Top: mass ratio, M_hydro/M_Nbody versus redshift from the ADIAB simulation. The halo mass is defined as the total mass of particles included in each fof halo. Bottom: baryon fraction, f_b as a function of redshift from the ADIAB simulation. The baryon fraction is defined as the mass ratio of the baryonic particles to all particles linked by the fof halo finder. The result using top hat properties (see the top left panels of Figs 1 and 3) is also shown here with dashed lines for comparison. In this work, we extract structures using a fof halo finder [Springel2001] and adopt a spherical top-hat profile to calculate mass. To check whether this introduces any bias, in Fig. A1, we plot instead the mass ratio, M_hydro/M_Nbody between the ADIAB and N-body simulations using the fof halo mass. This is the total mass of particles in each fof halo, modified to correct for the bias introduced by the fof halo finder as described in [Watson2013]. We find that the fof mass ratio evolves similarly to the top-hat mass ratio, supporting the conclusion that the suppressed growth of dark matter haloes is indeed due to the inclusion of baryons.


Top panel: the average normalized cumulative mass versus radius in the unit of virial radius. The radial profiles of dark matter, gas and total components from the ADIAB simulation are shown with black dash-dotted, dashed and solid lines, respectively for comparison with the N-body simulation shown with red solid line. Bottom panel: the ratio of the total components from ADIAB and N-body simulations shown in the top panel. The vertical and horizontal dotted lines indicate the radius and the value of ratio where R = R_vir. However, the offset between the fof mass and the top-hat mass cases suggests that a fixed overdensity cut (approximately 18*pi^2) may unfairly miss more mass at large radius in the simulations with baryons. Therefore, we plot the average radial profile for haloes with masses higher than 10^7.5 solar masses at z=5. In the top panel of Fig. A2, the average normalized cumulative mass (normalized by the fof halo mass) is shown as a function of radius in units of virial radius (defined as the top-hat radius) for the N-body simulation and the dark matter, gas and total components from the ADIAB simulation, respectively.


We see that the radial profile becomes flat at R is approximately 1.5*R_vir for each component, indicating that the fof halo finder includes the majority of the particles belonging to each halo. However, at R = R_vir, the ratio of mass between the ADIAB and N-body simulations (the bottom panel of Fig. A2) is approximately 0.96, which results in the offset between the mass ratios shown in Fig. A1 (at z=5). Additionally, baryon fractions calculated using all particles linked by the fof halo finder are shown in the bottom panel of Fig. A1, compared to the calculation with particles within the top-hat radius. Although for larger objects, the fraction of baryons within the entire fof halo is slightly smaller than within the virial radius, using fof properties generally shows larger baryon fractions for haloes less massive than 10^9 solar masses, indicating that the concentration of baryons is smaller than the collisionless component in those haloes due to the hydrostatic pressure. Moreover, while the growth rate of baryons is close to the dark matter component within the virial radius (see Section 2.3), baryon fractions within the entire fof halo decreases with time suggesting a slower growth of baryons compared to dark matter, and a baryon flow towards the inner region of haloes.


The results presented in this work are based on a suite of simulations, each of which has n x 512^3 (n=1 for DMONLY and n=2 for the baryon-included simulations) particles within a cube of comoving side 10 h^-1 Mpc. We present convergence tests of our results (mass ratio, baryon fraction and halo mass function) in Fig. B1 with simulations having the same volume, but different numbers of particles (N_part = n x 256^3 and 128^3). The left panels of Fig. B1 show the evolution of mass ratio and of baryon fraction for haloes with masses in the range of 10^9 to 10^10 solar masses, while the right panel shows the halo mass functions from ADIAB and N-body simulations. We see that the simulations are not formally converged in mass ratio for masses less than or equal to 10^9.5 solar masses. However, the baryon fraction and halo mass function show better convergence. In particular, the baryon fraction of massive objects is converged to 90 per cent as shown in the bottom left panel of Fig. B1. This is in agreement with [Crain2007], who studied the baryon fraction in a suite of non-radiative gas-dynamical simulations with different resolutions and also found that the baryon fraction of dark matter haloes can only reach 90 per cent of the cosmic mean with a decreasing value towards low mass end.


