--- { "path": "/en/theory/cosmology/dark-matter", "title": "Dark matter research: candidate spatial gravitational responses", "category": "Cosmology", "summary": "Explain smoothing, finite disk confinement, nonlinear polarization, coherent modes and finite-propagation memory, separating calculated results, constitutive inputs and joint observational tasks.", "related": [ "/en/theory/cosmology", "/en/research/data" ], "chapter": "第五章", "id": "ASD-DOC-THEORY-COSMOLOGY-DARK-MATTER-EN", "template": "theory", "version": "1.1", "updated": "2026-10-24", "canonical": false, "source_tier": "website-guide", "depends_on": [ "/en/theory/gravity", "/en/theory/cosmology" ], "open_problems": [], "equations": [], "sources": [ { "label": "Final combined manuscript — adopted scope documented on this page", "kind": "final-combined-manuscript", "url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json" }, { "label": "Previous source: CH5-ca737ec4e327 · Chapter 5 publication manuscript · Part V", "kind": "historical-source" }, { "label": "Current space dynamics source manifest (identifies the same Chapter 5 file; module scopes remain separate)", "kind": "source-identity", "url": "/downloads/ASD_Space_Dynamics_Source_Manifest.json" } ], "theory_status": "HYPOTHESIS", "evidence_status": "UNTESTED", "title_en": "Dark matter research: candidate spatial gravitational responses", "locale": "en", "translation_of": "/theory/cosmology/dark-matter", "translation_path": "/theory/cosmology/dark-matter", "translation_version": "1.1", "canonical_url": "https://asd-physics.org/en/theory/cosmology/dark-matter", "outline": [ "Starting with gravitational response", "What higher derivative smoothing calculates", "Response confined to a disk over a finite range", "Nonlinear polarization and the low acceleration limit", "How a gapped coherent mode can be nearly pressureless", "Finite propagation and spatial memory after a collision", "Joint tests and conventions for constants", "Sources and review scope" ], "section_anchors": [ "response-question", "smoothing-kernel", "disk-response", "nonlinear-polarization", "coherent-mode", "finite-memory", "joint-observables", "sources" ], "legacy_anchors": { "s0": "response-question", "s1": "response-question", "s2": "smoothing-kernel", "s3": "joint-observables", "s4": "joint-observables", "s5": "nonlinear-polarization", "s6": "joint-observables", "s7": "response-question", "s8": "sources", "s9": "joint-observables" }, "source_version": "ASD-BOOK-ddbade997b45", "source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86", "source_reference": "§§5.5–5.10, (5.5.15)–(5.5.31)", "source_records": [ { "id": "ASD-BOOK-ddbade997b45", "title": "各向异性空间动力学（最终合订稿）", "filename": "各向异性空间动力学.docx", "format": "docx", "size_bytes": 503564, "sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e" } ], "final_source": { "source_id": "ASD-BOOK-ddbade997b45", "chapter": "Chapter 5", "reference": "§§5.5–5.10, (5.5.15)–(5.5.31)", "locators": [ "p2155", "p2178", "p2190", "p2199", "p2207" ], "adopted_scope": "Candidate responses; exact power-law branch and leading coherent average", "conditions": [ "Each candidate keeps its own background, constitutive inputs and energy normalization; no combined observational fit" ], "unresolved": [], "migration_status": "reviewed", "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e", "source_manifest": "final-manuscript-source-manifest.json", "body_review": "updated" }, "editorial_date": "2026-10-24", "previous_source": { "source_reference": "Chapter 5, Part V §§5–10, (5.5.15)–(5.5.31); Part I variable conventions.", "source_version": "CH5-ca737ec4e327", "source_sha256": "ca737ec4e327d72c5507bbc8e00295b6e56029c7583ebad8e505cdb9c68c6262", "source_records": [ { "id": "CH5-ca737ec4e327", "title": "第五章-出版整理稿.docx", "sha256": "ca737ec4e327d72c5507bbc8e00295b6e56029c7583ebad8e505cdb9c68c6262", "format": "docx" } ], "sources": [ { "label": "CH5-ca737ec4e327 · Chapter 5 publication manuscript · Part V", "kind": "locked-manuscript" }, { "label": "Current space dynamics source manifest (identifies the same Chapter 5 file; module scopes remain separate)", "kind": "source-identity", "url": "/downloads/ASD_Space_Dynamics_Source_Manifest.json" } ] }, "source_chapter": "Chapter 5", "source_kind": "manifest", "source_manifest": "final-manuscript-source-manifest.json", "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json" } --- ## Starting with gravitational response The dark matter problem concerns comparable gravitational effects in matter motion, light