Padé Approximants for cosmic Dispersion Measures

Technical Summary: Padé Approximants for Cosmic Dispersion Measures

Problem Statement
Fast Radio Bursts (FRBs) have emerged as critical tools for investigating the "missing baryons," the ionization history of the Universe, and cosmological parameters. These studies rely on analyzing the diffuse dispersion measure ($DM_{\text{diff}}$) as a function of redshift. The theoretical calculation of $DM_{\text{diff}}$ involves a complex integral over the line of sight that depends on cosmological parameters (such as matter density $\Omega_m$ and the dark energy equation of state $w$). As the volume of FRB data grows, cosmological inference requires repeated evaluations of this integral within likelihood frameworks (e.g., Markov Chain Monte Carlo, MCMC). Standard numerical integration methods are computationally expensive, creating a bottleneck that hinders the scalability of FRB cosmology and limits accessibility for researchers with limited computational resources.

Methodology
The authors propose an analytical approximation for the cosmic dispersion measure integral using Padé approximants, a technique known for providing precise rational function approximations that often outperform standard power series expansions, even in divergent regimes.

1. Derivation for Flat $\Lambda$CDM: The authors derive an approximation for a flat universe with a cosmological constant. By transforming the integral variable from redshift $z$ to the scale factor $a = 1/(1+z)$, they define a function $F(a)$ and expand it as a power series in the limit of large redshift ($a \to 0$). They fit this series to a Padé approximant of order $(3,3)$, resulting in an analytical formula involving specific coefficients ($b_0$ through $b_3$ and $c_1$ through $c_3$) that depend on $\Omega_m$.
2. Derivation for Flat $w$CDM: The method is extended to flat universes with a general dark energy equation of state parameter $w$. Following a similar transformation and expansion strategy, they derive a $(3,3)$ Padé approximant for the $w$CDM case. The coefficients for this model are more complex, expressed as high-order polynomials in $w$.
3. Validation: The accuracy of these approximations is tested against numerical integration (using Python's quad function) across the parameter ranges $0.01 \le z \le 2$, $0.2 \le \Omega_m \le 1.0$, and $-3.0 \le w \le -0.5$.

Key Results

• Accuracy: The Padé approximants demonstrate high accuracy within the specified parameter ranges. The maximum relative error ($\Delta E$) is found to be approximately 3.5% in the worst-case scenario (low redshift $z=0.01$ and low matter density $\Omega_m=0.2$). For the concordance $\Lambda$CDM cosmology ($\Omega_m \approx 0.31, w=-1$), the error drops significantly, remaining below 0.5% across the entire redshift range. The authors note that even the worst-case error is smaller than the intrinsic scatter of observed FRB dispersion measures.
• Computational Speed: The analytical formulas offer substantial speed improvements over numerical integration.
  • For $\Lambda$CDM, the approximation is approximately 17 times faster than numerical integration.
  • For $w$CDM, it is approximately 2.5 times faster.
  • In a full MCMC cosmological inference context, the timing improvement reaches a factor of 27 for Gaussian PDFs and 3 for more complex Macquart PDFs. The authors identify the complexity of the PDF modeling itself as the primary bottleneck, rather than the DM integral calculation, once the approximation is applied.
• Cosmological Inference Robustness: Using simulated FRB data ($N=50$ events), the authors performed Bayesian inference to constrain $H_0$, $\Omega_m$, and $w$. They found that the posterior distributions derived using the Padé approximation are indistinguishable from those derived using numerical integration. Crucially, this holds true even when the inference is performed in regions of parameter space where the approximation's relative error exceeds 1% (and approaches 3.5%). The approximation yields unbiased results and recovers input cosmological parameters correctly.
• PDF Modeling Sensitivity: The study also investigated the impact of using mismatched probability density functions (PDFs) for data generation versus inference. The results indicate that for current levels of observational accuracy, the specific choice of PDF (Gaussian vs. complex Macquart model) does not significantly bias the cosmological constraints, provided the inference model is consistent with the data generation model.

Significance and Claims
The paper claims to present the first derivation of a Padé approximation specifically for cosmic dispersion measures. The significance of this work lies in providing a tool that makes FRB cosmological analysis computationally feasible for large datasets without sacrificing accuracy.

The authors assert that their approximation is:

1. Robust: It provides unbiased cosmological constraints even in parameter regions where the mathematical relative error is non-negligible, as these errors do not propagate into the final posterior distributions.
2. Efficient: It drastically reduces computational time, enabling more complex likelihood analyses and making FRB cosmology accessible to researchers with limited computational infrastructure.
3. Sufficiently Accurate: The error margins are well below current observational uncertainties and the intrinsic scatter of FRB data, making the approximation a practical alternative to numerical integration for both current and near-future FRB surveys.

The authors conclude that this method is a useful tool for ongoing and future astrophysical studies involving FRBs, particularly as data volumes continue to increase.