Resolution test on the mass ratio (top left), the baryon fraction (bottom left) and the halo mass function (right). On the left panels, results for halo mass around 10^9 to 10^10 solar masses are shown with different colours. On the right panel, mass functions from N-body and ADIAB simulations are indicated with different colours as well. Solid, dashed and dash-dotted lines illustrate simulations with 512^3, 256^3 and 128^3 particles within a cube of comoving side 10h^-1 Mpc, respectively. The convergence of the baryon fraction gives us confidence in our conclusions regarding mass ratio. As pointed out by [Schaye2014], due to the sub-grid baryonic physics applied, hydrodynamic simulations show different levels of the hydrostatic suppression. In particular, with decreasing resolutions, mass ratios and baryon fractions become smaller at a given redshift. Additionally, resolution also has an impact on the growth rate of baryons. While baryon fractions slightly increase in the simulation with the highest resolution, they decrease in the simulations with lower resolutions in the studied mass range. The rate of decrease increases if resolution is reduced. In addition, better convergence can be observed within larger objects. Since the offset results from the sub-grid physics of baryons, we also expect differences in the mass ratio and baryon fraction in simulations with more complex physics when different resolutions are adopted (see Section 2.2.1).


Although the qualitative conclusions presented in this paper do not change, the weak convergence shown in Fig. B1 indicates that one might need to pay attention to specific resolution requirements when calculating quantitative modifications of halo masses and baryon fractions to be used for running SAMs on halo merger trees constructed from collisionless simulations.


The hydrodynamic simulations indicate that baryons have a significant impact on halo growth for the galaxies thought to drive reionization, owing to pressure gradients that impede the growth of the gravitational potential well [Sawala2013,Schaller2014,Bocquet2016]. Motivated by these results, we create a simple analytic model that describes the growth of matter fluctuations and illustrates the difference of halo growth due to the inclusion of hydrostatic pressure of the baryons. In a collisionless universe, the linear overdensity (delta_t' = (rho/rho_crit)-1, where rho_crit is the critical background density) evolution in Fourier k-space is described by Equation 5 [Pace2010], where the expansion factor, a, is the independent variable. In the LambdaCDM model for redshifts of interest to this study, E(a) = sqrt(Omega_m*a^-3 + Omega_Lambda). We see that delta_t' increases with time, and only depends on the cosmological parameters and initial conditions. Moreover, in the absence of baryons, the evolution of the overdensity is independent of scale, owing to the lack of explicit dependence on the spatial wavenumber, k.


In order to evaluate the co-evolution of baryonic and dark matter density fields, we write delta_dm and delta_b as the dark matter and baryonic overdensities, respectively, and set rho = a^-3 * (rho_dm0*(1+delta_dm) + rho_b0*(1+delta_b)), where rho_dm0 = rho_crit*(1-(Omega_b/Omega_m)) and rho_b0 = rho_crit*(Omega_b/Omega_m) are the current (i.e. a=1) background densities for dark matter and baryons, respectively. With these definitions, the equations for dark matter and baryonic overdensities are, respectively given by Equation 6 (e.g. [Barkana2000]), where k_b, mu=0.59, m_p and H_0 are, respectively the Boltzmann constant, the mean molecular weight of ionized primordial gas in atomic units, the mass of the proton and the current Hubble constant. The quantity T is the temperature of gas as a function of the expansion factor. We assume a uniform temperature for all gas in the universe and adopt two piecewise functions shown in the top left panel of Fig. C1.


Prior to z of approximately 200, the gas temperature is coupled to the cosmic microwave background (CMB) through Compton scattering and decreases as (1+z). At later times (30 less than or equal to z less than or equal to 200), the gas has decoupled from the CMB and cools adiabatically, with temperature decreasing as (1+z)^2. When the first stars form and heat their environment (the heating is likely through X-rays), the gas temperature rises as approximately (1+z)^-4.9 and finally reaches 10^4 K following reionization at z of approximately 7 [Pritchard2008]. We note that the temperature evolution parametrized from the work of [Pritchard2008] is illustrative of the gas evolution. However, there is significant uncertainty regarding the role of X-ray sources in the pre-reionization epoch (e.g. intermediate-mass black holes, [Madau2004]; high-mass X-ray binaries, [Mineo2012]). This introduces uncertainties in the scaling of the temperature with redshift, since it depends on the evolution of the X-ray emissivity [Mesinger2014].


In order to make a direct comparison with simulations, we incorporate the IGM temperature from the ADIAB simulation, in which gas only cools adiabatically. However, due to shock-heating from structure growth in the simulation, the temperature evolution does not strictly follow (1+z). Therefore, for z greater than or equal to 60, we adopt the [Pritchard2008] model for the analytic calculation. The dark matter and baryonic overdensities in Equation (6) are coupled, and there is a mass dependence in the pressure term in Equation (6). The mass of a collapsed halo is related to the scale, k, following Equation 7, where lambda is the size of the density fluctuation and k=2*pi/lambda. This makes the pressure term in Equation (6) smaller at high masses, so that the hydrostatic suppression of baryons on halo growth becomes weaker. Below, we use these equations to illustrate the effect of the pressure term on the evolution of dark matter and baryonic overdensities.


