  We prove the first explicit rate of convergence to the Tracy-Widom
distribution for the fluctuation of the largest eigenvalue of sample covariance
matrices that are not integrable. Our primary focus is matrices of type $ X^*X
$ and the proof follows the Erd\"{o}s-Schlein-Yau dynamical method. We use a
recent approach to the analysis of the Dyson Brownian motion from [5] to obtain
a quantitative error estimate for the local relaxation flow at the edge.
Together with a quantitative version of the Green function comparison theorem,
this gives the rate of convergence.
  Combined with a result of Lee-Schnelli [26], some quantitative estimates also
hold for more general separable sample covariance matrices $ X^* \Sigma X $
with general diagonal population $ \Sigma $.
