  It is likely that young protostellar disks undergo a self-gravitating phase.
Such systems are characterized by the presence of a spiral pattern that can be
either in a quasi-steady state or in a nonlinear unstable condition. This
spiral wave affects both the gas dynamics and kinematics, resulting in
deviations from the Keplerian rotation. Recently, a lot of attention has been
devoted to kinematic studies of planet-forming environments, and we are now
able to measure even small perturbations of velocity field ($\lesssim 1 \%$ of
the Keplerian speed) thanks to high spatial and spectral resolution
observations of protostellar disks. In this work, we investigate the kinematic
signatures of gravitational instability: we perform an analytical study of the
linear response of a self-gravitating disk to a spiral-like perturbation,
focusing our attention on the velocity field perturbations. We show that
unstable disks have clear kinematic imprints into the gas component across the
entire disk extent, due to the GI spiral wave perturbation, resulting in
deviations from Keplerian rotation. The shape of these signatures depends on
several parameters, but they are significantly affected by the cooling factor:
by detecting these features, we can put constraints on protoplanetary disk
cooling.
