  For future surveys of fast radio bursts (FRBs), we clarify information
available from cosmic dispersion measures (DMs) through cross-correlation
analyses of foreground dark matter haloes (hosting galaxies and galaxy
clusters) with their known redshifts. With a halo-model approach, we predict
that the cross-correlation with cluster-sized haloes is less affected by the
details of gastrophysics, providing robust cosmological information. For less
massive haloes, the cross-correlation at angular scales of $<10\,
\mathrm{arcmin}$ is sensitive to gas expelled from the halo centre due to
galactic feedback. Assuming $20000$ FRBs over $20000 \, {\rm deg}^2$ with a
localisation error being 3 arcmin, we expect that the cross-correlation signal
at halo masses of $10^{12}-10^{14}\, M_\odot$ can be measured with a level of
$\sim 1\%$ precision in a redshift range of $0<z<1$. Such precise measurements
enable one to put a 1.5\% level constraint on $\sigma_8\,
(\Omega_\mathrm{M}/0.3)^{0.5}$ and a 3\% level constraint on
$(\Omega_\mathrm{b}/0.049)(h/0.67)(f_\mathrm{e}/0.95)$ ($\sigma_8$,
$\Omega_\mathrm{M}$, $\Omega_\mathrm{b}$, $h$ and $f_\mathrm{e}$ are the linear
mass variance smoothed at $8\, h^{-1}\mathrm{Mpc}$, mean mass density, mean
baryon density, the present-day Hubble parameter and fraction of free electrons
in cosmic baryons today), whereas the gas-to-halo mass relation in galaxies and
clusters can be constrained with a level of $10\%-20\%$. Furthermore the
cross-correlation analyses can break the degeneracy among $\Omega_\mathrm{b}$,
$h$ and $f_\mathrm{e}$, inherent in the DM-redshift relation. Our proposal
opens new possibilities for FRB cosmology, while it requires extensive galaxy
redshift catalogues and further improvement of the halo model.
