Gravitational wave background from binary systems

Basic aspects of the background of gravitational waves and its mathematical characterization are reviewed. The spectral energy density parameter $\Omega (f)$, commonly used as a quantifier of the background, is derived for an ensemble of many identical sources emitting at different times and locations. For such an ensemble, $\Omega (f)$ is generalized to account for the duration of the signals and of the observation, so that one can distinguish the resolvable and unresolvable parts of the background. The unresolvable part, often called confusion noise or stochastic background, is made by signals that cannot be either individually identified or subtracted out of the data. To account for the resolvability of the background, the overlap function is introduced. This function is a generalization of the duty cycle, which has been commonly used in the literature, in some cases leading to incorrect results. The spectra produced by binary systems (stellar binaries and massive black hole binaries) are presented over the frequencies of all existing and planned detectors. A semi-analytical formula for $\Omega (f)$ is derived in the case of stellar binaries (containing white dwarfs, neutron stars or stellar-mass black holes). Besides a realistic expectation of the level of background, upper and lower limits are given, to account for the uncertainties in some astrophysical parameters such as binary coalescence rates. One interesting result concerns all current and planned ground-based detectors (including the Einstein Telescope). In their frequency range, the background of binaries is resolvable and only sporadically present. In other words, there is no stochastic background of binaries for ground-based detectors.

Figure 1

Penrose diagram of a universe described by the metric (1). The straight black line crossing $z=0$ and $z=z_{\max }$ contains all the gravitons that we observe today.

Figure 2

Redshift versus observed frequency. The spectral function of the total background of a certain ensemble has support only within the shaded area.

Figure 3

Observed frequency versus time. Each line represents the evolution of one signal (like the one produced by a binary). Closer signals, as well as the superposition of many signals, are plotted darker than distant individual signals. Three frequency bins are distinguished: in $(\Delta f{)}_1$ the total background is completely resolvable, in $(\Delta f{)}_2$ there is an unresolvable part and a dominating resolvable part, and in $(\Delta f{)}_3$ there is a dominating unresolvable part and a resolvable part.

Figure 4

Redshift versus observed frequency. Each horizontal line contains the possible observed frequencies of a signal. The light-shaded (dark-shaded) area represents the resolvable (unresolvable) part of the background. The redshift functions $z_{\mathrm{low}}(f)$, $\overline{z}(f,\Delta f,{\mathcal{N}}_0)$, and $z_{\mathrm{upp}}(f)$ are shown with dashed, dotted, and solid lines, respectively. The frequencies $f_{p,\min }$ and $f_{p,\max }$ delimit the interval where the unresolvable part is present. The frequencies $f_{d,\min }$ and $f_{d,\max }$ delimit the interval where the unresolvable part dominates.

Figure 5

Spectral function of the total background versus observed frequency. The contributions of the different ensembles are calculated with the most likely values of masses and coalescence rates. No restrictions in the duration of the signals are assumed in this plot, which means that also very short and sporadic signals are taken into account. As discussed in the text, the spectral function in such circumstances should not be compared to the sensitivity curves of a detector.

Figure 6

Spectral function of the continuous background versus observed frequency. The contributions of the different ensembles are calculated with the most likely values of masses and coalescence rates. The sensitivity curves of LISA (obtained using 85 with the standard parameters), ET (from 86), BBO (from 4) and the complete Parkes PTA (from 16) are plotted for comparison.

Figure 7

Spectral function of the unresolvable background versus observed frequency, using ${\mathcal{N}}_0=1$ and $\Delta f=1{\mathrm{yr}}^{-1}$. The contributions of the different ensembles are calculated with the most likely values of masses and coalescence rates.

Figure 8

Spectral function of the total, continuous and unresolvable backgrounds of the different ensembles, versus observed frequency. In each plot there are nine curves: three of them are calculated with the highest values of coalescence rates, three with the most likely, and three with the lowest values possible. Three of the curves represent the total background, three the continuous part, and three the unresolvable part. In the case of NS-WD, the total and continuous curves are almost indistinguishable. The same occurs for WD-WD with the total, continuous and unresolvable curves.

Figure 9

Spectral function of the total (top), continuous (middle) and unresolvable (bottom) background versus observed frequency. The contributions of the different ensembles are calculated with the most likely values of masses and coalescence rates of Table , except for the case of BH-BH. The high rate of BH-BH, taken from the recent paper in 73, is $R={0.36}_{-0.26}^{+0.50}{\mathrm{Mpc}}^{-3}{\mathrm{Myr}}^{-1}$.

Figure 10

Spectral function of the total, continuous, and unresolvable backgrounds of the ensemble of BH-BH, versus observed frequency. This plot is analogous to that in Fig. 8, but using the rate from 73 of $R={0.36}_{-0.26}^{+0.50}{\mathrm{Mpc}}^{-3}{\mathrm{Myr}}^{-1}$, instead of the one in Table .

Figure 11

Spectral function of the resolvable and unresolvable parts of the background of NS-NS, versus observed frequency. The values of masses and coalescence rates adopted are the most likely ones. The resolvable and unresolvable parts calculated with the one- and with the eight-bin rule are compared.

Figure 12

Redshift versus observed frequency (left plot, analogous to that in Fig. 4) and spectral function versus observed frequency (right plot, analogous to that in Fig. 8, corresponding to NS-NS with the most likely values of masses and coalescence rates). The difference between these plots and the ones in Figs. 4, 8 is that here, as soon as the unresolvable part of the background dominates, all signals become unresolvable.

Figure 13

Redshift versus observed frequency (left plot, analogous to that in Fig. 4) and spectral function versus observed frequency (right plot, analogous to that in Fig. 8, corresponding to NS-NS with the most likely values of masses and coalescence rates). These plots (unlike those in Figs. 4, 8) are calculated assuming an unrealistic time resolution of $\Delta t=600\mathrm{s}$. With such a large time resolution, an unresolvable background would be present in the frequency band of ground-based detectors.

Figure 14

Spectral function of the unresolvable background of MBH-MBH (dotted line) and of all stellar binaries (dashed line), versus observed frequency. These curves are compared with previous predictions from the literature, which correspond to the unresolvable backgrounds of: (a) MBH-MBH from 16, (b) MBH-MBH from 15 and (c) extragalactic stellar binaries from 11.

Figure 15

Spectral function of the unresolvable background of MBH-MBH, calculated with the four different models (LE, LC, SE and SC), versus observed frequency. For comparison we include the curves of the unresolvable backgrounds of 15, 16.

Figure 16

Spectral function of the total, continuous and unresolvable backgrounds of the ensemble of NS-NS (with the most likely values of masses and coalescence rates), versus observed frequency. We compare these curves with those given in 13. Curves (a) and (b) correspond to what in that paper is called shot noise and Gaussian background, respectively. In the text we explain the origin of the discrepancy between (b) and our continuous or unresolvable backgrounds.