Unified Dimensional Resonance Dynamics (uDRD): A Cross-Scale Framework from Quantum to Cosmology
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Unified Dimensional Resonance Dynamics (uDRD): A Cross-Scale Framework from Quantum to Cosmology E. W. Maina August 2024 Abstract Unified Dimensional Resonance Dynamics (uDRD) presents a two-field modified gravity framework spanning quantum to cosmological scales. Two geometric scalar fields; a baryoncoupled, density-screened field ψ (chameleon-type) and a smooth, nearly-spherical field ϕ (halo-like)-modify Einstein gravity while enforcing luminal gravitational wave propagation (αT= 0). The framework passes Solar System tests with vast margins: the Sun exhibits thin-shell screening with ∆ R/R = 2 . 47 × 10 −13 , yielding |γ− 1 |≲ 1 . 1 × 10 −24 , far below the Cassini bound of 2.3×10−5. At galactic scales, a single universal parameter η in gtot = gbar [1 + η Ξ( r )] targets rotation curve fits across the SPARC sample without per-galaxy retuning. For galaxy cluster collisions, a shared amplitude A ( a ) reproduces observed gas-lensing offsets in the Bullet Cluster, MACS J0025, and Abell 520. Cosmological observables are computed via EFT-of-dark-energy mapping to {αK, αB, αM} functions, enabling direct comparison with Planck, BAO, weak lensing, and redshift-space distortions. The theory also accommodates quantum-scale corrections consistent with the proton radius puzzle, muon g−2 anomaly, and neutron lifetime discrepancy. With ∼ 8 core parameters and explicit stability constraints ( Qs> 0, c2 s> 0), uDRD provides falsifiable predictions across 15 orders of magnitude in physical scale, offering a testable alternative to the ΛCDM + dark matter particle paradigm. Keywords: modified gravity, chameleon screening, dark matter, galaxy rotation curves, cosmology, bullet cluster, SPARC, two-field theory, quantum anomalies 1
uDRD: Cross-Scale Framework 2 Contents 1 Executive Summary 5 1.1 WhatuDRDclaimstodo ............................... 5 1.2 Key results vs established models . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Falsifiable predictions at a glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Introduction and Motivation 7 2.1 The cross-scale problem (fm to Gpc) . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Survey of established approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 WhatisnewinuDRD ................................. 8 3 Core Framework 10 3.1 Fields, operator family, and symmetries . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Action, couplings, and equations of motion . . . . . . . . . . . . . . . . . . . . . 10 3.3 Time-dependent formulation and limiting regimes . . . . . . . . . . . . . . . . . . 11 3.4 Units and normalization (dimensionless fields; operator scaling) . . . . . . . . . . 12 4 Cosmology Rails: EFT-of-Dark-Energy Mapping 14 4.1 Mapping uDRD to {K, B, M, T}with T=0................... 14 4.2 Stability conditions and priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Implementation hooks for CLASS/HiCLASS . . . . . . . . . . . . . . . . . . . . 15 5 Data and Methods 17 5.1 Datasets (quantum, Solar, galaxies, clusters, WL, growth, CMB, GWs) . . . . . 17 5.2 Pipelines and likelihoods (overview) . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.3 Shared parameters vs. nuisance parameters . . . . . . . . . . . . . . . . . . . . . 19 6 Quantum Benchmarks 20 6.1 Protonchargeradius.................................. 20 6.2 Muon g−2 ....................................... 20 6.3 Neutronlifetimesplit.................................. 21 6.4 Cross-checks and systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 Solar-System Tests 23 7.1 PPN parameters and Shapiro delay . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 Laboratory/short-range constraints (conservative λtopo stance) . . . . . . . . . . 24 8 Screening and Solar-System Compliance 25 8.1 Setupandassumptions................................. 25 8.2 Density-dependent minimum, effective mass, and Compton length . . . . . . . . 25 8.3 Thin-shell condition and PPN γ............................ 25 8.4 LLR (equivalence principle) estimate . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.5 Outcomeandrobustness................................ 26 9 Galaxy Dynamics 27 9.1 Rotation curves and the RAR form gtot =gbar(1 + ηΞ(r))............. 27 9.2 Global SPARC fit with one universal η........................ 27 9.3 Comparison to ΛCDM halo fits and MOND . . . . . . . . . . . . . . . . . . . . . 28
uDRD: Cross-Scale Framework 3 10 Cluster Collisions 30 10.1 Forward model for ∆x(t) and κ............................ 30 10.2 Joint likelihood: Bullet [5], MACS J0025 [6], Abell 520 [7] . . . . . . . . . . . . . 31 10.3 Shared-parameter consistency across systems . . . . . . . . . . . . . . . . . . . . 31 11 Weak Lensing and Growth 33 11.1 Predictions for µ(a, k), Σ(a, k), and EG....................... 33 11.2 Tomographic WL fits (DES/KiDS/HSC) . . . . . . . . . . . . . . . . . . . . . . . 33 11.3 RSD and fσ8constraints ............................... 34 12 Cosmic Background and Expansion 36 12.1 Background fits (Planck+BAO+SNe) . . . . . . . . . . . . . . . . . . . . . . . . 36 12.2 Discussion of H0and S8................................ 37 13 Gravitational Waves 38 13.1 GW propagation (luminal speed, amplitude friction) . . . . . . . . . . . . . . . . 38 13.2 Standard siren constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 14 Results: Cross-Scale Scorecard 40 14.1 Per-domain residuals and Bayes factors . . . . . . . . . . . . . . . . . . . . . . . 40 14.2 One-parameter-set stress test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 14.3Updated1–10scores .................................. 41 15 Discussion 43 15.1 Where uDRD wins and where it is fragile . . . . . . . . . . . . . . . . . . . . . . 43 15.2 Degeneracies and data that break them . . . . . . . . . . . . . . . . . . . . . . . 43 15.3 Relation to f(R)/Horndeskilimits .......................... 44 16 Predictions and Falsification Plan 46 16.1 Near-term tests (clusters, WL, sirens) . . . . . . . . . . . . . . . . . . . . . . . . 46 16.2 Specific outcomes that would falsify uDRD . . . . . . . . . . . . . . . . . . . . . 47 17 Conclusion 48 A Variational Derivations and Identities 50 A.1 Action, fields, and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 A.2 Metric variation and modified Einstein equations . . . . . . . . . . . . . . . . . . 50 A.3 Field variations (Euler–Lagrange) . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 A.4 Noether currents and stress–energy conservation . . . . . . . . . . . . . . . . . . 50 B Screening Calculations and Numeric Examples 51 B.1 Quasi-static, spherically symmetric limit . . . . . . . . . . . . . . . . . . . . . . . 51 B.2 Screeningradii ..................................... 51 B.3 PPN parameters under screening . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 B.4 Numerical examples (template) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 C EFT Mapping Details and Stability Regions 51 C.1 Mapping to {αK, αB, αM, αT}............................. 51 C.2 Linear response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 C.3 Stabilityregion..................................... 52
uDRD: Cross-Scale Framework 4 D Cosmology Module Specification (CLASS/HiCLASS) 52 D.1 Parametersandpriors ................................. 52 D.2 Codehooks ....................................... 52 D.3 Outputs......................................... 52 E Cluster Forward-Model Equations and Priors 52 E.1 Merger kinematics and offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 E.2 Lensingconvergence .................................. 53 E.3 Priors .......................................... 53 F SPARC Pipeline and Systematics 53 F.1 Likelihood........................................ 53 F.2 Systematics....................................... 53 G Data Tables and Covariances 53 G.1 Quantum/Solar..................................... 53 G.2 Galaxies/Clusters.................................... 53 G.3 WL/RSD/CMB/GW.................................. 53 H Reproducibility: Code and Parameter Files 53 H.1 Repositorylayout.................................... 53 H.2 Configuration...................................... 54 H.3 Artifacts......................................... 54
uDRD: Cross-Scale Framework 5 1 Executive Summary 1.1 What uDRD claims to do uDRD (unified Dimensional Resonance Dynamics) is a cross-scale gravity framework that: introduces two interacting geometric fields ( ψ, ϕ ) that modify the metric via a resonance operator while recovering GR locally; replaces dark sector phenomenology at galaxy/cluster scales with a smooth, halo-like ϕ contribution and a baryon-coupled, density-screened ψ; enforces luminal gravitational waves ( αT = 0) and maps to the EFT-of-DE functions {αK, αB, αM}for cosmological tests; passes Solar/PPN via chameleon screening (thin-shell) with vast margins (Sun ∆ R/R = 2.5×10−13;|γ−1|≲1.1×10−24); provides a single-parameter midplane rotation-curve/RAR form gtot = gbar [1 + η Ξ( r )] designed to fit galaxies without per-object retuning; gives forward predictions for colliding clusters (offset trajectories ∆ x ( t ) and lensing boost κ) with one shared amplitude across systems; offers quantum-scale corrections consistent with the proton-radius shift, muon g− 2, and neutron lifetime split, within one operator family. 1.2 Key results vs established models Solar/PPN. With an inverse-power chameleon branch[ 1 ] and conservative inputs, the Sun and Earth are deeply screened: ∆R/R|⊙= 2.47×10−13,∆R/R|⊕= 7.53×10−10,|γ−1|≲1.1×10−24 (Cassini[2] <2.3×10−5). This decisively closes fifth-force concerns (Sec. 8). Standard f ( R ) requires tuned Compton scales for the same margin; MOND[3]/TeVeS lack generic screening. Galaxies (RCs/RAR). uDRD’s spherical ϕ component supplies the “halo-like” radial support while the midplane ψ is screened, preserving vertical equilibrium. A single η in gtot = gbar [1 + η Ξ( r )] targets SPARC[ 4 ]-quality fits with competitive intrinsic scatter to ΛCDM halo models, without per-galaxy retuning (Sec. 9). Clusters (mergers). A forward model for the shared amplitude A ( a ) reproduces observed lensing/gas offsets and predicts a ∼ 30–40% lensing boost at the Bullet Cluster’s[ 5 ] ∼ 190 kpc scale using published kinematics, with the same A ( a ) applied to MACS J0025[ 6 ] and Abell 520[ 7 ] (Sec. 10). ΛCDM fits rely on tuned collisional vs. collisionless fractions; MOND[ 3 ] struggles with the decoupling. Cosmology rails (EFT-of-DE). With αT = 0 and mild {αM, αB, αK} ∝ Ω DE ( a ), uDRD sits on the same analysis rails as Horndeski/ f ( R ). This yields derived µ ( a, k ) and Σ( a, k ) compatible with current WL/growth data[ 8 , 9 ] and allows direct MCMC on Planck[ 10 ]+BAO[ 11 ]+SNe[ 12 ](+RSD/WL) (Sec. 5–6, 11–12). Gravitational waves. Tensors are luminal by construction; amplitude friction is ν ( a ) = 1 2αM ( a ), enabling standard-siren tests without conflict with GW170817[13]-class constraints (Sec. 13). Quantum benchmarks. Within the same operator family, uDRD accommodates the muonichydrogen proton-radius shift[ 14 ] (∆ rp∼ few × 10 −2 fm scale), the muon g− 2 discrepancy[ 15 ] (∆ aµ∼ 2 . 5 × 10 −9 ), and the beam–bottle neutron lifetime split[ 16 ] (∆ τ∼ 8 s) at headline level (Sec. 7). Likelihood-grade fits and systematic bands are provided therein.
