>Magister Scientiae - MSc / The Cosmological Principle forms part of one of the most fundamental hypotheses of modern Cosmology. So it is very important to assess whether it holds true using observational data, or whether it consists of a mathematical simplification. We probe the statistical isotropy of the Universe using the existing radio continuum data, by means of a local variance estimator. In order to investigate this, we analyse the number count variance of the radio catalog by looking at patches of approximately 10, 15, 20 & 25 degrees in radii, and thus comparing it to mock catalogs which reproduce the matter density power spectrum, as well as the same sky coverage of the real data. We establish criteria for accepting patches that have more than 90%, 70% & 50% of their pixels not masked. We make use of the NRAO VLA Sky Survey (NVSS), whose operational frequency is 1.4 GHz. We perform statistical tests for detecting possible departures from statistical isotropy using galaxy number counts with flux limits of 20 < SNVSS < 1000 mJy. We also compare the real data to the mock catalogs of the radio data in order to assess the statistical significance of our results. We use the local variance estimator for testing the statistical isotropy of our data sample. We find that the statistical properties of our sample are in reasonable agreement with the standard cosmological model. The mean of the distribution for the data falls well within the 95% confidence interval of the average of the simulated mocks. For all the radii and acceptance criteria for the patches, we found no significant deviations beyond those allowed by the standard model. As expected there were no large discrepancies between our mocks and the data. The results are consistent with statistical isotropy.

>Magister Scientiae - MSc / In the standard (‘concordance’) model of Cosmology, there is a fundamental assumption that the Universe is statistically isotropic and homogeneous on large scales, known as the Cosmological Principle. The Cosmological Principle requires that the dipole anisotropy apparent in the CMB should also be observed in galaxy number counts if this signal occurs due to the aberration and Doppler effects from our peculiar motion. This thesis will investigate the accuracy with which the cosmic kinematic dipole can be determined by comparing real data from NRAO VLA Sky Survey (NVSS) catalog with the simulated sky maps following its specifications. The mock maps are generated using FLASK code which assumes a lognormal distribution for the radio count density field from z=0 to z =4 and taking as an entry an angular power spectrum from CAMB which assumed a flat ΛCDM cosmology and a redshift distribution. After analising the kinematic dipole, we turn to the analysis of statistical isotropy in the catalog. We used ANalysis Of Variance (ANOVA) test on patches in the sky of different radii as one of the statistical tools for the analysis. We found that as we go to a higher radius for the patches, we have a better agreement between the theory and the observation as expected. We also saw that the more we are rigorous on the rejection criteria, the smaller is the discrepancy between the observed and simulated number count distribution in the sky. We found an optimum choice of 25◦ as patch size, and if the accepted patches have a maximum of 30% of their pixels masked. Therefore, we find that the NVSS data agrees with the fundamental assumption of statistical isotropy at angular scales > 20◦.

