Non-linearities in the Lyman-$\alpha$ forest and in its
  cross-correlation with dark matter halos
  Three-dimensional correlations of the Lyman-$\alpha$ (Ly$\alpha$) forest and
cross correlations between the Ly$\alpha$ forest and quasars have been measured
on large scales, allowing a precise measurement of the baryon acoustic
oscillation (BAO) feature at redshifts $z>2$. These 3D correlations are often
modelled using linear perturbation theory, but full-shape analyses to extract
cosmological information beyond BAO will require more realistic models capable
of describing non-linearities present at smaller scales. We present a
measurement of the Ly$\alpha$ forest flux power spectrum from large
hydrodynamic simulations -- the Sherwood simulations -- and compare it to
different models describing the small-scale deviations from linear theory. We
confirm that the model presented in Arinyo-i-Prats et al. (2015) fits the
measured 3D power up to $k=10\, h\rm{Mpc^{-1}}$ with an accuracy better than
5%, and show that the same model can also describe the 1D correlations with
similar precision. We also present, for the first time, an equivalent study for
the cross-power spectrum of halos with the Ly$\alpha$ forest, and we discuss
different challenges we face when modelling the cross-power spectrum beyond
linear scales. We make all our measured power spectra public in
\url{https://github.com/andreufont/sherwood_p3d}. This study is a step towards
joint analyses of 1D and 3D flux correlations, and towards using the
quasar-Ly$\alpha$ cross-correlation beyond BAO analyses.
