Cosmic Microwave Background Data Analysis Research Articles

Enhancing CMB map reconstruction and power spectrum estimation with convolutional neural networks

The accurate reconstruction of Cosmic Microwave Background (CMB) maps and the measurement of its power spectrum are crucial for studying the early universe. In this paper, we implement a convolutional neural network to apply the Wiener Filter to CMB temperature maps, and use it intensively to compute an optimal quadratic estimation of the power spectrum. Our neural network has a UNet architecture as that implemented in WienerNet, but with novel aspects such as being written in python 3 and TensorFlow 2. It also includes an extra channel for the noise variance map, to account for inhomogeneous noise, and a channel for the mask. The network is very efficient, overcoming the bottleneck that is typically found in standard methods to compute the Wiener Filter, such as those that apply the conjugate gradient. It scales efficiently with the size of the map, making it a useful tool to include in CMB data analysis. The accuracy of the Wiener Filter reconstruction is satisfactory, as compared with the standard method. We heavily use this approach to efficiently estimate the power spectrum, by performing a simulation-based analysis of the optimal quadratic estimator. We further evaluate the quality of the reconstructed maps in terms of the power spectrum and find that we can properly recover the statistical properties of the signal. We find that the proposed architecture can account for inhomogeneous noise efficiently. Furthermore, increasing the complexity of the variance map presents a more significant challenge for the convergence of the network than the noise level does.

Analyzing Earth's Position based on the Anisotropic Characteristics of Cosmic Microwave Background Radiation

This paper elaborates on the results of an exhaustive study regarding the Earth's position in the universe based on the Cosmic Microwave Background (CMB) radiation map, which is the latest discovery in modern astronomy. CMB radiation provides crucial insights into the early distribution of mass and energy in the universe. The aim of this research is to understand the theory and mechanisms behind the formation of polarity structures in the CMB and analyze their correlation with Earth's position in the overall structure of the universe. Intensity measurement data of CMB radiation published by COBE, WMAP, and Planck present temperature distribution data in coordinates in FITS (Flexible Image Transport System) format files. Subsequently, a spherical harmonic transformation is performed to obtain spherical harmonic coefficients a_lm as equations that represent dipole, quadrupole, octupole models, and various other multipole models. The analysis of the correlation in the temperature distribution of CMB radiation involves detailing various patterns found in the dipole, quadrupole, and octopole models, demonstrating quasi-symmetry characteristics with Earth at its center. An analysis of anisotropic CMB data yields an interesting hypothesis that the Earth's position plays a role in shaping the structure of the universe on a certain scale. Even more extremely, it can be said that Earth is at the center of the universe. This finding prompts profound contemplation about Earth's position in the structure of the cosmos, opening the door for further research in this field.

Retrieving cosmological information from small-scale CMB foregrounds

Recent results of ground-based telescopes, giving high-quality measurements of the cosmic microwave background (CMB) temperature power spectrum on small scales (below 1 deg) motivate the need for an accurate model of foregrounds, which dominate the primary signal at these multipoles. In a previous work, we have shown that cosmological information could be retrieved from the power spectrum of the thermal Sunyaev Zel’dovich (SZ) effect. In this work, we introduce a physically motivated model of the Epoch of Reionisation in the cosmological analysis of CMB data, which is coherent on all scales. In particular, at high multipoles, the power spectrum of the kinetic SZ (kSZ) effect is inferred from a set of cosmological and reionisation parameters by a machine-learning algorithm. First including an asymmetric parameterisation of the reionisation history in the Planck 2018 data analysis, we retrieve a value of the Thomson optical depth consistent with previous results, but stemming from a completely different history of reionisation in which the first luminous sources light up as early as z = 15. Considering the latest small-scale data from the South Pole telescope (SPT) and letting the cosmology free to vary, we find that including the new cosmology-dependent tSZ and kSZ spectra helps tighten the constraints on their amplitudes by breaking their degeneracy. We report a 5σ measurement of the kSZ signal at ℓ = 3000, D3000kSZ = 3.4−0.3+0.5 μK2 at the 68% confidence level (C.L.), marginalised over cosmology, as well as an upper limit on the patchy signal from reionisation D3000pkSZ < 1.6 μK2 (95% C.L.). Additionally, we find that the SPT data favour slightly earlier reionisation scenarios than Planck, leading to τ = 0.062−0.015+0.012 and a reionisation midpoint zre = 7.9−1.3+1.1 (68% C.L.), which is in line with constraints from high-redshift quasars and galaxies.

