Partial geometrization of dark matter in metric f(R) gravity: Exact reconstruction and growth constraints

The gist The fraction of dark matter that can be accounted for by geometry in a stable metric f(R) theory, while keeping expansion history identical to ΛCDM, is negligible.

Background Reconstruction

The exact background reconstruction reduces to a linear equation for F ≡ fR, whose solution is F − 1 = (1 − ξ) Θ(a), with Θ independent of ξ <ref:3pg3. The departure from general relativity is exactly proportional to the geometrized fraction <ref:2610.30201#pg5>. This equation is solved in closed form in the matter+Λ and radiation+matter eras, yielding an elementary particular solution plus two hypergeometric modes <ref:2610.30201#pg6>. Stability requires F to decrease monotonically in time, meaning F cannot equal 1 both at early and at late times <ref:2610.30201#pg7>. The effective gravitational coupling G/F therefore necessarily runs, changing by at least 0.837 (1 − ξ) between the radiation era and the de Sitter attractor <ref:2610.30201#pg7>.

Constraints on Dark Matter Fraction

The constraints derived from structure formation are stringent; for linear growth at k ∼ 0.1 h Mpc−1 within 10% of its ΛCDM value, one requires 1 − ξ ≲ 10−3 for every stable reconstruction <ref:2610.30201#pg8>. Requiring σ8 within 3% necessitates a tighter bound of 2 × 10−4 (5 × 10−5) <ref:2610.30201#pg8>. Growth-rate data from DESI DR1 and SDSS DR16 give bounds of 1 − ξ ≲ 10−3–10−2 <ref:2610.30201#pg8>. The Cassini bound imposes an even stronger constraint, requiring 1 − ξ ≲ 10−10 <ref:2610.30201#pg8>.

Stability and Boundary Conditions

Stability, defined by fRR > 0, implies that F decreases monotonically in time, meaning F cannot equal 1 both at early and at late times <ref:2610.30201#pg7>. Once regularity at the de Sitter attractor is imposed, stable reconstructions form a one-parameter family labeled by the present value F0 <ref:2610.30201#pg7>. The two natural members are N1 (F0 = 1) and N2 (F → 1 in the radiation era) <ref:2610.30201#pg7>. Only N2 recovers general relativity at high curvature, while local gravity excludes all others <ref:2610.30201#pg7>.

Perturbations and Growth

The geometric density perturbation is entirely determined by the clustering matter, reaching −δρcl/(3F) for Q ≫ 1 <ref:2610.30201#pg9>. The effective sound speed squared of the geometric sector, δpgeom/δρgeom → −2/3 on small scales, indicating that the geometric sector has no independent dynamics and does not cluster like dust it replaces <ref:2610.30201#pg10>. The growth ratio RD is shown to depend on 1 − ξ <ref:2610.30201#pg9>, with N2 being the most favorable stable reconstruction <ref:2610.30201#pg9>.

Local Gravity and Recombination

Local gravity tests select N2, the reconstruction that recovers general relativity at high curvature <ref:2610.30201#pg13>. The Cassini bound requires 1 − ξ ≲ 7 × 10−11 on N2 <ref:2610.30201#pg13>. Recombination constraints suggest 1−ξ ≲ a few percent <ref:2610.30201#pg7>. The running of the effective Planck mass, αM = d ln F/d ln a = (1 − ξ)ΘN /F [40], is bounded by αM ≲ 10−6 today at our bounds <ref:2610.30201#pg13>.

Conclusion

The clustering, dust-like component must therefore constitute essentially all of the dark matter <ref:2610.30201#pg8>. The paper concludes that the clustering, dust-like component must therefore constitute essentially all of the dark matter <ref:2610.30201#pg8>. This parallels the conclusion reached for a geometric dark-radiation component in Ref. [20], and it is the f(R) counterpart of a lesson familiar from unified darkmatter models: a component that behaves as dust in the background can replace cold dark matter only if it also clusters like dust <ref:2610.30201#pg8>. The main limitations are using the quasi-static approximation <ref:2610.30201#pg14>. This reconstruction, N2, is the most favorable member of the whole stable family. This reconstruction, N2, is the most favorable member of the whole stable family. The clustering, dust-like component must therefore constitute essentially all of the dark matter <ref:2610.30201#pg8>.

