On the regularity of the covariance matrix of a discretized scalar field on the sphere

We present a comprehensive study of the regularity of the covariance matrix of a discretized field on the sphere. In a particular situation, the rank of the matrix depends on the number of pixels, the number of spherical harmonics, the symmetries of the pixelization scheme and the presence of a mask. Taking into account the above mentioned components, we provide analytical expressions that constrain the rank of the matrix. They are obtained by expanding the determinant of the covariance matrix as a sum of determinants of matrices made up of spherical harmonics. We investigate these constraints for five different pixelizations that have been used in the context of Cosmic Microwave Background (CMB) data analysis: Cube, Icosahedron, Igloo, GLESP and HEALPix, finding that, at least in the considered cases, the HEALPix pixelization tends to provide a covariance matrix with a rank closer to the maximum expected theoretical value than the other pixelizations. The effect of the propagation of numerical errors in the regularity of the covariance matrix is also studied for different computational precisions, as well as the effect of adding a certain level of noise in order to regularize the matrix. In addition, we investigate the application of the previous results to a particular example that requires the inversion of the covariance matrix: the estimation of the CMB temperature power spectrum through the Quadratic Maximum Likelihood algorithm. Finally, some general considerations in order to achieve a regular covariance matrix are also presented.Comment: 36 pages, 12 figures; minor changes in the text, matches published versio

QUIJOTE scientific results – IV. A northern sky survey in intensity and polarization at 10–20 GHz with the multifrequency instrument

J. A. Rubiño-Martín et al.We present QUIJOTE intensity and polarization maps in four frequency bands centred around 11, 13, 17, and 19 GHz, and covering approximately 29 000 deg2, including most of the northern sky region. These maps result from 9000 h of observations taken between May 2013 and June 2018 with the first QUIJOTE multifrequency instrument (MFI), and have angular resolutions of around 1°, and sensitivities in polarization within the range 35–40 µK per 1° beam, being a factor ∼2–4 worse in intensity. We discuss the data processing pipeline employed, and the basic characteristics of the maps in terms of real space statistics and angular power spectra. A number of validation tests have been applied to characterize the accuracy of the calibration and the residual level of systematic effects, finding a conservative overall calibration uncertainty of 5 per cent. We also discuss flux densities for four bright celestial sources (Tau A, Cas A, Cyg A, and 3C274), which are often used as calibrators at microwave frequencies. The polarization signal in our maps is dominated by synchrotron emission. The distribution of spectral index values between the 11 GHz and WMAP 23 GHz map peaks at β = −3.09 with a standard deviation of 0.14. The measured BB/EE ratio at scales of ℓ = 80 is 0.26 ± 0.07 for a Galactic cut |b| > 10°. We find a positive TE correlation for 11 GHz at large angular scales (ℓ ≲ 50), while the EB and TB signals are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150. The maps discussed in this paper are publicly available.Partial financial support was provided by the Spanish Ministry of Science and Innovation under the projects AYA2007-68058-C03-01,AYA2007-68058-C03-02, AYA2010-21766-C03-01, AYA2010-21766-C03-02, AYA2014-60438-P, ESP2015-70646-C2-1-R, AYA2017-84185-P, ESP2017-83921-C2-1-R, AYA2017-90675-REDC (co-funded with EU FEDER funds), PGC2018-101814-B-I00, PID2019-110610RB-C21,PID2020-120514GB-I00, IACA13-3E-2336, IACA15-BE-3707, EQC2018-004918-P, the Severo Ochoa Programs SEV-2015-0548 and CEX2019-000920-S, the Maria de Maeztu Program MDM-2017-0765, and by the Consolider-Ingenio project CSD2010-00064 (EPI: Exploring the Physics of Inflation). We acknowledge support from the ACIISI, Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference ProID2020010108. This project has received funding from the European Union’s Horizon 2020 - Research and Innovation Framework Programme under grant agreement number 687312 (RADIOFOREGROUNDS). FP acknowledges support from the Spanish State Research Agency (AEI) under grant number PID2019-105552RB-C43.With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2019-000920-S).Peer reviewe

QUIJOTE scientific results - IV. A northern sky survey in intensity and polarization at 10-20 GHz with the multifrequency instrument

International audienceWe present QUIJOTE intensity and polarization maps in four frequency bands centred around 11, 13, 17, and 19 GHz, and covering approximately 29 000 deg2, including most of the northern sky region. These maps result from 9000 h of observations taken between May 2013 and June 2018 with the first QUIJOTE multifrequency instrument (MFI), and have angular resolutions of around 1°, and sensitivities in polarization within the range 35-40 µK per 1° beam, being a factor ~2-4 worse in intensity. We discuss the data processing pipeline employed, and the basic characteristics of the maps in terms of real space statistics and angular power spectra. A number of validation tests have been applied to characterize the accuracy of the calibration and the residual level of systematic effects, finding a conservative overall calibration uncertainty of 5 per cent. We also discuss flux densities for four bright celestial sources (Tau A, Cas A, Cyg A, and 3C274), which are often used as calibrators at microwave frequencies. The polarization signal in our maps is dominated by synchrotron emission. The distribution of spectral index values between the 11 GHz and WMAP 23 GHz map peaks at β = -3.09 with a standard deviation of 0.14. The measured BB/EE ratio at scales of ℓ = 80 is 0.26 ± 0.07 for a Galactic cut |b| > 10°. We find a positive TE correlation for 11 GHz at large angular scales (ℓ ≲ 50), while the EB and TB signals are consistent with zero in the multipole range 30 ≲ ℓ ≲ 150. The maps discussed in this paper are publicly available