Abstract
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 Lyman-$\alpha$ forest transmitted flux at redshifts
$4.2z 5$. 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 $10$ 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 Lyman-$\alpha$ forest power spectrum at $4.1z 5.4$ in a form
that can be easily applied within other parameter inference frameworks. 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 Lyman-$\alpha$
forest data at $z>4$, $0.006 \,km^-1\,sk 0.2 \rm\, km^-1\,s$,
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, $1\sigma$ shifts
between the estimated and true parameters occur.
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