Limits on inflationary models of the universe

It is shown that there are severe limits on any model in which the universe undergoes a period of exponential expansion in the early stages. If one requires that the exponential expansion is long enough to account for the spatial flatness of the universe and that it should not create larger density fluctuations than are observed, it follows that the Hubble constant during the exponential expansion cannot be greater than 6 × 10^?5 of the Planck mass. This rules out all models in which the Hubble constant is of the order of the Planck mass. It is shown that one can satisfy the limits with a model containing a massive scalar field if the mass of the field is less than about 10¹⁴ GeV.