Cosmological parameter inference with Bayesian statistics

Bayesian statistics and Markov Chain Monte Carlo (MCMC) algorithms have found their place in the field of Cosmology. They have become important mathematical and numerical tools, especially in parameter estimation and model comparison. In this paper, we review some fundamental concepts to understand Bayesian statistics and then introduce MCMC algorithms and samplers that allow us to perform the parameter inference procedure. We also introduce a general description of the standard cosmological model, known as the $\Lambda$CDM model, along with several alternatives, and current datasets coming from astrophysical and cosmological observations. Finally, with the tools acquired, we use an MCMC algorithm implemented in python to test several cosmological models and find out the combination of parameters that best describes the Universe.Comment: 30 pages, 17 figures, 5 tables; accepted for publication in Universe; references adde

On the detection of spectral ripples from the Recombination Epoch

Photons emitted during the epochs of Hydrogen ($500 \lesssim z \lesssim 1600$) and Helium recombination ($1600 \lesssim z \lesssim 3500$ for HeII $\rightarrow$ HeI, $5000 \lesssim z \lesssim 8000$ for HeIII $\rightarrow$ HeII) are predicted to appear as broad, weak spectral distortions of the Cosmic Microwave Background. We present a feasibility study for a ground-based experimental detection of these recombination lines, which would provide an observational constraint on the thermal ionization history of the Universe, uniquely probing astrophysical cosmology beyond the last scattering surface. We find that an octave band in the 2--6 GHz window is optimal for such an experiment, both maximizing signal-to-noise ratio and including sufficient line spectral structure. At these frequencies the predicted signal appears as an additive quasi-sinusoidal component with amplitude about $8$ nK that is embedded in a sky spectrum some nine orders of magnitude brighter. We discuss an algorithm to detect these tiny spectral fluctuations in the sky spectrum by foreground modeling. We introduce a \textit{Maximally Smooth} function capable of describing the foreground spectrum and distinguishing the signal of interest. With Bayesian statistical tests and mock data we estimate that a detection of the predicted distortions is possible with 90\% confidence by observing for 255 days with an array of 128 radiometers using cryogenically cooled state-of-the-art receivers. We conclude that detection is in principle feasible in realistic observing times; we propose APSERa---Array of Precision Spectrometers for the Epoch of Recombination---a dedicated radio telescope to detect these recombination lines.Comment: 33 pages, 16 figures, submitted to ApJ, comments welcom

Bayesian interpolation

Although Bayesian analysis has been in use since Laplace, the Bayesian method of model-comparison has only recently been developed in depth. In this paper, the Bayesian approach to regularization and model-comparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other data modeling problems. Regularizing constants are set by examining their posterior probability distribution. Alternative regularizers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. “Occam's razor” is automatically embodied by this process. The way in which Bayes infers the values of regularizing constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling

Combining cosmological datasets: hyperparameters and Bayesian evidence

A method is presented for performing joint analyses of cosmological datasets, in which the weight assigned to each dataset is determined directly by it own statistical properties. The weights are considered in a Bayesian context as a set of hyperparameters, which are then marginalised over in order to recover the posterior distribution as a function only of the cosmological parameters of interest. In the case of a Gaussian likelihood function, this marginalisation may be performed analytically. Calculation of the Bayesian evidence for the data, with and without the introduction of hyperparameters, enables a direct determination of whether the data warrant the introduction of weights into the analysis; this generalises the standard likelihood ratio approach to model comparison. The method is illustrated by application to the classic toy problem of fitting a straight line to a set of data. A cosmological illustration of the technique is also presented, in which the latest measurements of the cosmic microwave background power spectrum are used to infer constraints on cosmological parameters.Comment: 12 pages, 6 figures, submitted to MNRA