  We show that charged-particles decaying in the early Universe can induce a
scale-dependent or `running' spectral index in the small-scale linear and
nonlinear matter power spectrum and discuss examples of this effect in minimal
supersymmetric models in which the lightest neutralino is a viable
cold-dark-matter candidate. We find configurations in which the neutralino
relic density is set by coannihilations with a long-lived stau, and the late
decay of staus partially suppresses the linear matter power spectrum. Nonlinear
evolution on small scales then causes the modified linear power spectrum to
evolve to a nonlinear power spectrum similar (but different in detail) to
models parametrized by a constant running $\alpha_{s}=d n_{s}/d{\rm ln} k$ by
redshifts of 2 to 4. Thus, Lyman-$\alpha$ forest observations, which probe the
matter power spectrum at these redshifts, might not discriminate between the
two effects. However, a measurement of the angular power spectrum of primordial
21-cm radiation from redshift $z \approx 30$--$200$ might distinguish between
this charged-decay model and a primordial running spectral index. The direct
production of a long-lived charged particle at future colliders is a dramatic
prediction of this model.
