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SAGE (Semi-Analytic Galaxy Evolution) is a new model described by Croton et al. (2016), which updates the earlier model of Croton et al. (2006).  SAGE revamps many prescriptions for the treatment of baryons, including the suppression of cooling within halos from active galactic nucleus (AGN) feedback, reincorporation of ejected gas, and the stripping of gas from satellite systems.  While semi-analytic models have historically been designed and calibrated for a single N-body simulation, SAGE is designed to be run on any simulation, so long as the merger trees are provided in an appropriate format.  The code is publicly available on GitHub and is on the Astrophysics Source Code Library.  Additional details and results concerning the model can be found in the thesis of Stevens (2016).  Currently on TAO, SAGE galaxies are attached to the following N-body simulations:

Bolshoi (Planck)

Model physics

Gas infall

As haloes grow in SAGE, their baryonic content is topped up by ensuring the cosmic baryon fraction is respected.  For smaller haloes and at higher redshift, the effective baryon fraction is reduced to account for photoionization heating.  When haloes acquire new baryons, they are added to the `hot gas' component and are assumed to be pristine (metal-free).  Strictly speaking, this reservoir represents the gas in the circumgalactic medium, which does not necessarily need to be hot.

Gas cooling and accretion

The hot gas in haloes is assumed to be at the virial temperature and follow the density profile of a singular isothermal sphere.  A `cooling radius' is identified in each halo, where the cooling time-scale of the gas matches the dynamical time of the halo.  If the cooling radius is within the virial radius, then the halo is in the `hot mode' of accretion, where the mass deposition rate onto the galaxy is taken as the mass flux rate through the cooling radius.  Otherwise, the halo is in the `cold mode' of accretion, where the hot gas mass is assumed to be deposited onto the galaxy linearly over the dynamical time.  These gross cooling rates of haloes are then modulated by radio mode feedback (see below).

Star formation

Star formation takes place in the cold gas disks of SAGE galaxies provided the average surface density of the disk gas is above a threshold.  The disk is always assumed to have an exponentially declining surface density profile. The scale radius of the disk is fixed by assuming the disk's rotational velocity meets the virial velocity and that the specific angular momentum of the disk matches that of the halo.  The passive star formation rate of a galaxy is then taken as the mass of cold gas above the threshold, divided by the dynamical time of the disk, multiplied by the star formation efficiency parameter.  Starbursts are also induced in the model as a result of disk instabilities and mergers.

To model the effects of stellar mass loss, the instantaneous recycling approximation is used to return a fraction (43%) of the star-forming gas mass immediately back to the cold gas reservoir.  Stellar feedback is modelled by assuming that cold gas will also be reheated out of the disk at a rate directly proportional to the star formation rate.  If the energy budget from supernovae following a star formation episode is greater than the energy required to reheat the prescribed amount of cold gas, the remaining energy unbinds hot gas from the halo, placing it in the `ejected mass' reservoir.

Active galactic nuclei

Each galaxy in SAGE is assumed to carry a black hole, which can grow to become supermassive.  Black holes grow in SAGE through two accretion channels, which each result in a different mode of feedback, and are instantly merged when their host galaxies merge.  

In the quasar mode, cold gas is accreted as a result of a merger or disk instability.  If the energy released by this accreted gas exceeds the binding energy of the remaining cold gas in the galaxy, then it is all shifted to the ejected reservoir.  If there is further energy left to unbind the hot gas, then that will occur as well.  

In the radio mode, black holes quiescently accrete gas from the hot reservoir.  In doing so, they release energy which is used to directly suppress cooling rates.  In addition, a `heating radius' is established, which gives the radius whose internal mass equals the mass of gas that was prevented from cooling.  The radius is maintained for a halo and can only ever increase, should radio mode become more efficient.  Radio mode feedback thus has a long-lasting effect.  The ratio of the heating radius to the cooling radius at any time describes the fraction of gas that is suppressed from cooling due to radio mode feedback.

Satellite galaxies

Satellite galaxies (i.e. those associated with a non-primary subhalo) are subject to cooling and other internal galaxy processes like any galaxy in SAGE.  Typically, subhaloes lose mass as they fall into larger haloes.  Hot gas is stripped from satellites at a rate proportional to the dark matter when this happens. Additionally, any reheated material from feedback is deposited in the central galaxy's hot/ejected reservoir.

SAGE does not have orphan galaxies.  If a subhalo disappears from the merger tree, which will happen when the subhalo drops below the minimum number of particles (e.g. 20 for Millennium, but Bolshoi has 2-particle subhaloes), the time of its existence as a subhalo is compared against the dynamical friction time-scale of the galaxy within the halo.  If it has lived beyond this, the satellite is instantly merged with the central.  If it hasn't lived this long, the satellite will be disrupted, where its stars go to the `intracluster stars' component of the central, and all its gas goes to the central's hot gas reservoir.


SAGE was calibrated to reproduce several statistical features and scaling relations of galaxies at z=0, including the stellar mass function, the black hole–bulge mass relation, the stellar mass–gas metallicity relation, and the Baryonic Tully–Fisher relation.  The model was also loosely constrained by the Madau–Lilly diagram.  SAGE was calibrated such that a single parameter set worked well for each of the Millennium, Bolshoi, and GiggleZ-MR simulations, although the greatest emphasis was placed on Millennium.  For some simulations, the baryon fraction was modified to ensure a realistic galaxy population at z=0.  For example, Millennium and GiggleZ-MR used fb = 0.17, whereas Bolshoi used fb = 0.13, and Vishnu used fb = 0.15.  In the case of MultiDark, we used the true baryon fraction from the Planck cosmology, but increased the star formation efficiency parameter from 0.05 to 0.07.  These small changes help to combat effects of resolution and merger tree construction.  See Croton et al. (2016) for full details.


Stellar mass function for SAGE galaxies for each of the Millennium, Bolshoi, and GiggleZ-MR simulations at z=0.  These were calibrated to match the observed mass function of Baldry et al. (2008).  Compared is the mass function from the Croton et al. (2006) model as run on Millennium.
Credit: Fig. 1 of Croton et al. (2016).

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