Testing the cosmological principle through topological properties of cosmic microwave background fluctuations

Cosmology is transitioning from a theoretical discipline towards one with increased focus on observations, resulting in a massive surge of data that demands increasingly more sophisticated methods to glean meaningful information. In a related development, geometry and topology have witnessed a tilt from purely theoretical fields towards strong focus on application. A foray into "big data" quickly brings to front two of the central statistical challenges of our times -- detection and classification of structure in extremely large, high-dimensional, data sets. Among the most intriguing new approaches to this challenge is "TDA," or "topological data analysis," the primary aim of which is providing topologically informative pre-analyses of data, which serve as input to more quantitative analyses at a later stage. Algebraic and computational topology at the level of homology and persistent homology are the foundational pillars of TDA. These developments on the topological side are recent, and add value to the already existing geometric tools and methodologies employed in investigating the cosmological fields.

In the first part of my talk, I will present a summary of the theoretical and computational aspects of geometry and topology from the view point of data analysis. Subsequently, I will present an analysis of the topological properties of the temperature and polarization maps of the cosmic microwave background (CMB) radiation obtained by the Planck satellite. The CMB radiation represents the earliest visible light in the Universe, and contains a treasure trove of information about the initial conditions of the Universe. The CMB also represents the largest canvas on which to test the fundamental tenets  of the cosmological principle.  Within this context, I will discuss some of the anomalies that the CMB temperature and polarization maps exhibit with respect to the simulations based on the standard cosmological model, which assumes the initial fluctuation field to be an instance of an isotropic and homogeneous Gaussian random field.