Abstract
The aberration and Doppler coupling effects of the Cosmic Microwave Background (CMB) were recently measured by the Planck satellite. The most straightforward interpretation leads to a direct detection of our peculiar velocity β, consistent with the measurement of the well-known dipole. In this paper we discuss the assumptions behind such interpretation. We show that Doppler-like couplings appear from two effects: our peculiar velocity and a second order large-scale effect due to the dipolar part of the gravitational potential. We find that the two effects are exactly degenerate but only if we assume second-order initial conditions from single-field Inflation. Thus, detecting a discrepancy in the value of β from the dipole and the Doppler couplings implies the presence of a primordial non-Gaussianity. We also show that aberration-like signals likewise arise from two independent effects: our peculiar velocity and lensing due to a first order large-scale dipolar gravitational potential, independently on Gaussianity of the initial conditions. In general such effects are not degenerate and so a discrepancy between the measured β from the dipole and aberration could be accounted for by a dipolar gravitational potential. Only through a fine-tuning of the radial profile of the potential it is possible to have a complete degeneracy with a boost effect. Finally we discuss that we also expect other signatures due to integrated second order terms, which may be further used to disentangle this scenario from a simple boost.
Highlights
We summarize this section by saying that in order to reproduce Planck [16] measurements on aberration couplings, a dipolar potential has to be negligible with respect to our peculiar velocity or it has to satisfy to the integral condition eq (3.27)
The Planck satellite has detected couplings between multipoles in the CMB at all scales which are consistent with Doppler and aberration effects due to our peculiar velocity [16]
In this paper we have tried to check whether a large scale dipolar gravitational potential could produce or not the same observational signatures
Summary
In what follows we will make use of the following notation: vectors and tensors will be written in boldface; their components will not be in boldface, and will carry Latin letters super-scripts which run from 0 (denoting the time component) to 3; second order perturbation quantities will always carry a subscript “2”. (2.5) and (2.6) were obtained in appendix A by rewriting previous results of [19] including scalar, vector and tensor contributions They are fully general and valid in all gauges. The second order anisotropies due to δr in eq (2.15) can be interpreted in the Poisson gauge as time-delay. Note that it is α that is relevant in the Poisson gauge for the discussion of lensing effects on CMB, because one is interested in the total deflection (the angular excursion) of a photon as it travels from the LSS to our observation point, and not in the change in its direction (see [21], pp.). In what follows we will restrict ourselves to the case of pure scalar perturbations in the Poisson gauge
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