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.Comment: LaTEX, 14 pp. + 1 postscript figure appende

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We consider a supersymmetric model of inflation in which the primordial density fluctuations are nearly scale invariant (spectral index n ≈ 0.98) with amplitude proportional to (M/M P lanck ) 2 , where M ∼ 10 16 GeV denotes the scale of the gauge symmetry breaking associated with inflation. The 60 or so e-foldings take place when all relevant scales are close to M , which helps suppress supergravity corrections. The gravitino and baryogenesis (via leptogenesis) constraints help determine the two heaviest right handed neutrino masses to be ≈ 2 × 10 13 GeV and 6 × 10 9 GeV. 98.80 Cq, 12.60 Jv, 95.35 +d