Probing Dark Low-mass Halos and Primordial Black Holes with
  Frequency-dependent Gravitational Lensing Dispersions of Gravitational Waves
  We explore the possibility of using amplitude and phase fluctuations of
gravitational waves due to gravitational lensing as a probe of the small-scale
matter power spectrum. The direct measurement of the small-scale matter power
spectrum is made possible by making use of the frequency dependence of such
gravitational lensing dispersions originating from the wave optics nature of
the propagation of gravitational waves. We first study the small-scale behavior
of the matter power spectrum in detail taking the so-called halo model approach
including effects of baryons and subhalos. We find that the matter power
spectrum at the wavenumber $k\sim 10^6h{\rm Mpc}^{-1}$ is mainly determined by
the abundance of dark low-mass halos with mass $1h^{-1}M_\odot \lesssim M
\lesssim 10^4h^{-1}M_\odot$ and is relatively insensitive to baryonic effects.
The matter power spectrum at this wavenumber is probed by gravitational lensing
dispersions of gravitational waves at frequencies of $f\sim 0.1-1$~Hz with
predicted signals of $\mathcal{O}(10^{-3})$. We also find that primordial black
holes (PBHs) with $M_{\rm PBH} \gtrsim 0.1~M_\odot$ can significantly enhance
the matter power spectrum at $k \gtrsim 10^5h{\rm Mpc}^{-1}$ due to both the
enhanced halo formation and the shot noise from PBHs. We find that
gravitational lensing dispersions at $f\sim 10-100$~Hz are particularly
sensitive to PBHs and can be enhanced by more than an order of magnitude
depending on the mass and abundance of PBHs.
