Probing Primordial black holes with the distortion of Stochastic Gravitational Wave Background

TL;DR

This study develops an analytical framework to quantify how gravitational lensing by primordial black holes (PBHs) affects the stochastic gravitational-wave background (SGWB) from binary black-hole mergers. By modeling PBHs as dark-matter lenses and employing wave-optics lensing, the authors derive the lensed SGWB spectrum and its dependence on PBH mass $M_{\mathrm{PBH}}$ and abundance $f_{\mathrm{PBH}}$, including the optical depth $\tau(z_s)$ and diffraction features. They show that PBH lensing can produce relative spectral deviations up to $\sim 10^{-1}$, with the peak amplitude mainly set by $f_{\mathrm{PBH}}$ and the peak frequency set by $M_{\mathrm{PBH}}$, offering a potential route to constrain PBH-DM scenarios with future SGWB detections. The framework combines a physically motivated BBH merger-rate model, a wave-optics treatment of lensing, and a statistical approach to PBH lensing, providing predictions for how PBH properties imprint on the SGWB spectrum and guiding observational strategies.

Abstract

The stochastic gravitational-wave background (SGWB), arising from the incoherent superposition of numerous compact binary coalescences, serves as a powerful probe of both astrophysical populations and fundamental physics. In this work, we investigate the influence of gravitational lensing on the SGWB, focusing on primordial black holes (PBHs) as potential lenses. Assuming PBHs as dark matter candidates with a broad cosmic distribution, we show that their lensing optical depth can be significantly enhanced, producing pronounced effects with relative deviations at the 10^-1 level. By systematically varying the PBH mass (M_PBH) and abundance (f_PBH), we demonstrate that the mass predominantly determines the frequency-dependent diffraction features of the spectrum, while the abundance primarily amplifies the overall lensing-induced deviation. Although the SGWB from binary black holes has not yet been observed, our analytical results provide theoretical insight into the possible imprint of lensing on its spectrum and suggest that future detections could offer a novel avenue to constrain dark matter scenarios.

Figures (7)

• Figure 1: The normalized star formation rate (SFR) density. This curve illustrates the evolutionary trend of the BBH merger rate, denoted as $R_{\mathrm{V}}$ in this work.
• Figure 2: Monte Carlo sampling of source positions within a circle. The probability that a point falls within an annular ring between $y$ and $y + dy$ is proportional to the area of the ring, i.e., $2\pi y\,dy$. Therefore, the normalized probability density function is given by Eq. \ref{['monte']}.
• Figure 3: The upper and lower panels correspond to the optical depth for PBH lensing as a function of source redshift and to its probability density function (PDF) as a function of lens redshift, respectively. As shown, the PDF curve first increases and then decreases, exhibiting a distinct peak. This behavior indicates that our assumption in Eq. \ref{['defpro']} is well justified in this context.
• Figure 4: In the upper panel, the purple solid line and the red dashed line represent the stochastic gravitational-wave spectra without lensing and with lensing, respectively, where we adopt a uniform value of $t_{\min}=50\,\mathrm{Myr}$. The lower panel shows the difference between the lensed and unlensed spectra, quantified by Eq. \ref{['delta_f']}. The adopted lensing parameters are $f_{\mathrm{PBH}}=1.0$ and $M_{\mathrm{PBH}}=90\,M_{\odot}$.
• Figure 5: As shown in the figure, in the first and third row, we fix the PBH abundance at $f_{\mathrm{PBH}}=1$ and vary the lens mass as $M=1\,,10\,,50\,,250\,,500\,,1000\,M_{\odot}$, these two rows show how the lens mass influences the relative difference between the lensed and unlensed spectra. For the second row, we fix the PBH mass at $M_{\mathrm{PBH}}=100\,M_{\odot}$ and vary the PBH abundance as $f_{\mathrm{PBH}}=0.1\,,0.5\,,1.0$, this row shows how the PBH abundance affects the relative difference between the lensed and unlensed spectra, with higher values of $f_{\mathrm{PBH}}$ leading to a more pronounced lensing effect.