Euclid: The linear-construction covariance and cosmology

Abstract

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 $Ω_{\rm m}$ and the amplitude of the matter density fluctuations $σ_8$ - 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 $Λ$CDM cosmology, with $Ω_{\rm m} = 0.30711$ and $σ_8=0.8288$. 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 $Ω_{\rm m} = 0.307 \pm 0.003$ and $σ_8 = 0.826\pm 0.009$ (sample covariance), and $Ω_{\rm m} = 0.308 \pm 0.003$ and $σ_8 = 0.825 \pm 0.009$ (LC covariance). The posterior widths are the same, and the difference in the median values is less than $0.16\,σ$ for both parameters.

Figures (5)

• Figure 1: Cumulative distribution of $\chi^2$ values. Top panels: distribution of the $\chi^2_i$ values of Eq. \ref{['eq:chisq']} for four flavours of inverse covariance: Hartlap-corrected sample covariance (magenta), the uncorrected LC covariance (orange), the de-biased inverse-LC covariance from Eq. \ref{['eq:biascorr2']} (red), the de-biased inverse-LC covariance after iterating Eq. \ref{['eq:biascorr1']} 14--36 times (blue), and the uncorrected sample covariance (grey). The smooth black line is the theoretical distribution. Bottom panels: the difference of each measured CDF with respect to the theoretical prediction. Left column: the $z=0.0$--$0.4$ shell. Right column: the $z=0.8$--$1.2$ shell.
• Figure 2: Comparison of the modelled 2PCF, $\xi(r)$, and the mean measured from 1000 simulations. Top panel: the mean measured (points) and model (solid lines) 2PCF in four redshift shells. Bottom panel: the relative difference between the model and the measurement. The grey box shows the 10% region.
• Figure 3: Comparison of the covariance model and the numerical LC covariance. Top panels: the first three diagonals of the LC covariance (solid lines), and the model covariance (dashed lines) from Eq. \ref{['eq:cov-model2']} with parameters $p_k$ for LC given in Table \ref{['tab:params-cov']}. Bottom panels: the relative difference of the model with respect to the LC covariance (solid lines), in addition to the difference of the "raw" covariance model (the free parameters $p_k$ set to unity in Eq. \ref{['eq:cov-model2']} or Eq. \ref{['eq:cov-model3']}) with respect to the numerical LC covariance (dotted lines). Left column: the $z=0.0$--$0.4$ shell. Right column: the $z=0.8$--$1.2$ shell.
• Figure 4: Comparison of covariance models fitted to either the LC covariance or the sample covariance. Top panels: the first three diagonals of the model corresponding to the LC covariance (solid lines) and the model corresponding to the sample covariance (dashed lines). Bottom panels: the relative difference of the LC-covariance model with respect to the sample-covariance model. Left column: the $z=0.0$--$0.4$ shell. Right column: the $z=0.8$--$1.2$ shell.
• Figure 5: Posterior distribution for $\Omega_{\rm m}$ and $\sigma_8$ obtained from four redshift shells. The LC covariance is shown in orange, and the sample covariance in red. The grey crosshair shows the values used in the simulation. The blue contours show the posterior in the case of the sample covariance, when constructing a likelihood from a single 2PCF realisation instead of the mean over the 1000 light cones. Crosses of corresponding colours show the median of each distribution. We have also plotted medians from two additional individual realisations to further illustrate their spread. The 2D contours correspond to the 68% and 95% confidence regions.