Creating Jackknife and Bootstrap estimates of the covariance matrix for the two-point correlation function

Abstract: We present correction terms that allow delete-one Jackknife and Bootstrap methods to be used to recover unbiased estimates of the data covariance matrix of the two-point correlation function $\xi\left(\mathbf{r}\right)$. We demonstrate the accuracy and precision of this new method using a large set of 1000 QUIJOTE simulations that each cover a comoving volume of $1\rm{\left[h^{-1}Gpc\right]^3}$. The corrected resampling techniques recover the correct amplitude and structure of the data covariance matrix as represented by its principal components to within $\sim10$\%, the level of error achievable with the size of the sample of simulations used for the test. Our corrections for the internal resampling methods are shown to be robust against the intrinsic clustering of the cosmological tracers both in real- and redshift space using two snapshots at $z=0$ and $z=1$ that mimic two samples with significantly different clustering. We also analyse two different slicing of the simulation volume into $n_{\rm sv}=64$ or $125$ sub-samples and show that the main impact of different $n_{\rm sv}$ is on the structure of the covariance matrix due to the limited number of independent internal realisations that can be made given a fixed $n_{\rm sv}$.