Gone

Prepared for submission to JCAP A Minimal Stochastic-Curvature Framework for Dark-Sector Phenomenology Keith A. Schneider University of Delaware E-mail: k[email protected] Abstract. The standard cosmological model accounts for structure formation and cosmic acceleration by introducing cold dark matter and dark energy as separate components with empirically assigned properties. While phenomenologically successful, these constructs do not explain the origin of the gravitational behavior they describe. We propose a minimal geometric reinterpretation in which both phenomena arise from stochastic curvature dipoles in spacetime itself. In a semiclassical gravitational regime, localized pairs of attractive and repulsive curvature—“wells” and “towers”—emerge and selforganize through gravitational interaction. The attractive components reproduce the clustering phenomenology attributed to cold dark matter, while the repulsive components generate large-scale cosmic acceleration as the dipole population coarsens over time. The framework introduces no new particles, fields, or modifications of the Einstein equations, and contains no freely tunable dark-sector parameters. Once the statistics of curvature fluctuations are specified, both structure growth and expansion history follow. The model is therefore falsifiable: failure to recover the inferred coarsening dynamics from observations would rule it out. In this sense, the dark sector is reinterpreted not as new substance, but as emergent spacetime geometry. Contents 1 Introduction 1 2 Conceptual Framework: Curvature Dipoles 2 3 Relation to Quantum Gravity and Gravitons 2 4 Coarsening Dynamics and Structure Formation 3 4.1 Field-level parametrization 3 4.2 Coarsening scale and correlators 4 4.3 From scale separation to “dark-matter-like” clustering 4 5 Emergent Cosmic Expansion 5 5.1 Coarse-grained stress–energy scaling 5 5.2 Why acceleration turns on late 6 5.3 Parameter inference and observational closure 6 5.4 Distinction from modified–gravity approaches 7 6 Observational Consequences 7 6.1 A minimal quantitative prediction 7 7 Comparison with Alternative Theories 8 8 Why This Was Not Seen Earlier 8 9 Conclusion 8 1 Introduction The discovery that most of the gravitational dynamics of the Universe cannot be explained by visible matter alone has led to the introduction of two dominant theoretical constructs: cold dark matter (CDM) [1] and dark energy. Together, these components account for galactic rotation curves, gravitational lensing, large-scale structure formation, and the observed accelerated expansion of the Universe. [2–8] Despite their empirical success, neither CDM nor dark energy is predicted by established microphysical theory; their properties are instead inferred a posteriori to match observations. [6,9–12] Over the past several decades, the CDM paradigm has accumulated a growing set of auxiliary assumptions designed to reconcile simulations with observations on small and intermediate scales, including baryonic feedback prescriptions, self-interactions, and environmentdependent phenomenology. [13–16] While each extension improves agreement in specific regimes, the overall pattern is reminiscent of historical frameworks in which descriptive flexibility substituted for explanatory depth. Historically, similar situations have arisen when descriptive additions compensated for gaps in theoretical understanding. Epicycles in Ptolemaic astronomy and the luminiferous ether in nineteenth-century electrodynamics successfully organized data but were ultimately discarded once deeper principles were identified. [17] We argue that the modern dark sector – 1 – occupies an analogous conceptual role: a highly successful bookkeeping device whose elements are not derived from first principles. This work advances the thesis that no additional substances are required. Instead, the phenomena attributed to CDM and dark energy emerge naturally if spacetime itself exhibits stochastic curvature fluctuations. When these fluctuations occur as paired attractive and repulsive perturbations, their collective evolution reproduces both structure formation and cosmic acceleration without introducing new degrees of freedom beyond those already implicit in gravitational dynamics. Why this matters now. A growing set of observational tensions—including small-scale challenges to CDM, void lensing anomalies, and persistent discrepancies in the Hubble constant— suggest that the standard dark-sector parametrization may be incomplete. [13,18–26] At the same time, advances in precision cosmology and gravitational probes increasingly constrain ad hoc extensions. A geometric reinterpretation that unifies dark matter and cosmic acceleration within spacetime dynamics offers a timely alternative that is conservative in assumptions yet predictive in its large-scale consequences. 