity. Fourier transforming Eq. (318) gives ¨ Φres(k, t) + Ω2 0(t) Φres(k, t)=4πG ¯ρb(t)δb(k, t),(319) showing that Φ res isaforced, linear oscillator sourced by the baryonic overdensity. Importantly, because σ = 0, there is no self-energy, no renormalisation, and no feedback loop: Eq. (319) adds no nonlinear gravitational effect. It is important to emphasise that Eq. (319) does not introduce an additional gravitational potential beyond the usual Newtonian metric perturbations (Φ N, Ψ N ). Rather, Φ res is a linear curvature response field driven by the same baryon fluctuations that also generate the Newtonian potentials. There is no independent source term and no modification of the Einstein equations in Regime 1. Consequently, the combination ΦN(k, t)−→ ΦN(k, t) + ϵ(t) Φres(k, t) (320) should be viewed merely as a bookkeeping device for how curvature–phonon oscillations produce a tiny correction to the effective driving of the photon Boltzmann hierarchy. It does not represent new gravitational physics or an additional source term. The dimensionless prefactor ϵ(t)≡Ω2 0(t) k2c2(321) 117
quantifies the relative size of this correction. It is possible to perform a basic parametric estimate at recombination. Using ¯ρb(z∼1100) ∼10−21 kg m−3gives Ω0(t∗)∼pG¯ρb∼10−17 s−1.(322) The comoving acoustic scale corresponds to k∼0.01–0.1 Mpc−1, i.e. k c ∼10−13–10−12 s−1.(323) Thus ϵ(t∗) = Ω2 0 k2c2∼10−8–10−10,(324) many orders of magnitude below unity. Therefore: ϵΦres ≪ΦN,(recombination era),(325) so the curvature–phonon contribution is far too small to affect the acoustic oscillations or the resulting CMB temperature and polarisation spectra. It is automatically consistent with Planck-precision constraints and cannot lead to double counting of metric perturbations. In summary, the Regime 1 limit of the model maps seamlessly onto the standard linear CMB hierarchy. The curvature–phonon field acts only as a tiny, strictly linear response driven by the same baryonic fluctuations already present in Φ N , and its contribution is O (10 −8 ) relative to the dominant gravitational potential. Only once δ→ O (1) does σ (x) become nonzero and the renormalised effective medium of Sections 5–9.3.4 begin to form. 13 Conclusion This work has shown that a small but physically essential feature of linearised general relativity—the full retarded Green–function response—combined with the ordinary damped–oscillator susceptibility of baryonic matter, yields a self–consistent and observationally successful extension of weak–field gravitational dynamics. No new fields, particles, or modifications of GR are introduced. When the usual quasistatic truncation of the retarded kernel is avoided, the delayed interaction between baryons and curvature generates a collective oscillatory mode whose cycle–averaged energy behaves as an effective dark–mass density. A central outcome of this work is that the entire phenomenology can be organised into a clear, four–level physical structure: (1) Existence. The Spacetime Resonance Theorem guarantees that any coarse–grained baryonic mass distribution supports a long–wavelength curvature mode (the gal–mode) with natural frequency Ω2∝G¯ρb. 118
(2) Driving. Time–varying baryonic structure feeds this resonant mode. In spirals the driver is coherent, phase–locked m = 1 sloshing; in ellipticals it is random microscopic agitation whose fluctuations are made locally coherent by the long–wavelength retarded Green function. Both mechanisms inject curvature energy into the resonant channel. (3) Gain. Multiple scattering of the curvature field produces self–energy, renormalising the propagation parameters (Ω , Γ , ceff ) and enabling amplification through Foldy–Lax loops and radiative transport. Where structure is coherent (spirals) or dense (ellipticals and clusters), this gain is positive; where the baryonic medium is smooth, the gain vanishes. (4) Storage and feedback. The resonant channel stores its energy over the long memory time τmem = Γ −1 , producing the quasi–static component ρeff . This stored component enters the Newtonian channel exactly as an effective mass density, shapes the gravitational potential, and in turn regulates the resonant frequency, the driver, and the gain environment. These four ingredients, assembled within a single renormalised curvature equation, reproduce flat rotation curves in spirals, exponential elliptical envelopes, cluster–scale NFW–like halos, filamentary coherence, and the observed silence of the early Universe and the CMB. The mechanism that enables this unification is minimal. Baryons slosh, the retarded kernel responds with finite memory, and multiple scattering shapes the collective mode. The resulting cycle–averaged curvature energy accumulates slowly but persistently into ρeff , which feeds the Poisson channel while the oscillatory component remains locally silent. This guarantees exact agreement with laboratory, Solar–System, and stellar tests of gravity. At the microscopic level the energy that ultimately populates ρeff originates entirely from baryonic motion. In spirals, coherent orbital sloshing excites the mode globally. In ellipticals, the rapidly fluctuating speckle agitation drives the mode statistically, with the retarded Green function enforcing the local field coherence required to sustain the resonant channel. In all systems, it is this continual baryonic agitation that supplies the energy later stored as effective dark mass. In summary, the causal–resonant framework shows that the observed dark–mass phenomena need not signal new gravitational laws or unseen particles. They emerge naturally from the interplay between the retarded response of general relativity, the dynamical susceptibility of baryonic matter, and the self–consistent feedback between coherent driving, resonant gain, and long–time storage. The large–scale gravitational structure of the Universe—its halos, clusters, and filaments—can thus be understood as a manifestation of the persistent memory and collective resonant dynamics already present within weak–field GR. 119