MAPPRAISER: A massively parallel map-making framework for multi-kilo pixel CMB experiments

Forthcoming cosmic microwave background (CMB) polarized anisotropy experiments have the potential to revolutionize our understanding of the Universe and fundamental physics. The sought-after, tale-telling signatures will be however distributed over voluminous data sets which these experiments will collect. These data sets will need to be efficiently processed and unwanted contributions due to astrophysical, environmental, and instrumental effects characterized and efficiently mitigated in order to uncover the signatures. This poses a significant challenge to data analysis methods, techniques, and software tools which will not only have to be able to cope with huge volumes of data but to do so with unprecedented precision driven by the demanding science goals posed for the new experiments.A keystone of efficient CMB data analysis is solvers of very large linear systems of equations. Such systems appear in very diverse contexts throughout CMB data analysis pipelines, however they typically display similar algebraic structures and can therefore be solved using similar numerical techniques. Linear systems arising in the so-called map-making problem are one of the most prominent and common ones.In this work we present a massively parallel, flexible and extensible framework, comprised of a numerical library, MIDAPACK, and a high level code, MAPPRAISER, which provide tools for solving efficiently such systems. The framework implements iterative solvers based on conjugate gradient techniques: enlarged and preconditioned using different preconditioners. We demonstrate the framework on simulated examples reflecting basic characteristics of the forthcoming data sets issued by ground-based and satellite-borne instruments, executing it on as many as 16,384 compute cores. The software is developed as an open source project freely available to the community at: https://github.com/B3Dcmb/midapack.

Retrieving cosmological information from small-scale CMB foregrounds

We propose a new analysis of small-scale cosmic microwave background (CMB) data by introducing the cosmological dependency of the foreground signals, focussing first on the thermal Sunyaev-Zel’dovich (tSZ) power spectrum, derived from the halo model. We analyse the latest observations by the South Pole Telescope (SPT) of the high-ℓ power (cross) spectra at 95, 150, and 220 GHz, as the sum of CMB and tSZ signals, both depending on cosmological parameters and remaining contaminants. In order to perform faster analyses, we propose a new tSZ modelling based on machine learning algorithms (namely Random Forest). We show that the additional information contained in the tSZ power spectrum tightens constraints on cosmological and tSZ scaling relation parameters. We combined for the first time the Planck tSZ data with SPT high-ℓ to derive new constraints. Finally, we show how the amplitude of the remaining kinetic SZ power spectrum varies depending on the assumptions made on both tSZ and cosmological parameters. These results show the importance of a thorough modelling of foregrounds in the cosmological analysis of small-scale CMB data. Reliable constraints on cosmological parameters can only be achieved once other significant foregrounds, such as the kinetic SZ and the cosmic infrared background (CIB), are also properly accounted for.

Euclid preparation

The combination and cross-correlation of the upcoming Euclid data with cosmic microwave background (CMB) measurements is a source of great expectation since it will provide the largest lever arm of epochs, ranging from recombination to structure formation across the entire past light cone. In this work, we present forecasts for the joint analysis of Euclid and CMB data on the cosmological parameters of the standard cosmological model and some of its extensions. This work expands and complements the recently published forecasts based on Euclid-specific probes, namely galaxy clustering, weak lensing, and their cross-correlation. With some assumptions on the specifications of current and future CMB experiments, the predicted constraints are obtained from both a standard Fisher formalism and a posterior-fitting approach based on actual CMB data. Compared to a Euclid-only analysis, the addition of CMB data leads to a substantial impact on constraints for all cosmological parameters of the standard Λ-cold-dark-matter model, with improvements reaching up to a factor of ten. For the parameters of extended models, which include a redshift-dependent dark energy equation of state, non-zero curvature, and a phenomenological modification of gravity, improvements can be of the order of two to three, reaching higher than ten in some cases. The results highlight the crucial importance for cosmological constraints of the combination and cross-correlation of Euclid probes with CMB data.