2 Conceptual Framework: Curvature Dipoles We consider a semiclassical regime in which spacetime geometry exhibits stochastic fluctuations around a classical background. [27,28] Rather than treating these fluctuations as isolated perturbations, we assume that they arise predominantly as dipolar structures: a localized attractive curvature well paired with a nearby repulsive curvature tower. The pair carries no net monopole curvature, preserving large-scale consistency with general relativity. We emphasize that the present framework is intentionally agnostic about the microscopic origin of these stochastic curvature fluctuations. The construction is not proposed as a theory of quantum gravity, but rather as a coarse–grained, effective description of emergent geometric structure in a semiclassical regime, analogous in spirit to other phenomenological approaches used in cosmology. [29] The attractive component deepens gravitational potential wells and captures baryonic matter, while the repulsive component produces an effective outward acceleration in surrounding regions. Individually, neither component dominates globally; collectively, their spatial organization determines large-scale dynamics. 3 Relation to Quantum Gravity and Gravitons These curvature dipoles are geometric rather than particulate in origin. They should not be interpreted as new particle–antiparticle species, nor as simple graviton–antigraviton pairs. [30,31] Instead, they represent localized configurations of spacetime curvature analogous to vacuum polarization effects in quantum field theory. Repulsive gravitational behavior is not foreign to general relativity: a positive cosmological constant, inflationary stress–energy, and certain effective stress tensors already generate repulsion without invoking negative masses. [32–34] In this sense, curvature towers extend known gravitational behavior into a stochastic, localized regime. – 2 – 4 Coarsening Dynamics and Structure Formation The emphasis throughout this section is on statistical, large–scale properties of structure formation. The framework is not intended to model detailed galactic rotation curves or individual halo profiles at the present stage, but rather to capture the collective influence of stochastic geometric structure on clustering, void formation, and late–time cosmic dynamics. 4.1 Field-level parametrization We introduce a minimal mathematical language sufficient to (i) define what is meant by a “well” and “tower” in a coordinate-independent way at the level of perturbations, and (ii) connect the coarsening scale to observables. Let gµν = ¯gµν +hµν be a perturbation about an approximately Friedmann–Robertson– Walker background ¯gµν . In the weak-field, sub-horizon regime one may describe scalar perturbations by a Newtonian-gauge potential Φ(x, t), ds2=−(1 + 2Φ) dt2+a2(t)(1 −2Φ) dx2,(4.1) with ∇2Φ=4πGa2δρb+S(x, t). Here δρbis the baryonic density perturbation and S is an effective geometric “source” encoding curvature dipoles. We emphasize that Sis not introduced as a new field with independent dynamics; it is a phenomenological representation of stochastic geometric structure in the semiclassical regime. A convenient parametrization is to decompose Sinto two signed components, S(x, t)=4πGa2ρw(x, t)−ρt(x, t).(4.2) Formal status of curvature dipoles What is fundamental. The only fundamental degrees of freedom assumed are the spacetime metric and quantum matter fields, governed by Einstein gravity coupled to quantum field theory in curved spacetime. What is emergent. In the Einstein–Langevin formulation of stochastic semiclassical gravity, stress–energy fluctuations source stochastic metric perturbations described by a noise kernel [27,28]. Upon coarse–graining over subhorizon scales, these metric fluctuations generate correlated regions of attractive and repulsive curvature. What is phenomenological. The quantities ρwand ρtintroduced in this paper are coarse–grained curvature correlators, not new matter fields. They provide a compact phenomenological encoding of the stochastic curvature response and carry no independent dynamical degrees of freedom. where ρw≥0corresponds to attractive “wells” and ρt≥0to repulsive “towers”. The dipole hypothesis is that on sufficiently large scales the monopole cancels, ⟨ρw−ρt⟩ ≈ 0,(4.3) while locally the two components are spatially separated by a typical distance that grows with time. – 3 – 4.2 Coarsening scale and correlators Define the signed geometric density δρg≡ρw−ρtand its two-point function C(r, t)≡δρg(x, t)δρg(x+r, t).