The Closed k = +1 FRW Model in Loop Quantum Cosmology

**The** **Closed** k = **+1** **FRW** **Model** **in** **Loop** **Quantum** **Cosmology**

Kev**in** Vandersloot

Institute for **Cosmology** and Gravitation

University of Portsmouth

IGC Inaugural Conferance

August 9, 2007

**Loop** quantum cosmology of k=1 **FRW** models

A. Ashtekar, T. Pawlowski, P. S**in**gh, K. Vandersloot

Phys.Rev. D75 (2007) 024035; gr-qc/0612104

Introduction

Prize of quantum gravity theory: Cure short distance problems near s**in**gularities while transition**in**g to

classical GR at large scales

Such results successfully achieved **in** loop quantum cosmology for k = 0 **FRW** model [Ashtekar, Pawlowski,

S**in**gh, 2006]:

Full control of physical sector of quantum theory (observables, **in**ner product, semi-classical states)

Big bang replaced with big bounce for massless scalar field model

Modifications to Friedmann equation

H 2 = 8πG

3 ρ

1 − ρ

ρc

Sub Planckian energies (ρ ≪ ρc) classical dynamics, high energies (ρ ≈ ρc) triggers bounce at

ρc ≈ ρPL

Current observations **in**dicate universe nearly flat but not necessarily k = 0, do LQC results hold for

k = ±1 models?

Today: focus on closed k = **+1** model [Ashtekar, Pawlowski, S**in**gh, KV PRD 2007]

[See KV PRD 2007; Szulc 2007 for progress **in** k = −1 model ]

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

**Closed** model k = **+1** LQC

Initial loop quantization of k = **+1** model [Bojowald, KV PRD 2003]

Simplified quantization with less connection to full theory of LQG

Based on holonomies of extr**in**sic curvature **in**stead of connection

Non self-adjo**in**t Hamiltonian constra**in**t

No physical **in**ner product, Dirac observables - physical sector not understood

Not based on improved (“µ”) Hamiltonian operator from [Ashtekar, Pawlowski, S**in**gh, 2006] - bad

semi-classical limit expected

Green and Unruh analysis - bad semi-classical limit, no classical recollapse for massless scalar field

But is based on above quantization

Can systematic loop quantization be performed? Do Green, Unruh problems persist? Correct

semi-classical limit? Big bounce?

Answer: Yes!

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

Classical Framework

k = **+1** **FRW** model, constant positive spatial curvature

ds 2 = −dt 2 + a(t) 2 dχ 2 + s**in** 2 χdΩ 2

Consider massless scalar field φ(t): ρφ = P2

φ

2a 6, dynamics from Friedmann, Kle**in**-Gordon equations

˙a

a

2

= 8πG

3 ρφ − 1

a 2

¨φ = −3 ˙a

a ˙ φ

All solutions s**in**gular, start**in**g with big bang, recollaps**in**g, and collaps**in**g **in**to big crunch

Note: φ(t) monotonic - will use as “**in**ternal clock” **in** quantum

dynamics

Classical big bang at φ = −∞, big crunch at φ = +∞

φ

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

0 1*10 4 2*10 4 3*10 4 4*10 4 5*10 4 6*10 4 7*10 4 8*10 4 9*10 4 1.0*10 5

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

v

WDW

classical

**Loop** Classical Framework

**Loop** quantization based on Hamiltonian with connection-triad variables

Phase space: triad p, connection c (triad “momentum”), scalar field φ, scalar momentum Pφ

p(t) ∝ a(t) 2

Hamiltonian H(p,c,φ,Pφ) quadratic **in** momenta Pφ and c

Hamilton’s equations give back Friedmann and Kle**in**-Gordon equation

Note: convenient **in** quantum dynamics to use dimensionless v ∝ p 3/2 /l 3 P

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

**Loop** Quantization

**Quantum** E**in**ste**in** equation: HΨ(v,φ) = 0 leads to:

∂ 2 Ψ(v,φ)

∂φ 2

= − ΘΨ(v,φ)

Similar to static Kle**in**-Gordon equation with φ as “time”

Θ is discrete difference operator **in** LQC (differential operator **in** Wheeler-DeWitt quantization)

Θ Ψ(v,φ) = C + (v)Ψ(v + 4,φ) + C 0 (v)Ψ(v,φ) + ...

