Forecasts for WEAVE-QSO: 3D clustering and connectivity of critical
  points with Lyman-$\alpha$ tomography
  The upcoming WEAVE-QSO survey will target a high density of quasars over a
large area, enabling the reconstruction of the 3D density field through
Lyman-$\alpha$ tomography over unprecedented volumes smoothed on intermediate
scales ($\approx$ 16 Mpc/$h$). We produce mocks of the Lyman-$\alpha$ forest
using LyMAS, and reconstruct the 3D density field between sightlines through
Wiener filtering in a configuration compatible with the future WEAVE-QSO
observations. The fidelity of the reconstruction is assessed by measuring one-
and two-point statistics from the distribution of critical points in the cosmic
web. In addition, initial Lagrangian statistics are predicted from first
principles, and measurements of the connectivity of the cosmic web are
performed. The reconstruction captures well the expected features in the auto-
and cross-correlations of the critical points. This remains true after a
realistic noise is added to the synthetic spectra, even though sparsity of
sightlines introduces systematics, especially in the cross-correlations of
points with mixed signature. Specifically, for walls and filaments, the most
striking clustering features could be measured with up to 4 sigma of
significance with a WEAVE-QSO-like survey. Moreover, the connectivity of each
peak identified in the reconstructed field is globally consistent with its
counterpart in the original field, indicating that the reconstruction preserves
the geometry of the density field not only statistically, but also locally.
Hence the critical points relative positions within the tomographic
reconstruction could be used as standard rulers for dark energy by WEAVE-QSO
and similar surveys.
