{
  "abstract": "We formulate the cosmic equation of motion in the Non-Equilibrium Dynamics (NED)\n\nframework, including the time variation of mass, and derive its exact analytic solution.\n\nUsing the exact mass scaling M(R) = (Mnow/R2\n\nnow)R(R − ℓp) with R ≥ 2ℓp, the equation\n\nof motion M′v + M a = F0 reduces to the total derivative form\n\nd\n\ndt\n\n[\n\nR(R − ℓp)R˙\n\n]\n\n= K, K =\n\nF0R2\n\nnow\n\nMnow\n\n.\n\nIntegrating twice yields the implicit solution\n\n2R\n\n3 − 3ℓpR\n\n2 = 3Kt2 + 12ℓ\n\n2\n\np\n\nv0 t + 4ℓ\n\n3\n\np\n\n.\n\nThe explicit solution is obtained via Cardano’s formula:\n\nR(t) = ℓp\n\n2\n\n+\n\n3\n\n√\n\nA +\n\n√\n\n∆ + 3\n\n√\n\nA −\n\n√\n\n∆,\n\nA =\n\n3K\n\n4\n\nt\n\n2 + 3ℓ\n\n2\n\np\n\nv0 t +\n\n9ℓ\n\n3\n\np\n\n8\n\n, ∆ = A\n\n2 −\n\nℓ\n\n6\n\np\n\n64\n\n.\n\nThis exact solution describes, without approximation, the entire cosmic history from the\n\nPlanck scale (R = 2ℓp) through a radiation-like era (R ∝ t\n\n1/2\n\n), a matter-dominated era\n\n(R ∝ t\n\n2/3\n\n), and the eternal future (v → 0, a → 0). The cosmic age is given by\n\ntnow =\n\n−2ℓ\n\n2\n\np\n\nv0 +\n\n√\n\n4ℓ\n\n4\n\np\n\nv\n\n2\n\n0 −\n\nKc\n\n3\n\nK\n\n,\n\nwhich is essentially independent of the initial velocity v0. For Rnow = 4.4 × 1026 m, we\n\nobtain tnow ≈ 1.39 × 1010 years, in agreement with observation. Introducing the time\n\nvariable t\n\n′ = tnow + t, all physical quantities are expressed in the unified form\n\nX(t) = Xnow (\n\ntnow + t\n\ntnow )n\n\n,\n\nwhere n is a constant specific to each quantity. This representation naturally explains why\n\nthe time variation of physical constants is not observed: the rate of change near the present is\n\nextremely small (1/tnow ∼ 10−18 s\n\n−1\n\n), and moreover the measuring instruments themselves\n\nscale with the same time variable, so that the variation cancels out in any measurement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "takashi ueda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5t0c6h7mcj7x1.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23422841",
  "fulltext_url": null,
  "id": "5t0c6h7mcj7x1",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23422841",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23422841/files/%E5%8E%B3%E5%AF%86%E5%AE%87%E5%AE%99%E6%96%B9%E7%A8%8B%E5%BC%8F.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5t0c6h7mcj7x1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:32Z",
      "native_id": "23422841",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23422841"
    }
  ],
  "subjects": [],
  "title": "Exact Solution of the Cosmic Equation in Non-Equilibrium Dynamics: Cosmic Age, Time Variation of Physical Constants, and the Observational Difficulty",
  "title_text": "Exact Solution of the Cosmic Equation in Non-Equilibrium Dynamics: Cosmic Age, Time Variation of Physical Constants, and the Observational Difficulty",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/5t0c6h7mcj7x1",
  "venue": "Zenodo",
  "versions": []
}