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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/G
-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=RxS , and is endowed with RW
metric
ds  dt
2
where
2
 a(t )d
2
 f (  )(d  sin d )
f ( )   , sin  , sinh  
2
2
2

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:
E 3 -Euclidean space with infinity volume
S  S 3 -spherical space with finite volume
H -hyperbolic space with infinity volume
3
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
~
S  S G  ( E 3 G, S 3 G, H 3 G)
where G is a discrete and fixed point-free group of isometries of
~
the covering space S .
In forming the quotient manifolds S the essential point is that they
~
are obtained from S by identifying points which are equivalent
under the action of the discrete groups G.
~
Or the action of G tessellates S into
identical cells or domains which are
copies of what is known as
fundamental polyhedron .
SE G
3
SS G
3
SH G
3
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:
S  RS
The quantum creation of the flat universe with
4
3-torus space topology has been done by
E
Zeldovich & Starobinsky and others:
/ G  R T
While the quantum creation of the compact
hyperbolic universe was recently studied by H 4
Gibbons, Ratcliffe & Tschantz and others:
G  R  H Gˆ
4
3
3
3
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 E n, whereas an
n-orbifold generalizes this notion by allowing the space to be
modeled on quotient of E n G by finite group action.
En
En G
Let M be a manifold and G be a discrete group with an action
G  M  M . We say that G acts freely, if for all x  M , x  x implies
  1 , then the quotient space is another manifold with nontrivial
topology.
but if some elements on G have fixed points, points for which
x  x for any   G with   1 , then the quotient space is an
orbifold O.
orbifold instanton
4
Instead of S we construct a more general instanton S 4 G , which
we proceed to describe.
It is known that the natural inclusion of (n)  (n  1) naturally
extends any isometric group action on S n 1 to an isometric action
on S n , in which the original action is now an action on an equator
of S n . This induced group action fixes the two antipodal points of
S n which lie on the line in E n 1 perpendicular to this equator.
n

Since S  { X  ,   0  n; X X   1}, we take its equator (n-1)-sphere
n1
n 1
n 1
i
S
S0  S ( X 0  0)  {X i , i  1  n; X X i  1} and let G act on 0 . This action is
naturally extended to all parallel (n-1)-spheres S n 1 X 0 , X 0  1 .
0
X
X

1
If 0
and for any  G
N
Sn G
wen1obtain n1
X
0
S
(S ,N )
S
S n 1 X 0
(S ,N )
Thus action of G on
n
S n is not free and S G
1
X
is an orbifold with 2
cone points at N and S.
S0
S
n 1
S n 1 G
III.- Euclidean functional integral on orbifolds
Following Wheelers and Hawking seminal ideas, we formulate a
Euclidean approach to quantum gravity on orbifolds: t  i
-the transition probability amplitude from an initial 3-manifold to a
final 3-manifold is given by
K (S f , h f ;S i , hi )    D( g  ) exp{  I (O, g )}
O
now the sum formally includes any 4-dimensional compact
orbifold O with metric g, and the Euclidean orbifold action is given
by I (O, g )   1 ( R  2)d ( g )  1 Kd (h)
16G O
8G S
-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.
Rd ( g )  Rd ( g )
R ~ l 2 ,  ~ l n

O

OS
IV.- Hartle-Hawking wave functions for RP3 and S 3 G
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:
 (hij )  N  An exp(  Bn )
n
where
N is the normalization constant
An -denotes the fluctuations about the classical solutions
Bn -are the actions of the Euclidean classical solutions
The manifold RP3 can be constructed from S3 by
identifying oposite points, so locally both manifolds a  H 1
have the same metrics
ds 2  dt 2  H 2 cosh( Ht )[d 2  sin 2  (d 2  sin 2 d 2 )]
0     ,0     ,0    
Locally, the equation of motion are the same as for S 4 Euclidean
de Sitter solution:
ds 2  d 2  a 2 ( )d2 , a 2 ( )  H 1 sin( H ).
This metric is of the cone over RP3 the corresponding
range of  is 0     H
 2
3V
a 3 
I
d  a a  a 


8G
3


The corresponding wave function for a < H-1
 

2 3/ 2
3
2

 ( RP ) ~ exp 
(1  H a0 )
2
 6H G

1 

In general, the volume of the space section is given by:
2
2

V ( S 3 G) 
G
RP3
and the final Euclidean action for a < H-1
I 

3 G GH
1  (1  H a
2
2
0
2 3/ 2
)

the wave function of the
multiply connected universe is

 
2 3/ 2
2
 ~ N 0 exp 
1

(
1

H
a
0 )
2
 3 G GH





and its analitical continuation
for a > H-1
3/ 2
2
2



 ( H a0  1)


 ~ exp 
cos 
 
2 
2
3
G
GH
3
G
GH
4




V.- Tunneling wave functions
-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
1
1
4
3
S (M , g ) 
(
R

2

)

g
dx

K
h
dx
16G M
8G S
for a closed universe the action reduces to
 2
3V
a 3 
S
dt  a a  a 


8G
3


The momentum and Hamiltonian
3V
pa  
aa
4G
a 2 
 3V  2 


 a 1 
3 
 4G 

2
H   pa
2
The wave function satisfies the Wheeler-De Witt equation
  2  3V  2 2  a 2 
  (a)  0
 a 1 
 2 
3 
 a  4G  
2
 3V  2  a 
U (a)  
 a 1 

 4G   3 
2
In the classical allowed region a  H
solutions is
1
the WKB
a


1 / 2
~
~
 (a )   p (a ) exp  i  p (a )da  i 
4 
 H 1
the under-barrier solutions are
H


~
1/ 2
~
~
 (a)   p(a) exp   p(a ) da 
 a

1
aH
1
with
p ( a )   U ( a )
1/ 2
2 2
V ( S G) 
G
using
3
The tunneling wave function is given by




3

~
  exp  

T (a  H ) ~  (a)  exp   2
2

 GGH 
8
G
G

v




1
The fundamental polyhedron of superspace
~
~ Riem S
SS 
~
Diff S



Riem S 
S S  
Diff S 
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 G, while in the
tunneling approach it increases with |G|
-it is necessary to extend nontrivial topologies and fundamental
polyhedrons into the superspace