Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets

Over the last few years, needlets have a emerged as a useful tool for the analysis of Cosmic Microwave Background (CMB) data. Our aim in this paper is first to introduce in the CMB literature a different form of needlets, known as Mexican needlets, first discussed in the mathematical literature by Geller and Mayeli (2009a,b). We then proceed with an extensive study of the properties of both standard and Mexican needlets; these properties depend on some parameters which can be tuned in order to optimize the performance for a given application. Our second aim in this paper is then to give practical advice on how to adjust these parameters in order to achieve the best properties for a given problem in CMB data analysis. In particular we investigate localization properties in real and harmonic spaces and propose a recipe on how to quantify the influence of galactic and point source masks on the needlet coefficients. We also show that for certain parameter values, the Mexican needlets provide a close approximation to the Spherical Mexican Hat Wavelets (whence their name), with some advantages concerning their numerical implementation and the derivation of their statistical properties.Comment: 40 pages, 11 figures, published version, main modification: added section on more realistic galactic and point source mask

Wilkinson Microwave Anisotropy Probe 7-yr constraints on fNL with a fast wavelet estimator

A new method to constrain the local non-linear coupling parameter fNL based on a fast wavelet decomposition is presented. Using a multiresolution wavelet adapted to the HEALPix pixelization, we have developed a method that is 10^2 times faster than previous estimators based on isotropic wavelets and 10^3 faster than the KSW bispectrum estimator, at the resolution of the Wilkinson Microwave Anisotropy Probe (WMAP) data. The method has been applied to the WMAP 7-yr V+W combined map, imposing constraints on fNL of -69 < fNL < 65 at the 95 per cent CL. This result has been obtained after correcting for the contribution of the residual point sources which has been estimated to be fNL = 7 +/- 6. In addition, a Gaussianity analysis of the data has been carried out using the third order moments of the wavelet coefficients, finding consistency with Gaussianity. Although the constrainsts imposed on fNL are less stringent than those found with optimal estimators, we believe that a very fast method, as the one proposed in this work, can be very useful, especially bearing in mind the large amount of data that will be provided by future experiments, such as the Planck satellite. Moreover, the localisation of wavelets allows one to carry out analyses on different regions of the sky. As an application, we have separately analysed the two hemispheres defined by the dipolar modulation proposed by Hoftuft et al. (2009). We do not find any significant asymmetry regarding the estimated value of fNL in those hemispheres.Comment: 8 pages, 5 figures. Submitted and Accepted for publication in MNRA

Improved constraints on primordial non-Gaussianity for the Wilkinson Microwave Anisotropy Probe 5-yr data

We present new constraints on the non-linear coupling parameter fnl with the Wilkinson Microwave Anisotropy Probe (WMAP) data. We use an updated method based on the spherical Mexican hat wavelet (SMHW) which provides improved constraints on the fnl parameter. This paper is a continuation of a previous work by Curto et al. where several third order statistics based on the SMHW were considered. In this paper, we use all the possible third order statistics computed from the wavelet coefficient maps evaluated at 12 angular scales. The scales are logarithmically distributed from 6.9 arcmin to 500 arcmin. Our analysis indicates that fnl is constrained to -18 < fnl < +80 at 95% confidence level (CL) for the combined V+W WMAP map. This value has been corrected by the presence of undetected point sources, which adds a positive contribution of Delta_fnl = 6 +- 5. Our result excludes at ~99% CL the best-fitting value fnl=87 reported by Yadav & Wandelt. We have also constrained fnl for the Q, V and W frequency bands separately, finding compatibility with zero at 95 % CL for the Q and V bands but not for the W band. We have performed some further tests to understand the cause of this deviation which indicate that systematics associated to the W radiometers could be responsible for this result. Finally we have performed a Galactic North-South analysis for fnl. We have not found any asymmetry, i.e. the best-fitting fnl for the northern pixels is compatible with the best-fitting fnl for the southern pixels.Comment: 6 pages, 4 figures, 4 tables. Accepted for publication in Ap

Non-Gaussian bubbles in the sky

We point out a possible generation mechanism of non-Gaussian bubbles in the sky due to bubble nucleation in the early universe. We consider a curvaton scenario for inflation and assume that the curvaton field phi, whose energy density is subdominant during inflation but which is responsible for the curvature perturbation of the universe, is coupled to another field sigma which undergoes false vacuum decay through quantum tunneling. For this model, we compute the skewness of the curvaton fluctuations due to its interaction with sigma during tunneling, that is, on the background of an instanton solution that describes false vacuum decay. We find that the resulting skewness of the curvaton can become large in the spacetime region inside the bubble. We then compute the corresponding skewness in the statistical distribution of the cosmic microwave background (CMB) temperature fluctuations. We find a non-vanishing skewness in a bubble-shaped region in the sky. It can be large enough to be detected in the near future, and if detected it will bring us invaluable information about the physics in the early universe.Comment: 6 pages, 6 figure

Foreground influence on primordial non-Gaussianity estimates: needlet analysis of WMAP 5-year data

We constrain the amplitude of primordial non-Gaussianity in the CMB data taking into account the presence of foreground residuals in the maps. We generalise the needlet bispectrum estimator marginalizing over the amplitudes of thermal dust, free-free and synchrotron templates. We apply our procedure to WMAP 5 year data, finding fNL= 38\pm 47 (1 \sigma), while the analysis without marginalization provides fNL= 35\pm 42. Splitting the marginalization over each foreground separately, we found that the estimates of fNL are positively cross correlated of 17%, 12% with the dust and synchrotron respectively, while a negative cross correlation of about -10% is found for the free-free component.Comment: Submitted to MNRA

