On Nonlinear Functionals of Random Spherical Eigenfunctions

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.Comment: 24 page

Angular Spectra for non-Gaussian Isotropic Fields

Cosmic Microwave Background (CMB) Anisotropies is a subject of intensive research in several fields of sciences. In this paper we start a systematic development of basic notions and theory in statistics according to the application for CMB. The main result of this paper is the necessary and sufficient condition for isotropy of a non-Gaussian field in terms of spectra. Clear formulae for bi-, tri- and polyspectra and bi-, tri-, and higher order covariances are also given. Keywords: Bispectrum, Trispectrum, Angular poly-Spectra, Cosmic microwave background radiation; Gaussianity; spherical random fieldsComment: 53 page

The Integrated Bispectrum and Beyond

The position-dependent power spectrum has been recently proposed as a descriptor of gravitationally induced non-Gaussianity in galaxy clustering, as it is sensitive to the "soft limit" of the bispectrum (i.e. when one of the wave number tends to zero). We generalise this concept to higher order and clarify their relationship to other known statistics such as the skew-spectrum, the kurt-spectra and their real-space counterparts the cumulants correlators. Using the {\em Hierarchical Ansatz} (HA) as a toy model for the higher order correlation hierarchy, we show how in the soft limit, polyspectra at a given order can be identified with lower order polyspectra with the same geometrical dependence but with {\em renormalised} amplitudes expressed in terms of amplitudes of the original polyspectra. We extend the concept of position-dependent bispectrum to bispectrum of the divergence of the velocity field Θ and mixed multispectra involving δ and Θ in the 3D perturbative regime. To quantify the effects of transients in numerical simulations, we also present results for lowest order in Lagrangian perturbation theory (LPT) or the Zel'dovich approximation (ZA). Finally, we discuss how to extend the position-dependent spectrum concept to encompass cross-spectra. And finally study the application of this concept to two dimensions (2D), for projected galaxy maps, convergence κ maps from weak-lensing surveys or maps of CMB secondaries e.g. the frequency cleaned y - parameter maps of thermal Sunyaev-Zel'dovich (tSZ) effect from CMB surveys

Phase Correlations in Cosmic Microwave Background Temperature Maps

We study the statistical properties of spherical harmonic modes of temperature maps of the cosmic microwave background. Unlike other studies, which focus mainly on properties of the amplitudes of these modes, we look instead at their phases. In particular, we present a simple measure of phase correlation that can be diagnostic of departures from the standard assumption that primordial density fluctuations constitute a statistically homogeneous and isotropic Gaussian random field, which should possess phases that are uniformly random on the unit circle. The method we discuss checks for the uniformity of the distribution of phase angles using a non-parametric descriptor based on the use order statistics, which is known as Kuiper's statistic. The particular advantage of the method we present is that, when coupled to the judicious use of Monte Carlo simulations, it can deliver very interesting results from small data samples. In particular, it is useful for studying the properties of spherical harmonics at low l for which there are only small number of independent values of m and which therefore furnish only a small number of phases for analysis. We apply the method to the COBE-DMR and WMAP sky maps, and find departures from uniformity in both. In the case of WMAP, our results probably reflect Galactic contamination or the known variation of signal-to-noise across the sky rather than primordial non-Gaussianity.Comment: 18 pages, 4 figures, accepted for publication in MNRA

Resumming Cosmological Perturbations via the Lagrangian Picture: One-loop Results in Real Space and in Redshift Space

We develop a new approach to study the nonlinear evolution in the large-scale structure of the Universe both in real space and in redshift space, extending the standard perturbation theory of gravitational instability. Infinite series of terms in standard Eulerian perturbation theory are resummed as a result of our starting from a Lagrangian description of perturbations. Delicate nonlinear effects on scales of the baryon acoustic oscillations are more accurately described by our method than the standard one. Our approach differs from other resummation techniques recently proposed, such as the renormalized perturbation theory, etc., in that we use simple techniques and thus resulting equations are undemanding to evaluate, and in that our approach is capable of quantifying the nonlinear effects in redshift space. The power spectrum and correlation function of our approach are in good agreement with numerical simulations in literature on scales of baryon acoustic oscillations. Especially, nonlinear effects on the baryon acoustic peak of the correlation function are accurately described both in real space and in redshift space. Our approach provides a unique opportunity to analytically investigate the nonlinear effects on baryon acoustic scales in observable redshift space, which is requisite in constraining the nature of dark energy, the curvature of the Universe, etc., by redshift surveys.Comment: 18 pages, 12 figures, replaced to match the published versio

On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics

We prove a Central Limit Theorem for the Critical Points of Random Spherical Harmonics, in the High-Energy Limit. The result is a consequence of a deeper characterizations of the total number of critical points, which are shown to be asymptotically fully correlated with the sample trispectrum, i.e., the integral of the fourth Hermite polynomial evaluated on the eigenfunctions themselves. As a consequence, the total number of critical points and the nodal length are fully correlated for random spherical harmonics, in the high-energy limit