# Thin and thick bubble walls I: vacuum phase transitions

## TL;DR

This paper investigates the dynamics of vacuum bubble walls during phase transitions, extending traditional models by relaxing assumptions like thin walls and spherical symmetry, and introduces methods for more accurate descriptions.

## Contribution

It develops improved analytical and numerical methods for modeling bubble wall dynamics beyond standard thin-wall and spherical assumptions, including arbitrary deformations.

## Key findings

- Derived equations of motion for bubble walls without shape assumptions
- Proposed iterative and perturbative methods for wall profile calculations
- Compared approximations with numerical solutions to validate approaches

## Abstract

This is the first in a series of papers where we study the dynamics of a bubble wall beyond usual approximations, such as the assumptions of spherical bubbles and infinitely thin walls. In this paper, we consider a vacuum phase transition. Thus, we describe a bubble as a configuration of a scalar field whose equation of motion depends only on the effective potential. The thin-wall approximation allows obtaining both an effective equation of motion for the wall position and a simplified equation for the field profile inside the wall. Several different assumptions are involved in this approximation. We discuss the conditions for the validity of each of them. In particular, the minima of the effective potential must have approximately the same energy, and we discuss the correct implementation of this approximation. We consider different improvements to the basic thin-wall approximation, such as an iterative method for finding the wall profile and a perturbative calculation in powers of the wall width. We calculate the leading-order corrections. Besides, we derive an equation of motion for the wall without any assumptions about its shape. We present a suitable method to describe arbitrarily deformed walls from the spherical shape. We consider concrete examples and compare our approximations with numerical solutions. In subsequent papers, we shall consider higher-order finite-width corrections, and we shall take into account the presence of the fluid.