14 Dedication and Acknowledgement I would like to dedicate this book to the late Prof. John T. Sheridan, my former PhD supervisor and friend, who introduced me to wave optics and is sadly missed. I acknowledge with gratitude the work of Prof. Joseph Goodman, whose seminal texts Fourier Optics and Statistical Optics have remained page-turners for me for more than twenty years. His contributions underpin much of my understanding of wave theory and coherence. I am indebted to the work of Sheng, Derode, Roux, Fink, Weaver, van Tiggelen, and others in wave transport and multiple-scattering theory, an area I was not very familiar with when I began this project, yet essential for deriving the behaviours of elliptical galaxies and clusters. I sincerely hope I have not misinterpreted their theories. I gratefully acknowledge the support of Research Ireland / Science Foundation Ireland and the European Union, whose funding over the past twenty years enabled my research in optics and allowed me to deepen my understanding of physics. Last and most importantly, I thank my wife Karen and my daughters for their unwavering support as I wrestled with this theory over these past several months. 120
A Appendix: Bessel and Hankel Functions Used in the Main Text This appendix is provided purely for convenience and reference. It contains no new results or derivations but simply summarizes the definitions and basic properties of the Bessel and Hankel functions employed throughout Section 8.4–Section 9.1.4. Two distinct families appear in the resonant–curvature formalism: 1. the cylindrical Bessel functions Jm ( z ) and Ym ( z ), which solve the two–dimensional Helmholtz equation relevant to thin–disk geometries, and 2. the spherical (or “normal”) Bessel functions jℓ ( z ), which arise in three–dimensional interference or isotropic–propagation problems. Both sets are standard special functions with well–known analytic forms. A.1 Cylindrical Bessel and Hankel Functions The cylindrical Bessel functions of the first and second kind, Jm ( z ) and Ym ( z ), are the two linearly independent solutions of Bessel’s differential equation, d2 dz2+1 z d dz +1−m2 z2um(z)=0.(A.1) Jm ( z ) is finite at z = 0 and represents the regular solution, while Ym ( z ) diverges logarithmically and represents the singular one. For small arguments (z≪1), Jm(z)≃1 m!z 2m, Ym(z)≃ −(m−1)! π2 zm ,(A.2) and for large arguments (z≫1), Jm(z)≃r2 πz cosz−mπ 2−π 4, Ym(z)≃r2 πz sinz−mπ 2−π 4. (A.3) The Hankel functions of the first and second kind, H(1) m(z) = Jm(z) + iYm(z), H(2) m(z)=Jm(z)−iYm(z),(A.4) represent outgoing and incoming cylindrical waves, respectively. They satisfy the same differential equation as Eq. (A.1) and are particularly convenient for enforcing causal (Sommerfeld) boundary conditions. Their large–argument forms, H(1) m(z)≃r2 πz ei(z−mπ 2−π 4), H(2) m(z)≃r2 πz e−i(z−mπ 2−π 4),(A.5) show explicitly that H(1) m corresponds to an outward–propagating (radiative) wave and H(2) m to an inward–propagating one. In the main text, H(1) 1 ( kΩr ) represents the outward, causally damped response of the resonant curvature field in Eq. (163). 121
195 axs[1,0].plot(r_kpc[limit], np.sqrt(G*M_eff[limit]/r[limit])/1e3, ,→label=r"$v_{\rm eff}$", color=’blue’) 196 axs[1,0].plot(r_kpc[limit], ,→np.sqrt(G*(M_b[limit]+M_eff[limit])/r[limit])/1e3, label=r"$v_{\rm ,→tot}$", color=’black’) 197 axs[1,0].set_title("(c) Rotation curves (50 kpc)") 198 axs[1,0].set_xlabel("r [kpc]") 199 axs[1,0].set_ylabel("Velocity [km s$^{-1}$]") 200 axs[1,0].legend(loc=’center right’, frameon=True, facecolor=’white’, ,→framealpha=0.3, edgecolor=’none’, fontsize=16) 201 202 # (d) Mass ratio 203 axs[1,1].plot(r_kpc, M_eff / np.maximum(M_b, 1e-30), color=’purple’) 204 axs[1,1].set_title("(d) Enclosed mass ratio") 205 axs[1,1].set_xlabel("r [kpc]") 206 axs[1,1].set_ylabel(r"$M_{\rm eff}/M_{\rm b}$") 207 208 fig.suptitle("3D Helmholtz Halo Model (m = 1 Mode)", 209 fontsize=26, fontweight=’bold’, y=0.98) 210 plt.tight_layout(rect=[0, 0, 1, 0.96]) 211 plt.savefig("C:/Users/BryanH/Downloads/fig1_spiral_profiles.png", dpi=300, ,→facecolor=’white’) 212 plt.show() 213 214 # ================================================================ 215 # FIGURE 2 2D Midplane Maps (Amplitude, Phase, and Densities) 216 # ================================================================ 217 218 Nmap = 400 219 x = np.linspace(-50, 50, Nmap) * kpc 220 y = np.linspace(-50, 50, Nmap) * kpc 221 X, Y = np.meshgrid(x, y) 222 R = np.sqrt(X**2 + Y**2) 223 TH = np.arctan2(Y, X) 224 225 A_radial = np.interp(R, r, A_r) 226 rho_b_2D = np.interp(R, r, rho_b) 227 # --- corrected factor of 2 for consistency with rho_eff_mid --- 228 rho_eff_2D = (2 * Omega**2 / (8.0 * np.pi * G * c_light**2)) * ,→np.abs(A_radial)**2 229 A_2D = A_radial * np.exp(1j * TH) 230 231 amp_map = np.abs(A_2D) 232 phase_map = (np.angle(A_2D) + np.pi) / (2.0 * np.pi) 233 rho_tot_2D = rho_b_2D + rho_eff_2D 234 235 amp_norm = amp_map / np.max(amp_map) 236 phase_norm = phase_map 237 rho_b_norm = rho_b_2D / np.max(rho_b_2D) 238 rho_eff_norm = rho_eff_2D / np.max(rho_eff_2D) 239 rho_tot_norm = rho_tot_2D / np.max(rho_tot_2D) 240 241 fig, axs = plt.subplots(2, 2, figsize=(10, 10)) 242 128