(4.4) The coarsening length ξ(t)can be defined operationally as the first zero-crossing (or decorrelation) scale of C(r, t), e.g. C(ξ, t) = 0 (or C(ξ, t)=C(0, t)/e; the convention is not important here). Coarsening corresponds to the growth of ξ(t)and to an approximate dynamic scaling form C(r, t)≃C(0, t)fr ξ(t),(4.5) with fa slowly varying shape function. The physical content is that the geometry organizes into like-signed domains of typical size ξ(t). In the simplest phenomenological model, ξ(t)grows as a power law, ξ(t)∝tα, α > 0,(4.6) with repulsive evacuation generically increasing αrelative to purely attractive gravitational clustering. 4.3 From scale separation to “dark-matter-like” clustering On scales ℓ≪ξ(t), attractive domains dominate: baryons fall into wells and the gravitational potential is well approximated by an effective Poisson equation with a positive source. In this regime, the model reproduces the phenomenology usually attributed to CDM halos (rotation curves, lensing masses, and hierarchical merging), with the detailed mapping depending on the small-scale statistics of ρw. [35–37] On scales ℓ≫ξ(t), the signed source δρgaverages toward zero while spatial segregation implies that repulsive regions occupy an increasing fraction of volume. This is the origin of the large-scale acceleration discussed next. Initially, curvature dipoles are assumed to be distributed nearly homogeneously, with small amplitudes and short correlation lengths set by microscopic or semiclassical physics. At this stage, the net gravitational influence of the dipole population averages to near zero on large scales, preserving consistency with an approximately Friedmann–Robertson–Walker background. As the Universe evolves, gravitational interactions cause attractive wells to cluster and deepen, analogously to standard gravitational instability. The presence of repulsive towers, however, qualitatively alters the dynamics: matter evacuation from tower-dominated regions enhances density contrasts and accelerates the growth of structure relative to purely attractive clustering. A convenient coarse-grained description is obtained by introducing a characteristic correlation length ξ(t)describing the typical separation of like-signed curvature regions. As coarsening proceeds, ξ(t)grows monotonically, reflecting the merger of wells into increasingly massive bound structures and the aggregation of repulsive curvature into extended domains. This process is illustrated schematically in Fig. 1and, at later stages, by the smooth curvature field shown in Fig. 2. The key outcome of coarsening is scale separation. Attractive curvature becomes increasingly localized within bound systems, while repulsive curvature dominates the interstitial volume. This separation underlies the simultaneous emergence of clustered structure and large-scale expansion without invoking distinct physical components. – 4 – Early: mixed dipoles Mid: segregation Late: localized wells, coherent tower regions Wells Towers Figure 1. Discrete curvature-dipole picture. Localized pairs of attractive curvature wells and repulsive curvature towers arise stochastically in spacetime. Gravitational clustering aggregates wells while towers evacuate surrounding regions, increasing the separation scale. The paired structure enforces cancellation of net monopole curvature. Early: local cancellation Mid: domain growth Late: well halos + repulsive tower domains 1.2 0.8 0.4 0.0 0.4 0.8 1.2 1.6 Signed curvature (wells + / towers ) Figure 2. Coarse–grained limit after coarsening. Attractive curvature becomes confined to compact regions associated with bound structure, while repulsive curvature fills most of the volume and drives accelerated expansion. Large–scale repulsion emerges from local geometric organization. 5 Emergent Cosmic Expansion 5.1 Coarse-grained stress–energy scaling In the coarse-grained limit, the dipole ensemble defines an effective stress–energy tensor T(eff) µν obtained by averaging on domains larger than ξ(t). [38–40] The contribution of towers is not a fundamental vacuum energy; it is an emergent, time-dependent effect of geometric segregation. A minimal way to encode this is to write ¯ Gµν = 8πG T(b) µν +T(eff) µν ,(5.1) where ¯ Gµν is the Einstein tensor of the smoothed metric and T(b) µν is the baryonic stress– energy. We take the effective component to be approximately homogeneous and isotropic at large scales, T(eff) µ ν= diag(−ρeff , peff, peff, peff).