the Green function comparison theorem, which is a slight extension of the classical result from [@GreenFunctionComparison].

As discussed in [@pillai2014universality Section 3], though we discuss the problems in the context of covariance matrices, the proof should also work for more generalized problems such as the quantitative edge universality of correlation matrices (see [@pillai2012edge]).

This paper is organized as follows. The main part of the paper (Section \[s.prelim\]-\[s.Comparison\]) is devoted to the null case $ X^*X $. In Section \[s.prelim\] we rephrase the classical results on the eigenvalues of sample covariance matrices to the version for singular values, including Dyson Brownian motion, local laws and rigidity estimates. In Section \[s.LocalRelaxationFlow\], we define an observable that describes the evolution of singular values, and then prove the error estimate for the local relaxation flow at the edge by studying the dynamics of the observable. In Section \[s.Comparison\] we prove the quantitative Green function comparison theorem and use it to derive the rate of convergence to the Tracy-Widom distribution. Finally, in Section \[s.GeneralPopulation\] we generalize our result to separable sample covariance matrices with diagonal population by studying the interpolation between general covariance matrices with the null case.

Notations
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Throughout this paper, we use 