  We investigate the quantum estimation on the Hubble parameter of an expanding
de Sitter space by quantum metrological techniques. By exploring the dynamics
of a freely falling Unruh-DeWitt detector, which interacts with a scalar field
coupling to curvature, we calculate the Fisher information (FI) and quantum
Fisher information (QFI) for the detector, which bound the highest precision of
the estimation on Hubble parameter. In standard Bunch-Davies vacuum, we show
that the maxima of FI/QFI are located for particular initial state of probe.
Beside its dependence on the evolving time of detector and the energy spacing
of atom $\omega$, we show that the maxima of FI/QFI can be significantly
enhanced once a proper coupling of scalar field to curvature is chosen. For
instance, we show numerically that the estimation in the scenario with
minimally/nearly minimally coupling scalar field can always outperform that
with conformally coupling scalar field, corresponding to a higher FI/QFI in
estimation. Moreover, we find that for general $\alpha-$vacua of de Sitter
space, a further improvement of estimation can be achieved, attributed to the
squeezed nature of $\alpha-$vacua that heavily constrains the measurement
uncertainty. Some implications of our results are also discussed.