243 axs[0,0].imshow(amp_norm, extent=[-50, 50, -50, 50], origin=’lower’, ,→cmap=’viridis’) 244 axs[0,0].set_title("(a) |A| amplitude\n(normalized)", fontsize=20) 245 axs[0,0].axis(’off’) 246 247 axs[0,1].imshow(phase_norm, extent=[-50, 50, -50, 50], origin=’lower’, ,→cmap=’gray’) 248 axs[0,1].set_title("(b) Phase arg(A)\n(normalized)", fontsize=20) 249 axs[0,1].axis(’off’) 250 251 rgb = np.zeros((*rho_eff_norm.shape, 3)) 252 rgb[..., 0] = rho_b_norm 253 rgb[..., 1] = rho_eff_norm 254 axs[1,0].imshow(rgb, extent=[-50, 50, -50, 50], origin=’lower’, vmin=0, ,→vmax=1) 255 axs[1,0].set_title("(c) Baryonic (red)\n and resonant (green)", ,→fontsize=20) 256 axs[1,0].axis(’off’) 257 258 axs[1,1].imshow(rho_tot_norm, extent=[-50, 50, -50, 50], origin=’lower’, ,→cmap=’gray’) 259 axs[1,1].set_title("(d) Combined density\n (normalized)", fontsize=20) 260 axs[1,1].axis(’off’) 261 262 fig.suptitle("Spiral Helmholtz Solutions\nNormalized Results (100 kpc 100 ,→kpc)", 263 fontsize=22, fontweight=’bold’, y=0.93) 264 plt.tight_layout(rect=[0, 0, 1, 0.95]) 265 plt.subplots_adjust(wspace=-0.3, hspace=0.25) 266 plt.savefig("C:/Users/BryanH/Downloads/fig2_spiral_maps.png", dpi=300, ,→facecolor=’white’) 267 plt.show() 268 269 # ================================================================ 270 # FIGURE 3 Dark Mass Density Slices (XY and XZ) 271 # ================================================================ 272 # The vertical XZ slice shows exponential decay exp(-2|z|/L_z), 273 # where L_z represents the geometric confinement scale. The vertical 274 # contrast therefore illustrates the halos stratification typically 275 # a few 10 kpc thick and is not rescaled by any normalization factor. 276 # ================================================================ 277 278 Nxy = 600 279 WIDTH = 200 280 extent_xy = [-WIDTH/2, WIDTH/2, -WIDTH/2, WIDTH/2] 281 extent_xz = [-WIDTH/2, WIDTH/2, -WIDTH/2, WIDTH/2] 282 283 x = np.linspace(-WIDTH/2, WIDTH/2, Nxy) * kpc 284 y = np.linspace(-WIDTH/2, WIDTH/2, Nxy) * kpc 285 z = np.linspace(-WIDTH/2, WIDTH/2, Nxy) * kpc 286 287 # --- Horizontal (XY) slice at z = 0 --- 288 X, Y = np.meshgrid(x, y) 289 R_xy = np.sqrt(X**2 + Y**2) 129
290 rho_eff_xy = np.interp(R_xy, np.real(r), np.real(rho_eff_mid)) 291 rho_eff_xy_norm = rho_eff_xy / np.max(rho_eff_xy) 292 293 # --- Vertical (XZ) slice using true L_z(r) profile --- 294 Xz, Z = np.meshgrid(x, z) 295 R_xz = np.abs(Xz) 296 297 # --- Ensure L_z and r are real 1D arrays --- 298 r_real = np.atleast_1d(np.real(r)) 299 rho_mid_real = np.atleast_1d(np.real(rho_eff_mid)) 300 301 # --- Case 1: L_z is an array --- 302 if np.ndim(L_z) > 0 and len(np.atleast_1d(L_z)) > 1: 303 Lz_real = np.atleast_1d(np.real(L_z)) 304 rho_mid_map = np.interp(R_xz.ravel(), r_real, ,→rho_mid_real).reshape(R_xz.shape) 305 Lz_map = np.interp(R_xz.ravel(), r_real, Lz_real).reshape(R_xz.shape) 306 else: 307 # --- Case 2: L_z is scalar --- 308 Lz_scalar = float(np.real(L_z)) if np.ndim(L_z) == 0 else ,→float(np.real(L_z[0])) 309 rho_mid_map = np.interp(R_xz.ravel(), r_real, ,→rho_mid_real).reshape(R_xz.shape) 310 Lz_map = np.full_like(R_xz, Lz_scalar) 311 312 # --- Apply vertical exponential envelope exp(-2|z|/L_z(r)) --- 313 rho_eff_xz = rho_mid_map * np.exp(-2.0 * np.abs(Z) / Lz_map) 314 rho_eff_xz_norm = rho_eff_xz / np.max(rho_eff_xz) 315 316 # --- Plot results --- 317 fig, axs = plt.subplots(1, 2, figsize=(12, 6), constrained_layout=False) 318 319 # (a) Horizontal slice XY 320 axs[0].imshow(rho_eff_xy_norm, extent=extent_xy, origin=’lower’, ,→cmap=’gray’, vmin=0, vmax=1) 321 axs[0].set_title("(a) Dark mass density\nHorizontal slice XY", fontsize=20) 322 axs[0].axis(’off’) 323 axs[0].set_aspect(’equal’, adjustable=’box’) 324 325 # (b) Vertical slice XZ (true or scalar L_z) 326 axs[1].imshow(rho_eff_xz_norm, extent=extent_xz, origin=’lower’, ,→cmap=’gray’, vmin=0, vmax=1) 327 axs[1].set_title("(b) Dark mass density\nVertical slice XZ", fontsize=20) 328 axs[1].axis(’off’) 329 axs[1].set_aspect(’equal’, adjustable=’box’) 330 331 fig.suptitle("Spiral 3D Halo Normalized |A| (200 kpc 200 kpc)", 332 fontsize=22, fontweight=’bold’, y=0.97) 333 fig.subplots_adjust(left=0.04, right=0.96, bottom=0.05, top=0.80, ,→wspace=0.10) 334 335 plt.savefig("C:/Users/BryanH/Downloads/fig3_darkmass_xy_xz.png", 336 dpi=300, facecolor=’white’) 337 plt.show() 130
Listing 1: Python code for the full 3D Helmholtz resonant–halo simulation. 131
C Appendix: Code for Simulating Elliptical Galaxies - Transport Theory This appendix provides the exact numerical implementation used to generate the ensemble–averaged resonant–halo predictions shown in Section 9.2.9. The simulation realizes the closed transport–law curvature–phonon halo derived analytically in Section 9.2.5 onwards, including the multiplicative gradient–energy contribution of Section 9.2.8. The stellar mass distribution is modeled as a Hernquist sphere with total stellar population N⋆ , individual mass m⋆ , and scale length a = Re/ 1 . 8153, giving the effective baryonic volume Vgal = 2πa3.(C.1) The elliptical transport closure is governed by four control parameters: •the transport damping length Ldamp,eff =ceff Γeff ,(C.2) fixed directly from the observed halo extent; •the resonant curvature wavelength λres,(C.3) which sets both the carrier frequency Ω = 2πceff λres (C.4) and the effective quality factor Qeff =Ω 2Γeff ; (C.5) • the speckle–shape factor α∼ 1, which controls the mean geometric coherence of the stellar sources; and • the mean coupling efficiency |X|2∼ 1, which sets the relative weighting of individual sources in the coherent ensemble field. Here Γ eff is the transport–damping rate associated with multiple stellar rescattering, and ceff is the curvature–phonon propagation speed; both are defined exactly as in the main text. 132
