Hierarchical Bayesian estimation of population-level torque law parameters from anomalous pulsar braking indices

TL;DR

This paper develops a hierarchical Bayesian method to estimate the distribution of pulsar braking indices from observed data, accounting for stochastic timing noise, and validates it with synthetic populations.

Contribution

It introduces a Bayesian framework combining a variance formula with hierarchical inference to determine population-level pulsar braking index distributions.

Findings

Accurately recovers population parameters from synthetic data.

Performs well across different population sizes and prior assumptions.

Infers mean braking index close to injected values with credible intervals.

Abstract

Abridged. Stochastic fluctuations in the spin frequency $\nu$ of a rotation-powered pulsar affect how accurately one measures the power-law braking index, $n_{\rm pl}$, defined through $\dot{\nu}=K\nu^{n_{\rm pl}}$, and can lead to measurements of anomalous braking indices, with $\vert n \vert = \vert \nu \ddot{\nu}/ \dot{\nu}^{2} \vert \gg1$, where the overdot symbolizes a derivative with respect to time. Previous studies show that the variance of the measured $n$ obeys the predictive, falsifiable formula $\langle n^{2} \rangle = n_{\rm pl}^{2}+\sigma^{2}_{\ddot{\nu}}\nu^{2}\gamma_{\ddot{\nu}}^{-2}\dot{\nu}^{-4}T_{\rm obs}^{-1}$ for $\dot{K}=0$, where $\sigma_{\ddot{\nu}}$ is the timing noise amplitude, $\gamma_{\ddot{\nu}}^{-1}$ is a stellar damping time-scale, and $T_{\rm obs}$ is the total observing time. Here we combine this formula with a hierarchical Bayesian scheme to infer the population-level distribution of $n_{\rm pl}$ for a pulsar population of size $M$. The scheme is validated using synthetic data. For a plausible test population with $M=100$ and injected $n_{\rm pl}$ values drawn from a population-level Gaussian with mean $\mu_{\rm pl}=4$ and standard deviation $\sigma_{\rm pl}=0.5$, intermediate between electromagnetic braking and mass quadrupole gravitational radiation reaction, the Bayesian scheme infers $\mu_{\rm pl}=3.89^{+0.24}_{-0.23}$ and $\sigma_{\rm pl}=0.43^{+0.21}_{-0.14}$. The $M=100$ per-pulsar posteriors for $n_{\rm pl}$ and $\sigma^{2}_{\ddot{\nu}}\gamma_{\ddot{\nu}}^{-2}$ contain $87\%$ and $69\%$, respectively, of the injected values within their $90\%$ credible intervals. Comparable accuracy is achieved for (i) population sizes spanning the range $50 \leq M \leq 300$, and (ii) wide priors satisfying $\mu_{\rm pl} \leq 10^{3}$ and $\sigma_{\rm pl} \leq 10^{2}$, which accommodate plausible spin-down mechanisms with $\dot{K}\neq0$.