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A universal scaling law for gravitational waves induced during inflation

Martin Teuscher, Ruth Durrer, Killian Martineau, Aurélien Barrau

TL;DR

This work establishes a universal scaling law for gravitational waves induced during inflation by arbitrary amplified source fields. By modeling the source as a Gaussian, quadratic anisotropic stress built from a field with a power-law spectrum, the authors derive a closed-form relation for the tensor tilt: $n_T=2(n-2)(q+1)$ with $q=2/(1+3w)$, which reduces to a near scale-invariant spectrum in slow-roll inflation ($w\approx -1$) for any non-red source ($n\ge 0$). The predictions show that for $n=2$ one obtains exact scale invariance, while other values yield different tilts depending on $w$; infrared issues arise for $n<0$ and are controlled by a cutoff. The framework is illustrated with scalar-induced and gauge-field-induced GWs, highlighting how the source spectrum shapes the induced SGWB and providing a tool to distinguish sourced GWs from vacuum-generated ones. Overall, the paper offers a concise, widely applicable scaling law that links the microphysics of amplified fields during inflation to observable stochastic gravitational waves.

Abstract

We consider the stochastic gravitational wave background induced by arbitrary source fields that are amplified during cosmological inflation. The associated tensor spectral index is shown to be given, under minimal assumptions, by a simple formula easy to use in most situations of accelerated expansion. For slow-roll inflation, the induced spectrum is nearly scale-invariant, with an index that slightly deviates from the standard outcome of vacuum generated gravitational waves. Remarkably, we demonstrate that scale invariance remains true regardless of the original spectrum of the source.

A universal scaling law for gravitational waves induced during inflation

TL;DR

This work establishes a universal scaling law for gravitational waves induced during inflation by arbitrary amplified source fields. By modeling the source as a Gaussian, quadratic anisotropic stress built from a field with a power-law spectrum, the authors derive a closed-form relation for the tensor tilt: with , which reduces to a near scale-invariant spectrum in slow-roll inflation () for any non-red source (). The predictions show that for one obtains exact scale invariance, while other values yield different tilts depending on ; infrared issues arise for and are controlled by a cutoff. The framework is illustrated with scalar-induced and gauge-field-induced GWs, highlighting how the source spectrum shapes the induced SGWB and providing a tool to distinguish sourced GWs from vacuum-generated ones. Overall, the paper offers a concise, widely applicable scaling law that links the microphysics of amplified fields during inflation to observable stochastic gravitational waves.

Abstract

We consider the stochastic gravitational wave background induced by arbitrary source fields that are amplified during cosmological inflation. The associated tensor spectral index is shown to be given, under minimal assumptions, by a simple formula easy to use in most situations of accelerated expansion. For slow-roll inflation, the induced spectrum is nearly scale-invariant, with an index that slightly deviates from the standard outcome of vacuum generated gravitational waves. Remarkably, we demonstrate that scale invariance remains true regardless of the original spectrum of the source.

Paper Structure

This paper contains 7 sections, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sourced stochastic gravitational waves
  3. General ansatz for the anisotropic stress tensor
  4. Universal scaling of the tensor power spectrum
  5. Applications
  6. Conclusion
  7. Proof that is a positive kernel

Figures (1)

  • Figure 1: GW spectral index $n_T$ from Eq. \ref{['eq:nT-main-formula']}. For proper visualization, all $n_T<-4$ are displayed with the same color. Dotted lines indicate $n_T=-1,0,1$. Solid (resp. dashed) black lines mark out regions where the inequality $2n(1+w)\geqslant 5+7w$ (resp. $n\geqslant3(1+w)$) fails. As an example, sources corresponding to $n=2$ lead to a nearly scale-invariant SGWB for all $w<-1/3$. For completeness we have included the region $w<-1$ where our analysis holds, although this case is somewhat exotic.