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 $c_w^2=\dot P/\dot\rho$. 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 $\zeta$.
Highlights
It is a well-known fact that in single-field slow-roll inflation [1,2,3], the comoving curvature perturbation Rc and the uniform density curvature perturbation ζ coincide and are conserved
This is in agreement with the well-known coincidence of ζ and Rc on super-horizon scales for slow roll-models in general relativity, since in this case cs = cw and δPnad ≈ 0 on superhorizon scales
The usual interpretation of the above equation is that for adiabatic perturbations, ζ is conserved on superhorizon scales, as long as the gradient terms can be neglected
Summary
It is a well-known fact that in single-field slow-roll inflation [1,2,3], the comoving curvature perturbation Rc and the uniform density curvature perturbation ζ coincide and are conserved. In the seminal works [4, 5], it was shown that requiring just energy conservation is enough to show the superhorizon conservation of ζ given that the non-adiabatic pressure δPnad vanishes, under the assumption that gradient terms are negligible It was shown in [4] that for adiabatic perturbations, on superhorizon scales the comoving slicing coincides with the uniform density slicing, as long as ∂V /∂φ = 0. As long as we are interested in superhorizon scale perturbations, the adiabaticity conditions for both of the previous two cases will be approximately satisfied if the universe is in the adiabatic limit Both δPnad and δPc,nad will be of O (k/H) and vanish in the superhorizon limit
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