A quick review on

Loop Qunatum Cosmology

Yi Ling Center for Gravity and Relativistic AstrophysicsNanchang University, ChinaNov. 5, 2007

KITPC & CCAST Workshop, Beijing

Outlines

• The framework of loop quantum cosmology

1. The classical framework 2. Quantum theory• The resolution of cosmological

singularity

• Effective formalism and inflation

• gr-qc/0702030, Ashtekar

• gr-qc/0304074, Ashtekar, Bojowald, Lewandowski

• gr-qc/0601085, Bojowald

• The WDW theory1. Good semi-classical limit.

2. No improvement on the classical short distance disasters like cosmological singularity.

• The key differences from WDW theory in LQC1. The classical framework is constructed based on the holo

nomy of SU(2) connection .

2. In quantum theory, Bohr compactification of the configuration space is employed in order to construct the representation of the holonomy algebras

3. The differential equation is replaced by the difference equation.

x

( )abq x

exp j aa jh A A ds

P

The WDW theory LQC

The Classical framework

2 2 2 2 2 2 22

1( )

1ds N dt a t dr r d

kr

31det ( )

16EHS dt d x gR gG

2

2 2 2 2 36

a a k a NR

N a N a a a N

23 30 0

2

3

16 8EH

V V aaS dt d xNa R dtN ka

G G N

• A quick view on standard FRW cosmology

30V dx

The Classical framework

EH a gravS dt ap NH

2

2

80

3grav m m

a G kH H H

a a

2

00

2 3

3 8a

grav

pGH V ak

V a G

03, 0

4a N

VL aa Lp p

a G N N

Conjugate momenta

Where

In general constraints become

3 2 30

0

1( )

2mH H a p a V VV

30

Lp a V

:i i i i ba a a a ab iA K K K e

,i aa iA E

1det deta b a a b ab

i j i i iE e e E E q q

The Classical framework

Ashtekar-Sen variables:

SU(2) connection

Barbero-Immirzi parameter

A triplet vector field with density weight one

iaA

, ababq

aiE

2 3 20 0 02 2 16a V a

Identify with symmetry group (2)SU

0 0 0 : Cartan-Killing metric on su(2)i jab a b ij ijq k k

1/30 0l V

3Space time , 1 SM R k

The Classical frameworkIn the present isotropic and homogeneous setting

Fiducial metric:

Physical metric: 02

4ab

ab

qq a

0 0 0 0 0 0Triad ; s.t. = , =a i a j j a i ai a i a i i b be e e

8,

3

Gc p

dimensionless dimension of areac p

0 0

2 20

( )2 2

4

l lc a k a

a lp

1 0 2 0 00 0 i i a a

a a i iA c l E p l q e

The Classical framework

3 2 3 1: i aj bkgrav grav ijk abv v

C d xNH d xN F e E E

( ) 0grav mH H H p

3/ 2 3/ 221( )

2mH H p p p V

The Classical framework

The Hamiltonian constraint in full theory:

In cosmological setting, it reduces to

2 2 23( )

8gravH c pG

Thus, the total Hamiltonian constraint reads as

Where

2= , ( ) :SG Bohr d cRH L

1 21 2 ,

The Quantum theory•The phase space of gravity part ( , )c p

•The Hilbert space

The almost periodic functions

/ 2( ) i cc c e N

constitute an ortho-normal basis in SGH

The Quantum theory

( ) exp( / 2)f c f i c

1( ) ( ) lim ( )

2Bohr

T

R TTf c d c f c dc

T

1 2 1 2 1 2

1( ) ( ) ( ) lim ( ) ( )

2Bohr

T

R TTc c d c c c dc

T

• Almost periodic functions

2 21 1ˆ ˆ :

3 6p p

dp i l p l p

dc

ˆ

2

i c

e

The Quantum theory•The action of the conjugate momentum

3/ 2V̂ p

3/ 2

3ˆ V :6 pl V

Another well-defined operator:

•The eigenbras and eigenvalues of volume operator:

The Quantum theory•The operator is well defined unitary operator, but fails to be continuous with respect to

/ 2 ˆ, ce V

•There is no operator corresponding to c on the Hilbert space

•The well defined fundamental operators

( )cN

Related to the holonomy of connection.

