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 δPnadδPnad, another is for a general matter field δPc,nadδPc,nad, and the last one is valid only on superhorizon scales. The first two definitions coincide if cs2=cw2 where cscs is the propagation speed of the perturbation, while cw2=P˙/ρ˙. Assuming the adiabaticity in the general sense, δPc,nad=0δPc,nad=0, we derive a relation between the lapse function in the comoving slicing AcAc and δPnadδPnad valid for arbitrary matter field in any theory of gravity, by using only momentum conservation. The relation implies that as long as cs≠cwcs≠cw, the uniform density, comoving and the proper-time slicings coincide approximately for any gravity theory and for any matter field if δPnad=0δPnad=0 approximately. In the case of general relativity this gives the equivalence between the comoving curvature perturbation RcRc and the uniform density curvature perturbation ζ on superhorizon scales, and their conservation. This is realized on superhorizon scales in standard slow-roll inflation.We then consider an example in which cw=cscw=cs, where δPnad=δPc,nad=0δPnad=δPc,nad=0 exactly, but the equivalence between RcRc and ζ   no longer holds. Namely we consider the so-called ultra slow-roll inflation. In this case both RcRc and ζ are not conserved. In particular, as for ζ, we find that it is crucial to take into account the next-to-leading order term in ζ  '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 RcRc or ζ.