This thesis contributes to the field of dark energy and observational cosmology. We have investigated possible direction dependent systematic signal and non-Gaussian features in the supernovae (SNe) Type Ia data. To detect these effects we propose a new method of analysis. Although We have used this technique on SNe Ia data, it is quite general and can be applied to other data sets as well. SNe Ia are the most precise known distance indicators at the cosmological distances. Their constant peak luminosity(after correction) makesthem standard candles and hence one can measure the distances in the universe using SNe Ia. This distance measurement can determine various cosmological parameters such as the Hubble constant, various components of matter density and dark energy from, the SNe Ia observations. Recent SNe Ia observations have shown that the expansion of the universe is currently accelerating. This recent acceleration is explained by invoking a component in the universe having negative pressure and is termed as dark energy. It can be described by a homogeneous and isotropic fluid with the equation of state P = wρ, where w is allowed to be negative. A constant(Λ) in the Einstein equation(known as cosmological constant) can explain the acceleration, in the fluid model it can be modeled with w = -1. Other models of dark energy with w = -1 can also explain the acceleration, however the precise nature of this mysterious component remains unknown. Although there exist a wide range of dark energy models, cosmological constant provides the simplest explanation to the acceleration of the expansion of the Universe. The equation of state parameter w has been investigated by recent surveys but the results are still consistent with a wide range of dark energy models. In order to discriminate among various cosmological models we need an even more precise measurement of distance and error bars in the SNe Ia data. From the central limit theorem we expect Gaussian errors in any experiment that is free from systematic noise. However in astronomy we do not have a control over the observed phenomena and thus can not control the systematic errors (due to some physical processes in the Universe) in the observed data. The only possible way to deal with such data is by using appropriate statistical techniques. Among these systematic features the direction dependent features are more dangerous ones since they may indicate a preferred direction in the Universe. To address the issue of direction dependent features we have developed a new technique(Δ statistic henceforth) which is based on the extreme value theory. We have applied this technique to the available high-z SNe Ia data from Riess et al.(2004)and Riess et al.(2007). In addition we have applied it to the HST data from HST key project for H0 measurement. Below we summarize the material presented in the thesis. Chapter wise summary of the thesis In the first chapter we present an introductory discussion of the various basic cosmological notions eg. Cosmological Principle (CP), observational evidence in support of CP and departures from it, distance measures and large scale structure. The observed departures from the CP could be present due to the systematic errors and/or non-Gaussian error bars in the data. We discuss the errors involved in the measurement process Basics of statistical techniques : In the next two chapters we discuss basics of the statistical techniques used in this thesis and extreme value theory. Extreme value theory describes how to calculate the distribution of extreme events. The simplest of the distributions of the extremes is known as the Gumbel distribution. We discuss features of the Gumbel distribution since it is used extensively in our analysis. Δ statistic and features in the SNe data : In the fourth chapter we derive Δ statistic and apply it to the SNe Ia data sets. An outline of the Δ statistic is as follows : a) We define a plane which cuts the sky into hemispheres. This plane will divide the data into two subsets, one in each hemisphere. b) Now we calculate the χ2 in each hemisphere for an FRW universe assuming a flat geometry. c) The difference of χ2 in the two hemisphere is calculated and maximized by rotating the plane. This maximum should follow the Gumbel distribution. Since it is difficult to calculate the analytic form of Gumbel distribution we calculate it numerically assuming Gaussian error bars. This gives the theoretical distribution for the above calculated maximum of difference of χ2 . The results indicate that GD04 shows systematic effects as well non-Gaussian features while the set GD07 is better in terms of systematic effects and non-Gaussian features. Non-Gaussian features in the H0 data : HST key project measures the value of Hubble constant at the level of 10% accuracy, which requires precise measurement of the distances. It uses various methods to measure distance for instance SNe Ia, Tully-Fisher relation, surface-brightness fluctuations etc. In the fifth chapter we apply Δ statistic to the HST Key Project data in order to check the presence of non-Gaussian and direction dependent features. Our results show that although this data set seems to be free of direction dependent features, it is inconsistent with the Gaussian errors. Analytic Marginalization : The quantities of real interest in cosmology are ΩM and ΩΛ, Hubble constant could in principle be treated as a nuisance parameter. It would be useful to marginalize over the nuisance parameter. Although it can be done numerically using Bayesian method, Δ statistic does not allow it. In chapter six we propose a method to marginalize over H0 analytically. The χ2 in this case is a complicated function of errors in the data. We compare this analytic method with the Bayesian marginalization method and results show that the two methods are quite consistent. We apply the Δ statistic to the SNe data after the analytic marginalization. Results do not change much indicating the insensitivity of the direction de-pendent features to the Hubble constant. A variation to the Δ statistic: As has been discussed earlier that, it is difficult to calculate the theoretical distribution of Δ in general. However if the parent distribution follows certain conditions it is possible to derive the analytic form for the Gumbel distribution for Δ. In the seventh chapter we derive a variation to the Δ statistic in a way that allows us to calculate the analytic distribution. The results in this case are different from those presented earlier, but they confirm the same direction dependence and non-Gaussian features in the data.