  We investigate causality constraints on the time evolution of entanglement
entropy after a global quench in relativistic theories. We first provide a
general proof that the so-called tsunami velocity is bounded by the speed of
light. We then generalize the free particle streaming model of
arXiv:cond-mat/0503393 to general dimensions and to an arbitrary entanglement
pattern of the initial state. In more than two spacetime dimensions the spread
of entanglement in these models is highly sensitive to the initial entanglement
pattern, but we are able to prove an upper bound on the normalized rate of
growth of entanglement entropy, and hence the tsunami velocity. The bound is
smaller than what one gets for quenches in holographic theories, which
highlights the importance of interactions in the spread of entanglement in
many-body systems. We propose an interacting model which we believe provides an
upper bound on the spread of entanglement for interacting relativistic
theories. In two spacetime dimensions with multiple intervals, this model and
its variations are able to reproduce intricate results exhibited by holographic
theories for a significant part of the parameter space. For higher dimensions,
the model bounds the tsunami velocity at the speed of light. Finally, we
construct a geometric model for entanglement propagation based on a tensor
network construction for global quenches.
