  We introduce simplified models for enhancements in the matter power spectrum
at small scales and study their implications for dark matter substructure and
gravitational observables. These models capture the salient aspects of a
variety of early universe scenarios that predict enhanced small-scale
structure, such as axion-like particle dark matter, light vector dark matter,
and epochs of early matter domination. We use a model-independent,
semi-analytic treatment to map bumps in the matter power spectrum to
early-forming sub-solar mass dark matter halos and estimate their evolution,
disruption, and contribution to substructure of clusters and galaxies at late
times. We discuss the sensitivity of gravitational observables, including
pulsar timing arrays and caustic microlensing, to both the presence of bumps in
the power spectrum and variations in their basic properties.