In order to solve Equations (5) and (6), we require initial conditions for delta_t', delta_dm, delta_b and their first-order derivatives in the linear regime. In the LambdaCDM model, the linear critical overdensity for a dark matter halo collapsing at redshift z_col is delta_c = 3/20*(12*pi)^(2/3). Using the growth factor D(z), where D(z=0)=1, we set the initial conditions for the overdensity of collisionless fluctuations at z = z_i to be given by Equations 8 and 9.


Top left: chosen parameterizations of the uniform temperature for the gas in the universe. Before z of approximately 200, the gas temperature is coupled to the CMB through Compton scattering and decreases as (1+z). At later times, gas cools adiabatically and its temperature drops as (1+z)^2. When the first stars form and heat their environment (the heating is likely from X-rays from the first galaxies), the gas temperature rises as approximately (1+z)^-4.9 and finally reaches 10^4 K following reionization at z of approximately 7 ([Pritchard2008], PL08 shown with the solid line). In the ADIAB model, the IGM temperature decrease as shown with the dash-dotted line instead of (1+z)^2 due to the shock-heating. For z greater than or equal to 60, the gas temperature evolution is extrapolated with the PL08 curve. Top right, Bottom left and Bottom right: the evolution of dark matter (delta_dm), baryonic (delta_b) and total overdensities (delta_t), normalized by the overdensity in a collisionless universe (delta_t'). Halos in this example are assumed to collapse at z = 7 with masses between 10^8.0 solar masses and 10^10.0 solar masses, which are indicated with different colours.


In order to investigate the effect of baryons on halo growth at the collapse redshift z_col, we set the initial overdensities of dark matter delta_dm_i and baryons delta_b_i to the same value as the collisionless case delta_t_i', and also set their derivatives equal to the derivative of delta_t_i'. In this work, we adopt a large initial redshift (z_i = 999) to ensure the initial overdensity is within the linear regime. The top right and bottom panels of Fig. C1 show the resulting evolutions of dark matter (delta_dm), baryonic (delta_b) and total (delta_t = Omega_b/Omega_m * delta_dm + (1-Omega_b/Omega_m) * delta_b) overdensities normalized by the corresponding dark matter overdensity in a collisionless universe (delta_t'). The spatial frequency scale, k, is varied corresponding to halo masses from 10^8 solar masses to 10^10 solar masses (see Equation 7), for a collapse redshift of z_col = 7.


We see that baryonic overdensities collapse slower than the corresponding dark matter overdensity due to the hydrostatic pressure from the baryons. This effect increases when the halo mass is smaller. Comparing different models, we see that when the IGM becomes heated by X-rays from the first galaxies, resulting in more hydrostatic pressure against baryonic accretion, baryonic overdensities are suppressed significantly [Naoz2005, Naoz2012]. Fig. C1 also illustrates that the inclusion of baryons causes the dark matter halo overdensity to increase more slowly than in the collisionless case. This effect is larger for smaller haloes as expected from the scale dependence in Equation (6).


We next investigate the delay in dark matter halo formation (when delta_t reaches delta_c) due to the inclusion of baryons. From Fig. C1, it is clear that it takes longer for the total matter overdensities to evolve to the linear critical overdensity delta_c owing to the suppressed collapse of baryons, especially when the system is smaller. Equivalently, a halo that formed with mass M_t' at z_col' in a collisionless universe will only reach that same mass later at z_col < z_col' if baryons are included such that M_t(z=z_col) = M_t'(z=z_col'). Given this delay and with the halo mass accretion history, one is able to estimate the halo mass, M_t at z_col', and predict the suppression of halo mass, M_t(z=z_col') / M_t'(z=z_col').


[Correa2015a,Correa2015b,Correa2015c] introduced an analytic calculation of accretion history for dark matter based on the extended Press-Schechter theory ([Press1974,Bond1991,Lacey1993], see Section 3.2) given by Equation 10, where alpha and beta are given by Equations 11 and 12, and q is given by Equation 13. [Correa2015a] also provide a fitting function for the relation between halo formation redshift z_f and the halo mass M given by Equation 14. With these equations, we estimate the total mass M_t(z=z_col') of the corresponding halo whose mass is equal to M_t'(z=z_col') in the dark matter only universe. Fig. C2 shows the ratio of M_t to M_t' as a function of redshift, for five halo masses (from M_t'=10^8 to 10^10 solar masses), where M_t and M_t' are the halo mass at z=z_col' in the scenario with and without baryons. Calculations using the [Pritchard2008] and ADIAB gas temperature evolutions (see the top left panel of Fig. C1) are shown with solid and dash-dotted lines, respectively.


