Observational constraints on the spectral index of the cosmological curvature perturbation

We evaluate the observational constraints on the spectral index $n$, in the context of the $\mathrm{\Lambda }$CDM hypothesis which represents the simplest viable cosmology. We first take $n$ to be practically scale-independent. Ignoring reionization, we find at a nominal 2-$\sigma$ level $n\simeq 1.0\pm 0.1$. If we 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 is unchanged while the 1-$\sigma$ bound rises slightly. These constraints are compared with the prediction of various inflation models. Then we investigate the two-parameter scale-dependent spectral index, predicted by running-mass inflation models, and find that present data allow significant scale-dependence of $n$, which occurs in a physically reasonable regime of parameter space.