  We calculate the lowest-order non-linear contributions to the power spectrum,
two-point correlation function, and smoothed variance of the density field, for
Gaussian initial conditions and scale-free initial power spectra, $P(k) \sim
k^n$. These results extend and in some cases correct previous work in the
literature on cosmological perturbation theory. Comparing with the scaling
behavior observed in N-body simulations, we find that the validity of
non-linear perturbation theory depends strongly on the spectral index $n$. For
$n<-1$, we find excellent agreement over scales where the variance $\sigma^2(R)
\la 10$; however, for $n \geq -1$, perturbation theory predicts deviations from
self-similar scaling (which increase with $n$) not seen in numerical
simulations. This anomalous scaling suggests that the principal assumption
underlying cosmological perturbation theory, that large-scale fields can be
described perturbatively even when fluctuations are highly non-linear on small
scales, breaks down beyond leading order for spectral indices $n \geq -1$. For
$n < -1$, the power spectrum, variance, and correlation function in the scaling
regime can be calculated using dimensional regularization.
