[Paper Review] Signals of merging supermassive black holes in pulsar timing arrays

Paul Frederik Depta, Kai Schmidt-Hoberg+1 more|arXiv (Cornell University)|Jan 1, 2023

Pulsars and Gravitational Waves ResearchPhysics and Astronomy28 citations

TL;DR

This paper investigates whether the stochastic gravitational wave background detected by pulsar timing arrays (PTAs) could originate from merging primordial supermassive black hole binaries. It finds that homogeneously distributed primordial black holes (PBHs) cannot explain the signal due to cosmological constraints, but clustered PBHs with enhanced merger rates offer a viable path—if µ-distortion constraints from PBH formation are evaded.

ABSTRACT

In this letter we evaluate whether the gravitational wave background recently observed by a number of different pulsar timing arrays could be due to merging primordial supermassive black hole binaries. We find that for homogeneously distributed primordial black holes this possibility is inconsistent with the strong cosmological and astrophysical constraints on their total abundance. If the distribution exhibits some clustering, however, the merger rate will in general be enhanced, opening the window for a consistent interpretation of the PTA data in terms of merging primordial black holes.

Motivation & Objective

• To assess whether the nHz gravitational wave background observed by pulsar timing arrays (PTAs) could originate from merging primordial supermassive black hole (PBH) binaries.
• To evaluate the consistency of PBH merger scenarios with existing cosmological and astrophysical constraints, particularly on PBH abundance and µ-distortions.
• To explore how spatial clustering of PBHs affects merger rates and the resulting gravitational wave energy density spectrum.
• To determine whether the observed PTA signal amplitude can be matched under realistic PBH formation and clustering models.
• To clarify the role of emission time delays in gravitational wave frequency redshifting and the correct rate formalism for GWB power spectra.

Proposed method

• Uses a frequency-dependent merger rate formalism, where the rate R(tr + τfr) accounts for GW emission at frequency f today, emitted at earlier cosmic times due to redshifting.
• Applies the energy density parameter h²ΩGW(f) derived from the integral of GW power per logarithmic frequency, incorporating the time delay τfr for GW emission from a binary merging at tr.
• Adapts the merger rate formula from [44] for clustered PBHs, including the density contrast δdc and three-body configuration statistics to model binary formation efficiency.
• Incorporates hierarchical merging by summing contributions from multiple merger steps, with mass growth and shorter coalescence times in high-clustering regimes.
• Uses the PBH binary GW spectrum from [43] and computes h²ΩGW(f) via Eq. (2), comparing results using the correct emission-time rate R(tr + τfr) versus the naive merger rate R(tr).
• Imposes constraints from µ-distortions in the cosmic microwave background and PBH abundance limits from astrophysical surveys to test viability.

Experimental results

Research questions

• RQ1 Can the observed nHz gravitational wave background in pulsar timing arrays be explained by merging primordial supermassive black hole binaries?
• RQ2 Is the PBH merger rate sufficient to produce the observed GWB amplitude under homogeneous PBH distribution, given cosmological constraints?
• RQ3 How does spatial clustering of PBHs affect the merger rate and the resulting GW energy density spectrum?
• RQ4 What are the implications of using the correct emission-time rate R(tr + τfr) instead of the merger rate R(tr) for GWB power spectrum predictions?
• RQ5 Can the PBH scenario remain viable if µ-distortion constraints from PBH formation are evaded?

Key findings

• Homogeneously distributed primordial black holes (PBHs) cannot explain the NANOGrav signal, as they fail to produce a continuous or stochastic GWB at the required amplitude.
• For PBHs with masses around 10⁵ M⊙ and fPBH = 1 (100% of dark matter), the signal from homogeneous distributions becomes discontinuous and non-stochastic at high redshifts.
• Clustering of PBHs enhances the local density and merger rate, enabling a consistent explanation of the PTA data if µ-distortion constraints are evaded.
• The correct use of the emission-time rate R(tr + τfr) leads to a significantly different GW energy density spectrum compared to using the merger rate R(tr), with the former showing earlier emission peaks.
• For mPBH = 10⁵ M⊙ and δdc = 1, the model predicts a GWB amplitude consistent with NANOGrav 15-year data when using the correct rate formalism.
• Hierarchical merging in high-clustering environments reduces the required PBH abundance by up to ~10% compared to single-step models, improving viability.