The cluster abundance in cosmic string models for structure formation

We use the present observed number density of large X-ray clusters to constrain the amplitude of matter density perturbations induced by cosmic strings on the scale of $8h^{-1}$Mpc (${\sigma }_8$), in both open cosmologies and flat models with a non-zero cosmological constant. We find a slightly lower value of ${\sigma }_8$ than that obtained in the context of primordial Gaussian fluctuations generated during inflation. This lower normalization of ${\sigma }_8$ results from the mild non-Gaussianity on cluster scales, where the one point probability distribution function is well approximated by a ${\chi }^2$ distribution. We use our estimate of ${\sigma }_8$ to constrain the string linear energy density $\mu$ and show that it is consistent with the COBE normalization.