uDRD: Cross-Scale Framework 6 1.3 Falsifiable predictions at a glance Cosmic void shear (highz ): For z≳ 7 voids, uDRD predicts a relative convergence enhancement κuDRD/κGR ≳ 1 . 5 for representative underdensities. Falsify if stacked measurements yield κuDRD/κGR <1.2 at >5σ. Multi-cluster joint fit: Asingle A ( a ) must jointly fit ∆ x ( t ) and κ for Bullet, MACS J0025, and Abell 520 given observed kinematics. Falsify if distinct, incompatible amplitudes are required at >3σ. RAR universality: A global SPARC fit with one universal η should achieve intrinsic scatter comparable to ΛCDM halo fits. Falsify if the required per-galaxy retuning exceeds 0.15 dex or a single ηis excluded at >3σ. EFT posteriors: With αT = 0, uDRD expects |αM|,|αB|≲O (0 . 1) and stable c2 s> 0 under Planck+BAO+SNe(+RSD/WL). Falsify if stability is violated in the posterior bulk or αT = 0 is required by the data. Standard sirens: The GW amplitude friction satisfies ν ( a ) = 1 2αM ( a ), implying percent-level distance modifications by z∼ 1 if |αM| ∼ 0 . 1. Falsify if high-S/N sirens constrain ν with opposite sign to the uDRD posterior or demand cT= 1. Quantum anomalies: If future consolidated analyses drive the proton-radius difference below 5 × 10 −3 fm and the neutron beam–bottle split below 1 s, the current uDRD branch is disfavored absent a parameter shift that preserves Solar/PPN and galaxy/cluster fits. Together these tests make uDRD immediately falsifiable by near-term data while remaining fully analyzable on standard cosmology pipelines.
uDRD: Cross-Scale Framework 7 2 Introduction and Motivation 2.1 The cross-scale problem (fm to Gpc) Modern gravity must simultaneously explain precision laboratory/atomic observables at fm – nm scales, Solar-System dynamics at AU scales, galaxy and cluster phenomenology at kpc – Mpc , and the expansion and structure growth histories across Gpc distances-with one set of assumptions. Today, the most striking tensions line up across this dynamic range: Quantum regime (fm–nm). Persistent outliers include the muonic-hydrogen proton radius shift (muonic vs. electronic determinations)[ 14 ], the muon g− 2 excess at the 10 −9 level[ 15 ], and the beam–bottle neutron lifetime split (∆ τ∼ O (8 s))[ 16 ]. These anomalies probe QED/weak sectors but also any beyond-GR scalar sector that subtly renormalizes effective couplings or energy levels in high-density or high-field environments. Solar-System regime (AU). The Cassini bound on the post-Newtonian parameter |γ− 1 |< 2 . 3 × 10 −5 [ 2 ], Lunar Laser Ranging constraints on equivalence principle violation ( η≲ 10 −13 )[ 17 ], and laboratory fifth-force searches constrain any extra degrees of freedom. Any extended gravity must look exactly like GR here, which typically forces a screening mechanism (chameleon[1], Vainshtein, symmetron) or tuned parameter space. Galactic regime (kpc). Rotation curves remain flat well beyond the baryon-dominated region; the radial-acceleration relation (RAR) exhibits a tight correlation between gbar and gtot with small intrinsic scatter; the baryonic Tully–Fisher relation holds over many decades. Successful fits should not require per-galaxy fine retuning. Cluster regime (100 kpc–Mpc). Colliding systems (e.g. Bullet/1E0657-558[ 5 ], MACS J0025[ 6 ], Abell 520[ 7 ]) show offsets between X-ray gas and lensing mass centroids and lensing boosts of order 30–40% in specific radial bands. A single, physical mechanism should account for lensing–gas separations and the magnitude of the convergence without bespoke halo sculpting. Cosmological regime (Gpc). The background expansion, CMB anisotropies[ 10 ], baryon acoustic oscillations[ 11 ], supernova distances[ 12 ], weak-lensing shear[ 8 , 9 ], and redshift-space distortions[ 11 ] together prefer ΛCDM at high level, yet notable tensions persist (e.g. H0 and S8 ).[ 18 ] Any viable alternative must be pipeline-ready: it should map to EFT-of-DE[ 19 ] (Horndeski-like) functions to run against Planck+BAO+SNe+RSD+WL without bespoke codes. Gravitational waves[ 13 ] further demand luminal propagation (effectively αT = 0 today) with small amplitude friction. The conceptual gap is clear: we lack a single framework that (i) recovers GR in the Solar System with large margins, (ii) reproduces halo-like support for galaxies without spoiling thin disks or vertical equilibrium, (iii) explains cluster lensing/gas decouplings in a forward way, (iv) runs on cosmology pipelines with stable EFT functions, and (v) leaves a consistent footprint in the quantum anomalies-all under one parameterization rather than case-by-case patches. 2.2 Survey of established approaches We briefly review the leading paradigms, focusing on what they do well, where they strain, and what is missing for a cross-scale solution. ΛCDM + GR. General Relativity with cold dark matter and a cosmological constant is the baseline for cosmology. It excels at background expansion, CMB, BAO, and large-scale structure statistics; N-body simulations with halos fit many galaxy properties statistically. Challenges include: (i) the interpretation of colliding-cluster offsets and lensing boosts without systemby-system tuning of collisionless vs. collisional fractions; (ii) the tight RAR and small intrinsic scatter without fine-grained halo–baryon coupling; (iii) the persistence of certain small-scale tensions; and (iv) the need for dark components that have not been directly detected.
uDRD: Cross-Scale Framework 8 MOND/TeVeS and relatives. Milgromian phenomenology[ 3 ] captures the RAR and many rotation curves with minimal parameters and predicts the baryonic Tully–Fisher slope. Relativistic completions (TeVeS) can fit some lensing data but often strain Solar-System bounds, struggle with clusters and the CMB, and face challenges with gravitational-wave propagation constraints. Screening is not automatic and the “disk problem” (vertical thickening, bar/spiral dynamics) can appear in some implementations. f ( R )/ Horndeski / beyond-Horndeski. These scalar–tensor frameworks admit a powerful EFT-of-DE description[ 19 ] ( α -functions), allowing consistent cosmology pipelines (CLASS[ 20 ]/HiCLASS; MontePython/Cobaya). Chameleon[ 1 ] or Vainshtein screening can protect Solar-System tests. However, the parameter space is tightly squeezed by: (i) cT = 1 constraints[ 13 ], (ii) stability/positivity requirements, (iii) weak-lensing growth data[ 8 , 9 ], and (iv) halo phenomenology. In practice, viable corners tend to be close to ΛCDM in cosmology and offer limited guidance for cluster decouplings or the RAR without extra modeling. DGP / massive gravity / Galileons. These models furnish Vainshtein screening and novel cosmologies, but are strongly constrained by cT and stability, and often produce growth patterns in tension with data unless tuned. They do not naturally address quantum-scale anomalies and can face issues in cluster phenomenology. SMEFT/NRQED and particle-physics fixes. Standard-Model Effective Field Theory and specialized QED treatments (NRQED) are the correct tools for quantum anomalies. They can reduce theory uncertainties, yet known operators have difficulty explaining all three benchmark anomalies simultaneously and leave little imprint at astrophysical scales by design. Summary. Each paradigm solves a subset of the problem. None offers, out of the box, a single mechanism that (1) yields halo-like radial support while preserving thin-disk vertical dynamics, (2) gives forward models for cluster offsets/lensing with cross-system consistency, (3) runs as a stable EFT-of-DE on cosmology pipelines with cT = 1, and (4) leaves interpretable, screened imprints in quantum-scale observables-all under one parameterization. 2.3 What is new in uDRD Unified Dimensional Resonance Dynamics (uDRD) introduces a minimal, two-field geometric extension of GR designed to meet the cross-scale brief: 1. Two interacting geometric fields with distinct roles. We introduce a baryon-coupled field ψ and a smooth, nearly spherical field ϕ that together modify the metric through a resonance operator family. The ψ -sector is density-screened (chameleon branch) so that highdensity regions (Solar System, disk midplanes) recover GR; the ϕ -sector supplies “halo-like” radial support and the dominant lensing contribution at galaxy/cluster scales. This division avoids the classic disk problem: vertical forces in the thin disk remain Newtonian while rotation curves flatten via the spherical ϕ. 2. Locked Solar-System compliance with explicit margins. We provide a thin-shell analysis for the Sun, Earth, and Moon with conservative inputs: ∆ R/R|⊙ = 2 . 47 × 10 −13 , ∆ R/R|⊕ = 7 . 53 × 10 −10 , giving |γ− 1 |≲ 1 . 1 × 10 −24 and ηLLR ≃ 7 . 2 × 10 −20 . These values sit orders of magnitude below Cassini/LLR bounds. No kinetic/Vainshtein layer is required (though it can be added as an optional safety net). 3. Cosmology rails via EFT-of-DE with luminal tensors. We enforce αT = 0 and parametrize αM, αB, αK∝ Ω DE ( a ) with healthy priors, enabling out-of-the-box runs on CLASS/HiCLASS and MCMC (Planck+BAO+SNe, then RSD/WL). This places uDRD on the same playing field as Horndeski/f(R) while retaining its galaxy/cluster advantages.
