Abstract:
We have evaluated the observational constraints on the spectral index $n$, in the context of a $\Lambda$CDM model. For $n$ scale-independent, as predicted by most models of inflation, present data require $n\simeq 1.0 \pm 0.1$ at the 2-$\sigma$ level. We have also studied the two-parameter scale-dependent spectral index, predicted by running-mass inflation models. Present data allow significant variation of $n$ in this case, within the theoretically preferred region of parameter space.

Abstract:
We realize and study a model of hybrid inflation in the context of softly broken supersymmetry. The inflaton is taken to be a flat direction in the superfield space and, due to unsuppressed couplings, its soft supersymmetry breaking mass runs with scale. Both gauge and Yukawa couplings are taken into account and different inflationary scenarios are investigated depending on the relative strenght of the couplings and the mass spectrum.

Abstract:
We review the connection between inflationary models and observations and concentrate to describe models based on softly broken supersymmetry, in particular running mass models, and their predictions. We then present a fit of the spectral index of the curvature perturbation, assuming a flat $\Lambda$CDM cosmology.

Abstract:
We study models of inflation where the inflaton corresponds to a flat direction in field space and its mass term is generated by gravity mediated soft supersymmetry breaking at high scale. Assuming the inflaton to have non negligible couplings to other fields, its mass runs with scale and can reach the small value required by slow roll inflation at a lower scale, even if its initial value is too large. Slow roll inflation can therefore take place in such a regime, as long as the mass remains small, with a spectral index that is then scale dependent. We explore the parameter space of this kind of models to find the region compatible with the present observations.

Abstract:
For gravitino dark matter with conserved R-parity and mass in the GeV range, very strong constraints from Big Bang Nucleosynthesis exclude the popular NLSP candidates like neutralino and charged sleptons. In this letter we therefore draw attention to the case of a sneutrino NLSP, that is naturally realised in the context of gaugino mediation. We find interesting collider signatures, characterised by soft jets or leptons due to the small sneutrino--stau mass splitting. Moreover, the lightest neutralino can have visible decays into staus, and in some part of the parameter space also into selectrons and smuons. We also show the importance of coannihilation effects for the evaluation of the BBN constraints.

Abstract:
We review here the status of different dark matter candidates in the context of supersymmetric models, in particular the neutralino as a realization of the WIMP-mechanism and the gravitino. We give a summary of the recent bounds in direct and indirect detection and also of the LHC searches relevant for the dark matter question. We discuss also the implications of the Higgs discovery for the supersymmetric dark matter models and give the prospects for the future years.

Abstract:
The identification of dark matter in our particle physics model is still a very open question. Here we will argue that axinos can be successful dark matter candidates in models with supersymmetry and the axion solution of the strong CP problem. Axinos can be the lightest supersymmetric particle (LSP), or can be heavier than the LSP. Axinos can be produced in the right abundance by thermal scatterings and if they are the LSP also by out of equilibrium decays of the lightest superpartner of SM fields (LSPSMs). On the other hand heavier (not LSP) axinos can generate a part of the neutralino LSP dark matter. Depending on the nature of the supersymmetric spectrum, and if R-parity is strictly conserved or slightly broken, very different signals of the LSP axino scenario can arise at colliders and in astrophysics.

Abstract:
We evaluate the observational constraints on the spectral index $n$, in the context of the $\Lambda$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.

Abstract:
If the inflaton sector is described by softly broken supersymmetry, and the inflaton has unsuppressed couplings, the inflaton mass will run strongly with scale. Four types of model are possible. The prediction for the spectral index involves two parameters, while the COBE normalization involves a third, all of them calculable functions of the relevant masses and couplings. A crude estimate is made of the region of parameter space allowed by present observation.