Statistical techniques in cosmology

In these lectures I cover a number of topics in cosmological data analysis. I concentrate on general techniques which are common in cosmology, or techniques which have been developed in a cosmological context. In fact they have very general applicability, for problems in which the data are interpreted in the context of a theoretical model, and thus lend themselves to a Bayesian treatment. We consider the general problem of estimating parameters from data, and consider how one can use Fisher matrices to analyse survey designs before any data are taken, to see whether the survey will actually do what is required. We outline numerical methods for estimating parameters from data, including Monte Carlo Markov Chains and the Hamiltonian Monte Carlo method. We also look at Model Selection, which covers various scenarios such as whether an extra parameter is preferred by the data, or answering wider questions such as which theoretical framework is favoured, using General Relativity and braneworld gravity as an example. These notes are not a literature review, so there are relatively few references

Generalisations of Fisher Matrices

Fisher matrices play an important role in experimental design and in data analysis. Their primary role is to make predictions for the inference of model parameters - both their errors and covariances. In this short review, I outline a number of extensions to the simple Fisher matrix formalism, covering a number of recent developments in the field. These are: (a) situations where the data (in the form of (x,y) pairs) have errors in both x and y; (b) modifications to parameter inference in the presence of systematic errors, or through fixing the values of some model parameters; (c) Derivative Approximation for LIkelihoods (DALI) - higher-order expansions of the likelihood surface, going beyond the Gaussian shape approximation; (d) extensions of the Fisher-like formalism, to treat model selection problems with Bayesian evidence.Comment: Invited review article for Entropy special issue on 'Applications of Fisher Information in Sciences'. Accepted versio

Statistical techniques in cosmology

In these lectures I cover a number of topics in cosmological data analysis. I concentrate on general techniques which are common in cosmology, or techniques which have been developed in a cosmological context. In fact they have very general applicability, for problems in which the data are interpreted in the context of a theoretical model, and thus lend themselves to a Bayesian treatment. We consider the general problem of estimating parameters from data, and consider how one can use Fisher matrices to analyse survey designs before any data are taken, to see whether the survey will actually do what is required. We outline numerical methods for estimating parameters from data, including Monte Carlo Markov Chains and the Hamiltonian Monte Carlo method. We also look at Model Selection, which covers various scenarios such as whether an extra parameter is preferred by the data, or answering wider questions such as which theoretical framework is favoured, using General Relativity and braneworld gravity as an example. These notes are not a literature review, so there are relatively few references.Comment: Typos corrected and exercises adde

Intrinsic Galaxy Alignments and Weak Gravitational Lensing

Gravitational lensing causes background galaxy images to become aligned, and the statistical characteristics of the image alignments can then be used to constrain the power spectrum of mass fluctuations. Analyses of gravitational lensing assume that intrinsic galaxy alignments are negligible, but if this assumption does not hold, then the interpretation of image alignments will be in error. As gravitational lensing experiments become more ambitious and seek very low-level alignments arising from lensing by large-scale structure, it becomes more important to estimate the level of intrinsic alignment in the galaxy population. In this article, I review the cluster of independent theoretical studies of this issue, as well as the current observational status. Theoretically, the calculation of intrinsic alignments is by no means straightforward, but some consensus has emerged from the existing works, despite each making very different assumptions. This consensus is that a) intrinsic alignments are a small but non-negligible (< 10%) contaminant of the lensing ellipticity correlation function, for samples with a median redshift z = 1; b) intrinsic alignments dominate the signal for low-redshift samples (z = 0.1), as expected in the SuperCOSMOS lensing survey and the Sloan Digital Sky SurveyComment: 8 pages. Invited talk at Yale Workshop on `The Shapes of Galaxies and their halos', May 200

Weak gravitational lensing: reducing the contamination by intrinsic alignments

Intrinsic alignments of galaxies can mimic to an extent the effects of shear caused by weak gravitational lensing. Previous studies have shown that for shallow surveys with median redshifts z_m = 0.1, the intrinsic alignment dominates the lensing signal. For deep surveys with z_m = 1, intrinsic alignments are believed to be a significant contaminant of the lensing signal, preventing high-precision measurements of the matter power spectrum. In this paper we show how distance information, either spectroscopic or photometric redshifts, can be used to down-weight nearby pairs in an optimised way, to reduce the errors in the shear signal arising from intrinsic alignments. Provided a conservatively large intrinsic alignment is assumed, the optimised weights will essentially remove all traces of contamination. For the Sloan spectroscopic galaxy sample, residual shot noise continues to render it unsuitable for weak lensing studies. However, a dramatic improvement for the slightly deeper Sloan photometric survey is found, whereby the intrinsic contribution, at angular scales greater than 1 arcminute, is reduced from about 80 times the lensing signal to a 10% effect. For deeper surveys such as the COMBO-17 survey with z_m = 0.6, the optimisation reduces the error from a largely systematic 220% error at small angular scales to a much smaller and largely statistical error of only 17% of the expected lensing signal. We therefore propose that future weak lensing surveys be accompanied by the acquisition of photometric redshifts, in order to remove fully the unknown intrinsic alignment errors from weak lensing detections.Comment: 10 pages, 6 figures, MNRAS accepted. Minor changes to match accepted version. RCS and ODT predictions are modifie

Objective Bayesian analysis of neutrino masses and hierarchy

Given the precision of current neutrino data, priors still impact noticeably the constraints on neutrino masses and their hierarchy. To avoid our understanding of neutrinos being driven by prior assumptions, we construct a prior that is mathematically minimally informative. Using the constructed uninformative prior, we find that the normal hierarchy is favoured but with inconclusive posterior odds of 5.1:1. Better data is hence needed before the neutrino masses and their hierarchy can be well constrained. We find that the next decade of cosmological data should provide conclusive evidence if the normal hierarchy with negligible minimum mass is correct, and if the uncertainty in the sum of neutrino masses drops below 0.025 eV. On the other hand, if neutrinos obey the inverted hierarchy, achieving strong evidence will be difficult with the same uncertainties. Our uninformative prior was constructed from principles of the Objective Bayesian approach. The prior is called a reference prior and is minimally informative in the specific sense that the information gain after collection of data is maximised. The prior is computed for the combination of neutrino oscillation data and cosmological data and still applies if the data improve.Comment: 15 pages. Minor changes to match accepted version in JCA