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We evaluate the observational constraints on the spectral index $n$, in thecontext of the $\Lambda$CDM hypothesis which represents the simplest viablecosmology. We first take $n$ to be practically scale-independent. Ignoringreionization, we find at a nominal 2-$\sigma$ level $n\simeq 1.0 \pm 0.1$. Ifwe make the more realisitic assumption that reionization occurs when a fraction$f\sim 10^{-5}$ to 1 of the matter has collapsed, the 2-$\sigma$ lower bound isunchanged while the 1-$\sigma$ bound rises slightly. These constraints arecompared with the prediction of various inflation models. Then we investigatethe two-parameter scale-dependent spectral index, predicted by running-massinflation models, and find that present data allow significant scale-dependenceof $n$, which occurs in a physically reasonable regime of parameter space.Comment: ReVTeX, 15 pages, 5 figures and 3 tables, uses epsf.sty Improved  treatment of reionization and small bug fixed in the constant n case; more  convenient parameterization and better treatment of the n dependence in the  CMB anisotropy for the running mass case; conclusions basically unchanged;  references adde

Observational constraints on the spectral index of the cosmological curvature perturbation