Accurate cosmic microwave background covariance matrices: Exact calculation and approximations

Context. A reliable estimation of cosmological parameters from pseudo-power spectrum estimators requires accurate covariance matrices.Aims. We focus on the analytical calculation of covariance matrices. We consider the case of observations of the cosmic microwave background (CMB) in temperature and polarization on a small footprint such as in the South Pole Telescope third-generation (SPT-3G) experiment, which observes 4% of the sky. Power spectra evaluated on small footprints are expected to have strong correlations between modes, and these need to be accurately modeled.Methods. We present for the first time an algorithm that allows an efficient (but computationally expensive) exact calculation of analytic covariance matrices. Using it as our reference, we tested the accuracy of existing fast approximations of the covariance matrix. Furthermore, we propose a new approximation that is designed to be more precise. Finally, we derived the covariance matrices for mask-corrected power spectra estimated by the PolSpice code. In particular, in the case of a small sky fraction, we included the effect of the apodization of the large-scale modes.Results. We find that when the power spectrum is binned in wide bandpowers, current approximations of the covariance matrix are correct up to the 5% level on the SPT-3G small sky footprint. Our new approximation improves the previous approximations and reaches a precision of 1% for the wide bandpowers. It is generally more than four times more accurate than current approaches.Conclusions. While we considered the specific case of the CMB, our results are applicable to any other cosmological probe that requires the calculation of pseudo-power spectrum covariance matrices.