We present ECLIPSE (Efficient Cmb poLarization and Intensity Power Spectra Estimator), an optimized implementation of the Quadratic Maximum Likelihood (QML) method for the estimation of the power spectra of the Cosmic Microwave Background (CMB). This approach allows one to reduce significantly the computational costs associated to this technique, allowing to estimate the power spectra up to higher multipoles than previous implementations. In particular, for a resolution of $N_\mathrm{side}=64$, $\ell_{\mathrm{max}}=192$ and a typical Galactic mask, the number of operations can be reduced by approximately a factor of 1000 in a full analysis including intensity and polarization with respect to an efficient direct implementation of the method. In addition, if one is interested in studying only polarization, it is possible to obtain the power spectra of the E and B modes with a further reduction of computational resources without degrading the results. We also show that for experiments observing a small fraction of the sky, the Fisher matrix becomes singular and, in this case, the standard QML can not be applied. To solve this problem, we have developed a binned version of the method that is unbiased and of minimum variance. We also test the robustness of the QML estimator when the assumed fiducial model differs from that of the sky and show the performance of an iterative approach. Finally, we present a comparison of the results obtained by QML and a pseudo-$C_{\ell}$ estimator (NaMaster) for a next-generation satellite, showing that, as expected, QML produces significantly smaller errors at low multipoles. The ECLIPSE fast QML code developed in this work will be made publicly available.
We present a new Planck CMB lensing-CMB temperature cross-correlation likelihood that can be used to constrain cosmology via the Integrated Sachs-Wolfe (ISW) effect. CMB lensing is an excellent tracer of ISW, and we use the latest PR4 Planck data maps and lensing reconstruction to produce the first public Planck likelihood to constrain this signal. We demonstrate the likelihood by constraining the CMB background temperature from Planck data alone, where the ISW-lensing cross-correlation is a powerful way to break the geometric degeneracy, substantially improving constraints from the CMB and le…
We assess the strengths and weaknesses of several likelihood formalisms, including the XFaster likelihood. We compare the performance of the XFaster likelihood to that of the Offset Lognormal Bandpower likelihood on simulated data for the Planck satellite. Parameters estimated with these two likelihoods are in good agreement. The advantages of the XFaster likelihood can therefore be realized without compromising performance.
We extend the ILC method in harmonic space to include the error in its CMB estimate. This allows parameter estimation routines to take into account the effect of the foregrounds as well as the errors in their subtraction in conjunction with the ILC method. Our method requires the use of a model of the foregrounds which we do not develop here. The reduction of the foreground level makes this method less sensitive to unaccounted for errors in the foreground model. Simulations are used to validate the calculations and approximations used in generating this likelihood function.
Fast robust methods for calculating likelihoods from CMB observations on small scales generally rely on approximations based on a set of power spectrum estimators and their covariances. We investigate the optimality of these approximation, how accurate the covariance needs to be, and how to estimate the covariance from simulations. For a simple case with azimuthal symmetry we compare optimality of hybrid pseudo-C_l CMB power spectrum estimators with the exact result, indicating that the loss of information is not negligible, but neither is it enough to have a large effect on standard parameter…
Non-Gaussian correlations induced in CMB power spectra by gravitational lensing must be included in likelihood analyses for future CMB experiments. We present a simple but accurate likelihood model which includes these correlations and use it for Markov Chain Monte Carlo parameter estimation from simulated lensed CMB maps in the context of $Λ$CDM and extensions which include the sum of neutrino masses or the dark energy equation of state $w$. If lensing-induced covariance is not taken into account for a CMB-S4 type experiment, the errors for one combination of parameters in each case would be …
Accurate and efficient likelihood modeling for large-scale CMB data
Università degli Studi di Ferrara（费拉拉大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara（意大利国家核物理研究所费拉拉分部）；ICSC, Centro Nazionale “High Performance Computing, Big Data and Quantum Computing”（国家高性能计算、大数据和量子计算中心）；Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay（法国国家科学研究中心巴黎第十一大学巴黎萨克雷大学空间天体物理学研究所）
