Constraints on the induced gravitational wave background from primordial black holes

TL;DR

This work links small-scale primordial curvature perturbations to both PBH production and a second-order induced GW background by computing PBH mass spectra within a Press-Schechter framework and evaluating the resulting GWB from peaked perturbation spectra. It finds that the maximal induced GW amplitude compatible with PBH constraints is largely insensitive to the width of the primordial peak and extends GW constraints over a broad frequency range, mapping PBH mass scales to GW frequencies via horizon-entry relations. By translating PBH abundance limits into bounds on the energy density of induced GWs, the authors show that pulsar timing data can strongly constrain PBH densities in the $M_{BH}\sim 0.03-10\,M_\odot$ range, with current interferometers like LIGO providing complementary limits and future detectors (LISA, BBO, SKA) expected to probe well below these bounds. Overall, the results provide robust, width-insensitive constraints on the induced GW background arising from small-scale primordial perturbations, highlighting the power of PBH observations to inform high-frequency gravitational-wave physics.

Abstract

We perform a consistent calculation of primordial black hole (PBH) mass spectrum and second-order induced gravitational wave (GW) background produced from primordial scalar perturbations in radiation era of the early Universe. It is shown that the maximal amplitudes of the second order GW spectrum that can be approached without conflicting with the PBH data do not depend significantly on the shape of primordial perturbation spectrum. The constraints on the GW background obtained in previous works are extended to a wider GW frequency range. We discuss the applicability of the currently available pulsar timing limits for obtaining the constraints on scalar power spectrum and PBH abundance and show that they can be used for strongly constraining the PBH number density in the PBH mass range $\sim (0.03 - 10) M_{\odot}$.

Figures (4)

• Figure 1: The calculation of PBH mass spectra for a peaked ${\cal P}_{\cal R}(k)$-spectrum, for $M_h^0=10^{33}\;$g and different values of $\Sigma$. The value of ${\cal P}_{\cal R}^0$ was chosen so that for all cases $\Omega_{PBH} \approx 0.024$ (for $\Sigma=1$, ${\cal P}_{\cal R}^0=0.0488$; for $\Sigma=3$, ${\cal P}_{\cal R}^0=0.0294$; For $\Sigma=5$, ${\cal P}_{\cal R}^0=0.0267$; For $\Sigma=7$, ${\cal P}_{\cal R}^0=0.0258$.) The dashed curve shows, for comparison, the form of the classical Carr's $M_{BH}^{-5/2}$-mass spectrum Carr:1975qj.
• Figure 2: The limits on the value of ${\cal P}_{\cal R}^0$ from PBH non-observation, for three different values of $\Sigma$. The constraints based on the $\beta$-function approach are shown by thin lines.
• Figure 3: a) The calculation of the induced GWB corresponding to primordial power spectra and PBH mass spectra given in Fig. \ref{['fig-pbh-sp']} (cases of $\Sigma=1, 3, 5$). The pulsar timing limit for $\Omega_{GW}$ is also shown. b) The calculation of the induced GWB for $M_h^0=10^{17}\;$g (for all three cases, $\Omega_{PBH}\approx 0.24$, the corresponding sets of parameters are ${\cal P}_{\cal R}^0=0.028$ for $\Sigma=1$, ${\cal P}_{\cal R}^0=0.0172$ for $\Sigma=3$, and ${\cal P}_{\cal R}^0=0.0159$ for $\Sigma=5$).
• Figure 4: The limits on the 2-nd order GWB from primordial black holes, obtained in this paper. Solid thin line shows the result assuming PBH formation threshold is $\delta_{th}=1/3$. Dashed line corresponds to the assumption $\delta_{th}=0.45$. Thin line shows the maximum values of GWB that can be reached for $\Omega_{PBH}=10^{-5}$ (and $\delta_{th}=1/3$). Also shown are current pulsar timing limit, LIGO S5 limit, Advanced LIGO planned sensitivity to $\Omega_{GW}$, and frequency ranges in which other future experiments (LISA, BBO, SKA) will operate. The estimate sensitivity of all future experiments, in the corresponding frequency ranges, is much better than the PBH bound shown.