Gravitational Lensing Limits on Cold Dark Matter and Its Variants

Standard $\Omega_0=1$ cold dark matter (CDM) needs $0.27 < \sigma_8 < 0.63$ ($2\sigma$) to fit the observed number of large separation lenses, and the constraint is nearly independent of $H_0=100h^{-1}\kms$ Mpc$^{-1}$. This range is strongly inconsistent with the COBE estimate of $\sigma_8=(2.8\pm0.2)h$. Tilting the primordial spectrum $\propto k^n$ from $n=1$ to $0.3 \ltorder n \ltorder 0.7$, using an effective Hubble constant of $0.15 \ltorder \Gamma=h \ltorder 0.30$, or reducing the matter density to $0.15 \ltorder \Omega_0 h \ltorder 0.3$ either with no cosmological constant ($\lambda_0=0$) or in a flat universe with a cosmological constant ($\Omega_0+\lambda_0=1$) can bring the lensing estimate of $\sigma_8$ into agreement with the COBE estimates. The models and values for $\sigma_8$ consistent with both lensing and COBE match the estimates from the local number density of clusters and correlation functions. The conclusions are insensitive to systematic errors except for the assumption that cluster core radii are singular. If clusters with $\rho\propto(r^2+s^2)^{-1}$ have core radii exceeding $s = 15h^{-1}\sigma_3^2$ kpc for a cluster with velocity dispersion $\sigma=10^3\sigma_3 \kms$ then the estimates are invalid. There is, however, a fine tuning problem in making the cluster core radii large enough to invalidate the estimates of $\sigma_8$ while producing several lenses that do not have central or ``odd images.'' The estimated completeness of the current samples of lenses larger than $5\parcs0$ is 20\%, because neither quasar surveys nor lens surveys are optimized to this class of lenses.