Scalar-Induced Gravitational Waves in the Multi-Messenger Era: From Fundamental Physics in the Early Universe to PBH Evolution

Among the various types of gravitational waves that can arise in the cosmos, some are generated by the interactions of density fluctuations, or scalar perturbations, in the early universe. These waves, known as scalar-induced gravitational waves (SIGWs), provide a unique window into the universe’s first moments, revealing information about both its initial conditions and the fundamental laws of gravity. Unlike gravitational waves produced by astrophysical events such as merging black holes or neutron stars, SIGWs are sourced by primordial density variations as the universe evolves, carrying imprints of the physics that shaped the early cosmos. This thesis develops a comprehensive theoretical framework to study the generation, evolution, and observable properties of SIGWs. The analysis begins within the standard framework of general relativity, where the predicted gravitational wave signals exhibit a characteristic peak corresponding to the scale of horizon re-entry of the density fluctuations. This spectrum falls within the sensitivity range of current and planned gravitational wave detectors, including ground-based interferometers, space-based observatories, and pulsar timing arrays, highlighting the observational relevance of the study. One important theoretical result is the identification of a subtle effect in the generation of SIGWs that is usually ignored in simpler models. In standard general relativity, this effect vanishes, but in certain alternative theories of gravity—such as f(R) gravity—it can be nonzero, slightly altering the predicted gravitational wave signal. To study this, the analysis focuses on a specific f(R) model and computes how the changes in the underlying gravitational dynamics affect the SIGWs. The resulting gravitational wave spectrum shows a modest reduction at low frequencies, while the overall amplitude and shape remain very similar to the predictions of general relativity. This indicates that, although alternative gravity models can introduce small corrections, their impact on observable SIGWs is subtle and would not dramatically change what detectors are likely to see. The thesis also explores SIGW generation during non-standard cosmological phases, such as an early matter-dominated era potentially driven by primordial black holes. These scenarios reveal additional features in the gravitational wave spectrum and suggest novel ways to connect SIGWs to the formation of primordial black holes and the dynamics of the early universe. Furthermore, this thesis explores how the predicted SIGWs can depend on the choice of coordinate system, or gauge, used to describe spacetime. Calculations in common gauges—including Poisson, synchronous, and uniform curvature—show that while the predictions generally agree on smaller scales, noticeable differences can appear on the largest scales. This highlights the importance of developing a gauge-independent approach, ensuring that the predicted gravitational wave signals reflect real physical effects rather than artifacts of the mathematical framework. Overall, this research advances the theoretical understanding of SIGWs, clarifies the impact of alternative gravitational theories on their signals, and establishes their potential as tools for probing fundamental processes in the early universe. These results provide a solid foundation for future observational efforts, connecting theory to upcoming gravitational wave measurements and offering a pathway to test both the physics of the early universe and the nature of gravity itself.