Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol

Abstract

Consistency checks of cosmological data sets are an important tool because they may suggest systematic errors or the type of modifications to ΛCDM necessary to resolve current tensions. In this work, we derive an analytic method for calculating the level of correlations between model parameters from two correlated cosmological data sets, which complements more computationally expensive simulations. This method is an extension of the Fisher analysis that assumes a Gaussian likelihood and a known data covariance matrix. We apply this method to the SPTpol temperature and polarization CMB spectra (TE and EE). We find weak correlations between ΛCDM parameters with a 9% correlation between the TE-only and EE-only constraints on H 0 and a 25% and 32% correlation for log(A s ) and n s respectively. Despite the negative correlations between the TE and EE power spectra, the correlations in the parameters are positive. The TE-EE parameter differences are consistent with zero, with a PTE of 0.53, in contrast to the PTE of 0.017 reported by SPTpol for the consistency of the TE and EE power spectra with ΛCDM. Using simulations we find that the results of these two tests are independent and that this difference can arise simply from statistical fluctuations. Ignoring correlations in the TT-TE and TE-EE comparisons biases the χ 2 low, artificially making parameters look more consistent. Therefore, we conclude that these correlations need to be accounted for when performing internal consistency checks of the TT vs TE vs EE power spectra for future CMB analyses.

High $H_0$ Values from CMB E-mode Data: A Clue for Resolving the Hubble Tension?

“…Over the multipole range accessible to Planck, however, the correlation coefficients between ΛCDM parameters from TT and their EE counterparts have magnitude 0.1 or smaller, even for ideal noiseless data (see Figs. 4 and 5 of Kable et al 2020). Given the significant noise levels in the Planck polarization data the TT and EE constraints are therefore well approximated as independent.…”

candl: cosmic microwave background analysis with a differentiable likelihood

“…We now use candl and CosmoPower-JAX to calculate the covariance of best-fit parameters of correlated data sets. We follow the prescription of Kable et al (2020):…”

“…where C XX is block the band power covariance matrix for data set X and F XX the Fisher matrix when only considering data set X. The derivatives of the model spectra are typically evaluated at the best-fit point of the full data set (Kable et al 2020).…”

“…A differentiable inference pipeline makes this exercise fast, reliable, and easy. We note that an alternative approach to calculating the above parameter correlations exists, in which one simulates mock band powers and find their best-fit values; this was done by Kable et al (2020) to verify the above procedure. We note that this approach is also accelerated by candl thanks to access to gradient-based minimisers (see Sect.…”