[Paper Review] Complete analysis of the background and anisotropies of scalar-induced gravitational waves: primordial non-Gaussianity $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$ considered

Junpeng Li, Sai Wang+2 more|arXiv (Cornell University)|Sep 14, 2023

Pulsars and Gravitational Waves ResearchPhysics and Astronomy201 references3 citations

TL;DR

This paper presents the first complete analysis of scalar-induced gravitational waves (SIGWs) by simultaneously incorporating local-type primordial non-Gaussianities $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$, using a novel Feynman-like diagrammatic technique to derive semi-analytic formulas for both the energy-density fraction spectrum and the angular power spectrum. It reveals that SIGW anisotropies follow a $\tilde{C}_{\ell} \propto [\ell(\ell+1)]^{-1}$ dependence and can reach $\tilde{C}_{\ell} \sim 10^{-3}$, offering a powerful probe to break parameter degeneracies and test early-universe physics via future space-based detectors and pulsar timing arrays.

ABSTRACT

Investigation of primordial non-Gaussianity holds immense importance in testing the inflation paradigm and shedding light on the physics of the early Universe. In this study, we conduct the complete analysis of scalar-induced gravitational waves (SIGWs) by incorporating the local-type non-Gaussianity $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$. We develop Feynman-like diagrammatic technique and derive semi-analytic formulas for both the energy-density fraction spectrum and the angular power spectrum. For the energy-density fraction spectrum, we analyze all the relevant Feynman-like diagrams, determining their contributions to the spectrum in an order-by-order fashion. As for the angular power spectrum, our focus lies on the initial inhomogeneities, giving rise to anisotropies in SIGWs, that arise from the coupling between short- and long-wavelength modes due to primordial non-Gaussianity. Our analysis reveals that this spectrum exhibits a typical multipole dependence, characterized by $ ilde{C}_{\ell}\propto[\ell(\ell+1)]^{-1}$, which plays a crucial role in distinguishing between different sources of gravitational waves. Depending on model parameters, significant anisotropies can be achieved. We also show that the degeneracies in model parameters can be broken. The findings of our study underscore the angular power spectrum as a robust probe for investigating primordial non-Gaussianity and the physics of the early Universe. Moreover, our theoretical predictions can be tested using space-borne gravitational-wave detectors and pulsar timing arrays.

Motivation & Objective

• To develop a comprehensive theoretical framework for scalar-induced gravitational waves (SIGWs) that includes both $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$ primordial non-Gaussianities.
• To resolve the limitations of prior studies that considered only one non-Gaussianity parameter at a time.
• To derive semi-analytic expressions for the energy-density fraction spectrum and the angular power spectrum of SIGWs using a systematic diagrammatic approach.
• To identify distinct spectral signatures—particularly the $\tilde{C}_{\ell} \propto [\ell(\ell+1)]^{-1}$ dependence—that can distinguish SIGWs from other gravitational wave sources.
• To demonstrate that the angular power spectrum enables breaking of degeneracies in model parameters, enhancing its utility for probing early-universe physics.

Proposed method

• Developed a Feynman-like diagrammatic technique to systematically compute all relevant contributions to the SIGW energy-density fraction spectrum in an order-by-order manner.
• Derived semi-analytic formulas for the energy-density fraction spectrum by evaluating all relevant diagrams arising from nonlinear interactions of curvature perturbations.
• Formulated the angular power spectrum by analyzing initial inhomogeneities induced by coupling between short- and long-wavelength modes due to primordial non-Gaussianities.
• Used the line-of-sight solution to the Boltzmann equation to express the present density contrast of SIGWs in terms of the primordial potential $\Phi$ and perturbations $\Gamma$, incorporating both Sachs-Wolfe and integrated Sachs-Wolfe effects.
• Expressed the density contrast $\delta_{\mathrm{gw},0}$ in configuration space using the background $\bar{\Omega}_{\mathrm{gw}}$ and spectral index $n_{\mathrm{gw}}(\nu)$, enabling quantitative predictions.
• Applied the formalism to compute the angular power spectrum $\tilde{C}_{\ell}$, showing its characteristic $[\ell(\ell+1)]^{-1}$ scaling and quantifying anisotropy amplitudes up to $\sim 10^{-3}$.

Experimental results

Research questions

• RQ1 How do simultaneous contributions from $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$ non-Gaussianities affect the energy-density fraction spectrum of scalar-induced gravitational waves?
• RQ2 What is the angular power spectrum of SIGWs induced by primordial non-Gaussianities, and what spectral signature does it exhibit?
• RQ3 Can the angular anisotropy of SIGWs break degeneracies in model parameters that are otherwise indistinguishable in the isotropic power spectrum?
• RQ4 What is the maximum amplitude of anisotropy achievable in the SIGW angular power spectrum under realistic early-universe models?
• RQ5 How can future space-borne and ground-based gravitational-wave detectors exploit the predicted anisotropy to probe primordial non-Gaussianity?

Key findings

• The angular power spectrum of scalar-induced gravitational waves exhibits a characteristic $\tilde{C}_{\ell} \propto [\ell(\ell+1)]^{-1}$ dependence, which is a distinctive signature of primordial non-Gaussianity-induced anisotropies.
• Anisotropies in the SIGW background can reach amplitudes of $\tilde{C}_{\ell} \sim 10^{-3}$, depending on model parameters, making them potentially detectable with future observatories.
• The semi-analytic framework based on a Feynman-like diagrammatic technique successfully captures all order-by-order contributions to the energy-density fraction spectrum, enabling precise predictions.
• The inclusion of both $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$ allows for breaking of degeneracies in model parameters that would otherwise be indistinguishable in isotropic analyses.
• The theoretical predictions are directly testable with upcoming space-based missions like LISA and pulsar timing arrays such as NANOGrav, offering a new probe into early-universe physics.