Nearby galaxy clusters can potentially induce sub-microkelvin polarization signals in the cosmic microwave background (CMB) at characteristic scales of a few arcminutes. We explore four such polarization signals induced in a rich nearby fiducial cluster and calculate the likelihood of their detection by a telescope project with capabilities such as those of the Simons Observatory (SO). In our feasibility analysis, we include instrumental noise, primordial CMB anisotropy, statistical thermal Sunyaev-Zeldovich (SZ) cluster signal, and point source confusion, assuming a few percent of the nominal…
We present a CMB large-scale polarization dataset obtained by combining WMAP Ka, Q and V with Planck 70 GHz maps. We employ the legacy frequency maps released by the WMAP and Planck collaborations and perform our own Galactic foreground mitigation technique, which relies on Planck 353 GHz for polarized dust and on Planck 30 GHz and WMAP K for polarized synchrotron. We derive a single, optimally-noise-weighted, low-residual-foreground map and the accompanying noise covariance matrix. These are shown, through $χ^2$ analysis, to be robust over an ample collection of Galactic masks. We use this da…
We develop a Maximum Likelihood estimator (MLE) to measure the masses of galaxy clusters through the impact of gravitational lensing on the temperature and polarization anisotropies of the cosmic microwave background (CMB). We show that, at low noise levels in temperature, this optimal estimator outperforms the standard quadratic estimator by a factor of two. For polarization, we show that the Stokes Q/U maps can be used instead of the traditional E- and B-mode maps without losing information. We test and quantify the bias in the recovered lensing mass for a comprehensive list of potential sys…
The Atacama Cosmology Telescope has measured the angular power spectra of microwave fluctuations to arcminute scales at frequencies of 148 and 218 GHz, from three seasons of data. At small scales the fluctuations in the primordial Cosmic Microwave Background (CMB) become increasingly obscured by extragalactic foregounds and secondary CMB signals. We present results from a nine-parameter model describing these secondary effects, including the thermal and kinematic Sunyaev-Zel'dovich (tSZ and kSZ) power; the clustered and Poisson-like power from Cosmic Infrared Background (CIB) sources, and thei…
We present a cross-spectra based approach for the analysis of CMB data at large angular scales to constrain the reionization optical depth $τ$, the tensor to scalar ratio $r$ and the amplitude of the primordial scalar perturbations $A_s$. With respect to the pixel-based approach developed so far, using cross-spectra has the unique advantage to eliminate spurious noise bias and to give a better handle over residual systematics, allowing to efficiently combine the cosmological information encoded in cross-frequency or cross-dataset spectra. We present two solutions to deal with the non-Gaussiani…
The angular power spectra of the cosmic microwave background (CMB) temperature anisotropies reconstructed from Planck data seem to present too much gravitational lensing distortion. This is quantified by the control parameter $A_L$ that should be compatible with unity for a standard cosmology. With the Class Boltzmann solver and the profile-likelihood method, for this parameter we measure a 2.6$σ$ shift from 1 using the Planck public likelihoods. We show that, owing to strong correlations with the reionization optical depth $τ$ and the primordial perturbation amplitude $A_s$, a $\sim2σ$ tensio…
We investigate the performance of the parametric Maximum Likelihood component separation method in the context of the CMB B-mode signal detection and its characterization by small-scale CMB suborbital experiments. We consider high-resolution (FWHM=8') balloon-borne and ground-based observatories mapping low dust-contrast sky areas of 400 and 1000 square degrees, in three frequency channels, 150, 250, 410 GHz, and 90, 150, 220 GHz, with sensitivity of order 1 to 10 micro-K per beam-size pixel. These are chosen to be representative of some of the proposed, next-generation, bolometric experiments…
We apply the Karhunen-Loéve methods to cosmic microwave background (CMB) data sets, and show that we can recover the input cosmology and obtain the marginalized likelihoods in $Λ$ cold dark matter cosmologies in under a minute, much faster than Markov chain Monte Carlo methods. This is achieved by forming a linear combination of the power spectra at each multipole $l$, and solving a system of simultaneous equations such that the Fisher matrix is locally unchanged. Instead of carrying out a full likelihood evaluation over the whole parameter space, we need evaluate the likelihood only for the p…
One of the major targets for next-generation cosmic microwave background (CMB) experiments is the detection of the primordial B-mode signal. Planning is under way for Stage-IV experiments that are projected to have instrumental noise small enough to make lensing and foregrounds the dominant source of uncertainty for estimating the tensor-to-scalar ratio $r$ from polarization maps. This makes delensing a crucial part of future CMB polarization science. In this paper we present a likelihood method for estimating the tensor-to-scalar ratio $r$ from CMB polarization observations, which combines th…