Cooling Improves Cosmic Microwave Background Map-making when Low-frequency Noise is Large

Abstract In the context of cosmic microwave background data analysis, we study the solution to the equation that transforms scanning data into a map. As originally suggested in “messenger” methods for solving linear systems, we split the noise covariance into uniform and nonuniform parts and adjust their relative weights during the iterative solution. With simulations, we study mock instrumental data with different noise properties, and find that this “cooling” or perturbative approach is particularly effective when there is significant low-frequency noise in the timestream. In such cases, a conjugate gradient algorithm applied to this modified system converges faster and to a higher fidelity solution than the standard conjugate gradient approach. We give an analytic estimate for the parameter that controls how gradually the linear system should change during the course of the solution.

Accelerating linear system solvers for time-domain component separation of cosmic microwave background data

Component separation is one of the key stages of any modern cosmic microwave background data analysis pipeline. It is an inherently nonlinear procedure and typically involves a series of sequential solutions of linear systems with similar but not identical system matrices, derived for different data models of the same data set. Sequences of this type arise, for instance, in the maximization of the data likelihood with respect to foreground parameters or sampling of their posterior distribution. However, they are also common in many other contexts. In this work we consider solving the component separation problem directly in the measurement (time-) domain. This can have a number of important benefits over the more standard pixel-based methods, in particular if non-negligible time-domain noise correlations are present, as is commonly the case. The approach based on the time-domain, however, implies significant computational effort because the full volume of the time-domain data set needs to be manipulated. To address this challenge, we propose and study efficient solvers adapted to solving time-domain-based component separation systems and their sequences, and which are capable of capitalizing on information derived from the previous solutions. This is achieved either by adapting the initial guess of the subsequent system or through a so-called subspace recycling, which allows constructing progressively more efficient two-level preconditioners. We report an overall speed-up over solving the systems independently of a factor of nearly 7, or 5, in our numerical experiments, which are inspired by the likelihood maximization and likelihood sampling procedures, respectively.

Wiener filtering and pure $\mathcal {E}/\mathcal {B}$ decomposition of CMB maps with anisotropic correlated noise

ABSTRACTWe present an augmented version of our dual messenger algorithm for spin field reconstruction on the sphere, while accounting for highly non-trivial and realistic noise models such as modulated correlated noise. We also describe an optimization method for the estimation of noise covariance from Monte Carlo simulations. Using simulated Planck polarized cosmic microwave background (CMB) maps as a showcase, we demonstrate the capabilities of the algorithm in reconstructing pure $\mathcal {E}$ and $\mathcal {B}$ maps, guaranteed to be free from ambiguous modes resulting from the leakage or coupling issue that plagues conventional methods of $\mathcal {E}/\mathcal {B}$ separation. Due to its high speed execution, coupled with lenient memory requirements, the algorithm can be optimized in exact global Bayesian analyses of state-of-the-art CMB data for a statistically optimal separation of pure $\mathcal {E}$ and $\mathcal {B}$ modes. Our algorithm, therefore, has a potentially key role in the data analysis of high-resolution and high-sensitivity CMB data, especially with the range of upcoming CMB experiments tailored for the detection of the elusive primordial $\mathcal {B}$-mode signal.

Convolutional neural networks on the HEALPix sphere: a pixel-based algorithm and its application to CMB data analysis

We describe a novel method for the application of convolutional neural networks (CNNs) to fields defined on the sphere, using the Hierarchical Equal Area Latitude Pixelization scheme (HEALPix). Specifically, we have developed a pixel-based approach to implement convolutional and pooling layers on the spherical surface, similarly to what is commonly done for CNNs applied to Euclidean space. The main advantage of our algorithm is to be fully integrable with existing, highly optimized libraries for NNs (e.g., PyTorch, TensorFlow, etc.). We present two applications of our method: (i) recognition of handwritten digits projected on the sphere; (ii) estimation of cosmological parameter from simulated maps of the cosmic microwave background (CMB). The latter represents the main target of this exploratory work, whose goal is to show the applicability of our CNN to CMB parameter estimation. We have built a simple NN architecture, consisting of four convolutional and pooling layers, and we have used it for all the applications explored herein. Concerning the recognition of handwritten digits, our CNN reaches an accuracy of ∼95%, comparable with other existing spherical CNNs, and this is true regardless of the position and orientation of the image on the sphere. For CMB-related applications, we tested the CNN on the estimation of a mock cosmological parameter, defining the angular scale at which the power spectrum of a Gaussian field projected on the sphere peaks. We estimated the value of this parameter directly from simulated maps, in several cases: temperature and polarization maps, presence of white noise, and partially covered maps. For temperature maps, the NN performances are comparable with those from standard spectrum-based Bayesian methods. For polarization, CNNs perform about a factor four worse than standard algorithms. Nonetheless, our results demonstrate, for the first time, that CNNs are able to extract information from polarization fields, both in full-sky and masked maps, and to distinguish betweenEandB-modes in pixel space. Lastly, we have applied our CNN to the estimation of the Thomson scattering optical depth at reionization (τ) from simulated CMB maps. Even without any specific optimization of the NN architecture, we reach an accuracy comparable with standard Bayesian methods. This work represents a first step towards the exploitation of NNs in CMB parameter estimation and demonstrates the feasibility of our approach.

