  In this work, we explore the application of intensity mapping to detect
extended Ly$\alpha$ emission from the IGM via cross-correlation of PAUS images
with Ly$\alpha$ forest data from eBOSS and DESI. Seven narrow-band (FWHM=13nm)
PAUS filters have been considered, ranging from 455 to 515 nm in steps of 10
nm, which allows the observation of Ly$\alpha$ emission in a range $2.7<z<3.3$.
The cross-correlation is simulated first in an area of 100 deg$^2$ (PAUS
projected coverage), and second in two hypothetical scenarios: a deeper PAUS
(complete up to $i_{\rm AB}<24$ instead of $i_{\rm AB}<23$, observation time
x6), and an extended PAUS coverage of 225 deg$^2$ (observation time x2.25). A
hydrodynamic simulation of size 400 Mpc/h is used to simulate both extended
Ly$\alpha$ emission and absorption, while the foregrounds in PAUS images have
been simulated using a lightcone mock catalogue. Using an optimistic estimation
of uncorrelated PAUS noise, the total probability of a non-spurious detection
is estimated to be 1.8\% and 4.5\% for PAUS-eBOSS and PAUS-DESI , from a run of
1000 simulated cross-correlations with different realisations of instrumental
noise and quasar positions. The hypothetical PAUS scenarios increase this
probability to 15.3\% (deeper PAUS) and 9.0\% (extended PAUS). With realistic
correlated noise directly measured from PAUS images, these probabilities become
negligible. Despite these negative results, some evidences suggest that this
methodology may be more suitable to broad-band surveys.