Mass ratio between haloes collapsing in the universe with and without baryons, M_t/M_t' as a function of redshift, z=z_col'. The calculation is shown with different colours for 4 halo masses, M_t(z=z_col') from 10^8 to 10^10 solar masses. Calculations with [Pritchard2008] and ADIAB gas temperature evolutions are shown with solid and dash-dotted lines, respectively. The PL08 result shows that at early times, the suppression is less than or equal to 5 per cent. However, when heating becomes important, the mass ratio is significantly suppressed. For example, we find the mass ratio is approximately 60 per cent at z=5 for haloes around 10^8 solar masses. This mass suppression becomes more dramatic in less massive haloes. From Equation (10), we can calculate the specific halo mass growth rate given by Equation 15. Since beta increases with decreasing mass [Correa2015a], the specific halo mass growth rate at a given redshift decreases towards less massive haloes, and hence the mass ratio becomes higher.


The analytic calculation of mass ratio for the ADIAB simulation IGM temperature evolution results in a similar trend to the simulation (mass ratios are smaller for less massive haloes and at higher redshift, see the top left panel of Fig. 1). However, it predicts far less suppression of halo masses (e.g. the mass ratio is less than or equal to 5 per cent for 10^8 solar masses haloes) than the ADIAB simulation (e.g. the mass ratio is around 50 per cent for 10^8 solar masses haloes at z=14 in the simulation). There are several possible reasons for this quantitative difference. Compared to the analytic model, which uses a simple IGM temperature evolution to calculate the pressure of baryons, the numerical simulation is much more complicated due to non-linear physics. Although the temperature decreases to less than 10^2K for gas particles identified as IGM, the gas temperature can be heated to more than 10^4-10^5K once falling close by or into haloes due to shocks, resulting in stronger pressure gradients and more dramatic mass suppression shown in the simulation.


In order to prevent gas particles being too close to each other, resulting in an extremely small calculation time step and dramatically increasing the computational cost, there is an existing minimum gas temperature in the simulation, which is approximately 5K. This will enhance the mass suppression due to a minimum effective hydrostatic pressure in the simulation (although typically this at very small scales only). There are also many limitations in the analytic model. For instance, the hypothesis of uniform gas temperature may lead to an underestimation of the baryonic overdensity [Naoz2005], while setting the same initial conditions for the baryonic and dark matter overdensities overestimates the baryonic fluctuations on small scales [Naoz2011]. Nonetheless, the analytic calculation provides a frame work to interpret the mass reduction observed in simulations.


Motivated by recent measurements of the number density of faint AGN at high redshift, we investigate the contribution of quasars to reionization by tracking the growth of central supermassive black holes in an update of the Meraxes semi-analytic model. The model is calibrated against the observed stellar mass function at redshift z from approximately 0.6 to 7, the black hole mass function at redshift z less than or approximately equal to 0.5, the global ionizing emissivity at redshift z from approximately 2 to 5 and the Thomson scattering optical depth. The model reproduces a Magorrian relation in agreement with observations at redshift z < 0.5 and predicts a decreasing black hole mass towards higher redshifts at fixed total stellar mass. With the implementation of an opening angle of 80 deg for quasar radiation, corresponding to an observable fraction of approximately 23.4 per cent due to obscuration by dust, the model is able to reproduce the observed quasar luminosity function at redshift z from approximately 0.6 to 6. The stellar light from galaxies hosting faint AGN contributes a significant or dominant fraction of the UV flux.


At high redshift, the model is consistent with the bright end quasar luminosity function and suggests that the recent faint redshift z~4 AGN sample compiled by [Giallongo2015] includes a significant fraction of stellar light. Direct application of this luminosity function to the calculation of AGN ionizing emissivity consequently overestimates the number of ionizing photons produced by quasars by a factor of 3 at redshift z~6. We conclude that quasars are unlikely to make a significant contribution to reionization. The epoch of reionization (EoR) is the phase of the Universe when neutral hydrogen in the intergalactic medium (IGM) was reionized. Star-forming galaxies at high redshift are believed to be one of the dominant sources of ionizing UV photons provided one assumes a high average escape fraction of Lyman continuum radiation (an escape fraction for stars greater than or approximately equal to 10 per cent) and extends the UV luminosity function to faint dwarf galaxies [Kuhlen2012, duffy2014low, Feng2016, Mesinger2016]. However, the value of the escape fraction for stars is very uncertain.


