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 $8 h^{-1}$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.