# Loop quantum cosmology of k=1 FRW models

## Abstract

The closed, k=1, FRW model coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k=0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the big-bang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semiclassical at some late time, the big bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the 'difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k=0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k=0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists.

- Authors:

- Institute for Gravitational Physics and Geometry, Physics Department, Penn State, University Park, Pennsylvania 16802 (United States)
- (Netherlands)
- (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 21010905

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.024035; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; COSMOLOGY; GENERAL RELATIVITY THEORY; HILBERT SPACE; QUANTUM GRAVITY; SCALAR FIELDS; SEMICLASSICAL APPROXIMATION; SINGULARITY

### Citation Formats

```
Ashtekar, Abhay, Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, Pawlowski, Tomasz, Singh, Parampreet, Vandersloot, Kevin, and Institute of Cosmology and Gravitation, Mercantile House, Portsmouth University, Portsmouth PO1 2EG.
```*Loop quantum cosmology of k=1 FRW models*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.024035.

```
Ashtekar, Abhay, Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, Pawlowski, Tomasz, Singh, Parampreet, Vandersloot, Kevin, & Institute of Cosmology and Gravitation, Mercantile House, Portsmouth University, Portsmouth PO1 2EG.
```*Loop quantum cosmology of k=1 FRW models*. United States. doi:10.1103/PHYSREVD.75.024035.

```
Ashtekar, Abhay, Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, Pawlowski, Tomasz, Singh, Parampreet, Vandersloot, Kevin, and Institute of Cosmology and Gravitation, Mercantile House, Portsmouth University, Portsmouth PO1 2EG. Mon .
"Loop quantum cosmology of k=1 FRW models". United States.
doi:10.1103/PHYSREVD.75.024035.
```

```
@article{osti_21010905,
```

title = {Loop quantum cosmology of k=1 FRW models},

author = {Ashtekar, Abhay and Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, 3584 CC Utrecht and Pawlowski, Tomasz and Singh, Parampreet and Vandersloot, Kevin and Institute of Cosmology and Gravitation, Mercantile House, Portsmouth University, Portsmouth PO1 2EG},

abstractNote = {The closed, k=1, FRW model coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k=0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the big-bang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semiclassical at some late time, the big bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the 'difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k=0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k=0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists.},

doi = {10.1103/PHYSREVD.75.024035},

journal = {Physical Review. D, Particles Fields},

number = 2,

volume = 75,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}