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Analytic Integration of the Lambert W Cosmic Fluid Model H(z) Formula

Ken Roberts, S. R. Valluri

math-pharXiv:2608.07878

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, (H(z)/H0), 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)/H0 may be convenient for exploration of the LW model.

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