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 $\Lambda$CDM model. Only a fraction $\xi$ of dark matter clusters as dust, whereas $f(R)$ supplies the rest, as well as $\Lambda$. We show that the background reconstruction reduces to a linear equation for $F\equiv f_R$, whose solution is $F-1=(1-\xi )\Theta (a)$, with $\Theta$ independent of $\xi$: the departure from general relativity is exactly proportional to the geometrized fraction. We solve this equation in closed form in the matter$+\Lambda$ 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 $0.837(1-\xi )$. 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 $F\to 1$ at early times: it recovers general relativity at high curvature and attains the minimal running of $F$. Keeping the linear growth factor at $k=0.1h{\mathrm{Mpc}}^{-1}$ within $10\%$ of its $\Lambda$CDM value requires $1-\xi \lesssim 10^{-3}$, the weakest bound among all stable reconstructions; keeping ${\sigma }_8$ within $3\%$ requires $1-\xi \lesssim 5\times 10^{-5}$. Growth-rate data from DESI DR1 and SDSS DR16 give $1-\xi \lesssim 10^{-3}$--$10^{-2}$, and the Cassini bound gives $1-\xi \lesssim 10^{-10}$. A smoothly evolving metric $f(R)$ background can therefore geometrize only a negligible fraction of dark matter.

arXiv comment14 pages, 7 figures, 2 tables