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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 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 find that the computationally most efficient basis is the signal-to-noise eigenvector basis, which is closely related to the Karhunen-Loeve and Principal Component transforms, in agreement with previous suggestions. For this basis, the information in 6836 unmasked WMAP sky map pixels can be compressed into a smaller set of 3102 modes, with a maximum error increase of any single multipole of 3.8% at $ellle32$, and a maximum shift in the mean values of a joint distribution of an amplitude--tilt model of 0.006$sigma$. This compression reduces the computational cost of a single likelihood evaluation by a factor of 5, from 38 to 7.5 CPU seconds, and it also results in a more robust likelihood by implicitly regularizing nearly degenerate modes. Finally, we use the same compression framework to formulate a numerically stable and computationally efficient variation of the Quadratic Maximum Likelihood implementation that requires less than 3 GB of memory and 2 CPU minutes per iteration for $ell le 32$, rendering low-$ell$ QML CMB power spectrum analysis fully tractable on a standard laptop.
We present the Planck likelihood, a complete statistical description of the two-point correlation function of the CMB temperature fluctuations. We use this likelihood to derive the Planck CMB power spectrum over three decades in l, covering 2 <= l <= 2500. The main source of error at l <= 1500 is cosmic variance. Uncertainties in small-scale foreground modelling and instrumental noise dominate the error budget at higher ls. For l < 50, our likelihood exploits all Planck frequency channels from 30 to 353 GHz through a physically motivated Bayesian component separation technique. At l >= 50, we employ a correlated Gaussian likelihood approximation based on angular cross-spectra derived from the 100, 143 and 217 GHz channels. We validate our likelihood through an extensive suite of consistency tests, and assess the impact of residual foreground and instrumental uncertainties on cosmological parameters. We find good internal agreement among the high-l cross-spectra with residuals of a few uK^2 at l <= 1000. We compare our results with foreground-cleaned CMB maps, and with cross-spectra derived from the 70 GHz Planck map, and find broad agreement in terms of spectrum residuals and cosmological parameters. The best-fit LCDM cosmology is in excellent agreement with preliminary Planck polarisation spectra. The standard LCDM cosmology is well constrained by Planck by l <= 1500. For example, we report a 5.4 sigma deviation from n_s /= 1. Considering various extensions beyond the standard model, we find no indication of significant departures from the LCDM framework. Finally, we report a tension between the best-fit LCDM model and the low-l spectrum in the form of a power deficit of 5-10% at l <~ 40, significant at 2.5-3 sigma. We do not elaborate further on its cosmological implications, but note that this is our most puzzling finding in an otherwise remarkably consistent dataset. (Abridged)
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 systematic errors. Using realistic simulations, we examine the cluster mass uncertainties from CMB-cluster lensing as a function of an experiments beam size and noise level. We predict the cluster mass uncertainties will be 3 - 6% for SPT-3G, AdvACT, and Simons Array experiments with 10,000 clusters and less than 1% for the CMB-S4 experiment with a sample containing 100,000 clusters. The mass constraints from CMB polarization are very sensitive to the experimental beam size and map noise level: for a factor of three reduction in either the beam size or noise level, the lensing signal-to-noise improves by roughly a factor of two.
The maximum likelihood estimator plays a fundamental role in statistics. However, for many models, the estimators do not have closed-form expressions. This limitation can be significant in situations where estimates and predictions need to be computed in real-time, such as in applications based on embedded technology, in which numerical methods can not be implemented. This paper provides a modification in the maximum likelihood estimator that allows us to obtain the estimators in closed-form expressions under some conditions. Under mild conditions, the estimator is invariant under one-to-one transformations, consistent, and has an asymptotic normal distribution. The proposed modified version of the maximum likelihood estimator is illustrated on the Gamma, Nakagami, and Beta distributions and compared with the standard maximum likelihood estimator.
The Epoch of Reionization (EoR) depends on the complex astrophysics governing the birth and evolution of the first galaxies and structures in the intergalactic medium. EoR models rely on cosmic microwave background (CMB) observations, and in particular the large-scale E-mode polarization power spectra (EE PS), to help constrain their highly uncertain parameters. However, rather than directly forward-modelling the EE PS, most EoR models are constrained using a summary statistic -- the Thompson scattering optical depth, $tau_e$. Compressing CMB observations to $tau_e$ requires adopting a basis set for the EoR history. The common choice is the unphysical, redshift-symmetric hyperbolic tangent (Tanh) function, which differs in shape from physical EoR models based on hierarchical structure formation. Combining public EoR and CMB codes, 21cmFAST and CLASS, here we quantify how inference using the $tau_e$ summary statistic impacts the resulting constraints on galaxy properties and EoR histories. Using the last Planck 2018 data release, we show that the marginalized constraints on the EoR history are more sensitive to the choice of the basis set (Tanh vs physical model) than to the CMB likelihood statistic ($tau_e$ vs PS). For example, EoR histories implied by the growth of structure show a small tail of partial reionization extending to higher redshifts. However, biases in inference using $tau_e$ are negligible for the Planck 2018 data. Using EoR constraints from high-redshift observations including the quasar dark fraction, galaxy UV luminosity functions and CMB EE PS, our physical model recovers $tau_e=0.0569^{+0.0081}_{-0.0066}$.