Dark Matter as Information Islands: Charge-less Topological Knots and Gravitational Lensing in QCA Networks

Dark Matter as Information Islands: Charge-less Topological Knots and Gravitational Lensing in QCA Networks Anonymous Author November 26, 2025 Abstract Astrophysical observations reveal that approximately one-quarter of the universe’s total energy density exists in a form that does not emit light, does not participate in electromagnetic interactions, but influences structure formation through gravity—known as dark matter. Precision cosmological data (e.g., Planck 2018 cosmological parameter measurements) indicate that the dark matter energy density parameter is about five times that of visible baryonic matter, and existed in a cold, non-relativistic form in the early universe [?]. Although candidates such as weakly interacting massive particles (WIMPs) and axions have been extensively studied at the particle physics level, no direct detection has provided a confirmation signal to date, merely continuing to raise the lower limits on new particle interaction strengths and mass parameter space [?]. This paper provides a purely topological and information-theoretic interpretation of dark matter within the unified framework of quantum cellular automata (QCA) and the optical path conservation/Information-Gravity Variational Principle (IGVP), without introducing any new fundamental particle species. Building on our previously established picture of “mass as topological knot” and “gravity as information congestion”, we first rigorously characterize the topological origin of mass in Dirac-type QCA: the effective mass of local excitations is determined by the winding number of the evolution operator on the Brillouin zone; simultaneously, we characterize charge as the representation type of the local Hilbert space under the U(1) gauge group. We then introduce a natural Hilbert space decomposition Hcell =Hvis ⊗Hhid, where the visible sector Hvis carries standard model gauge charges, while the topological winding on the hidden sector Hhid is trivial under the electromagnetic U(1)EM representation. We call the local excitations supported by the latter information islands: they possess nonzero topological index in momentum space, hence have rest mass and inertia, but appear chargeless in all observable electromagnetic processes. Within the IGVP framework, the gravitational field is not directly determined by components of the energy-momentum tensor, but controlled by the spatial distribution of total information processing density ρinfo(x). The internal evolution of both visible and hidden sectors consumes the update budget of the underlying QCA, thus contributing equally to ρinfo(x). We prove: in the weak-field limit, the effective optical metric derived from IGVP is fully equivalent to the Newton–Poisson limit in general relativity determined by total mass density ρvis +ρhid, hence information islands are indistinguishable from cold dark matter in gravitational lensing and galaxy rotation curves on galactic scales. In particular, the information island sector does not couple to electromagnetic or other “collisional” fields, thus appearing as nearly frictionless collision-independent matter in galaxy cluster collisions (such as the Bullet Cluster), with the lensing mass center naturally separated from the hot baryon gas X-ray peak, consistent with observational results [?]. On galactic scales, we treat information islands as a cold, dissipationless, collisionindependent particle species satisfying the collisionless Boltzmann equation in the QCA continuum limit. Using phase-space volume conservation and the maximum entropy principle, we derive that spherically symmetric, virialized halo structures have approximately 1 isothermal sphere density distribution ρ(r)∝r−2, yielding flat rotation curves v(r)≈const, consistent with observed late-type galaxy rotation curves [?]