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

Abstract

Abridged. Stochastic fluctuations in the spin frequency $\nu$ of a rotation-powered pulsar affect how accurately one measures the power-law braking index, $n_{\mathrm{pl}}$, defined through $\dot{\nu }=K{\nu }^{n_{\mathrm{pl}}}$, and can lead to measurements of anomalous braking indices, with $|n|=|\nu \ddot{\nu }/{\dot{\nu }}^2|\gg 1$, 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 $⟨n^2⟩=n_{\mathrm{pl}}^2+{\sigma }_{\ddot{\nu }}^2{\nu }^2{\gamma }_{\ddot{\nu }}^{-2}{\dot{\nu }}^{-4}{T}_{\mathrm{obs}}^{-1}$ for $\dot{K}=0$, where ${\sigma }_{\ddot{\nu }}$ is the timing noise amplitude, ${\gamma }_{\ddot{\nu }}^{-1}$ is a stellar damping time-scale, and $T_{\mathrm{obs}}$ is the total observing time. Here we combine this formula with a hierarchical Bayesian scheme to infer the population-level distribution of $n_{\mathrm{pl}}$ for a pulsar population of size $M$. The scheme is validated using synthetic data. For a plausible test population with $M=100$ and injected $n_{\mathrm{pl}}$ values drawn from a population-level Gaussian with mean ${\mu }_{\mathrm{pl}}=4$ and standard deviation ${\sigma }_{\mathrm{pl}}=0.5$, intermediate between electromagnetic braking and mass quadrupole gravitational radiation reaction, the Bayesian scheme infers ${\mu }_{\mathrm{pl}}=3.89_{-0.23}^{+0.24}$ and ${\sigma }_{\mathrm{pl}}=0.43_{-0.14}^{+0.21}$. The $M=100$ per-pulsar posteriors for $n_{\mathrm{pl}}$ and ${\sigma }_{\ddot{\nu }}^2{\gamma }_{\ddot{\nu }}^{-2}$ 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 $50\le M\le 300$, and (ii) wide priors satisfying ${\mu }_{\mathrm{pl}}\le 10^3$ and ${\sigma }_{\mathrm{pl}}\le 10^2$, which accommodate plausible spin-down mechanisms with $\dot{K}\ne 0$.

16 pages, 7 figures

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