Cosmological scaling of precursor domain walls

TL;DR

This paper studies the scaling behavior of precursor domain walls in cosmology, deriving how the correlation length evolves with time under different expansion regimes and confirming a universal transition pattern.

Contribution

It provides an explicit computation of the scaling exponent for precursor domain walls in expanding universes across different dimensions, revealing a universal transition behavior.

Findings

The correlation length scales as t^{1/2} for slow expansion (α ≤ 1/2).

For faster expansion (α ≥ 1/2), the scaling exponent matches the scale factor power α.

The transition in scaling behavior is dimension-independent, suggesting applicability to 3+1 dimensions.

Abstract

Domain wall (DW) networks have a large impact on cosmology and present interesting dynamics that can be controlled by various scaling regimes. In the first stage after spontaneous breaking of the discrete symmetry, the network is seeded with `DW precursors', the zeros of a tachyonic field. At sufficiently weak coupling, this stage can be quite long. The network is then driven to a non-relativistic scaling regime: in flat spacetime the correlation length grows like $L\sim t^{\kappa}$ with $\kappa=1/2$. We focus on the precursor regime in cosmology, assuming a power-law scale factor $a\propto t^\alpha$. We obtain the scaling exponent as a function of the external parameter, $\kappa(\alpha)$, by explicit computation in $1+1$ and $2+1$ dimensions, and find a smooth transition from nonrelativistic scaling with $\kappa\simeq 1/2$ for $\alpha\lesssim1/2$ to DW gas regime $\kappa \simeq \alpha$ for $\alpha\gtrsim1/2$, confirming previous arguments. The precise form of the transition $\kappa(\alpha)$ is surprisingly independent of dimension, suggesting that similar results should also be valid in $3+1$ dimensions.