. The main results of this paper can be summarized at three levels: (i) In QCA models satisfying local unitarity and finite-dimensional local Hilbert space, there exist “charge-less topological knot” subspaces with nontrivial topological mass but trivial under visible gauge groups; (ii) Under IGVP, the internal evolution of this subspace couples to the effective optical metric with weight equivalent to baryonic matter, thus appearing as dark matter in all gravity-probed observations; (iii) On galactic and cluster scales overall dynamics, such information islands automatically form diffuse, hard-to-dissipate halo structures without collapsing into thin disks. Thus, within the unified quantum information ontology, dark matter can be interpreted as information topological knots in QCA networks that “have only core, no interface”, without introducing any additional fundamental particles. Keywords: Dark matter; Quantum cellular automaton; Topological winding; Information islands; Information-Gravity Variational Principle; Gravitational lensing; Galaxy rotation curves 1 Introduction & Historical Context 1.1 Observational Origin of the Dark Matter Problem Since the mid-20th century, systematic observations of galaxy and galaxy cluster dynamics have revealed serious deviations from gravitational potentials explained solely by visible stars and gas: rotation curves of late-type galaxies tend to flatten at large radii rather than decay as r−1/2as expected from Newtonian potential; velocity dispersion of galaxies in clusters and the virial mass implied by hot X-ray gas far exceed the total visible mass [?]. Modern precision cosmological measurements (such as Planck 2018) further provide the dark matter energy density parameter Ωc≃0.26 through cosmic microwave background anisotropy and large-scale structure power spectra, while baryonic matter is only about Ωb≃0.05 [?]. This means that in the standard ΛCDM model, dark matter is the universe’s dominant nonrelativistic component, playing a key role in both linear and nonlinear stages of structure formation. Strong gravitational lensing provides almost geometrical-optical direct evidence for dark matter: in galaxy cluster collision systems such as the Bullet Cluster, the mass distribution obtained from general relativistic lensing inversion is significantly offset from the hot X-ray gas position and more consistent with galaxy distribution [?]. This “gravitational potential center separated from baryon mass center” phenomenon is difficult to explain by schemes that merely modify gravity laws, but naturally fits the intuitive picture of “existence of collision-independent matter species decoupled from baryons”. 1.2 Traditional Routes: Particle Dark Matter and Hidden Sectors At the particle physics level, dark matter is typically viewed as some new particle beyond the standard model: weakly interacting massive particles (WIMPs), axions, sterile neutrinos, dark photons, etc. These candidates can naturally produce cold, stable, non-relativistic cosmological backgrounds and obtain the correct order-of-magnitude relic density through “freeze-out” or misalignment mechanisms [?]. In recent reviews, “hidden sector dark matter” has become an important paradigm: dark matter exists in a set of quantum fields interacting with the visible sector only through gravity or extremely weak portals [?]. This line of thought formally resembles our starting point: dark matter need not share the same gauge charges as the standard model, only coupling to us through gravity. However, even in hidden sector schemes, one typically still assumes the existence of a family of new particle fields, whose masses and interaction forms are controlled by additional free parameters in the Lagrangian. 2 Meanwhile, topological defects (such as cosmic strings, domain walls, axionic string networks) produced in early universe phase transitions have been widely discussed as potential sources of dark matter or dark energy [?]. In these works, topological properties are mainly manifested in field configuration space, rather than directly connected to ontological quantities of “information processing”. 