[Paper Review] Implications of Pulsar Timing Array Data for Scalar-Induced Gravitational Waves and Primordial Black Holes: Primordial Non-Gaussianity $f_{\mathrm{NL}}$ Considered

Sai Wang, Zhichao Zhao+2 more|arXiv (Cornell University)|Jul 2, 2023

Pulsars and Gravitational Waves ResearchPhysics and Astronomy84 references19 citations

TL;DR

The paper analyzes pulsar timing array data under the scalar-induced gravitational wave framework with primordial non-Gaussianity, constraining $f_{ m NL}$ and the primordial black hole mass range. It also studies the angular power spectrum of SIGWs to break parameter degeneracies and assess PBH viability.

ABSTRACT

Multiple pulsar-timing-array collaborations have reported strong evidence for the existence of a gravitational-wave background. We study physical implications of this signal for cosmology, assuming that it is attributed to scalar-induced gravitational waves. By incorporating primordial non-Gaussianity $f_{\mathrm{NL}}$, we specifically examine the nature of primordial curvature perturbations and primordial black holes. We find that the signal allows for a primordial non-Gaussianity $f_{\mathrm{NL}}$ in the range of $-4.1\lesssim f_{\mathrm{NL}} \lesssim 4.1$ (68\% confidence intervals) and a mass range for primordial black holes $m_{\mathrm{pbh}}$ spanning from $\sim10^{-5}M_{\odot}$ to $\sim10^{-2}M_{\odot}$. Furthermore, we find that the signal favors a negative non-Gaussianity, which can suppress the abundance of primordial black holes. We also demonstrate that the anisotropies of scalar-induced gravitational waves serve as a powerful tool to probe the non-Gaussianity $f_{\mathrm{NL}}$. We conduct a comprehensive analysis of the angular power spectrum within the nano-Hertz band. Looking ahead, we anticipate that future projects, such as the Square Kilometre Array, will have the potential to measure these anisotropies and provide further insights into the primordial universe.

Motivation & Objective

• Assess the implications of NG15 PTA data for the primordial curvature perturbations and PBH formation when SIGWs are the cosmological source.
• Incorporate local-type primordial non-Gaussianity characterized by $f_{NL}$ into the SIGW energy-density framework.
• Infer constraints on the curvature power spectrum parameters and map them to PBH abundance and mass ranges.
• Demonstrate that SIGW anisotropies can break degeneracies and provide a potential observable for $f_{NL}$ in the PTA band.

Proposed method

• Express the primordial curvature perturbation as $\,\zeta(\mathbf{q})=\zeta_{g}(\mathbf{q})+\tfrac{3}{5}f_{NL}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}\zeta_{g}(\mathbf{k})\zeta_{g}(\mathbf{q}-\mathbf{k})$ and define $F_{NL}=\tfrac{3}{5}f_{NL}$.
• Decompose the SIGW energy-density spectrum into three contributions $ar{\Omega}_{gw}^{(0)}$, $\bar{\Omega}_{gw}^{(1)}$, $\bar{\Omega}_{gw}^{(2)}$ proportional to $A_S^2 (A_S F_{NL}^2)^n$ with $n=0,1,2$ (Gaussian and NG terms).
• Adopt a log-normal primordial power spectrum $\Delta_g^2(q)=\frac{A_S}{\sqrt{2\pi\sigma^2}}\exp[-\ln^2(q/q_*)/(2\sigma^2)]$ and relate wavenumber to frequency via $q=2\pi u$.
• Perform Bayesian inference (NG15 data) to constrain $F_{NL}$, $A_S$, $\sigma$, and $\nu_*$.
• Translate constraints on the power spectrum to PBH mass via $m_{PBH}/M_\odot \simeq m_H/0.31M_\odot \simeq (\nu_*/5\ \text{nHz})^{-2}$ and compute PBH abundance $f_{PBH}$ using $f_{PBH}\simeq 2.5\times10^8 \beta (g_{*,\rho}(T_f)/10.75)^{-1/4} (m_{PBH}/M_\odot)^{-1/2}$.
• Analyze angular power spectrum $\tilde{C}_\ell(\nu)$ and $C_\ell(\nu)$ for SIGW anisotropies, including Sachs–Wolfe contributions, to assess sensitivity to $f_{NL}$.

Experimental results

Research questions

• RQ1 What constraints do NG15 PTA data place on the local-type primordial non-Gaussianity parameter $f_{NL}$ when SIGW is the source of the GWB?
• RQ2 How do the primordial power-spectrum parameters ($A_S$, $\sigma$, $\nu_*$) translate into PBH mass ranges and abundances in light of NG15 data?
• RQ3 Can the anisotropies of scalar-induced GWs in the PTA band break degeneracies, especially the sign degeneracy of $f_{NL}$?
• RQ4 What are the prospects for future measurements, such as SKA, to detect SIGW anisotropies and constrain early-universe physics?
• RQ5 Do negative $f_{NL}$ values alleviate PBH overproduction within the inferred parameter space?

Key findings

• NG15 data constrain $|f_{NL}|$ to be less than about 4.1 at 68% CL.
• The inferred PBH mass range is roughly $m_{PBH}\sim 10^{-5}$ to $10^{-2}\ M_\odot$, with PBH abundance overproduced for sizable positive $f_{NL}$.
• A negative $f_{NL}$ can suppress PBH abundance and potentially prevent PBH overproduction.
• The angular power spectrum of SIGWs in the nano-Hertz band can break sign degeneracies of $f_{NL}$ and other parameter degeneracies.
• Future SKA-level measurements could measure SIGW anisotropies and provide tighter constraints on primordial non-Gaussianity.
• The study emphasizes the importance of considering both the energy-density spectrum and the angular power spectrum to infer early-universe physics.