1 Universe with cosmological constant in Loop Quantum Cosmology IGC INAUGURAL CONFERENCE, PENN STATE, AUG Tomasz Pawlowski (work by Abhay Ashtekar, El...

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Tomasz Pawlowski (work by Abhay Ashtekar, Eloisa Bentivegna, TP)

– p.1

Purpose of the talk Isotropic flat universe with massless scalar field in LQC: Recent results for Λ = 0 (A Ashtekar, P Singh, TP gr-qc/0607039): change of dynamics due to quantum geometric effects. Existence of large semiclassical (contracting) universe preceding expanding one. Bounce at energy density ρ = ρc ≈ 0.82ρPl . Presented work: Extension to the case of nonvanishing cosmological constant (preliminary investigation in gr-qc/0607039). Questions: Does the qualitative picture (bounce, preexisting branch) remain ? If yes, does ρc still play fundamental role ? What new properties the models with Λ possess ? Due to distinct mathematical properties of an evolution operator +ve and −ve Λ have to be investigated separately.

– p.2

LQC quantization scheme Considered model: flat isotropic (FRW) universe Matter content: massless scalar field Basic variables: geometry: Aia , Eia in isotropic situation reexpressed in terms of coefficients c, p. matter: field φ and conjugate momentum pφ . Quantization method following LQG: Geometric DOF: triads p and holonomies h raised to operators. Matter DOF: standard (Schrodinger) quantization. Kinematical Hilbert space: ¯ Bohr , dµBohr ) ⊗ L2 (R, dφ) Hkin = Hg ⊗ Hφ =: L2 (R Basis of Hg : eigenstates of pˆ for convenience labeled by v s.t. 1 2 6 3 pˆ |vi = 2 · 3 πγ sgn(v)|v| |vi Quantization of Hamiltonian constraint Cgrav + Cmatt = 0: Its geometric components reexpressed in terms of holonomies (Thiemann method), next raised to operators. – p.3

Evolution operator The quantized constraint similar to Klein-Gordon equation: ∂φ2 Ψ(v, φ) = −ΘΨ(v, φ) Θ is a difference operator ΘΨ(v, φ) = C + (v)Ψ(v + 4, φ) + C o (v)Ψ(v, φ) + C − (v)Ψ(v − 4, φ), Λ enters C o only, approximately acts like v 2 potential, Θ is symmetric on the domain D of finite combinations of |vi. System reinterpreted as free one evolving with respect to φ. Few important details: No C-symmetry violation interactions ⇒ states symmetric with respect to reflection Π in v. Domain of v naturally splits onto family of sets preserved by action of Θ and Π: Lǫ := {v ∈ R : v = ±ǫ + 4n, n ∈ Z}. In consequence Hg = ⊕Hǫ , where Hǫ contains functions supported on Lǫ only.

– p.4

Λ<0 Work by: E Bentivegna, TP

Classically recollapsing system. Recollapse when energy density of φ satisfies: ρφ + Λ/8πG = 0. Λ acts approximately like +ve v 2 potential. Θ is positively definite, essentially self-adjoint, its spectrum is discrete (Lewandowski, Kaminski, Szulc). Normalizable eigenfunctions singled out numerically. Each normalizable eigenspace 1-dimensional. Basis en (v) of physical Hilbert space selected out of normalized eigenfunctions. P ˜ Physical states: Ψ(v, φ) = n Ψn en (v) exp[iωn (φ − φo )]. Choice: Gaussian states sharply peaked about ω ⋆ = ~−1 p⋆φ and some large v ⋆ for some initial φ = φo ˜ n = exp(−(ωn − ω ⋆ )2 /2σ 2 ) Ψ Dirac observables: pˆφ , |ˆ v |φ .

– p.5

Λ < 0: classical trajectory 3 classical

2.5

2

φ 1.5

1

0.5

0 0

1*104

2*104

3*104

4*104

5*104

|v| – p.6

Λ < 0: wave function

|Ψ(v,φ)|

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.4 0.3 0.2 0.1 0 1*104 2*104

|v|

3*104 4*104

3

5*104

2

1.5 4

6*10

0

0.5

1

2.5

φ

– p.7

Λ < 0: quantum trajectory 3 LQC classical

2.5

2

φ 1.5

1

0.5

0 0

5*103

1.0*104

1.5*104

2.0*104

2.5*104

3.0*104

3.5*104

4.0*104

4.5*104

5.0*104

|v| – p.8

Λ < 0 – results State remains sharply peaked throughout the evolution. Expectation values follow classical trajectory till (total) energy density becomes comparable to ρc . In particular classical recollapse at the size predicted by classical theory. Bounce exactly at ρφ + Λ/8πG = ρc joins two large semiclassical sectors. Singularities are resolved - replaced by a quantum bounce. Resulting evolution is periodic (with period depending on Λ).

