{
  "abstract": "The Lambert $W$ (LW) model for the cosmic fluid equation of state was proposed by S. Saha and K. Bamba in 2019-2020. A recent (early 2026) paper by Dubey, et al, carries out a new fit of the LW model to observational data, in order to estimate the model's parameters. That paper exhibits a formula for the logarithm of the relative Hubble factor at redshift $z$, $\\ln(H(z)/H_0)$, which is based upon a numerical integration. That integral can be evaluated analytically, and we present the details in this working paper. The resulting analytic expression for $H(z)/H_0$ may be convenient for exploration of the LW model. We present two approximation formulas for the Lambert $W$ function, highly accurate for redshifts $z$ between 0 and 3, which may be useful for researchers who do not have ready access to a function library with the Lambert $W$ function. We also present a simplified proof of the analytic formula for another logarithmic integral which may be useful for exploring some variants of the LW model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.07878",
  "author_total": 4,
  "authors": [
    {
      "name": "Ken Roberts"
    },
    {
      "name": "Francisco Javier Garcia-Sánchez"
    },
    {
      "name": "Subhajit Saha"
    },
    {
      "name": "S. R. Valluri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/16af60kywrpmy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.07878",
  "fulltext_url": null,
  "id": "16af60kywrpmy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.07878",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3103|Cosmology and Gravitation Theories"
    ]
  ],
  "openalex": "W7202176531",
  "pdf_url": "https://arxiv.org/pdf/2608.07878",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/16af60kywrpmy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2608.07878",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.07878"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Analytic Integration of the Lambert W Cosmic Fluid Model H(z) Formula",
  "title_text": "Analytic Integration of the Lambert W Cosmic Fluid Model H(z) Formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/16af60kywrpmy",
  "venue": "arXiv",
  "versions": []
}