Adiabaticity and gravity theory independent conservation laws for
  cosmological perturbations
  We carefully study the implications of adiabaticity for the behavior of
cosmological perturbations. There are essentially three similar but different
definitions of non-adiabaticity: one is appropriate for a thermodynamic fluid
$\delta P_{nad}$, another is for a general matter field $\delta P_{c,nad}$, and
the last one is valid only on superhorizon scales. The first two definitions
coincide if $c_s^2=c_w^2$ where $c_s$ is the propagation speed of the
perturbation, while $c_w^2=\dot P/\dot\rho$. Assuming the adiabaticity in the
general sense, $\delta P_{c,nad}=0$, we derive a relation between the lapse
function in the comoving sli\-cing $A_c$ and $\delta P_{nad}$ valid for
arbitrary matter field in any theory of gravity, by using only momentum
conservation. The relation implies that as long as $c_s\neq c_w$, the uniform
density, comoving and the proper-time slicings coincide approximately for any
gravity theory and for any matter field if $\delta P_{nad}=0$ approximately. In
the case of general relativity this gives the equivalence between the comoving
curvature perturbation $R_c$ and the uniform density curvature perturbation
$\zeta$ on superhorizon scales, and their conservation.
  We then consider an example in which $c_w=c_s$, where $\delta P_{nad}=\delta
P_{c,nad}=0$ exactly, but the equivalence between $R_c$ and $\zeta$ no longer
holds. Namely we consider the so-called ultra slow-roll inflation. In this case
both $R_c$ and $\zeta$ are not conserved. In particular, as for $\zeta$, we
find that it is crucial to take into account the next-to-leading order term in
$\zeta$'s spatial gradient expansion to show its non-conservation, even on
superhorizon scales. This is an example of the fact that adiabaticity (in the
thermodynamic sense) is not always enough to ensure the conservation of $R_c$
or $\zeta$.