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Partial geometrization of dark matter in metric f(R) gravity: Exact reconstruction and growth constraints<ref:2610.30201#pg3> We ask what fraction of dark matter can be accounted for by the geometry of a stable metric f(R) theory, requiring the expansion history to be exactly equal to that of the ΛCDM model<ref:2610.30201#pg3>. Only a fraction ξ of dark matter clusters as dust, whereas f(R) supplies the rest, as well as Λ<ref:2610.30201#pg3>. We show that the background reconstruction reduces to a linear equation for F ≡ fR, whose solution is F − 1 = (1 − ξ) Θ(a), with Θ independent of ξ: the departure from general relativity is exactly proportional to the geometrized fraction <ref:2610.30201#pg4>. We solve this equation in closed form in the matter+Λ and radiation+matter eras: in both, the solution is an elementary particular solution plus two hypergeometric modes<ref:2610.30201#pg6>. Stability requires F to decrease monotonically in time, so F cannot take its general-relativity value F = 1 both at early and at late times <ref:2610.30201#pg7>. The effective gravitational coupling G/F therefore necessarily runs: between the radiation era and the de Sitter attractor, F changes by at least 0.837 (1 − ξ)<ref:2610.30201#pg7>. In all stable reconstructions but one, F − 1 grows as R 1/3 at high curvature, i.e., in dense environments such as the Sun, and local tests of gravity exclude them<ref:2610.30201#pg7>. The surviving member has F → 1 at early times: it recovers general relativity at high curvature and attains the minimal running of F<ref:2610.30201#pg7>. Keeping the linear growth factor at k = 0.1 h Mpc−1 within 10% of its ΛCDM value requires 1 − ξ ≲ 10−3, the weakest bound among all stable reconstructions; keeping σ8 within 3% requires 1 − ξ ≤5 × 10−5<ref:2610.30201#pg8>. Growth-rate data from DESI DR1 and SDSS DR16 give 1 − ξ ≤–10−3–10−2, and the Cassini bound gives 1 − ξ ≤–8 × 10−4<ref:2610.30201#pg8>. A smoothly evolving metric f(R) background can therefore geometrize only a negligible fraction of dark matter<ref:2610.30201#pg8>.

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2IN I INTRODUCTION Metric f(R) gravity propagates an extra scalar degree of freedom, the scalaron, and the freedom in the choice of f is enough to reproduce essentially any expansion history H(a) [1–5]<ref:2610.30201#pg2>. This reconstruction is not unique: for a given matter content, a whole class of f(R) models shares the same expansion history [9]<ref:2610.30201#pg2>. For the ΛCDM background, the reconstruction has been worked out in closed form [6–8]<ref:2610.30201#pg2>. Reference [7] concluded that, with dust alone, the only real-valued f(R) that reproduces this background is general relativity with a cosmological constant, f = R − 2Λ, whereas Ref. [8] later found nontrivial real-valued solutions in terms of hypergeometric functions, contradicting this conclusion<ref:2610.30201#pg2>. The viability of such reconstructions has been examined [10]<ref:2610.30201#pg2>, and in the “designer” approach f(R) is obtained by integrating the background equations numerically for a prescribed H(a), leaving a single free parameter that sets the present range of the scalaron [11, 12]<ref:2610.30201#pg2>. The same freedom has motivated the idea that curvature may play the role of dark matter, both in the cosmological background [13]<ref:2610.30201#pg2> and on galactic scales, where f(R) halo solutions have been proposed to reproduce flat rotation curves without particle dark matter [14–16] (but see Ref. [17])<ref:2610.30201#pg2>. Reproducing H(a), however, constrains only the total dark stress-energy [19]<ref:2610.30201#pg2>. How this is split between “dark∗ dcampanellagalanti@uninsubria.it† matter” and “geometry” can be determined by investigating how small perturbations evolve, since structure formation requires a pressureless component that clusters<ref:2610.30201#pg2>. In metric f(R), a smoothly evolving scalaron modifies the Poisson equation and generates a gravitational slip, but it does not provide an independent source of clustering<ref:2610.30201#pg2>. The question we address is therefore quantitative<ref:2610.30201#pg2>. Suppose that a fraction 1 − ξ of the dark matter density is traded for geometry, i.e.

1. Precise constraint derivation for dark matter fraction by solving linear equations for F ≡ fR, enabling AI systems to calculate exactly what fraction of dark matter can be geometrized while maintaining an exact ΛCDM expansion history.

2. Accurate reconstruction of the running effective gravitational coupling G/F, allowing AI systems to predict how the gravitational constant changes between the radiation era and the de Sitter attractor based on stability requirements (F decreasing monotonically).

3. Identification and selection of stable metric f(R) reconstructions by using boundary conditions like regularity at the de Sitter attractor, which allows AI to distinguish between viable models (N1 vs N2) based on local gravity tests.

4. Calculation of growth rate deviation bounds by integrating the linear perturbation equation (31), enabling AI systems to quantify how much structure growth departs from ΛCDM for specific dark matter fractions and scales k.

5. Prediction of scale dependent deviations in the effective clustering source, specifically calculating the ratio (ρX − ρΛ)/[(1 − ξ)ρdm], which allows AI to determine if the geometric sector clusters like dust or not across different eras.

6. Quantitative assessment of local gravity constraints, enabling AI systems to calculate necessary bounds on 1 - ξ based on screening criteria like the Cassini bound and the thin-shell criterion for solar system stability.