Spontaneously Broken Symmetries: Positivity Bounds beyond Lorentz Invariance and Cosmological solutions of dRGT Massive Gravity

Effective Field Theories (EFTs) offer a systematic framework for describing low-energy phenomena without requiring full knowledge of the underlying ultraviolet (UV) completion. In Lorentz-invariant settings, general principles such as causality, unitarity, and locality ensure that scattering amplitudes are analytic functions of the Mandelstam invariants, enabling dispersion relations that link infrared EFT coefficients to ultraviolet physics. This connection leads to powerful positivity bounds that sharply constrain the space of consistent low-energy theories. While this approach has been extensively developed for Lorentz-invariant systems, many physical contexts of interest, ranging from relativistic field theories at finite density to cosmological backgrounds, feature spontaneous breaking of boosts. In such cases, the standard S-matrix framework faces conceptual challenges: the altered analytic structure of amplitudes may obstruct direct application of positivity arguments, and the link between UV and IR physics becomes more subtle. In this thesis, we address these challenges by pursuing two complementary strategies. First, we investigate the fate of the S-matrix in a simple, fully UV-complete Lorentz violating model: the complex λ|Φ|^4 theory at finite charge density. This model allows explicit computation of Goldstone scattering amplitudes, revealing new non-analytic features tied to boost breaking and highlighting the limitations of standard positivity methods in such settings. Second, motivated by these limitations, we turn to observables whose analyticity properties are still clearly readable without Lorentz invariance: retarded correlation functions. We analyze their analytic structure and derive multidimensional dispersion relations for their Fourier transform G(ω, k) and for the function F(ω, k) = ωG(ω, k). Combining causality, unitarity, and passivity, we prove strict positivity of Im F in its domain of analyticity, both with and without rotational invariance. These properties lead to new constraints on response functions in passive systems and imply, in particular, that ImG cannot have compact support in frequency–momentum space. Finally, as a complementary application, we extend symmetry-based classification methods to identify admissible flat FLRW cosmologies in dRGT massive gravity, imposing background symmetries directly on the theory’s building blocks. This approach yields novel perspectives on self-accelerating and fluid-like branches, as well as insights into their perturbative stability. Together, these results provide new conceptual and technical tools for constraining Lorentz-violating EFTs, bridging S-matrix and correlator-based approaches, and applying them to modified gravity scenarios of cosmological relevance.