Nonminimally coupled Scalar Field in Loop Quantum Cosmology

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Nonminimally coupled Scalar Field in Loop Quantum Cosmology

Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Nonminimally coupled Scalar Field

in

Loop Quantum Cosmology

Michal Artymowski, Andrea Dapor, Tomasz Pawlowski

Stockholm, 6th July 2012


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

1 Introduction

2 Quantization

3 Analysis

Outline


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

1 Introduction

2 Quantization

3 Analysis

Outline


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

S[gµν, φ] = 1


8πG

Nonminimal coupling:

Nonminimal Scalar Field

d 4 x √

−g −U(φ)R + 1

2 gµν

∂µφ∂νφ − V(φ)

U = 1

2 (1 + ξφ2 )


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

S[gµν, φ] = 1


8πG

Nonminimal coupling:

Nonminimal Scalar Field

d 4 x √

−g −U(φ)R + 1

2 gµν

∂µφ∂νφ − V(φ)

U = 1

2 (1 + ξφ2 )

• Reduction to Cosmology: gµν → a


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

S[gµν, φ] = 1


8πG

Nonminimal coupling:

Nonminimal Scalar Field

d 4 x √

−g −U(φ)R + 1

2 gµν

∂µφ∂νφ − V(φ)

U = 1

2 (1 + ξφ2 )

• Reduction to Cosmology: gµν → a

• Canonical analysis: Hamiltonian Constraint

H = −6U ′ ˙φ˙aa 2 − 6U ˙a 2 a + a 3 8πGρ = 0


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Equation of motion for φ:

Why nonminimal?

¨φ + 3H ˙φ = 2U′ V − UV ′ − U ′ ˙φ 2 (3U ′′ + 1

2 )

U + 3U ′2


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Equation of motion for φ:

Minimal case:

Why nonminimal?

¨φ + 3H ˙φ = 2U′ V − UV ′ − U ′ ˙φ 2 (3U ′′ + 1

2 )

U + 3U ′2

¨φ + 3H ˙φ = −V ′ = −m 2 φ

inflation drived by a fine-tuned heavy field: φin ≈ mPl, m ≈ 10 −6 mPl


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Equation of motion for φ:

Minimal case:

Why nonminimal?

¨φ + 3H ˙φ = 2U′ V − UV ′ − U ′ ˙φ 2 (3U ′′ + 1

2 )

U + 3U ′2

¨φ + 3H ˙φ = −V ′ = −m 2 φ

inflation drived by a fine-tuned heavy field: φin ≈ mPl, m ≈ 10 −6 mPl

Nonminimal case:

U(φ)R + V = 1 1

R +

2 2 φ2 (ξR + m 2 ) = Lgr + 1

2 φ2m 2

eff

⇒ Inflation can be driven by a light scalar field.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Higgs-based model:

with

Quantization Scheme

V = λ

4 φ4

ξ ≈ 47000 √ λ, λ = 0.5


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Higgs-based model:

with

Quantization Scheme

V = λ

4 φ4

ξ ≈ 47000 √ λ, λ = 0.5

→ canonical transformation to the minimal form


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Higgs-based model:

with

Quantization Scheme

V = λ

4 φ4

ξ ≈ 47000 √ λ, λ = 0.5

→ canonical transformation to the minimal form

→ standard loop quantization


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Higgs-based model:

with

Quantization Scheme

V = λ

4 φ4

ξ ≈ 47000 √ λ, λ = 0.5

→ canonical transformation to the minimal form

→ standard loop quantization

→ effective limit


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

1 Introduction

2 Quantization

3 Analysis

Outline


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Canonical Transformation

˜gµν = 2Ugµν,

d ˜φ


2

= U + 3U′2

2U 2


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

For this choice

Canonical Transformation

˜gµν = 2Ugµν,

2 d ˜φ

=


U + 3U′2

2U2 SNM[gµν, φ] = SM[˜gµν, ˜φ]

where the potential of field ˜φ in SM is given by

˜V = V

4U 2

The ”non-minimality” is absorbed in the effective potential ˜V.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Change to variables

sgn(˜v)˜v =

ã 3

2πγ √ ∆ℓ2 ,

Pl

Loop Quantization

˜b = −γ √ ∆ 1 dã

ã d˜t

where γ is Barbero-Immirzi parameter and ∆ = 4πγ √ 3ℓ2 Pl is ”area gap”.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Change to variables

sgn(˜v)˜v =

ã 3

2πγ √ ∆ℓ2 ,

Pl

Loop Quantization

˜b = −γ √ ∆ 1 dã

ã d˜t

where γ is Barbero-Immirzi parameter and ∆ = 4πγ √ 3ℓ2 Pl is ”area gap”.

