# Bachelor Thesis With Prof. Lesgourgues in 2022

1. Formal comparison between the emission of electromagnetic and gravitational radiation

In Electrodynamics, you have studied the analytical formalism showing how a distribution of electric charges can emit electromagnetic waves, at leading order in a multipole expansion. Next year, in the Master of Physics, some of you will study gravitational waves. The goal of this bachelor thesis is to learn the basic equations governing gravitational radiation, and then, to do a similar calculation, that is, to find how a distribution of masses can emit gravitational waves, at leading order in a multipole expansion. You will try to highlight the formal similarities and differences between these two calculations. As an application, you may carry an estimation of the amplitude of gravitational radiation emitted in the early universe during phase transitions.

**Skills required:** analytical skills, end of the course on Electrodynamics

**Skills learned: **a few basic aspect of General Relativity

**Main supervisor:** Prof. Julien Lesgourgues

2. Playing with the map of CMB anisotropies

The Planck satellite has measured some beautiful maps of temperature and polarisation fluctuations all over the sky, that are often called “pictures of the baby universe”. Our group has worked on the interpretation of this map under the assumption that our universe is statistically homogeneous on very large scales. In this project, we will try to go beyond this assumption. For this purpose, the student will learn the basics of HEALPIX, the favourite astrophysicist's software for playing with pixelised maps of the sky. They will learn how to decompose the entire sky map in smaller maps, and how to extract some statistical properties from each individual patch. Then, they will use a numerical pipeline developed by our group to test the assumption of homogeneity of the universe on very large scales.

**Skills required:** light numerical skills to learn and play with a software, HELPIX, that does image processing

**Skills learned: **several basic aspect of cosmology, in particular CMB physics

**Main supervisor:** Prof. Julien Lesgourgues

3. N-body simulations with decaying modes / modified gravity

N-body simulations are commonly employed numerical tools that accurately describe the non-linear gravitational formation of structures in the Universe. Starting from small initial perturbations, they describe the growth of inhomogeneities and eventually the formation of galaxies and voids that constitute the cosmic web. However, those simulations usually assume an idealised Universe that is governed by Newtonian gravity only. In this project, we want to explore some simple modifications and study their impact on structure formation. Theories of modified gravity usually require complex modifications far beyond the scope of this project. However, in some cases, a good approximation can be found such that the dynamics is still Newtonian, only with a time-dependant gravitational constant or other minor changes. We want to explore the features such changes induce and understand which modifications would lead to observationally interesting signatures that can be tested in upcoming surveys. The task of the student is to learn using N-body simulations and implement some simple modifications. The goal is to perform a simple data analysis on the level of the power-spectrum.

**Skills required:** numerical methods, and to a lesser degree statistical methods for data analysis

**Skills learned: **working with N-body simulations, analysing numerical data, cosmology of the large scale structure

**Main supervisor: **Priv. Doz. Christian Fidler

4. Machine Learning techniques to emulate non-linear cosmological observables

In cosmology, many observable quantities can only be computed with the help of computationally-expensive N-body simulations, which make them unpractical for data analysis. These observables, such as the nonlinear matter power spectrum, convergence maps, lensing peaks and others, need to be computed for a large set of different input cosmological parameters, but since their calculation is very time-consuming, only few realizations exist as open source data. The idea of this project is to use Machine Learning techniques to emulate the changes of these quantities as a function of the input parameters, such that new predictions can be made at points in which the observables have not been computed directly. These evaluations should be fast and noise-free. The general code for this project has been written by the supervisor almost completely, so that the task for the student is mostly to understand the Machine Learning techniques: PCA, Gaussian Processes and Dictionary Learning and then use the code and apply it on different sources of data and finally cross-validate and train hyperparameters. Also, it would involve code documentation, testing and interfacing to other cosmological codes.

**Skills required:** Programming in python, understanding of basic statistical techniques and numerical methods

**Skills learned: **Machine Learning techniques, training and validating models, familiarity with cosmological nonlinear simulations and data

**Main supervisor: **Dr. Santiago Casas

5. Study of variational techniques for Bayesian parameter estimation

In cosmology we currently use Bayesian statistical techniques to fit our models to observational data and obtain probabilistic confidence contours on the model parameters. The most used technique for this inference problem is Markov-Chain-Monte-Carlo (MCMC), in which random samples from a model distribution are compared to a data vector and on the process the posterior is estimated. However, this process is relatively inefficient and time-consuming. If on the other hand, the derivatives of the model can be computed exactly, this process can be greatly improved. This is called Variational Inference. In this project, the student is expected to use already existing codes to fit very simple analytical models using MCMC and then compare with the same simple model using variational inference. There are several open source codes in the community that can serve this purpose. At the end of the project, the student should have a good understanding on the applicability of each method and its advantages/disadvantages. If time permits, the student can advance to more realistic models used in astrophysics, such as cosmological distance measurements using supernovae data and related observations.

**Skills required:** Python programming, basic knowledge on statistics, differential numerical methods

**Skills learned: **Understanding of Bayesian statistics, probabilistic programming, model fitting

**Main supervisor: **Dr. Santiago Casas