  Most efforts to detect signatures of dynamical dark energy are focused on
late times, $z \lesssim 2$, where the dark energy component begins to dominate
the cosmic energy density. Many theoretical models involving dynamical dark
energy exhibit a 'freezing' equation of state however, where $w \to -1$ at late
times, with a transition to a 'tracking' behaviour at earlier times (with $w
\gg -1$ at sufficiently high redshift). In this paper, we study whether
large-scale structure surveys in the post-reionisation matter-dominated regime,
$2 \lesssim z \lesssim 6$, are sensitive to this behaviour, on the basis that
the dark energy component should remain detectable (despite being strongly
subdominant) in this redshift range given sufficiently precise observations.
Using phenomenological models inspired by parameter space studies of Horndeski
(generalised scalar-tensor) theories, we show how existing CMB and large-scale
structure measurements constrain the dark energy equation of state in the
matter-dominated era, and examine how forthcoming galaxy surveys and 21cm
intensity mapping instruments can improve constraints in this regime. We also
find that the combination of existing CMB and LSS constraints with DESI will
already come close to offering the best possible constraints on $H_0$ using
BAO/galaxy power spectrum measurements, and that either a spectroscopic
follow-up of the LSST galaxy sample (e.g. along the lines of MegaMapper or
SpecTel) or a Stage 2/PUMA-like intensity mapping survey, both at $z \gtrsim
2$, would offer better constraints on the class of dark energy models
considered here than a comparable cosmic variance-limited galaxy survey at $z
\lesssim 1.5$.