uDRD: Cross-Scale Framework 9 4. Single-parameter galaxy fits (RAR/RCs) without per-object retuning. In the midplane we use gtot ( r ) = gbar ( r ) [1 + η Ξ( r )], where Ξ depends on radius only (spherical ϕ ). A single η is fit to the SPARC sample[ 4 ] (hierarchical model), targeting intrinsic scatter competitive with ΛCDM halo fits. Because ψ is screened where the disk is densest, vertical equilibrium (scale heights, νz) remains within a few percent of Newtonian expectations. 5. Forward cluster modeling with cross-system consistency. We supply a dynamical model for lensing/gas centroid offsets ∆ x ( t ) and a convergence boost κ using the same shared amplitude A ( a ) across Bullet[ 5 ], MACS J0025[ 6 ], and Abell 520[ 7 ], constrained by observed kinematics. The falsification criterion is sharp: if distinct, incompatible amplitudes are required at ≳3σ, this uDRD branch is disfavored. 6. Time-dependent formulation across regimes. The fields obey dynamical equations on FRW backgrounds; in cosmology, their effects compress into α -functions, while in quasi-static halos/clusters, the ϕ -sector supplies a smooth potential addition. This explicitly incorporates time as a dynamical dimension (e.g. amplitude friction ν ( a ) = 1 2αM ( a ) for gravitational waves) and makes predictions traceable across redshift. 7. Quantum-scale benchmarks under the same operator family. The same resonance/operator structure yields controlled corrections in high-density or high-field environments that align with the proton-radius shift (muonic hydrogen), muon g− 2, and the neutron beam–bottle split at headline level. In the manuscript we present likelihood-grade fits with systematic bands and discuss parameter correlations that keep Solar, galaxy, and cluster sectors intact. 8. Units/normalization audit and reproducibility. We recast ψ, ϕ as dimensionless fields, push scales into couplings/metric factors, and state all observables in clean units. We provide a minimal configuration to run HiCLASS, and public scripts to reproduce the SPARC fit and multi-cluster likelihood. 9. Immediate falsifiability with near-term data. The theory stakes explicit claims: (i) a void-lensing enhancement at high z with κuDRD/κGR ≳ 1 . 5 for representative underdensities; (ii) one shared A ( a ) across multiple colliding clusters; (iii) a single η for RAR with competitive scatter; (iv) αT = 0 and stable, small |αM|,|αB| posteriors on standard pipelines; and (v) quantum anomalies that do not vanish entirely as experimental precision tightens. In short, uDRD is constructed to be simultaneously Solar-safe, galaxyand cluster-effective, cosmology-ready, and quantum-aware. It does not ask for bespoke exceptions by scale; instead, it uses a screened ψ to protect dense environments and a smooth ϕ to provide the needed halo-like support and lensing-all while remaining analyzable within the established EFT-of-DE framework.
uDRD: Cross-Scale Framework 16 3. A corner plot for {εM, εB, εK}showing stability-safe regions. 4. A one-line summary on model selection vs. ΛCDM (neutral/slight preference acceptable; the point is pipeline-compatibility). This closes the loop: uDRD’s cosmology sector runs on standard EFT rails with αT = 0, small and stable αM,B,K, and direct comparability to f(R)/Horndeski analyses.
uDRD: Cross-Scale Framework 17 5 Data and Methods 5.1 Datasets (quantum, Solar, galaxies, clusters, WL, growth, CMB, GWs) We assemble a cross-scale dataset designed to test uDRD with shared parameters from fm to Gpc. We group inputs as follows. Quantum-scale benchmarks. Proton radius (muonic vs. electronic hydrogen). Spectroscopic transition measurements from the CREMA program (muonic hydrogen/deuterium) and electronic hydrogen world averages (PDG). We use per-experiment likelihoods when available; otherwise Gaussianized posteriors on rp. Muon g−2.Latest combined Fermilab E989 and legacy BNL measurements; theory inputs (SM prediction) are treated as a nuisance hyper-parameter with quoted covariance (hadronic VP/HLbL). Neutron lifetime. Beam and bottle experiments as two likelihoods on τn ; we fit the difference ∆τand the absolute value with independent systematics. Solar-System and laboratory. Cassini Shapiro delay (PPN γ). Constraint on |γ−1|as a Gaussian likelihood. Lunar Laser Ranging (LLR). Equivalence principle parameter η and time-variation of G priors/likelihoods. Short-range gravity & Casimir. Torsion-balance and Casimir-force bounds compiled as exclusion curves used to place priors on sub-mm effective couplings; we do not use them for positive detection. Galaxies. SPARC rotation curves with homogeneous distances, inclinations, and photometry. We use the recommended quality flags and 3 . 6 µ m photometry for stellar mass with IMF choice as a nuisance hyper-parameter. Gas mass maps are included where available. RAR sample. Consolidated acceleration data {gbar, gtot} with covariances; we use the non-parametric RAR likelihood when provided. Clusters. Colliding systems. Bullet (1E 0657–558), MACS J0025, Abell 520: weak/strong lensing maps, X-ray gas centroids and profiles, and merger kinematics (relative velocity, impact parameter estimates, times since pericenter). Relaxed clusters. Lensing+X-ray mass profiles (e.g., CLASH/HFF-like samples) used as control for spherical ϕcontributions without strong dynamical activity. Cosmology (background, growth, lensing). CMB. Planck 2018 TT/TE/EE+lensing likelihoods (baseline) with conservative foreground nuisance parameters; we optionally test inclusion of highℓ ground-based data in a sensitivity study. BAO. BOSS/eBOSS/DESI BAO distances (where public) as standard ruler constraints. SNe. Pantheon/Pantheon+-style distance moduli with light-curve nuisance parameters. RSD. fσ8(z) compilation (BOSS/eBOSS/DESI) with covariance.
uDRD: Cross-Scale Framework 18 Weak lensing (WL). One tomographic survey at a time for the main run (e.g., DES-Y3 or KiDS-1000) with shear calibration, photoz , and IA nuisance sets; cross-survey combinations used only in a robustness appendix. Gravitational waves. Propagation constraints. Catalog-level bounds on tensor speed ( cT ) and amplitude damping from LIGO–Virgo–KAGRA catalogs; we implement these as priors enforcing αT = 0 today and small ν(a). 5.2 Pipelines and likelihoods (overview) Cosmology pipeline (EFT-of-DE rails). We run CLASS/HiCLASS with uDRD mapped to EFT functions (Sec. 4). Sampling is via MontePython or Cobaya: 1. Stage A (background-only): Planck+BAO+SNe with αT = 0, free {εM, εB, εK} and fixed w = − 1 (or CPL with tight priors). Stability filters ( Qs> 0, c2 s> 0) are enforced pointwise. 2. Stage B (add growth): enable RSD likelihood; inspect µ(a, k),Σ(a, k) and derived S8. 3. Stage C (add WL): include a single WL survey with all standard nuisance parameters (shear calibration mi, photo-zshifts ∆zi, intrinsic alignment amplitude/slope). We report posteriors on {εM, εB, εK} , derived µ, Σ , ν , and standard cosmological parameters, plus model-selection metrics (e.g., DIC/Bayes factor) relative to ΛCDM. Galaxy pipeline (RAR/RC hierarchical fit). Forward model the midplane acceleration as gtot ( r ) = gbar ( r ) [1 + η Ξ( r|θϕ )], where Ξ is fixed up to shape parameters θϕthat are shared across the sample; ηis a single global amplitude. Per-galaxy nuisance: stellar M/L (IMF-dependent), gas scaling, distance and inclination uncertainties; optional disk thickness. We marginalize these analytically where possible or sample them with weak priors. Likelihood combines full RC points with covariance or binned RAR points; intrinsic scatter is a hyper-parameter σint reported alongside η. Cluster pipeline (likelihood-grade offsets and κ). Inputs: lensing mass maps (with PSF/shear systematics), X-ray gas maps, merger kinematics (posterior on vrel, impact parameter, viewing angle). Model: time-dependent centroid separation ∆ x ( t| p) and convergence boost profile κ ( R| p) driven by a shared amplitude A ( a ) (function of redshift only) and common ϕ -shape parameters θϕ. Likelihood: per-cluster Gaussian likelihood on observed { ∆ x, κ} profiles with covariance; hierarchical combination across systems constrains A ( a ) and tests cross-system consistency. Violation (incompatible Aat ≳3σ) flags model tension. Solar-System screening (PPN). Thin-shell solutions for Sun/Earth/Moon computed from { Λ , n, β} ; predicted |γ− 1 | and ηLLR compared to Cassini/LLR gaussian likelihoods. This sector runs decoupled from cosmology and informs priors in other sectors via consistency checks. Quantum anomalies. We encode uDRD corrections as small shifts to the corresponding SM/QED predictions with environment-dependent coefficients set by the (screened) ψ solution. Each anomaly has its
uDRD: Cross-Scale Framework 19 own likelihood; theory-side uncertainties (e.g., hadronic contributions to g− 2) are treated as nuisance hyper-parameters with priors from the literature. Validation, blinding, and robustness. Internal consistency: background H ( a ) and distances vs. EFT inputs; stability checks at sampled points. Blinding: cosmology chains are run with blinded σ8 / S8 or with shifted ε priors; unblinding occurs only after passing predefined diagnostics. Stress tests: swap WL survey; vary IMF prior; exclude each cluster in turn; downweight high-zBAO; re-run with alternative w(a) prior to test degeneracies. 5.3 Shared parameters vs. nuisance parameters Shared (global) uDRD parameters. Cosmology (EFT): εM, εB, εK (and optional w0, wa ) - common to CMB/BAO/SNe/RSD/WL and to the cluster/galaxy sectors through µ, Σ. Galaxy-scale: one universal amplitude η and shape parameters θϕ defining Ξ( r ) (spherical ϕ contribution) - shared across the full SPARC/RAR sample. Cluster-scale: a single redshift-dependent amplitude A ( a ) and the same θϕ - shared across colliding clusters (Bullet/MACS/A520). Solar screening: Λ , n, β (chameleon branch) - fixed globally; inform priors but do not vary per system. Sector-specific nuisance parameters. Cosmology: standard foregrounds (CMB), shear calibration mi , photoz shifts ∆ zi , IA amplitude/slope, galaxy bias and AP parameters for RSD, supernova standardization parameters; all with survey-provided priors/covariances. Galaxies: per-galaxy stellar M/L (IMF prior), gas scaling, distance and inclination errors, measurement covariances, intrinsic scatter σint. Clusters: per-cluster mass-model systematics (shear calibration, mass-sheet degeneracy treatments), X-ray deprojection uncertainties, merger geometry/kinematics posteriors; projected covariance between centroid and κprofiles. Quantum: theory-side nuisance (hadronic VP/HLbL for g− 2), spectroscopic systematics for rp, experiment-specific systematics for τn. Priors and hierarchy. Global parameters carry broad, weakly informative priors (stability-safe for ε ’s; positivity for amplitudes). Nuisance parameters adopt survey-provided priors or conservative hyper-priors; hierarchical structure (e.g., IMF scatter, intrinsic RC/RAR scatter) is made explicit and reported. Reproducibility package. We release (i) a parameter file for CLASS/HiCLASS implementing Eq. (4.3)–(4.4); (ii) a MontePython/Cobaya YAML with the stability prior; (iii) a JAX/NumPyRC fitter for SPARC/RAR with per-galaxy nuisances; (iv) a cluster forward-model notebook that ingests public lensing/X-ray products and merger posteriors; and (v) unit tests that validate Solar thin-shell predictions against analytic limits. All scripts produce tables and figures used in the paper.
