Promise of Future Searches for Cosmic Topology

The shortest distance around the Universe through us is unlikely to be much larger than the horizon diameter if microwave background anomalies are due to cosmic topology. We show that observational constraints from the lack of matched temperature circles in the microwave background leave many possibilities for such topologies. We evaluate the detectability of microwave background multipole correlations for sample cases. Searches for topology signatures in observational data over the large space of possible topologies pose a formidable computational challenge.

Figure 1

Portions of rescaled CMB temperature correlation matrices for a half-turn space ($E_2$). $L_{\text{circle}}$ is the length scale below which matched circles would be detected; $L_B=\{0.9,1.1\}{L}_{\text{circle}}$ is the length along the corkscrew axis; $L_A=1.4L_{\mathrm{LSS}}$ is the other topological length scale. The observer is off-axis at $x_0=(0.35,0,0)L_{\mathrm{LSS}}$.

Figure 2

Regions of topology parameter space where observers would or would not see matched circle pairs. For each topology ($E_1–E_5$), for values of $L_A$ and $L_B$ in the regions below and to the left of the associated curve, $\ge 95\%$ of observers have a clone closer than $L_{\mathrm{LSS}}$. (In the region to the right of $L_A/L_{\mathrm{LSS}}=1$ and above $L_B/L_{\mathrm{LSS}}=1$, no topology will produce any matched circles.) For the $E_2$ manifold we choose the right rectangular prism. This figure is identical to Fig. 5 of Ref. [51].

Figure 3

For $E_3$ with $L_A=2L_{\mathrm{LSS}}$, locations $(x/L_{\mathrm{LSS}},y/L_{\mathrm{LSS}})$ of observers in planes of constant $z$ (the corkscrew axis) in which matched circle pairs would be detected, as a function of $L_B$, the length along the corkscrew axis; $L_A$ is the other topological length scale. Even for $L_B=0.4L_{\mathrm{LSS}}$ 20% of observers would still not see matched circles (white regions). This figure is identical to Fig. 3(a) of Ref. [51].

Figure 4

Top: The Kullback-Leibler (KL) divergence as a function of $L_B/L_{\text{circle}}$, the length along the corkscrew axis, for $E_2$ and a simple untilted three-torus ($E_1$). $L_{\text{circle}}$ is the length scale below which matched circles would be detected. Bottom: The KL divergence as a function of ${\ell }_{\max }$ for $E_2$.