Solving linear equations with messenger-field and conjugate gradient techniques: An application to CMB data analysis

We discuss linear system solvers invoking a messenger-field and compare them with (preconditioned) conjugate gradient approaches. We show that the messenger-field techniques correspond to fixed point iterations of an appropriately preconditioned initial system of linear equations. We then argue that a conjugate gradient solver applied to the same preconditioned system, or equivalently a preconditioned conjugate gradient solver using the same preconditioner and applied to the original system, will in general ensure at least a comparable and typically better performance in terms of the number of iterations to convergence and time-to-solution. We illustrate our conclusions with two common examples drawn from the cosmic microwave background (CMB) data analysis: Wiener filtering and map-making. In addition, and contrary to the standard lore in the CMB field, we show that the performance of the preconditioned conjugate gradient solver can depend significantly on the starting vector. This observation seems of particular importance in the cases of map-making of high signal-to-noise ratio sky maps and therefore should be of relevance for the next generation of CMB experiments.

Iterative map-making with two-level preconditioning for polarized cosmic microwave background data sets

Context. An estimation of the sky signal from streams of time ordered data (TOD) acquired by the cosmic microwave background (CMB) experiments is one of the most important steps in the context of CMB data analysis referred to as the map-making problem. The continuously growing CMB data sets render the CMB map-making problem progressively more challenging in terms of computational cost and memory in particular in the context of ground-based experiments with their operational limitations as well as the presence of contaminants. Aims. We study a recently proposed, novel class of the Preconditioned Conjugate Gradient (PCG) solvers which invoke two-level preconditioners in the context of the ground-based CMB experiments. We compare them against the PCG solvers commonly used in the map-making context considering their precision and time-to-solution. Methods. We compare these new methods on realistic, simulated data sets reflecting the characteristics of current and forthcoming CMB ground-based experiments. We develop a divide-and-conquer implementation of the approach where each processor performs a sequential map-making for a subset of the TOD. Results. We find that considering the map level residuals, the new class of solvers permits us to achieve a tolerance that is better than the standard approach by up to three orders of magnitude, where the residual level often saturates before convergence is reached. This often corresponds to an important improvement in the precision of the recovered power spectra in particular on the largest angular scales. The new method also typically requires fewer iterations to reach a required precision and therefore shorter run times are required for a single map-making solution. However, the construction of an appropriate two-level preconditioner can be as costly as a single standard map-making run. Nevertheless, if the same problem needs to be solved multiple times, for example, as in Monte Carlo simulations, this cost is incurred only once, and the method should be competitive, not only as far as its precision is concerned but also its performance.

CMB lensing bispectrum as a probe of modified gravity theories

Cosmological structures grow differently in theories of gravity which are modified as compared to Einstein's General relativity (GR). Cosmic microwave background (CMB) fluctuation patterns at the last scattering surface are lensed by these structures along the photon path to the observer. The observed CMB pattern therefore keeps trace of the growth history of structures. We show that observations of the CMB lensing bi-spectrum offer an interesting way to constrain deviations from GR in a broad class of scalar-tensor theories of gravity called "beyond Horndeski". We quantify how the constraints on generic parameters describing the deviations from GR depend on the effective multipole range of the analysis. Our results further indicate that an accurate nonlinear correction of the matter bi-spectrum in the modified gravity considered is necessary when the bi-spectrum is used to probe scales beyond a multipole $\ell_{\rm max} \gtrsim 1500$. We also found that the results are insensitive to details of the implementation of the screening mechanism, at very small scales. We finally demonstrate the potential of the lensing bi-spectrum to provide a blind reconstruction of the redshift evolution of our modified gravity parameters by combining the analysis of CMB and low-z source lensing data.

