Circles-in-the-sky searches and observable cosmic topology in the inflationary limit

While the topology of the Universe is at present not specified by any known fundamental theory, it may in principle be determined through observations. In particular, a nontrivial topology will generate pairs of matching circles of temperature fluctuations in maps of the cosmic microwave background, the so-called circles-in-the-sky. A general search for such pairs of circles would be extremely costly and would therefore need to be confined to restricted parameter ranges. To draw quantitative conclusions from the negative results of such partial searches for the existence of circles we need a concrete theoretical framework. Here we provide such a framework by obtaining constraints on the angular parameters of these circles as a function of cosmological density parameters and the observer’s position. As an example of the application of our results, we consider the recent search restricted to pairs of nearly back-to-back circles with negative results. We show that assuming the Universe to be very nearly flat, with its total matter-energy density satisfying the bounds $0<|{\Omega }_0-1|\lesssim {10}^{-5}$, compatible with the predictions of typical inflationary models, this search, if confirmed, could in principle be sufficient to exclude a detectable nontrivial cosmic topology for most observers. We further relate explicitly the fraction of observers for which this result holds to the cosmological density parameters.

Figure 1

Depiction of a screw motion isometry $\gamma$, whose action takes the point $P$ to $P^{\prime }$. This amounts to a translation by $L$ along the axis of isometry (dotted line), and a rotation by $\alpha$ around the same axis. The LSS sphere (thick solid line) centered at $P$ intersects its images (also in thick solid lines) centered at $\gamma P$ and ${\gamma }^{-1}P$ along two matched circles-in-the-sky.

Figure 2

This figure depicts the circles at the intersection of the LSS with its images. The parameter $\theta$ is a measure of the deviation of the circles from being antipodal, and $\varphi$ measures the phase difference between the circles of radius $\nu$.

Figure 3

Upper bound ${\theta }_{\max }$ for the maximum deviation from antipodicity as a function of the fraction of potential observers for which $\theta \le {\theta }_{\max }$ in the inflationary limit. This figure shows, for example, that for 98.7% of all observers in spherical manifolds, $\theta \le 10°$. It also shows that for all observers in hyperbolic manifolds, $\theta \le 0.8°$. In this figure, we have taken ${\Omega }_{m0}=0.28$ and circles with radii $\nu \ge 18°$, but for circles with arbitrarily small radii, the value of ${\theta }_{\max }$ would increase by a factor of at most 1.05.

Figure 4

Upper bound ${\theta }_{\max }$ for the maximum deviation from antipodicity as a function of the total density parameter ${\Omega }_0$, calculated for the sets of 90%, 99% and 99.9% of observers (respectively, $\mathcal{R}=0.1$, 0.01 and 0.001), and for the totality of observers in hyperbolic manifolds. The shaded region corresponds to the inflationary limit as defined in the text. Again, we have taken ${\Omega }_{m0}=0.28$ and circles with radii $\nu \ge 18°$.