## Relativistic effects of our galaxy's motion on circles-in-the-sky in CMB maps

##### Abstract

For an observer in the Hubble flow, the last scattering surface (LSS) is well approximated by a 2-sphere. If a non-trivial topology of space is detectable, then this sphere intersects some of its topological images, eventually giving rise, in cosmic microwave background (CMB) radiation maps (which are projections of the LSS onto the observer's sky sphere), to circles-in-the-sky, i.e., pairs of matching circles of equal radii, centred at different points on the sky. We examine the geometric effects due to our galaxy's peculiar motion on circles-in-the-sky in CMB maps, and show that their shape remains circular, as detected by a local observer with arbitrary peculiar velocity. In general, a circle is detected as a circle of different radius, displaced relative to its original position, and centred at a point which does not correspond to its detected centre in the comoving frame. Further, there is an angular displacement of points on the circles. These effects all arise from aberration of cosmic microwave background radiation, exhausting the purely geometric effects due to the peculiar motion of our galaxy, and are independent of the large scale curvature of space, the expansion of the universe and the acceleration of the observer, since aberration is a purely local phenomenon. For a Lorentz-boosted observer with the speed of our entire galaxy, the maximum changes in the angular radius of a circle, its maximum centre displacement as well as the maximum angular distortion are all shown to be of order β = (v/c) ≃ 1.23 × 10-3 radians. In particular, two back-to-back matching circles in a finite universe will have an upper bound of 2|β| in the variation of either their radii, the angular position of their centres, or the angular distribution of points. Although below current WMAP's resolution, these results are within Planck's and other more accurate missions' resolution. © 2005 IOP Publishing Ltd.