Energy of string loops and thermodynamics of dark energy

Abstract

We discuss the thermodynamic aspects of a simple model of cosmic string loops, whose energy is nonlinearly related to their lengths. We obtain in a direct way an equation of state having the form $p=-(1+\alpha)\rho/3$, with $\rho$ the energy density and $1+\alpha$ the exponent which relates the energy $u_l$ of a loop with its length $l$ as $u_l\sim l^{1+\alpha}$. In the linear situation ($\alpha=0$) one has $p=-\rho/3$, in the quadratic one ($\alpha=1$) $p=-2\rho/3$, and in the cubic case ($\alpha =2$), $p=-\rho$. For all values of $\alpha$ the entropy goes as $S\sim(2-\alpha)L^{3/2}$ ($L$ being the string length density). The expression of $S$ is useful to explore the behavior of such string loops under adiabatic expansion of the universe. Thermodynamic stability suggests that the gas of string loops must coexist with several long strings, longer than the horizon radius.