The ensemble–averaged effective gravitating density implemented in the code follows the closed transport–halo law, ρeff(r)=4πG c2 eff 2Qeff 2πΞ∇Genv,eff r, r < Rmatch, e−2r/Ldamp,eff (4π)2r2, r ≥Rmatch. (C.6) where Genv,eff is the ensemble environmental gain, Qeff = Ω / (2Γ eff ) is the effective quality factor, and Ξ ∇ is the transport–enhanced gradient–energy boost all defined in Section 9.2.8. For numerical regularity, the small–radius linear core ⟨ρeff(r)⟩∝ris enforced for r < Rmatch (see Section 9.2.7). For numerical consistency and physical regularity, a small–radius linear core, ⟨ρeff(r)⟩∝r(r < Rmatch),(C.7) is enforced as in Section 9.2.7. All symbols in Eqs. (C.1) – (C.7) retain the exact same meaning and notation as in the main text; no new variables are introduced in the numerical implementation. The code uses α=|X|2=1 for the baseline models. The code below: •constructs ρeff(r), enclosed Meff(r), and circular velocities; •compares directly to the stellar mass distribution; and • produces both midplane composite maps and full–scale XY/XZ slices of the resonant dark halo. 1 2# ===================================================================== 3# Elliptical Galaxy Transport-Law Curvature-Phonon Speckle Halo 4# Written by: Bryan Hennelly 28 October 2025 5# ===================================================================== 6# 7# This script computes the ensemble-averaged effective halo density 8# generated by curvature-phonon transport in a densely populated, 9# phase-mixed stellar medium (elliptical galaxy). 10 # 11 # Physics summary: 12 # 13 # Stars are both sources and scatterers of curvature phonons. 14 # Multiple resonant scattering increases the coherent path length 15 # far beyond single-pass damping: _eff << . 16 # The coherent transport tail decays as exp(-2r/L_damp_eff)/r. 17 # The ensemble-stored curvature energy manifests as an 18 # effective gravitating density _eff(r). 19 # 20 # There are FOUR principal control parameters: 133
21 # 22 # (1) L_damp_eff transport damping length of the coherent tail 23 # (DATA-DRIVEN from observed halo extent) 24 # 25 # (2) _res structural resonant wavelength of curvature 26 # (PHYSICAL MICRO-INPUT controlling amplitude) 27 # 28 # (3) speckle-shape factor (~1), controls phase-averaged 29 # geometric coherence of stellar sources 30 # 31 # (4) |X| mean stellar coupling efficiency (~1), sets 32 # relative weighting of each source in the coherent ,→field 33 # 34 # From _res we obtain: 35 # 36 # = 2 c_eff / _res (carrier frequency) 37 # Q_eff = / (2 _eff) (effective quality factor) 38 # 39 # where _eff = c_eff / L_damp_eff is implied by transport theory, 40 # and c_eff is the curvature-phonon propagation speed we take to be 2e5 m/s 41 # 42 # FINAL transport-enhanced halo law: 43 # 44 # _eff(r) = 45 # _nabla * 46 # [ G v /(2 c c_eff) ] * 47 # L_damp_eff * Q_eff * 48 # (N_* m_* / V_gal) * |X| * 49 # { r , r<R_match ; exp[-2r/L_damp_eff]/r , rR_match } 50 # 51 # with: 52 # _eff = c_eff / L_damp_eff 53 # Q_eff = / (2 _eff) 54 # _nabla = 1 + (2 L_damp_eff / _res) 55 # G_env,eff = |X| (N_*/V_gal) (_res/2) (v / 2_eff) 56 # 57 # Interpretation: 58 # Tail SHAPE is dictated solely by L_damp_eff (observed). 59 # Tail AMPLITUDE is dictated solely by _res (microphysics). 60 # and |X| enter only as weak, O(1) normalization factors. 61 # 62 # Parameter-economical closure: 63 # One observational input halo size fixed 64 # One physical parameter halo mass fixed 65 # 66 # A small-r linear core is enforced by regularity of coherent flux. 67 # 68 # Outputs: 69 # _eff(r), circular velocities, enclosed masses 70 # 2D composite baryonic vs dark mass maps 71 # large XY/XZ slices of the full halo field 72 # 73 # All variable names follow the books notation for 1:1 correspondence 134
74 # between analytic expressions and numerical quantities. 75 # ===================================================================== 76 77 78 79 # ===================================================================== 80 # Elliptical Galaxy Transport-Law Statistical Resonant Halo 81 # With Multiple-Scattering Effective Damping + Gradient Boost 82 # ===================================================================== 83 84 import numpy as np 85 import matplotlib.pyplot as plt 86 87 # ---------- Constants ---------- 88 G = 6.67430e-11 89 c = 2.99792458e8 90 Msun = 1.98847e30 91 pc = 3.085677581491367e16 92 kpc = 1.0e3 * pc 93 Mpc = 1.0e6 * pc 94 pi = np.pi 95 96 # ============================================================ 97 # TRANSPORT MEDIUM (effective) PARAMETERS [DATA-DRIVEN] 98 # ============================================================ 99 100 L_damp_eff = 636.0 * kpc # [m] SET by observed halo e-folding 101 c_eff = 2.0e5 # [m/s] TUNABLE within plausible range 102 Gamma_eff = c_eff / L_damp_eff # [1/s] 103 tau_coh_eff = 1/(2 * Gamma_eff) # [s] 104 105 print("\n--- Transport Medium (effective) ---") 106 print(f"L_damp_eff = {L_damp_eff/kpc:.1f} kpc [SET]") 107 print(f"c_eff = {c_eff:.3e} m/s [TUNE]") 108 print(f"Gamma_eff = {Gamma_eff:.3e} s^-1 [IMPLIED]") 109 print(f"tau_coh_eff = {tau_coh_eff/3.15576e13:.2f} Myr") 110 111 # ============================================================ 112 # RESONANT MICROPHYSICS 113 # ============================================================ 114 115 lambda_res = 27.0 * pc # [m] 116 Omega = 2*np.pi*c_eff/lambda_res # [1/s] 117 Q_eff = Omega/(2*Gamma_eff) # dimensionless 118 Xi_nabla = 1.0 + (2.0*pi*L_damp_eff / lambda_res)**2 # gradient-energy ,→boost 119 120 print("\n--- Resonant Microphysics ---") 121 print(f"lambda_res = {lambda_res/pc:.2f} pc") 122 print(f"Omega = {Omega:.3e} s^-1") 123 print(f"Q_eff = {Q_eff:.3e}") 124 print(f"Xi_nabla = {Xi_nabla:.3e}") 125 126 # ============================================================ 135
