The effect of covariance estimator error on cosmological parameter constraints

Extracting parameter constraints from cosmological observations requires accurate determination of the covariance matrix for use in the likelihood function. We show here that uncertainties in the elements of the covariance matrix propagate directly to increased uncertainties in cosmological parameters. When the covariance matrix is determined by simulations, the resulting variance of the each parameter increases by a factor of order $1+N_b/N_s$, where $N_b$ is the number of bands in the measurement and $N_s$ is the number of simulations.

Figure 1

The coefficient $B$ from Eq. (26) as a function of the number of simulation realizations using the lensing two-point correlation function (“corr”) and power spectrum (“power”) from Refs. 18, 19 (with a delta-function source distribution at $z=1$). There are 1000 realizations of the simulated two-point functions. We take the sample covariance using all 1000 simulations as a reference and compare them with the sample covariance using subsets of $N_s$ simulations. The dashed black line shows the prediction from Eq. (28) with $N_b=24$. The error bars indicate the standard error on $B$ as a regression coefficient fit simultaneously to the $N_b^2\times N_b^2$ components of $\mathrm{Cov}(\Delta \Psi )$.