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

Equivalence of cosmological observables in conformally related scalar tensor theories