Wavelets Applied to CMB Maps: a Multiresolution Analysis for Denoising

Analysis and denoising of Cosmic Microwave Background (CMB) maps are performed using wavelet multiresolution techniques. The method is tested on $12^{\circ}.8\times 12^{\circ}.8$ maps with resolution resembling the experimental one expected for future high resolution space observations. Semianalytic formulae of the variance of wavelet coefficients are given for the Haar and Mexican Hat wavelet bases. Results are presented for the standard Cold Dark Matter (CDM) model. Denoising of simulated maps is carried out by removal of wavelet coefficients dominated by instrumental noise. CMB maps with a signal-to-noise, $S/N \sim 1$, are denoised with an error improvement factor between 3 and 5. Moreover we have also tested how well the CMB temperature power spectrum is recovered after denoising. We are able to reconstruct the $C_{\ell}$'s up to $l\sim 1500$ with errors always below $20$ in cases with $S/N \ge 1$.Comment: latex file 9 pages + 5 postscript figures + 1 gif figure (figure 6), to be published in MNRA

Contributions of point extragalactic sources to the Cosmic Microwave Background bispectrum

All the analyses of Cosmic Microwave Background (CMB) temperature maps up--to--date show that CMB anisotropies follow a Gaussian distribution. On the other hand, astrophysical foregrounds which hamper the detection of the CMB angular power spectrum, are not Gaussian distributed on the sky. Therefore, they should give a sizeable contribution to the CMB bispectrum. In fact, the first year data of the Wilkinson Microwave Anisotropy Probe (WMAP) mission have allowed the {\it first} detection of the extragalactic source contribution to the CMB bispectrum at 41 GHz and, at the same time, much tighter limits than before to non--Gaussian primordial fluctuations. In view of the above and for achieving higher precision in current and future CMB measurements of non--Gaussianity, in this paper we discuss a comprehensive assessment of the bispectrum due to either uncorrelated and clustered extragalactic point sources in the whole frequency interval around the CMB intensity peak. Our calculations, based on current cosmological evolution models for sources, show that the reduced angular bispectrum due to point sources, $b_{ps}$, should be detectable in all WMAP and Planck frequency channels. We also find agreement with the results on $b_{ps}$ at 41 GHz coming from the analysis of the first year WMAP data. Moreover, by comparing $b_{ps}$ with the primordial reduced CMB bispectrum, we find that only the peak value of the primordial bispectrum (which appears at $l\simeq 200$) results greater than $b_{ps}$ in a frequency window around the intensity peak of the CMB. The amplitude of this window basically depends on the capability of the source detection algorithms (i.e., on the achievable flux detection limit, $S_{lim}$, for sources).Comment: 26 pages, 6 Figures, use AasTex5.0, ApJ, in press, Oct. 10, 2003 Issu

Filter Design for the Detection/Estimation of the Modulus of a Vector. Application to Polarization Data

We consider a set of M images, whose pixel intensities at a common point can be treated as the components of a M-dimensional vector. We are interested in the estimation of the modulus of such a vector associated to a compact source. For instance, the detection/estimation of the polarized signal of compact sources immersed in a noisy background is relevant in some fields like Astrophysics. We develop two different techniques, one based on the Maximum Likelihood Estimator (MLE) applied to the modulus distribution, the modulus filter (ModF) and other based on prefiltering the components before fusion, the filtered fusion (FF), to deal with this problem. We present both methods in the general case of M images and apply them to the particular case of three images (linear plus circular polarization). Numerical simulations have been performed to test these filters considering polarized compact sources immersed in stationary noise. The FF performs better than the ModF in terms of errors in the estimated amplitude and position of the source, especially in the low signal-to-noise case. We also compare both methods with the direct application of a matched filter (MF) on the polarization data. This last technique is clearly outperformed by the new methods.Comment: 18 pages, 3 figures, accepted for publication in Signal Processin

Isotropic Wavelets: a Powerful Tool to Extract Point Sources from CMB Maps

It is the aim of this paper to introduce the use of isotropic wavelets to detect and determine the flux of point sources appearing in CMB maps. The most suited wavelet to detect point sources filtered with a Gaussian beam is the Mexican Hat. An analytical expression of the wavelet coefficient obtained in the presence of a point source is provided and used in the detection and flux estimation methods presented. For illustration the method is applied to two simulations (assuming Planck Mission characteristics) dominated by CMB (100 GHz) and dust (857 GHz) as these will be the two signals dominating at low and high frequency respectively in the Planck channels. We are able to detect bright sources above 1.58 Jy at 857 GHz (82% of all sources) and above 0.36 Jy at 100 GHz (100% of all) with errors in the flux estimation below 25%. The main advantage of this method is that nothing has to be assumed about the underlying field, i.e. about the nature and properties of the signal plus noise present in the maps. This is not the case in the detection method presented by Tegmark and Oliveira-Costa 1998. Both methods are compared producing similar results.Comment: 6 pages. Accepted for publication in MNRA

Analysis of CMB maps with 2D wavelets

We consider the 2D wavelet transform with two scales to study sky maps of temperature anisotropies in the cosmic microwave background radiation (CMB). We apply this technique to simulated maps of small sky patches of size 12.8 \times 12.8 square degrees and 1.5' \times 1.5' pixels. The relation to the standard approach, based on the cl's is established through the introduction of the scalogram. We consider temperature fluctuations derived from standard, open and flat-Lambda CDM models. We analyze CMB anisotropies maps plus uncorrelated Gaussian noise (uniform and non-uniform) at idfferent S/N levels. We explore in detail the denoising of such maps and compare the results with other techniques already proposed in the literature. Wavelet methods provide a good reconstruction of the image and power spectrum. Moreover, they are faster than previously proposed methods.Comment: latex file 7 pages + 5 postscript files + 1 gif file; accepted for publication in A&A

