Gravitational-Wave Constraints on the Abundance of Primordial Black Holes

TL;DR

This work examines second-order gravitational waves induced by large-amplitude primordial density fluctuations that form primordial black holes (PBHs). By modeling the scalar power spectrum with a peaked shape and analyzing both delta-function and finite-width (top-hat) cases, the authors derive how the induced GW spectrum encodes PBH abundance and mass, including the dependence on peak width $\Delta$. They show that, for small $\Delta$, there is a near one-to-one mapping between PBH parameters and the GW signal, while larger widths modify the amplitude and create plateau features that still enable inference of the underlying scalar power. Observationally, pulsar timing arrays and space-based GW detectors (LISA, DECIGO, BBO) can probe PBHs across the IMBH and DM mass ranges, with CMB constraints limiting the supermassive PBH regime. Overall, induced GWs provide a powerful indirect probe of PBH formation and abundance across a wide mass spectrum.

Abstract

We investigate features of Gravitational Waves (GWs) induced by primordial density fluctuations with a large amplitude peak associated with formation of Primordial Black Holes (PBHs). It is shown that the spectrum of induced GW is insensitive to the width of the peak in wavenumber space provided it is below a certain value, but the amplitude of the spectrum reduces at the peak frequency and decreases faster at low frequencies for a larger width. A correspondence between the GW amplitude and PBH abundance is also investigated incorporating the peak width. We find that PBHs with masses 10^{20-26}g can be probed by space-based laser interferometers and atomic interferometers irrespective of whether the peak width is small or not. Further we obtain constraints on the abundance of the supermassive PBHs by comparing a low frequency tail of the GW spectrum with CMB observations.

Figures (3)

• Figure 1: The spectrum of the function $I_d^2$ given by Eq.(\ref{['eq:irad']}) (left) and the energy density of induced GWs for the power spectrum (\ref{['eq:delta']}) (right). In the left panel, the function $I_d^2$ estimated at $k_p\eta=1.0 \times 10^2$ (solid line) and $k_p\eta=1.0 \times 10^3$ (broken line) are plotted. As mentioned in the text, ignoring oscillations, $I_d^2$ is time-independent. A time averaged one is also plotted (thick solid line). In the right panel, the plotted energy density is time averaged and normalized by the $\mathcal{A}^4\Omega_{\mathrm{rad}}$.
• Figure 2: The energy density of the induced GWs for the power spectrum (\ref{['eq:top-hat']}) for $\Delta=0.0,1.0 \times 10^{-3}, 1.0 \times 10^{-1},1.0$ (left) and its value at $k=k_p$ normalized by that for the delta-function type spectrum as a function of $\Delta$ (right). Both are estimated at $k_p\eta=1.0\ \times 10^{3}$.
• Figure 3: Energy density of induced GWs (solid lines and broken lines) with a limit by pulsar timing observations and planned sensitivities of space-based interferometers (dotted lines). Thick solid lines indicate the energy density with the parameters $(\Omega_{\mathrm{PBH}}h^2,M_{\mathrm{PBH}})=(10^{-5},10^2M_{\odot})$ (left), $(10^{-1},10^{20}\mathrm{g})$ (right) for sufficiently small $\Delta$. We have depicted those at the peak frequency as thick broken lines for $\Omega_{\mathrm{PBH}}h^2=10^{-5}$ (below), $10^{-1}$ (above). Energy densities with $\Delta=1.0$ are also shown by thin lines. In estimating them, we have assumed that $g_{\ast,p}$ is approximately $10$ for the IMBH mass scales and $10^2$ for the DM mass scales. Sensitivities of the space-based interferometers are depicted by using Sensitivity with the instrumental parameters used in Kudoh:2005as for DECIGO/BBO. Sensitivities of the ground-based interferometer, LIGO LIGO are also plotted as a reference. The right vertical axis represents the corresponding power of scalar modes with small $\Delta$ and $g_{\ast,p}=10$.