[Paper Review] Detecting the Stochastic Gravitational Wave Background from Primordial Black Hole Formation

Sébastien Clesse, J. García-Bellido+1 more|arXiv (Cornell University)|Dec 28, 2018

Cosmology and Gravitation TheoriesPhysics and Astronomy79 references41 citations

TL;DR

The paper investigates how PBH formation from peaks in the curvature power spectrum induces a second-order stochastic gravitational wave background, computes its spectrum for various power-spectrum shapes, and assesses detectability by PTA and future interferometers.

ABSTRACT

Primordial Black Holes (PBH) from peaks in the curvature power spectrum could constitute today an important fraction of the Dark Matter in the Universe. At horizon reentry, during the radiation era, order one fluctuations collapse gravitationally to form black holes and, at the same time, generate a stochastic background of gravitational waves coming from second order anisotropic stresses in matter. We study the amplitude and shape of this background for several phenomenological models of the curvature power spectrum that can be embedded in waterfall hybrid inflation, axion, domain wall, and boosts of PBH formation at the QCD transition. For a broad peak or a nearly scale invariant spectrum, this stochastic background is generically enhanced by about one order of magnitude, compared to a sharp feature. As a result, stellar-mass PBH from Gaussian fluctuations with a wide mass distribution are already in strong tension with the limits from Pulsar Timing Arrays, if they constitute a non negligible fraction of the Dark Matter. But this result is mitigated by the uncertainties on the curvature threshold leading to PBH formation. LISA will have the sensitivity to detect or rule out light PBH down to $10^{-14} M_{\odot}$. Upcoming runs of LIGO/Virgo and future interferometers such as the Einstein Telescope will increase the frequency lever arm to constrain PBH from the QCD transition. Ultimately, the future SKA Pulsar Timing Arrays could probe the existence of even a single stellar-mass PBH in our Observable Universe.

Motivation & Objective

• Motivate PBH as a dark matter candidate formed from peaks in the curvature power spectrum.
• Quantify the amplitude and shape of the induced stochastic gravitational wave background from PBH formation.
• Explore how different plausible primordial power spectra affect GW production and PBH mass distributions.
• Assess current PTA limits on the GW background and future detector prospects for PBH-related signals.

Proposed method

• Compute PBH formation from Gaussian and other peak-shaped curvature perturbations using the collapse threshold formalism.
• Relate PBH mass to formation scale and horizon entry time via m_PBH ~ M_p^2/H_end and exponential factors.
• Calculate the second-order GW power spectrum from scalar perturbations using the integral formulation P_h(k,t_k) with P_zeta.
• Apply transfer functions to obtain the present-day GW energy density Omega_GW,0 from P_h(k,t_k).
• Evaluate GW spectra for Gaussian, sharp peak, broken power-law, and nearly flat/QCD-boosted power spectra.

Experimental results

Research questions

• RQ1 What is the amplitude and spectral shape of the stochastic GW background induced by PBH formation for different curvature power spectrum shapes?
• RQ2 Can future GW detectors (PTA, LISA, ET, SKA) detect or rule out PBH scenarios that account for all dark matter?
• RQ3 How do uncertainties in PBH formation thresholds (zeta_c) and power-spectrum peaks affect GW predictions and constraints?
• RQ4 To what extent do wide PBH mass distributions enhance the GW background compared to monochromatic PBH scenarios?
• RQ5 How can GW measurements distinguish among inflationary formation models (waterfall inflation, axion inflation, QCD-boost scenarios) based on GW spectra?

Key findings

• The stochastic GW background from PBH formation is enhanced by about an order of magnitude for broad or nearly scale-invariant spectra compared with sharp features.
• For broad peaks with PBHs making all DM, Omega_GW,0 h^2 peaks around 1.5e-9 in the nanoHertz PTA range.
• Current PTA limits (NANOGrav, PPTA) already constrain wide-PBH scenarios; EPTA can accommodate some parameter choices.
• Future PTA (IPTA, SKA) could probe or rule out even a single stellar-mass PBH in the observable universe under Gaussian fluctuations.
• Space-based detectors like LISA can detect or constrain PBH scenarios down to m_PBH ~ 1e-14 Msun and provide complementary constraints to ground-based interferometers.
• The paper emphasizes that the GW signal’s amplitude scales roughly with the square of the peak power P_p and that PBH abundance is extremely sensitive to zeta_c^2/ P_p, shaping detectability.