127 # STELLAR BARYONIC INPUT 128 # ============================================================ 129 130 M_tot = 1.0e11 * Msun 131 m_star = 1.0 * Msun 132 N_star = M_tot/m_star 133 Re = 5*kpc 134 a_h = Re/1.8153 135 rho0 = M_tot/(2*np.pi*a_h**3) 136 V_gal = 2*np.pi*a_h**3 137 138 v_rms = 200e3 139 alpha = 1.0 140 X = 1.0 # |X| = 1 for now 141 142 print("\n--- Galaxy (baryons) ---") 143 print(f"M_tot = {M_tot/Msun:.2e} Msun") 144 print(f"N_star = {N_star:.2e}") 145 print(f"V_gal = {V_gal/(kpc**3):.2e} kpc^3") 146 print(f"v_rms = {v_rms/1e3:.1f} km/s") 147 print(f"alpha = {alpha:.2f}") 148 149 # ============================================================ 150 # TRANSPORTENHANCED HALO DENSITY (grouped analytic form) 151 # ============================================================ 152 153 # Environmental gain (for diagnostic only; not changing normalization) 154 G_env_eff = (X**2 155 * (N_star**2 / V_gal) 156 * alpha 157 * (lambda_res / (2*pi))**2 158 * (v_rms / (2*Gamma_eff))) 159 160 # Working prefactor (kept identical to original numerics) 161 K_base = (G * alpha * v_rms) / (2.0 * c**2 * c_eff**3) \ 162 * L_damp_eff * Q_eff \ 163 * (N_star**2 * m_star**2 / V_gal) * X**2 164 165 def rho_eff_far(r): 166 return Xi_nabla * K_base * np.exp(-2.0*r/L_damp_eff) / ((4.0*pi)**2 * ,→r**2) 167 168 # ============================================================ 169 # INNER-CORE REGULARIZATION ( r for r < R_match) 170 # ============================================================ 171 172 R_match = 5.0 * kpc 173 rho_match = rho_eff_far(R_match) 174 slope = rho_match / R_match 175 176 def rho_eff_piecewise(r): 177 r = np.asarray(r, dtype=float) 178 return np.where(r < R_match, slope*r, rho_eff_far(r)) 179 136
180 # ============================================================ 181 # OPTIONAL: Hernquist baryons (for comparison) 182 # ============================================================ 183 184 def rho_b_hernquist(r): 185 r = np.asarray(r, dtype=float) 186 return (M_tot / (2.0 * np.pi)) * (a_h / (r * (r + a_h)**3)) 187 188 def M_b_enclosed(r): 189 r = np.asarray(r, dtype=float) 190 return M_tot * (r**2) / (r + a_h)**2 191 192 # ============================================================ 193 # RADIAL GRID + ENSEMBLE MASS INTEGRATION 194 # ============================================================ 195 196 r = np.geomspace(0.05 * kpc, 2.0e3 * kpc, 2000) 197 rho_eff_arr = rho_eff_piecewise(r) 198 rho_b_arr = rho_b_hernquist(r) 199 200 M_eff = np.zeros_like(r) 201 for iin range(1, len(r)): 202 r0, r1 = r[i-1], r[i] 203 y0 = 4.0 * np.pi * r0**2 * rho_eff_arr[i-1] 204 y1 = 4.0 * np.pi * r1**2 * rho_eff_arr[i] 205 M_eff[i] = M_eff[i-1] + 0.5 * (y0 + y1) * (r1 - r0) 206 207 M_b = M_b_enclosed(r) 208 209 ratio_rho = rho_eff_arr / np.maximum(rho_b_arr, 1e-99) 210 ratio_M = M_eff / np.maximum(M_b, 1e-99) 211 212 # ============================================================ 213 # DIAGNOSTICS 214 # ============================================================ 215 216 print("\n--- Diagnostics ---") 217 print(f"G_env_eff = {G_env_eff:.3e}") 218 print(f"Q_eff = {Q_eff:.3e}") 219 print(f"Xi_nabla = {Xi_nabla:.3e}") 220 print(f"K_base = {K_base:.3e} [SI, identical to original]") 221 print(f"rho_eff(R_match)= {rho_eff_piecewise(R_match):.3e} kg m^-3") 222 idx_100kpc = np.searchsorted(r, 100*kpc) 223 print(f"M_eff(100 kpc) = {float(M_eff[idx_100kpc])/Msun:.3e} Msun") 224 225 # ============================================================ 226 # Plots 227 # ============================================================ 228 229 plt.rcParams.update({ 230 ’font.size’: 21, 231 ’axes.titlesize’: 23, 232 ’axes.labelsize’: 21, 233 ’xtick.labelsize’: 19, 137
54 G = 6.67430e-11 # m^3 kg^-1 s^-2 55 c = 2.99792458e8 # m/s 56 Msun= 1.98847e30 # kg 57 pc = 3.085677581491367e16 # m 58 kpc = 1.0e3 * pc 59 Mpc = 1.0e6 * pc 60 pi = np.pi 61 62 # --------------------------------------------------------------------- 63 # Cluster-scale parameters 64 # --------------------------------------------------------------------- 65 L_bar = 636.0 * kpc # main transport length [m] 66 sigma_lnL = 2.5 # lognormal scatter in L 67 R_cl = 2.0 * Mpc # cluster radius [m] 68 N_gal = 300 # number of elliptical galaxies 69 70 # Structural (cluster-scale) parameters 71 R_sp = 2.5 * Mpc # splashback radius [m] 72 Delta_sp = 0.5 * Mpc # splashback transition width [m] 73 r_s = 1.0 * Mpc # satellite NFW scale radius [m] 74 n0_sat = 2.0e-63 # satellite number-density scale [m^-3] 75 76 # --------------------------------------------------------------------- 77 # Elliptical-halo microphysics Sbar_base 78 # --------------------------------------------------------------------- 79 L_damp_eff = L_bar 80 c_eff_E = 2.0e5 81 Gamma_eff = c_eff_E / L_damp_eff 82 lambda_res = 27.0 * pc 83 Omega_E = 2.0 * pi * c_eff_E / lambda_res 84 Q_eff = Omega_E / (2.0 * Gamma_eff) 85 Xi_nabla = 1.0 + (2.0 * pi * L_damp_eff / lambda_res)**2 86 87 M_tot_E = 2.0e11 * Msun # total mass of one elliptical 88 m_star_E = Msun 89 N_star_E = M_tot_E / m_star_E 90 Re_E = 5.0 * kpc 91 a_h_E = Re_E / 1.8153 92 V_gal_E = 2.0 * pi * a_h_E**3 93 v_E = 200.0e3 94 alpha_E = 1.0 95 96 K_base = (G * alpha_E * v_E) / (2.0 * c**2 * c_eff_E**3) \ 97 * L_damp_eff * Q_eff * (N_star_E**2 * m_star_E**2 / V_gal_E) 98 Sbar_base = Xi_nabla * K_base 99 100 print("=== Single Elliptical Kernel Parameters ===") 101 print(f"L_bar = {L_bar/kpc:.1f} kpc, Q_eff = {Q_eff:.3e}, Xi_nabla = ,→{Xi_nabla:.3e}") 102 print(f"Sbar_base = {Sbar_base:.3e} [kg/m^3]\n") 103 104 # --------------------------------------------------------------------- 105 # Dimensionless transport kernel (shape only) 106 # --------------------------------------------------------------------- 144
