Euclid: The linear-construction covariance and cosmology

We study the properties of galaxy cluster 2-point correlation function covariance matrices estimated using the linear-construction (LC) method, which is computationally up to 20 times faster than the standard sample-covariance method. Our goal is to assess how well the LC method performs in cosmological parameter estimation compared to the sample covariance. We use a set of 1000 mock dark matter halo catalogues to compute both the LC-covariance and the sample-covariance estimates in four redshift shells. These numerical matrices are used to fit a theoretical four-parameter model for the covariance. We then use the two fitted covariance models in a likelihood function to estimate two cosmological parameters - the matter density parameter $\Omega_{\rm m}$ and the amplitude of the matter density fluctuations $\sigma_8$ - from the simulated mock catalogues. The purpose of this is to validate the LC-covariance-based model against the sample-covariance model. The catalogues were simulated assuming the spatially flat $\Lambda$CDM cosmology, with $\Omega_{\rm m} = 0.30711$ and $\sigma_8=0.8288$. We find that the parameter posteriors obtained using the sample- and LC-covariance models agree well with each other and with the simulation cosmology. The two pairs of marginalized constraints are $\Omega_{\rm m} = 0.307 \pm 0.003$ and $\sigma_8 = 0.826\pm 0.009$ (sample covariance), and $\Omega_{\rm m} = 0.308 \pm 0.003$ and $\sigma_8 = 0.825 \pm 0.009$ (LC covariance). The posterior widths are the same, and the difference in the median values is less than $0.16\,\sigma$ for both parameters.