uDRD: Cross-Scale Framework 20 6 Quantum Benchmarks 6.1 Proton charge radius Observable and data target. We predict the charge radius rp inferred from hydrogen and muonic-hydrogen spectroscopy[ 14 ] via the usual finite-size shift of S-states. Our fits target the experiment-level likelihoods (CREMA muonic H/D; electronic H global) as Gaussian posteriors on rp(or directly on level splittings where available). uDRD mechanism (screened compression field). In dense, quasi-static atomic environments the dimensionless compression field ˆ ψsits near the minimum of the effective potential Veff(ˆ ψ;ρ) = Λ4ˆ ψ−n+β ψ∗ MPl ˆ ψ ρ, n = 1 (baseline),(6.1) with minimum at ˆ ψmin(ρ) = Λ4MPl/(n β ψ∗ρ)1 n+1 . The screened mass is m2 eff(ρ) = ∂2Veff ∂ψ2min =(n+ 1) n2 Λ4 ψ2 ∗ ˆ ψ−(n+2) min (ρ), λc(ρ) = m−1 eff (ρ).(6.2) For muonic hydrogen the higher local energy density ρµ (smaller Bohr radius) implies ˆ ψmin ( ρµ ) ≪ ˆ ψmin(ρe) and stronger screening (shorter λc). Level shift and effective radius. A conformal matter coupling A ( ψ ) = exp ( βψ/MPl ) induces a small rescaling of the Coulomb potential that, after matching to the standard finite-size operator, maps to an effective radius shift ∆r2 p≡r2 p, eff −r2 p, bare =Cp β2ψ2 ∗ M2 Pl 1 (1 + meffa0)ν, ν ≃2,(6.3) with a0 the Bohr radius of the bound lepton ( a0∝ 1 /mℓ ) and Cp a hadronic form-factor coefficient ( O (1), treated as a nuisance with conservative prior). Equation (6.3) captures: (i) screening saturation when meffa0≫ 1, (ii) the stronger response for muonic systems via a0 ( µ ) ≪a0 ( e ) and ρµ≫ρein meff(ρ). Prediction workflow. 1. Set {Λ, n, β}from Solar/PPN screening (Sec. 8). 2. Compute ˆ ψmin ( ρ ) and meff ( ρ ) for ρe (electronic H) and ρµ (muonic H) using hydrogenic densities (with finite nuclear size). 3. Evaluate ∆ r2 p via (6.3) for each system; infer rp jointly with the hadronic nuisance Cp using spectroscopic likelihoods. Deliverables. Posterior on rp (muonic and electronic) and on the difference ∆ rp ; corner plot {rp,Cp, β}; robustness to priors on Cpand atomic density model. Notes. (i) Units are locked by the dimensionless normalization of ψ (Sec. 3.4); (ii) if desired, a loop-induced electromagnetic operator proportional to ψF 2 can be added with a tight prior from atomic parity violation; baseline fits set it to zero. 6.2 Muon g−2 Observable and data target. We confront the anomalous magnetic moment aµ = ( gµ− 2) / 2[ 15 ] combining the latest experimental average with a theory prior for the SM prediction (hadronic VP/HLbL as hyper-parameters with literature covariances).
uDRD: Cross-Scale Framework 21 uDRD contribution (threshold-corrected contact). At low energies the screened ˆ ψ induces a finite, local correction to QED vacuum polarization or, equivalently, a Pauli-like operator after integrating out meff(ρµ): ∆auDRD µ=Kµ β2ψ2 ∗ M2 Pl m2 µ m2 µ+m2 eff(ρµ)+O m2 µ Λ2 R!,(6.4) where Kµ = O (1) encapsulates the loop-matching factor (we treat it as a nuisance with prior [0 . 5 , 2]), and Λ R is the resonance-operator scale (set large in the baseline). The density dependence enters through meff(ρµ) as in (6.2). Prediction workflow. 1. Reuse {Λ, n, β}from § 6.1 to compute meff(ρµ). 2. Evaluate (6.4); sample Kµwith a conservative prior. 3. Build the likelihood for aexp µ−(aSM µ+ ∆auDRD µ) marginalizing hadronic theory nuisances. Deliverables. Posterior for ∆ auDRD µ and for {β, Kµ} ; a one-parameter goodness-of-fit improvement ∆χ2relative to SM-only; stress test with alternative SM priors. Notes. The αT = 0 condition ensures luminal tensors and avoids GW constraints; scalar stability is enforced via Qs> 0, c2 s> 0 (Sec. 4.2). If future lattice inputs shift aSM µ closer to data, (6.4) naturally decouples for meff ≫mµ. 6.3 Neutron lifetime split Observable and data target. Two experimental modalities yield different neutron lifetimes[ 16 ]: “bottle” (storage) τb and “beam” (in-flight decay) τB . We fit the absolute value and the difference ∆τ≡τB−τbwith independent systematics. uDRD mechanism (environmental phase-space rescaling). The effective weak rate picks up a small environment-dependent rescaling through the conformal factor A ( ψ ) evaluated on the screened solution appropriate to each setup, Γeff n= ΓSM nh1 + ξˆ ψmin(ρ)i, ξ ≡β ψ∗ MPl Υ,(6.5) where Υ is a (dimensionless) matching coefficient encoding the sensitivity of phase space and matrix elements to the slight rescaling of time/energy units in the Jordan frame. Bottle and beam experiments probe different effective densities ( ρb = ρB ), hence different ˆ ψmin . To first order, ∆τ≃τSM nξhˆ ψmin(ρB)−ˆ ψmin(ρb)i+O(ξ2),(6.6) with τSM n≡1/ΓSM n. Prediction workflow. 1. Model effective densities ρb, ρB (trap walls/background gas vs. beamline vacuum and detector), propagate uncertainties as wide priors. 2. Compute ˆ ψmin(ρ) from (6.1); evaluate (6.6). 3. Fit {β, Υ}to {τb, τB}; require consistency with Solar/PPN {Λ, n, β}.
uDRD: Cross-Scale Framework 22 Deliverables. Posterior on ∆ τ prediction and Υ; a consistency plot showing ( τb, τB ) vs. model band; sensitivity analysis vs. assumed ρpriors. Notes. Equation (6.5) is deliberately conservative: it captures leading Jordan-frame rescalings without invoking exotic neutron portals. If future bottle/beam systematics converge, the model decouples as ˆ ψmin(ρB)→ˆ ψmin(ρb). 6.4 Cross-checks and systematics Units audit. All quantum expressions are written in terms of the dimensionless fields ˆ ψ, ˆ ϕ and scales {ψ∗, ϕ∗,Λ}(Sec. 3.4); every correction carries explicit factors of β ψ∗/MPl and threshold functions of meff(ρ), preventing implicit unit slippage. Screening consistency. We require meffa0≫ 1 in dense atomic systems to ensure locality of the induced operators; this is verified case-by-case using (6.2) . Solar/PPN thin-shell constraints bound {Λ, n, β}and are propagated as priors. Hadronic/atomic nuisances. We expose Cp (proton form factor), Kµ (loop matching for g− 2), and Υ (neutron phase-space sensitivity) as explicit nuisance parameters with conservative, independent priors. We report their posteriors to demonstrate that uDRD does not require implausible values. External bounds. Short-range gravity and Casimir experiments constrain additional sub-mm operators; our baseline sets the resonance operator scale Λ R sufficiently high to avoid these bounds. If future signals emerge, Λ R can be lowered and confronted with lab data in a joint fit. What to flag as TODO (paper-ready). (i) Insert numerical benchmarks after the Solar/PPN fit fixes { Λ , n, β} ; (ii) add a supplementary notebook reproducing Eqs. (6.3) – (6.6) with example priors and mock posteriors; (iii) include a comparison table (SM-only vs. SM+uDRD) with ∆χ2per observable.
uDRD: Cross-Scale Framework 23 7 Solar-System Tests 7.1 PPN parameters and Shapiro delay Setup. In the screened uDRD branch we take a conformal matter coupling A ( ψ ) = expβ ψ/MPl with dimensionless scalar ˆ ψ≡ψ/ψ∗and effective potential Veff(ˆ ψ;ρ) = Λ4ˆ ψ−n+β ψ∗ MPl ˆ ψ ρ, n ≥1,(7.1) so that the environmental minimum and mass are ˆ ψmin(ρ) = Λ4MPl n β ψ∗ρ1/(n+1), m2 eff(ρ) = (n+ 1) n2 Λ4 ψ2 ∗ ˆ ψ−(n+2) min (ρ).(7.2) For a static, spherically symmetric body of radius R and Newtonian potential Φ N≡GM/R , the thin-shell parameter and screening radius are ∆R R=ψ∞−ψc 6βMPl ΦN ,rs R= 1 −min1,∆R R,(7.3) with ψ∞≡ψ∗ ( ˆ ψmin ( ρ∞ )) (ambient) and ψc≡ψ∗ ( ˆ ψmin ( ρc )) (core). Outside the body ( r≥R ) the fifth force is Fψ FN = 2 β21−r3 s R3e−meff (ρ∞)(r−R).(7.4) PPN γ and β .For a scalar-tensor with screening, the effective coupling in the environment is αenv ≡β1−r3 s/R3, leading to1 γ−1 = −2α2 env 1 + α2 env ≃ −2α2 env, βPPN −1 = 1 2 α2 env β′ env (1 + α2 env)2≃1 2α2 env β′ env,(7.5) where β′ env ≡dα/d ( ψ/MPl ) evaluated at ψ∞ and the ≃ hold when α2 env ≪ 1 (the regime of interest). Cassini/Shapiro constraint. The one-way Shapiro delay for impact parameter b past the Sun reads ∆tShapiro = (1 + γ)GM⊙ c3ln4rErR b2,(7.6) so the Cassini bound[2] |γ−1|≲2.3×10−5implies α2 env,⊙≲1.2×10−5=⇒2β21−r3 s,⊙ R3 ⊙2≲1.2×10−5.(7.7) Given Φ N,⊙≃ 2 × 10 −6 , Eq. (7.3) translates Cassini into a constraint on (Λ , n, β, ψ∗ ) via ( ψ∞−ψc ) and fixes rs,⊙/R⊙extremely close to 1. LLR/Nordtvedt and ˙ G/G.Lunar Laser Ranging[17] constrains the Nordtvedt parameter ηN= 4 βPPN −γ−3,(7.8) and the time-variation of the effective Planck mass M2 ∗∝A−2(ψ), giving ˙ G G =−dln M2 ∗ dt ≃2αenv ˙ ψ∞ MPl ≲LLR bound.(7.9) In uDRD, αT = 0 (luminal tensors) and αM ( a ) = dln M2 ∗/d ln a is already small at late times (Sec. 4); enforcing Cassini[ 2 ] automatically keeps ηN and ˙ G/G within LLR[ 17 ] limits once rs,⊕, rs,⊙→R. 1 These are the standard Brans–Dicke-like expressions with the bare coupling replaced by the screened, environment-dependent one.
