W-˄ cosmological parametrization from a new time-dependent deceleration parameter: observational constraints from H(z) data

W-˄ cosmological parametrization from a new time-dependent deceleration parameter: observational constraints from H(z) data | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article W-˄ cosmological parametrization from a new time-dependent deceleration parameter: observational constraints from H (z) data Rohit Madhukar Patne This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9605368/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 7 You are reading this latest preprint version Abstract We propose a new time-dependent deceleration parameter \((q(t) = \dfrac{1 - 21\beta (t/t_0)^2 - 18\beta^2 (t/t_0)^4}{2[1+3\beta (t/t_0)^2]^2})\) ( \((\beta>0)\) ), which smoothly describes a transition from early-time deceleration ( \((q>0)\) ) to late-time acceleration ( \((q<0)\) ).Integration yields the scale factor \((R(t) = (t/t_0)^{2/3}\exp[\beta((t/t_0)^2-1)])\) , normalised to unity today.The corresponding Hubble parameter can be inverted exactly in terms of the Lambert \((W)\) function, giving the closed-form \((H(z) = H_0\,\frac{\sqrt{3\beta}}{1+3\beta}\,\frac{1+W\!\big(3\beta e^{3\beta}(1+z)^{-3}\big)}{\sqrt{W\!\big(3\beta e^{3\beta}(1+z)^{-3}\big)}})\) .We call this the W- \((\Lambda)\) parametrisation .We test the model against a compilation of 57 cosmic-chronometer \((H(z))\) measurements.With the shape parameter fixed to \((\beta=0.173)\) and the Hubble constant set to \((H_0=71.12\;\mathrm{km\,s^{-1}\,Mpc^{-1}})\) , the model yields \((\chi^2=44.79)\) (no free parameters), outperforming a flat \((\Lambda)\) CDM model with Planck \((H_0=67\;\mathrm{km\,s^{-1}\,Mpc^{-1}})\) and a fitted \((\Omega_m)\) ( \((\chi^2=51.30)\) with one free parameter).The corresponding Akaike Information Criterion values are 44.79 and 53.30, respectively.While the parameter settings are chosen a priori and a full simultaneous fit of \((H_0)\) and \((\beta)\) is still needed, the results already demonstrate that the W- \((\Lambda)\) parametrisation provides an excellent description of the observed expansion history. deceleration parameter Lambert W function Hubble parameter cosmological parameters observations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 18 May, 2026 Reviews received at journal 17 May, 2026 Reviewers agreed at journal 07 May, 2026 Reviewers invited by journal 06 May, 2026 Editor assigned by journal 05 May, 2026 Submission checks completed at journal 05 May, 2026 First submitted to journal 04 May, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9605368","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":636097592,"identity":"99b0012f-6fd0-4728-bc2b-dba108c451a3","order_by":0,"name":"Rohit Madhukar Patne","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIie3PIU8DMRTA8dc02UzZ7MOMr1ByybLvgtk89RXjUnUzI9gTy/gKYDbbpslhGvgAmCUY5BHMBIJ3J8gMvckl6180ec37pXcAqdQJ1uN01BIFgAALmibOTQfhwEo9GbXEhoawOKEN4CLorCHgiuamgwz6Q/95UeBsu7h3/mt9dzNcENnrTezDepNLIqvwOrVu86JKzwxbhvcokddESryVRCpliHBWdJDZH1lV6vEIwnc2YIYtMXP1dAQZM6NxhCJIGyqrnom42L9cPbiP7x+ZC+wvs1rPc7V+82631/+T5iE8GHx72tg+xeuDIe9YTqVSqXPsF636W4DrR9jwAAAAAElFTkSuQmCC","orcid":"","institution":"JD COLLEGE OF ENGINEERING AND MANAGEMENT","correspondingAuthor":true,"prefix":"","firstName":"Rohit","middleName":"Madhukar","lastName":"Patne","suffix":""}],"badges":[],"createdAt":"2026-05-04 07:38:50","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9605368/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9605368/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109320793,"identity":"9d661474-b891-4440-8048-985c8af98bb2","added_by":"auto","created_at":"2026-05-15 13:32:17","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":263192,"visible":true,"origin":"","legend":"","description":"","filename":"LastPaper1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9605368/v1_covered_b765067e-9821-44b9-a44f-2d9afb1ed929.