Back Reaction of Strings in Self-Consistent String Cosmology

Abstract

We compute the string energy-momentum tensor and {\bf derive} the string equation of state from exact string dynamics in cosmological spacetimes. $1+1,~2+1$ and $D$-dimensional universes are treated for any expansion factor $R$. Strings obey the perfect fluid relation $ p = (\gamma -1) \rho $ with three different behaviours: (i) {\it Unstable} for $ R \to \infty $ with growing energy density $ \rho \sim R^{2-D} $, {\bf negative} pressure, and $ \gamma =(D-2)/(D-1) $; (ii){\it Dual} for $ R \to 0 $, with $ \rho \sim R^{-D} $, {\bf positive} pressure and $\gamma = D/(D-1) $ (as radiation); (iii) {\it Stable} for $ R \to \infty $ with $ \rho \sim R^{1-D} $, {\bf vanishing} pressure and $\gamma = 1 $ (as cold matter). We find the back reaction effect of these strings on the spacetime and we take into account the quantum string decay through string splitting. This is achieved by considering {\bf self-consistently} the strings as matter sources for the Einstein equations, as well as for the complete effective string equations. String splitting exponentially suppress the density of unstable strings for large $R$. The self-consistent solution to the Einstein equations for string dominated universes exhibits the realistic matter dominated behaviour $ R \sim (X^0)^{2/(D-1)}\; $ for large times and the radiation dominated behaviour $ R \sim (X^0)^{2/D}\; $ for early times. De Sitter universe does not emerge as solution of the effective string equations. The effective string action (whatever be the dilaton, its potential and the central charge term) is not the appropriate framework in which to address the question of string driven inflation.