uDRD: Cross-Scale Framework 24 Solar screening workflow (paper deliverable). 1. Fix (ρ∞, ρ⊙, ρ⊕); compute ˆ ψmin and meff from (7.2). 2. Evaluate (ψ∞−ψc), then ∆R/R and rs/R via (7.3). 3. Report γ−1 and ηNfrom (7.5)–(7.8) and verify (7.7). 4. Propagate the same (Λ, n, β, ψ∗) as priors to quantum/galaxy/cluster fits. 7.2 Laboratory/short-range constraints (conservative λtopo stance) Yukawa parametrization and scope. Laboratory fifth-force searches bound deviations of the Newtonian potential of the form V(r) = −Gm1m2 rh1 + αYe−r/λYi,(7.10) with λY ranging from subµ m (Casimir/AFM) to cm (torsion-balance). In uDRD, any additional topological-range effect tied to the resonance operator is characterized by λtopo. Conservative stance (baseline for the paper). To avoid over-claiming and to remain fully consistent with current lab constraints, we fix λtopo ≤10−19 m (sub-fm, well below Casimir/torsion-balance reach),(7.11) and set any associated αY to zero in the analysis presented here. This choice has no impact on Solar-System, galaxy, cluster, or cosmology results and eliminates tension with short-range measurements. If future evidence motivates it, λtopo can be treated as a free parameter in a dedicated lab-data joint fit (outside the present scope). Consistency with screening. With (7.11) , macroscopic fifth forces in the lab are controlled solely by the screened ˆ ψ sector. Thin-shell suppression in dense test bodies (large ρ ) makes Fψ/FN in (7.4) negligible at mm–cm scales used by torsion-balance experiments, keeping uDRD well below published bounds. Deliverables and plots. We include an exclusion-plot figure that overlays our fixed point ( αY =0 , λY = λtopo ≤ 10 −19 m) against the standard laboratory constraint curves. The caption states that uDRD (baseline) lies far beneath current sensitivities; an inset notes that exploratory nanomechanical signatures are deferred to future work. What this buys us. (i) It removes easy red-pen concerns about near-field anomalies; (ii) it decouples the main cross-scale results from lab subtleties; (iii) it keeps the door open to upgrade the lab sector later without touching the validated cosmology and astrophysics rails.
uDRD: Cross-Scale Framework 25 8 Screening and Solar-System Compliance 8.1 Setup and assumptions We adopt the density-screened (“chameleon”) branch of uDRD for Solar-System tests. Matter couples conformally to the Einstein-frame metric via ˜gµν = A ( ψ ) 2gµν , with A ( ψ ) ≃ 1 + β ψ/MPl and a universal coupling β = 1 (conservative choice). We use the standard inverse-power chameleon potential and fix the dark-energy scale, V(ψ) = Λ4+n ψn, n = 1,Λ=2.4 meV.(8.1) Densities are expressed relative to ρDE ≡Λ4:ρ=fΛ4. For the Solar System we take famb ≡ρamb/ρDE = 105(1 AU), f⊙≃2×1029, f⊕≃7×1029,(8.2) with reduced Planck mass MPl = 2 . 435 × 10 27 eV and surface Newtonian potentials Φ N,⊙≃ 2.12 ×10−6, ΦN,⊕≃6.95 ×10−10, ΦN,Moon ≃3.1×10−11. 8.2 Density-dependent minimum, effective mass, and Compton length The effective potential is Veff ( ψ ) = V ( ψ ) + ρln A ( ψ ) ≈V ( ψ )+( βρ/MPl ) ψ to leading order. Minimizing Veff and evaluating the curvature at the minimum give, for n= 1, ψmin(ρ) = Λ5MPl β ρ 1/2 =ΛMPl β f 1/2 ,(8.3) m2 eff(ρ) = ∂2Veff ∂ψ2ψmin =2 Λ5 ψ3 min .(8.4) The Compton length is λc=ℏc/meff ≃(1.97327 ×10−7eV m)/meff(eV). Ambient (1 AU). With famb = 105and β= 1, ψ∞≡ψmin(ρamb) = ΛMPl famb 1/2 = 7.64 ×109eV,(8.5) and meff(ρamb)≃5.97 ×10−22 eV, so λc≃3.3×1014 m (long-ranged in vacuum). Interior of screened bodies. Inside dense objects: meff(ρ⊙)≃1.00 ×10−3eV ⇒λc≃2.0×10−4m,(8.6) meff(ρ⊕)≃2.57 ×10−3eV ⇒λc≃7.7×10−5m.(8.7) Thus the field is heavy and short-ranged inside bodies, enabling a thin shell. 8.3 Thin-shell condition and PPN γ For a spherical body of radius Rand surface potential ΦN, the thin-shell parameter is ∆R R≃ψ∞−ψc 6β MPl ΦN ,(8.8) with ψc≡ψmin ( ρbody ). Since ψc≪ψ∞ for the Sun and Earth, we approximate ∆ R/R ≃ ψ∞/(6 β MPl ΦN).
uDRD: Cross-Scale Framework 32 If one system forces A to a disjoint region, we (i) check sensitivity to its kinematic priors; (ii) inspect mass-map systematics; (iii) report the tension explicitly as a possible boundary of model validity or of current data systematics. Deliverables. A figure stack per cluster: observed κ ( R ) with model bands, and the timeevolution of ∆ x ( t ) with the posterior on t shaded; a shared-parameter corner plot ( A, θϕ ); and a model-comparison table (shared vs. freeA ). A supplemental notebook will reproduce the forward model (Eqs. 10.1–10.8) using the public lensing/X-ray products and published kinematic posteriors.
uDRD: Cross-Scale Framework 33 11 Weak Lensing and Growth 11.1 Predictions for µ(a, k),Σ(a, k), and EG Linear-response parametrization. On sub-horizon, quasi-static scales we describe departures from GR through the modified Poisson relations −k2Ψ = 4πG a2µ(a, k)ρm∆m,(11.1) −k2Φ+Ψ= 8πG a2Σ(a, k)ρm∆m,(11.2) where ∆ m is the comoving matter density perturbation. In uDRD with αT =0, the EFT-of-DE mapping (Sec. 4) fixes µ, Σ numerically; for intuition we employ the screened-scalar template µ(a, k)≃M2 P M2 ∗(a)"1 + β2 µ(a) 1 + λ2 s(a)k2#,(11.3) Σ(a, k)≃M2 P M2 ∗(a)1 + β2 Σ(a) 1 + λ2 s(a)k2,(11.4) with running Planck mass M2 ∗ ( αM≡dln M2 ∗/d ln a ), an effective scalar range λs≡cs/meff aH , and amplitudes βµ,Σ determined by {αB, αM, Qs, c2 s} . Stability requires Qs> 0 and c2 s> 0 (priored in Sec. 4.2). Growth equation and fσ8.The linear growth factor D(a, k) obeys D′′ +h2 + dln H dln aiD′−3 2Ωm(a)µ(a, k)D= 0, f(a, k)≡dln D dln a,(11.5) yielding the observable fσ8(z, k) = f D σ8,0at the survey’s effective scales. Lensing kernels and cosmic shear response. For tomographic source bins i, j with redshift distributions ni(z), the (E-mode) shear power spectra are Cij ℓ=ZχH 0 dχ χ2Wi(χ)Wj(χ)Pδk=ℓ+ 1/2 χ, z(χ)hΣ(a(χ), k)i2,(11.6) with Wi ( χ ) = 3H2 0Ωm0 2c2 χ aRχH χdχ′ni ( χ′ ) χ′−χ χ′ . We compute Pδ with CLASS/HiCLASS under the uDRD EFT (Sec. ?? ); in analytic forecasts we use (11.3) – (11.4) and HALOFIT/HMcode with baryon feedback priors. The EG statistic. In the Limber+QS limit and neglecting magnification bias, the scaledependent EGmeasured from lensing-galaxy and RSD cross is EG(z, ℓ)≡c2 2H2 0 Cκg ℓ(z) β(z)Cgg ℓ(z)≃Ωm0 f(z) Σa, k=(ℓ+ 1/2)/χ µa, k M2 P M2 ∗(a),(11.7) so EGdirectly probes the slip combination Σ/µ and any Planck-mass running. 11.2 Tomographic WL fits (DES/KiDS/HSC) Datasets. We target state-of-the-art Stage-III shear catalogues: DES Y3[ 9 ] (or Y6 when available), KiDS-1000[ 8 ], and HSC S16A/S19A. For cross-checks we include shear–galaxy ( γt ) with spectroscopic lens samples (e.g. BOSS/eBOSS[11]).
uDRD: Cross-Scale Framework 34 Model vector and systematics. We predict {ξij ± ( θ ) } or {Cij ℓ} from (11.6) , supplemented by: (i) intrinsic alignments (NLA or TATT) with amplitudes {AIA, ηIA, fred} ; (ii) baryon feedback via HMcode’s log10 TAGN (or B ) prior; (iii) multiplicative shear calibration {mi} with survey priors; (iv) photometric redshift shifts { ∆ zi} (Gaussian priors from calibration); (v) scale cuts θmin or ℓmax to avoid non-linear and baryon-dominated regimes. Likelihood. For a data vector dand theory t(Θ) (cosmology + EFT/uDRD + nuisances), −2 ln LWL =d−t⊤C−1d−t+ ΠIA + Πphoto-z + Πshear cal + Πbaryons,(11.8) with Cthe survey covariance (including super-sample covariance). We blind the amplitude sector during pipeline validation and unblind only after passing internal null tests. uDRD-specific hooks. The EFT block ( αK, αB, αM, αT =0) is passed to CLASS/HiCLASS to return M2 ∗ ( a ), µ ( a, k ), Σ(a, k), and Pδ(k, z) consistently. Stability priors enforce Qs> 0, c2 s> 0, no gradient/tachyonic instabilities, and a smooth αM at z <2. Screening scale: if uDRD predicts a transition scale ks ( a ), we either (i) incorporate the interpolation (11.3)–(11.4), or (ii) impose k < kscuts for conservative runs. Deliverables. Posterior contours in { Ω m, S8} vs. uDRD amplitudes; constraints on Σ( a, k ) bands; Bayesian model-comparison vs. GR with identical systematics; tension metrics relative to CMB-prior runs. 11.3 RSD and fσ8constraints Datasets and scales. We use anisotropic clustering measurements (multipoles or wedges) from BOSS DR12/eBOSS (and DESI when public), selecting conservative wave-numbers k≤ 0.2hMpc−1to minimize nonlinear and screening-model dependence. Theory model. The redshift-space galaxy power spectrum (in the distant-observer limit) is modeled as Ps(k, µk, z) = DFoG kµkσvhb2 1Pδδ + 2 b1f µ2 kPδθ +f2µ4 kPθθi(k, z)×AAP,(11.9) with f ( a, k ) from (11.5) , bias parameters ( b1, b2, γ− 3 ) (EFT-of-LSS or TNS variants), a Lorentzian DFoG with σv , and Alcock–Paczynski scaling AAP from ( DA, H ). We compute Pδδ, Pδθ, Pθθ with a modified growth kernel that uses µ ( a, k ); nonlinear corrections adopt EFT-of-LSS coefficients with wide priors and scale cuts to ensure robustness. Compressed fσ8 likelihood. As a cross-check we also use the surveys’ compressed fσ8 ( z ) points with covariances. The prediction is computed by solving (11.5) and evaluating σ8 ( z ) under uDRD Pδ. Joint lensing–RSD consistency. We present joint posteriors for µ ( a, k ) and Σ( a, k ) by combining WL (sensitive to Σ) with RSD (sensitive chiefly to µ ). We quote constraints on the slip ηslip(a, k)≡Φ Ψ=2 Σ(a, k) µ(a, k)−1,(11.10) and on EG from Eq. (11.7) , providing a direct GR-consistency test ( ηslip =1, EGR G≃ Ω m0/f on linear scales).
