How would we know if we are living in a compact hyperbolic universe? Local
observations can only measure the geometry of the universe. We need to look
out beyond our own fundamental domain to begin to see the effects of
topology. Most cosmologists trying to look for observational signature of
topology have attempted to find replicas of our own Galaxy or other familar
objects such as rich clusters (Gott 1980; Lehoucq,
Lachieze-Rey & Luminet 1996;
Roukema & Edge 1997). In a typical small volume
compact hyperbolic universe, the expectation value for the length of the
shortest closed geodesic in generic small volume manifolds (volumes less
than 10 say) is between 0.5 and 1 (Thurston, private communication). That
is, if we randomly select a point in such a manifold, this is the most
probable value for the conformal distance to our nearest copy.
In a matter dominated universe, the redshift of an object is related to the
conformal distance by
where is the conformal lookback time, and is the present conformal time. For an universe, and the nearest image of ourselves is likely to lie between a redshift of 0.9 and 2.9. Similarly, if , we find and the nearest copy will typically lie between a redshift of 1.0 and 3.8. These numbers indicate that it will be very difficult to constrain the topology of a hyperbolic universe using direct searches for ghost images of any astrophysical objects. This difficulty is compounded by the evolution of astrophysical objects on much shorter timescales.
Fortunately, observations of the microwave background are potentially a very powerful tool for probing the topology of the universe. MAP and Planck will measure millions of independent points on the surface of last scatter. If we live in a universe where the distance to our nearest copy is less than the diameter of the last scattering surface, then we will see the same position on the surface of last scatter at multiple points on the microwave sky (Cornish, Spergel & Starkman 1996a,b). Since the surface of last scatter is a sphere, it will intersect itself along circles. This will lead to pairs of matched circles across the sky. Note that the temperature is not constant along these circles, but rather there are pairs of points along each circle with identical temperatures.
There are several effects that make this signature very difficult to detect in the COBE data. If the universe is negatively curved, then most of the large-scale fluctuations are not due to physics at the surface of last scatter, but rather due to the decay of potential fluctuations along the line of sight (Kamionkowski & Spergel 1994). These fluctuations are mostly generated at z<2, generally within our fundamental domain. Thus, large angular scale measurements are not sensitive to topology. Even without this effect, we will likely need higher resolution maps with higher signal-to-noise to definitively detect the matched circles. The combination of high signal-to-noise and large numbers of independent pixels along each circle significantly reduces the chances of false detections.
Can we detect topology with the MAP data? The number of matched circle pairs
is equal to the number of copies of the fundamental cell that can fit inside
a ball of proper radius (we are approximating the radius of
the LSS by the radius of the particle horizon). A fair estimate of the
number of cells, needed to tile this hyperbolic ball is
given by the volume ratio,
where Vol is the volume of the fundamental domain in curvature units. In a universe with this is a whopping ratio of Vol. Considering that there are an infinite number of hyperbolic three manifolds with volumes less that 3, we have no shortage of model universes that can be seen by this method. Even if was as high as 0.95, we would still have an infinite number of manifolds to chose from that would produce at least some matched circle pairs.
How big are these matched circles? Lets consider an infinite hyperbolic
universe tiled with copies of our fundamental domain. We can draw the last
scattering sphere centered not only around our own location, but also around
all of our images. The circles arise at the intersections of these spheres.
The angular radius, of the circles is set by the radius of the
last scattering sphere (, and the conformal distance, to the appropriate image of the MAP satellite:
If there will be over 4000/Vol( circles of radius between 10 and 15 degrees. At its highest frequency, MAP will have a resolution of 0.21, thus it will measure over 300 independent points along each of these circles. Even at a resolution of 0.5 where we can use the three highest frequency maps, there are still over 125 independent points along each circle. We are currently simulating the analysis of MAP data using synthetically generating skies: our current results are promising, it appears that MAP has sufficient signal-to-noise and resolution to detect topology in a universe even if the volume of the fundamental domain of several tens of curvature volumes.
Once we have detected the circles, J. Weeks showed us that we can use them to determine the topology of the universe. Each pair of circles lie on matched faces of the fundamental cell, thus, each pair of circles gives us an element of the fundamental group . From the list of elements, we can construct the generators of the group. Most small volume universe have only two or three generators. Since we expect to have many hundreds if not thousands of circle pairs, constructing the generators from the elements is a highly overdetermined problem. Thus, producing a consistent solution is an extremely powerful check that will help demonstrate that the circles are not random events or artifacts of some pernicious instrumental effect. The Mostow-Prasad rigidity theorem implies that a given topology has a fixed volume measured in curvature units, thus, once we know the topology and the angular size of the circles, we have an independent topological determination of the radius of last scattering surface in units of the curvature radius. This yields an independent measurement of that does not rely on any assumptions about the nature of the primordial fluctuations.
After determining the topology of the universe, we can then proceed onwards and reconstruct the temperature of the photons and the velocity of the baryon photon fluid throughout the entire fundamental domain from the microwave sky. The surface area of the microwave sky is thus, in an universe, we see Vol slices through the fundamental domain. Hence, the characteristic distance between slices is only a few co-moving Megaparsecs. We should be able to reconstruct not only the statistical properties of the initial fluctuations, but the actual amplitude and phases of the initial fluctuations on the scale of galaxy clustering. These initial fluctuations can, of course, be integrated forward and compared to the local observed universe. At the end of the analysis, we, in principle, should be able to identify the cold spot that eventually collapsed to form the Virgo supercluster and other familiar parts of our local universe. Cosmologists are used to thinking of looking out at the universe and measuring the prehistory of other regions of the universe. If we are fortunate enough to live in a compact hyperbolic universe, we can look out and see our own beginnings.