Induced gravitational waves -- beyond linear cosmological perturbation theory

TL;DR

This work systematically extends the theory of induced gravitational waves beyond linear perturbation theory by including linear tensor perturbations as additional second order sources, unveiling two new classes of iGWs: scalar–tensor and tensor–tensor induced GWs. It demonstrates that scalar–tensor iGWs can dominate when first order tensor perturbations are enhanced, while tensor–tensor iGWs remain subdominant, and it identifies an unphysical UV enhancement in the scalar–tensor sector for non sharply peaked scalar spectra. To address this, the study investigates third order induced GWs and their correlation with primordial GWs, finding that third order terms suppress the total signal but do not cancel the UV issue, motivating further theoretical work. The analysis also explores how primordial scalar non Gaussianity affects scalar–tensor iGWs and compares Gaussian and non Gaussian contributions to assess potential observational signatures in upcoming GW surveys. Overall, the results provide detailed predictions for the stochastic GW background in scenarios with PBH formation, highlighting key signatures and cautions related to gauge choices, UV behavior, and non Gaussianity that impact the interpretation of future detections.

Abstract

This thesis focuses on gravitational waves (GWs) that arise beyond linear order in cosmological perturbation theory. In recent years, scalar-induced GWs have attracted significant attention because they may serve as the observational signature of primordial black holes (PBHs) formed in the early universe. The formation of PBHs requires large density perturbations, which can naturally emerge in some models of inflation. When these large density fluctuations couple, they act as a source for scalar-induced GWs at second order. In this work, we extend the existing formalism by including linear tensor fluctuations as an additional source term. This gives rise to two new classes of second-order GWs: those sourced by scalar-tensor couplings (scalar-tensor induced GWs) and those quadratic in tensor modes (tensor-tensor induced GWs). We find that the scalar-tensor contribution becomes significant if first-order tensor modes are enhanced, whilst the tensor-tensor contribution remains subdominant. Moreover, we demonstrate that the spectrum of scalar-tensor induced GWs exhibits an unphysical enhancement in the UV limit when the primordial scalar power spectrum is insufficiently peaked. To investigate whether this can be resolved, we study third-order induced GWs and their correlation with primordial GWs. We find that this new contribution suppresses the overall signal but does not cancel the unphysical enhancement. Possible explanations for this behaviour are discussed and left for future work. Finally, we explore the effect of primordial scalar non-Gaussianity on the spectrum of scalar-tensor induced GWs, building on previous results showing its impact on scalar-induced GWs.

Figures (29)

• Figure 1: Plot of the characteristic strain against frequency. We have plotted the sensitivity curves of current ground-based detectors (sensitive to higher frequencies), i.e. the LVK collaboration. Future ground-based detectors include the Cosmic Explorer cosmicexplorer and the Einstein Telescope einsteintelecope. At intermediate frequencies, future space-based detectors include the Laser Interferometer Space Antenna (LISA) LISA:2017pwj, the DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) decigo and the Big Bang Observer (BBO) bboCrowder:2005nr. At lower frequencies, PTA experiments detect GWs by monitoring the arrival times of radio pulses from many millisecond pulsars and looking for tiny, correlated deviations across the sky caused by passing GWs between Earth and the pulsars. Shaded areas correspond to the first GW detection from two black hole mergers (red) and the stochastic background which would come from unresolved sources (orange). Credit: This plot was generated via the http://gwplotter.com/ (the website was last accessed on the 09/09/2025), see Ref. Moore:2014lga for details.
• Figure 2: Spectral density of GWs against frequency. The spectral density is a measure of how much energy GWs carry. Some typical cosmological sources of GWs are plotted with different detectors also represented. We note that the grey dashed line corresponds to GWs generated during inflation, and it assumes we can extrapolate the Planck data Planck:2018jri to all scales, whilst the cyan line represents GWs produced in a different model of inflation (axion inflation) Garcia-Bellido:2016dkw. We further note that the authors have included at very low frequencies the sensitivity of the Planck Planck:overview and the future LiteBIRD telescope LiteBIRD:2022cnt: GWs interact with the CMB by imprinting a distinct curl-like polarisation pattern, known as B-modes Kamionkowski:2015yta. Credit: This plot was taken from Ref. LISACosmologyWorkingGroup:2022jok.
• Figure 3: (Left) Plot of density (normalised to the critical density, introduced in the next sub-section) against the scale factor for different values of $w$. (Right) How the scale factor evolves throughout the history of our universe, normalised such that $a(t_0)=1$, where $t_0$ is the age of our universe. We have labelled on the two plots $a_{eq}$ and $t_{eq}$, which respectively correspond to the value of the scale factor and time at the time of radiation-matter equality. For both plots, we use the values in Eq. \ref{['omegaLCDM']}.
• Figure 4: Plot of the transfer function for tensor modes in Eq. \ref{['tensortransfer']} against $x=k\eta$, normalised by their initial value. We have plotted the behaviour of tensor modes in an RD universe (blue line), MD universe (orange line) and the $x=1$ line (dashed, faded black). Modes are constant and around $x\approx 1$ start to decay and oscillate.
• Figure 5: Plot of the transfer functions for scalar modes against $x=k\eta$, normalised by their initial value. We have plotted the behaviour of scalar modes in an RD universe (blue line), MD universe (orange line) and the $x=1$ line (dashed, faded black). Modes are constant and around $x\approx 1$ start to decay and oscillate only in an RD universe.