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In order to relate the observed evolution of the galaxy stellar mass function and the luminosity function of active galactic nuclei (AGN), we explore a co-evolution scenario in which AGN are associated only with the very last phases of the star-forming life of a galaxy. We derive analytically the connections between the parameters of the observed quasar luminosity functions and galaxy mass functions. The $(m_{rm bh}/m_{*})_{Qing}$ associated with quenching is given by the ratio of the global black hole accretion rate density (BHARD) and star-formation rate density (SFRD) at the epoch in question. Observational data on the SFRD and BHARD suggests $(m_{rm bh}/m_{*})_{Qing} propto (1+z)^{1.5}$ below redshift 2. This evolution reproduces the observed mass-luminosity plane of SDSS quasars, and also reproduces the local $m_{rm bh}/m_{*}$ relation in passive galaxies. The characteristic Eddington ratio, $lambda^*$, is derived from both the BHARD/SFRD ratio and the evolving $L^*$ of the AGN population. This increases up to $z sim 2$ as $lambda^* propto (1+z)^{2.5}$ but at higher redshifts, $lambda^*$ stabilizes at the physically interesting Eddington limit, $lambda^* sim 1$. The new model may be thought of as an opposite extreme to our earlier co-evolution scenario in Caplar et al. 2015. The main observable difference between the two co-evolution scenarios, presented here and in Caplar et al. 2015, is in the active fraction of low mass star-forming galaxies. We compare the predictions with the data from deep multi-wavelength surveys and find that the quenching scenario developed in the current paper is much to be preferred.
We explore the connections between the evolving galaxy and AGN populations. We present a simple phenomenological model that links the evolving galaxy mass function and the evolving quasar luminosity function, which makes specific and testable predict
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