  Several recent works have considered Lieb-Robinson bounds (LRBs) for
Bose-Hubbard-type Hamiltonians. For certain special classes of initial states
(e.g., states with particle-free regions or perturbations of stationary
states), the velocity of information propagation was bounded by a constant in
time, $v\leq C$, similarly to quantum spin systems. However, for the more
general class of bounded-density initial states, the first-named author
together with Vu and Saito derived the velocity bound $v\leq C t^{D-1}$, where
$D$ is the spatial lattice dimension. For $D\geq 2$, this bound allows for
accelerated information propagation. It has been known since the work of Eisert
and Gross that some systems of lattice bosons are capable of accelerated
information propagation. It is therefore a central question to understand under
what conditions the bound $v\leq C t^{D-1}$ can be enhanced. Here, we prove
that additional physical constraints, translation-invariance and a $p$-body
repulsion of the form $n_x^p$ with $p>D+1$, lead to a LRB with $v\leq C
t^{\frac{D}{p-D-1}}$ for any initial state of bounded energy density. We also
identify examples of quantum states which show that no further enhancement is
possible without using additional dynamical constraints.