Inner product evaluated at some “time” φ0:

〈Ψ1|Ψ2〉 = dvB(v) Ψ1(v,φ0)Ψ2(v,φ0)

Dirac observable of **in**terest - value of v at some “time” φ0: v|φ0

Technical notes:

Used the improved µ quantization - leads to difference operator Θ **in** discrete steps of v

Also Θ constructed us**in**g holonomies of c - Hamiltonian constra**in**t from closed holonomy loop

us**in**g SU(2) right and left **in**variant vector fields [**in**dependently developed by Szulc, Kam**in**ski,

Lewandowski 2006]

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

**Loop** **Quantum** Dynamics

**Quantum** dynamics program:

|Ψ(v,φ)|

30

20

10

0

0

Consider semi-classical state approximat**in**g large universe, i.e. Ψ(v,φ0) peaked around a large

value of v at **in**itial time φ0

Evolve backwards/forwards **in** time us**in**g quantum E**in**ste**in** equation

Evaluate expectation value of v|φ, compare with classical expectation

Wheeler-DeWitt results:

Wave-packets follows classical trajectory - big bang, recollapse, big crunch

S**in**gularity not resolved

2*10 4

4*10 4

v

6*10 4

8*10 4

1.0*10 5

1.2*10 5

1

1.5

0

0.5

-1

-0.5

φ

30

25

20

15

10

5

0

φ

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

0 1*10 4 2*10 4 3*10 4 4*10 4 5*10 4 6*10 4 7*10 4 8*10 4 9*10 4 1.0*10 5

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

v

WDW

classical

**Loop** **Quantum** Dynamics

LQC quantum dynamics:

0.8

0.6

0.4

0.2

0

2*10 4

v

Wave-packet no longer follows classical trajectory near s**in**gularity

Universe bounces at f**in**ite value of v

Classical recollapse still present - cyclic behavior

Bounce occurs when matter density ρφ ≈ ρPL, agrees with k = 0 model

|Ψ(v,φ)|

4*10 4

6*10 4

8*10 4

1.0*10 5

1.2*10

-4

5

-3.5

-3

-2.5

-1.5

-2

φ

-1

-0.5

0

v

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

φ

0

-1

-2

-3

-4

-5

0 2*10 4

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

4*10 4

v

6*10 4

LQC

Effective

Classical

8*10 4

1.0*10 5

**Loop** **Quantum** Dynamics

Notes about results

Recollapse occurs at the classical value of v

Correct description of recollapse - problems of Green and Unruh no longer relevant

Bounce is not result of exotic matter - only **in**cluded w = **+1** massless scalar field

LQC **in**verse volume “dj” effects negligible - bounce occurs at volumes above the critical value

**Quantum** dynamics described accurately by effective Friedmann equation

2 ˙a

a

**Quantum** bounce when ρφ ≈ ρc ≈ ρPL

Classical recollapse when ρφ ≈ 3/(8πGa 2 max )

≈

v

8πG

3 ρφ − 1

a 2

1 − ρφ

φ

0

-1

-2

-3

-4

-5

0 2*10 4

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot

ρc

4*10 4

v

6*10 4

LQC

Effective

Classical

8*10 4

1.0*10 5

Conclusion/Outlook

Successful rigorous description of physical sector of LQC for closed model - self-adjo**in**t improved

constra**in**t built from holonomies of connection, physical **in**ner product, observables

Big bang/Big crunch resolved, replaced with big bounce

Massless scalar field leads to cycles of contraction/expansion connected by bounces and recollapses

Caveats of orig**in**al LQC k = **+1** quantization handled properly

Problems of Green, Unruh no longer persist - quantum theory implies recollapse at classical value and

correct semi-classical limit

Further evidence that loop quantum gravity resolves classical s**in**gularities **in** a natural fashion!

ICG **Closed** k = **+1** **Model** **in** **Loop** **Quantum** **Cosmology** - Kev**in** Vandersloot