Follow standard LQC:

−i∂˜tΨ(˜v, ˜φ) =

3πG



|ˆ˜v| ш 2 − ˆ

−22

Ñ |ˆ˜v| + 1


where ” ˆ Ñ = e iˆ˜b/2 ” and α = 2πγ √ ∆ℓ 2

Pl .


|˜v| −1 ˆπ 2 ˜φ

α

+

|ˆ˜v|


˜V Ψ(˜v, ˜φ)


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Effective Limit

Substitute the fundamental (observable) operators ˆ˜φ, ˆπ ˜φ, ˆ˜v, ˆ Ñ with their

expectation values:

Heff = − 3πG

2α |˜v| sin2 π

(˜b) +

2 ˜φ α

+

2α|˜v| |˜v| ˜V


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Effective Limit

Substitute the fundamental (observable) operators ˆ˜φ, ˆπ ˜φ, ˆ˜v, ˆ Ñ with their

expectation values:

Heff = − 3πG

2α |˜v| sin2 π

(˜b) +

2 ˜φ α

+

2α|˜v| |˜v| ˜V

The effective dynamics (with respect to ˜t) is obtained from Hamilton

equations, with initial conditions

˜b = π/2, ˜v = 1, ˜φ = ˜φin, π ˜φ = π ˜φ(˜v, ˜b, ˜φ)

The solutions are parametrized by a single value: ˜φin.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Effective Limit

Substitute the fundamental (observable) operators ˆ˜φ, ˆπ ˜φ, ˆ˜v, ˆ Ñ with their

expectation values:

Heff = − 3πG

2α |˜v| sin2 π

(˜b) +

2 ˜φ α

+

2α|˜v| |˜v| ˜V

The effective dynamics (with respect to ˜t) is obtained from Hamilton

equations, with initial conditions

˜b = π/2, ˜v = 1, ˜φ = ˜φin, π ˜φ = π ˜φ(˜v, ˜b, ˜φ)

The solutions are parametrized by a single value: ˜φin.

Numerically simulate the evolution, and transform back to non-tilded

quantities.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

1 Introduction

2 Quantization

3 Analysis

Outline


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

History of a Toy-Universe (1/2)

Toy model: ξ = 47 √ λ, symmetric trajectory (φin = 0).

0.2

0.1

0.0

0.1

H

t 0.0

0.2

0.2

10 000 5000 0 5000 10 000 100 50 0 50 100

10

5

0

5

10

0.2 0.1 0.0 0.1 0.2

H

t

0.2

0.1

0.1

10

5

0

5

10

H

H

0.004 0.002 0.000 0.002 0.004

t

t


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

100

80

60

40

20

History of a Toy-Universe (2/2)

a

1.0

0.8

0.6

0.4

0.2

0

0.0

100 50 0 50 100 1.0 0.5 0.0 0.5 1.0

a


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

100

80

60

40

20

History of a Toy-Universe (2/2)

a

1.0

0.8

0.6

0.4

0.2

0

0.0

100 50 0 50 100 1.0 0.5 0.0 0.5 1.0

40

30

20

10

0

15 10 5 0 5 10 15

Φ'

0.004

0.002

0.000

0.002

0.004

a

0 2 4 6 8 10

Φ


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

0.004

0.002

0.000

H

0.002

0.004

φ = 0.055

in

φ

in=

0

φ = -0.055

in

2000 1000 0 1000 2000

t

10

8

6

4

2

H

0

-2

-4

-6

-8

-10

-1.0*10 -5

-5*10 -6

Real Universe (1/3)

100

50

0

H

-50

-100

φ = 0.055

in

φ

in=

0

φ = -0.055

in

-0.10 -0.05 0.00

t

0.05 0.10

0 5*10 -6

t

1.0*10 -5


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

10

8

6

4

2

0

a

4000 2000 0 2000 4000

Real Universe (2/3)

0.5

0.4

0.3

0.2

0.1

0.0

0.2 0.1 0.0 0.1 0.2

a


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

10

8

6

4

2

0

7

6

5

4

dphi

3

2

1

0

a

4000 2000 0 2000 4000

-1 -0.5 0 0.5 1

phi

Real Universe (2/3)

0.5

0.4

0.3

0.2

0.1

0.0

0.2 0.1 0.0 0.1 0.2

0.5

0.4

0.3

dphi

0.2

0.1

0

a

-10 -5 0 5 10

phi


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

N

10 6

10 5

10 4

1000

100

inflation

denflation

Real Universe (3/3)

-0.04 -0.02 0.00 0.02 0.04

φ

in


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Conclusions


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Conclusions

• The very successful model of non-minimal scalar field admits a

simple LQC analogue.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Conclusions

• The very successful model of non-minimal scalar field admits a

simple LQC analogue.

• The LQC-corrected model preserves the usual properties of the

classical one: highly probable inflation, driven from light scalar

field.


Nonminimally

coupled Scalar

Field

in

Loop Quantum

Cosmology

Michal

Artymowski,

Andrea Dapor,

Tomasz Pawlowski

Introduction

Quantization

Analysis

Conclusions

• The very successful model of non-minimal scalar field admits a

simple LQC analogue.

• The LQC-corrected model preserves the usual properties of the

classical one: highly probable inflation, driven from light scalar

field.

• The LQC-corrected model presents the usual properties of LQC

models: resolution of singularity, which is substituted by a

(qualitatively new form of) Big Bounce.

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