  Weak gravitational lensing is now established as a powerful method to measure
mass fluctuations in the universe. It relies on the measurement of small
coherent distortions of the images of background galaxies. Even low-level
correlations in the intrinsic shapes of galaxies could however produce a
significant spurious lensing signal. These correlations are also interesting in
their own right, since their detection would constrain models of galaxy
formation. Using $3\times 10^{4} - 10^5$ halos found in N-body simulations, we
compute the correlation functions of the intrinsic ellipticity of spiral
galaxies assuming that the disk is perpendicular to the angular momentum of the
dark matter halo. We also consider a simple model for elliptical galaxies, in
which the shape of the dark matter halo is assumed to be the same as that of
the light. For deep lensing surveys with median redshifts $\sim 1$, we find
that intrinsic correlations of $\sim 10^{-4}$ on angular scales $\theta \sim
0.1-10'$ are generally below the expected lensing signal, and contribute only a
small fraction of the excess signals reported on these scales. On larger scales
we find limits to the intrinsic correlation function at a level $\sim 10^{-5}$,
which gives a (model-dependent) range of separations for which the intrinsic
signal is about an order of magnitude below the ellipticity correlation
function expected from weak lensing. Intrinsic correlations are thus negligible
on these scales for dedicated weak lensing surveys. For wider but shallower
surveys such as SuperCOSMOS, APM and SDSS, we cannot exclude the possibility
that intrinsic correlations could dominate the lensing signal. We discuss how
such surveys could be used to calibrate the importance of this effect, as well
as study spin-spin correlations of spiral galaxies.