Non-blind catalogue of extragalactic point sources from the Wilkinson Microwave Anisotropy Probe (WMAP) first 3--year survey data

We have used the MHW2 filter to obtain estimates of the flux densities at the WMAP frequencies of a complete sample of 2491 sources, mostly brighter than 500 mJy at 5 GHz, distributed over the whole sky excluding a strip around the Galactic equator (b < 5 degrees). After having detected 933 sources above the 3 sigma level in the MHW2 filtered maps - our New Extragalactic WMAP Point Source (NEWPS_3sigma) Catalogue - we are left with 381 sources above 5 sigma in at least one WMAP channel, 369 of which constitute our NEWPS_5sigma catalogue. It is remarkable to note that 98 (i.e. 26%) sources detected above 5 sigma are `new', they are not present in the WMAP catalogue. Source fluxes have been corrected for the Eddington bias. Our flux density estimates before such correction are generally in good agreement with the WMAP ones at 23 GHz. At higher frequencies WMAP fluxes tend to be slightly higher than ours, probably because WMAP estimates neglect the deviations of the point spread function from a Gaussian shape. On the whole, above the estimated completeness limit of 1.1 Jy at 23 GHz we detected 43 sources missed by the blind method adopted by the WMAP team. On the other hand, our low-frequency selection threshold left out 25 WMAP sources, only 12 of which, however, are 5 sigma detections and only 3 have fluxes S at 23 GHz > 1.1 Jy. Thus, our approach proved to be competitive with, and complementary to the WMAP one.Comment: 18 pages, 6 figures, 5 tables. Accepted for publication in ApJ

Goodness-of-Fit Tests to study the Gaussianity of the MAXIMA data

Goodness-of-Fit tests, including Smooth ones, are introduced and applied to detect non-Gaussianity in Cosmic Microwave Background simulations. We study the power of three different tests: the Shapiro-Francia test (1972), the uncategorised smooth test developed by Rayner and Best(1990) and the Neyman's Smooth Goodness-of-fit test for composite hypotheses (Thomas and Pierce 1979). The Smooth Goodness-of-Fit tests are designed to be sensitive to the presence of ``smooth'' deviations from a given distribution. We study the power of these tests based on the discrimination between Gaussian and non-Gaussian simulations. Non-Gaussian cases are simulated using the Edgeworth expansion and assuming pixel-to-pixel independence. Results show these tests behave similarly and are more powerful than tests directly based on cumulants of order 3, 4, 5 and 6. We have applied these tests to the released MAXIMA data. The applied tests are built to be powerful against detecting deviations from univariate Gaussianity. The Cholesky matrix corresponding to signal (based on an assumed cosmological model) plus noise is used to decorrelate the observations previous to the analysis. Results indicate that the MAXIMA data are compatible with Gaussianity.Comment: MNRAS, in pres

Spherical Mexican Hat wavelet: an application to detect non-Gaussianity in the COBE-DMR maps

The spherical Mexican Hat wavelet is introduced in this paper, with the aim of testing the Gaussianity of the Cosmic Microwave Background temperature fluctuations. Using the information given by the wavelet coefficients at several scales, we have performed several statistical tests on the COBE-DMR maps to search for evidence of non-Gaussianity. Skewness, kurtosis, scale-scale correlations (for two and three scales) and Kolmogorov-Smirnov tests indicate that the COBE-DMR data are consistent with a Gaussian distribution. We have extended the analysis to compare temperature values provided by COBE-DMR data with distributions (obtained from Gaussian simulations) at each pixel and at each scale. The number of pixels with temperature values outside the 95% and the 99% is consistent with that obtained from Gaussian simulations, at all scales. Moreover, the extrema values for COBE-DMR data (maximum and minimum temperatures in the map) are also consistent with those obtained from Gaussian simulations.Comment: to be published in MNRA

Forecasts on the contamination induced by unresolved point sources in primordial non-Gaussianity beyond Planck

In this paper we present forecasts of the contamination on different shapes of the primordial non-Gaussianity fnl parameter \u2013 detectable on future Cosmic Microwave Background (CMB) high\u2013resolution anisotropy maps \u2013 produced by unresolved ex- tragalactic point sources at frequencies of cosmological interest (45\u2013375 GHz). We consider two scenarios: an ideal (noiseless) mission and a possible future space-borne satellite, with instrumental characteristics similar to the ones proposed for the Cos- mic Origins Explorer (COrE). The local, equilateral, orthogonal and flat shapes are considered in both temperature (intensity) and polarized emission data. The angular power spectrum and bispectrum of extragalactic point sources are estimated by state- of-the-art models of source number counts. The impact of all the most relevant (far\u2013IR and radio selected) source populations on these shapes at COrE frequencies is studied. The results of this analysis show that unresolved extragalactic point sources should not induce a very relevant non-Gaussian signal in the frequency range 100\u2013200GHz, thus not preventing a correct estimate of the CMB primordial fnl parameter. Polariza- tion information allows one to significantly reduce the error\u2013bars in the fnl parameter and the bias induced by unresolved sources and, hence, to widen the range of frequen- cies for fnl studies. On the contrary, at \u3bd 225GHz, important non-Gaussian deviations in CMB anisotropy maps are expected due to unresolved extragalactic sources