Constraints on the topology of the Universe: Extension to general geometries

We present an update to the search for a nontrivial topology of the Universe by searching for matching circle pairs in the cosmic microwave background [N. J. Cornish et al. Phys. Rev. Lett. 92, 201 302 (2004)] using the WMAP 7-year data release. We extend the existing bounds to encompass a wider range of possible topologies by searching for matching circle pairs with opening angles $10°\le \alpha \le 90°$ and separation angles $11°\le \theta \le 180°$. The extended search reveals two small anomalous regions in the cosmic microwave background sky. Numerous pairs of well-matched circles are found where both circles pass through one or the other of those regions. As this is not the signature of any known manifold, but is a likely consequence of contamination in those sky regions, we repeat the search excluding circle pairs where both pass through either of the two regions. We then find no statistically significant pairs of matched circles, and so no hints of a nontrivial topology. The absence of matched circles increases the lower limit on the length of the shortest closed null geodesic that self-intersects at our location in the Universe (equivalently the injectivity radius at our location) to 98.5% of the diameter of the last scattering surface or approximately 26 Gpc. It extends the limit to any manifolds in which the intersecting arcs of said geodesic form an angle greater than 10°.

Figure 1

Schematic geometry of the circles-in-the-sky method. If the topology of the Universe is a torus, the observer, located at the center of the 2-sphere of the surface of last scattering, sees matching circles of temperature fluctuations on opposite sides of the CMB sky.

Figure 2

A circle of opening angle $\alpha$ centered at pixel $i$, and a circle centered at pixel $i$ with its phase shifted by $\beta$.

Figure 3

Doppler contribution to the matching circle signal. Small circles (bottom) have opposite Doppler contributions, whereas large circles (top) obtain similar Doppler contributions. Thus large circles show a smaller deviation from $S_{ij}^{\max }=1$.

Figure 4

Searching for orientable topologies: The statistics $S$ (solid blue line) as defined in Eq. (1) as a function of opening angle $\alpha$ and separation angle $\theta$ (indicated by the numbers in the center of each plot). The red (green) line corresponds to the $2\sigma$ ($3\sigma$) C.L. The spikes come from circles osculating at the galactic anticenter and at $l\approx 29°$, $b\approx -19°$ (see Fig. 7). Removing circles that touch these regions removes all features (see Fig. 8).

Figure 5

Searching for orientable topologies: For separation angle $\theta =50°$, a plot of the statistics $S$ as a function of opening angle $\alpha$. Without cutting anything, the dashed black line shows an extended feature at $\alpha =\frac{1}{2}\theta =25°$. Ignoring all circle pairs that overlap within 2.5° of the galactic anticenter (GAC), the feature mostly disappears (solid blue line). The red (green) solid line shows the 95.4% (99.7%) C.L. on $S$, obtained by scrambling the $a_{\ell m}$ (see the main text). See Fig. 7 for an explanation of the systematic spikes in $S$ as well as Fig. 8 for the final result after excising the offending circles.

Figure 6

As soon as the separation angle $\theta$ between the two circles becomes approximately twice their opening angle $\alpha$, $\theta <2\alpha$, the circles start partially overlapping, creating matching patterns in the statistics $S^{\max }$ as for orientable manifolds; the “kissing” segments match.

Figure 7

For the search for orientable manifolds: Location of the circles pairs with maximal statistics $S$ that lies above the $3\sigma$ C.L., colored by opening angle $\alpha$. Note that the osculating circles at the galactic anticenter and at $l\approx 29°$, $b\approx -19°$ do not hint at a nontrivial topology. (a) The highest signal comes from circle pairs osculating at the galactic anticenter and at $l\approx 29°$, $b\approx -19°$. (b) Same as (a) but rotated by $180°$ such that the galactic anticenter is in the middle of the plot. (c) Removing circle pairs that osculate at the galactic anticenter, the highest signal comes from pairs that osculate at $l\approx 29°$, $b\approx -19°$. (d) Removing circles that osculate either at the galactic anticenter or at $l\approx 29°$, $b\approx -19°$, no special position on the sky is apparent.

Figure 8

Searching for orientable topologies, final result: The statistics $S$ (solid blue line) as defined in Eq. (1) as a function of opening angle $\alpha$ and separation angle $\theta$ (indicated by the numbers in the center of each plot). The red (green) line corresponds to the $2\sigma$ ($3\sigma$) C.L. We disregard circles that touch either the galactic anticenter or $l\approx 29°$, $b\approx -19°$. The peaks for separation angles $\theta >170°$ and opening angle $\alpha \approx 90°$ are caused by the fact that circles with $\alpha =90°$ have an angular diameter of 180°. Thus, two such circles, when separated by $\theta \approx 180°$ will start overlapping, until for $\theta =180°$, they coincide, independent of the topology of the Universe. No signs of a nontrivial orientable topology are found.

Figure 9

Searching for nonorientable topologies: The statistics $S$ (solid blue line) as defined in Eq. (1) as a function of opening angle $\alpha$ and separation angle $\theta$ (indicated by the numbers in the center of each plot). The red (green) line corresponds to the $2\sigma$ ($3\sigma$) C.L. The spikes come from circles osculating in the galactic anticenter and at $l\approx 29°$, $b\approx -19°$ (see Fig. 10). Removing circles that touch these regions removes all features (see Fig. 11).

Figure 10

For the search for nonorientable manifolds: Location of the circle pairs with maximal statistics $S$ that lies above the $3\sigma$ C.L., colored by opening angle $\alpha$. Note that the circles that touch either the galactic anticenter or $l\approx 29°$, $b\approx -19°$ do not hint at a nontrivial topology. (a) The highest signal comes from circle pairs osculating at the galactic anticenter and at $l\approx 29°$, $b\approx -19°$. (b) Same as (a) but rotated by 180° such that the galactic anticenter is in the middle of the plot. (c) Removing circle pairs that touch the galactic anticenter, the highest signal comes from pairs that touch $l\approx 29°$, $b\approx -19°$. (d) Removing circles that touch either the galactic anticenter or $l\approx 29°$, $b\approx -19°$, no special position on the sky is apparent.

Figure 11

Searching for nonorientable topologies, final result: The statistics $S$ (solid blue line) as defined in Eq. (1) as a function of opening angle $\alpha$ and separation angle $\theta$ (indicated by the numbers in the center of each plot). The red (green) line corresponds to the $2\sigma$ ($3\sigma$) C.L. We disregard circles that touch either the galactic anticenter or $l\approx 29°$, $b\approx -19°$. No signs of a nontrivial nonorientable topology are found.