  We investigate general scalar field potentials \hbox{$V\left(\phi\right)$}
for inflationary cosmology arising from spontaneous symmetry breaking. We find
that potentials which are dominated by terms of order $\phi^m$ with \hbox{$m >
2$} can satisfy observational constraints at an arbitrary symmetry breaking
scale. Of particular interest, the spectral index of density fluctuations is
shown to be independent of the specific form of the potential, depending only
on the order $m$ of the lowest non-vanishing derivative of $V(\phi)$ near its
maximum. The results of a model with a broken ${\rm SO(3)}$ symmetry illustrate
these features.
