It is likely that young protostellar discs undergo a self-gravitating phase. Such systems are characterised by the presence of a spiral pattern that can be either in a quasi-steady state or in a non-linear 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 thanks to high spatial and spectral resolution observations of protostellar discs. In this work, we investigate the kinematic signatures of gravitational instability: we perform an analytical study of the linear response of a self-gravitating disc to a spiral-like perturbation, focusing our attention on the velocity field perturbations. We show that unstable discs have clear kinematic imprints into the gas component across the entire disc 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 discs cooling.
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