Pad\'e Approximants for Fast Radio Bursts Diffuse Dispersion Measure

Fast Radio Bursts (FRBs) have become an indispensable tool for studying the Universe's ionisation properties, as well as its cosmological parameters. This is achieved by analysing their diffuse dispersion measure (${\rm DM}_{\rm diff}$) as a function of redshift. However, this requires an integration along the line-of-sight, which is time-consuming. In this work, we derive an analytical approximation formula for ${\rm DM}_{\rm diff}$ for flat, $\Lambda$CDM and $w$CDM universes. 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 fractional error to a numerically evaluated ${\rm DM}_{\rm diff}$ always smaller than $3.5\ \%$, in the worst case scenario, and in most cases smaller than $0.5\ \%$. Moreover, the approximation is more than $15$ ($2$) times faster than the numerical solution of $\Lambda$CDM ($w$CDM). Therefore, we hope that it could be a useful tool for the FRB community.