– p.9

Λ>0 Work by: A Ashtekar, E Bentivegna, TP

Classically two distinct classes: ever-expanding and ever-contracting. In both classes v reaches infinity for finite φ = φo . Solutions parametrized by proper time end there. Can they be extended ? Yes: One can introduce new variable such that domain of v is compact in it and equation of motion (wrt. φ) is analytic in it. In consequence EOM can be analytically extended. Behavior of energy density shows that ’new’ regions are copies of existing ones. Extension simply identifies v = +∞ with v = −∞. Extended solutions: at infinity universe transits from expanding to contracting phase. On the quantum level: contribution from cosmological constant acts approximately as ∝ −v 2 potential (unbounded from below). Hamiltonians of such system are usually not self-adjoint. Is that the case here ?

– p.10

Λ > 0: classical trajectory 1 expanding contracting

0.5

φ

0

-0.5

-1 0

2*104

4*104

6*104

8*104

1.0*105

|v| – p.11

Self-adjoint extensions: general theorem Consider general operator Θ symmetric on the domain D. Deficiency subspaces K± : spaces of normalizable solutions ϕ± to equation hϕ± | Θ⋆ ∓ iI |ψi = 0,

ψ∈D

If dim(K+ ) = dim(K− ) 6= 0 domain of Θ has many extensions. All of them are defined by unitary transformations Uα : K+ → K− as follows: Dα = {ψ + a(ϕ+ + Uα (ϕ+ )); ψ ∈ D, a ∈ C}

– p.12

Self-adjoint extensions: Λ > 0 Method presented on previous slide applied directly to case Λ > 0. For simplicity we focus on the case ǫ = 0. The analysis: Elements of K± – normalizable solutions to difference equation Θϕ± = ±iϕ± – found numerically. Symmetric ones are unique up to normalization. dim(K+ ) = dim(K− ) = 1. For normalized ϕ± all Uα are of the form Uα ϕ+ = eiα ϕ− . One parameter family of extensions. Since D – finite combinations of |vi the behavior at v → ∞ of wave functions is for each extension given by a(ϕ+ + Uα ϕ+ ). Basis of Dα can be selected out of eigenfunctions converging to a(ϕ+ + Uα ϕ+ ) at v → ∞. Results: All extensions Θα of Θ have discrete spectra. The physical states have form analogous ones for Λ < 0. We can repeat the construction + analysis done for that case (Gaussian states). – p.13

Λ > 0: wave function

|Ψ(v,φ)|

70

80 60 40 20 0

60 50 40 30 20

2*104

10 0

4*104

|v|

6*104 8*104 1.0*105

-2

-1.5

-1

-0.5

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0

1

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2

φ

– p.14

Λ > 0: quantum trajectory 2 LQC classical asymptotics 1.5

1

0.5

φ

0

-0.5

-1

-1.5

-2 102

103

|v|

104

105 – p.15

Λ > 0: energy density 1

10-1

10-2

ρ 10-3

10-4

LQC classical -2

-1.5

-1

-0.5

0

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1

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φ – p.16

Λ > 0 – results The results are the same for all extensions: States remain sharply peaked through the evolution. States follow classical trajectory until total energy density approaches critical one, when gravity becomes repulsive and state bounces. Bounce joins deterministically contracting and expanding sectors. Evolution is nonsingular, bounce replaces singularitites. For all extensions the expanding universe after reaching infinite volume (or, equivalently ρ = Λ/8πG) reflects back into contracting one. Due to quantum bounce and reflection at infinity we again have cyclic evolution. Comment: Results are analogous for other values of ǫ. For ǫ = 2 one parameter family of extensions. For ǫ 6= 0, 2 – four parameter (work in progress). – p.17

Summary For both signs of Λ the results of Λ = 0 hold. Constructed semiclassical wave packets have the following properties: They follow appropriate classical trajectories till energy density becomes Planckian, in which case we observe bounce. Bounce at ρφ + Λ/8πG always joins two large semiclassical sectors (epochs) of considered universe. For Λ < 0 evolution operator has unique self-adjoint extension, whereas for Λ > 0 (in the sector of ǫ = 0, 2) there exists one-parameter family of extensions. Latter property is expected to be shared by eg. hyperbolic k = −1 model. Due to bounce and, either classical recollapse or reflection at infinity evolution is periodic in φ.

– p.18

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