Fast simulations of cosmological fields

Forthcoming galaxy surveys are going to provide a wealth of data that will allow us to study the Universe to an unprecedented precision. We have the opportunity to make decisive tests of gravity and of the standard model of cosmology. Nonetheless, analytical modelling of structure formation from first principles falls short of the requirements of future observations, because we need to model the large scale structure in a nonlinear regime. For this reason, N-body simulations have become the cornerstone to test and calibrate our predictions. On the other hand, N-body simulations are computationally expensive, so only a handful of them are available. It is hard to exploit them in actual analysis which require statistics on a very broad parameter space. In addition, relying only on simulations limit our comprehension of the underlying physics. For this reason, this thesis has been devoted in developing semi-analytical methods to produce fast surrogate simulations of the matter distribution, which maintain some of the statistical properties of the result from the full numerical code. We believe that the results of this thesis aid our understanding of the physics of structure formation, of the statistics of the matter distribution, and can be used to run Monte Carlo simulations to compare with data. After a brief introduction on the standard cosmological model (chapter 1), and a summary of the theoretical background (chapter 2), in chapter 3 we present the first result of this work. Digital simulations of arbitrary stochastic fields can only be done by means of monotonic transformations where a Gaussian field is mapped onto a new target field. The most famous example of this is the lognormal approximation to describe the dark matter distribution at low redshift. However this procedure is limited: the mapping has control only over the amplitudes of the distribution, while a desirable property would be to have control over the resulting correlation function of the field as well. This can be achieved with the iterative translation approximation method (itam), which optimizes the power spectrum to generate the Gaussian field to map onto a target field, such that the resulting field has prescribed probability distribution function and correlation function. We apply itam to the case of the matter density field smoothed at a fewh −1Mpc, showing that we can generate a surrogate density field with exact probability distribution function, as measured in N-body, and a power spectrum that matches the target up to nonlinear scales. The remainder of this thesis is focused on algorithms to approximate the evolution of the particles in simulations, rather than working at the level of the Eulerian density. This approach to perform fast simulations is Lagrangian, since it is based on the displacement field from which the density field is derived. In fact, the displacement eld informs us on how to move particles from their initial positions to their final positions. Over the years, considerable e ort has been invested to derive approximations of this field by means of perturbation theories and non perturbative models. Lagrangian perturbation theory at first order shows that the displacement field can be approximated as a map for the initial Gaussian density field. Hence in the last part of chapter 3, we apply itam to the Lagrangian displacement field measured from a reference simulation. Ultimately this is a test of semi-analytical schemes; we can check whether there is information encoded in the probability distribution function and correlation function of the exact displacement field that they cannot capture. The result is surprising: we find that techniques based on non-local information in the linear density already provide a better displacement field than the one generated through itam. This suggests that further improvement in Lagrangian algorithms should be based on some non-local method, beyond the simple two-point correlation of the displacement. In the final part of this thesis (chapter 4), we propose such a multi-scale Lagrangian scheme, dubbed muscle-ups. It starts from the realization that present Lagrangian algorithms fail to describe the evolution of particles around overdense regions in the initial density field. This is because particles in proto-halos have been modelled locally, despite being highly correlated. muscleups accounts for this correlation by implementing a modification of the extended Press and Schechter formalism adapted to present Lagrangian algorithms. It can naturally group particles into halos in the initial conditions, and build a halo catalogue that matches a target halo mass function as measured in full N-body. Finally, it displaces these halo particles to match halo density pro les, so that it effectively provides a semi-analytical Lagrangian implementation of the halo model. muscle-ups improves the description of the displacement of particles in correspondence of overdensities in the initial density field, while yielding the expected results from perturbation theories on the larger and less dense scales. This translates into better statistics of the dark matter field as well with respect to present algorithms: muscle-ups recovers the high density tail of the probability distribution function, remedies to the decline of the power spectrum in the quasi linear regime and increases the cross correlation with the density field from N-body.