Abstract
(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.
Highlights
The formation of cosmic structures occurred via gravitational instability
In order to show the degeneracy of the models output of our analysis with respect to Cosmic Microwave Background radiation (CMB) observables, in Figure 2 we report the CMB temperature power spectra for the ΛCDM cosmology and the model with running spectral index, both in the case of absence or presence of primordial gravitational waves
It should be kept in mind that this increment is not solely due to the running itself, and to the fact that, e.g., the normalization σ8 in the model with running is slightly larger than in the ΛCDM cosmology
Summary
Small density fluctuations in the dark matter fluid grew up in time, and eventually detached from the overall expansion of the Universe, collapsing and giving rise to dark matter halos This kind of process is relatively well understood, thanks to both perturbation theory in the linear and quasi-linear regimes, and numerical simulations in the fully non-linear regime. The evolution with redshift of cosmic structures is mainly determined, in a statistical sense, by the initial conditions and by the subsequent expansion history of the Universe The latter depends critically on the matter and energy content of the Universe itself, and this dependence is important in order to gauge the dynamical evolution of dark energy (Cunha, Huterer, & Frieman 2009; Grossi & Springel 2009; Samushia & Ratra 2009). Concerning the initial conditions, the simplest models of inflation predict that the primordial density fluctuations should be distributed according to a nearly Gaussian statistics, with a nearly
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