1.3 QCA Universe Ontology and Re-characterization of Mass/Gravity Quantum cellular automata provide a radically different microscopic picture of the universe: the world consists of local quantum systems on discrete lattice sites, evolving in discrete time according to strictly unitary, local, translation-invariant update rules. Under appropriate continuum limits (lattice spacing a→0, time step ∆t→0), standard field theory equations such as Dirac, Weyl, and Maxwell can emerge [?]. We proposed two complementary structural principles in our previous work: 1. Mass as topological knot: In Dirac-type QCA, the mass term is not an arbitrary parameter but a function of the homotopy class (winding number W) of the evolution operator U(k) mapping S1→SU(2) on the Brillouin zone. Modes with nonzero winding number must maintain continuous internal oscillations on local scales, with nonzero information update rate vint, corresponding to effective mass and inertia. 2. Gravity as information congestion (IGVP): The gravitational field is no longer viewed as metric tensor on a continuous manifold given a priori, but interpreted as the geometric response of the QCA network to inhomogeneity of local information processing density ρinfo(x) under the condition of maintaining overall information optical path conservation. Local information overdensity causes “information congestion”, equivalent to reduced effective light speed or increased refractive index n(x), thus bending light rays and worldlines. In this picture, mass and gravity are respectively manifestations of “self-referential topological loop” and “global information budget balance”, not additional entities introduced externally. 1.4 Goals and Contributions of This Paper Following the above QCA–IGVP ontological direction, this paper reformulates the dark matter problem as: In a discrete unitary universe based on QCA, must there necessarily exist a class of topologically massive but electromagnetically decoupled local excitations that remain “dark” in all photon-mediated observations yet are gravitationally equivalent to baryonic matter? We provide an affirmative answer and prove that in a very broad family of models, such charge-less topological knots not only exist but naturally occupy a finite fraction of the universe’s energy budget statistically, making the dark matter phenomenon an inevitable byproduct of QCA network topological structure rather than additional Lagrangian assumptions. 2 Model & Assumptions 2.1 Dirac-Type QCA and Topological Mass Consider a one-dimensional translation-invariant Dirac-type QCA with local Hilbert space dimension d, lattice sites labeled x∈Z, one-step evolution given by unitary operator ˆ Usatisfying ˆ U=Y x Ux,x+1, 3 where Ux,x+1 acts only on the local degrees of freedom of adjacent cells and is translationinvariant on the lattice. Performing Fourier transform, we can write in momentum representation ˆ U=Zπ/a −π/a dk 2π|k⟩⟨k|⊗U(k), U(k)∈U(d), where ais the lattice spacing. For Dirac-type QCA, we may choose d= 2 and write under suitable parametrization U(k) = exp−iω(k)n(k)·σ, where σare Pauli matrices, n(k) is a unit vector, ω(k) is the dispersion relation. If in the small-momentum limit there exists ω(k)≈p(ck)2+ (mc2/ℏ)2, then the long-wavelength limit of this QCA satisfies the Dirac equation, with parameter m interpreted as effective mass [?]. From the topological perspective, the mapping k∈S17→ U(k)∈SU(2) is classified by K1(S1)∼ =Z, with topological invariant being the winding number W=1 2πiZπ/a −π/a dk ∂klog det U(k). In Dirac-type QCA, W = 0 corresponds to a “topological mass” sector with an energy gap, whose band structure completes a nontrivial winding within the Brillouin zone. We define mass as the image of this winding number mapped to effective field theory mass parameters through the continuum limit renormalization, thereby realizing the characterization of “mass as topological knot”. 