Detecting the gravitational wave background from primordial black hole dark matter

TL;DR

This paper investigates the stochastic gravitational-wave background generated by binaries of primordial black holes that could constitute dark matter, incorporating PBH clustering in halos, velocity distributions, and a broad PBH mass spectrum. It develops a synthetic PBH binary population to compute the ensemble GW spectrum, demonstrates that eccentricity effects are negligible due to rapid circularization, and shows that a broad mass distribution can substantially boost the background, with significant implications for detectability by LISA and PTAs. Analytic and semi-analytic treatments of redshift integration and merger rates across monochromatic, broad, and extended halo-mass-function models reveal that LISA could detect backgrounds from PBH-DM models compatible with AdvLIGO rates, while SKA-era PTAs could probe broad-spectrum scenarios; current PTA limits already constrain some broad-mass models. A key result is that the PBH capture process imposes a minimal emission frequency, and the distinctive frequency dependence of the PBH background—driven by clustering and mass spread—offers a robust discriminator from astrophysical BH binaries, linking GW observations to PBH formation scenarios in the early Universe.

Abstract

The black hole merging rates inferred after the gravitational-wave detection by Advanced LIGO/VIRGO and the relatively high mass of the progenitors are consistent with models of dark matter made of massive primordial black holes (PBH). PBH binaries emit gravitational waves in a broad range of frequencies that will be probed by future space interferometers (LISA) and pulsar timing arrays (PTA). The amplitude of the stochastic gravitational-wave background expected for PBH dark matter is calculated taking into account various effects such as initial eccentricity of binaries, PBH velocities, mass distribution and clustering. It allows a detection by the LISA space interferometer, and possibly by the PTA of the SKA radio-telescope. Interestingly, one can distinguish this background from the one of non-primordial massive binaries through a specific frequency dependence, resulting from the maximal impact parameter of binaries formed by PBH capture, depending on the PBH velocity distribution and their clustering properties. Moreover, we find that the gravitational wave spectrum is boosted by the width of PBH mass distribution, compared with that of the monochromatic spectrum. The current PTA constraints already rule out broad-mass PBH models covering more than three decades of masses, but evading the microlensing and CMB constraints due to clustering.

Figures (4)

• Figure 1: PDF of initial orbit parameters $a_0$, $1-e_0$, $r_{\mathrm p}$ and $f_{\rm min}$ for a synthetized population of $10^6$ PBH binaries and halo Virial velocities $v_{\rm vir } = 2$ (green), $20$ (blue) and $200$ (red) km/s.
• Figure 2: The function $R(\sigma)$ rescaling the stochastic GW background for a broad PBH mass spectrum, compared to the monochromatic $(\sigma = 0)$ case.
• Figure 3: Top panels. Left: Stochastic GW spectrum $h_{\mathrm c}$ for a close to monochromatic ($\sigma = 0.1$) PBH mass distribution with $\mu = 30 \, M_\odot$ and $v_{\rm vir} = 2$ (green), $20$ (blue) and $200 \, \mathrm{km/s}$ (red). Density contrasts are normalized to produce a constant merger rate $\tau = 50$ yr$^{-1}$ Gpc$^{-3}$, which corresponds respectively to $10^{-8} \delta_{\rm PBH}^{\rm loc} = 0.2 / 8.0 / 290$. The expected GW background for Bird et al. model Bird:2016dcv (extended halo mass function) is also represented (dotted brown). Right: GW spectrum for $\mu = 30 M_\odot$ and $v_{\rm vir} = 20 \, \mathrm{km/s}$ in the broad mass spectrum case, with $\sigma= 0.1 / 1 / 2$ (solid, dashed and dotted line respectively). Bottom panels. The GW background $\Omega_{\rm GW} h^2$ for the same parameters. Solid and Dotted black curves are the expected sensitivities for LISA respectively for the best and worse experimental designs. Black dashed curve represents the sensitivity of the SKA through PTA observations. The grey curves on the right represent the sensitivity of Advanced LIGO (O1 Run - dashed, O2 Run - dotted, O5 Run - solid).
• Figure 4: Sketch displaying the expected limits for future GW experimentsas well as the stochastic GW background for various astrophysical and cosmological processes. The mauve band correspond to the expectations for the PBH-DM model considered in this paper, where PBH are regrouped in dense sub-halos, for merging rates consistent with the ones inferred by AdvLIGO, i.e. between 8 and 240 events per year per Gpc$^3$, and mean PBH masses in the range $10_\odot \mathrel{\hbox{$\sim$$<$}} m_{\rm PBH} \mathrel{\hbox{$\sim$$<$}}100 M_\odot$. Broad mass spectra allow to enhance the spectrum up to the present EPTA PTAs and LIGO limits. For these merging rates, the stochastic GW background reaches the level of detectability of LISA and the SKA. For comparison, the green band represents the region covered by the model of Bird et al. Bird:2016dcv extrapolated to lower frequencies.