Small-scale shear: Peeling off diffuse subhalos with gravitational waves

Subhalos at subgalactic scales ($M\lesssim 10^7 M_\odot$ or $k\gtrsim 10^3 \,{\rm Mpc}^{-1}$) are pristine test beds of dark matter (DM). However, they are too small, diffuse and dark to be visible, in any existing observations. In this paper, we develop a complete formalism for weak and strong diffractive lensing, which can be used to probe such subhalos with chirping gravitational waves (GWs). Also, we show that Navarro-Frenk-White(NFW) subhalos in this mass range can indeed be detected individually, albeit at a rate of ${\cal O}(10)$ or less per year at BBO and others limited by small merger rates and large required SNR $\gtrsim 1/\gamma(r_0) \sim 10^3$. It becomes possible as NFW scale radii $r_0$ are of the right size comparable to the GW Fresnel length $r_F$, and unlike all existing probes, their lensing is more sensitive to lighter subhalos. Remarkably, our formalism further reveals that the frequency dependence of weak lensing (which is actually the detectable effect) is due to shear $\gamma$ at $r_F$. Not only is it consistent with an approximate scaling invariance, but it also offers a new way to measure the mass profile at a successively smaller scale of chirping $r_F \propto f^{-1/2}$. Meanwhile, strong diffraction that produces a blurred Einstein ring has a universal frequency dependence, allowing only detections. These are further demonstrated through semianalytic discussions of power-law profiles. Our developments for a single lens can be generalized and will promote diffractive lensing to a more concrete and promising physics in probing DM and small-scale structures.