We revisit the problem of exact CMB likelihood and power spectrum estimation with the goal of minimizing computational cost through linear compression. This idea was originally proposed for CMB purposes by Tegmark et al.\ (1997), and here we develop it into a fully working computational framework for large-scale polarization analysis, adopting \WMAP\ as a worked example. We compare five different linear bases (pixel space, harmonic space, noise covariance eigenvectors, signal-to-noise covariance eigenvectors and signal-plus-noise covariance eigenvectors) in terms of compression efficiency, and…
The maximum likelihood estimator for CMB map-making is optimal and unbiased as long as the data model is correct, but in practice it rarely is, with model errors including sub-pixel structure and instrumental problems like time-variable gain and pointing errors. In the presence of such errors, the solution is biased, with the local error in each pixel leaking outwards along the scanning pattern by a noise correlation length. The most important sources of such leakage are strong point sources, and for common scanning patterns the leakage manifests as an X around each such source. I discuss why …
University of Trento（特伦托大学）；Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN（特伦托基础物理与应用研究所（TIFPA）-意大利国家核物理研究所）；University of Hawai‘i（夏威夷大学）；Shanghai Jiao Tong University（上海交通大学）；University of Sheffield（谢菲尔德大学）
We investigate sets of random variables that can be arranged sequentially such that a given variable only depends conditionally on its immediate predecessor. For such sets, we show that the full joint probability distribution may be expressed exclusively in terms of uni- and bivariate marginals. Under the assumption that the CMB power spectrum likelihood only exhibits correlations within a banded multipole range, Δl, we apply this expression to two outstanding problems in CMB likelihood analysis. First, we derive a statistically well-defined hybrid likelihood estimator, merging two independent…
A great deal of experimental effort is currently being devoted to the precise measurements of the cosmic microwave background (CMB) sky in temperature and polarisation. Satellites, balloon-borne, and ground-based experiments scrutinize the CMB sky at multiple scales, and therefore enable to investigate not only the evolution of the early Universe, but also its late-time physics with unprecedented accuracy. The pipeline leading from time ordered data as collected by the instrument to the final product is highly structured. Moreover, it has also to provide accurate estimates of statistical and s…
Every experiment is affected by systematic effects that hamper the data analysis and have the potential to ultimately degrade its performance. In the case of probes of the cosmic microwave background (CMB) radiation, a minimal set of issues to consider includes asymmetric beam functions, correlated noise, and incomplete sky coverage. Presuming a simplified scanning strategy that allows for an exact analytical treatment of the problem, we study the impact of systematic effects on the likelihood function of the CMB power spectrum. We use the Fisher matrix, a measure of the information content of…
A Quadratic Maximum Likelihood (QML) estimator is applied to the WMAP 5 year low resolution maps to compute the CMB angular power spectra at large scales for both temperature and polarization. Estimates and error bars for the six angular power spectra are provided and compared, when possible, to those obtained by the WMAP team, without finding any inconsistency. The conditional likelihood slices are also computed for the $C_{\ell}$ of all the six power spectra from $\ell =2$ to 10 through a pixel based likelihood code. Both the codes treat the covariance for $(T,Q,U)$ in a single matrix withou…
We use the 2015 Planck likelihood in combination with the Bicep2/Keck likelihood (BKP and BK14) to constrain the chirality, $χ$, of primordial gravitational waves in a scale-invariant scenario. In this framework, the parameter $χ$ enters theory always coupled to the tensor-to-scalar ratio, $r$, e.g. in combination of the form $χ\cdot r$. Thus, the capability to detect $χ$ critically depends on the value of $r$. We find that with present data set $χ$ is \textit{de facto}unconstrained. We also provide forecasts for $χ$ from future CMB experiments, including COrE+, exploring several fiducial valu…
We propose an efficient Bayesian MCMC algorithm for estimating cosmological parameters from CMB data without use of likelihood approximations. It builds on a previously developed Gibbs sampling framework that allows for exploration of the joint CMB sky signal and power spectrum posterior, P(s,Cl|d), and addresses a long-standing problem of efficient parameter estimation simultaneously in high and low signal-to-noise regimes. To achieve this, our new algorithm introduces a joint Markov Chain move in which both the signal map and power spectrum are synchronously modified, by rescaling the map ac…
