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

TL;DR

This paper introduces an analytical approximation for the diffuse dispersion measure of Fast Radio Bursts, significantly speeding up calculations while maintaining high accuracy, aiding cosmological studies using FRBs.

Contribution

It provides a new Pade approximant-based formula for ${ m DM}_{ m diff}$ that is faster and highly accurate across relevant parameter ranges for flat, $ m\Lambda$CDM, and $w$CDM universes.

Findings

Approximation has less than 3.5% error compared to numerical solutions.

Speeds up calculations by over 15 times for $ m ext\\Lambda$CDM.

Applicable for redshift 0.01 to 2 and various cosmological parameters.

Abstract

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.