exp cos 2 sin2 2

j ai a i j i

c ch A X IP

The Quantum theory

1 0 2 0 00 0 i i a a

a a i iA c l E p l q e

•The holonomy along the segment of length in the i-th direction0l

1 1

2i ihi

I

The Quantum theoryClassical constraints in full theory :

3 2 3 1: i aj bkgrav grav ijk abv v

C d xNH d xN F e E E

0 0 32 2

0

1( )iji i j

ab i a b

hF O c

l

1 1( ) ( )ij i j i jh h h h h

11 1 002(8 ) ,i aj bk abc k

ijk c k ke E E G l h h V

After regularization

1 1 13 3 1 34(8 ) , 0( )ijkgrav i j i j k k

ijk

C G tr h h h h h h V c

0lij

1i 1j

(a)

(b)

v

The Quantum theoryThe constraint in terms of well-defined fundamental variables:

1 1 13 3 2 1

3 3 2 1

4( ) ,

ˆ ˆ96 ( ) sin cos cos sin2 2 2 2

ijkgrav pl i j i j k k

ijk

pl

C l tr h h h h h h V

c c c ci l V V

3 3 2 1ˆ 3( ) 4 2 4grav plC l V V

The resolution of cosmological singularity

The physical state ( , )

5 3

3 3 23 5

( , 4 ) 2 ( , )

8 ˆ( , 4 ) ( , )3 pl matter

V V V V

GV V l C

†ˆ ˆ 0grav matterC C

2 2 2ˆ ˆmatterC p

Big bang corresponds to the state 0

Given initial states ( , 4 ) and ( , ( 4 4) )N N One may determine all ( , ( 4 4 ) ) for 1N n n

• Cosmological singularity

Closed universe ： k=1,

Scale factor

Originated from a big-bang

3S

( )a t

32

1

( )R

a t

( ) 0a t 3classicallyR

The resolution of cosmological singularity

32

1 1R

a p

3 34: / 10pl G c m

2 2

1 1p

p

l l Rll

3 2 32

11R a R

a Only valid at classical level

The resolution of cosmological singularity

32

1ˆˆ

Ra

Effective formalism and inflation

The effective or “semi-classical” Friedmann equations from LQC receive corrections from the following two facts :

1. The replacement of the inverse of scale factor:

2. The holonomy corrections.

3/ 2 3/ 221( )

2mH H p p p V

63

1

1

1( ) ,

3 I I II

d p tr h h VG

1ˆ ˆ,

ˆx p

p

33/ 2 1/3ˆˆ ˆ,d p c V

Effective formalism and inflation

The operator corresponding to the inverse of scale factor

3/ 2 3/ 221( )

2mH H p p p V

In standard quantum mechianics:

1

2j

6

1 2 ˆ ˆ( ) 8 sin cos cos sin2 2 2 2p

c c c cd p i l V V

Effective formalism and inflation

62

1 1( ) 4 pd p l V V

Ambiguities at semi-classical limit:

1. The representations of SU(2) for holonomy. 2. The operator ordering.

jl

3/ 2 3/ 2 3/(2 2 ) 2,( ) ( ) 3 /( )lj l l pp d p p P q q p jl

3/ 2 3/ 221( )

2H p p p V

2 2 1 113 1 1( ) 1 1 1 ( 1) 1

2 2 1l l l ll

lP q q q q q q Sgn q ql l l

3/ 2 2* *

,

1 for :

( ) 30 for 0

pj l

p p p p j ld p

p

3/ 22,

1( ) ( )

2eff

j lH d p p p V

Effective formalism and inflation

In general case

{ , }c c H

23/ 2 2

,2

8 1( ) ( )

3 2 j l

a k Gp d p p V

a a

2 2 23( ) ( ) 0

8effH c p H p

G

3/ 2 3/ 21 2, ,

8( ) 1 log( ( ) ) ( )

3 4j l j l

a G a dp d p p d p V

a da

Effective formalism and inflation

Effective Friedmann euqations:

Effective formalism and inflation

2

1 2

8 ( ) ( )

3 crit

a G

a

1 2 1 , ( ) 0 ( ) 1

28

13 crit

a G

a

2. The holonomy corrections

2 2 33/(16 ) 0.82crit plG l

Effective formalism and inflationFrom these effective equations, the following relevant phenomena have been investigated:

1. Super-inflation and inflations due to quantum geometry.

2. The big bounce universe.

3. The cosmological perturbation theory and scale invariance .

4. The resolution of the big rip in phantom cosmology.