MADmap is a software application used to produce maximum-likelihood images of the sky from time-ordered data which include correlated noise, such as those gathered by Cosmic Microwave Background (CMB) experiments. It works efficiently on platforms ranging from small workstations to the most massively parallel supercomputers. Map-making is a critical step in the analysis of all CMB data sets, and the maximum-likelihood approach is the most accurate and widely applicable algorithm; however, it is a computationally challenging task. This challenge will only increase with the next generation of gr…
Constructing a fast and efficient estimator for the B-mode power spectrum of cosmic microwave background (CMB) is of critical importance for CMB science. For a general CMB survey, the Quadratic Maximum Likelihood (QML) estimator for CMB polarization has been proved to be the optimal estimator with minimal uncertainties, but it is computationally very expensive. In this article, we propose two new QML methods for B-mode power spectrum estimation. We use the Smith-Zaldarriaga approach to prepare pure-B mode map, and E-mode recycling method to obtain a leakage free B-mode map. We then use the sca…
The kinetic Sunyaev Zel'dovich (kSZ) effect, cosmic microwave background (CMB) temperature anisotropies induced by the scattering of CMB photons from free electrons, will be measured by near-term CMB experiments at high significance. By combining CMB temperature anisotropies with a tracer of structure, such as a galaxy redshift survey, previous literature introduced a number of techniques to reconstruct the radial velocity field. This reconstructed radial velocity field encapsulates the majority of the cosmological information contained in the kSZ temperature anisotropies, and can provide powe…
We have made a Markov Chain Monte Carlo (MCMC) analysis of primordial non-Gaussianity (f_NL) using the WMAP bispectrum and power spectrum. In our analysis, we have simultaneously constrained f_NL and cosmological parameters so that the uncertainties of cosmological parameters can properly propagate to the f_NL estimation. Investigating the parameter likelihoods deduced from MCMC samples, we find slight deviation from Gaussian shape, which makes a Fisher matrix estimation less accurate. Therefore, we have estimated the confidence interval of f_NL by exploring the parameter likelihood without us…
We present a new estimation method for mapping the gravitational lensing potential from observed CMB intensity and polarization fields. Our method uses Bayesian techniques to estimate the average curvature of the potential over small local regions. These local curvatures are then used to construct an estimate of a low pass filter of the gravitational potential. By utilizing Bayesian/likelihood methods one can easily overcome problems with missing and/or non-uniform pixels and problems with partial sky observations (E and B mode mixing, for example). Moreover, our methods are local in nature wh…
Estimating the cross-correlation power spectra of cosmic microwave background (CMB), in particular, the T B and EB spectra, is important for testing parity symmetry in cosmology and diagnosing insidious instruments systematics. The Quadratic Maximum Likelihood (QML) estimator provides the optimal estimates of power spectra, but it is computationally very expensive. The hybrid pseudo-Cl estimator is computationally fast but performs poorly on large scales. As a natural extension of previous work (Chen et al. 2021), in this article, we present a new unbiased estimator based on the Smith-Zaldarri…
The new generation of CMB polarization experiments will reach limits of sensitivity never achieved before to detect the primordial B-mode signal. However, all these efforts will be futile if we lack a tight control of systematics. Here, we focus on the systematic that arises from the uncertainty on the calibration of polarization angles. Miscalibrated polarization angles induce a mixing of E- and B-modes that obscures the primordial B-mode signal. We introduce an iterative angular power spectra maximum likelihood-based method to calculate polarization angles from the multi-frequency signal. Th…
In the context of cosmic microwave background (CMB) data analysis, we compare the efficiency at large scale of two angular power spectrum algorithms, implementing, respectively, the quadratic maximum likelihood (QML) estimator and the pseudo spectrum (pseudo-Cl) estimator. By exploiting 1000 realistic Monte Carlo (MC) simulations, we find that the QML approach is markedly superior in the range l=[2-100]. At the largest angular scales, e.g. l < 10, the variance of the QML is almost 1/3 (1/2) that of the pseudo-Cl, when we consider the WMAP kq85 (kq85 enlarged by 8 degrees) mask, making the pseu…
We introduce a new estimator of the CMB lensing power spectrum, together with its likelihood, based on iterative lensing reconstruction. Despite the increased complexity of the lensing maps, this estimator shares similarities with the standard quadratic estimator. Most importantly, it is unbiased towards the assumptions done on the noise and cosmology for the lensing reconstruction. This new spectrum estimator can double the constraints on the lensing amplitude compared to the quadratic estimator, while keeping numerical cost under control and being robust to errors.
