  It has been recently shown that the astrophysics of reionization can be
extracted from the Ly$\alpha$ forest power spectrum by marginalizing the memory
of reionization over cosmological information. This impact of cosmic
reionization on the Ly$\alpha$ forest power spectrum can survive cosmological
time scales because cosmic reionization, which is inhomogeneous, and subsequent
shocks from denser regions can heat the gas in low-density regions to $\sim
3\times10^4$ K and compress it to mean-density. Current approach of
marginalization over the memory of reionization, however, is not only
model-dependent, based on the assumption of a specific reionization model, but
also computationally expensive. Here we propose a simple analytical template
for the impact of cosmic reionization, thereby treating it as a broadband
systematic to be marginalized over for Bayesian inference of cosmological
information from the Ly$\alpha$ forest in a model-independent manner. This
template performs remarkably well with an error of $\leq 6 \%$ at large scales
$k \approx 0.19$ Mpc$^{-1}$ where the effect of the memory of reionization is
important, and reproduces the broadband effect of the memory of reionization in
the Ly$\alpha$ forest correlation function, as well as the expected bias of
cosmological parameters due to this systematic. The template can successfully
recover the morphology of forecast errors in cosmological parameter space as
expected when assuming a specific reionization model for marginalization
purposes, with a slight overestimation of tens of per cent for the forecast
errors on the cosmological parameters. We further propose a similar template
for this systematic on the Ly$\alpha$ forest 1D power spectrum.
