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We present a new modeling of the X-ray luminosity function (XLF) of ActiveGalactic Nuclei (AGN) out to z$\sim$3, dissecting the contribution ofmain-sequence (MS) and starburst (SB) galaxies. For each galaxy population, weconvolved the observed galaxy stellar mass (M$_{\star}$) function with a gridof M$_{\star}$-independent Eddington ratio ($\lambda_{\rm EDD}$) distributions,normalised via empirical black hole accretion rate (BHAR) to star formationrate (SFR) relations. Our simple approach yields an excellent agreement withthe observed XLF since z$\sim$3. We find that the redshift evolution of theobserved XLF can only be reproduced through an intrinsic flattening of the$\lambda_{\rm EDD}$ distribution, and with a positive shift of the break$\lambda^{*}$, consistent with an anti-hierarchical behavior. The AGN accretionhistory is predominantly made by massive (10$^{10}<$M$_{\star}<$10$^{11}$M$_{\odot}$) MS galaxies, while SB-driven BH accretion, possibly associatedwith galaxy mergers, becomes dominant only in bright quasars, at $\log$(L$_{\rmX}$/erg s$^{-1}$)$>$44.36 + 1.28$\cdot$(1+z). We infer that the probability offinding highly-accreting ($\lambda_{\rm EDD}>$ 10%) AGN significantly increaseswith redshift, from 0.4% (3.0%) at z=0.5 to 6.5% (15.3%) at z=3 for MS (SB)galaxies, implying a longer AGN duty cycle in the early Universe. Our resultsstrongly favor a M$_{\star}$-dependent ratio between BHAR and SFR, as BHAR/SFR$\propto$ M$_{\star}^{0.73[+0.22,-0.29]}$, supporting a non-linear BH builduprelative to the host. Finally, this framework opens potential questions onsuper-Eddington BH accretion and different $\lambda_{\rm EDD}$ prescriptionsfor understanding the cosmic BH mass assembly.

The evolving AGN duty cycle in galaxies since z$\sim$3 as encoded in the X-ray luminosity function