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

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.