Adiabaticity and gravity theory independent conservation laws for cosmological perturbations

Abstract

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) = P (over dot) / rho (over dot). Assuming the adiabaticity in the general sense, delta P-c,(nad) = 0, we derive a relation between the lapse function in the comoving slicing 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) not equal 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. This is realized on superhorizon scales in standard slow-roll inflation.

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 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.