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 ₄₅₅, is conserved up to at least the second post-Newtonian order. The measured distribution of ₄₅₅ 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 ₄₅₅ values which is symmetric does not necessarily lead to a symmetric distribution for the detected effective spin values, leading to a selection bias; and (ii) the posterior distribution of ₄₅₅ for individual events is asymmetric and it cannot usually be treated as a Gaussian. We find that the posterior distributions for ₄₅₅ systematically show fatter tails toward larger positive values, unless the total mass is large or the mass ratio m₂/m₁ is smaller than 1/2. Finally we show that uncertainties in the measurement of ₄₅₅ 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 100 events and should be taken into account when using gravitational-wave measurements to characterize astrophysical populations.