(5.2) Acceleration requires ρeff + 3peff <0. – 5 – To connect this to coarsening, suppose that the volume fraction occupied by towerdominated regions is ft(t), increasing as ξ(t)grows. A minimal scaling ansatz is ρeff(t)∼ρ∗ft(t), peff (t)∼ −w(t)ρeff(t),(5.3) with w(t)an emergent equation-of-state parameter approaching (but not necessarily equaling) unity as the tower network becomes smoother and more volume-filling. Late-time acceleration occurs when w(t)>1/3and ft(t)becomes large enough. A concrete (still phenomenological) mapping is to relate ftto the domain scale by a percolation-style relation ft(t)∼1−exp[−(ξ/ξ0)γ]with γ > 0. Then the expansion history H(t)constrains αand γthrough H2(t) = 8πG 3ρb(t)+ρeff(t),¨a a=−4πG 3ρb+ρeff + 3peff.(5.4) The important point is not the specific functional choice, but the logical structure: once ξ(t) and the mapping ft(ξ)are specified, acceleration is predicted rather than inserted. [6–8] 5.2 Why acceleration turns on late Early times correspond to small ξ(t), limited segregation, and negligible ρeff. As wells merge and attraction becomes confined to compact regions, ft(t)increases and the repulsive contribution becomes dynamically important. In this sense, the expansion history encodes information about the initial fluctuation spectrum and the subsequent coarsening dynamics. 5.3 Parameter inference and observational closure Although the discussion above has been framed phenomenologically, the structure of the theory admits a clear inference strategy rather than arbitrary parameter fitting. The coarsening exponent αgoverning ξ(t)is constrained by the redshift dependence of the expansion rate H(z), while the mapping between ξand the tower volume fraction ftis constrained by the onset epoch and strength of late-time acceleration. On smaller scales, the statistics of ρwdetermine halo mass functions, clustering amplitudes, and lensing profiles in a manner directly comparable to CDM-based analyses. Conversely, the spatial distribution of ρtcontrols void expansion and lensing signals, providing an independent handle on the repulsive sector. Joint fits to growth-of-structure data, void lensing measurements, and background expansion therefore overconstrain the model. Crucially, the parameters inferred in this way describe geometric organization rather than properties of new substances. Successful inference would reconstruct the statistics of spacetime curvature fluctuations themselves, while failure would falsify the framework. In this sense, the model replaces the descriptive freedom of the dark sector with a closed, testable chain from geometry to observation. Gravitons, repulsion, and stability. The repulsive “towers” appearing in this framework are not anti–gravitons, negative–mass particles, or new propagating modes. No modification of Einstein gravity is introduced at the microscopic level. Repulsion arises only in the effective stress–energy tensor obtained after coarse–graining stochastic curvature fluctuations, in direct analogy with inflationary vacuum energy, a cosmological constant, or averaged cosmological backreaction. Any violation of classical energy conditions therefore occurs only in this effective sense and does not imply ghost instabilities or dynamical pathologies. – 6 – The effective nature of the repulsive contribution invites comparison with a wide range of modified–gravity proposals. It is therefore important to be explicit about what this framework does, and does not, assume. 5.4 Distinction from modified–gravity approaches This framework does not modify the Einstein field equations, introduce new gravitational degrees of freedom, or alter the coupling between curvature and stress–energy. General relativity is assumed to hold at all scales, with baryonic matter sourcing curvature in the standard way. The phenomena usually attributed to dark components arise here from coarse–grained stochastic structure in spacetime geometry rather than from modified force laws, additional fields, or nonlocal dynamics. A detailed comparison with modified gravity, dark-sector extensions, and emergent gravity proposals is given in Section 7. 6 Observational Consequences 6.1 A minimal quantitative prediction The stochastic coarsening of curvature dipoles implies a scaling relation between the coarsening exponent α, the redshift zacc marking the onset of cosmic acceleration, and the weak– lensing convergence associated with cosmic voids: κvoid(z)∝(1+zacc)−α.