  The intergalactic medium (IGM) in the vicinity of galaxy protoclusters are
interesting testbeds to study complex baryonic effects such as gravitational
shocks and feedback. Here, we utilize hydrodynamical simulations from the SIMBA
and The Three Hundred suites to study the mechanisms influencing large-scale
Lyman-$\alpha$ transmission in $2<z<2.5$ protoclusters. We focus on the matter
overdensity-Lyman-$\alpha$ transmission relation $(\delta_m-\delta_F)$ on
Megaparsec-scales in these protoclusters, which is hypothesized to be sensitive
to the feedback implementations. The lower-density regions represented by the
SIMBA-100 cosmological volume trace the power-law $\delta_m-\delta_F$
relationship often known as the fluctuating Gunn-Peterson approximation. This
trend is continued into higher-density regions covered by simulations that
implement stellar feedback only. Simulations with AGN thermal and AGN jet
feedback , however, exhibit progressively more Lyman-$\alpha$ transmission at
fixed matter overdensity. Compared with the 7 protoclusters observed in the
COSMOS field, only 2 display the excess absorption expected from protoclusters.
The others exhibit deviations: 4 show some increased transparency suggested by
AGN X-ray thermal feedback models while the highly transparent COSTCO-I
protocluster appears to reflect intense jet feedback. Discrepancies with the
stellar-feedback-only model suggests processes at play beyond gravitational
heating and/or stellar feedback as the cause of the protocluster
transparencies. Some form of AGN feedback is likely at play in the observed
protoclusters, and possibly long-ranged AGN jets in the case of COSTCO-I. While
more detailed and resolved simulations are required to move forward, our
findings open new avenues for probing AGN feedback at Cosmic Noon.
