Gravitational wave background from binary systems

TL;DR

The paper develops a rigorous, cosmology-aware framework to quantify the gravitational-wave background from coalescing binaries, distinguishing resolvable and unresolvable components via an overlap function that generalizes the duty cycle. It derives a semi-analytic expression for the spectral function $\Omega(f)$ for stellar binaries and extends the analysis to MBH binaries, incorporating realistic mass ranges and coalescence-rate uncertainties. The results show that ground-based detectors operate in a band free of confusion noise from binaries, while space-based and PTA bands may encounter a stochastic background from WD-WD and MBH binaries, respectively. The work provides robust prescriptions for computing the background across detectors and highlights the importance of resolvability and continuity in interpreting potential primordial signals.

Abstract

Basic aspects of the background of gravitational waves and its mathematical characterization are reviewed. The spectral energy density parameter $Ω(f)$, 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, $Ω(f)$ 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 $Ω(f)$ 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.

Figures (16)

• Figure 1: Penrose diagram of a universe described by the metric (\ref{['eq:metric']}). The straight black line crossing $z=0$ and $z=z_{\text{max}}$ 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 $z_{\text{low}}(f)$, $\overline{z}(f,\Delta f,\mathcal{N}_0)$, and $z_{\text{upp}}(f)$ are shown with dashed, dotted, and solid lines, respectively. The frequencies $f_{\text{p,min}}$ and $f_{\text{p,max}}$ delimit the interval where the unresolvable part is present. The frequencies $f_{\text{d,min}}$ and $f_{\text{d,max}}$ 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.