Beyond Perturbation Theory in Cosmology

Over many years, our current understanding of the Universe has been extremely relied on perturbation theory (PT) both theoretically and experimentally. There are, however, many situations in cosmology in which the analysis beyond PT is required. In this thesis we study three examples: the resonant decay of gravitational waves (GWs), the dark energy (DE) instabilities induced by GWs, and the tail of the primordial field distribution function. The first two cases are within the context of the Effective Field Theory (EFT) of DE, whereas the last one is within inflation. We first review the construction of the EFT of DE, which is the most general Lagrangian for the scalar and tensor perturbations around the flat FLRW metric. Specifically, this EFT can be mapped to the covariant theories, known as Horndeski and Beyond Horndeski theories. We then study the implications on the dark energy theories coming from the fact that GWs travel with the speed $c_T = 1$ at LIGO/Virgo frequencies. After that, we consider the perturbative decay of GWs into DE fluctuations ($gamma ightarrow pipi$) due to the $ ilde{m}_4^2$ operator. This process is kinematically allowed by the spontaneous breaking of Lorentz invariance. Therefore, having no perturbative decay of gravitons together with $c_T = 1$ at LIGO/Virgo, rules out all quartic and quintic beyond Horndeski theories. As the first non-perturbative regime in this thesis, we study the decay of GWs into DE fluctuations $pi$, taking into account the large occupation numbers of gravitons. When the $m_3^3$ (cubic Horndeski) and $ ilde{m}_4^2$ (beyond Horndeski) operators are present, the GW acts as a classical background for $pi$ and modifies its dynamics. In particular, $pi$ fluctuations are described by a Mathieu equation and feature instability bands that grow exponentially. In the regime of small GW amplitude which corresponds to narrow resonance, we calculate analytically the produced $pi$, its energy and the change of the GW signal. Eventually, the resonance is affected by $pi$ self-interactions in a way that we cannot describe analytically. The fact that $pi$ self-couplings coming from the $m_3^3$ operator become quickly comparable with the resonant term affects the growth of $pi$ so that the bound on $alpha_{ m B}$ remains inconclusive. However, in the case of the $ ilde{m}_4^2$ operator self-interactions can be neglected at least in some regimes. Therefore, our resonant analysis improves the perturbative bounds on $alpha_{ m H}$, ruling out quartic Beyond Horndeski operators. In the second non-perturbative regime we show that $pi$ may become unstable in the presence of a GW background with sufficiently large amplitude. We find that dark-energy fluctuations feature ghost and/or gradient instabilities for GW amplitudes that are produced by typical binary systems. Taking into account the populations of binary systems, we conclude that the instability is triggered in the whole Universe for $|alpha_{ m B}| gtrsim 10^{-2}$, i.e. when the modification of gravity is sizable. The fate of the instability and the subsequent time-evolution of the system depend on the UV completion, so that the theory may end up in a state very different from the original one. In conclusion, the only dark-energy theories with sizable cosmological effects that avoid these problems are $k$-essence models, with a possible conformal coupling with matter. In the second part of the thesis we consider physics of inflation. Inflationary perturbations are approximately Gaussian and deviations from Gaussianity are usually calculated using in-in perturbation theory. This method, however, fails for unlikely events on the tail of the probability distribution: in this regime non-Gaussianities are important and perturbation theory breaks down for $|zeta| gtrsim |f_{ m NL}|^{-1}$. We then show that this regime is amenable to a semiclassical treatment, $hbar ightarrow 0$. In this limit the wavefunction of the Universe can be calculated in saddle-point, corresponding to a resummation of all the tree-level Witten diagrams. The saddle can be found by solving numerically the classical (Euclidean) non-linear equations of motion, with prescribed boundary conditions. We apply these ideas to a model with an inflaton self-interaction $propto lambda dot{zeta}^4$. Numerical and analytical methods show that the tail of the probability distribution of $zeta$ goes as $exp(-lambda^{1/4}zeta^{3/2})$, with a clear non-perturbative dependence on the coupling. Our results are relevant for the calculation of the abundance of primordial black holes.