Constraining dark energy using the cross correlations of weak lensing
  with post-reionization probes of neutral hydrogen
  We investigate the prospects of detecting the cross correlation of CMBR
weak-lensing convergence field with the large scale tracers of the underlying
dark matter distribution in the post-reionization epoch. The cross-correlation
is then used to make error projections for dark energy equation of state
(EoS)for models with a time evolving dark energy. We study the
cross-correlation angular power spectrum of the weak-lensing field with the
Lyman-$\alpha$ forest and the redshifted HI 21 cm signal from the post
reionization epoch. The angular power spectra is expressed as a line of sight
average over the tomographic slices. We find that on using multiple $400$ hrs
observation with an extended uGMRT like instrument or with a BOSS like survey
with quasar (QSO) density of $16 {\rm deg}^{-2}$ the cross-correlation with
weak-lensing convergence field covering half the sky can be detected at a very
high SNR ($>20$). The cross-correlation of weak-lensing with Lyman-$\alpha$
forest allows the $1-\sigma$ errors on the dark energy EoS parameters for
different parametrizations to be constrained at a level of precision comparable
to combined Planck+SNIa+BAO+HST projections. The 21-cm weak-lensing
cross-correlation is found to provide strong constraints on the present value
of the dark energy EoS parameter at $4\%$ for the 7CPL model. The constraints
on $w_a$ is comparable ($\sim 12\%$) for models other than the 7CPL model. We
also find that the CPL parametrization may not be the best constrained
parametrization for dark energy evolution. The cross-correlation of CMBR
weak-lensing with the post-reionization probes of neutral hydrogen thus holds
the potential to give us valuable understanding about the nature of evolving
dark energy.
