Imprint of the merger and ring-down on the gravitational wave background from black hole binaries coalescence

We compute the gravitational wave background (GWB) generated by a cosmological population of black hole-black hole (BH-BH) binaries using hybrid waveforms recently produced by numerical simulations of (BH-BH) coalescence, which include the inspiral, merger, and ring-down contributions. A large sample of binary systems is simulated using the population synthesis code SeBa, and we extract fundamental statistical information on (BH-BH) physical parameters (primary and secondary BH masses, orbital separations and eccentricities, formation, and merger time scales). We then derive the binary birth and merger rates using the theoretical cosmic star formation history obtained from a numerical study which reproduces the available observational data at redshifts $z<8$. We evaluate the contributions of the inspiral, merger, and ring-down signals to the GWB, and discuss how these depend on the parameters which critically affect the number of coalescing (BH-BH) systems. We find that Advanced LIGO/Virgo have a chance to detect the GWB signal from the inspiral phase with a $(S/N)=10$ only for the most optimistic model, which predicts the highest local merger rate of $0.85{\mathrm{Mpc}}^{-3}{\mathrm{Myr}}^{-1}$. Third generation detectors, such as the Einstein Telescope (ET), could reveal the GWB from the inspiral phase predicted by any of the considered models. In addition, ET could sample the merger phase of the evolution at least for models which predict local merger rates between $[0.053–0.85]{\mathrm{Mpc}}^{-3}{\mathrm{Myr}}^{-1}$, which are more than a factor 2 lower than the upper limit inferred from the analysis of the LIGO S5 run [J. Abadie et al., Phys. Rev. D 83, 122005 (2011)]. The frequency dependence and amplitude of the GWB generated during the coalescence is very sensitive to the adopted core mass threshold for BH formation. This opens up the possibility to better understand the final stages of the evolution of massive stellar binaries using observational constraints on the associated gravitational wave emission.

Figure 1

Mass of the secondary stellar progenitors as a function of the corresponding primary mass. Empty grey circles indicate stellar binaries which lead to (BH-BH) systems; black squares show the subsample of these progenitors which form (BH-BH) binaries with merger times smaller than the Hubble time (see text).

Figure 2

Same as Fig. 1 but for black hole primary and secondary masses. The various regions in the parameter space which are overpopulated or underpopulated originate from the decision making process in SeBa, and can be related to the chain of events in the population synthesis.

Figure 3

Upper panel: Formation time scales for compact black hole binaries as a function of the corresponding semimajor axis. For clarity, only systems with $\mathrm{sma}<1600R_{\odot }$ are shown (empty grey circles). Black squares are (BH-BH) pairs with merger times smaller than the Hubble time (see text). Lower panel: Merger time scales as a function of the chirp mass of merging systems. The dashed horizontal and vertical lines indicate threshold values, ${\tau }_{\mathrm{m},\mathrm{thre}}$ and ${\mathcal{M}}_{\mathrm{thre}}$, so that 70% of the systems have ${\tau }_{\mathrm{m}}>{\tau }_{\mathrm{m},\mathrm{thre}}$ and 75% have $\mathcal{M}<{\mathcal{M}}_{\mathrm{thre}}$.

Figure 4

Eccentricity as a function of the black hole mass ratio for merging systems.

Figure 5

Single source signal as a function of frequency for a (BH-BH) binary with total mass $M=20M_{\odot }$, symmetric mass ratio $\eta =0.25$, located at a distance of 10 Mpc.

Figure 6

Redshift evolution of the total binary birth rate (solid red line) and of (BH-BH) birth and merger rates (dotted and dashed, respectively).

Figure 7

The closure energy density, ${\Omega }_{\mathrm{GW}}$, generated by (BH-BH) coalescing binaries in model A, plotted as a function of the observational frequency. The three contributions coming from the inspiral (dashed), merger (solid), and ring-down (dotted) phases are plotted separately. The black solid line is the total GWB signal. The shaded regions in the three panels indicate the foreseen sensitivities for different interferometers and integration times (see labels on the figures).

Figure 8

${\Omega }_{\mathrm{GW}}$ generated during the inspiral (upper panel), merger (central panel), and ring-down (lower panel) phases in models F3 (dashed), B5 (dotted), and A (solid). In all panels, the two shaded regions indicate the foreseen sensitivities of ALIGO and ET-B assuming three years of integration.

Figure 9

Redshift evolution of (BH-BH) merger rates for models F3 (solid line), A (dotted line), and B5 (dashed line). The upper limit labeled “LIGO S5” shows the constraint recently derived in 73 (see text).

Figure 10

The closure energy density, ${\Omega }_{\mathrm{GW}}$, for (BH-BH) binaries predicted by models F3 (dashed line) and B5 (dotted line) is compared to that of model A (solid line). The shaded region can be considered a measure of the uncertainty on ${\Omega }_{\mathrm{GW}}$. The two black solid lines indicate the foreseen sensitivities of ALIGO and ET-B assuming three years of integration.