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Primordial Gravitational Wave Background as a Probe of the Primordial Black Holes

Utkarsh Kumar

TL;DR

The paper investigates whether primordial gravitational waves (PGWs) can seed primordial black holes (PBHs) through second-order density perturbations, while remaining compatible with the stochastic GW background detected by pulsar timing arrays. It develops a framework for tensor-induced density perturbations, analyzes both power-law and log-normal tensor spectra, and performs a joint NG15+EPTA2 Bayesian analysis to constrain tensor amplitudes under PBH and $N_{ m eff}$ constraints; it also connects the log-normal shape to an inflationary axion-gauge model, mapping LN template parameters to fundamental parameters. The results show that the PTA interpretation of the SGWB can correspond to PBH production in the mass range $[10^{-12},10^{-3}] M_{}$, with PL scenarios peaking near $M_{ m PBH}\sim 10^{-3} M_{}$ and modest reheating temperatures, while LN scenarios allow broader PBH mass windows. This work provides a cohesive framework linking PTA GW observations to PBH formation, constraining tensor amplitudes with PBH data and predicting SGWB signals within reach of future detectors like SKA, LISA, and DECIGO, with potential implications for PBH dark matter or SMBH seeding.

Abstract

We study the formation of primordial black holes (PBHs) from the collapse of density perturbations induced by primordial gravitational waves (PGWs). The PGWs' interpretation of the stochastic gravitational wave background (SGWB) detected by the Pulsar Timing Array (PTA) corresponds to PBHs formation in the mass range $[10^{-12}-10^{-3}] M_{\odot}$. Importantly, our analysis shows that PGWs' interpretation of recent PTA data remains viable, as it does not lead to PBH overproduction. We derive the amplitude of PGWs by leveraging existing constraints on the PBH abundance across a wide mass range. Notably, these constrained amplitudes predict SGWB signals that would be detectable by future gravitational wave observatories.

Primordial Gravitational Wave Background as a Probe of the Primordial Black Holes

TL;DR

The paper investigates whether primordial gravitational waves (PGWs) can seed primordial black holes (PBHs) through second-order density perturbations, while remaining compatible with the stochastic GW background detected by pulsar timing arrays. It develops a framework for tensor-induced density perturbations, analyzes both power-law and log-normal tensor spectra, and performs a joint NG15+EPTA2 Bayesian analysis to constrain tensor amplitudes under PBH and constraints; it also connects the log-normal shape to an inflationary axion-gauge model, mapping LN template parameters to fundamental parameters. The results show that the PTA interpretation of the SGWB can correspond to PBH production in the mass range , with PL scenarios peaking near and modest reheating temperatures, while LN scenarios allow broader PBH mass windows. This work provides a cohesive framework linking PTA GW observations to PBH formation, constraining tensor amplitudes with PBH data and predicting SGWB signals within reach of future detectors like SKA, LISA, and DECIGO, with potential implications for PBH dark matter or SMBH seeding.

Abstract

We study the formation of primordial black holes (PBHs) from the collapse of density perturbations induced by primordial gravitational waves (PGWs). The PGWs' interpretation of the stochastic gravitational wave background (SGWB) detected by the Pulsar Timing Array (PTA) corresponds to PBHs formation in the mass range . Importantly, our analysis shows that PGWs' interpretation of recent PTA data remains viable, as it does not lead to PBH overproduction. We derive the amplitude of PGWs by leveraging existing constraints on the PBH abundance across a wide mass range. Notably, these constrained amplitudes predict SGWB signals that would be detectable by future gravitational wave observatories.

Paper Structure

This paper contains 5 sections, 34 equations, 5 figures, 2 tables.

