We use the Velocity-dependent One Scale Model for topological defect evolution to explore and classify the possible scaling solutions for string networks with time-varying tension, in cosmological and non-cosmological settings and under two different phenomenological assumptions for the behavior of these variations, which rely on different stretching and damping contributions to the string dynamics. We discuss how these assumptions impact the standard scaling solutions, as well as the evolution of the string network density. In addition to simple power-law cosmological epochs solutions, we also discuss the behavior of the network during the radiation-to-matter and matter-to-acceleration transitions. Overall, our results show that for the same amount of tension variation, a change in the stretching length scale tends to have a more significant impact on the network than a change in the damping length.
Anonymous (Tue,) studied this question.