# Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations

## Abstract

This is the first paper in a series where we study collisions of nucleated bubbles taking into account the effects of small initial (quantum) fluctuations in a fully 3+1-dimensional setting. In this paper, we consider the evolution of linear fluctuations around highly symmetric though inhomogeneous backgrounds. In particular, we demonstrate that a large degree of asymmetry develops over time from tiny initial fluctuations superposed upon planar and SO(2,1) symmetric backgrounds. These fluctuations are inevitable consequences of zero-point vacuum oscillations, so excluding them by enforcing a high degree of spatial symmetry is inconsistent in a quantum treatment. To simplify the analysis we consider the limit of two colliding planar walls, with mode functions for the fluctuations characterized by the wavenumber transverse to the collision direction and a longitudinal shape along the collision direction x, which we solve for. In the linear regime, the fluctuations obey a linear wave equation with a time- and space-dependent mass m{sub eff}(x,t). In situations where the walls collide multiple times, m{sub eff} oscillates in time. We use Floquet theory to study the evolution of the fluctuations and generalize the calculations familiar from the preheating literature to the case with many coupled degrees of freedom. The inhomogeneousmore »

- Authors:

- CITA, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8 (Canada)
- (Canada)
- Department of Physics and Astronomy, University of North Carolina-Chapel Hill, NC 27599-3255 (United States)

- Publication Date:

- Sponsoring Org.:
- SCOAP3, CERN, Geneva (Switzerland)

- OSTI Identifier:
- 22454515

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2015; Journal Issue: 03; Other Information: PUBLISHER-ID: JCAP03(2015)007; OAI: oai:repo.scoap3.org:9417; Article funded by SCOAP3. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AMPLIFICATION; COSMOLOGICAL MODELS; DEGREES OF FREEDOM; FLUCTUATIONS; MASS; MONOPOLES; NONLINEAR PROBLEMS; OCCUPATION NUMBER; OSCILLATIONS; PHASE TRANSFORMATIONS; QUASI PARTICLES; RESONANCE; SO GROUPS; SPACE DEPENDENCE; WAVE EQUATIONS

### Citation Formats

```
Braden, Jonathan, Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Bond, J. Richard, and Mersini-Houghton, Laura.
```*Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations*. United States: N. p., 2015.
Web. doi:10.1088/1475-7516/2015/03/007.

```
Braden, Jonathan, Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Bond, J. Richard, & Mersini-Houghton, Laura.
```*Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations*. United States. doi:10.1088/1475-7516/2015/03/007.

```
Braden, Jonathan, Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Bond, J. Richard, and Mersini-Houghton, Laura. Tue .
"Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations". United States.
doi:10.1088/1475-7516/2015/03/007.
```

```
@article{osti_22454515,
```

title = {Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations},

author = {Braden, Jonathan and Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8 and Bond, J. Richard and Mersini-Houghton, Laura},

abstractNote = {This is the first paper in a series where we study collisions of nucleated bubbles taking into account the effects of small initial (quantum) fluctuations in a fully 3+1-dimensional setting. In this paper, we consider the evolution of linear fluctuations around highly symmetric though inhomogeneous backgrounds. In particular, we demonstrate that a large degree of asymmetry develops over time from tiny initial fluctuations superposed upon planar and SO(2,1) symmetric backgrounds. These fluctuations are inevitable consequences of zero-point vacuum oscillations, so excluding them by enforcing a high degree of spatial symmetry is inconsistent in a quantum treatment. To simplify the analysis we consider the limit of two colliding planar walls, with mode functions for the fluctuations characterized by the wavenumber transverse to the collision direction and a longitudinal shape along the collision direction x, which we solve for. In the linear regime, the fluctuations obey a linear wave equation with a time- and space-dependent mass m{sub eff}(x,t). In situations where the walls collide multiple times, m{sub eff} oscillates in time. We use Floquet theory to study the evolution of the fluctuations and generalize the calculations familiar from the preheating literature to the case with many coupled degrees of freedom. The inhomogeneous case has bands of unstable transverse wavenumbers k{sub ⊥} whose corresponding mode functions grow exponentially. By examining the detailed spatial structure of the mode functions in x, we identify both broad and narrow parametric resonance generalizations of the homogeneous m{sub eff}(t) case of preheating. The unstable k{sub ⊥} modes are longitudinally localized, yet can be described as quasiparticles in the Bogoliubov sense. We define an effective occupation number and show they are created in bursts for the case of well-defined collisions in the background. The transverse-longitudinal coupling accompanying nonlinearity radically breaks this localized particle description, with nonseparable 3D modes arising that will be studied in subsequent papers.},

doi = {10.1088/1475-7516/2015/03/007},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 03,

volume = 2015,

place = {United States},

year = {Tue Mar 03 00:00:00 EST 2015},

month = {Tue Mar 03 00:00:00 EST 2015}

}