  We use smoothed particle hydrodynamics simulations of massive protostellar
discs to investigate the predicted broadening of molecular lines from discs in
which self-gravity is the dominant source of angular momentum transport. The
simulations include radiative transfer, and span a range of disc-to-star mass
ratios between 0.25 and 1.5. Subtracting off the mean azimuthal flow velocity,
we compute the distribution of the in-plane and perpendicular peculiar velocity
due to large scale structure and turbulence induced by self-gravity. For the
lower mass discs, we show that the characteristic peculiar velocities scale
with the square root of the effective turbulent viscosity parameter, as
expected from local turbulent-disc theory. The derived velocities are
anisotropic, with substantially larger in-plane than perpendicular values. As
the disc mass is increased, the validity of the locally determined turbulence
approximation breaks down, and this is accompanied by anomalously large
in-plane broadening. There is also a high variance due to the importance of
low-m spiral modes. For low-mass discs, the magnitude of in-plane broadening
is, to leading order, equal to the predictions from local disc theory and
cannot constrain the source of turbulence. However, combining our results with
prior evaluations of turbulent broadening expected in discs where the
magnetorotational instability (MRI) is active, we argue that self-gravity may
be distinguishable from the MRI in these systems if it is possible to measure
the anisotropy of the peculiar velocity field with disc inclination.
Furthermore, for large mass discs, the dominant contribution of large-scale
modes is a distinguishing characteristic of self-gravitating turbulence versus
MRI driven turbulence.
