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The WMAP results on the scalar spectral index n and its running with scale,though preliminary, open a very interesting window to physics at very highenergies. We address the problem of finding inflaton potentials well motivatedby particle physics which can accomodate WMAP data. We make a model independentanalysis of a large class of models: those with flat tree-level potentiallifted by radiative corrections, which cause the slow rolling of the inflatonand the running of n. This includes typical hybrid inflation models. In thesmall-coupling regime the predictions for the size and running of n areremarkably neat, e.g. -dn/dln k=(n-1)^2 << 1, and n does not cross n=1,contrary to WMAP indications. On the other hand, n can run significantly if thecouplings are stronger but at the price of having a small number of e-folds,Ne. We also examine the effect of mass thresholds crossed during inflation.Finally, we show that the presence of non-renormalizable operators for theinflaton, suppressed by a mass scale above the inflationary range, is able togive both dn/dln k ~ O(-0.05) and Ne ~ 50.Comment: 33 pages, LaTeX, 5 ps figures. Minor errata fixed, refs. added.  Matches published versio

The running spectral index as a probe of physics at high scales