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Physical and invariant models for defect network evolution

We revisit the velocity-dependent one-scale model for topological defect evolution, and present a new alternative formulation in terms of a physical (rather than invariant) characteristic length scale. While the two approaches are equivalent (as we explicitly demonstrate), the new one is particularly relevant when studying the evolution of ultra-relativistic defects. Moreover, a comparison of the two provides further insight on the interpretation of the model's two phenomenological parameters, $c$ related to energy losses and $k$ related to the curvature of the defects. As an illustration of the relevance of the new formulation, we use it to study the evolution of cosmic string and domain wall networks in contracting universes. We show that these networks are ultra-relativistic and conformally contracted, with the physical length scale behaving as $L_{ph}\propto a$ and the density as $ρ\propto a^{-4}$ (as in a radiation fluid) in both cases. On the other hand the velocity and invariant length respectively behave as $(γv)\propto a^{-n}$ and $L_{inv}\propto a^{\frac{4}{4-n}}$, where $n$ is the number of dimensions of the defect's worldsheet. Finally we also study an alternative friction-dominated scenario and show that the stretching and Kibble regimes identified in the case of expanding universes can also occur for contracting ones.