Fast directional continuous spherical wavelet transform algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE Trans. Sig. Pro

Limits on non-Gaussianities from WMAP data

We develop a method to constrain the level of non-Gaussianity of density perturbations when the 3-point function is of the "equilateral" type. Departures from Gaussianity of this form are produced by single field models such as ghost or DBI inflation and in general by the presence of higher order derivative operators in the effective Lagrangian of the inflaton. We show that the induced shape of the 3-point function can be very well approximated by a factorizable form, making the analysis practical. We also show that, unless one has a full sky map with uniform noise, in order to saturate the Cramer-Rao bound for the error on the amplitude of the 3-point function, the estimator must contain a piece that is linear in the data. We apply our technique to the WMAP data obtaining a constraint on the amplitude f_NL^equil of "equilateral" non-Gaussianity: -366 < f_NL^equil < 238 at 95% C.L. We also apply our technique to constrain the so-called "local" shape, which is predicted for example by the curvaton and variable decay width models. We show that the inclusion of the linear piece in the estimator improves the constraint over those obtained by the WMAP team, to -27 < f_NL^local < 121 at 95% C.L.Comment: 20 pages, 12 eps figure

Estimators for local non-Gaussianities

We study the Likelihood function of data given f_NL for the so-called local type of non-Gaussianity. In this case the curvature perturbation is a non-linear function, local in real space, of a Gaussian random field. We compute the Cramer-Rao bound for f_NL and show that for small values of f_NL the 3-point function estimator saturates the bound and is equivalent to calculating the full Likelihood of the data. However, for sufficiently large f_NL, the naive 3-point function estimator has a much larger variance than previously thought. In the limit in which the departure from Gaussianity is detected with high confidence, error bars on f_NL only decrease as 1/ln Npix rather than Npix^-1/2 as the size of the data set increases. We identify the physical origin of this behavior and explain why it only affects the local type of non-Gaussianity, where the contribution of the first multipoles is always relevant. We find a simple improvement to the 3-point function estimator that makes the square root of its variance decrease as Npix^-1/2 even for large f_NL, asymptotically approaching the Cramer-Rao bound. We show that using the modified estimator is practically equivalent to computing the full Likelihood of f_NL given the data. Thus other statistics of the data, such as the 4-point function and Minkowski functionals, contain no additional information on f_NL. In particular, we explicitly show that the recent claims about the relevance of the 4-point function are not correct. By direct inspection of the Likelihood, we show that the data do not contain enough information for any statistic to be able to constrain higher order terms in the relation between the Gaussian field and the curvature perturbation, unless these are orders of magnitude larger than the size suggested by the current limits on f_NL.Comment: 26 pages. v2: added comments about the approximations used, published JCAP versio

Non-Gaussianity detections in the Bianchi VIIh corrected WMAP 1-year data made with directional spherical wavelets

Many of the current anomalies reported in the Wilkinson Microwave Anisotropy Probe (WMAP) 1-year data disappear after `correcting' for the best-fit embedded Bianchi type VII_h component (Jaffe et al. 2005), albeit assuming no dark energy component. We investigate the effect of this Bianchi correction on the detections of non-Gaussianity in the WMAP data that we previously made using directional spherical wavelets (McEwen et al. 2005a). As previously discovered by Jaffe et al. (2005), the deviations from Gaussianity in the kurtosis of spherical Mexican hat wavelet coefficients are eliminated once the data is corrected for the Bianchi component. This is due to the reduction of the cold spot at Galactic coordinates (l,b)=(209^\circ,-57\circ), which Cruz et al. (2005) claim to be the source of non-Gaussianity introduced in the kurtosis. Our previous detections of non-Gaussianity observed in the skewness of spherical wavelet coefficients are not reduced by the Bianchi correction. Indeed, the most significant detection of non-Gaussianity made with the spherical real Morlet wavelet at a significant level of 98.4% remains (using a very conservative method to estimate the significance). We make our code to simulate Bianchi induced temperature fluctuations publicly available.Comment: 11 pages, 8 figures, replaced to match version accepted by MNRA

A high-significance detection of non-Gaussianity in the WMAP 1-year data using directional spherical wavelets

