# Bachelor thesis with Prof. Lesgourgues in 2019

Analytical and numerical calculation of the bispectrum of the Cosmic Microwave Background

The anisotropies of the Cosmic Microwave Background (CMB) are a still picture of the seeds of all the structures in the present universe. We can use the statistical properties of those anisotropies to characterise the evolution of the Universe during its early history. The simplest measure of those statistical properties, the /power spectrum/, accounts for the first Gaussian approximation of those anisotropies and can already provide us with a good picture of the early universe. The next-order, non-Gaussian approximation would be captured by the /bispectrum/ of the CMB anisotropies. The bispectrum, if detected by future experiments, may provide us with a whole new batch of cosmological information. However, to use that information to test actual models of the early universe, we need to perform a series of expensive calculations.

In this project, the student will review the usual approximations used to compute the CMB bispectrum and compare it to real data. She/he will try to make some progress along the less travelled paths, involving complicated integrals of special functions.

N-body simulations and new method for interpolating their results

N-body simulations describe the process of non-linear structure formation from the tiny initial density fluctuations to the structures (galaxies, clusters) that we observe today. They are computationally expensive and can therefore only be performed for a very limited set of cosmological parameters. However, when analysing data and comparing to different models we need to sample the allowed model parameter space to identify the combination of cosmological parameters that best fit the data. One important task is thus interpolating between the few N-body runs available. We propose an idea for boosting the accuracy of these interpolations with very little efforts. Indeed, using cosmological perturbation theory, we could post-process the results of each simulation in order to get almost for free the first derivative of the results with respect to each cosmological parameter.

The task of the bachelor student is to study previous interpolation methods and to understand the new technique that we propose. She/he will try to estimate by how much the accuracy could be improved, and if she/he has enough time, she/he will try to implement it concretely in a simplified case.

New methods for solving oscillatory integrals in cosmology

In our era of highly accurate cosmological measurements, very precise theoretical predictions are required to constrain new and exotic models of dark matter. Speeding up the calculation of these predictions would enable us to test more new dark matter models and with better precision. For this purpose, highly oscillatory differential equations and integration problems have to be solved. The goal of this project the interface between theoretical cosmology and applied mathematics is to further investigate several possible mathematical tricks and numerical methods. If time permits she/he will try to implement them in existing codes and check their efficiency.