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Padé Approximants for Fast Radio Bursts Diffuse Dispersion Measure

Marios Kalomenopoulos, Jiaming Zhuge

TL;DR

The paper tackles the computational bottleneck of evaluating the diffuse FRB dispersion measure ${\rm DM}_{\rm diff}$ by deriving analytical Padé approximants for flat $\Lambda$CDM and flat $w$CDM cosmologies. It produces closed-form expressions with a [3,3] Padé for a rational function $\Phi$ that, combined with a compact redshift-dependent variable $x$, yields ${\rm DM}_{\rm diff}$ in terms of ${\rm DM}_{\rm diff}^c$, $\Omega_m$, and $w$; these approximants achieve relative errors below $3.5\%$ (often $<0.5\%$) across $0.01 \leq z \leq 2$, $0.2 \leq \Omega_m \leq 1.0$, and $-3.0 \leq w \leq -0.5$. The method delivers substantial speedups, over $15\times$ faster than numerical integration for $\Lambda$CDM and more than $2\times$ for $w$CDM, enabling efficient FRB-involved analyses and MCMC workflows. The results provide a practical tool for cosmological and astrophysical studies using FRB DMs, with forthcoming code releases and emphasis on accuracy in the relevant parameter space. Overall, the work offers a scalable, analytic alternative to time-consuming line-of-sight integrals in FRB cosmology.

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, $Λ$CDM and $w$CDM universes. We show that our approximation works well for the ranges $0.01 \leq z \leq 2$, $0.2 \leq Ω_m \leq 1.0$ 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 $Λ$CDM ($w$CDM). Therefore, we hope that it could be a useful tool for the FRB community.

Padé Approximants for Fast Radio Bursts Diffuse Dispersion Measure

TL;DR

The paper tackles the computational bottleneck of evaluating the diffuse FRB dispersion measure by deriving analytical Padé approximants for flat CDM and flat CDM cosmologies. It produces closed-form expressions with a [3,3] Padé for a rational function that, combined with a compact redshift-dependent variable , yields in terms of , , and ; these approximants achieve relative errors below (often ) across , , and . The method delivers substantial speedups, over faster than numerical integration for CDM and more than for CDM, enabling efficient FRB-involved analyses and MCMC workflows. The results provide a practical tool for cosmological and astrophysical studies using FRB DMs, with forthcoming code releases and emphasis on accuracy in the relevant parameter space. Overall, the work offers a scalable, analytic alternative to time-consuming line-of-sight integrals in FRB cosmology.

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 () 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 for flat, CDM and CDM universes. We show that our approximation works well for the ranges , and , with relative fractional error to a numerically evaluated always smaller than , in the worst case scenario, and in most cases smaller than . Moreover, the approximation is more than () times faster than the numerical solution of CDM (CDM). Therefore, we hope that it could be a useful tool for the FRB community.

Paper Structure

This paper contains 8 sections, 22 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. FRBs Dispersion Measure
  3. Padé Approximant for ${\rm DM}_{\rm diff}$ - Flat $\Lambda$CDM
  4. Padé Approximant for ${\rm DM}_{\rm diff}$ - Flat $w$CDM
  5. Accuracy and Timing Performance
  6. Conclusions
  7. Software:
  8. Comparison to the hypergeometric function

Figures (2)

  • Figure 1: (Left) Fractional error $\Delta E$ for the ${\rm DM}$ Padé Approximant in a flat, $\Lambda$CDM universe, for the redshift range $0.01 \leq z \leq 2$, and for two values of $\Omega_m$: ($\Omega_m = 0.2$ - blue, crosses) and ($\Omega_m = \Omega_m^{\Lambda \textrm{CDM}}$ - red, dots). Increasing $z$ and $\Omega_m$ reduces the fractional error. (Right) Fractional error $\Delta E$ for the ${\rm DM}$ Padé Approximant in a flat, $w$CDM universe, for the redshift range $0.01 \leq z \leq 2$, and for two values of $w$: ($w = -0.5$ - blue, crosses) and ($w = -3.0$ - red, dots). Increasing $z$ and decreasing $w$ reduces the fractional error. Here $\Omega_m = \Omega_m^{\Lambda \textrm{CDM}}$.
  • Figure 2: Fractional error $\Delta E$ for the ${\rm DM}$ Padé Approximant in a flat, $w$CDM universe, for a fixed redshift $z=0.01$, i.e. the most conservative choice. The other cosmological parameters vary in the ranges: $-3 \leq w \leq -0.5$ and $0.1 \leq \Omega_m \leq 1$. We also denote with a red, dashed line the limit where the fractional error is $\Delta E = 1 \%$, while the white, dotted line corresponds to the cases where the parameters take their $\Lambda$CDM values. We observe that the value of $\Omega_m$ plays the more crucial role in our Padé approximants, compared to $w$.