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.