Pia Mukherjee

A NESTED SAMPLING ALGORITHM FOR COSMOLOGICAL MODEL SELECTION

The abundance of new cosmological data becoming available means that a wider range of cosmological models are testable than ever before. However, an important distinction must be made between parameter fitting and model selection. While parameter fitting simply determines how well a model fits the data, model selection statistics, such as the Bayesian Evidence, are now necessary to choose between these different models, and in particular to assess the need for new parameters. We implement a new evidence algorithm known as nested sampling, which combines accuracy, generality of application and computational feasibility, and apply it to some cosmological datasets and models. We find that a five-parameter model with Harrison–Zel’dovich initial spectrum is currently preferred. Subject headings: cosmology: theory

Model-independent Constraints on Dark Energy Density from Flux-averaging Analysis of Type Ia Supernova Data

We reconstruct the dark energy density X(z) as a free function from current Type Ia supernova (SN Ia) data, together with the cosmic microwave background (CMB) shift parameter from CMB data from the Wilkinson Microwave Anisotropy Probe (WMAP), Cosmic Background Imager (CBI), and Arcminute Cosmology Bolometer Array Receiver (ACBAR), and the large-scale structure (LSS) growth factor from the Two-Degree Field (2dF) galaxy survey data. We parameterize X(z) as a continuous function, given by interpolating its amplitudes at equally spaced z-values in the redshift range covered by SN Ia data, and a constant at larger z [where X(z) is only weakly constrained by CMB data]. We assume a flat universe and use the Markov Chain Monte Carlo (MCMC) technique in our analysis. We find that the dark energy density X(z) is constant for 0<~z<~0.5 and increases with redshift z for 0.5<~z<~1 at a 68.3% confidence level, but is consistent with a constant at a 95% confidence level. For comparison, we also give constraints on a constant equation of state for the dark energy. Flux averaging of SN Ia data is required to yield cosmological parameter constraints that are free of the bias induced by weak gravitational lensing. We set up a consistent framework for flux-averaging analysis of SN Ia data, based on the work of Wang. We find that flux averaging of SN Ia data leads to slightly lower m and smaller time variation in X(z). This suggests that a significant increase in the number of SNe Ia from deep SN surveys on a dedicated telescope is needed to place a robust constraint on the time dependence of the dark energy density.

Wavelets and WMAP non-Gaussianity

We study the statistical properties of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data on different scales using the spherical Mexican hat wavelet transform. Consistent with the 2004 results of Vielva et al., we find a deviation from Gaussianity in the form of kurtosis of wavelet coefficients on 3°-4° scales in the southern Galactic hemisphere. This paper extends the work of Vielva et al. as follows: We find that the non-Gaussian signal shows up more strongly in the form of a larger than expected number of cold pixels and also in the form of scale-scale correlations among wavelet coefficients. We establish the robustness of the non-Gaussian signal under more wide-ranging assumptions regarding the Galactic mask applied to the data and the noise statistics. This signal is unlikely to be due to the usual quadratic term parameterized by the nonlinearity parameter fNL. We use the skewness of the spherical Mexican hat wavelet coefficients to constrain fNL with the first-year WMAP data. Our results constrain fNL to be 50 ± 80 at 68% confidence and less than 280 at 99% confidence.We study the statistical properties of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data on different scales using the spherical Mexican hat wavelet transform. Consistent with the 2004 results of Vielva et al., we find a deviation from Gaussianity in the form of kurtosis of wavelet coefficients on 3-4 scales in the southern Galactic hemisphere. This paper extends the work of Vielva et al. as follows: We find that the non-Gaussian signal shows up more strongly in the form of a larger than expected number of cold pixels and also in the form of scale-scale correlations among wavelet coefficients. We establish the robustness of the non-Gaussian signal under more wide-ranging assumptions regarding the Galactic mask applied to the data and the noise statistics. This signal is unlikely to be due to the usual quadratic term parameterized by the nonlinearity parameter fNL. We use the skewness of the spherical Mexican hat wavelet coefficients to constrain fNL with the first-year WMAP data. Our results constrain fNL to be 50+/-80 at 68% confidence and less than 280 at 99% confidence.