Observations of star-forming galaxies at low redshift usually indicate a much lower escape fraction. For example, by measuring the ratio of Lyman alpha to H-beta line emission, [Ciardullo2014] derived an escape fraction of 4.4 per cent while [Matthee2016] measured a median escape fraction of 1.6 per cent using stacking of H-alpha-selected galaxies. Moreover, some theoretical works also suggest a low escape fraction for stars [Gnedin2007, Hassan2016, Sun2015]. In order to reconcile the difference between low-redshift observations and the photon budget at high redshift, some propose a rapid increase of the escape fraction for stars with redshift [Haardt2012, Khaire2016, Price2016, Sharma2016] and with decreasing mass [Paardekooper2013, Kimm2014, Wise2014]. This is supported by identifying local analogues of high-redshift galaxies and extrapolating the observed escape fraction for stars using indicators such as the [OIII]/[OII] ratio to high redshift ([Faisst2016] and references therein).


On the other hand, additional contributors to reionization may also be present. For example, [Ma2016] included a binary population into a set of radiative transfer cosmological simulations and found that they produced significantly more ionizing photons among the old stellar population compared to a model without binaries, reducing the requirement of high escape fraction. In addition, these photons produced at later times can escape from galaxies more easily since the local feedback efficiently clears out nearby gas, leading to a lower required escape fraction on average. With a high escape fraction (an escape fraction for quasars of approximately 1, [Barkana2000]), quasars (AGN) are potential contributors to reionization [Volonteri2009, Fontanot2014, Madau2015, Mitra2015], despite their relatively low number. Recently, [Giallongo2015] identified faint AGN candidates in the CANDELS GOODS-South field and suggested that there is a high number density of faint AGN at redshift z=4-6. These faint quasars provide a new source of reionization [Madau2015].


However, it is still debated whether there are enough luminous quasars at high redshift to make a significant contribution [Bouwens2015, Weigel2015, Manti2017, Parsa2017] and whether the escape fraction of high-redshift low-luminosity AGN is of order of unity [Cristiani2016, Micheva2016]. To explore the consequences for galaxy formation and reionization from theses faint quasars, this paper describes the addition of a population of evolving black holes to the Meraxes semi-analytic model of galaxy formation and reionization [Mutch2016]. This new model enables a detailed exploration of the relative role of quasars during the EoR. The paper is organized as follows. We begin with a description of the semi-analytic model in Section 2, in which the black hole growth model is introduced in detail. We present the black hole properties in Section 3 and explore reionization from quasars in Sections 4 and 5. Conclusions are given in Section 6. In this work, we adopt cosmological parameters from the Planck 2015 results (Omega_m, Omega_b, Omega_Lambda, h, sigma_8, n_s = 0.308, 0.0484, 0.692, 0.678, 0.815, 0.968; [PlanckCollaboration2015]).


Built on halo merger trees constructed from the Tiamat collisionless N-body simulation [Poole2015], the Meraxes semi-analytic model [Mutch2016] was specifically designed to study galaxy formation and reionization at high redshift. The model computes galaxy properties according to different astrophysical processes including gas infall, cooling, star formation, supernova feedback, metal enrichment, mergers and reionization. In order to properly track the evolution of galaxies during reionization, Tiamat provide 100 snapshots between redshift z=35 and 5 with a time interval of approximately 11.1 Myr and 64 additional snapshots between redshift z=5 and 2 separated equally in units of Hubble time. The mass resolution of Tiamat is approximately 2.64x10^6 per h solar masses and the box size is 67.8 per h Mpc. Additionally, in order to obtain information of more massive objects and lower redshifts, we also take advantages of the dark matter halo merger trees generated from the Tiamat-125-HR simulation (Poople et al. in prep).


The Tiamat-125-HR simulation shares identical cosmology with Tiamat but has a lower mass resolution of 0.12x10^9 per h solar masses in a bigger simulation volume, with side lengths equal to 125 per h Mpc. The Tiamat-125-HR trees are constructed down to redshift z=0.56 with the same snapshot separation strategy as the Tiamat trees. In the following subsections, we briefly describe the galaxy formation in Meraxes and introduce the new implementation of black hole growth and feedback in detail. More details about the implemented galaxy formation physics can be found in [Mutch2016]. In the Meraxes semi-analytic model, cooling and star formation are assumed to be negligible in haloes below the atomic cooling mass threshold, approximately 10^8 solar masses. Thus, once a halo grows larger than the atomic cooling limit it is designated as a galaxy, which has three baryonic components: gas, stars and a central black hole.


