Constraining the delay time distribution of compact binary objects from the stochastic gravitational wave background searches

The initial separation of massive star binaries sets the timescale over which their compact remnants merge through the emission of gravitational waves. We show that the delay time distribution (DTD) of binary neutron stars or black holes can be inferred from the stochastic gravitational wave background (SGWB). If the DTD of a population is long, most of the mergers take place at low redshifts and the background would be rather quiet compared to a scenario in which the DTD is short leading to few individual detections at low redshift but a rather loud background. We show that different DTDs predict a factor of 5 difference in the magnitude of the gravitational wave background energy density ($\Omega_{\rm GW}$) and have the dominant effect on $\Omega_{\rm GW}$ over other factors such as the mass function of the primary BH mass, $p(m_1)$, the maximum considered BH mass ($M_{\rm max}$), and the effective spin of the black hole ($\chi_{\rm eff}$). A non-detection of such a background can rule out the short DTD scenario. We show that SGWB searches can rule out the short DTD scenario for the BBHs within about four years of observing time at advanced LIGO design sensitivty for a local merger rate of 30 $\rm Gpc^{-3} yr^{-1}$ assuming $p(m_1)\propto m_1^{-1}$, and $M_{\rm max}=50 M_{\odot}$.