Model-independent reconstruction of the primordial power spectrum from WMAP data

Reconstructing the shape of the primordial power spectrum in a model-independent way from cosmological data is a useful consistency check on what is usually assumed regarding early universe physics. It is also our primary window to unknown physics during the inflationary era. Using a power-law form for the primordial power spectrum Pin(k) and constraining the scalar spectral index and its running, in 2003 Peiris and coworkers found that the first-year Wilkinson Microwave Anistropy Probe (WMAP) data seem to indicate a preferred scale in Pin(k). We use two complementary methods, the wavelet band power method of Mukherjee & Wang and the top-hat binning method of Wang, Spergel, & Strauss, to reconstruct Pin(k) as a free function from cosmic microwave background (CMB) data alone (WMAP, CBI, and ACBAR), or from CMB data together with large-scale structure data (2dFGRS and PCSz). The shape of the reconstructed Pin(k) is consistent with scale invariance, although it allows some indication of a preferred scale at k~0.01 Mpc-1. While consistent with the possible evidence for a running of the scalar spectral index found by the WMAP team, our results highlight the need of more stringent and independent constraints on cosmological parameters (the Hubble constant in particular) in order to more definitively constrain deviations of Pin(k) from scale invariance without making assumptions about the inflationary model.

Bayesian methods in cosmology

In recent years cosmologists have advanced from largely qualitative models of the Universe to precision modelling using Bayesian methods, in order to determine the properties of the Universe to high accuracy. This timely book is the only comprehensive introduction to the use of Bayesian methods in cosmological studies, and is an essential reference for graduate students and researchers in cosmology, astrophysics and applied statistics. The first part of the book focuses on methodology, setting the basic foundations and giving a detailed description of techniques. It covers topics including the estimation of parameters, Bayesian model comparison, and separation of signals. The second part explores a diverse range of applications, from the detection of astronomical sources (including through gravitational waves), to cosmic microwave background

Bayesian model selection analysis of WMAP3

We present a Bayesian model selection analysis of WMAP3 data using our code CosmoNest. We focus on the density perturbation spectral index nS and the tensor-to-scalar ratio r, which define the plane of slow-roll inflationary models. We find that while the Bayesian evidence supports the conclusion that nS 6= 1, the data are not yet powerful enough to do so at a strong or decisive level. If tensors are assumed absent, the current odds are approximately 8 to 1 in favour of nS 6= 1 under our assumptions, when WMAP3 data is used together with external data sets. WMAP3 data on its own is unable to distinguish between the two models. Further, inclusion of r as a parameter weakens the conclusion against the Harrison–Zel’dovich case (nS = 1, r = 0), albeit in a prior-dependent way. In appendices we describe the CosmoNest code in detail, noting its ability to supply posterior samples as well as to accurately compute the Bayesian evidence. We make a first public release of CosmoNest, now available at www.cosmonest.org.

Designing a space-based galaxy redshift survey to probe dark energy

A space-based galaxy redshift survey would have enormous power in constraining dark energy and testing general relativity, provided that its parameters are suitably optimized. We study viable space-based galaxy redshift surveys, exploring the dependence of the Dark Energy Task Force (DETF) figure-of-merit (FoM) on redshift accuracy, redshift range, survey area, target selection and forecast method. Fitting formulae are provided for convenience. We also consider the dependence on the information used: the full galaxy power spectrum P(k), P(k) marginalized over its shape, or just the Baryon Acoustic Oscillations (BAO). We find that the inclusion of growth rate information (extracted using redshift space distortion and galaxy clustering amplitude measurements) leads to a factor of ∼3 improvement in the FoM, assuming general relativity is not modified. This inclusion partially compensates for the loss of information when only the BAO are used to give geometrical constraints, rather than using the full P(k) as a standard ruler. We find that a space-based galaxy redshift survey covering ∼20 000 deg2 over 0.5≲z≲2 with σz/(1 +z) ≤ 0.001 exploits a redshift range that is only easily accessible from space, extends to sufficiently low redshifts to allow both a vast 3D map of the universe using a single tracer population, and overlaps with ground-based surveys to enable robust modelling of systematic effects. We argue that these parameters are close to their optimal values given current instrumental and practical constraints.