During one time step, Delta t, additional gas falls into the hot gas component of a galaxy from the IGM when the mass fraction of baryons in the halo is lower than the cosmic mean value, f_b = Omega_b / Omega_m: The change in hot gas mass, Delta m_hot, is calculated as the maximum of zero or the difference between the required baryon mass (chi_r times f_b times M_vir) and the existing baryonic mass (stars + cold gas + hot gas + ejected gas), where m_star, m_cold, m_hot and m_eject are the masses of the stellar component, cold gas, hot gas and ejected gas, respectively, and M_vir is the virial mass of the host halo in which the galaxy forms. chi_r is a baryon fraction modifier to take account of reionization feedback and will be introduced later in Section 4.1. Some of the hot gas, m_cool, cools and collapses on to the cold disc. Assuming the cooling process is in quasi-static thermal equilibrium, one can calculate the cooling time by the cooling time t_cool(r) equals (3 times mu bar times m_p times k times T_hot) divided by (2 times rho_hot(r) times Lambda(T_hot, Z_hot)).


Following [croton2006many], we calculate the cooling radius, r_cool, at which the cooling time is equal to the halo dynamical time... Cooling is sufficient within r_cool and we estimate the cooling mass by the cooling mass m_cool is defined as m_hot times the minimum of [1, min(1, r_cool / R_vir) times (Delta t / t_cool)], which is removed from the hot gas reservoir and redistributed into the cold gas disc, Delta m_hot = -Delta m_cold = -m_cool. From this, we see that depending on the ratio of r_cool to R_vir, cooling is separated into two regimes: static hot halo (r_cool > R_vir) and rapid cooling (r_cool < R_vir, see more in [croton2006many]). When the galaxy collects enough cold gas, m_cold > m_crit, based on [kennicutt1998global] and [kauffmann1996disc], it forms new stars, Delta m_star, through a burst.


The change in stellar mass Delta m_star equals the minimum of [(alpha_sf times max(0, m_cold - m_crit) times Delta t) / t_dyn,disc, m_cold], where t_dyn,disc is the dynamical time of the cold gas disc and alpha_sf is a free parameter corresponding to the star formation efficiency. This mass is removed from the cold gas reservoir. The new stellar mass is assumed to form following a [Salpeter1955ApJ...121..161S] initial mass function (IMF). Ultimately some of these stars recycle their mass back to the interstellar medium (ISM) through type-II supernovae explosion. The energy produced by these supernovae heats the ISM and converts some of the cold gas to the hot phase or, if there is sufficient energy, even ejects a fraction of hot gas from the galaxy. Assuming that the efficiency for supernovae energy coupling with the ISM scales with mass and is in proportion to a free parameter [guo2011dwarf], alpha_energy, then the total energy released by supernovae that couples to the ISM is calculated.


Since a massive star (greater than or approximately equal to 8 solar masses) takes approximately 40 million years, or 4 snapshots, before reaching its type-II supernova stage, Meraxes accounts for supernovae not only from the current snapshot, j, but also from the stars formed in the previous 4 snapshots. Therefore, the total energy released during one snapshot is the sum of the total energy from snapshots i=j to i=j-4. Since the hot gas shares the same virial temperature as the host halo, assuming the mass loading factor for reheating cold gas is a free parameter, alpha_mass, the energy utilized in gas heating can be calculated by the reheating energy E_reheat equals 1/2 times alpha_mass times Delta m_star times V_vir^2. Depending on the available energy, E_total, and the required energy for re-heating, E_reheat, the reheated mass is calculated. This mass is removed from the cold gas reservoir and redistributed in the hot gas component.


If there is still some energy left after reheating, the supernovae feedback will further remove hot gas from the galaxy, adding it to the ejected component. In addition, the metals produced by supernovae enrich the environment, which then enhances the cooling rate. Moreover, mergers drive strong turbulence and hence increase the possibility of star formation. Major mergers generally introduce more energetic bursts than minor mergers since they induce strong inflows and easily trigger bar-like instabilities in the cloud [Somerville2001]. Therefore, following mergers, Meraxes also includes a starburst mechanism. Reionization feedback will be introduced in Section 4.1. There are more details of the semi-analytic model in [Mutch2016]. This work extends the current model with a detailed black hole growth prescription based on [Croton2016] and is described in the following subsection.


In the updated model, every newly formed galaxy is seeded with a central black hole of mass 1000 per h solar masses. The two gas reservoirs in the galaxy (i.e. hot and cold) lead to two different black hole growth scenarios, termed radio and quasar modes [Croton2016]. In the normal quiescent state, black holes only accrete mass from the hot gas reservoir, resulting in radio emission in the centre of galaxy. However, mergers trigger rapid accretion on to the black hole from the cold gas disc, causing them to radiate as quasars. In this work, we do not distinguish AGN with different types and refer to them all as quasars unless specified otherwise. Whenever there is a static hot gas reservoir, m_hot, around the galaxy, some of it will cool, m_cool, and form a cold gas disc, m_cold, while some will be directly accreted by the central black hole.