Table of Contents

  1. Evolution of density power spectrum
  2. Power-Law Spectrum:
  3. Log-Normal Spectrum:
  4. Results for PGW energy spectrum from NANOGrav and PTA
  5. Derivation of Log-Normal Power spectrum from a Inflationary model

Figures (5)

  • Figure 1: Left panel: Two dimensional contour plot for inferred posterior values of $n_t$ and present day GW energy density evaluated at reference frequency $f_{\rm yr}$ obtained from the joint analysis of NG-15 and EPTA-DR2 datasets. Black curves correspond to the tensor-to-scalar ratio as shown. We include the allowed values of reheating temperatures that satisfy the $N_{\rm eff}$ bound (shown in dotted lines) and the corresponding viable parameter space (on the left) to avoid the overproduction of PBHs, given that $f_{\rm PBH} \geq 1$. Right panel: same as left panel but for the LN model. Again, $N_{\rm eff} = 0.3$ constraint given in black dashed line. The parameter space lying below blue ($\Delta$ = 1.68), green ($\Delta$ = 1.42), and orange ($\Delta$ = 1.16) respectively survives the $f_{\rm pbh} \geq 1$.
  • Figure 2: Left panel: GW power spectra obtained from PL and LN spectra in orange and blue respectively. Shown are the violin plots in green and red for EPTA2 and NG15 measurements respectively, and the sensitivity curves for SKA, $\mu$Ares, LISA, and DECIGO in colors indicated. We also have included the updated SKA sensitivity curve using the SGWB as a foreground Babak:2024yhuCecchini:2025oks in brown. Right panel: Mass function resulting from the PL (orange) and LN spectrum for the values of parameters described in the text. The PBHs abundance constrained by variety of observations including PBH evaporationKatz:2018zrnMontero-Camacho:2019jteArbey:2019vqxBoudaud:2018hqbDeRocco:2019fjqLaha:2020ivkCarr:2009jmBallesteros:2019exrLaha:2019ssqPoulter:2019oooKatz:2018zrnSaha:2024iesSaha:2021pqf, gravitational lensingNiikura:2019kqiNiikura:2017zjdBarnacka:2012bmWilkinson:2001vvZumalacarregui:2017qqdOguri:2017ockMacho:2000nvdEROS-2:2006ryyGriest:2013aaaGriest:2013esaSmyth:2019whb, dynamical effectsGraham:2015apaCapela:2013yfCapela:2012jzCarr:2018ridMonroy-Rodriguez:2014ulaMUSE:2020qboKoushiappas:2017chw are also shown. We have chosen $\delta_c = 0.42$ for our analysis.
  • Figure S.1: Evolution of total density power spectrum for Power-law type spectrum. We have numerically integrated \ref{['eq:anaPLPSD']} using the expression given in \ref{['eq:PL']} (see main text). Due to the Heaviside function, we note that slope of the resulting expression differs from exact result slighting around its peak. We have fixed the scalar power spectrum parameters $A_s = 2.1 \times 10^{-9}\,,\, k_s = 0.05 \,\text{Mpc}^{-1}\,,$ and $n_s = 0.96$ according to CMB measurements Planck:2018vyg. We fix the tensor spectral index $n_t = 1.0$ and vary to tensor-to-scalar ration $r \in [10^{-5}\,,\,10^{-2}\,]$ in left panel and vary $n_t \in [1.8\,,\,2.2]$ and $r = 10^{-12}$ in right panel.
  • Figure S.2: Evolution of total density power spectrum for Log-Normal type spectrum. We show the impact of the spectrum width $\Delta$ and peak frequency $k_p$ on the total power spectrum in left and right panel respectively. Note that for $\Delta = 0.5$ the spectrum shows the similar slope as in \ref{['eq:IRLN']}.
  • Figure S.3: Triangle plot for posterior distribution for PL (top) and LN (bottom) spectrum. Note that $\Omega_{\rm GW}(f_{\rm yr})$ is a derived quantity derived using the posterior values for $\log_{10} r$ and $n_t$. The brown and cyan points corresponds to PBHs with mass function dominated at $10^{-12} M_{\odot}$ and $10^{-10} M_{\odot}$ shown in main text.