The first objects to arise in a cold dark matter universe present a daunting challenge for models of structure formation . In the ultra small-scale limit , CDM structures form nearly simultaneously across a wide range of scales . Hierarchical clustering no longer provides a guiding principle for theoretical analyses and the computation time required to carry out credible simulations becomes prohibitively high . To gain insight into this problem , we perform high-resolution ( N = 720 ^ { 3 } -1584 ^ { 3 } ) simulations of an Einstein-de Sitter cosmology where the initial power spectrum is P ( k ) \propto k ^ { n } , with -2.5 \leq n \leq - 1 . Self-similar scaling is established for n = -1 and n = -2 more convincingly than in previous , lower-resolution simulations and for the first time , self-similar scaling is established for an n = -2.25 simulation . However , finite box-size effects induce departures from self-similar scaling in our n = -2.5 simulation . We compare our results with the predictions for the power spectrum from ( one-loop ) perturbation theory and demonstrate that the renormalization group approach suggested by McDonald ( 20 ) improves perturbation theory ’ s ability to predict the power spectrum in the quasilinear regime . In the nonlinear regime , our power spectra differ significantly from the widely used fitting formulae of Peacock & Dodds ( 26 ) and Smith et al . ( 36 ) and a new fitting formula is presented . Implications of our results for the stable clustering hypothesis vs. halo model debate are discussed . Our power spectra are inconsistent with predictions of the stable clustering hypothesis in the high- k limit and lend credence to the halo model . Nevertheless , the fitting formula advocated in this paper is purely empirical and not derived from a specific formulation of the halo model .