Partial geometrization of dark matter in metric f(R) gravity: Exact reconstruction and growth constraints
Abstract
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. Only a fraction ξ of dark matter clusters as dust, whereas f(R) supplies the rest, as well as Λ. 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. 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. 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. 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). In all stable reconstructions but one, F−1 grows as R1/3 at high curvature, i.e., in dense environments such as the Sun, and local tests of gravity exclude them. The surviving member has F→1 at early times: it recovers general relativity at high curvature and attains the minimal running of F. 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. Growth-rate data from DESI DR1 and SDSS DR16 give 1−ξ≲10−3--10−2, and the Cassini bound gives 1−ξ≲10−10. A smoothly evolving metric f(R) background can therefore geometrize only a negligible fraction of dark matter.
Related
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BibTeX
@misc{galanti2026partial,
title = {{Partial geometrization of dark matter in metric $f(R)$ gravity: Exact reconstruction and growth constraints}},
author = {Davide Campanella Galanti and Oliver F. Piattella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30201},
eprint = {2610.30201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30201},
}
RIS
TY - UNPB
TI - Partial geometrization of dark matter in metric f(R) gravity: Exact reconstruction and growth constraints
AU - Davide Campanella Galanti
AU - Oliver F. Piattella
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30201
UR - https://arxiv.org/abs/2610.30201
AB - 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. Only a fraction ξ of dark matter clusters as dust, whereas f(R) supplies the rest, as well as Λ. We show that the background reconstruction reduces to a linear equation for F≡ f_R, 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. 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. 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. 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). 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. The surviving member has F→1 at early times: it recovers general relativity at high curvature and attains the minimal running of F. 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 σ_8 within 3% requires 1−ξ≲5×10^−5. Growth-rate data from DESI DR1 and SDSS DR16 give 1−ξ≲10^−3--10^−2, and the Cassini bound gives 1−ξ≲10^−10. A smoothly evolving metric f(R) background can therefore geometrize only a negligible fraction of dark matter.
ER -