Equivalence of cosmological observables in conformally related scalar tensor theories

TL;DR

This paper demonstrates that cosmological observables, including CMB temperature anisotropies, are equivalent in conformally related scalar-tensor theories when proper transformations are applied, confirming the consistency between Einstein and Jordan frames.

Contribution

The study provides a detailed comparison and analytical proof that cosmological equations and observables are equivalent in conformally related frames, supported by a numerical example.

Findings

Field and conservation equations are equivalent in both frames.

Line-of-sight integration yields identical CMB results in both frames.

Correct redshift definition is crucial for equivalence.

Abstract

Scalar tensor theories can be expressed in different frames, such as the commonly-used Einstein and Jordan frames, and it is generally accepted that cosmological observables are the same in these frames. We revisit this by making a detailed side-by-side comparison of the quantities and equations in two conformally related frames, from the actions and fully covariant field equations to the linearised equations in both real and Fourier spaces. This confirms that the field and conservation equations are equivalent in the two frames, in the sense that we can always re-express equations in one frame using relevant transformations of variables to derive the corresponding equations in the other. We show, with both analytical derivation and a numerical example, that the line-of-sight integration to calculate CMB temperature anisotropies can be done using either Einstein frame or Jordan frame quantities, and the results are identical, provided the correct redshift is used in the Einstein frame ($1+z\neq1/a$).