Large Scale Structure and Supersymmetric Inflation without Fine Tuning

We explore constraints on the spectral index $n$ of density fluctuations and the neutrino energy density fraction ${\Omega }_{\nu }$ from observations of large scale structure. The best fits imply $n\approx 1$ and ${\Omega }_{\nu }\approx 0.1-0.3$, for Hubble constants 40-60 km ${\mathrm{s}}^{-1\mathrm{}}$ ${\mathrm{Mpc}}^{-1\mathrm{}}$. We present a new class of inflationary models based on realistic supersymmetric grand unified theories (GUTs) which do not have the usual "fine tuning" problems. The amplitude of primordial density fluctuations is found to be $\propto {\left(\frac{M_X}{M_P}\right)}^2$, where $M_X$ ($M_P$) denote the GUT (Planck) scale. The spectral index $n=0.98\mathrm{}$, in excellent agreement with the observations.