The circular loop equation of a cosmic string with time-varying tension   in de Sitter spacetimes
  In this work the equation of circular loops of cosmic string possessing time-dependent tension is studied in the de Sitter spacetime. We find that the cosmic string loops with initial radius $r(t_{0})>0.707L$, L de Sitter radius, should not collapse to form a black holes. It is also found that in the case of $r(t_{0})<0.707L$ a loop of cosmic string whose tension depends on some power of cosmic time can not become a black hole if the power is lower than a critical value which is associated with the initial size of the loop. 
Yunqi Liu, Hongbo Cheng