# Cosmological particle production and the precision of the WKB approximation

## Abstract

Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples.

- Authors:

- Department of Physics, Ludwig-Maximilians University, 80333 Munich (Germany)

- Publication Date:

- OSTI Identifier:
- 20711564

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 72; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.72.104011; (c) 2005 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; GRAVITATIONAL FIELDS; PARTICLE PRODUCTION; PERTURBATION THEORY; QUANTUM FIELD THEORY; SPACE-TIME; TIME DEPENDENCE; VACUUM STATES; WKB APPROXIMATION

### Citation Formats

```
Winitzki, Sergei.
```*Cosmological particle production and the precision of the WKB approximation*. United States: N. p., 2005.
Web. doi:10.1103/PhysRevD.72.104011.

```
Winitzki, Sergei.
```*Cosmological particle production and the precision of the WKB approximation*. United States. doi:10.1103/PhysRevD.72.104011.

```
Winitzki, Sergei. Tue .
"Cosmological particle production and the precision of the WKB approximation". United States.
doi:10.1103/PhysRevD.72.104011.
```

```
@article{osti_20711564,
```

title = {Cosmological particle production and the precision of the WKB approximation},

author = {Winitzki, Sergei},

abstractNote = {Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples.},

doi = {10.1103/PhysRevD.72.104011},

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

number = 10,

volume = 72,

place = {United States},

year = {Tue Nov 15 00:00:00 EST 2005},

month = {Tue Nov 15 00:00:00 EST 2005}

}