We present an analysis of errors on the tensor-to-scalar ratio due to residual diffuse foregrounds. We use simulated observations of a CMB polarization satellite, the Cosmic Origins Explorer, using the specifications of the version proposed to ESA in 2010 (COrE). We construct a full pipeline from microwave sky maps to $r$ likelihood, using two models of diffuse Galactic foregrounds with different complexity, and assuming component separation with varying degrees of accuracy. Our pipeline uses a linear mixture (Generalized Least Squares) solution for component separation, and a hybrid approach …
We extend our maximum likelihood method for reconstructing the cluster-mass cross-correlation from cosmic microwave background (CMB) temperature anisotropies and develop new estimators that utilize six different quadratic combinations of CMB temperature and polarization fields. Our maximum likelihood estimators are constructed with delensed CMB temperature and polarization fields by using an assumed model of the convergence field and they can be iteratively applied to a set of clusters, approaching to the optimal condition for the lensing reconstruction as the assumed initial model is refined.…
The cosmic microwave background provides constraints on the annihilation and decay of light dark matter at redshifts between 100 and 1000, the strength of which depends upon the fraction of energy ending up in the form of electrons and photons. The resulting constraints are usually presented for a limited selection of annihilation and decay channels. Here we provide constraints on the annihilation cross section and decay rate, at discrete values of the dark matter mass, for all the annihilation and decay channels whose secondary spectra have been computed using PYTHIA in arXiv:1012.4515 ("PPPC…
CMB constraints on dark matter-proton scattering: investigating prior-volume effects using profile likelihoods
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Chicago（芝加哥大学）；Instituto de Física de Cantabria (IFCA), CSIC-Univ. de Cantabria（坎塔布里亚物理研究所（IFCA），西班牙国家研究委员会-坎塔布里亚大学）
Signals from the early Universe: a comprehensive search for primordial features in Planck CMB datasets
Università degli Studi di Ferrara（费拉拉大学）；INFN, Sezione di Ferrara（意大利国家核物理研究所费拉拉分部）；INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna（博洛尼亚天文物理与空间科学研究所）；University of the Western Cape（西开普大学）；Instituto de Física Corpuscular (IFIC), CSIC‐Universitat de València（瓦伦西亚大学西班牙国家科学研究委员会粒子物理研究所）
We show that the impact of energy injection by dark matter annihilation on the cosmic microwave background power spectra can be apprehended via a residual likelihood map. By resorting to convolutional neural networks that can fully discover the underlying pattern of the map, we propose a novel way of constraining dark matter annihilation based on the Planck 2018 data. We demonstrate that the trained neural network can efficiently predict the likelihood and accurately place bounds on the annihilation cross-section in a $\textit{model-independent}$ fashion. The machinery will be made public in t…
We present algorithms for searching for azimuthally symmetric features in CMB data. Our algorithms are fully optimal for masked all-sky data with inhomogeneous noise, computationally fast, simple to implement, and make no approximations. We show how to implement the optimal analysis in both Bayesian and frequentist cases. In the Bayesian case, our algorithm for evaluating the posterior likelihood is so fast that we can do a brute-force search over parameter space, rather than using a Monte Carlo Markov chain. Our motivating example is searching for bubble collisions, a pre-inflationary signal …
We demonstrate how to obtain optimal constraints on a primordial gravitational wave component in lensed Cosmic Microwave Background (CMB) data under ideal conditions. We first derive an estimator of the tensor-to-scalar ratio, $r$, by using an error-controlled close approximation to the exact posterior, under the assumption of Gaussian primordial CMB and lensing deflection potential. This combines fast internal iterative lensing reconstruction with optimal recovery of the unlensed CMB. We evaluate its performance on simulated low-noise polarization data targeted at the recombination peak. We c…
Cross-spectra likelihood for robust $τ$ constraints from all satellite polarisation data
Università degli Studi di Ferrara（费拉拉大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara（意大利国家核物理研究所，费拉拉分部）；Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay（法国国家科学研究中心天体物理学研究所，巴黎南大学，巴黎萨克雷大学）
We use a Quadratic Maximum Likelihood (QML) method to estimate the angular power spectrum of the cross-correlation between cosmic microwave background and large scale structure maps as well as their individual auto-spectra. We describe our implementation of this method and demonstrate its accuracy on simulated maps. We apply this optimal estimator to WMAP 7-year and NRAO VLA Sky Survey (NVSS) data and explore the robustness of the angular power spectrum estimates obtained by the QML method. With the correction of the declination systematics in NVSS, we can safely use most of the information co…
We show that the cosmic birefringence and miscalibrated polarisation angles can be determined simultaneously by cosmic microwave background (CMB) experiments using the cross-correlation between $E$- and $B$-mode polarisation data. This is possible because polarisation angles of the CMB are rotated by both the cosmic birefringence and miscalibration effects,whereas those of the Galactic foreground emission only by the latter. Our method does not require prior knowledge of the $E$- and $B$-mode power spectra of the foreground emission, but uses only the knowledge of the CMB polarisation spectra.…
