{
  "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 $Λ$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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30201",
  "author_total": 2,
  "authors": [
    {
      "name": "Davide Campanella Galanti"
    },
    {
      "name": "Oliver F. Piattella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0g9jpb3gmp35t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30201",
  "fulltext_url": null,
  "id": "0g9jpb3gmp35t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30201",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "astro-ph.CO"
    ],
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "openalex",
      "3103|Cosmology and Gravitation Theories"
    ]
  ],
  "openalex": "W7223279490",
  "pdf_url": "https://arxiv.org/pdf/2610.30201",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0g9jpb3gmp35t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2610.30201",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30201"
    }
  ],
  "subjects": [
    {
      "field": "astronomy",
      "origin": "arxiv:astro-ph.CO",
      "subfield": "cosmology"
    },
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Partial geometrization of dark matter in metric $f(R)$ gravity: Exact reconstruction and growth constraints",
  "title_text": "Partial geometrization of dark matter in metric f(R) gravity: Exact reconstruction and growth constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:34Z",
  "url": "https://research.draconitic.com/paper/0g9jpb3gmp35t",
  "venue": "arXiv",
  "versions": []
}