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Studying the smallest self-bound dark matter structure in our Universe canyield important clues about the fundamental particle nature of dark matter.Galaxy-scale strong gravitational lensing provides a unique way to detect andcharacterize dark matter substructures at cosmological distances from the MilkyWay. Within the cold dark matter (CDM) paradigm, the number of low-masssubhalos within lens galaxies is expected to be large, implying that theircontribution to the lensing convergence field is approximately Gaussian andcould thus be described by their power spectrum. We develop here a generalformalism to compute from first principles the substructure convergence powerspectrum for different populations of dark matter subhalos. As an example, weapply our framework to two distinct subhalo populations: a truncatedNavarro-Frenk-White subhalo population motivated by standard CDM, and atruncated cored subhalo population motivated by self-interacting dark matter(SIDM). We study in detail how the subhalo abundance, mass function, internaldensity profile, and concentration affect the amplitude and shape of thesubstructure power spectrum. We determine that the power spectrum is mostlysensitive to a specific combination of the subhalo abundance and moments of themass function, as well as to the average tidal truncation scale of the largestsubhalos included in the analysis. Interestingly, we show that the asymptoticslope of the substructure power spectrum at large wave number reflects theinternal density profile of the subhalos. In particular, the SIDM powerspectrum exhibits a characteristic steepening at large wave number absent inthe CDM power spectrum, opening the possibility of using this observable, if atall measurable, to discern between these two scenarios.

On the Power Spectrum of Dark Matter Substructure in Strong Gravitational Lenses