(6.1) This relation fixes the sign and redshift dependence of the effect without introducing new free parameters. A statistically significant violation of this scaling in future void–lensing surveys would directly falsify the curvature–dipole framework at the level of its core coarsening hypothesis. If future surveys were to find no statistically significant scale–dependent deviations in large– scale clustering, void statistics, or weak–lensing correlations beyond those predicted by ΛCDM, the curvature–dipole framework proposed here would be ruled out. No–Free–Parameters Clarification. The curvature–dipole framework does not introduce tunable particle properties, interaction strengths, or screening scales. All phenomenological behavior follows from geometric organization. Fixed ingredients: Einstein gravity; baryonic stress–energy; stochastic curvature fluctuations with zero net monopole. Emergent quantities: the coarsening length ξ(t), the tower volume fraction ft(t), and the effective equation of state w(t). These are not adjustable inputs but observables to be inferred. The expansion history H(z)constrains the growth rate of ξ(t); halo statistics constrain the small–scale distribution of wells; void lensing constrains the spatial organization of towers. A single set of geometric statistics must satisfy all three simultaneously. Failure to do so falsifies the framework. There is no parameter freedom analogous to particle mass, cross section, or vacuum energy density. – 7 – Figure-guided interpretation. Figure 1illustrates the microscopic picture underlying the curvature-dipole framework: a nearly homogeneous population of localized dipoles whose internal monopole cancels. Figure 2shows the coarse-grained limit in which attraction localizes into bound structure while repulsion becomes volume-filling. The transition between these regimes is the mechanism that unifies “dark-matter-like” clustering and accelerated expansion. The curvature-dipole framework reproduces the leading phenomenology attributed to cold dark matter at galactic and cluster scales while predicting characteristic deviations in environments dominated by repulsive curvature. Because attraction and repulsion arise from the same geometric degrees of freedom, observables that are usually treated independently become correlated. At small scales (ℓ≪ξ), the statistics of ρwgovern halo formation, internal density profiles, and merger histories. At intermediate scales, the evacuation of matter from towerdominated regions sharpens density contrasts and alters void profiles, giving a direct weaklensing handle on repulsive curvature. On the largest scales (ℓ≫ξ), the tight coupling between structure growth and expansion implies environment-dependent expansion histories. 7 Comparison with Alternative Theories Modified gravity models [41–46] alter the Einstein equations by introducing new fields, screening mechanisms, or nonlocal operators. The curvature-dipole framework preserves Einstein gravity at the fundamental level and introduces no new propagating degrees of freedom. All departures from standard behavior arise from the statistical organization of curvature itself. Concrete alternatives that have been pursued in the literature include empirical and relativistic MOND-like frameworks and a broad family of modified-gravity models (e.g. scalar– tensor, f(R), braneworld, and massive-gravity constructions). These approaches are valuable as phenomenological baselines, but they typically introduce additional fields, screening mechanisms, or model-specific tunings to recover general relativity in high-density environments. [41–43,47–52] 8 Why This Was Not Seen Earlier The conceptual barrier is the implicit assumption that gravitational phenomena must be sourced either by matter-like components or by homogeneous vacuum energy. Allowing repulsion to appear in localized, stochastic form admits a third possibility: emergent large-scale behavior from geometric self-organization. 9 Conclusion We have presented a unified geometric framework in which the phenomena conventionally attributed to cold dark matter and dark energy arise from stochastic curvature dipoles in spacetime. In this picture, localized attractive and repulsive curvature fluctuations self-organize through gravitational interaction, producing clustered structure on small scales and accelerated expansion on large scales through a single physical mechanism. Crucially, the framework introduces no new particles, fields, interaction scales, or vacuum energies. There are no dark-sector masses, cross sections, or equation-of-state parameters to tune. Once the statistical properties of curvature fluctuations are specified, the subsequent coarsening dynamics determine both the growth of structure and the expansion – 8 –