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 nonlinear 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. astroph/0601594). So is this equation correct, and if so what does it mean?
[astroph/0604416] CMB Anisotropies at Second Order I
Authors:  N. Bartolo (ICTP, Trieste), S. Matarrese (Univ. of Padova), A. Riotto (CERN) 
Abstract:  We present the computation of the full system of Boltzmann equations at secondorder describing the evolution of the photon, baryon and cold dark matter fluids. These equations allow to follow the time evolution of the Cosmic Microwave Background (CMB) anisotropies at secondorder at all angular scales from the early epoch, when the cosmological perturbations were generated, to the present through the recombination era. This paper sets the stage for the computation of the full secondorder radiation transfer function at all scales and for a a generic set of initial conditions specifying the level of primordial nonGaussianity. In a companion paper, we will present the computation of the threepoint correlation function at recombination which is so relevant for the issue of nonGaussianity in the CMB anisotropies. 
[PDF] [PS] [BibTex] [Bookmark] 

 Posts: 1696
 Joined: September 23 2004
 Affiliation: University of Sussex
 Contact:

 Posts: 2
 Joined: September 08 2005
 Affiliation: University of Bielefeld
[astroph/0604416] CMB Anisotropies at Second Order I
I am very surprised that there is no reference to the works of Kenji Tomita.
Especially in astroph/0501663, Tomita derived the second order version of the SachsWolfe 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 secondorder calculation is limited to intermediate angular scales:
at the largest angular scales, we have shown that local nonlinear effects like the one discussed recently in astroph/0601445 (we looked at the local ReesSciama 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 secondorder calculation as has been pointed out by Antony.
Especially in astroph/0501663, Tomita derived the second order version of the SachsWolfe 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 secondorder calculation is limited to intermediate angular scales:
at the largest angular scales, we have shown that local nonlinear effects like the one discussed recently in astroph/0601445 (we looked at the local ReesSciama 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 secondorder calculation as has been pointed out by Antony.