Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Western University, King’s University College
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202176531