Relativistic Effects of our Galaxy's Motion on Circles-in-the-sky

TLDR: In this paper, the geometric effects of our galaxy's peculiar motion on the circles-in-the-sky were studied and an explicit expression for the radius and center position of such an observed circle in the sky, as well as the angular displacement of points on the circle, were derived.

Abstract: We study the geometric effects of our galaxy's peculiar motion on the circles-in-the-sky. We show that the shape of these circles-in-the-sky remains circular, as detected by a local observer with arbitrary peculiar velocity. Explicit expressions for the radius and center position of such an observed circle-in-the-sky, as well as for the angular displacement of points on the circle, are derived. In general, a circle is detected as a circle of different radius, displaced relative to its original position, and centered at a point which does not correspond to its detected center 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 both the large scale curvature of space and the expansion of the universe, since aberration is a purely local phenomenon. For a Lorentz-boosted observer with the speed of our entire galaxy, the maximum (detectable) changes in the angular radius of a circle, its maximum center displacement, as well as the maximum angular distortion are shown all to be of order $\beta=(v/c)$ radians. In particular, two back-to-back matching circles in a finite universe will have an upper bound of $2|\beta|$ in the variation of either their radii, the angular position of their centers, or the angular distribution of points.