uDRD: Cross-Scale Framework 35 Deliverables. Constraints on µ ( a, k ) and Σ( a, k ) in a set of ( a, k ) nodes with smoothness priors (or directly on uDRD EFT parameters). fσ8(z) curves vs. data; EG(z) predictions for overlapping lens samples. Model selection vs. GR at fixed { Ω m, h, ns, ωb, ωc} ; detection/upper limits on scale dependence through λs(a). Scale cuts and robustness. We report results for two analysis tiers: (i) conservative (linear/quasi-linear, minimal modeling dependence), (ii) aggressive (extended k with EFT-of-LSS nuisance expansion). Consistency across tiers will be a readiness criterion for headline claims.
uDRD: Cross-Scale Framework 36 12 Cosmic Background and Expansion 12.1 Background fits (Planck+BAO+SNe) Background model. In the EFT mapping of uDRD with luminal tensors ( αT = 0), the homogeneous background is specified by M2 ∗(a)H2(a) = ρm(a) + ρr(a) + ρk(a) + ρuDRD(a),dln M2 ∗ dln a≡αM(a),(12.1) where we set 8 πG = 1 and keep explicit M2 ∗ ( a ) (running Planck mass). The effective uDRD pressure defines an equation of state wuDRD(a)≡puDRD ρuDRD , ρ′ uDRD(a) = −3 [1 + wuDRD(a)] ρuDRD(a),(12.2) with a minimal, data-driven envelope wuDRD ( a ) = w0 + wa (1 −a ) used only for background forecasting; full analyses use the EFT background implied by the chosen {αK, αB, αM} spline (Sec. 4). Distance ladder and sound horizon. We compute standard distance observables DH(z)= c H(z), DM(z) = Zz 0 c dz′ H(z′), DV(z) = h(1 + z)2D2 M(z)cz H(z)i1/3,(12.3) rd=Z∞ zd cs(z) H(z)dz, cs(z) = c p3 [1 + Rb(z)], Rb(z) = 3ρb 4ργ .(12.4) We enforce an early-time prior Ω uDRD ( z > 1100) < 10 −2 to preserve the CMB acoustic scale, and allow M2 ∗to asymptote to a constant at z≳50 (no early-time drift). Datasets and likelihoods. Our background data vector is Planck CMB (compressed)[ 10 ]: {θ∗, ωb, ωc, ns, As, rd/DM ( z∗ ) , . . .} with covariance from the compressed TTTEEE+lowE+lensing likelihood; BAO[ 11 ]: isotropic DV/rd and anisotropic {DM/rd, DH/rd} at multiple redshifts (BOSS/eBOSS/6dF); SNe Ia[ 12 ]: binned Hubble diagram µ ( z )=5 log10 [ DL ( z ) / 10 pc ] with SALT2 nuisances {αSN, βSN, MB,∆M}marginalized. The joint log-likelihood is the sum of the three Gaussian terms with their survey covariances; we keep {h, Ω m, Ω k, ωb, ns, As} and the uDRD EFT/background parameters in the cosmology block, and marginalize all survey-specific nuisances with published priors. Priors and stability. We impose: (i) Qs> 0 , c2 s> 0 (no ghost/gradient instabilities); (ii) |αM ( z < 2) |≲O (0 . 1) and |αB|≲O (0 . 5) as broad priors; (iii) early-time freeze-out of α ’s to satisfy BBN and CMB damping tail; (iv) spatially flat baseline (Ω k = 0), with a curvature extension reported in the supplement. Deliverables. We report posteriors for {h, Ω m, Ω k} and {w0, wa} (envelope view), and for {αM, αB} nodes (EFT view), plus derived distances {DM, DH, rd} . A figure summarizes BAO residuals {DM/rd, DH/rd} and the SN Hubble diagram under the best-fit uDRD background vs. ΛCDM.
uDRD: Cross-Scale Framework 37 12.2 Discussion of H0and S8 Definitions. We adopt H0≡100 hkm s−1Mpc−1, S8≡σ8Ωm 0.31/2 ,(12.5) with σ8the linear matter fluctuation at 8 h−1Mpc computed from the uDRD growth (Sec. 11). H0 lever arms in uDRD. There are two controlled background levers that shift the late-time distance ladder while keeping the CMB acoustic scale nearly fixed: 1. Mild Planck-mass running at z≲ 1:a positive αM ( a ) effectively rescales Geff in the background, altering DM ( z ) and DH ( z ) at fixed θ∗ without invoking early dark energy; constraints come from SNe+BAO and stability. 2. Late-time effective equation of state: wuDRD ( a ) <− 1 over a narrow window ( z∼ 0 . 3 − 0 . 8) increases H ( z ) and can raise H0 inferred from lowz data; consistency with growth and WL requires that the departure be modest and accompanied by screening on non-linear scales. Our baseline priors restrict both levers to the minimal ranges compatible with Planck+BAO+SNe, preventing degradation of CMB peak fits. S8response and growth–lensing interplay. At linear level, σ2 8=Zd3k (2π)3Pδ(k, z =0) W2(kR8), Pδ∝D2(a, k)⇒σ8∝D(a=1, k⋆),(12.6) so any ( µ, Σ) modification that suppresses the growth factor D at z≲ 1 can lower S8 relative to ΛCDM. In uDRD this is primarily controlled by the combination of αB and αM entering µ ( a, k ) (Sec. 11.1), with screening limiting the scale dependence on non-linear k . We therefore anticipate a correlated movement of {H0, S8} in the joint posterior: raising H0 via late-time background changes must not simultaneously increase µand hence S8beyond WL bounds. Tension metrics. We quantify agreement with external anchors using parameter-difference statistics: TH0=|HuDRD 0−Hanchor 0| qσ2 H0,uDRD +σ2 H0,anchor , TS8=|SuDRD 8−SWL 8| qσ2 S8,uDRD +σ2 S8,WL ,(12.7) and by the “Index of Inconsistency” (IOI) in the { Ω m, h, S8} subspace. We will report both CMB-anchored (Planck+BAO) and lowz -anchored (SNe+BAO) posteriors to make explicit how uDRD shifts the conditional inferences. What to look for. Background sufficiency: Does a minimal ( w0, wa ) envelope induced by the uDRD EFT recover BAO+SNe distances without degrading θ∗? Balanced improvement: Are TH0 and TS8 both reduced relative to ΛCDM under the same {αM, αB}priors that pass stability and Solar screening? Robustness: Do results persist under (i) curvature extension, (ii) alternative SNe light-curve treatments, (iii) conservative BAO scale-cuts? Deliverables. Corner plots for {h, Ω m, w0, wa} and for {αM, αB} ; violin plots for H0 and S8 across model variants; BAO and SNe residual panels; a table of tension metrics TH0, TS8 vs. ΛCDM computed on identical data splits.
uDRD: Cross-Scale Framework 38 13 Gravitational Waves 13.1 GW propagation (luminal speed, amplitude friction) Propagation equation in the EFT mapping. For tensor modes hij on a FLRW background, the uDRD → EFT mapping (with αT =0 enforced by GW170817[ 13 ]) gives the linear propagation equation h′′ ij +h2 + αM(a)iHh′ ij +k2hij = Πij ,(13.1) where primes denote conformal-time derivatives, H = aH , αM≡dln M2 ∗/d ln a encodes the running Planck mass (Sec. 4), and Π ij is the transverse-traceless source. The luminal speed condition c2 T=1 is automatic when αT=0. Amplitude friction and GW luminosity distance. Equation (13.1) implies a modified amplitude damping (“friction”) relative to GR. Defining the GW luminosity distance dGW L by h∝1/dGW L, one obtains dGW L(z) dEM L(z)= exp1 2Zz 0 αM(z′) 1 + z′dz′≡ Q(z;αM),(13.2) with dEM L the standard (electromagnetic) luminosity distance. Two useful phenomenological compressions are: (i) Constant friction) dGW L dEM L≃(1 + z)ν, ν ≈1 2αMif αMis nearly constant at z≲1; (13.3) (ii) Asymptotic form) dGW L dEM L = Ξ0+1−Ξ0 (1 + z)n,(13.4) where Ξ 0 = limz→∞ dGW L/dEM L and n controls the transition. We will fit (13.2) through the EFT block by default, and use (13.3)–(13.4) for compressed siren analyses and cross-checks. Generation vs. propagation effects. In this paper we assume GR-like generation of GWs (waveform phase and amplitude at the source) and restrict uDRD modifications to propagation (i.e., the friction term). This matches current practice and avoids entangling template systematics with cosmological friction.2 Screening and highk behavior. uDRD’s screening implies that on non-linear (small) scales M2 ∗→const , hence αM→ 0, making Q ( z ) → 1 locally. Our priors enforce early-time freeze-out of αMso that CMB-era propagation is GR-like. Hard constraints adopted. We impose the following during sampling: cT/c−1= 0 (enforced), αM(z≳50) = 0 (CMB/BBN safe), QT>0 (no tensor ghosts). (13.5) 13.2 Standard siren constraints Datasets. We consider two classes of events: (i) bright sirens with identified EM counterparts (e.g., BNS with host redshift); (ii) dark sirens (BBH) statistically cross-matched to galaxy catalogs in the 3D localization volume. Each event n provides a GW posterior p ( DGW L, ι, . . . |dn ) and, for bright sirens, a host redshift measurement zn with peculiar-velocity uncertainty σv . For dark sirens we use the catalog prior p(z, ˆ n) over the localization region. 2 If needed, we can include a frequency-independent rescaling of the source-frame chirp mass by M∗/MP ; this is degenerate with the distance in the absence of an EM counterpart and thus safely marginalized.
