  Inspired by dense contractile tissues, where cells are subject to periodic
deformation, we formulate and study a generic hydrodynamic theory of pulsating
active liquids. Combining mechanical and phenomenological arguments, we
postulate that the mechanochemical feedback between the local phase, which
describes how cells deform due to autonomous driving, and the local density can
be described in terms of a free energy. We argue that such a feedback is
consistent with the coarse-graining of a broad class of microscopic models. Our
hydrodynamics captures the three main states emerging in its particle-based
counterparts: a globally cycling state, a homogeneous arrested state with
constant phase, and a state with propagating radial waves. Remarkably, we show
that the competition between these states can be rationalized intuitively in
terms of an effective landscape, and argue that waves can be regarded as
secondary instabilities. Linear stability analysis of the arrested and cycling
states, including the role of fluctuations, leads to predictions for the phase
boundaries. Overall, our results demonstrate that our minimal, yet non-trivial
model provides a relevant platform to study the rich phenomenology of pulsating
liquids.
