CosmoCoffee

This paper attempts to continue a study of the CMB at second order on all scales. This is a complex and subtle task!

I have one general comment: we know that both CMB lensing and SZ are significantly non-linear effects (i.e. they cannot be accurately modelled at only second order). They can however be studied separately in ways which isolate the relevant physicical processes (and in the case of CMB lensing modelled very accurately). Wouldn't it be better to attempt a second order calculation without these effects (assuming a way can be found to define this consistently), then put them in (at above second order) later?

Also Eq 3.21 looks rather like a lensing effect (and this was said explicitly in a talk I heard). However the sign in the second term looks odd: if the signs in the first and second terms were the same it would basically be a perpendicular gradient which is what is expected for lensing. Furthermore for it to be interpreted as lensing the result should be conformally invariant, which implies only a function of \Psi + \Phi (see e.g. astro-ph/0601594). So is this equation correct, and if so what does it mean?

I am very surprised that there is no reference to the works of Kenji Tomita.
Especially in astro-ph/0501663, Tomita derived the second order version of the Sachs-Wolfe formula for the LambdaCDM cosmology.
The work of Bartolo et al goes beyond Tomitas work, but an essential part is already present in Tomitas paper.

Regarding the claim that the calculation holds for all scales, I am very sceptic.
It seems to me that the applicability of a full second-order calculation is limited to intermediate angular scales:
at the largest angular scales, we have shown that local non-linear effects like the one discussed recently in astro-ph/0601445 (we looked at the local Rees-Sciama effect from the local 100 Mpc structure) might be of order 10^(-5), and thus are likely to dominate the effects under consideration. At the smallest angular scales, SZ and weak lensing cannot be fully included in a second-order calculation as has been pointed out by Antony.