The boost operator for frequency-dependent sky signals equals the aberration kernel evaluated with the Doppler weight replaced by -nu d/dnu, allowing direct computation to second order in v/c.
The CMB Dipole: Eppur Si Muove
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abstract
The largest temperature anisotropy in the cosmic microwave background (CMB) is the dipole. The simplest interpretation of the dipole is that it is due to our motion with respect to the rest frame of the CMB. As well as creating the $\ell$=1 mode of the CMB sky, this motion affects all astrophysical observations by modulating and aberrating sources across the sky. It can be seen in galaxy clustering, and in principle its time derivative through a dipole-shaped acceleration pattern in quasar positions. Additionally, the dipole modulates the CMB temperature anisotropies with the same frequency dependence as the thermal Sunyaev-Zeldovich (tSZ) effect and so these modulated CMB anisotropies can be extracted from the tSZ maps produced by Planck. Unfortunately, this measurement cannot determine if the dipole is due to our motion, but it does provide an independent measure of the dipole and a validation of the y maps. This measurement, and a description of the first-order terms of the CMB dipole, are outlined here.
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The Boost Operator: Properties, Computation and Applications
The boost operator for frequency-dependent sky signals equals the aberration kernel evaluated with the Doppler weight replaced by -nu d/dnu, allowing direct computation to second order in v/c.