Induced gravitational wave background and primordial black holes

TL;DR

The paper analyzes the stochastic gravitational-wave background produced at second order by scalar perturbations, focusing on two scalar-power scenarios: a peaked spectrum and a running-mass inflation model. It derives the induced GW spectrum through second-order perturbation theory, maps frequency to horizon mass, and computes $\Omega_{\rm GW}$ for both delta-like and finite-width peaks, then confronts the predictions with primordial black hole constraints and current/future LIGO bounds. It finds that PBH constraints currently dominate the limits on small-scale curvature perturbations, but Advanced LIGO has the potential to surpass these bounds in the gravitational-wave band, with running-mass models offering potentially detectable signals in the ground-based detector range depending on reheating and running parameters. The results illustrate a strong interplay between PBH physics and stochastic gravitational-wave searches as a probe of sub-Hubble-scale perturbations.

Abstract

We calculate the frequency dependence of gravitational wave background arising at second order of cosmological perturbation theory due to mixing of tensor and scalar modes. The calculation of the induced gravitational background is performed for two special cases: for the power spectrum of scalar perturbations which has a peak at some scale and for the scalar spectrum predicted by the inflationary model with the running mass potential. We show that the amplitudes of the induced gravitational background, in the frequency region $\sim 10^{-3} - 10^3 $Hz, are effectively constrained by results of studies of primordial black hole production in early universe. We argue that though today's LIGO bound on ${\cal P}_{\cal R}(k)$ is weaker than the PBH one, Advanced LIGO will be able to set a stronger bound, and in future the ground-based interferometers of LIGO type will be suitable for obtaining constraints on PBH number density in the mass range $\sim 10^{11} - 10^{15}$ g.

Figures (7)

• Figure 1: The dependence of ${\cal P}_h(k)$ on the scale factor $a$ for several wave numbers. As an input, we used here a delta-function power spectrum for ${\cal P}_\Psi$, with $\tilde{P}_0=10^{-3}$ for each case. For curves from top to bottom, $k=k_0=6 k_{eq}, 20 k_{eq}, 3 \times 10^3 k_{eq}$.
• Figure 2: GW spectrum from a delta-function peak in ${\cal P}_\Psi$ ($\tilde{P}_0=1.2 \times 10^{-3}$, $k_0=8\times 10^{16}$ Mpc$^{-1}$). Thin solid line - calculation for $\tau_{calc}=10^3 k_0^{-1}$, thin dashed line - for $\tau_{calc}=(50+N_{\rm rnd})k^{-1}$, thick line is the envelope.
• Figure 3: Calculation of $\Omega_{GW}(k)$ at the present epoch for finite width curvature perturbation power spectra of the form (\ref{['PRparam']}) (from bottom to top, $\Sigma=0.1, 0.3, 0.8$ and ${\cal P}_{\cal R}^0 = 0.01$; upper curve is for scale-invariant input spectrum with ${\cal P}_{\cal R}(k) = 0.01$). We assumed that $g_*(k_{0})\approx 100$.
• Figure 4: Calculation of $\Omega_{GW}(k)$ at the present epoch for the case $f_0=100$ Hz, $\Sigma=3$, ${\cal P}_{\cal R}^0 = 0.032$ (upper curve) and ${\cal P}_{\cal R}^0 = 0.016$ (lower curve). Such parameters are maximal allowed from PBH constraints. Also shown are experimental limits on $\Omega_{GW}$ obtained in the LIGO experiment and bound range expected in the future.
• Figure 5: Power spectrum ${\cal P}_{\cal R}(k)$, calculated for the running mass model, with $T_{RH}=10^{10}$ GeV and $n_0=0.96$; $n_0'=4.5\times 10^{-3}$ for the upper curve and $n_0'=4.0 \times 10^{-3}$ for the lower curve.