A directional spherical wavelet analysis is performed to examine the Gaussianity of the WMAP 1-year data. Such an analysis is facilitated by the introduction of a fast directional continuous spherical wavelet transform. The directional nature of the analysis allows one to probe orientated structure in the data. Significant deviations from Gaussianity are detected in the skewness and kurtosis of spherical elliptical Mexican hat and real Morlet wavelet coefficients for both the WMAP and Tegmark et al. (2003) foreground-removed maps. The previous non-Gaussianity detection made by Vielva et al. (2003) using the spherical symmetric Mexican hat wavelet is confirmed, although their detection at the 99.9% significance level is only made at the 95.3% significance level using our most conservative statistical test. Furthermore, deviations from Gaussianity in the skewness of spherical real Morlet wavelet coefficients on a wavelet scale of 550 arcmin (corresponding to an effective global size on the sky of approximately 26 degrees and an internal size of 3 degrees) at an azimuthal orientation of 72 degrees, are made at the 98.3% significance level, using the same conservative method. The wavelet analysis inherently allows one to localise on the sky those regions that introduce skewness and those that introduce kurtosis. Preliminary noise analysis indicates that these detected deviation regions are not atypical and have average noise dispersion. Further analysis is required to ascertain whether these detected regions correspond to secondary or instrumental effects, or whether in fact the non-Gaussianity detected is due to intrinsic primordial fluctuations in the cosmic microwave background.Comment: 14 pages, 10 figures, references added, replaced to match version accepted by MNRA

A high-significance detection of non-Gaussianity in the WMAP 5-year data using directional spherical wavelets

We repeat the directional spherical real Morlet wavelet analysis, used to detect non-Gaussianity in the Wilkinson Microwave Anisotropy Probe (WMAP) 1-year and 3-year data (McEwen et al. 2005, 2006a), on the WMAP 5-year data. The non-Gaussian signal detected previously is present in the 5-year data at a slightly increased statistical significance of approximately 99%. Localised regions that contribute most strongly to the non-Gaussian signal are found to be very similar to those detected in the previous releases of the WMAP data. When the localised regions detected in the 5-year data are excluded from the analysis the non-Gaussian signal is eliminated.Comment: 4 pages, 4 figures, replaced to match version accepted by MNRAS, masks available for downloa

The inflationary trispectrum

We calculate the trispectrum of the primordial curvature perturbation generated by an epoch of slow-roll inflation in the early universe, and demonstrate that the non-gaussian signature imprinted at horizon crossing is unobservably small, of order tau_NL < r/50, where r < 1 is the tensor-to-scalar ratio. Therefore any primordial non-gaussianity observed in future microwave background experiments is likely to have been synthesized by gravitational effects on superhorizon scales. We discuss the application of Maldacena's consistency condition to the trispectrum.Comment: 23 pages, 2 diagrams drawn with feynmp.sty, uses iopart.cls. v2, replaced with version accepted by JCAP. Estimate of maximal tau_NL refined in Section 5, resulting in smaller numerical value. Sign errors in Eq. (44) and Eq. (48) corrected. Some minor notational change

Primordial Non-Gaussianities of General Multiple Field Inflation

We perform a general study of the primordial scalar non-Gaussianities in multi-field inflationary models in Einstein gravity. We consider models governed by a Lagrangian which is a general function of the scalar fields and their first spacetime derivatives. We use $\delta N$ formalism to relate scalar fields and curvature perturbations. We calculate the explicit cubic order perturbation action and the three-point function of curvature perturbation evaluated at horizon-crossing. Under reasonable assumptions, in the limit of small slow-varying parameters and a sound speed $c_s$ close to one, we find that the non-Gaussianity is completely determined by these slow-varying parameters and some other parameters determined by the structure of the inflationary models. Our work generalizes previous results, and would be useful to study non-Gaussianity in multi-field inflationary models that will be constructed in the future.Comment: 26 pages, no figure; v2, minor revision; v3 minor misprints corrected; v4 minor misprints correcte

Volume Modulus Inflation and the Gravitino Mass Problem

The Hubble constant during the last stages of inflation in a broad class of models based on the KKLT mechanism should be smaller than the gravitino mass, H <~ m_{3/2}. We point out that in the models with large volume of compactification the corresponding constraint typically is even stronger, H <~ m_{3/2}^{3/2}, in Planck units. In order to address this problem, we propose a class of models with large volume of compactification where inflation may occur exponentially far away from the present vacuum state. In these models, the Hubble constant during inflation can be many orders of magnitude greater than the gravitino mass. We introduce a toy model describing this scenario, and discuss its strengths and weaknesses.Comment: 24 pages, JHEP style; v2. refs adde

Planck 2013 results. XXVI. Background geometry and topology of the Universe

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