  Consider an $N$ by $N$ matrix $X$ of complex entries with iid real and
imaginary parts. We show that the local density of eigenvalues of $X^*X$
converges to the Marchenko-Pastur law on the optimal scale with probability
$1$. We also obtain rigidity of the eigenvalues in the bulk and near both hard
and soft edges. Here we avoid logarithmic and polynomial corrections by working
directly with high powers of expectation of the Stieltjes transforms. We work
under the assumption that the entries have a finite 4th moment and are
truncated at $N^{1/4}$. In this work we simplify and adapt the methods from
prior papers of G\"otze-Tikhomirov and Cacciapuoti-Maltsev-Schlein to
covariance matrices.
