  If an adiabatic density perturbation is responsible for large scale structure
and the cmb anisotropy, its spectral index $n$ will be measured in the
forseeable future with an accuracy $\Delta n\sim .01$. This is precisely the
kind of accuracy required to distinguish between many models of inflation. Most
of them have an inflationary potential $V\simeq V_0(1\pm\mu\phi^p)$ with the
constant term dominating. Except for $0\lsim p\lsim 2$, the prediction is
$n=1\pm (2/N)(p-1)/(p-2)$, where $N$ is the number of $e$-folds of inflation
after cosmological scales leave the horizon. It typically lies in the range
$0.9\lsim n\lsim 1.1$. For $p=2$ one has $n=1\pm 2\Mpl^2 \mu$ where $\Mpl=(8\pi
G)^{-1/2}$. A generic supergravity theory gives contributions of order $\pm
H^2$ to the inflaton mass-squared $m^2$ whereas inflation requires $|m^2| <<
H^2$. Proposals for keeping $m^2$ small are surveyed with emphasis on their
implications for $n$. Finally, the case of a multi-component inflaton (as in
`double' inflation) is discussed. In all cases, the best method of calculating
the spectrum of the adiabatic density perturbation starts with the assumption
that, after smoothing on a super-horizon scale, the evolution of the universe
along each comoving worldline will be practically the same as in an unperturbed
universe with the same initial inflaton field. The possible isocurvature
density perturbation is briefly discussed, with a simple derivation of the fact
that the low multipoles of the cmb anisotropy are six times as big as for an
adiabatic density perturbation.