2.2 Gauge Fields and Representation-Theoretic Characterization of Charge To introduce electromagnetic interaction in QCA, we endow each cell with local U(1)EM gauge freedom, whose gauge transformation acts as |ψx⟩ 7→ eiqαx|ψx⟩, where αxis the local gauge phase and qis the particle charge. Link fields are realized through link variables on lattice edges Ulink x,x+1 = expiaAx, and the evolution operator after coupling to the electromagnetic field obeys covariance U(k;A)7→ eiqα(k)Uk+∂α;Ae−iqα(k). In this language, charge is defined as the representation of U(1)EM on the local Hilbert space Hcell: if under some basis the gauge transformation acts as eiqαxIH, then the mode is electrically neutral; if the representation is a nontrivial diagonal or off-diagonal matrix, it corresponds to having charge or multiple charges. Electromagnetic interaction is essentially the non-commutative sensitivity of U(k) to Ulink x,x+1. 4 2.3 Visible/Hidden Sectors and Definition of Information Islands We introduce the following structural assumption. Assumption 2.1 (Local Hilbert Space Decomposition).There exists a tensor decomposition of the local Hilbert space Hcell =Hvis ⊗Hhid, satisfying: 1. Electromagnetic U(1)EM acts with nontrivial representation only on Hvis, acting as identity on Hhid, i.e., V(αx) = Vvis(αx)⊗Ihid. 2. In the ideal limit without other interactions, the total evolution operator can be written as ˆ U=ˆ Uvis ⊗ˆ Uhid, where ˆ Uvis supports the spectrum of standard model particles, while ˆ Uhid is an additional set of Dirac/QCA models. In momentum representation, this means there exists U(k) = Uvis(k)⊗Uhid(k), with topological property given by Wtot =Wvis +Whid. Definition 2.2 (Information Island Modes).A family of Bloch modes is called information islands if satisfying: 1. On the Hhid projection, its topological winding number Whid = 0, corresponding to nonzero rest mass mhid >0; 2. On Hvis, it resides in the electrically neutral subspace, i.e., Vvis(αx)acts as identity, hence producing no Aharonov–Bohm phase under U(1)EM transformation; 3. The total state can be written as |Ψ⟩=|ψ0 vis⟩⊗|ψhid⟩, where |ψ0 vis⟩is vacuum or neutral background. For such modes, mass is determined by the topological structure of ˆ Uhid, while all electromagnetic processes (scattering, radiation, absorption) are controlled by ˆ Uvis, thus appearing as completely “dark” local excitations in electromagnetic observations. 2.4 Information-Gravity Variational Principle and Effective Optical Metric Within the IGVP framework, the gravitational field is determined by the QCA network’s optimal response to local information density under total information rate conservation. We only need its simplified form in the weak-field, static limit. Define local information processing density ρinfo(x) = ρvis(x) + ρhid(x), where each term is given by the internal evolution frequency and occupation number of the corresponding sector. For approximately free quasiparticle excitations, we write ρhid(x)∼X i ni(x)ℏω(i) int, 5 where ni(x) is the number density of the i-th information island quasiparticle species at position x,ω(i) int is its internal oscillation frequency, related to mass through mic2=ℏω(i) int. In the weak-field, quasi-static limit, the effective optical metric derived from IGVP can be written as ds2=−1 + 2Φ(x)/c2c2dt2+1−2Φ(x)/c2dx2, where the effective potential Φ(x) satisfies a Poisson-type equation ∇2Φ(x) = 4πGeffρvis(x) + ρhid(x). With appropriate scale choice, Geff is equivalent to Newton’s constant G, making the weakfield limit of IGVP consistent with the Newton–Poisson limit of general relativity. Thus from the perspective of light rays, test particle orbits, and structure formation, information islands are completely equivalent to ordinary cold dark matter on gravitational scales. 2.5 Macroscopic Dynamics Approximation On galactic and cluster scales, we adopt the following dynamical assumptions: Assumption 2.3 (Collision-Independence and Dissipationlessness).1. Information islands interact only through gravity/IGVP, with two-body scattering cross-section far smaller than Coulomb cross-section of baryon gas, viewable as collision-independent on galaxy cluster collision timescales. 