Primordial density perturbations with running spectral index: impact on
  non-linear cosmic structures
  (abridged) We explore the statistical properties of non-linear cosmic
structures in a flat $\Lambda$CDM cosmology in which the index of the
primordial power spectrum for scalar perturbations is allowed to depend on the
scale. Within the inflationary paradigm, the running of the scalar spectral
index can be related to the properties of the inflaton potential, and it is
hence of critical importance to test it with all kinds of observations, which
cover the linear and non-linear regime of gravitational instability. We focus
on the amount of running $\alpha_{\mathrm{S},0}$ allowed by an updated
combination of CMB anisotropy data and the 2dF Galaxy Redshift Survey. Our
analysis constrains $\alpha_{\mathrm{S},0} = -0.051^{+0.047}_{-0.053}$
$(-0.034^{+0.039}_{-0.040})$ at 95% Confidence Level when (not) taking into
account primordial gravitational waves in a ratio as predicted by canonical
single field inflation, in agreement with other works. For the cosmological
models best fitting the data both with and without running we studied the
abundance of galaxy clusters and of rare objects, the halo bias, the
concentration of dark matter halos, the Baryon Acoustic Oscillation, the power
spectrum of cosmic shear, and the Integrated Sachs-Wolfe effect. We find that
counting galaxy clusters in future X-ray and Sunyaev-Zel'dovich surveys could
discriminate between the two models, more so if broad redshift information
about the cluster samples will be available. Likewise, measurements of the
power spectrum of cosmological weak lensing as performed by planned all-sky
optical surveys such as EUCLID could detect a running of the primordial
spectral index, provided the uncertainties about the source redshift
distribution and the underlying matter power spectrum are well under control.
