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

Stirring the cosmic pot: how black hole feedback shapes the matter power spectrum in the Fable simulations