2. Information islands cannot emit, absorb, or scatter photons, hence lacking dissipation channels similar to baryonic matter’s radiative cooling; the only energy loss mechanism is extremely weak gravitational wave radiation or network geometric perturbation waves, such processes having timescales far exceeding Hubble time. 3. In galaxy halo formation and evolution, information islands can be treated as cold, nonrelativistic particles satisfying the collisionless Boltzmann equation, macroscopically forming virialized self-gravitating systems. Under this approximation, information islands are nearly identical to standard cold dark matter (CDM) dynamics on large scales, while having radically different ontological interpretation on microscopic levels. 3 Main Results (Theorems and Alignments) This section presents the main theorems of this paper and their correspondence with observational phenomena. Rigorous proofs are developed in subsequent sections and appendices. Theorem 3.1 (Orthogonality of Mass and Charge in QCA).In Dirac-type QCA satisfying Assumption ??, topological mass and electromagnetic charge can achieve orthogonal separation under the Hilbert space decomposition Hcell =Hvis ⊗Hhid: 1. Mass is entirely determined by winding numbers Wvis,Whid of ˆ Uvis and ˆ Uhid in momentum space; 2. Electromagnetic charge is entirely determined by the representation of U(1)EM on Hvis, being trivial representation on Hhid. 6 Therefore there necessarily exists a family of modes satisfying Whid = 0,Wvis = 0,q= 0, corresponding to massive but chargeless “hidden topological knots”, i.e., information islands. Theorem 3.2 (Equivalent Gravitational Contribution of Information Islands to Optical Metric).In the weak-field limit of IGVP, if total information density is ρinfo(x) = ρvis(x) + ρhid(x), then the effective potential Φ(x)obtained from the variational principle satisfies ∇2Φ(x) = 4πGρvis(x) + ρhid(x), where Gis the gravitational constant after appropriate normalization. Thus information islands contribute to gravitational potential equivalently to ordinary cold dark matter, as long as their internal evolution frequency is related to effective mass through mc2=ℏωint. Any observation dependent on optical metric (such as gravitational lensing, orbital dynamics) cannot distinguish “baryon mass” from “information island mass”. Theorem 3.3 (Approximate Isothermal Sphere Distribution and Flat Rotation Curves of Galaxy Dark Halos).Under Assumption ??, treating information islands as cold, collisionindependent particle gas, when reaching virialized equilibrium in static, spherically symmetric gravitational potential Φ(r), its phase-space distribution function f(x,v) = f(E)satisfies the maximum entropy principle, yielding f(E)∝exp−E/σ2, E =1 2v2+ Φ(r), thus spatial density ρ(r)∝exp−Φ(r)/σ2. Combined with the Poisson equation, the large-radius asymptotic solution is ρ(r)≃σ2 2πGr2, with corresponding enclosed mass M(r)∝r, circular orbit velocity vrot(r) = rGM(r) r≈√2σ≈const. Therefore, information islands naturally form isothermal halo structures with ρ(r)∝r−2, yielding flat galaxy rotation curves without modifying Newtonian gravity or introducing specific potential function shapes. Theorem 3.4 (Mass–Baryon Separation in Galaxy Cluster Collisions).Consider two clusters each containing abundant information islands and baryon gas undergoing nearly head-on collision: 1. Information islands are collision-independent components with typical mean free path far exceeding cluster scales; 2. Baryon gas is strongly collisional, undergoing violent shocks, compression and heating, forming concentrated hot X-ray gas clouds at collision center [?]. Then shortly after collision, the mass distribution satisfies: gravitational potential main peak consistent with galaxy distribution, while X-ray gas peak located in the central region between the two. This “mass–baryon separation” phenomenon is the characteristic signature of observations like the Bullet Cluster, naturally derived in this model from the drastically different collisional properties of information islands and baryon gas. 7 Proposition 3.5 (Compatibility with CMB and Large-Scale Structure).If information islands are already in non-relativistic state (cold dark matter condition) in the early universe and couple to baryon–radiation fluid only through gravity during radiation-dominated era, then this model is equivalent to the standard ΛCDM model at the level of background expansion and linear fluctuation evolution. Its main contributions to CMB anisotropy spectrum, BAO scale, and large-scale structure power spectrum can be viewed as a cold, negligible-pressure, collision-independent dark matter component, compatible with existing observational constraints [?]