Lectures on Cosmology

Volume 800 of the series Lecture Notes in Physics pp 147-177


Statistical Methods in Cosmology

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The advent of large data set in cosmology has meant that in the past 10 or 20 years our knowledge and understanding of the Universe has changed not only quantitatively but also, and most importantly, qualitatively. Cosmologists are interested in studying the origin and evolution of the physical Universe. They rely on data where a host of useful information is enclosed, but is encoded in a non-trivial way. The challenges in extracting this information must be overcome to make the most of the large experimental effort. Even after having analyzed a decade or more of data and having converged to a standard cosmological model (the so-called and highly successful Λ CDM model) we should keep in mind that this model is described by 10 or more physical parameters and if we want to study deviations from the standard model the number of parameters is even larger. Dealing with such a high-dimensional parameter space and finding parameters constraints is a challenge on itself. In addition, as gathering data is such an expensive and difficult process, cosmologists want to be able to compare and combine different data sets both for testing for possible disagreements (which could indicate new physics) and for improving parameter determinations. Finally, because experiments are always so expansive, cosmologists in many cases want to find out a priori, before actually doing the experiment, how much one would be able to learn from it. For all these reasons, more and more sophisticated statistical techniques are being employed in cosmology, and it has become crucial to know some statistical background to understand recent literature in the field. Here, I will introduce some statistical tools that any cosmologist should know about in order to be able to understand recently published results from the analysis of cosmological data sets. I will not present a complete and rigorous introduction to statistics as there are several good books which are reported in the references. The reader should refer to those. I will take a practical approach and I will touch upon useful tools such as statistical inference, Bayesians vs Frequentists approach, chisquare and goodness of fit, confidence regions, likelihood, Fisher matrix approach, Monte Carlo methods, and a brief introduction to model testing. Throughout, I will use practical examples often taken from recent literature to illustrate the use of such tools. Of course this will not be an exhaustive guide: it should be interpreted as a “starting kit,” and the reader is warmly encouraged to read the references to find out more.