We adopt the Bondi-Hoyle accretion model proposed in [Croton2016] to describe this hot gas accretion. The Bondi-Hoyle accretion rate is given by the Bondi-Hoyle accretion rate, dot m_Bondi, equals (2.5 times pi times G^2 times m_bh^2 times rho_hot) divided by c_s^3, where G, m_bh, c_s and rho_hot are the gravitational constant, the black hole mass, the speed of sound and the density of the hot gas reservoir, respectively. Then, assuming the accretion rate does not change during one time step, Delta t, the accretion mass is the change in black hole mass from hot gas, Delta m_bh,hot, equals the minimum of (m_hot, m_Edd, k_h times dot m_Bondi times Delta t), where k_h is a free parameter, used to adjust the efficiency of black hole growth in the radio mode. This mass is removed from the hot gas reservoir. Assuming a fraction, eta, of the accreted mass is radiated, the black hole only grows by the change in black hole mass Delta m_bh equals (1-eta) times Delta m_bh,hot.


Moreover, the radiation acting outward is limited by the Eddington luminosity... By integrating this through one time step, this provides the second limitation... the Eddington mass limit m_Edd equals m_bh times [exp((epsilon times Delta t) / (eta times t_Edd)) - 1], where t_Edd is the Eddington accretion time-scale and m_bh is the black hole mass at the beginning of this time step. Assuming adiabatic heating and that a fraction of the radiated energy, kappa_r, is coupled to the surrounding gas and therefore can contribute to feedback, then the heated mass due to black hole feedback can be calculated through the heated mass m_heat equals (kappa_r times eta times Delta m_bh,hot times c^2) / (0.5 times V_vir^2), which is subtracted from the cooling flow. However, if the heating from black hole feedback is too strong... it will significantly suppress the cooling flow, which will consequently restrain the black hole accretion of hot gas. This suppression is referred to as radio mode feedback.


In this case, following [Croton2016], Delta m_bh,hot is rescaled to be the amount of mass within the cooling radius... and the heated mass consequently shrinks to be the cooling mass, resulting in a complete quenching of cooling. Radio mode feedback limits gas condensation in cluster cooling flows and regulates star formation in massive galaxies [croton2006many]. We note that this does not have a significant impact on the results from Tiamat at redshift z>=2 due to the limited number of massive objects in the box. However, black hole feedback is important in more massive objects at lower redshifts, which can be observed using the Tiamat-125-HR halo merger trees. We note that at high redshift, accreted hot gas does not contribute significantly to black hole growth, with the accreted mass during the radio mode typically being only approximately 0.1 per cent of the mass from the quasar mode. Additionally, because the radio mode accretion rate is approximately 3 orders of magnitude smaller than the Eddington accretion rate, we ignore the energy due to the radio mode for the calculation of the quasar luminosity.


Many binary AGN have been detected in merging galaxies [Shields2012, Comerford2013, Comerford2014, Comerford2015, Muller-Sanchez2015]. This suggests that galaxy mergers might trigger AGN activity, which is also supported by hydrodynamic simulations of galaxy mergers [Capelo2015, Volonteri2015b, Volonteri2015a, Steinborn2016]. Following [Croton2016], when a merger occurs between two galaxies, the central black holes coalesce and their masses combine. As mergers drive strong gas inflows towards the central region [Capelo2015], cold gas is funnelled to the central black hole of the resulting merged galaxy, significantly increasing its mass. The amount of accretion can be estimated by [Bonoli2009, Croton2016]... where m_cold is the amount of mass available in the cold gas disc, k_c is a free parameter used to modulate the strength of black hole accretion and gamma is the mass ratio between the two merging galaxies, respectively. The term Delta m_bh,max' corresponds to the accretion mass left from the quasar mode in the previous snapshot, which will be introduced later.


Unlike the radio mode, black holes grow dramatically during the quasar mode. The AGN activity lifetime is much longer than the 11Myr time step at redshift z>5. In some cases, this leaves the central black hole insufficient time to consume all of the newly accreted gas, Delta m_bh,max, at the Eddington limit. Therefore, the mass actually accreted by the black hole is the change in black hole mass from cold gas Delta m_bh,cold equals the minimum of (m_Edd, Delta m_bh,max), which is removed from the cold gas reservoir. In our model, during the quasar mode black holes are assumed to either accrete and radiate at the Eddington rate or stay quiescent if the accretion mass is not sufficient. Therefore, depending on the total available mass brought in, Delta m_bh,max, and the Eddington limit, m_Edd, there are two possible scenarios when the central black hole is undergoing a merger: 1) Delta m_bh,max < m_Edd... 2) Delta m_bh,max >= m_Edd.


