Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol. (arXiv:1908.01626v2 [astro-ph.CO] UPDATED)

<a href="http://arxiv.org/find/astro-ph/1/au:+Kable_J/0/1/0/all/0/1">Joshua A. Kable</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Addison_G/0/1/0/all/0/1">Graeme E. Addison</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Bennett_C/0/1/0/all/0/1">Charles L. Bennett</a>

Consistency checks of cosmological data sets are an important tool because

they may suggest systematic errors or the type of modifications to $Lambda$CDM

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 $Lambda$CDM 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 $Lambda$CDM. 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 $chi^2$ 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.

Consistency checks of cosmological data sets are an important tool because

they may suggest systematic errors or the type of modifications to $Lambda$CDM

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 $Lambda$CDM 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 $Lambda$CDM. 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 $chi^2$ 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.

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