[Paper Review] On the waveform of the scalar induced gravitational waves

Fengge Zhang, Arshad Ali+3 more|arXiv (Cornell University)|Aug 29, 2020

Cosmology and Gravitation Theories77 references4 citations

TL;DR

This paper proposes two parameterizations for scalar power spectra with sharp or broad spikes at small scales to study scalar-induced gravitational waves (SIGWs). It finds that SIGW waveforms closely mirror the scalar power spectrum shape, and away from the spike peak, the relation ΩGW(k) ∝ 𝒫ζ²(k) holds universally, regardless of the power spectrum's functional form, providing a robust tool for probing early-universe physics and inflationary models via SIGW detection.

ABSTRACT

The scalar induced gravitational waves (SIGWs) is a useful tool to probe the physics in the early universe. To study inflationary models with this tool, we need to know how the waveform of SIGWs is related to the shape of the scalar power spectrum. We propose two parameterizations to approximate the scalar power spectrum with either a sharp or a broad spike at small scales, and then use these two parameterizations to study the relation between the shapes of $Ω_{GW}$ and the scalar power spectrum. We find that the waveform of SIGWs has a similar shape to the power spectrum. Away from the peak of the spike, the frequency relation $Ω_{GW}(k)\sim \mathcal{P}_ζ^2(k)$ holds independent of the functional form of the scalar power spectrum. We also give a physical explanation for this general relationship. The general relation is useful for determining the scalar power spectrum and probing inflationary physics with the waveform of SIGWs.

Motivation & Objective

• To understand the relationship between the waveform of scalar-induced gravitational waves (SIGWs) and the shape of the primordial scalar power spectrum.
• To develop parameterizations for scalar power spectra with sharp or broad spikes at small scales to model non-Gaussian inflationary scenarios.
• To determine whether the frequency dependence ΩGW(k) ∝ 𝒫ζ²(k) holds universally, independent of the functional form of the scalar power spectrum.
• To provide a physical explanation for the observed waveform similarity between SIGWs and the scalar power spectrum.
• To enable the use of SIGW waveforms as a probe for reconstructing the primordial scalar power spectrum and constraining inflationary models.

Proposed method

• Two parameterizations are introduced: one for a sharp spike (approaching a δ-function) and another for a broad spike, both modeling the scalar power spectrum 𝒫ζ(k) with tunable peak width and amplitude.
• The SIGW energy density spectrum ΩGW(k) is computed numerically using second-order perturbation theory, accounting for nonlinear mode coupling.
• The analysis compares the shape of ΩGW(k) to the underlying 𝒫ζ(k), focusing on regions near and away from the spike peak.
• The paper derives and tests the relation ΩGW(k) ∝ 𝒫ζ²(k) for k ≫ kp and k ≪ kp, validating it across different power-law and peaked forms of 𝒫ζ(k).
• A physical explanation is provided based on the dominant nonlinear coupling: modes with k̃ ≈ √3/2 k contribute most to ΩGW(k), leading to the square-law scaling.
• The results are verified using numerical fits to the waveforms, with parameters for the SIGW envelope (e.g., ag, bg, cg, eg) derived from the scalar power spectrum parameters.

Experimental results

Research questions

• RQ1 How does the waveform of scalar-induced gravitational waves (SIGWs) relate to the shape of the primordial scalar power spectrum?
• RQ2 Does the relation ΩGW(k) ∝ 𝒫ζ²(k) hold universally across different functional forms of the scalar power spectrum, especially away from the spike peak?
• RQ3 What is the physical origin of the observed similarity between the SIGW waveform and the scalar power spectrum?
• RQ4 How do the parameters of the scalar power spectrum (e.g., peak width, amplitude, power-law indices) affect the resulting SIGW spectrum?
• RQ5 To what extent do nonlinear mode couplings distort the SIGW waveform near the spike, and when does the simple square-law relation break down?

Key findings

• The waveform of SIGWs closely mirrors the shape of the scalar power spectrum, particularly in the broad and sharp spike parameterizations studied.
• Away from the spike peak, the relation ΩGW(k) ∝ 𝒫ζ²(k) holds universally, independent of the functional form of the scalar power spectrum.
• For k > 2kp/√3 or k ≪ kp, the dominant contribution to ΩGW(k) arises from modes with k̃ ≈ √3/2 k, explaining the square-law scaling.
• The amplitude of the SIGW envelope is approximately related to the scalar power spectrum by ag ≈ 2a, cg ≈ 1/(2c), eg ≈ 1/(2e), and bg ≈ 2b + 4.1, with the constant 4.1 arising from Ωr,0 ≈ 10⁻⁴.
• Near the peak (k ≈ kp), the waveform becomes complex and may develop additional bumps due to nonlinear coupling, especially in extreme cases like a δ-function power spectrum.
• The simple square-law relation breaks down for the δ-function case, indicating that the universal scaling ΩGW ∝ 𝒫ζ²(k) is not valid in such singular limits.