  We explore constraints on the spectral index $n$ of density fluctuations and
the neutrino energy density fraction $\Omega_{HDM}$, employing data from a
variety of large scale observations. The best fits occur for $n\approx 1$ and
$\Omega_{HDM} \approx 0.15 - 0.30$, over a range of Hubble constants $40-60$ km
s$^{-1}$ Mpc$^{-1}$. We present a new class of inflationary models based on
realistic supersymmetric grand unified theories which do not have the usual
`fine tuning' problems. The amplitude of primordial density fluctuations, in
particular, is found to be proportional to $(M_X /M_P)^2$, where $M_X (M_P)$
denote the GUT (Planck) scale, which is reminiscent of cosmic strings! The
spectral index $n = 0.98$, in excellent agreement with the observations
provided the dark matter is a mixture of `cold' and `hot' components.
