The logarithmic equation of state for superconducting cosmic strings
  This investigation follows up the suggestion that the equation of state for superconducting cosmic strings provided by Witten's prototype biscalar field model can be well represented by an effective Lagrangian of simple logarithmic form depending on only 3 independent parameters. The numerical work described here confirms the validity of this approximation, and initiates the evaluation of the 3 required parameters, as functions of the masses and other parameters specifying the underlying U(1) x U(1) scalar field model in the limit for which the relevant gauge coupling constants are small.In this limit, subject to calibration of the relevant length and mass scales, the scalar field model is characterised by just 3 dimensionless ratios which (in order to provide conducting strings) must be subject to three inequalities (of which two have obvious analytic expressions). It is found here that when all three of these inequalities are satisfied by a reasonably large margin, there is a simple empirical formula that can be used to provide a fairly accurate prescription for the algebraic dependence on these 3 dimensionless ratios of the 3 parameters required for the logarithmic equation of state. 
Betti Hartmann and Brandon Carter