The effect of inhomogeneous reionisation on the Lyman-$\alpha$ forest
  power spectrum at redshift $z>4$: implications for thermal parameter recovery
  We use the Sherwood-Relics suite of hybrid hydrodynamical and radiative
transfer simulations to model the effect of inhomogeneous reionisation on the
1D power spectrum of the \Lya forest transmitted flux at redshifts $4.2\leq z
\leq 5$. Relative to models that assume a homogeneous UV background,
reionisation suppresses the power spectrum at small scales, $k \sim
0.1\rm\,km^{-1}\,s$ by $\sim 10$ per cent because of spatial variations in the
thermal broadening kernel and the divergent peculiar velocity field associated
with over-pressurised intergalactic gas. On larger scales,
$k<0.03\rm\,km^{-1}\,s$, the power spectrum is instead enhanced by $10$--$50$
per cent by large scale spatial variations in the neutral hydrogen fraction.
The effect of inhomogeneous reionisation must therefore be accounted for in
analyses of forthcoming high precision measurements. We provide a correction
for the Ly$\alpha$ forest power spectrum at $4.1\leq z \leq 5.4$ that can be
easily applied within other parameter inference frameworks using similar
reionisation models. We perform a Bayesian analysis of mock data to assess the
extent of systematic biases that may arise in measurements of the intergalactic
medium if ignoring this correction. At the scales probed by current high
resolution Ly$\alpha$ forest data at $z>4$, $0.006 \rm \,km^{-1}\,s\leq k \leq
0.2 \rm\, km^{-1}\,s$, we find inhomogeneous reionisation does not introduce
any significant bias in thermal parameter recovery for the current measurement
uncertainties of $\sim 10$ per cent. However, for $5$ per cent uncertainties,
$\sim 1\sigma$ shifts between the estimated and true parameters occur.