On the optimality of the spherical Mexican hat wavelet estimator for the primordial non-Gaussianity

We study the spherical Mexican hat wavelet (SMHW) as a detector of primordial non-Gaussianity of the local type on the Cosmic Microwave Background (CMB) anisotropies. For this purpose we define third order statistics based on the wavelet coefficient maps and the original map. We find the dependence of these statistics in terms of the non-linear coupling parameter fnl and the bispectrum of this type of non-Gaussianity. We compare the analytical values for these statistics with the results obtained with non-Gaussian simulations for an ideal full-sky CMB experiment without noise. We study the power of this method to detect fnl, i. e. the variance of this parameter, and compare it with the variance obtained from the primary bispectrum for the same experiment. Finally we apply our wavelet based estimator on WMAP-like maps with incomplete sky and inhomogeneous noise and compare with the optimal bispectrum estimator. The results show that the wavelet cubic statistics are as efficient as the bispectrum as optimal detectors of this type of primordial non-Gaussianity.Comment: 10 pages, 9 figures, 1 table. Minor revision, references added, accepted for publication in MNRA

Constraining the WMAP9 bispectrum and trispectrum with needlets

We develop a needlet approach to estimate the amplitude of general (including non-separable) bispectra and trispectra in the cosmic microwave background, and apply this to the WMAP 9-year data. We obtain estimates for the `orthogonal' bispectrum mode, yielding results which are consistent with the WMAP 7-year data. We do not observe the frequency-dependence suggested by the WMAP team's analysis of the 9-year data. We present 1-$\sigma$ constraints on the `local' trispectrum shape \gnl/10^5= -4.1\pm 2.3, the `$c1$' equilateral model \gnl^{c_1}/10^6= -0.8\pm 2.9, and the constant model \gnl^{\rm{const}}/10^6= -0.2\pm 1.8, together with a $95\%$ confidence-level upper bound on the multifield local parameter \taunl<22000. We estimate the bias on these parameters produced by point sources. The techniques developed in this paper should prove useful for other datasets such as Planck.Comment: 21 pages - matches published versio

The Weyl curvature conjecture and black hole entropy

The universe today, with structure such as stars, galaxies and black holes, seems to have evolved from a very homogeneous initial state. From this it appears as if the entropy of the universe is decreasing, in violation of the second law of thermodynamics. It has been suggested by Roger Penrose \cite{grossmann:penrose:wcc} that this inconsistency can be solved if one assigns an entropy to the spacetime geometry. He also pointed out that the Weyl tensor has the properties one would expect to find in a description of a gravitational entropy. In this article we make an attempt to use this so-called Weyl curvature conjecture to describe the Hawking-Bekenstein entropy of black holes and the entropy of horizons due to a cosmological constant.Comment: 10 pages, 2 figures, typesetting revtex

Constraining Primordial Non-Gaussianity with High-Redshift Probes

We present an analysis of the constraints on the amplitude of primordial non-Gaussianity of local type described by the dimensionless parameter $f_{\rm NL}$. These constraints are set by the auto-correlation functions (ACFs) of two large scale structure probes, the radio sources from NRAO VLA Sky Survey (NVSS) and the quasar catalogue of Sloan Digital Sky Survey Release Six (SDSS DR6 QSOs), as well as by their cross-correlation functions (CCFs) with the cosmic microwave background (CMB) temperature map (Integrated Sachs-Wolfe effect). Several systematic effects that may affect the observational estimates of the ACFs and of the CCFs are investigated and conservatively accounted for. Our approach exploits the large-scale scale-dependence of the non-Gaussian halo bias. The derived constraints on {$f_{\rm NL}$} coming from the NVSS CCF and from the QSO ACF and CCF are weaker than those previously obtained from the NVSS ACF, but still consistent with them. Finally, we obtain the constraints on $f_{\rm NL}=53\pm25$ ($1\,\sigma$) and $f_{\rm NL}=58\pm24$ ($1\,\sigma$) from NVSS data and SDSS DR6 QSO data, respectively.Comment: 16 pages, 8 figures, 1 table, Accepted for publication on JCA

Hunting for Primordial Non-Gaussianity in the Cosmic Microwave Background

Since the first limit on the (local) primordial non-Gaussianity parameter, fNL, was obtained from COBE data in 2002, observations of the CMB have been playing a central role in constraining the amplitudes of various forms of non-Gaussianity in primordial fluctuations. The current 68% limit from the 7-year WMAP data is fNL=32+/-21, and the Planck satellite is expected to reduce the uncertainty by a factor of four in a few years from now. If fNL>>1 is found by Planck with high statistical significance, all single-field models of inflation would be ruled out. Moreover, if the Planck satellite finds fNL=30, then it would be able to test a broad class of multi-field models using the four-point function (trispectrum) test of tauNL>=(6fNL/5)^2. In this article, we review the methods (optimal estimator), results (WMAP 7-year), and challenges (secondary anisotropy, second-order effect, and foreground) of measuring primordial non-Gaussianity from the CMB data, present a science case for the trispectrum, and conclude with future prospects.Comment: 33 pages, 4 figures. Invited review, accepted for publication in the CQG special issue on nonlinear cosmological perturbations. (v2) References added. More clarifications are added to the second-order effect and the multi-field consistency relation, tauNL>=(6fNL/5)^2