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.
