Stochastic Background of Gravitational Waves Generated by Black Hole MACHO Binaries in the Galaxy

Abstract

The interest in gravitational waves (GW) has increased since the recent detections of such signals announced by the LIGO Scientific Collaboration and the Virgo Collaboration. On the other hand, also interesting is the study of the stochastic backgrounds of GW (SBGW), which could in principle be generated by a variety of sources and astrophysical processes; particularly, in the present paper we are concerned with the SBGW from MACHO binaries in the Galaxy. In our scheme, we take into account only MACHOs which are black holes (BHMACHOs) and are located up to 50 kpc from the center of the Galaxy. Our spectra, which are calculated in the scenarios with $\Omega h^{2}=0.1$ and $\Omega h^{2}=1.0$, are parameterized by the fraction of the Galaxy’s dark halo mass which is in the form of BHMACHOs and are compared with some results found in the literature. Besides, we investigate in what situations the backgrounds are above the sensitivity curves of the Laser Interferometer Space Antenna (LISA), Einstein Telescope (ET), Advanced Virgo (AdV) and Advanced Laser Interferometer Gravitational-Wave Observatory Plus (LIGO A+).

Appendix 1

Assuming that the halo is a sphere of radius $R_\mathrm{m}$ with the Sun located at the distance $R_\mathrm{sun}$ from its center and being R and r, respectively, the radial distances from the center of the halo and from the Sun, we can write the volume element as

$$\begin{aligned} dV=2\pi R^{2}\sin \theta d\theta dR, \end{aligned}$$

(30)

with

$$\begin{aligned} \cos \theta =\frac{R^{2}_\mathrm{sun}+R^{2}-r^{2}}{2R_\mathrm{sun}R}. \end{aligned}$$

(31)

Using Eqs. (30) and (31) and bearing in mind that $R_\mathrm{m}>R_\mathrm{sun}$, Eq. (22) assumes the form

$$\begin{aligned} I&=\frac{2\pi R^{3}_\mathrm{s}(1\,{\mathrm{kpc}})^2}{R_\mathrm{sun}}\bigg \{\int ^{R_\mathrm{sun}}_{0}\int ^{R_\mathrm{sun}+R}_{R_\mathrm{sun}-R}\frac{drdR}{r(R_\mathrm{s}+R)^{2}}\,+ \nonumber \\& \int ^{R_\mathrm{m}}_{R_\mathrm{sun}}\int ^{R+R_\mathrm{sun}}_{R-R_\mathrm{sun}}\frac{drdR}{r(R_\mathrm{s}+R)^{2}}\bigg \} \therefore \nonumber \\&=\frac{2\pi R^{3}_\mathrm{s}(1\,{\mathrm{kpc}})^2}{R_\mathrm{sun}}\left\{ \int ^{R_\mathrm{m}}_{0}\frac{1}{(R_\mathrm{s}+R)^{2}}\ln {\frac{R+R_\mathrm{sun}}{|R-R_\mathrm{sun}|}}dR\right\} , \end{aligned}$$

(32)

where, for the sake of calculation, we divided the domain in two regions. Using the values $R_\mathrm{m}=50\,{\mathrm{kpc}}$, $R_\mathrm{sun}=8.29\,{\mathrm{kpc}}$ and $R_\mathrm{s}=18\,{\mathrm{kpc}}$ in Eq. (32) and solving the integral, we have $I\approx 213\,{\mathrm{kpc}^3}$.