  This paper shows that the largest eigenvalue of a real symmetric or complex
Hermitian Wigner matrix of size $N$ converges to the Tracy-Widom laws at a rate
$O(N^{-1/3+\omega})$, as $N$ tends to infinity. For Wigner matrices this
improves the previous rate $O(N^{-2/9+\omega})$ obtained by Bourgade [5] for
generalized Wigner matrices. Our result follows from a Green function
comparison theorem, originally introduced in [19] to prove edge universality,
on a finer spectral parameter scale with improved error estimates. The proof
relies on the continuous Green function flow induced by a matrix-valued
Ornstein-Uhlenbeck process. Precise estimates on leading contributions from the
third and fourth order moments of the matrix entries are obtained using
iterative cumulant expansions and recursive comparisons for correlation
functions, along with uniform convergence estimates for correlation kernels of
the Gaussian invariant ensembles.
