Eigen-decomposition of Covariance matrices: An application to the BAO Linear Point

The Baryon Acoustic Oscillation (BAO) feature in the two-point correlation function (TPCF) of discrete tracers such as galaxies is an accurate standard ruler. The covariance matrix of the TPCF plays an important role in determining how the precision of this ruler depends on the number density and clustering strength of the tracers, as well as the survey volume. An eigen-decomposition of this matrix provides an objective way to separate the contributions of cosmic variance from those of shot-noise to the statistical uncertainties. For the signal-to-noise levels that are expected in ongoing and next-generation surveys, the cosmic variance eigen-modes dominate. These modes are smooth functions of scale, meaning that: they are insensitive to the modest changes in binning that are allowed if one wishes to resolve the BAO feature in the TPCF; they provide a good description of the correlated residuals which result from fitting smooth functional forms to the measured TPCF; they motivate a simple but accurate approximation for the uncertainty on the Linear Point (LP) estimate of the BAO distance scale. This approximation allows one to quantify the precision of the BAO distance scale estimate without having to generate a large ensemble of mock catalogs and explains why: the uncertainty on the LP does not depend on the functional form fitted to the TPCF or the binning used; the LP is more constraining than the peak or dip scales in the TPCF; the evolved TPCF is less constraining than the initial one, so that reconstruction schemes can yield significant gains in precision.