Non Commutative Cosmic Topology & The Suggestion of a Multif

Spherical Space Forms

Cosmic topologies among the spherical space forms, were simulated by CMB skies and studied for the various candidate topologies.

These initial results suggested the Poincaré homology sphere (or dodecahedral space) account for missing Large angle correlation of 2 point angle correlation fxn of temp spectrum of CMB.

Cosmetology Topology

Einstein equations can only detect geometry but not distinguish between different topologies with the same local geometry.

Missing Large angle correlation of 2 point angle correlation fxn of temp spectrum of CMB.

We see these temperature fluctuations projected in a 2D spherical surface sky, and so it has become common

in the literature to expand the temperature field using spherical harmonics. The spherical harmonics form

a complete orthonormal set on the unit sphere and are defined as

Despite these different methods used to investigate, currently there have been no conclusive results pointing to the presence of a non-simply connected topology.

So maybe we should just give up....

The problem of cosmic topology can be stated as the question of identifying signatures of nontrivial topologies in the spatial sections of space-time.

The anisotropies of the CMB depend upon the geometry of the universe (flat, positively or negatively curved) and this information can be detected through Doppler peaks which depends on the value of the curvature. This made it possible to test the inflationary theory, which predicts a flat or nearly flat geometry. Running through result, shows that among the nearly flat cases, the slightly positively curve case is favored over the negatively curved one.

A first naive idea about detecting nonsimply connected topologies relies on direct observation of periodicities in the catalog of observable astronomical objects.

The list of candidate cosmic topologies with underlying positively curved spherical geometry is given by the spherical space forms, which are quotients of the round 3-sphere by the action of a finite subgroup of isometries:

Collectively we are trying to analyse the dynamical behaviour of

a photon-baryon fluid, and study how temperature fluctuations behave in this system. We took some very constraining assumptions (such as ignoring gravity and the baryons!) and worked on a system whose only force was given by radiation pressure gradients. What we found is that this pressure acts as a restoring force to initial perturbations and we are left with oscillations which propagate at the speed of sound. This behavior continues until we hit the temperature of recombination, at which time matter and radiation de-couple and any temperature fluctuations are essentially frozen into the photons’ temperature, which we measure (nearly unchanged!) today.

Spectral Action

• Statistical search for matching circles in the CMB sky: identify a nontrivial fundamental domain

• Anomalies of the CMB: quadrupole suppression, the small value of the two-point temperature correlation function at angles above 60 degrees, and the anomalous alignment of the quadrupole and octupole

• Residual gravity acceleration: gravitational effects from other fundamental domains

• Bayesian analysis of different models of CMB sky for different candidate topologies.

In the homogeneous and isotropic case, we can consider a model that is compactified and Euclidean. This model would be of the form SB1 X SA3 or S1 x Ya, where

is a spherical space form, with a = radius. Modifing this space by replacing the single sphere with an Apollonian packing P_ of 3-spheres , so that we have a space of the form

Similarly, we can look at fractal arrangements based on some nontrivial cosmic topology Y, instead of a 3-sphere. For example, one of the most promising candidates for a non-trivial cosmic topology is the Poincaré homology sphere, the dodecahedral space which has fundamental domains given by spherical dodecahedra.