  In this paper we compute some of the higher order terms in the large-t
asymptotic expansion of the Airy process two-point function, extending the
previous work of Adler and van Moerbeke and Widom. We prove that it is possible
to represent any order asymptotic approximation as a polynomial and integrals
of the Hastings-McLeod Painlev\'e II function and its first derivative.
Further, for up to tenth order we give this asymptotic approximation as a
linear combination of the Tracy-Widom GUE density function f_2 and its
derivatives. As a corollary to this, the asymptotic covariance is expressed up
to tenth order in terms of the moments of the Tracy-Widom GUE distribution.