107 def K_tail(s, L): 108 """Dimensionless transport kernel.""" 109 s_eps = 1.0 * pc 110 s_clp = np.maximum(s, s_eps) 111 return np.exp(-2.0 * s_clp / L) / (((4.0 * pi)**2) * s_clp**2) 112 113 def K_tail_ensemble(s, L_mean, sigma_ln): 114 """Lognormal-averaged kernel ensemble.""" 115 if sigma_ln <= 0.0: 116 return K_tail(s, L_mean) 117 xi = np.array([-2, -1, 0, 1, 2]) * sigma_ln 118 w = np.array([1, 4, 10, 4, 1], dtype=float) 119 w /= w.sum() 120 out = np.zeros_like(s) 121 for wi, Li in zip(w, L_mean * np.exp(xi)): 122 out += wi * K_tail(s, Li) 123 return out 124 125 Kfun = lambda s: K_tail_ensemble(s, L_bar, sigma_lnL) 126 127 # --------------------------------------------------------------------- 128 # Galaxy number-density profile (satellite population) 129 # --------------------------------------------------------------------- 130 def W_splash(r): 131 """Smooth splashback taper.""" 132 return 1.0 / (1.0 + np.exp((r - R_sp) / Delta_sp)) 133 134 def n_sat_NFW(r): 135 """Satellite population ~ 1/[x(1+x)^2].""" 136 x = np.maximum(r / r_s, 1e-9) 137 return W_splash(r) * n0_sat / (x * (1.0 + x)**2) 138 139 # --------------------------------------------------------------------- 140 # Reduced spherical convolution (~2.5D form) 141 # --------------------------------------------------------------------- 142 def angular_average_K(r, rp, Kfun): 143 """Angularly averaged kernel integral K(s) d.""" 144 mu, w = np.polynomial.legendre.leggauss(48) 145 r, rp = np.atleast_1d(r).reshape(-1,1), np.atleast_1d(rp).reshape(1,-1) 146 mu3, w3 = mu[:,None,None], w[:,None,None] 147 s = np.sqrt(r**2 + rp**2 - 2.0 * r * rp * mu3) 148 return np.sum(w3 * Kfun(s), axis=0) 149 150 def rho_from_population(n_pop): 151 """Compute reduced convolution _shape(r).""" 152 I = angular_average_K(r, r, Kfun) 153 integrand = n_pop * r 154 return 2.0 * pi * (I @ (integrand * dr)) 155 156 # --------------------------------------------------------------------- 157 # Radial grid and convolution 158 # --------------------------------------------------------------------- 159 r = np.geomspace(0.2 * kpc, 3.0 * Mpc, 480) 160 dr = np.gradient(r) 145
161 rho_eff_shape = rho_from_population(n_sat_NFW(r)) 162 163 # --------------------------------------------------------------------- 164 # (1) Geometric normalization _geom by mass conservation 165 # --------------------------------------------------------------------- 166 r_single = np.geomspace(0.1 * kpc, 100.0 * kpc, 400) 167 rho_single = Sbar_base * K_tail(r_single, L_bar) 168 M_DM_ellipt = 4.0 * np.pi * np.trapz(rho_single * r_single**2, r_single) 169 170 sum_M_DM_ellipticals = N_gal * M_DM_ellipt 171 M_cluster_shape = 4.0 * np.pi * np.trapz(rho_eff_shape * r**2, r) 172 Lambda_geom = sum_M_DM_ellipticals / (Sbar_base * M_cluster_shape) 173 174 print(f"Geometric normalization _geom = {Lambda_geom:.3e}") 175 176 # --------------------------------------------------------------------- 177 # (2) Reverberation gain factor G_rev = 1 + N_gal * 178 # --------------------------------------------------------------------- 179 eta = 0.00825 # fractional coherence (~1e-31e-2 typical) 180 G_rev = 1.0 + N_gal * eta 181 print(f"Reverberation gain G_rev = {G_rev:.2f}\n") 182 183 # --------------------------------------------------------------------- 184 # Final physical density 185 # --------------------------------------------------------------------- 186 rho_eff = G_rev * Lambda_geom * Sbar_base * rho_eff_shape 187 188 # --------------------------------------------------------------------- 189 # Enclosed mass and circular velocity 190 # --------------------------------------------------------------------- 191 M_eff = 4.0 * pi * np.cumsum(rho_eff * r**2 * dr) 192 v_c = np.sqrt(G * np.maximum(M_eff, 0.0) / r) / 1.0e3 # [km/s] 193 194 # --------------------------------------------------------------------- 195 # NFW benchmark 196 # --------------------------------------------------------------------- 197 H0 = 70.0 * 1000.0 / (1.0e6 * pc) 198 rho_crit = 3.0 * H0**2 / (8.0 * pi * G) 199 M200 = 1.0e15 * Msun 200 c200 = 4.0 201 R200 = (3.0 * M200 / (4.0 * pi * 200.0 * rho_crit))**(1/3) 202 r_s_DM = R200 / c200 203 rho_s = (200.0/3.0) * rho_crit * (c200**3) / (np.log(1+c200) - ,→c200/(1+c200)) 204 205 def rho_nfw(ri): 206 x = np.maximum(ri / r_s_DM, 1e-12) 207 return rho_s / (x * (1.0 + x)**2) 208 209 rho_nfw_vals = rho_nfw(r) 210 v_nfw = np.sqrt(G * (4.0*pi*np.cumsum(rho_nfw_vals*r**2*dr)) / r) / 1.0e3 211 212 # --------------------------------------------------------------------- 213 # Diagnostics 146
