Pad\'e Approximants for cosmic Dispersion Measures

Abstract

Fast Radio Bursts (FRBs) have become an indispensable tool for studying the ``missing baryons'', the Universe's ionisation properties, as well as the cosmological parameters. This is achieved by analysing the diffuse dispersion measure (${\rm DM}_{\rm diff}$) of FRBs as a function of redshift. However, the rapidly increasing data size requests more and more computational resources. In this work, we first develop an accelerated method for any cosmic dispersion measure by deriving an analytical approximation formula for flat, $\Lambda$CDM and $w$CDM universes. Focusing on FRBs, we show that our approximation works well for the ranges $0.01 \leq z \leq 2$, $0.2 \leq \Omega_m \leq 1.0$ and $-3.0 \leq w \leq -0.5$, with relative error to a numerically evaluated ${\rm DM}_{\rm diff}$ always smaller than $3.5 \%$ (in the worst case scenario). This error remains below observationally relevant ${\rm DM}$ scatter and is especially small near the concordance $\Lambda$CDM cosmology. Additionally, we perform a cosmological analysis of simulated FRB data and show that our approximation gives robust and unbiased results, even when applied in regions of parameter space where its relative error becomes larger than $3\%$. Finally, the approximation is more than $15$ ($2$) times faster than the numerical solution of $\Lambda$CDM ($w$CDM), and can reach a timing improvement of a factor of $25$ when used in an MCMC cosmological inference.