. 4 Proofs This section provides proof sketches and key steps for the main theorems, with more technical details arranged in appendices. 4.1 Proof of Theorem ??: Orthogonality of Mass and Charge in QCA Under Assumption ??, the evolution operator in momentum representation can be written as U(k) = Uvis(k)⊗Uhid(k). Consider the representation of U(1)EM V(α) = Vvis(α)⊗Ihid, where Vvis is a unitary representation on Hvis. 1. Topological origin of mass: The winding number is defined by W=1 2πiZπ/a −π/a dk ∂klog det U(k). Using det(U⊗V) = (det U)dim V(det V)dim U, we decompose Was W= dim Hhid ·Wvis + dim Hvis ·Whid, where Wvis =1 2πiZdk ∂klog det Uvis(k),Whid =1 2πiZdk ∂klog det Uhid(k). In the continuum limit, these two topological numbers correspond respectively to mass parameters of visible and hidden sectors. 2. Representation-theoretic origin of charge: Electromagnetic charge is determined by the irreducible decomposition of Vvis(α) on Hvis: if on some irreducible component Vvis(α) = eiqα representation, then its charge is q. For Hhid, the representation is identically identity, hence its charge must be zero. 3. Orthogonality: Mass depends on topological properties of Uvis, Uhid, while charge depends only on representation of Vvis. As long as Uhid can be chosen with parameter intervals having nonzero Whid, while Vvis is trivial representation on some subspace, then modes with Whid = 0, q = 0 exist in that subspace, i.e., information islands. Topological classification theory (e.g., 1D gapped band structure classified by K1(S1)∼ =Z) guarantees this choice is open and stable in parameter space. Therefore, mass and charge can achieve completely orthogonal degree-of-freedom allocation in the QCA framework, thus admitting existence of chargeless topologically massive modes. 8 4.2 Proof of Theorem ??: Information Island Contribution to Optical Metric IGVP in the weak-field limit can be written as an action functional SIGVP[g, ρinfo] = Zd4x√−gLgeom(g) + λκ(g)−ρinfo, where κ(g) is the unified time density defined by Wigner–Smith time-delay trace or scattering phase derivative, λis a Lagrange multiplier. Varying with respect to gµν and taking weak-field limit gµν =ηµν +hµν yields approximate field equation ∇2Φ(x)=4πGeffρinfo(x), where Φ is Newtonian potential, Geff determined by λand details of Lgeom. When both visible and hidden sectors exist, information density is ρinfo(x) = X j nvis j(x)ℏωvis int,j +X i nhid i(x)ℏωhid int,i. Using relation mc2=ℏωint, we define equivalent mass density ρmass(x) = ρvis(x) + ρhid(x), satisfying ρinfo(x) = c2ρmass(x). Therefore ∇2Φ(x)=4πGeffc2ρvis(x) + ρhid(x). By appropriately choosing normalization in the theory to make Geffc2=G, we obtain the standard Newton–Poisson equation. Hence, the refractive index in the optical metric n(x)≈1−2Φ(x) c2 is entirely controlled by total mass density, decomposed into “baryon + information island” contributions. This proves that information islands’ contribution to gravity and lensing effects is equivalent to ordinary cold dark matter in the weak-field limit. 4.3 Proof of Theorem ??: Isothermal Sphere and Flat Rotation Curves For galaxy halos composed of information islands, we adopt the collisionless Boltzmann equation ∂f ∂t +v·∇xf−∇Φ·∇vf= 0. In steady-state, spherically symmetric case, fdepends only on total particle energy E= 1 2v2+ Φ(r). Maximum entropy principle yields f(E)∝exp−E/σ2, corresponding to an isothermal family. Spatial density ρ(r) = Zf(E)d3v∝exp−Φ(r)/σ2. Substituting into spherically symmetric Poisson equation 1 r2 d drr2dΦ dr= 4πGρ0exp−Φ(r)/σ2, 9 same as visible sector, but since U(1)EM acts as identity on Hhid, all modes in hidden sector are electromagnetically neutral. Total evolution operator U(k) = Uvis(k)⊗Uhid(k) operates on Hvis ⊗Hhid ∼ =C4. Choosing basis {|e1⟩,|e2⟩}vis ⊗{|h1⟩,|h2⟩}hid, visible charge representation acts only on |ei⟩directions: Vvis(α) = eiqα 0 0 1. In the two-dimensional subspace generated by states |e2⟩⊗|hj⟩, charge is zero, but Uhid(k) topology still gives nonzero mass, i.e., information island modes. A.3 Genericity Under Random QCA Rules Consider randomly sampling local evolution operators U(k) (satisfying translation invariance and locality constraints) on finite-dimensional Hilbert space. In the vast majority of cases, its Bloch band structure will possess nonzero topological number, unless parameters happen to fall on critical hypersurfaces of measure zero. On the other hand, the choice of U(1)EM representation determines which bands correspond to charged visible modes and which to neutral hidden modes. Therefore, in broad families of random QCA models, “dark bands” with nonzero topological mass yet trivial under U(1)EM are statistically natural; dark matter sectors can appear without fine-tuning. B IGVP, Optical Metric and Lensing This appendix more systematically derives the optical metric and lensing formulas in the weakfield limit of IGVP. B.1 Unified Time Density and Information Density In scattering theory, the Wigner–Smith delay operator is defined as Q(ω) = −iS†(ω)∂S(ω) ∂ω , whose normalized trace gives the sum of Eisenbud–Wigner–Smith delay times. We introduce time density in unified time theory κ(ω) = 1 2πtr Q(ω), and define local information density as ρinfo(x) = Zdω κ(ω;x). In the QCA framework, κ(ω;x) can be defined through local spectral measure and scattering phase derivative, equivalent to “density of states per unit frequency interval”. 16 B.2 IGVP Variational Equation The IGVP action can be written as S[g, κ] = Zd4x√−gc4 16πGR(g) + λ(x)κ(g;x)−κmicro(x), where κmicro(x) is the unified time density determined by QCA microscopic degrees of freedom, λ(x) is Lagrange multiplier. Varying with respect to gµν and linearizing yields ∇2Φ(x) = 4πGρmass(x), where ρmass(x)∝κmicro(x) is proportional to local information density. This process illustrates the unified chain “time density–density of states–mass density”. B.3 Optical Metric and Lensing Formula In weak-field static case, take g00 =−(1 + 2Φ/c2), gij = (1 −2Φ/c2)δij. Light rays satisfy null geodesic condition ds2= 0. In thin lens approximation, gravitational lensing deflection angle can be written as α(ξ) = 2 c2Z∇⊥Φ(ξ, z)dz=4G c2Zξ−ξ′Σ(ξ′) |ξ−ξ′|2d2ξ′, where Σ(ξ) = Zρmass(ξ, z)dz is the projected mass density along the line of sight. Since ρmass =ρvis +ρhid, information island contribution directly enters the lensing kernel. This derivation is completely isomorphic to standard GR lensing theory, except that the composition of ρmass is ontologically reinterpreted as “information density” rather than traditional energy-momentum tensor source term. C Collisionless Virialization and Isothermal Halos This appendix provides detailed steps for deriving isothermal halos and flat rotation curves under the collisionless Boltzmann equation. C.1 Boltzmann Equation and Conserved Quantities In spherically symmetric, steady-state case, the collisionless Boltzmann equation is v·∇xf−∇Φ·∇vf= 0. Any distribution f(E) depending only on energy E=1 2v2+ Φ(r) is a solution of this equation, because v·∇xf(E) = f′(E)v·∇xE=f′(E)v·∇Φ, ∇Φ·∇vf(E) = f′(E)∇Φ·v, which cancel. 17 C.2 Maximum Entropy Principle and Maxwell–Boltzmann Distribution Under constraints of given total particle number and total energy, maximizing entropy S=−Zfln fd3xd3v yields optimal distribution f(E) = Aexp−E/σ2, where σ2is analogous to velocity dispersion squared. Spatial density ρ(r) = Zf(E)d3v= 4πA exp−Φ(r)/σ2Z∞ 0 v2exp−v2/2σ2dv=ρ0exp−Φ(r)/σ2. C.3 Poisson Equation and Asymptotic Solution Poisson equation 1 r2 d drr2dΦ dr= 4πGρ0exp−Φ(r)/σ2 can be linearized in the region where ris sufficiently large and |Φ(r)| ≪ σ2as 1 r2 d drr2dΦ dr≈4πGρ01−Φ(r) σ2. Seeking solution of form Φ(r)=2σ2ln(r/r0), substituting into left side 1 r2 d drr22σ2 r=1 r2 d dr(2σ2r) = 2σ2 r2, right side approximately constant 4πGρ0in asymptotic region. Setting them equal yields ρ0=σ2 2πGr2. 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