214 # --------------------------------------------------------------------- 215 print("=== Diagnostics ===") 216 print(f"L_bar = {L_bar/kpc:.1f} kpc") 217 print(f"sigma_lnL = {sigma_lnL:.2f} (lognormal width)") 218 print(f" (eta) = {eta:.4f} (reverberation coupling strength)") 219 print(f"G_rev = {G_rev:.2f} (1 + N_gal , nonlinear reverberation ,→gain)") 220 print(f"_geom = {Lambda_geom:.3e} (mass-conserving normalization)") 221 print(f"max _eff = {rho_eff.max():.3e} kg/m at r = ,→{r[np.argmax(rho_eff)]/kpc:.2f} kpc") 222 print(f"max v_c = {v_c.max():.2f} km/s at r = ,→{r[np.argmax(v_c)]/kpc:.2f} kpc\n") 223 224 225 r_Mpc = r / Mpc 226 # ================================================================ 227 # FIGURE 1 Density and Enclosed Mass (Linear + Log, raised titles) 228 # ================================================================ 229 230 from matplotlib.ticker import ScalarFormatter 231 formatter = ScalarFormatter(useMathText=True) 232 formatter.set_powerlimits((-2, 4)) 233 234 r_Mpc = r / Mpc 235 rho_eff_msun_pc3 = rho_eff * (pc**3 / Msun) 236 rho_nfw_msun_pc3 = rho_nfw_vals * (pc**3 / Msun) 237 238 # ---------- Figure setup ---------- 239 fig1, axs1 = plt.subplots(2, 2, figsize=(13, 11)) 240 plt.subplots_adjust(wspace=0.35, hspace=0.35) 241 242 # ---------- Font sizes ---------- 243 title_fs = 26 244 label_fs = 22 245 tick_fs = 18 246 legend_fs = 20 247 suptitle_fs = 32 248 title_pad = 28 # raised titles for all panels 249 250 # (a) Density (linear) 251 axs1[0,0].plot(r_Mpc, rho_eff_msun_pc3, color=’navy’, lw=2, ,→label=r"$\rho_{\rm eff}$") 252 axs1[0,0].plot(r_Mpc, rho_nfw_msun_pc3, ’--’, color=’darkorange’, lw=2, ,→label="NFW") 253 axs1[0,0].set_title("(a) Density (linear)", fontsize=title_fs, pad=title_pad) 254 axs1[0,0].set_xlabel("r [Mpc]", fontsize=label_fs) 255 axs1[0,0].set_ylabel(r"Density [$M_\odot\,{\rm pc^{-3}}$]", ,→fontsize=label_fs) 256 axs1[0,0].legend(fontsize=legend_fs) 257 axs1[0,0].tick_params(axis=’both’, which=’major’, labelsize=tick_fs) 258 axs1[0,0].xaxis.set_major_formatter(formatter) 259 axs1[0,0].yaxis.set_major_formatter(formatter) 260 261 # (b) Density (loglog) 147
262 axs1[0,1].loglog(r_Mpc, rho_eff_msun_pc3, color=’navy’, lw=2) 263 axs1[0,1].loglog(r_Mpc, rho_nfw_msun_pc3, ’--’, color=’darkorange’, lw=2) 264 axs1[0,1].set_title("(b) Density (loglog)", fontsize=title_fs, pad=title_pad) 265 axs1[0,1].set_xlabel("r [Mpc]", fontsize=label_fs) 266 axs1[0,1].set_ylabel(r"Density [$M_\odot\,{\rm pc^{-3}}$]", ,→fontsize=label_fs) 267 axs1[0,1].tick_params(axis=’both’, which=’major’, labelsize=tick_fs) 268 269 # (c) Enclosed mass (linear) 270 M_nfw = 4.0 * np.pi * np.cumsum(rho_nfw_vals * r**2 * dr) 271 axs1[1,0].plot(r_Mpc, M_eff/Msun, color=’crimson’, lw=2, label=r"$M_{\rm ,→eff}$") 272 axs1[1,0].plot(r_Mpc, M_nfw/Msun, ’--’, color=’orange’, lw=2, label="NFW") 273 axs1[1,0].set_title("(c) Enclosed mass (linear)", fontsize=title_fs, ,→pad=title_pad) 274 axs1[1,0].set_xlabel("r [Mpc]", fontsize=label_fs) 275 axs1[1,0].set_ylabel(r"$M(<r)\,[M_\odot]$", fontsize=label_fs) 276 axs1[1,0].legend(fontsize=legend_fs) 277 axs1[1,0].tick_params(axis=’both’, which=’major’, labelsize=tick_fs) 278 axs1[1,0].xaxis.set_major_formatter(formatter) 279 axs1[1,0].yaxis.set_major_formatter(formatter) 280 axs1[1,0].yaxis.get_offset_text().set_fontsize(tick_fs) 281 282 # (d) Enclosed mass (loglog) 283 axs1[1,1].loglog(r_Mpc, M_eff/Msun, color=’crimson’, lw=2) 284 axs1[1,1].loglog(r_Mpc, M_nfw/Msun, ’--’, color=’orange’, lw=2) 285 axs1[1,1].set_title("(d) Enclosed mass (loglog)", fontsize=title_fs, ,→pad=title_pad) 286 axs1[1,1].set_xlabel("r [Mpc]", fontsize=label_fs) 287 axs1[1,1].set_ylabel(r"$M(<r)\,[M_\odot]$", fontsize=label_fs) 288 axs1[1,1].tick_params(axis=’both’, which=’major’, labelsize=tick_fs) 289 290 # ---------- Main title ---------- 291 fig1.suptitle("Cluster Density and Enclosed Mass", 292 fontsize=suptitle_fs, fontweight=’bold’, y=0.98) 293 294 # ---------- Save & display ---------- 295 plt.tight_layout(rect=[0, 0, 1, 0.96]) 296 plt.savefig("C:/Users/BryanH/Downloads/fig1_density_mass.png", 297 dpi=300, facecolor=’white’) 298 plt.show() 299 300 # ================================================================ 301 # FIGURE 2 Circular Velocity Profile (Linear Only, matching Figure 1 font ,→sizes) 302 # ================================================================ 303 304 fig2, ax2 = plt.subplots(figsize=(14, 6)) 305 306 # Match Figure 1 font sizes 307 title_fs = 26 308 label_fs = 22 309 tick_fs = 18 310 legend_fs = 20 148
311 suptitle_fs = 32 312 313 ax2.plot(r_Mpc, v_c, color=’crimson’, lw=2, label="Transport overlap") 314 ax2.plot(r_Mpc, v_nfw, ’--’, color=’orange’, lw=2, label="NFW") 315 316 ax2.set_title("(a) Circular velocity (linear)", fontsize=title_fs, pad=20) 317 ax2.set_xlabel("r [Mpc]", fontsize=label_fs) 318 ax2.set_ylabel(r"$v_c$[km s$^{-1}$]", fontsize=label_fs) 319 ax2.legend(fontsize=legend_fs, loc=’best’) 320 ax2.tick_params(axis=’both’, which=’major’, labelsize=tick_fs) 321 ax2.xaxis.set_major_formatter(formatter) 322 ax2.yaxis.set_major_formatter(formatter) 323 324 fig2.suptitle("Circular Velocity Profile Transport vs NFW", 325 fontsize=suptitle_fs, fontweight=’bold’, y=0.97) 326 327 plt.tight_layout(rect=[0.02, 0, 0.98, 0.95]) 328 plt.savefig("C:/Users/BryanH/Downloads/fig2_velocity.png", 329 dpi=300, facecolor=’white’) 330 plt.show() 331 332 333 # ================================================================ 334 # FIGURE 3 2D Distributions over r_max (final with extra top space) 335 # ================================================================ 336 337 # ---------- Construct 2D grid ---------- 338 n = 700 339 r_max_Mpc = r.max() / Mpc 340 extent = r_max_Mpc 341 x2 = np.linspace(-extent, extent, n) 342 y2 = np.linspace(-extent, extent, n) 343 X, Y = np.meshgrid(x2, y2) 344 R2 = np.sqrt(X**2 + Y**2) * Mpc # convert to meters 345 346 # ---------- Interpolate 1D radial profiles ---------- 347 rho_2D = np.interp(R2, r, rho_eff, right=np.nan) 348 v_2D = np.interp(R2, r, v_c, right=np.nan) # [km/s] 349 350 # ---------- Normalize effective density ---------- 351 rho_log = np.log10(rho_2D + 1e-40) 352 rho_norm = (rho_log - np.nanmin(rho_log)) / (np.nanmax(rho_log) - ,→np.nanmin(rho_log)) 353 354 # ---------- Figure setup ---------- 355 fig3, axs3 = plt.subplots(1, 2, figsize=(10, 6)) 356 title_fs = 22 357 358 # (a) Effective density 359 im0 = axs3[0].imshow( 360 rho_norm, extent=[-r_max_Mpc, r_max_Mpc, -r_max_Mpc, r_max_Mpc], 361 origin=’lower’, cmap=’inferno’, vmin=0, vmax=1 362 ) 363 axs3[0].set_title(r"(a) $\log_{10}\rho_{\mathrm{eff}}$" "\n(normalized)", 149
