Quantum cosmology with nontrivial topologies

T. Vargas Center for Mathematics and Theoretical Physics National Central University

Outline

-A brief introduction to cosmology with nontrivial topologies -Orbifold -Euclidean functional integrals on orbifolds-HH wave functions for RP3 and S3/ -Tunneling wave functions-Discussion

I.- Cosmology with nontrivial topologies

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 , and is endowed with RW metric

)sin)(()( 222222 ddfdtadtds

sinh,sin,)( fwhere

depending on the sign of the constant

spatial curvature (k=0,1,-1), and a(t) is the scale factor. The spatial sections are usually assumed to be simply connected:

3

3

3

H

S

E

-Euclidean space with infinity volume

-spherical space with finite volume

-hyperbolic space with infinity volume

However each of these 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 is a discrete and fixed point-free group of isometries of the covering space . In forming the quotient manifolds the essential point is that they are obtained from by identifying points which are equivalent under the action of the discrete groups

),,(~ 333 HSE

~

~

Or the action of tessellates into identical cells or domains which are copies of what is known as fundamental polyhedron .

~

3E 3S 3H

For a complete review see: Lachieze-Rey & Luminet–Phys. Rep. 254 135 and Levin–Phys. Rep. 365 251.

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 argued 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:

34 SRS

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:

34 / TRE

ˆ34 HRH

II.- Orbifolds Now let us see what the orbifolds are.The notion of orbifold was first introduced by -Satake in 1957 J. Math. Soc. Japan 9 464, who used for it the term V-manifold, and -was rediscovered by Thrurston in 1978, where the term orbifold was coined. -also the orbifold appeared in string theory: Witten et. all in 1985 Roughly speaking, an n-dimensional manifold is a topological space locally modeled on Euclidean space , whereas an n-orbifold generalizes this notion by allowing the space to be modeled on quotient of by finite group action.

nEnE

nE

nE

Let M be a manifold and be a discrete group with an action . We say that acts freely, if for all implies , then the quotient space is another manifold with nontrivial topology. but if some elements on have fixed points, points for which for any with , then the quotient space is an orbifold O.

MM xxMx ,1

xx 1

orbifold instantonInstead of we construct a more general instanton , which we proceed to describe.It is known that the natural inclusion of naturally extends any isometric group action on to an isometric action on , in which the original action is now an action on an equator of . This induced group action fixes the two antipodal points of which lie on the line in perpendicular to this equator.

4S 4S

)1()( nn1nS

nSnS

nS 1nE

Since

}1;0,{

XXnXS n , we take its equator (n-1)-sphere }1;1,{)0( 0

110

ii

inn XXniXXSS and let act on . This action is

10

nS

naturally extended to all parallel (n-1)-spheres , . 0

1X

nS 10 X

If and for any we obtain

Thus action of on is not free and is an orbifold with 2cone points at N and S.

10 X

),(1

),(1

NSn

NSn SS

nS nS1X

10

nS

0

1X

nS

0XN

S

0XnS

1nS

III.- Euclidean functional integral on orbifoldsFollowing Wheelers and Hawking seminal ideas, we formulate a Euclidean approach to quantum gravity on orbifolds: -the transition probability amplitude from an initial 3-manifold to a final 3-manifold is given by

)},(exp{)(),( ,; gOIgDhhKO

iiff

now the sum formally includes any 4-dimensional compact orbifold O with metric g, and the Euclidean orbifold action is given by

-the induced metric and the curvature will have singularities at the fixed points of the orbifolds. -as observed by Schleich & Witt, the curvature singularity at these points is dimension dependent, so in dimension greater than 2 the integral leaves out only a set of zero measure, and the Einstein-Hilbert action is finite.

)(8

1)()2(

16

1),( hdK

GgdR

GgOI

O

it

SOO

gRdgRd )()( nllR ~,~ 2

Following the Hartle-Hawking proposal, in which -the initial boundary is absent -using the WKB semiclassical approximation the wave function of the universe is of the form:

)exp()( nn

nij BANh where N is the normalization constant -denotes the fluctuations about the classical solutions

-are the actions of the Euclidean classical solutionsnA

nB

The manifold RP3 can be constructed from S3 by identifying oposite points, so locally both manifolds have the same metrics

0,0,0

IV.- Hartle-Hawking wave functions for RP3 and 3S

)]sin(sin)[cosh( 22222222 dddHtHdtds

1Ha

).sin()(,)( 122222 HHadadds

This metric is of the cone over RP3 the corresponding range of is

The corresponding wave function for a < H-1

H 0

In general, the volume of the space section is given by:

1)1(6

exp~)( 2/320

22

3 aHGH

RP

Locally, the equation of motion are the same as for Euclidean de Sitter solution:

4S

23 2)(

SV

RP3

38

3 32 a

aaadG

VI

and the final Euclidean action for a < H-1

the wave function of the multiply connected universe is

and its analitical continuation

for a > H-1

2/320

22

)1(13

aHGH

I

2/32

02

20 )1(13

exp~ aHGH

N

43

)1(cos

3exp~

2/3

2

20

2

2

GH

aH

GH

V.- Tunneling wave functions

34

8

1)2(

16

1),( dxhK

GdxgR

GgMS

M

-Vilenkin considered the quantum creation of a closed universe in analogy with a tunneling effect in quantum mechanics. The Einstein-Hilbert action with the Lorentzian signature is

31

4

3)(

22

2a

aG

VaU

38

3 32 a

aaadtG

VS

0)(3

14

3 22

2

2

2

aa

aG

V

a

for a closed universe the action reduces to

The momentum and Hamiltonian

aaG

Vpa

43

31

4

3 22

22 a

aG

VpH a

The wave function satisfies the Wheeler-De Witt equation

In the classical allowed region the WKB solutions is

1 Ha

4

~)~(exp)()(1

2/1 iadapiapa

a

H

adapapa

H

a

~)~(exp)()(~

1

2/1

with 2/1)()( aUap

the under-barrier solutions are 1 Ha

22

1 exp8

3exp)(

~~)(

HGGaHa

vT

23 2)(

SVusing

The tunneling wave function is given by

~

~~

Diff

RiemS

Diff

RiemSThe fundamental polyhedron of superspace

V.- Discussions

- inclusion of nontrivial topologies into quantum cosmology need orbifolds-allowing orbifolds as extrema in semiclassical approximation to Euclidean functional integral does not change radically their properties, only enlarges the number of semiclassical amplitudes -these orbifolds are Euclidean and the singular points are not located in Lorentzian space-time- in non-boundary condition approach the probability of creation is maximum for minimum order of while in the tunneling approach it increases with -it is necessary to extend nontrivial topologies and fundamental polyhedrons into the superspace