Effective Field Theory of Dark Energy

This thesis has been written at the Lorentz Instituut of Leiden within a research project whose aim was the implementation of a new branch within the EFTCAMB code developed in [1][2][3]. The code is designed as a patch to the public Einstein-Boltzman solver CAMB [4][5] and it is based on the Effective Field Thoery approach [6][7]. It represents one of the most important code on the market for the analysis of Modified Theories of Gravity. Once a model is chosen, the code output spectra for all the observables of interest, and does so under a powerful check for the stability of the given model. The code can work in two different modes: the Pure EFT and the Full Mapping one. In the first one the code performs a model independent analysis starting from user specified parametrizations for the background evolution; while in the latter case a specific model is fully implemented with its own background. Our work is going to explore this last possibility. Along this thesis, we will describe the whole pipeline for the implementation of a specific model of f(R)-theory: the well known Hu-Sawicki f(R) [8]. In order to give a self-contained description, we will discuss and review all the theoretical tools that are needed in order to fully deal with the topic of the Effective Field Theory of Dark Energy, the title of this dissertation. The thesis is organized as follows. In the first chapter, we will introduce the reader to some basic aspects of General Relativity since this is the theory we will try to modify consistently. We will derive the equations of motion for a Friedman Robertson Walker metric providing some fundamental tools concerning the homogeneous background cosmology. After that, we will describe the issue of cosmic acceleration pointing to some observational results related to it. We will discuss how these challenge the Theory of General Relativity and lead theoretical cosmologists to explore alternative approaches including modification of Einstein’s Theory. In particular we will focus on one possible way of modifying gravity: f(R)-theories. Since the most important bounds on new models will come from the analysis of spectra and being them the main output of the code too, we will start the second chapter of this work making the reader familiar with these important cosmological observables. We will not just describe them, but we will also provide some details about calculations to be performed to get them. The Boltzmann Equations for different species filling the Universe will be shown and the Einstein perturbed ones as well. In particular these last ones, will be drawn within the EFT formalism, the one used by EFTCAMB. This will give the chance to perform a deep analysis of this formalism going through the Effective Lagrangian where all the information about background evolution and perturbations is stored for a generic theory of modified gravity once mapped into it. The third chapter will be devoted to a very detailed description of the scenario we will test: the Hu-Sawicki model. We will provide the Modified Friedman Equations for this model and we will solve them numerically. This is the original and most important part of our work. As a matter of fact, in order to properly implement the model we chose within our framework, we need to provide EFTCAMB the exact background evolution for this paradigm. The relevance of our approach is also related to the fact that other publicly available codes for testing Theories of Modified Gravity evolve perturbations assuming ΛCDM background [9][10][11]. In the end, in the fourth chapter we will provide all the information related to the new branch of the code. After the theoretical analysis of the previous chapters, we will deal at this point with practical issues related to the coding within this new procedure: strategies will be developed according to the accuracy we need to provide in our results compared with the forecasting for the next generations’ surveys [12]. In the end we will show spectra obtained from EFTCAMB via the implementation in it of the code developed. We will compare the results from the Hu-Sawicki model with those from a General Realitivistic scenario and the modifications to gravity will be finally disclosed.