Analytical scaling solutions for the evolution of cosmic domain walls in a parameter-free velocity-dependent one-scale model

TL;DR

This paper derives an analytical approximation for the evolution of cosmic domain wall networks in expanding universes, matching simulations and revealing the energy-loss parameter vanishes as the universe approaches a linear expansion.

Contribution

It provides a new analytical approximation for domain wall evolution in power-law universes, extending previous models and matching numerical simulations with high accuracy.

Findings

Approximation accurately predicts domain wall evolution for 0 ≤ λ < 1.

The energy-loss parameter approaches zero as λ approaches 1.

The model aligns well with numerical simulations, especially near λ=1.

Abstract

We derive an analytical approximation for the linear scaling evolution of the characteristic length $L$ and the root-mean-squared velocity $\sigma_v$ of standard frictionless domain wall networks in Friedmann-Lema\^itre-Robertson-Walker universes with a power law evolution of the scale factor $a$ with the cosmic time $t$ ($a \propto t^\lambda$). This approximation, obtained using a recently proposed parameter-free velocity-dependent one-scale model for domain walls, reproduces well the model predictions for $\lambda$ close to unity, becoming exact in the $\lambda \to 1^-$ limit. We use this approximation, in combination with the exact results found for $\lambda=0$, to obtain a fit to the model predictions valid for $\lambda \in [0, 1[$ with a maximum error of the order of $1 \%$. This fit is also in good agreement with the results of field theory numerical simulations, specially for $\lambda \in [0.9, 1[$. Finally, we explicitly show that the phenomenological energy-loss parameter of the original velocity-dependent one-scale model for domain walls vanishes in the $\lambda \to 1^-$ limit and discuss the implications of this result.