Improved constraints on lepton asymmetry from the cosmic microwave background

A cosmic lepton asymmetry affects the primordial helium abundance and the expansion rate of the early Universe. Both of these effects have an impact on the anisotropies of the cosmic microwave background (CMB). We derive constraints on the neutrino chemical potentials from the Planck 2015 data, assuming equal lepton flavour asymmetries and negligible neutrino masses. We find (95% CL) for the chemical potentials, which corresponds to . Our constraints on the lepton asymmetry are significantly stronger than previous constraints from CMB data analysis and we argue that they are more robust than those from primordial light element abundances. The resulting constraints on the primordial helium and deuterium abundances are consistent with those from direct measurements.

A multi-resolution HEALPix data structure for spherically mapped point data.

Data describing entities with locations that are points on a sphere are described as spherically mapped. Several data structures designed for spherically mapped data have been developed. One of them, known as Hierarchical Equal Area iso-Latitude Pixelization (HEALPix), partitions the sphere into twelve diamond-shaped equal-area base cells and then recursively subdivides each cell into four diamond-shaped subcells, continuing to the desired level of resolution. Twelve quadtrees, one associated with each base cell, store the data records associated with that cell and its subcells.HEALPix has been used successfully for numerous applications, notably including cosmic microwave background data analysis. However, for applications involving sparse point data HEALPix has possible drawbacks, including inefficient memory utilization, overwriting of proximate points, and return of spurious points for certain queries.A multi-resolution variant of HEALPix specifically optimized for sparse point data was developed. The new data structure allows different areas of the sphere to be subdivided at different levels of resolution. It combines HEALPix positive features with the advantages of multi-resolution, including reduced memory requirements and improved query performance.An implementation of the new Multi-Resolution HEALPix (MRH) data structure was tested using spherically mapped data from four different scientific applications (warhead fragmentation trajectories, weather station locations, galaxy locations, and synthetic locations). Four types of range queries were applied to each data structure for each dataset. Compared to HEALPix, MRH used two to four orders of magnitude less memory for the same data, and on average its queries executed 72% faster.

Statical Properties of CMB B-Mode Polarisation in a Partial Sky Analysis

Measuring the imprint of primordial gravitational waves in the cosmic microwave background (CMB) polarisation field is one of the main goals in modern cosmology. However, the so called [Formula: see text]-mode polarisation can be generated by different sources besides the primary one predicted by inflationary theories, known as secondary [Formula: see text]-mode signal. Among them, CMB lensing and astrophysical foregrounds play an important role. Moreover, a partial sky analysis leads to a leakage between [Formula: see text]-modes and [Formula: see text]-modes. In this article, we use the well known Minkowski functionals (MF) statistics to study the significance of this leakage in the CMB lensing [Formula: see text]-mode signal. We find that the MF can detect the [Formula: see text]-to-[Formula: see text] leakage contamination, thus it should not be neglected in future CMB data analysis.

Information gain on reheating: The one bit milestone

We show that the Planck 2015 and BICEP2/KECK measurements of the Cosmic Microwave Background (CMB) anisotropies provide together an information gain of 0.82 +- 0.13 bits on the reheating history over all slow-roll single-field models of inflation. This corresponds to a 40% improvement compared to the Planck 2013 constraints on the reheating. Our method relies on an exhaustive CMB data analysis performed over nearly 200 models of inflation to derive the Kullback-Leibler entropy between the prior and the fully marginalized posterior of the reheating parameter. This number is a weighted average by the Bayesian evidence of each model to explain the data thereby ensuring its fairness and robustness.

Neutrino constraints: what large-scale structure and CMB data are telling us?

