Estensione di metodi approssimati per la generazione di cataloghi di aloni di materia oscura a teorie di gravità modificata

The standard cosmological model requires an additional Dark Energy component to account for the accelerated expansion of the Universe, whose nature is still unknown. Its simplest interpretation is that of a cosmological constant, that requires however an unexplained fine-tuning. Alternative theories based on Modified Gravity (MG) have been proposed to avoid the introduction of a cosmological constant, though signatures of MG on cosmological observables are not detected in current data. My PhD was focused on extending approximate methods to generate dark matter distributions in the context of MG models. Such methods can quickly generate large sets of simulated halo catalogs, reproducing well the mildly non-linear scales of cosmic structures, and are therefore a key tool for the analysis of high precision data from future cosmological surveys, such as Euclid. I focused on extending the PINOCCHIO software (Monaco+2002) to MG theories, which involves formulating and implementing both Lagrangian perturbation theory (LPT) and ellipsoidal collapse (EC) with MG. LPT is adopted by several approximate methods because of its ability to separate the time evolution of the displacement field from the spatial part, allowing to readily compute particle displacements for any given redshift. In MG however the growth rate becomes scale dependent. Moreover, the second-order growth rate D2 depends on three wave-numbers, constrained to form a triangle in Fourier space. Solving the full equation for second-order displacements would be too demanding in terms of computational time. One possibility is to consider approximations for D2 that only depend on k, as proposed in Winther+2017. To find the proper approximation for D2 I develop a new numerical method based on FFTs, that consists in computing the full source term of the second-order displacement differential equation, and comparing to several triangle configurations to choose the one that best matches the full source term. The resulting approximated D2 is then implemented in a code to compute Lagrangian displacements, and tested against N-body simulations run with Hu-Sawicki f(R) (Giocoli+2018). The halo catalog is constructed by matching the particle membership to the simulation catalog. From the reconstructed catalog I compute the halo power spectrum and compare to N-body simulations, showing that our approximation allows to recover the halo power spectrum within 10% up to mildly-NL scales (~0.2hMpc^-1), with the same performance as in the LCDM case. These results are summarized in a paper, submitted to MNRAS (Moretti+2019). To construct halo catalogs, PINOCCHIO relies on the computation of collapse times (CT) obtained treating overdensities as homogeneous ellipsoids. In its standard version, PINOCCHIO computes CT taking advantage of the LPT formulation. The latter involves using D1 as time variable to evaluate collapse times. Such approach is not suitable in the MG case, since D1 is scale-dependent. Another approach to EC is described in Bond+1996 (BM), and involves the solution of integro-differential equations. BM was later reformulated in Nadkarni-Ghosh+2016 (NGS) to avoid integrals. The NGS formulation allows for a faster numerical solution respect to BM, making it suitable to implement in PINOCCHIO. Starting from the results of Ruan+2019, that extend the BM approach to f(R), I reformulate the NGS description to include both the gravity enhancement and the screening mechanism due to MG. With this formulation of EC, currently in the phase of implementation in PINOCCHIO, the code will be able to generate the large sets of realizations needed to properly compute covariance matrices for cosmological observables in MG. These results will appear in a paper, now in preparation (Moretti+2020). This thesis presents the optimal numerical techniques to implement 2LPT and EC with f(R) gravity in a fast approximate method, providing the theoretical framework for the extension of PINOCCHIO to MG.