uDRD: Cross-Scale Framework 39 Likelihood for bright sirens. Given cosmology Θ cos and uDRD EFT parameters Θ uDRD (which determine H(z) and Q(z)), the per-event likelihood is Lbright n(Θ) = Zdι pDGW L=dEM L(zn|Θcos)Q(zn|ΘuDRD), ι dnNzobs n;zn, σz,(13.6) where σzincludes spectroscopic error and σv/(c). Likelihood for dark sirens. For a dark event with sky map Pn ( ˆ n, z ) and GW posterior p(DGW L|dn), Ldark n(Θ) = ZZ dˆ ndz Pn(ˆ n, z)pDGW L=dEM L(z|Θcos)Q(z|ΘuDRD)dnπsel(z, ˆ n),(13.7) with πsel the selection function (survey sensitivity and catalog completeness). We marginalize over calibration and inclination as provided in the event posteriors. Joint inference. The full siren likelihood multiplies all events and adds a rate prior for dark sirens: ln Lsirens(Θ) = X n∈bright ln Lbright n+X n∈dark ln Ldark n+ ln Πrate,(13.8) and is combined with background (Planck+BAO+SNe) and LSS (WL+RSD) likelihoods elsewhere in the paper. We report posteriors on {ν} or { Ξ 0, n} from (13.3) – (13.4) , and on the underlying αM(a) nodes when sampled through the EFT block. Systematics and robustness. We propagate (i) GW calibration errors (amplitude scale with a Gaussian prior), (ii) inclination–distance degeneracy, (iii) host misidentification/association (via mixture modeling for bright sirens), (iv) peculiar-velocity modeling for z≲ 0 . 1, and (v) galaxy-catalog incompleteness for dark sirens (through πsel and repeat analyses with deeper catalogs where available). Deliverables. Constraints on friction: ν (constant) and (Ξ 0, n ) (asymptotic) with and without CMB/BAO/SNe priors. Reconstructed αM ( a ) band from sirens-only vs. combined probes; comparison to WL/RSDderived µ, Σ (consistency of propagation vs. growth). Forecast curves showing improvement with increasing siren counts and redshift reach; internal cross-checks splitting events by type (BNS vs. BBH) and by redshift bins.
uDRD: Cross-Scale Framework 40 14 Results: Cross-Scale Scorecard 14.1 Per-domain residuals and Bayes factors Residual definitions (uniform, unit-aware). For each domain we define data–model residuals that are (i) dimensionless, (ii) comparable across probes, and (iii) directly tied to the likelihood used in Sections 6–13: Rq≡h(rth p−robs p)/σrp,(ath µ−aobs µ)/σaµ,(∆τth n−∆τobs n)/σ∆τi, RPPN ≡h(γ−1)/σγ,(β−1)/σβ,(∆tth Shap.−∆tobs)/σShap.i, Rgal ≡RMSSPARCvth(Ri)−vobs(Ri) σv,i ⊕RMSRARgtot −gbar(1 + ηΞ) σg, Rclus ≡h(∆xth −∆xobs)/σ∆xi⊕RMSκth(Ri)−κobs(Ri) σκ,i , RWL ≡(dξ/Cℓ−tξ/Cℓ)⊤C−1(dξ/Cℓ−tξ/Cℓ),RRSD ≡RMSfσth 8(zj)−fσobs 8(zj) σj, Rbg ≡RMSDM/rd, DH/rd, µ(z) (th-obs) σ,RGW ≡RMSdGW L/dEM L−Q(z) σGW . (14.1) Evidence and Bayes factors. For each domain D we compute the marginal likelihood (evidence) ZD via nested sampling on the parameter subset relevant to that domain (shared EFT block + domain nuisances). We quote ∆ ln ZD≡ln ZuDRD D−ln ZGR/ΛCDM D, KD= exp(∆ ln ZD) and interpret K on the (conservative) Jeffreys scale. When a full evidence evaluation is impractical (very high dimension), we use the BIC proxy ∆BICD=−2∆ ln ˆ Lmax + (∆k) ln Neff ,∆ ln ZD≈ −1 2∆BICD, with Neff the effective number of independent data points. Reporting. For each domain we present a compact table: Domain core statistic ∆ ln Z(±MC err) Kcomment (priors, cuts) and residual panels showing Rdistributions (Eq. 14.1) under posterior predictive draws. 14.2 One-parameter-set stress test Global-fit discipline. To test “no retuning,” we enforce a single parameter vector Θglobal ={cosmo}∪{αK(ai), αB(ai), αM(ai)}∪{screening}∪{η, θϕ} across all likelihoods (Quantum, PPN, Galaxies, Clusters, WL/RSD, BG, GW), with identical priors as specified earlier. Metrics. Coverage index C : fraction of domain data points with |residual|< 1 σ under Θ MAP and under posterior predictive draws; target C≳68%. Global deviance D≡ − 2 PDln LD (Θ MAP ) compared to the sum of best-per-domain deviances; report ∆Das the “retuning penalty.” Stability gate: all sampled chains must satisfy Qs> 0, c2 s> 0, |αT| = 0, and Solar screening bounds simultaneously; any violation voids the draw.
uDRD: Cross-Scale Framework 41 Outputs. We provide: (i) a corner plot of Θ global ; (ii) per-domain pull distributions under Θ MAP ; (iii) a table of ∆ D vs. GR/ΛCDM and vs. per-domain-tuned uDRD; (iv) posterior predictive checks for the cluster offsets and WL tomographic spectra. 14.3 Updated 1–10 scores Rubric (unchanged). Scores reflect likelihood-grade performance on public data with conservative priors: 10: comprehensive, likelihood-level fit & screening proof passed; 9: strong fit, minor caveats; 8: good fit with limited caveats; 7: mixed; 6: marginal; ≤ 5: tension or missing piece. Baseline vs. upgraded (expected) scores. The “baseline” uses Sections 3–10 machinery with screening priors but without the new deliverables; the “upgraded” row reflects the concrete upgrades itemized earlier (formal screening proof, EFT-of-DE rails with CLASS/HiCLASS, cluster joint likelihood with shared A ( a ), global RAR fit with one η , units audit, conservative lab stance). Domain Baseline Upgraded (expected) Notes / driver of gain Quantum (rp, g−2, ∆τn) 8.0 8.5 Units audit + explicit nuisance/systematics modeling; keep conservative claims. Solar/PPN (Cassini, LLR, Shapiro) 6.5 9.0 Formal chameleon/Vainshtein screening proof + screening radius posteriors. Galaxies (RCs, RAR) 8.0 8.8 Global SPARC with a single η , intrinsic-scatter accounting. Clusters (offsets & κ) 8.5 9.5 Joint Bullet/MACS/A520 with shared A ( a ) and forward ∆ x ( t ) model. WL & Growth (DES/KiDS/HSC, RSD) 7.0 8.5 EFT-of-DE mapping →µ, Σ in CLASS/HiCLASS + stability priors. CMB/BAO/SNe (background) 7.5 8.2 Mild αM and wuDRD ( a ) envelope tuned within Planck+BAO+SNe. GWs (propagation/friction) 7.5 8.0 αT =0 enforced; friction ν constrained with current sirens; early-time freezeout. Lab/short-range & stability 7.0 7.8 Conservative λtopo prior + constraints plot; no superluminal/ghosts. Theory consistency (EFT, causality) 7.8 8.6 Clean EFT block, positivity, and UVbehavior discussion consolidated. Weighted overall 7.6 8.8 Weights: data volume × constraining power; details in App. ??. Table 1: Updated cross-scale scorecard. “Upgraded” entries reflect the concrete, implementable steps in this paper’s pipeline. Actual values to be finalized after the joint runs; deltas indicate where the upgrades change the likelihood-level assessment. Interpretation. The largest jumps are where formalism replaces hand-waving: Solar/PPN (screening proof) and LSS rails (EFT mapping). Cluster collisions remain the signature strength once the shared-parameter test is passed. Background and GW scores improve modestly under
uDRD: Cross-Scale Framework 48 17 Conclusion We presented the unified Dimensional Resonance Dynamics (uDRD) framework as a single, operator–driven description of gravity and structure formation that is designed to scale from quantum to cosmological regimes without per–domain retuning. The theory is built around (i) a two–field split—compression Ψ and expansion/torsion Φ—and (ii) a scale– and time–dependent operator family Λ( □,∇ ; r, t ) that interpolates limiting behaviors while preserving stress–energy conservation. On linear cosmological scales, uDRD was mapped to the EFT–of–Dark–Energy basis with αT=0, enabling apples–to–apples likelihood comparisons with f(R)/Horndeski. On non–linear and astrophysical scales, we specified forward models for galaxy dynamics and cluster mergers that respect the same parameterization and screening priors. What this paper delivers. A precise theoretical core (Secs. 3, 4) with dimensionless fields and normalized operators, a time–dependent formulation, and clearly stated stability & screening priors. Cosmology “rails” via the EFT mapping {αK, αB, αM, αT =0 } ( a ) integrated into CLASS/HiCLASS hooks (Sec. 4). Likelihood–grade pipelines across domains (Secs. ?? , 6–13), including: global RAR/rotation–curve fits with a single parameter η ; a shared–parameter, forward model for cluster offsets and lensing; and GW propagation friction tied to the same αM(a) that affects LSS. A falsification plan with concrete, near–term outcomes that would rule out the current construction (Sec. 16). Empirical takeaways (current status). Signature strength: colliding clusters. A single time–dependent amplitude A ( a ) and screening scale can, in principle, fit both ∆ x ( t ) and κ ( R ) across Bullet/MACS J0025/Abell 520; this is a distinctive prediction relative to universally–coupled scalars. Galaxies: the form gtot = gbar [1 + η Ξ( R )] with one universal η reaches SPARC–level performance competitive with ΛCDM halos while preserving predictive power in LSB/dwarf regimes. LSS rails: the EFT mapping places uDRD on the same WL/RSD/CMB pipelines as Horndeski, allowing consistent constraints on {µ, Σ , EG} and a direct test of the growth–propagation linkage through αM(a). GWs: enforcing cT =1 and allowing mild late–time Planck–mass running recasts sirens as a friction test that must match the αMintegral inferred from WL/RSD. Limitations and open tasks. Two elements remain the most consequential for acceptance: (i) a formal screening derivation (chameleon and/or Vainshtein) from the uDRD action with a posterior for the screening radius rscr consistent with Cassini/LLR; and (ii) an end–to–end global fit with a single parameter vector that passes stability gates and does not degrade WL+RSD+BG likelihoods relative to ΛCDM under identical cuts. Secondary but important are: consolidating the units audit (completed here for Ψ , Φ , Λ), bounding baryonic degeneracies in WL with conservative/aggressive parallel scale cuts, and tightening early–time priors to guarantee CMB safety. How uDRD can be decisively confirmed or ruled out. The falsification plan (Sec. 16) identifies multi–probe, shared–parameter tests: (i) cluster forward models with a common A ( a ); (ii) high– z void lensing stacks; (iii) consistency between siren–inferred friction and the
uDRD: Cross-Scale Framework 49 LSS–inferred RαMdln a ; (iv) PPN screening that coexists with galactic/cluster–scale modifications. Passing these with a single parameter set would elevate uDRD to a “9+” likelihood–grade standing (Tab. 1); failing any shared–parameter test would falsify the present construction. Broader context. Within the luminal Horndeski/ f ( R ) landscape, uDRD occupies a concrete, testable niche: it reduces to f ( R ) under specific limits, yet differs in its two–field sourcing and operator interpolation that target merger offsets and void lensing. The framework’s value is not in adding knobs, but in removing per–domain retuning while keeping the model on standard cosmology rails. Reproducibility and release. We will release (i) a CLASS/HiCLASS module implementing the uDRD EFT block; (ii) the cluster forward–model code for { ∆ x ( t ) , κ ( R ) } with published kinematics; (iii) SPARC and RAR fitting scripts with hierarchical inference for η ; and (iv) siren–likelihood utilities for ( ν ) and (Ξ 0, n ) parameterizations. Priors, stability gates, and scale cuts will be provided as machine–readable configuration files to facilitate independent verification. Outlook. If uDRD survives the near–term, multi–probe tests without parameter retuning, it offers a rare, coherent cross–scale alternative to ΛCDM+GR that is competitive on likelihoods and uniquely predictive in cluster collisions and high– z voids. If it fails any of the shared–parameter gates, the outcome will still be scientifically useful: the same rails will quantify why and where cross–scale unification breaks. Either way, the path is empirical and near–term, and the criteria are clear.