We discuss the reliability of neutrino mass constraints, either active or sterile, from the combination of different low redshift Universe probes with measurements of CMB anisotropies. In our analyses we consider WMAP 9-year or Planck Cosmic Microwave Background (CMB) data in combination with Baryonic Acoustic Oscillations (BAO) measurements from BOSS DR11, galaxy shear measurements from CFHTLenS, SDSS Ly α forest constraints and galaxy cluster mass function from Chandra observations. At odds with recent similar studies, to avoid model dependence of the constraints we perform a full likelihood analysis for all the datasets employed. As for the cluster data analysis we rely on to the most recent calibration of massive neutrino effects in the halo mass function and we explore the impact of the uncertainty in the mass bias and re-calibration of the halo mass function due to baryonic feedback processes on cosmological parameters. We find that none of the low redshift probes alone provide evidence for massive neutrino in combination with CMB measurements, while a larger than 2σ detection of non zero neutrino mass, either active or sterile, is achieved combining cluster or shear data with CMB and BAO measurements. Yet, the significance of the detection exceeds 3σ if we combine all four datasets. For a three active neutrino scenario, from the joint analysis of CMB, BAO, shear and cluster data including the uncertainty in the mass bias we obtain ∑ mν =0.29+0.18-0.21 eV and ∑ mν =0.22+0.17-0.18 eV 95%CL) using WMAP9 or Planck as CMB dataset, respectively. The preference for massive neutrino is even larger in the sterile neutrino scenario, for which we get mseff=0.44+0.28-0.26 eV and Δ Neff=0.78+0.60-0.59 95%CL) from the joint analysis of Planck, BAO, shear and cluster datasets. For this data combination the vanilla ΛCDM model is rejected at more than 3σ and a sterile neutrino mass as motivated by accelerator anomaly is within the 2σ errors. Conversely, the Ly α data favour vanishing neutrino masses and from the data combination Planck+BAO+Ly α we get the tight upper limits ∑ mν <0.14 eV and mseff<0.22 eV—Δ Neff<1.11 95%CL) for the active and sterile neutrino model, respectively. Finally, results from the full data combination reflect the tension between the σ8 constraints obtained from cluster and shear data and that inferred from Ly α forest measurements; in the active neutrino scenario for both CMB datasets employed, the full data combination yields only an upper limits on ∑ mν, while assuming an extra sterile neutrino we still get preference for non-vanishing mass, mseff=0.26+0.22-0.24 eV, and dark contribution to the radiation content, Δ Neff=0.82±0.55.

The scalar bi-spectrum in the Starobinsky model: the equilateral case

While a featureless, nearly scale invariant, primordial scalar power spectrum fits the most recent Cosmic Microwave Background (CMB) data rather well, certain features in the spectrum are known to lead to a better fit to the data (although, the statistical significance of such results remains an open issue). In the inflationary scenario, one or more periods of deviations from slow roll are necessary in order to generate features in the scalar perturbation spectrum. Over the last couple of years, it has been recognized that such deviations from slow roll inflation can also result in reasonably large non-Gaussianities. The Starobinsky model involves the canonical scalar field and consists of a linear inflaton potential with a sudden change in the slope. The change in the slope causes a brief period of departure from slow roll which, in turn, results in a sharp rise in power, along with a burst of oscillations in the scalar spectrum for modes that leave the Hubble radius just before and during the period of fast roll. The hallmark of the Starobinsky model is that it allows the scalar power spectrum to be evaluated analytically in terms of the three parameters that describe the model, viz. the two slopes that describe the potential on either side of the discontinuity and the Hubble scale at the time when the field crosses the discontinuity. In this work, we evaluate the bi-spectrum of the scalar perturbations in the Starobinsky model in the equilateral limit. Remarkably, we find that, just as the power spectrum, all the different contributions to the the bi-spectrum too can be evaluated completely analytically and expressed in terms of the three paramaters that describe the model. We show that the quantity fNL, which characterizes the extent of non-Gaussianity, can be expressed purely in terms of the ratio of the two slopes on either side of the discontinuity in the potential. Further, we find that, for certain values of the parameters, fNL in the Starobinsky model can be as large as the mean value that has been arrived at from the analysis of the recent CMB data. We also demonstrate that the usual hierarchy of contributions to the bi-spectrum can be altered for certain values of the parameters. Altogether, we find that the Starobinsky model represents a unique scenario wherein, even when the slow roll conditions are violated, the background, the perturbations as well as the corresponding two and three point correlation functions can be evaluated completely analytically. As a consequence, the Starobinsky model can also be used to calibrate numerical codes aimed at computing the non-Gaussianities.

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.