Analytic Integration of the Lambert W Cosmic Fluid Model H(z) Formula
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)/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. 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.
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BibTeX
@misc{roberts2026analytic,
title = {{Analytic Integration of the Lambert W Cosmic Fluid Model H(z) Formula}},
author = {Ken Roberts and Francisco Javier Garcia-Sánchez and Subhajit Saha and S. R. Valluri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.07878},
eprint = {2608.07878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07878},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Analytic Integration of the Lambert W Cosmic Fluid Model H(z) Formula
AU - Ken Roberts
AU - Francisco Javier Garcia-Sánchez
AU - Subhajit Saha
AU - S. R. Valluri
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.07878
UR - https://arxiv.org/abs/2608.07878
AB - 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.
ER -