propagation and background evolution. ASD asks whether longitudinal spatial states and their coupling to other degrees of freedom can supply these effects. Part V of Chapter 5 develops effective constructions for smoothing, response confined to a disk, nonlinear polarization and coherent modes. Each begins with explicit inputs and then calculates a result. They have not been combined into a model fitted to all observations. This page uses $\Phi_N=-c^2\varphi$ and $\mathbf g=-\nabla\Phi_N=c^2\nabla\varphi$. $\Phi_N$ has dimensions of energy per unit mass, $\varphi$ is a dimensionless stretching potential and $\mathbf g$ is vector acceleration; $g=|\mathbf g|$ denotes its magnitude. $G$ is the weak field gravitational coupling in use and $\rho_b$ is physical baryonic mass density. Sources, backgrounds and parameters must refer to the same conditions. A local density perturbation is not already a calculated long range potential. ## What higher derivative smoothing calculates First consider a static response with a fixed length $\ell_g>0$, manuscript equation 5.5.15: $$ (1-\ell_g^2\nabla^2)\nabla^2\Phi_N=4\pi G\rho_b. $$ The extra spatial derivatives introduce a short distance response scale. Fourier transformation over all space replaces $\nabla^2$ with $-k^2$, giving $$ \widetilde\Phi_N(\mathbf k)=-\frac{4\pi G\widetilde\rho_b(\mathbf k)}{k^2(1+\ell_g^2k^2)},\qquad \widetilde\rho_{\rm eff}=\frac{\widetilde\rho_b}{1+\ell_g^2k^2}. $$ This is equation 5.5.16. $\rho_{\rm eff}$ is the effective source obtained by rewriting this potential in ordinary Poisson form, not an independently measured new material component. For a point source $M\delta^{(3)}(\mathbf r)$, impose a potential tending to zero at infinity and add no singular homogeneous solution. Partial fraction decomposition of the kernel into $1/k^2$ and a Yukawa part, followed by inverse transformation, gives equations 5.5.17–5.5.18: $$ \begin{aligned} \Phi_N(r)&=-\frac{GM}{r}(1-e^{-r/\ell_g}),\\ \rho_{\rm eff}(r)&=\frac{Me^{-r/\ell_g}}{4\pi\ell_g^2r},\\ g(r)&=\frac{GM}{r^2}\left[1-\left(1+\frac r{\ell_g}\right)e^{-r/\ell_g}\right]\leq\frac{GM}{r^2}. \end{aligned} $$ The radial volume integral is $\int\rho_{\rm eff}d^3r=M$. The smoothing kernel changes the near field distribution without adding mass to the asymptotic Newtonian term. Defining the medium contribution as $\rho_{\rm med}=\rho_{\rm eff}-\rho_b$ gives a zero total integral. The calculation identifies what smoothing does and why additional long range attraction requires a different long wavelength constitutive law or dynamical degree of freedom. ## Response confined to a disk over a finite range If a physical binding, boundary or constitutive mechanism confines one spatial response to a disk, the manuscript adopts a quasi-two-dimensional approximation between source scale $R_d$ and three-dimensional leakage scale $R_c$. A disk-shaped baryonic source alone does not establish this confinement. In the source-free radial interval $R_d\ll r\ll R_c$, the two-dimensional Laplace equation is $$ \frac1r\frac d{dr}\left(r\frac{d\Phi_N}{dr}\right)=0. $$ One integration makes $r\Phi_N'$ constant. A second gives equation 5.5.19: $$ \Phi_N=v_f^2\ln(r/r_0),\qquad \varphi=-\frac{v_f^2}{c^2}\ln(r/r_0),\qquad \mathbf g=-\frac{v_f^2}{r}\widehat{\mathbf r}. $$ $r_0$ fixes the potential reference and $v_f^2$ is the integration coefficient representing source strength. Circular motion obeys $v_c^2/r=g$, so $v_c=v_f$ in this interval: constant speed is a result of the prescribed two-dimensional boundary problem. The manuscript then states a mass scaling that needs an additional explanation, $v_f^4=GM_ba_*$, equation 5.5.20. $M_b$ is total baryonic mass and $a_*$ an acceleration scale. This source-strength relation is not derived by integrating the logarithmic potential. Beyond $R_c$, matching to a three-dimensional exterior or a specified background must fix the potential and field-energy boundary. ## Nonlinear polarization and the low acceleration limit A second candidate allows spatial response to depend on field strength. The manuscript uses the existing [Bekenstein–Milgrom nonlinear Poisson structure](https://articles.adsabs.harvard.edu/pdf/1984ApJ...286....7B) as an effective input and requires ASD to explain its origin in microscopic longitudinal–transverse coupling. Equation 5.5.21 chooses the static functional $$ E_\varphi=\int d^3r\left[\frac{a_*^2}{8\pi G}F(X)-\rho_bc^2\varphi\right],\qquad X=\frac{c^4|\nabla\varphi|^2}{a_*^2}. $$ $X$ is dimensionless field strength and $F$ an undetermined constitutive function. The first term specifies nonlinear field energy; the second specifies source coupling. Use $\delta X=2c^4\nabla\varphi\cdot\nabla\delta\varphi/a_*^2$ and integrate by parts with the boundary variation set to zero. Equation 5.5.22 becomes $$ \delta E_\varphi=-\int d^3r\left[\frac{c^4}{4\pi G}\nabla\cdot(F'(X)\nabla\varphi)+\rho_bc^2\right]\delta\varphi. $$ With $\mu(x)=F'(x^2)$, stationarity gives $$ \nabla\cdot\left[\mu(g/a_*)\mathbf g\right]=-4\pi G\rho_b. $$ This is equation 5.5.23. The high acceleration condition $\mu\to1$ restores the Newtonian limit of this model. Outside a spherical source, a surface integral gives $\mu(g/a_*)g=GM_b/r^2$. Only after the additional deep low acceleration input $\mu(x)=x$ is imposed does equation 5.5.24 follow: $$ g(r)=\frac{\sqrt{GM_ba_*}}r,\qquad v_c^4=GM_ba_*. $$ The result depends on the chosen constitutive law. Its low field form corresponds to $F(X)=2X^{3/2}/3$ and its high field form to $F(X)=X$. Local ellipticity of the static equation requires at least $\mu>0$ and $\mu+x\mu'>0$. These are checkable mathematical conditions for the specified boundary problem. Causality and stability of time evolution require a dynamic action. ASD must still calculate $F$ and $a_*$ from its own degrees of freedom and solve with actual baryonic distributions and global boundaries; an input law is not an independent prediction. Here the equalities are exact for the selected deep-acceleration power-law branch $F(X)=2X^{3/2}/3$. A smooth constitutive function approaching this branch gives corresponding asymptotic relations instead; no global replacement of approximation signs is implied. ## How a gapped coherent mode can be nearly pressureless A neutral collective mode with positive energy is another possible carrier of additional gravitational response. Chapter 5 denotes the candidate coherent field by $s_c$, the dimensionless mean spatial scale by $a_U(t)$, and sets $H=\dot a_U/a_U$; $\mathbf x$ denotes comoving coordinates. Equation 5.5.25 chooses $$ S_c=\int dt\,d^3x\,a_U^3\left[\frac{\chi_c}{2}\dot s_c^{\,2}-\frac{\chi_cc_c^2}{2a_U^2}|\nabla s_c|^2-\frac{\chi_c\Omega_c^2}{2}s_c^2-\frac{\lambda_c}{4}s_c^4\right]. $$ $\chi_c>0$ is the field normalization, $\Omega_c$ a nonzero intrinsic angular frequency and $c_c$ a gradient speed parameter, distinct from vacuum propagation speed and the photon–baryon fluid sound speed. $\lambda_c$ is a self-coupling. Time variation acts on $a_U^3\chi_c\dot s_c$ and therefore produces the term $3H\dot s_c$, giving equation 5.5.26: $$ \ddot s_c+3H\dot s_c-\frac{c_c^2}{a_U^2}\nabla^2s_c+\Omega_c^2s_c+\frac{\lambda_c}{\chi_c}s_c^3=0. $$ For a homogeneous rapid oscillation with negligible gradient and quartic terms, $\Omega_c\gg H$ and $|\dot H|\ll\Omega_c^2$, the approximate solution is $s_c=A_0a_U^{-3/2}\cos(\Omega_ct+\delta)$. Energy density and pressure in the same quadratic model are equation 5.5.27: $$ \varepsilon_c=\frac{\chi_c}{2}(\dot s_c^{\,2}+\Omega_c^2s_c^2),\qquad p_c=\frac{\chi_c}{2}(\dot s_c^{\,2}-\Omega_c^2s_c^2). $$ At leading adiabatic order, averaging over the rapid phase with $a_U$ held fixed gives the final manuscript's (5.5.28): $$ \langle p_c\rangle_0=0,\qquad \langle\varepsilon_c\rangle_0=\frac12\chi_c\Omega_c^2A_0^2a_U^{-3},\qquad \rho_c=\frac{\chi_c\Omega_c^2A_0^2}{2c^2}a_U^{-3}. $$ The subscript 0 makes the averaging order explicit. Differentiating the slowly changing amplitude produces higher-order terms involving $H/\Omega_c$; thus the full evolving mode is nearly pressureless, not exactly pressureless at every order. Here $\rho_c$ is mass density, not the core position probability density in the quantum module. The nearly pressureless limit relies on a gap and quadratic-potential dominance; ordinary gapless longitudinal waves do not meet these conditions. Gradients, self-coupling, lifetime and perturbation pressure determine the scales on which the mode can cluster. Its effective mathematics resembles existing coherent scalar models, while its existence as a collective branch of ASD longitudinal–transverse fields remains to be established. ## Finite propagation and spatial memory after a collision A static field alone cannot determine dynamic peak locations in a colliding cluster. [Weak lensing observations by Clowe and colleagues](https://arxiv.org/abs/astro-ph/0608407) constrain the separation of lensing reconstruction and hot gas locations. The manuscript therefore examines additional spatial response with memory; it does not claim a completed