364 fontsize=title_fs, pad=10) 365 axs3[0].axis(’off’) 366 fig3.colorbar(im0, ax=axs3[0], fraction=0.046, pad=0.04) 367 368 # (b) Velocity field (linear) 369 im1 = axs3[1].imshow( 370 v_2D, extent=[-r_max_Mpc, r_max_Mpc, -r_max_Mpc, r_max_Mpc], 371 origin=’lower’, cmap=’viridis’ 372 ) 373 axs3[1].set_title(r"(b) Circular velocity $v_c$" "\n(linear)", 374 fontsize=title_fs, pad=10) 375 axs3[1].axis(’off’) 376 cbar1 = fig3.colorbar(im1, ax=axs3[1], fraction=0.046, pad=0.04) 377 cbar1.set_label(r"$v_c$[km s$^{-1}$]", fontsize=14) 378 379 # ---------- Main title ---------- 380 fig3.suptitle( 381 rf"2D Distributions over {r_max_Mpc:.1f} Mpc", 382 fontsize=28, fontweight=’bold’, y=0.995 383 ) 384 385 # ---------- Layout & spacing ---------- 386 plt.subplots_adjust(left=0.03, right=0.97, top=0.86, bottom=0.05, 387 wspace=0.20, hspace=0.25) 388 389 # ---------- Save & display ---------- 390 plt.savefig("C:/Users/BryanH/Downloads/fig3_2D_distributions.png", 391 dpi=300, facecolor=’white’, bbox_inches=’tight’) 392 plt.show() Listing 3: Python code for the quarter–wave spherical cluster resonator. References [1] Albert Einstein. Die grundlagen der allgemeinen. Relativitatsteorie, Annale der Physic, 49:769, 1916. [2] Robert M Wald. General relativity. University of Chicago press, 2024. [3] John P. Cox. Theory of Stellar Pulsation. Princeton Series in Astrophysics. Princeton University Press, 1980. [4] Arnaud Derode, Arnaud Tourin, and Mathias Fink. Random multiple scattering of ultrasound. i. coherent and ballistic waves. Physical Review E, 64(3):036605, 2001. [5] Arnaud Derode, Arnaud Tourin, and Mathias Fink. Random multiple scattering of ultrasound. ii. is time reversal a self-averaging process? Physical Review E, 64(3):036606, 2001. 150
[6] Leslie L Foldy. The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers. Physical review, 67(3-4):107, 1945. [7] Melvin Lax. Multiple scattering of waves. ii. the effective field in dense systems. Physical Review, 85(4):621, 1952. [8] Ping Sheng and Bart van Tiggelen. Introduction to wave scattering, localization and mesoscopic phenomena., 2007. [9] Arnaud Derode, Philippe Roux, and Mathias Fink. Robust acoustic time reversal with high-order multiple scattering. Physical review letters, 75(23):4206, 1995. [10] Emil Wolf. Introduction to the Theory of Coherence and Polarization of Light. Cambridge university press, 2007. [11] Joseph W Goodman. Statistical optics. John Wiley & Sons, 2015. [12] Anthony E Siegman. Lasers. University science books, 1986. [13] Joseph W Goodman. Introduction to Fourier optics. Roberts and Company publishers, 2005. [14] Alison M Yao and Miles J Padgett. Orbital angular momentum: origins, behavior and applications. Advances in optics and photonics, 3(2):161–204, 2011. [15] Richard L Weaver. Diffusivity of ultrasound in polycrystals. Journal of the Mechanics and Physics of Solids, 38(1):55–86, 1990. [16] Richard L Weaver. Anderson localization of ultrasound. Wave motion, 12(2):129–142, 1990. [17] Richard L Weaver and Oleg I Lobkis. Ultrasonics without a source: Thermal fluctuation correlations at mhz frequencies. Physical Review Letters, 87(13):134301, 2001. [18] Richard L Weaver and Oleg I Lobkis. Diffuse fields in open systems and the emergence of the green’s function (l). The Journal of the Acoustical Society of America, 116(5):2731–2734, 2004. [19] Eric Larose, Ludovic Margerin, Arnaud Derode, Bart van Tiggelen, Michel Campillo, Nikolai Shapiro, Anne Paul, Laurent Stehly, and Mickael Tanter. Correlation of random wavefields: An interdisciplinary review. Geophysics, 71(4):SI11–SI21, 2006. [20] MCW van van Rossum and Th M Nieuwenhuizen. Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion. Reviews of Modern Physics, 71(1):313, 1999. 151
[21] Ortwin Gerhard, Andi Kronawitter, RP Saglia, and Ralf Bender. Dynamical family properties and dark halo scaling relations of giant elliptical galaxies. The Astronomical Journal, 121(4):1936, 2001. [22] Julio F Navarro. The structure of cold dark matter halos. In Symposiuminternational astronomical union, volume 171, pages 255–258. Cambridge University Press, 1996. [23] Julio F Navarro, Carlos S Frenk, and Simon DM White. A universal density profile from hierarchical clustering. The Astrophysical Journal, 490(2):493, 1997. [24] Surhud More, Benedikt Diemer, and Andrey V Kravtsov. The splashback radius as a physical halo boundary and the growth of halo mass. The Astrophysical Journal, 810(1):36, 2015. [25] Sarah M Hansen, Timothy A McKay, Risa H Wechsler, James Annis, Erin Scott Sheldon, and Amy Kimball. Measurement of galaxy cluster sizes, radial profiles, and luminosity functions from sdss photometric data. The Astrophysical Journal, 633(1):122, 2005. 152