  Understanding the impact of baryonic physics on cosmic structure formation is
crucial for accurate cosmological predictions, especially as we usher in the
era of large galaxy surveys with the Rubin Observatory as well as the Euclid
and Roman Space Telescopes. A key process that can redistribute matter across a
large range of scales is feedback from accreting supermassive black holes. How
exactly these active galactic nuclei (AGN) operate from sub-parsec to
Mega-parsec scales however remains largely unknown. To understand this, we
investigate how different AGN feedback models in the Fable simulation suite
affect the cosmic evolution of the matter power spectrum (MPS).
  Our analysis reveals that AGN feedback significantly suppresses clustering at
scales $k \sim 10\,h\,cMpc^{-1}$, with the strongest effect at redshift $z = 0$
causing a reduction of $\sim 10\%$ with respect to the dark matter-only
simulation. This is due to the efficient feedback in both radio (low Eddington
ratio) and quasar (high Eddington ratio) modes in our fiducial Fable model. We
find that variations of the quasar and radio mode feedback with respect to the
fiducial Fable model have distinct effects on the MPS redshift evolution, with
the radio mode being more effective on larger scales and later epochs.
Furthermore, MPS suppression is dominated by AGN feedback effects inside haloes
at $z = 0$, while for $z \gtrsim 1$ the matter distribution both inside and
outside of haloes shapes the MPS suppression. Hence, future observations
probing earlier cosmic times beyond $z \sim 1$ will be instrumental in
constraining the nature of AGN feedback.
