Smooth hybrid inflation in supergravity with a running spectral index and early star formation

It is shown that in a smooth hybrid inflation model in supergravity adiabatic fluctuations with a running spectral index with $\ns >1$ on a large scale and $\ns <1$ on a smaller scale can be naturally generated, as favored by the first-year data of WMAP. It is due to the balance between the nonrenormalizable term in the superpotential and the supergravity effect. However, since smooth hybrid inflation does not last long enough to reproduce the central value of observation, we invoke new inflation after the first inflation. Its initial condition is set dynamically during smooth hybrid inflation and the spectrum of fluctuations generated in this regime can have an appropriate shape to realize early star formation as found by WMAP. Hence two new features of WMAP observations are theoretically explained in a unified manner.