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Transcript

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 . SE G 3 SS G 3 SH 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 RS 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 n1 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 wen1obtain n1 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) 16G O 8G 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 OS 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 ( )d2 , 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 8G 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 16G M 8G S for a closed universe the action reduces to 2 3V a 3 S dt a a a 8G 3 The momentum and Hamiltonian 3V pa aa 4G a 2 3V 2 a 1 3 4G 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 4G 2 3V 2 a U (a) a 1 4G 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 aH 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