Bianchi type-III quantum cosmology T Vargas National Central University. I.-Introduction In the standard cosmological model, the universe is described by a FLRW solution, where the universe is modeled by a 4-manifold M which is decomposed into M=Rx S , and is endowed with RW metric.
In the standard cosmological model, the universe is described by a FLRW solution, where the universe is modeled by a 4-manifold M which is decomposed into M=RxS , and is endowed with RW metric
depending on the sign of the constant
spatial curvature (k=0,1,-1), and a(t) is the scale factor.
The spatial sections S are usually assumed to be simply connected:
-Euclidean space with infinity volume
-spherical space with finite volume
-hyperbolic space with infinity volume
However each of this geometries can support many nontrivial topologies with finite volumes without altering the dynamics or the curvature.
These non-simply connected topologies may equally be any one of the possible quotient manifolds , where G is a discrete and fixed point-free group of isometries of the covering space .
In forming the quotient manifolds Sthe essential point is that they are obtained from by identifying points which are equivalent under the action of the discrete groups G.
The action of G tessellates into identical cells or domains which are copies of what is known as fundamental polyhedron
The aim of quantum cosmology is to study the universe in the Plank era, in which the main process would be the formation of space-time itself.
In fact, it was arguing by Fang & Mo and Gibbons & Hartle that the global topology of the present universe would be a relic of its quantum era, since the global topology would not have changed under evolution after the Plank era.
In the pioneering works on quantum creation
of closed universe, in both the tunneling from
nothing and non-boundary proposals:
The quantum creation of the flat universe with
3-torus space topology has been done by
Zeldovich & Starobinsky and others:
While the quantum creation of the compact
hyperbolic universe was recently studied by
Gibbons, Ratcliffe & Tschantz and others:
II.- Bianchi type-III
Here we consider the quantum creation of the anisotropic Bianchi type-III universe:
Two Remarks: - this model is not relevant for our own universe, however it is interesting because it gives us the intuitive idea about topology. -the quantum cosmology of this model was carefully analyzed by Louko and Halliwell & Louko, our aim here is only to note the relation of this model to topology.
where, is the closed disk in , and the line element of this Euclidean Bianchi model is given by
the coordinate r is to be periodic with period 2p, and the metric of the compact hyperbolic surface of genus g is
Using the WKB semiclassical approximation and restricted to minisuperspace, the wave function of the universe is of the form:
N is the normalization constant
-denotes the fluctuations about the classical solutions
-are the actions of the Euclidean classical solutions
The volume of space section is
and the Euclidean Einstein-Hilbert action is
The field equations are obtained by varying the action with respect to N, b and a in the gauge
Integrating we obtain
Putting C equal to zero, we obtain the solution to the field equation
-Classical solution must be everywhere regular
-4-metric has vanishing 3-volume at the bottom
The Euclidean action reduces to
and integrating we obtain
The semiclassical approximation to the HH wave function for
whereas for , the wave function can be obtained by analytical continuation
-the wave functions are proportional to the genus of the 2-dimens
compact hyperbolic manifold and
-in HH approach the probability of creation increases with the genus g