fit of this system. Let gravitational polarization $\mathbf P$ obey $\rho_{\rm pol}=-\nabla\cdot\mathbf P$, so its dimensions are mass per area. To retain a finite front in a local constant-coefficient model, equation 5.5.30 uses a second order time response: $$ \tau_2^2D_t^2\mathbf P+\tau_1D_t\mathbf P+\mathbf P-\ell_P^2\nabla^2\mathbf P=\chi_P(g)\mathbf g,\qquad D_t=\partial_t+\mathbf v_P\cdot\nabla. $$ $\tau_2>0$, $\tau_1\geq0$, $\ell_P$ is a spatial response scale and $\mathbf v_P$ is the response migration velocity. Relative to a constant migration flow, the principal characteristic speed is $\ell_P/\tau_2$. The condition $|\mathbf v_P|+\ell_P/\tau_2\leq c$ is sufficient for fronts in all directions to stay below $c$ in this local model. A slowly changing limit can reduce to first order relaxation–diffusion, but the first order equation with parabolic tails cannot serve as the fundamental strictly finite-propagation equation. For zero migration and constant coefficients, with boundary energy flux terms vanishing, multiply the equation by $\dot{\mathbf P}$ and integrate. Manuscript equation 5.5.31 gives $$ \begin{aligned} \frac d{dt}\frac12\int\left[\tau_2^2|\dot{\mathbf P}|^2+|\mathbf P|^2+\ell_P^2|\nabla\mathbf P|^2\right]d^3r &=\int\chi_P\mathbf g\cdot\dot{\mathbf P}\,d^3r\\ &\quad-\tau_1\int|\dot{\mathbf P}|^2d^3r. \end{aligned} $$ The bracketed quantity requires a positive normalization to express energy in joules. The right-hand terms represent driving work and energy transferred into a coarse-grained environment. A conserved quantity for a closed system must include that environment. Finite propagation and long term memory can coexist, but retention time, initial fields and migration response need a solution; the equation by itself does not calculate an observed peak location. ## Joint tests and conventions for constants If coherent, baryonic and polarization mass densities enter the weak field model in use, equation 5.5.29 is written in comoving coordinates as $$ \nabla_x^2\varphi=-\frac{4\pi Ga_U^2}{c^2}(\delta\rho_b+\delta\rho_c+\delta\rho_{\rm pol}). $$ Because $\varphi$ is dimensionless, a mass-density source still requires $c^{-2}$; an energy-density source uses $c^{-4}$. Contributions must be defined separately so one response is not counted as both coherent field and polarization energy. Before their microscopic relationship is established, smoothing, disk confinement, nonlinear polarization and coherent modes cannot be treated as four independently additive dark masses. Lensing must also connect to [light propagation in Chapter 2](/en/theory/gravity/light-deflection). In a weak deflection limit, the ray depends on the transverse gradient of effective refractive index. Its relation to $\varphi$, shear and background cannot be silently replaced by the source formula of another theory. Ultimately one parameter set must calculate rotation curves, lensing, collision response, structure growth, background temperature and polarization spectra. That closed joint fit is not currently available. The effective calculations on this page use $c,\hbar$ and normalization coefficients under their declared constant-coefficient conditions. Other proposals for background-dependent action scales cannot be implemented merely by substituting dimensional constants into these equations. They require variable-coefficient dynamics, a common reference process and comparable dimensionless quantities. A change in unit values alone is not an observational prediction. See [cosmic backgrounds](/en/theory/cosmology/cmb) and the [prediction registry](/en/tests/predictions) for the associated scope. ## Sources and review scope This page organizes the final collected manuscript, Chapter 5 (ASD-BOOK-ddbade997b45), Part V, subsections 5–10, equations 5.5.15–5.5.31. Variable conventions connect to Chapter 5 Part I and the current [gravity module](/en/theory/gravity). The original Bekenstein–Milgrom paper identifies the external effective structure being adopted; the original Clowe paper identifies an observational constraint on collisions. Neither independently confirms ASD. These are explicit candidates and conditional calculations. The page remains an exploratory guide with HYPOTHESIS and UNTESTED status. It supplies no raw-data fit of a galaxy or cluster and incorporates no constant or photon revision outside the locked source or awaiting approval. The [manuscript guide](/en/research/monograph) and [knowledge scope](/en/ai/knowledge-base) retain the source and publication boundaries.