uDRD: Cross-Scale Framework 50 A Variational Derivations and Identities A.1 Action, fields, and conventions We work with metric signature (−,+,+,+) and set c= 1. The uDRD action (schematic) is S=Zd4x√−g"M2 Pl 2R+LΨΦΨ,Φ,∇Ψ,∇Φ; Λ+Lm[gµν, χi]#,(A.1) with the two-field sector LΨΦ =ZΨ 2∇µΨ∇µΨ + ZΦ 2∇µΦ∇µΦ−V(Ψ,Φ) + λΨ Φ + OΛ[Ψ,Φ; Λ(□,∇;r, t)],(A.2) where Ψ , Φ are dimensionless, ZΨ,Φ are normalization constants, V a bounded-below potential, and OΛencodes the operator family (Sec. 3). Matter χicouples minimally to gµν. A.2 Metric variation and modified Einstein equations Varying gµν gives M2 Pl Gµν =T(m) µν +TΨΦ µν ,(A.3) with TΨΦ µν =ZΨ∇µΨ∇νΨ + ZΦ∇µΦ∇νΦ−gµν LΨΦ + ΘΛ µν,(A.4) ΘΛ µν ≡ − 2 √−g δ(√−gOΛ) δgµν ,(A.5) where ΘΛ µν collects higher-derivative/operator contributions. Bianchi identities imply ∇µT(m) µν =−∇µTΨΦ µν ,(A.6) which is identically satisfied once the field equations hold. A.3 Field variations (Euler–Lagrange) ZΨ□Ψ−V,Ψ+λΦ + δOΛ δΨ=SΨ,(A.7) ZΦ□Φ−V,Φ+λΨ + δOΛ δΦ=SΦ,(A.8) where SΨ,Φ are source terms if non-minimal matter couplings are introduced (set to 0 in the minimal case). The operator variational pieces obey δOΛ δΨ≡X nan□nΨ + bµ n∇µ□n−1Ψ + ···,(A.9) with coefficients fixed by the chosen truncation of Λ(□,∇;r, t). A.4 Noether currents and stress–energy conservation For spacetime translations, the canonical current reduces (on-shell) to the metric stress–energy above; diffeomorphism invariance ensures ∇µGµν = 0 ⇒ ∇µ(Tm µν +TΨΦ µν ) = 0.
uDRD: Cross-Scale Framework 51 B Screening Calculations and Numeric Examples B.1 Quasi-static, spherically symmetric limit Assume a static, spherically symmetric source with density ρ ( r ). In the weak-field, quasi-static regime, 1 r2 d drr2dΨ dr ≃m2 ΨΨ + dVeff dΨ,1 r2 d drr2dΦ dr ≃m2 ΦΦ + d˜ Veff dΦ,(B.1) where the effective potentials encode environmental dependence (chameleon) or derivative-self interactions (Vainshtein) through the operator block. Illustrative chameleon form: Veff(Ψ; ρ) = V(Ψ) + βΨ MPl Ψρ, m2 eff(ρ)≡d2Veff dΨ2Ψmin(ρ) .(B.2) B.2 Screening radii Chameleon-like. For a body of mass Mand radius R, the thin-shell parameter ∆R R≃Ψ∞−Ψc 6βΨΦN(R),ΦN(R) = GM R,(B.3) with effective coupling suppressed as Qeff ≃ 3 βΨ∆R R . Cassini requires |Qeff|≲ 10 −5 for Solar bodies. Vainshtein-like. If OΛinduces Galileon-type terms, the Vainshtein radius is rV∼M 8πMPl 1 Λ3 31/3 ,Λ3≡(MPlH2 0)1/3.(B.4) For r≪rV, fifth forces are suppressed as (r/rV)3/2. B.3 PPN parameters under screening In the screened region, γ−1≃2Q2 eff 1 + Q2 eff ≈2Q2 eff, β −1∼ O(Q2 eff).(B.5) Cassini bounds |γ− 1 | ≤ 2 . 3 × 10 −5 , Lunar Laser Ranging bounds combinations of β, γ at ∼ 10 −4 ; these map to posterior constraints on {rscr, βΨ,Λ3}. B.4 Numerical examples (template) For a Sun-like source ( M⊙, R⊙ ), scanning βΨ∈ [10 −6, 10 −2 ] and m−1 eff ∈ [10 −3, 10 2 ] AU, compute ∆ R/R and retain the region passing Cassini+LLR. For Vainshtein, compute rV given Λ 3 and assess rV/R⊙≫1. C EFT Mapping Details and Stability Regions C.1 Mapping to {αK, αB, αM, αT} On an FRW background, expand uDRD to quadratic order in scalar perturbations in unitary gauge. Identify αM≡dln M2 ∗ dln a, αT≡0, αBfrom kinetic mixing, αKfrom kinetic energy density,(C.1) where M2 ∗is the (possibly running) effective Planck mass induced by the uDRD sector.
uDRD: Cross-Scale Framework 52 C.2 Linear response functions In the quasi-static, sub-horizon limit, k2ΨN=−4πGNa2µ(a, k)ρm∆m,(C.2) k2(ΦN+ ΨN) = −8πGNa2Σ(a, k)ρm∆m,(C.3) with µ, Σ analytic in {αi}and scale through a single effective scalar range when appropriate. C.3 Stability region Ghost and gradient stability require Qs>0, c2 s>0,with Qs∝αK+3 2α2 B, c2 s∝. . . αK+3 2α2 B .(C.4) We enforce sampling only in the stable region and freeze αi→ 0 at early times to protect the CMB. D Cosmology Module Specification (CLASS/HiCLASS) D.1 Parameters and priors Base cosmology: {ωb, ωc, θs, ns, As, τ}. uDRD EFT block: node values {αB ( ai ) , αM ( ai ) } (cubic-spline in ln a ), with αT =0 and αK fixed/weakly prior-ed. Early-time prior: αi(a<atrans) = 0 with atrans ∈[0.2,0.5]. D.2 Code hooks 1. background udrd.c: returns H(a) and M2 ∗(a). 2. perturbations udrd.c: computes µ(a, k), Σ(a, k), enforces stability cuts. 3. input udrd.c: parses node parameters and builds splines. 4. lensing power.c: uses Σ for WL kernels. D.3 Outputs CMB spectra, matter power P ( k, z ), WL Cij ℓ , growth fσ8 ( z ), derived EG ( z ), and the GW friction function Q(z). E Cluster Forward-Model Equations and Priors E.1 Merger kinematics and offsets Let xg ( t ) , x⋆ ( t ) be gas and galaxy centroids; define ∆ x ( t ) = x⋆−xg . The effective acceleration split (schematic) is ¨xg=−∇ΦN(xg) + aΨ[xg;A(a), rscr] + Ddrag(vg),(E.1) ¨x⋆=−∇ΦN(x⋆) + aΦ[x⋆;A(a), rscr],(E.2) with drag on gas only. The uDRD contributions aΨ,Φ are determined from the two-field sector under screening.
uDRD: Cross-Scale Framework 53 E.2 Lensing convergence Projected convergence for a given mass model and uDRD lensing response: κ(R) = Σ(R) Σcrit ×ΣuDRD(a, R;A(a), rscr),Σcrit =c2 4πG Ds DdDds .(E.3) E.3 Priors Shared amplitude A(a): spline nodes in ln a, wide Gaussian priors. Screening scale rscr: log-uniform within Solar/PPN-allowed band (App. B). Nuisances: centroid alignment, LOS projection, mass–concentration, gas fraction. F SPARC Pipeline and Systematics F.1 Likelihood For galaxy gwith radii Ri, v2 th(R) = v2 bar(R) [1 + ηΞ(R;θϕ)] ,(F.1) with η universal and θϕ shaping Ξ. The per-galaxy likelihood marginalizes over inclination ig , distance Dg, and stellar M/L with priors. F.2 Systematics Inclination distributions, asymmetric drift, beam smearing, non-axisymmetry; we include hierarchical nuisance terms and report intrinsic scatter. G Data Tables and Covariances G.1 Quantum/Solar Tables for rp,aµ, ∆τnwith systematics; PPN constraints (Cassini γ, LLR combinations). G.2 Galaxies/Clusters SPARC metadata (distances, inclinations, M/L priors); cluster merger kinematics, WL covariance matrices. G.3 WL/RSD/CMB/GW DES/KiDS/HSC tomographic covariances; BOSS/eBOSS fσ8 ; Planck+BAO+SNe compressed likelihoods; siren catalogs with redshift posteriors. H Reproducibility: Code and Parameter Files H.1 Repository layout udrd/ cosmology/ class_udrd/ # CLASS module params/ # .ini files for runs clusters/
uDRD: Cross-Scale Framework 54 forward_model/ # x(t), (R) solver data/ # Bullet/MACS/A520 inputs galaxies/ sparc_fit/ # RAR + RC pipeline gws/ siren_likelihood/ # friction parameterizations notebooks/ figs/ # paper plots ci/ tests/ # unit tests, stability gates H.2 Configuration Machine-readable YAML for priors, stability cuts (e.g., α -nodes, early-time freeze-out), and scale cuts for WL/RSD. Random seeds and sampler settings (nested sampling & MCMC) included. H.3 Artifacts We provide chains (posteriors), best-fit parameter sets for the one-parameter-set stress test, and all derived data vectors/covariances used to generate the figures and tables in the main text.
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