Gravitational-wave astrophysics with effective-spin measurements: asymmetries and selection biases

Gravitational waves emitted by coalescing compact objects carry information about the spin of the individual bodies. However, with present detectors only the mass-weighted combination of the components of the spin along the orbital angular momentum can be measured accurately. This quantity, the effective spin $\chi_{\mathrm{eff}}$, is conserved up to at least the second post-Newtonian order. The measured distribution of $\chi_{\mathrm{eff}}$ values from a population of detected binaries, and in particular whether this distribution is symmetric about zero, encodes valuable information about the underlying compact-binary formation channels. In this paper we focus on two important complications of using the effective spin to study astrophysical population properties: (i) an astrophysical distribution for $\chi_{\mathrm{eff}}$ values which is symmetric does not necessarily lead to a symmetric distribution for the detected effective spin values, leading to a \emph{selection bias}; and (ii) the posterior distribution of $\chi_{\mathrm{eff}}$ for individual events is \emph{asymmetric} and it cannot usually be treated as a Gaussian. We find that the posterior distributions for $\chi_{\mathrm{eff}}$ systematically show fatter tails toward larger positive values, unless the total mass is large or the mass ratio $m_2/m_1$ is smaller than $\sim 1/2$. Finally we show that uncertainties in the measurement of $\chi_{\mathrm{eff}}$ are systematically larger when the true value is negative than when it is positive. All these factors can bias astrophysical inference about the population when we have more than $\sim 100$ events and should be taken into account when using gravitational-wave measurements to characterize astrophysical populations.