  In a recent paper, we suggested that the density fluctuation spectra arising
from power-law (or extended) inflation, which are tilted with respect to the
Harrison--Zel'dovich spectrum, may provide an explanation for the excess large
scale clustering seen in galaxy surveys such as the APM survey. In the light of
the new results from COBE, we examine in detail here cold dark matter
cosmogonies based on inflationary models predicting power-law spectra. Along
with power-law and extended inflation, this class includes natural inflation.
The latter is of interest because, unlike the first two, it produces a
power-law spectrum without significant gravitational wave production. We
examine a range of phenomena, including large angle microwave background
fluctuations, clustering in the galaxy distribution, bulk peculiar velocity
flows, the formation of high redshift quasars and the epoch of structure
formation. Of the three models, only natural inflation seems capable of
explaining the large scale clustering of optical galaxies. Such a model, though
at best marginal even at present, has some advantages over standard CDM and on
most grounds appears to perform at least as well. Power-law inflation's primary
interest may ultimately only be in permitting a larger bias parameter than
standard CDM; it appears unable to explain excess clustering. Most models of
extended inflation are ruled out at a high confidence level.
