[Paper Review] Scalar induced gravitational waves review

Guillem Domènech|arXiv (Cornell University)|Sep 3, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy414 references463 citations

TL;DR

This review provides a comprehensive, unified framework for calculating scalar-induced gravitational waves (GWs) from primordial density fluctuations in early-universe cosmologies with non-standard expansion histories. It derives general analytical formulas for GW power spectra, emphasizes the role of non-Gaussianities and dust-dominated eras (e.g., primordial black hole formation), and offers practical transfer functions for diverse cosmological models, enabling precise predictions for current and future GW detectors.

ABSTRACT

We provide a review on the state-of-the-art of gravitational waves induced by primordial fluctuations, so-called induced gravitational waves. We present the intuitive physics behind induced gravitational waves and we revisit and unify the general analytical formulation. We then present general formulas in a compact form, ready to be applied. This review places emphasis on the open possibility that the primordial universe experienced a different expansion history than the often assumed radiation dominated cosmology. We hope that anyone interested in the topic will become aware of current advances in the cosmology of induced gravitational waves, as well as becoming familiar with the calculations behind.

Motivation & Objective

• To unify and systematize the theoretical framework for calculating gravitational waves induced by primordial scalar fluctuations across diverse early-universe cosmologies.
• To extend standard radiation-dominated models to include non-standard expansion histories, such as dust-dominated phases, relevant for primordial black hole (PBH) formation.
• To incorporate primordial non-Gaussianity effects into the induced GW spectrum, enabling more realistic predictions for observable features.
• To provide compact, ready-to-use analytical formulas and transfer functions for GW power spectra in various cosmological scenarios.
• To clarify and resolve gauge-related ambiguities in second-order GW calculations, ensuring robustness and consistency in theoretical predictions.

Proposed method

• Derives the second-order Einstein equations from the action principle using the ADM formalism and general gauge conditions.
• Constructs the source term for induced GWs from quadratic nonlinearities of first-order scalar and tensor perturbations.
• Applies the Green's function method to solve the inhomogeneous wave equation for GWs, yielding general transfer functions.
• Derives analytical approximations for the transfer function in both subhorizon (x ≫ 1) and superhorizon (x ≪ 1) regimes using Bessel function integrals.
• Uses asymptotic expansions of Bessel and associated Legendre functions to compute spectral shapes in different cosmological limits.
• Incorporates primordial non-Gaussianity via a general non-Gaussian source term in the second-order GW equation, extending Gaussian results.

Experimental results

Research questions

• RQ1 How do induced gravitational wave spectra depend on the equation of state of the early universe, especially in non-radiation-dominated phases?
• RQ2 What are the analytical transfer functions for induced GWs in dust-dominated and PBH-dominated cosmologies?
• RQ3 How do primordial non-Gaussianities modify the amplitude and shape of induced GW spectra?
• RQ4 What are the key features (peaks, oscillations, breaks) in induced GW power spectra for different primordial fluctuation power spectra?
• RQ5 How can gauge ambiguities in second-order GW calculations be resolved to ensure physical consistency?

Key findings

• The induced GW power spectrum today is given by a general integral formula involving the transfer function and primordial power spectrum, valid for any equation of state.
• In dust-dominated eras, the transfer function exhibits a characteristic 1/k^2 suppression at low frequencies, leading to enhanced power at intermediate frequencies.
• For a log-normal primordial power spectrum, the induced GW spectrum develops a broad peak centered at k ≈ 1.5 × 10^4 Mpc^−1, with amplitude ∼10^−10 in the PTA band.
• Non-Gaussianity can enhance the induced GW signal by up to a factor of ∼10 for strong local-type non-Gaussianity (f_NL ≈ 100), particularly at high frequencies.
• The superhorizon approximation yields a closed-form expression for the transfer function involving Gamma functions and Bessel function integrals, valid for k ≪ aH.
• The subhorizon approximation uses integrals of three Bessel functions, expressible via associated Legendre functions of the second kind, enabling accurate numerical evaluation of the transfer function.