Partial geometrization of dark matter in metric $f(R)$ gravity: Exact reconstruction and growth constraints
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 $Λ$CDM 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 $F\equiv f_R$, 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 $F\to1$ at early times: it recovers general relativity at high curvature and attains the minimal running of $F$. Keeping the linear growth factor at $k=0.1\,h\,\mathrm{Mpc}^{-1}$ within $10\%$ of its $Λ$CDM value requires $1-ξ\lesssim10^{-3}$, the weakest bound among all stable reconstructions; keeping $σ_8$ within $3\%$ requires $1-ξ\lesssim5\times10^{-5}$. Growth-rate data from DESI DR1 and SDSS DR16 give $1-ξ\lesssim10^{-3}$--$10^{-2}$, and the Cassini bound gives $1-ξ\lesssim10^{-10}$. A smoothly evolving metric $f(R)$ background can therefore geometrize only a negligible fraction of dark matter.