The circular loop equation of a cosmic string with time-varying tension
  The equation of circular loops of cosmic string with time-dependent tension is studied in the Minkowski spacetime and Robertson-Walker universe. We find that, in the case where the tension depends on some power of the cosmic time, cosmic string loops with time-varying tension should not collapse to form a black hole if the power is lower than a critical value. 
Hongbo Cheng, Yunqi Liu