Constraining the Primoridal Black Hole Abundance with Space-Based Detectors

TL;DR

This paper assesses how future space-based gravitational wave detectors (LISA, Taiji, TianQin) can constrain the abundance of asteroid-mass primordial black holes (PBHs) through scalar-induced gravitational waves (SIGWs) generated by enhanced primordial curvature perturbations. By modeling the curvature spectrum with a log-normal form of width $\Delta$ and computing the resulting PBH mass function via peak theory and Press–Schechter methods, the authors connect PBH formation to the SIGW signal. They then evaluate detector sensitivities, including instrumental noise and the galactic binary foreground, using a two-step detection approach, and translate SIGW detectability into bounds on the PBH fraction $f_{\rm PBH}$ across PBH masses. The main finding is that LISA, Taiji, and TianQin can fully probe the asteroid-mass window, with signal-to-noise ratios in the range $\sim 10^3$–$10^4$ if PBHs constitute all dark matter, largely independent of the power-spectrum width. These results underscore the potential of space-based GW observatories to directly test the PBH dark matter scenario and highlight robustness to spectral shape assumptions, while also noting caveats from non-Gaussianity and PBH mass–horizon relations that warrant further study.

Abstract

Overdense regions can collapse into primordial black holes (PBHs) in the early universe, which are a compelling candidate for dark matter. Current constraints leave the asteroid-mass window the only possible one for PBH to account for all the dark matter, which can only be probed indirectly by the scalar-induced gravitational waves (GWs) sourced by the curvature perturbation which forms PBH. In this work, we explore the capabilities of future space-based gravitational wave detectors, including LISA, Taiji, and TianQin, to constrain such induced GWs as well as the PBH abundance. We systematically account for the width of the primordial curvature power spectrum, and find that the asteroid-mass window can be fully probed by all three space-based interferometers. If PBHs constitute the majority of dark matter, the induced GW leaves a strong signal in the mHz band with a signal-to-noise ratio of $10^3\sim10^4$.

Figures (7)

• Figure 1: The PBH mass functions corresponding to power spectra with widths $\Delta = 0$ (blue), $0.28$ (orange), $0.5$ (green), and $1$ (red), all normalized to a total PBH abundance of $f_{\rm PBH} = 1$.
• Figure 2: The strain sensitivity curves of LISA (red), Taiji (green), and TianQin (purple) for isotropic SGWB searches. The sensitivities are evaluated from the instrumental noise models \ref{['eq:Noise spectrum of ins']} summarized in Tab. \ref{['tab:detector param']}. Detectors with longer arm lengths achieve optimal sensitivity at lower frequencies. The low-frequency rise is dominated by residual acceleration noise, whereas the high-frequency degradation is caused by the reduced detector response.
• Figure 3: Noise spectrum for various space-based detectors, including the instrument noise (solid black) and the confusion foreground (dashed gray). The dashed gray line shows an envelop curve of $f_{\rm PBH} = 1$ obtained by shifting $\mathcal{A}_\zeta(k_p)$. Colored curves indicate the marginally detectable induced gravitational wave signals with SNR $\varrho_i=3$ for 3 years at different peak wavenumbers: $k_p = 10^{11}$ (blue), $10^{12}$ (orange), $10^{13}$ (green), $10^{14}$ (red), and $10^{15}$ (purple) $\mathrm{Mpc}^{-1}$. From top to bottom, the rows correspond to power spectrum widths $\Delta = 0$, 0.28, 0.5, and 1, respectively. From left to right, the columns show the results for TianQin, LISA, and Taiji. The corresponding power-law-integrated sensitivity curves for SNR $\varrho_i = 3$ and observation time $T = 3\,\mathrm{yr}$ are shown in gray solid curves for comparison.
• Figure 4: The constraints of detectable region on amplitude $\mathcal{A}_\zeta$ and peak wavenumber $k_p$ for power spectra with $\Delta = 0$ (upper left), $\Delta = 0.28$ (upper right), $\Delta = 0.5$ (lower left), $\Delta = 1$ (lower right). The solid curves represent the sensitivity of LISA (red), Taiji (green) and TianQin (purple), with a total duration 3 yr and SNR $\varrho = 3$. The gray dashed line represents $f_{\rm PBH} = 1$.
• Figure 5: Constraints from different observations on the faction of PBH in DM, $f_{\rm PBH}$, as a function of the PBH average mass $\langle M_{\rm PBH} \rangle$, for PBH mass function showed in Fig. \ref{['fig:PBH mass function']} with monochromatic spectrum $\Delta = 0$ (upper left) and log-normal spectrum $\Delta = 0.28$ (upper right), $\Delta = 0.5$ (lower left) and $\Delta = 1$ (lower right). The orange region on the left is excluded by evaporations of CMB anisotropies Acharya:2020jbv and extragalactic $\gamma$-ray Carr:2020gox. The right blue region by microlensing (blue shades) of Subaru HSC Niikura:2017zjd, EROS EROS-2:2006ryy, OGLE Mroz:2024mse. The region between colored dashed lines is the detectable parameters space of LISA (red), Taiji (green) and TianQin (purple), with a total duration 3 yr and SNR $\varrho = 3$.