Abstract
We investigate the gauge-invariant cosmological perturbations in the gravity and matter frames in the general scalar–tensor theory where two frames are related by the disformal transformation. The gravity and matter frames are the extensions of the Einstein and Jordan frames in the scalar–tensor theory where two frames are related by the conformal transformation, respectively. First, it is shown that the curvature perturbation in the comoving gauge to the scalar field is disformally invariant as well as conformally invariant, which gives the predictions from the cosmological model where the scalar field is responsible both for inflation and cosmological perturbations. Second, in case that the disformally coupled matter sector also contributes to curvature perturbations, we derive the evolution equations of the curvature perturbation in the uniform matter energy density gauge from the energy (non)conservation in the matter sector, which are independent of the choice of the gravity sector. While in the matter frame the curvature perturbation in the uniform matter energy density gauge is conserved on superhorizon scales for the vanishing nonadiabatic pressure, in the gravity frame it is not conserved even if the nonadiabatic pressure vanishes. The formula relating two frames gives the amplitude of the curvature perturbation in the matter frame, once it is evaluated in the gravity frame.
Highlights
The Concordance Model of Cosmology has succeeded in explaining the history of the universe [1]
We investigate whether the curvature perturbation in the comoving gauge to the scalar field is invariant under the disformal transformation as well as the conformal transformation [12,13,14]
Using (54), we find that when the nonadiabatic pressure of the (a)-th component in the matter frame vanishes, Γ(a) = 0, and the curvature perturbation in the uniform energy density gauge of the (a)-th component in the matter frame is conserved on superhorizon scales, ζ,(ta) = 0, the nonadiabatic pressure in the gravity frame is given by p(a)β p(,ta) ρ(,ta) ρ(a) p(a) p(a) ρ(a) ρ(a) p(a) p(,ta) ρ(,ta) ρ(a) p(a) p(a) ρ(a) δρ(a) φφ which does not vanish in general, meaning that the adiabaticity condition for the (a)-th component does not hold in the gravity frame
Summary
The Concordance Model of Cosmology has succeeded in explaining the history of the universe [1]. In the simplest case where there is no matter field disformally coupled to gravity and the only scalar field is responsible both for inflation and cosmological perturbations, the comoving curvature perturbation is conserved after the scale of interest crosses the horizon, which gives the final prediction from the given model. If it is disformally invariant, it may be evaluated in any disformally related frame as done in the Einstein frame in the gravitational theory with the nonminimal coupling of the scalar field to the Ricci scalar where two frames are related by the conformal transformation.
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