  We generalize previously derived analytic results for the one-loop power
spectrum (PS) in scale-free models (with linear PS $P(k) \propto k^n$) to a
broader class of such models in which part of the matterlike component driving
the Einstein de Sitter expansion does not cluster. These models can be
conveniently parametrized by $\alpha$, the constant logarithmic linear growth
rate of fluctuations (with $\alpha=1$ in the usual case). For $-3< n<-1$, where
the one-loop PS is both infrared and ultraviolet convergent and thus explicitly
self-similar, it is characterized conveniently by a single numerical
coefficient $c(n, \alpha)$. We compare the analytical predictions for $c(n=-2,
\alpha)$ with results from a suite of $N$-body simulations with $\alpha \in
[0.25, 1]$ performed with an appropriately modified version of the GADGET code.
Although the simulations are of small ($256^3$) boxes, the constraint of
self-similarity allows the identification of the converged PS at a level of
accuracy sufficient to test the analytical predictions for the $\alpha$
dependence of the evolved PS. Good agreement for the predicted dependence on
$\alpha$ of the PS is found. To treat the UV sensitivity of results which grows
as one approaches $n =-1$, we derive exact results incorporating a
regularization $k_c$ and obtain expressions for $c(n, \alpha, k_c/k)$. Assuming
that this regularization is compatible with self-similarity allows us to infer
a predicted functional form of the PS equivalent to that derived in effective
field theory (EFT). The coefficient of the leading EFT correction at one loop
has a strong dependence on $\alpha$, with a change in sign at $\alpha \approx
0.16$, providing a potentially stringent test of EFT.
