The two fundamental assumptions of the standard cosmological model -that the initial fluctuations are statistically isotropic and Gaussian -are rigorously tested using maps of the cosmic microwave background (CMB) anisotropy from the Planck satellite. The detailed results are based on studies of four independent estimates of the CMB that are compared to simulations using a fiducial ΛCDM model and incorporating essential aspects of the Planck measurement process. Deviations from isotropy have been found and demonstrated to be robust against component separation algorithm, mask choice, and frequency dependence. Many of these anomalies were previously observed in the WMAP data, and are now confirmed at similar levels of significance (about 3σ). However, we find little evidence of non-Gaussianity, with the exception of a few statistical signatures that seem to be associated with specific anomalies. In particular, we find that the quadrupole-octopole alignment is also connected to a low observed variance in the CMB signal. A power asymmetry is now found to persist on scales corresponding to about = 600 and can be described in the low-regime by a phenomenological dipole modulation model. However, any primordial power asymmetry is strongly scale-dependent and does not extend to arbitrarily small angular scales. Finally, it is plausible that some of these features may be reflected in the angular power spectrum of the data, which shows a deficit of power on similar scales. Indeed, when the power spectra of two hemispheres defined by a preferred direction are considered separately, one shows evidence of a deficit in power, while its opposite contains oscillations between odd and even modes that may be related to the parity violation and phase correlations also detected in the data. Although these analyses represent a step forward in building an understanding of the anomalies, a satisfactory explanation based on physically motivated models is still lacking.

We test the statistical isotropy and Gaussianity of the cosmic microwave background (CMB) anisotropies using observations made by the Planck satellite. Our results are based mainly on the full Planck mission for temperature, but also include some polarization measurements. In particular, we consider the CMB anisotropy maps derived from the multi-frequency Planck data by several component-separation methods. For the temperature anisotropies, we find excellent agreement between results based on these sky maps over both a very large fraction of the sky and a broad range of angular scales, establishing that potential foreground residuals do not affect our studies. Tests of skewness, kurtosis, multi-normality, N-point functions, and Minkowski functionals indicate consistency with Gaussianity, while a power deficit at large angular scales is manifested in several ways, for example low map variance. The results of a peak statistics analysis are consistent with the expectations of a Gaussian random field. The "Cold Spot" is detected with several methods, including map kurtosis, peak statistics, and mean temperature profile. We thoroughly probe the large-scale dipolar power asymmetry, detecting it with several independent tests, and address the subject of a posteriori correction. Tests of directionality suggest the presence of angular clustering from large to small scales, but at a significance that is dependent on the details of the approach. We perform the first examination of polarization data, finding the morphology of stacked peaks to be consistent with the expectations of statistically isotropic simulations. Where they overlap, these results are consistent with the Planck 2013 analysis based on the nominal mission data and provide our most thorough view of the statistics of the CMB fluctuations to date.

We study the power asymmetry between even and odd multipoles in the multipolar expansion of cosmic microwave background temperature data from the Wilkinson Microwave Anisotropy Probe (WMAP), recently reported in the literature. We introduce an alternate statistic which probes this effect more sensitively. We find that the data are highly anomalous and consistently outside the 2σ significance level in the whole multipole range l = [2, 101]. We examine the possibility that this asymmetry may be caused by the foreground cleaning procedure or by residual foregrounds. By direct simulations we rule out this possibility. We also examine several possible subdominant foregrounds, which might lead to such an asymmetry. However, in all cases we are unable to explain the signal seen in data. We next examine cleaned maps, using procedures other than that followed by the WMAP science team. In particular, we analysed the maps cleaned by the Internal Powar Spectrum Estimation (IPSE), needlets and the harmonic Internal Linear Combination (ILC) procedures. In all these cases, we also find a statistically significant signal of power asymmetry if the power spectrum is estimated from the masked sky. However, the significance level is found to be not as high as that in the case of the WMAP best‐fitting power spectrum. Finally, we test the contribution of low‐l multipoles to the observed power asymmetry. We find that if we eliminate the first six multipoles, l = [2, 7], the significance falls below the 2σ confidence level. Hence, we find that the signal gets dominant contribution from low‐l modes.

We relate the observed hemispherical anisotropy in the cosmic microwave radiation data to an inhomogeneous power spectrum model. The hemispherical anisotropy can be parameterized in terms of the dipole modulation model. This model leads to correlations between spherical harmonic coefficients corresponding to multipoles, l and l + 1. We extract the l dependence of the dipole modulation amplitude, A, by making a fit to the WMAP and PLANCK CMBR data. We propose an inhomogeneous power spectrum model and show that it also leads to correlations between multipoles, l and l + 1. The model parameters are determined by making a fit to the data. The spectral index of the inhomogeneous power spectrum is found to be consistent with zero.

We study several anisotropic inflationary models and their implications for the observed violation of statistical isotropy in the CMBR data. In two of these models the anisotropy decays very quickly during the inflationary phase of expansion. We explicitly show that these models lead to violation of isotropy only for low l CMBR modes. Our primary aim is to fit the observed alignment of l = 2, 3 multipoles to the theoretical models. We use two measures, based on the power tensor, which contains information about the alignment of each multipole, to quantify the anisotropy in data. One of the measures uses the dispersion in eigenvalues of the power tensor. We also define another measure which tests the overall correlation between two different multipoles. We perturbatively compute these measures of anisotropy and fix the theoretical parameters by making a best fit to l = 2, 3 multipoles. We show that some of the models studied are able to consistently explain the observed violation of statistical isotropy.

We show that perturbations generated during the anisotropic preinflationary stage of cosmic evolution may affect cosmological observations today for a certain range of parameters. Due to the anisotropic nature of the universe during such early times, it might explain some of the observed signals of large scale anisotropy. In particular we argue that the alignment of CMB quadrupole and octopole may be explained by the Sachs-Wolfe effect due to the large scale anisotropic modes from very early times of cosmological evolution. We also comment on how the observed dipole modulation of CMB power may be explained within this framework.approximation. This axis is found to point towards (l = 237.64 o , b = 62.95 o ), approximately towards the Virgo cluster of galaxies, and makes an angle of approximately 27 o with the galactic axis (Aluri et

We consider a generalization of Einstein's general theory of relativity such that it respects local scale invariance. This requires the introduction of a scalar and a vector field in the action. We show that the theory naturally displays both dark energy and dark matter. We solve the resulting equations of motion assuming an FRW metric.The solutions are found to be almost identical to those corresponding to the standard ΛCDM model.

The Cosmological Principle (CP) -- the notion that the Universe is spatially isotropic and homogeneous on large scales -- underlies a century of progress in cosmology. It is conventionally formulated through the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmologies as the spacetime metric, and culminates in the successful and highly predictive $\Lambda$-Cold-Dark-Matter ($\Lambda$CDM) model. Yet, tensions have emerged within the $\Lambda$CDM model, most notably a statistically significant discrepancy in the value of the Hubble constant, $H_0$. Since the notion of cosmic expansion determined by a single parameter is intimately tied to the CP, implications of the $H_0$ tension may extend beyond $\Lambda$CDM to the CP itself. This review surveys current observational hints for deviations from the expectations of the CP, highlighting synergies and disagreements that warrant further study. Setting aside the debate about individual large structures, potential deviations from the CP include variations of cosmological parameters on the sky, discrepancies in the cosmic dipoles, and mysterious alignments in quasar polarizations and galaxy spins. While it is possible that a host of observational systematics are impacting results, it is equally plausible that precision cosmology may have outgrown the FLRW paradigm, an extremely pragmatic but non-fundamental symmetry assumption.