Context. One of the main questions of modern cosmology and physics concerns the origin of the Universe as we know it today. Thanks to the tremendous progress made in the last decade, in big part driven by the Europe-led satellite mission Planck, the stage has been set to start addressing this question for the first time in a meaningful way. Our most promising observational probe in this context is cosmic microwave background (CMB) - the primordial light generated in the very early Universe - and in particular observations of its polarization properties. Many popular theories of the early Universe predict that gravitational waves were generated in its very first moments. These theories generally invoke new physics from beyond the standard model of particle physics, which determines properties of the primordial gravitational waves and which consequently carry clues about the nature of those new laws of physics. Detecting the primordial gravitational waves would thus have revolutionary impact on our understanding of cosmology and fundamental physics while at the same time explaining the origin of the observed structures of the Universe.
The most promising way to do so is by detecting a divergence-free pattern in CMB polarization, the so-called B-modes, which to first order could only be generated by the primordial gravitational waves. While the recent experiments have set tight constraints on their amplitudes, there is no convincing detection of such a signal as of today. However,  the most advanced, soon-to-be-deployed and currently planned efforts are expected to start reaching sensitivities sufficient for a detection of this signal as predicted by many of the currently popular theories. This promises that the next few years can be particularly exciting and fruitful for this area of research with evidence in favor or against some of those models finally becoming available.
The impact of the CMB polarization is larger than that. It will also shed a new light on many other key questions of modern physics, such as those concerning the nature of dark matter, total mass of neutrinos and their mass hierarchy, or the presence of unknown relativistic particle species.
In all these cases the telltale information is contained in a pattern of minuscule CMB polarization anisotropies, which have to be recovered from huge volumes of data collected by current and forthcoming CMB experiments, differentiate from other unwanted signals, and characterized with unprecedented precision and robustness.
Consequently, the success and prominence of the CMB polarisation as one of the key probes of modern cosmology is hinged on the success of the CMB data analysis which has emerged in the last decade as one of the main enabling technologies, alongside of and on the same footing as instrumental progress, for this scientific endeavor.

Thesis. The proposed thesis is in the area of the CMB data analysis and proposes to contribute to global, worldwide effort of exploiting the scientific potential of the CMB polarization by developing novel methods and techniques suitable for robust, statistical characterization of the CMB polarisation signals. These tools and methods will be then applied to analysis of data from some of the most exciting and promising  forthcoming ground-based CMB experiments as well as to design and scientific optimisation of the next generation of efforts.

The thesis will be performed in the context of the two major CMB experiments: Simons Observatory and CMB-S4. These are large-scale international undertakings in which the CMB group at APC is heavily involved and play important, coordinating roles. These cutting-edge experiments will bring unprecedented volumes of data and reach sensitivity that will allow to significantly advance the current state-of-the-art in this area reaching many of the goals mentioned above. As part of the proposed thesis work the student will develop new data analysis techniques appropriate for these forthcoming data sets, apply them to the actual data as they become available, and participate in the exploitation of their scientific potential.

The specific focus of the proposed thesis is on a development of novel, statistically robust and computationally-efficient algorithms for an estimation of the power spectra of polarised signals and in particular those of the B-mode polarization. These methods will be devised and optimized for ground-based data sets accounting for their typical features. These include: limited-sky coverage, highly inhomogeneous noise coverage, presence of  missing sky-modes due to filtering of atmospheric and ground-pick up contributions as well as instrumental contributions, presence of residuals signals of astrophysical, Galactic and extra-Galactic origins signals left over after component separation procedures which are typically applied to remove such signals.
The weakness of the B-mode polarization as compared to other signals makes this task very challenging but making even partial progress in this area will have huge impact on our ability to fully exploit the scientific potential of the CMB data sets and therefore the entire field and will establish the student as a true leader in this science area.

We will first consider so called pseudo-spectral methods and will generalise them taking into account the aforementioned sytematic effects. These methods are typically computationally very efficient but their performance tend to deteriorate quickly in the presence of the systematic effects. We will consider extensions of the existing methods, such as pure pseudo spectrum techniques, which will allow to correct for such effects without significant loss in precision.
We will also consider maximum likelihood approaches combining Monte Carlo sampling techniques with useful approximations derived from Machine Learning application in order to make the more computationally feasible. We will consider such techniques operating in the pixel- (sky) domain as well as in the domain of raw data as directly collected by the experiments. While these latter approaches tend to be very computationally-involved as they have to deal with the full volume of the collected data at the same time they offer the best prospects for accounting on the systematic effects in the data.

Simons Observatory is expected to conclude its first observational campaign in the second year of the proposed thesis and the techniques and tools developed during the course of this thesis will be applied to their analysis. They will also become building blocks of the data analysis pipeline for the CMB-S4 experiment which is expected to become operational at the end of this decade. This will be used to address some of the remaining design questions, concerning CMB-S4 hardware and operations.

Modern CMB data analysis is a multi-disciplinary effort, involving statistical methods, signal processing, machine learning, high performance scientific computing in addition to physics and cosmology. The student will have an opportunity to collaborate with experts in all those fields, capitalizing on active research collaborations developed by our group in France and internationally. This will allow the student to develop attractive, broad scientific background and research experience well beyond that of a typical physics thesis, preparing him/her for a future career in diverse contexts. In addition, the student will have opportunity to participate hands-on in the deployment and observational campaigns, and pay long term visits to our collaborators in Berkeley and Princeton.

The student will have an opportunity to become a full member of  two international collaborations,  Simons Observatory and CMB-S4, which gather some of the top experts in the field and to develop personal research projects with their members.