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eW-˄ cosmological parametrization from a new time-dependent deceleration parameter: observational constraints from \u003cem\u003eH\u003c/em\u003e(z) data\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"astrophysics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Astrophysics](https://link.springer.com/journal/10511)","snPcode":"10511","submissionUrl":"https://submission.springernature.com/new-submission/10511/3","title":"Astrophysics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"deceleration parameter, Lambert W function, Hubble parameter, cosmological parameters, observations","lastPublishedDoi":"10.21203/rs.3.rs-9605368/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9605368/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose a new time-dependent deceleration parameter\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((q(t) = \\dfrac{1 - 21\\beta (t/t_0)^2 - 18\\beta^2 (t/t_0)^4}{2[1+3\\beta (t/t_0)^2]^2})\\)\u003c/span\u003e\u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\beta\u0026gt;0)\\)\u003c/span\u003e\u003c/span\u003e), which smoothly describes a transition from early-time deceleration (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((q\u0026gt;0)\\)\u003c/span\u003e\u003c/span\u003e) to late-time acceleration (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((q\u0026lt;0)\\)\u003c/span\u003e\u003c/span\u003e).Integration yields the scale factor\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((R(t) = (t/t_0)^{2/3}\\exp[\\beta((t/t_0)^2-1)])\\)\u003c/span\u003e\u003c/span\u003e, normalised to unity today.The corresponding Hubble parameter can be inverted exactly in terms of the Lambert \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((W)\\)\u003c/span\u003e\u003c/span\u003e function, giving the closed-form\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((H(z) = H_0\\,\\frac{\\sqrt{3\\beta}}{1\u0026amp;#x002B;3\\beta}\\,\\frac{1\u0026amp;#x002B;W\\!\\big(3\\beta e^{3\\beta}(1\u0026amp;#x002B;z)^{-3}\\big)}{\\sqrt{W\\!\\big(3\\beta e^{3\\beta}(1\u0026amp;#x002B;z)^{-3}\\big)}})\\)\u003c/span\u003e\u003c/span\u003e.We call this the \u003cem\u003eW-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\Lambda)\\)\u003c/span\u003e\u003c/span\u003e parametrisation\u003c/em\u003e.We test the model against a compilation of 57 cosmic-chronometer \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((H(z))\\)\u003c/span\u003e\u003c/span\u003e measurements.With the shape parameter fixed to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\beta=0.173)\\)\u003c/span\u003e\u003c/span\u003e and the Hubble constant set to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((H_0=71.12\\;\\mathrm{km\\,s^{-1}\\,Mpc^{-1}})\\)\u003c/span\u003e\u003c/span\u003e, the model yields \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\chi^2=44.79)\\)\u003c/span\u003e\u003c/span\u003e (no free parameters), outperforming a flat \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\Lambda)\\)\u003c/span\u003e\u003c/span\u003eCDM model with Planck \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((H_0=67\\;\\mathrm{km\\,s^{-1}\\,Mpc^{-1}})\\)\u003c/span\u003e\u003c/span\u003e and a fitted \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\Omega_m)\\)\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\chi^2=51.30)\\)\u003c/span\u003e\u003c/span\u003e with one free parameter).The corresponding Akaike Information Criterion values are 44.79 and 53.30, respectively.While the parameter settings are chosen a priori and a full simultaneous fit of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((H_0)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\beta)\\)\u003c/span\u003e\u003c/span\u003e is still needed, the results already demonstrate that the W-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\Lambda)\\)\u003c/span\u003e\u003c/span\u003e parametrisation provides an excellent description of the observed expansion history.\u003c/p\u003e","manuscriptTitle":"W-˄ cosmological parametrization from a new time-dependent deceleration parameter: observational constraints from H(z) data","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-15 13:32:04","doi":"10.21203/rs.3.rs-9605368/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision 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