Lorentz-boosted circles-in-the-sky and cosmic topology

A topologically finite universe, smaller than the observable horizon, will have circles-in-the-sky: pairs of circles around which the temperature fluctuations in the cosmic microwave background are correlated. The circles occur along the intersection of copies of the spherical surface of last scattering. For any observer moving with respect to the microwave background, the circles will be deformed into ovals. The ovals will also be displaced relative to the direction they appear in a comoving frame. The displacement is the larger of the two effects, being proportional to the velocity. For the Earth’s motion, the effect is on the order of 0.14° at the very worst. This can affect all pattern-based searches for the topology of the universe. In particular, although the deviation is too small to impact the search for circles in the Wilkinson microwave anisotropy probe (WMAP) data, higher-resolution searches for circle pairs will need to compensate for this effect.

Figure 1

The surface of last scattering intersects itself in a finite universe smaller than the diameter of the surface. The self-intersection can most easily be visualized in a tiling picture, such as the above where a two-dimensional slice through space is shown. The intersection of the two spheres occurs along the dotted-line circle. When an observer at rest with respect to the expansion looks to the right, he sees temperature fluctuations along the dotted line. When he looks to the left he sees identical temperature fluctuations along the dotted line. Therefore he measures a correlated pair of circles, one to the right, the other to the left.

Figure 2

Another view of the correlated circle pair of Fig. 1. C sees one circle in front of him and one behind him. In general, circle pairs may be out of phase and not face-to-face.

Figure 3

Two points on opposite sides of a circle-in-the-sky from Fig. 2. The $z$-direction is suppressed.

Figure 4

O measures a Lorentz contraction of the distance between $P_1$ and $P_2$.

Figure 5

The points $P_1$ and $P_2$ subtend angles ${\theta }_1^{\prime }$ and ${\theta }_2^{\prime }$ respectively.

Figure 6

Observer C measures the dotted-line circle centered on $x=0$ while observer O measures the solid-line ellipse displaced by $\gamma \beta r$. The circle is displaced as well as distorted in shape. In the figure $r$ is taken to be $2L$ for illustration and $\beta =3/4$.

Figure 7

In Fig. 2 observer C sees one circle in front of him and one behind him. Unlike C, the Lorentz boosted observer O does not see one circle in front of him and one behind him. He sees the center of both ovals displaced by $\delta \sim \beta$ as drawn. Notice the oval pairs are opposite each other and face on but they are not on either side of the observer O.