Present and future evidence for evolving dark energy

We compute the Bayesian evidences for one- and two-parameter models of evolving dark energy, and compare them to the evidence for a cosmological constant, using current data from Type Ia supernova, baryon acoustic oscillations, and the cosmic microwave background. We use only distance information, ignoring dark energy perturbations. We find that, under various priors on the dark energy parameters, LambdaCDM is currently favoured as compared to the dark energy models. We consider the parameter constraints that arise under Bayesian model averaging, and discuss the implication of our results for future dark energy projects seeking to detect dark energy evolution. The model selection approach complements and extends the figure-of-merit approach of the Dark Energy Task Force in assessing future experiments, and suggests a significantly-modified interpretation of that statistic.

Effects of cosmological model assumptions on galaxy redshift survey measurements

The clustering of galaxies observed in future redshift surveys will provide a wealth of cosmological information. Matching the signal at different redshifts constrains the dark energy driving the acceleration of the expansion of the Universe. In tandem with these geometrical constraints, redshift-space distortions depend on the build up of large-scale structure. As pointed out by many authors, measurements of these effects are intrinsically coupled. We investigate this link and argue that it strongly depends on the cosmological assumptions adopted when analysing data. Using representative assumptions for the parameters of the Euclid survey in order to provide a baseline future experiment, we show how the derived constraints change due to different model assumptions. We argue that even the assumption of a Friedman–Robertson–Walker space–time is sufficient to reduce the importance of the coupling to a significant degree. Taking this idea further, we consider how the data would actually be analysed and argue that we should not expect to be able to simultaneously constrain multiple deviations from the standard Λ cold dark matter (ΛCDM) model. We therefore consider different possible ways in which the Universe could deviate from the ΛCDM model, and show how the coupling between geometrical constraints and structure growth affects the measurement of such deviations.

Wavelets and Wilkinson Microwave Anisotropy Probe Non-Gaussianity

We study the statistical properties of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data on different scales using the spherical Mexican hat wavelet transform. Consistent with the 2004 results of Vielva et al., we find a deviation from Gaussianity in the form of kurtosis of wavelet coefficients on 3°-4° scales in the southern Galactic hemisphere. This paper extends the work of Vielva et al. as follows: We find that the non-Gaussian signal shows up more strongly in the form of a larger than expected number of cold pixels and also in the form of scale-scale correlations among wavelet coefficients. We establish the robustness of the non-Gaussian signal under more wide-ranging assumptions regarding the Galactic mask applied to the data and the noise statistics. This signal is unlikely to be due to the usual quadratic term parameterized by the nonlinearity parameter fNL. We use the skewness of the spherical Mexican hat wavelet coefficients to constrain fNL with the first-year WMAP data. Our results constrain fNL to be 50 ± 80 at 68% confidence and less than 280 at 99% confidence.We study the statistical properties of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data on different scales using the spherical Mexican hat wavelet transform. Consistent with the 2004 results of Vielva et al., we find a deviation from Gaussianity in the form of kurtosis of wavelet coefficients on 3-4 scales in the southern Galactic hemisphere. This paper extends the work of Vielva et al. as follows: We find that the non-Gaussian signal shows up more strongly in the form of a larger than expected number of cold pixels and also in the form of scale-scale correlations among wavelet coefficients. We establish the robustness of the non-Gaussian signal under more wide-ranging assumptions regarding the Galactic mask applied to the data and the noise statistics. This signal is unlikely to be due to the usual quadratic term parameterized by the nonlinearity parameter fNL. We use the skewness of the spherical Mexican hat wavelet coefficients to constrain fNL with the first-year WMAP data. Our results constrain fNL to be 50+/-80 at 68% confidence and less than 280 at 99% confidence.