In the case of Delta m_bh,max > m_Edd, instead of consuming this instantaneously, causing a super-Eddington accretion event, some of the mass is accreted by the central black hole, limited by the Eddington rate, while the rest is stored in the accretion disc to be consumed in the next time step. Similarly, when this quasar is observed at t_obs, the bolometric luminosity can be calculated. It is suggested that during mergers, black holes undergo rapid accretion for a certain time period, which is followed by a long quiescent phase [Hopkins2005c, Hopkins2005a, Hopkins2005d, Hopkins2005b]. The assumption that black holes are either accreting at the Eddington rate (epsilon=1) or stay quiescent has been shown to provide a good description of black hole growth for the majority of black holes at high redshift [Bonoli2009]. The energy injected into galactic gas during the quasar mode is given by kappa_q times eta times Delta m_bh,cold times c^2, where kappa_q represents the mass coupling factor in the quasar mode.


Unlike the radio mode, this energy generates a wind, which heats the gas in the cold disc into the hot reservoir. Depending on the amount of energy provided by the quasar, the wind can further unbind and eject the hot gas in a manner similar to the stellar feedback prescription presented in Section 2.1. In order to compare the predicted black hole population in our model with observations, the intrinsic B-band and UV 1450 Angstrom band luminosities of quasars are calculated as follows: 1. We calculate the bolometric magnitude... 2. We calculate the B-band magnitude in the Vega magnitude system using the bolometric correction proposed by [Hopkins2007]... 3. We convert the B-band magnitude from the Vega system to the AB system following [Glikman2010]... 4. We extrapolate the B-band magnitude of which the effective wavelength is 4344 Angstrom [Blanton2007] to the 1450 Angstrom magnitude, assuming the quasar continuum between 1450 and 4344 Angstrom has a power-law slope of alpha_q,optical = 0.44 relative to wavelength [Schirber2003].


We summarize the relevant model parameters in Table 1 compared to the original value adopted in [Mutch2016]. In this work, we constrain our model against: the observed evolution of the galaxy stellar mass function between redshift z from approximately 0.6 and 7; the observed black hole mass function and Magorrian relation at low redshift (redshift z less than or approximately equal to 0.5); the latest integrated free electron Thomson scattering optical depth measurement [PlanckCollaboration2016]; the predicted global ionizing emissivity from Lyman alpha opacities [Becker2013]. The black hole mass functions at redshift z from approximately 8.0 to 0.6 are shown with different colours in the left-hand panel of Fig. 1. The results calculated using the Tiamat and Tiamat-125-HR trees are shown with thick and thin lines, respectively. The shaded regions represent the 1-sigma Poisson uncertainties.


Estimates of the local black hole mass function are shown with points and grey shaded regions. We see that the discrepancy between various observations is substantial. This is a result of inconsistent correlations between black hole mass and observable quantities, such as the Sersic indices, bulge velocity dispersion or luminosity and galaxy geometry, and from the intrinsic scatter of these adopted scaling relations. Extrapolations of these observed scaling relations have impacts on the black hole mass function at the high-mass end, while different treatments of the spiral galaxy bulge can significantly change the low-mass end [Shankar2009]. In this work, the model is therefore calibrated against the black hole mass function between 10^7.5 solar masses and 10^9 solar masses. In the bottom left-hand panel of Fig. 1, we see that the mass function converges at lower redshifts above a black hole mass of 10^6 solar masses (shown as the vertical dotted line).


The different mass resolutions of Tiamat and Tiamat-125-HR result in different merger rates, especially when approaching the resolution limit. At high redshift, because the growth of black hole is dominated by the merger triggered quasar mode, the number density of small black holes is relatively lower in the Tiamat-125-HR result (e.g. comparing the redshift z=8.0 thick and thin lines). At low redshift, redshift z~0.6, the model is in agreement with the observational estimations. The middle panel of Fig. 1 shows the relation between black hole mass and stellar mass (the Magorrian relation). The 2D histogram indicates the distribution of galaxies in logarithm from the fiducial model using the Tiamat-125-HR halo merger trees at redshift z~0.6 while the solid line represents the mean. The Magorrian relations at redshift z=2, 5 and 7 from Tiamat-125-HR are also shown with dash-dotted, dashed and dash-dot-dotted lines, respectively.


The right bottom subplot shows the redshift z=2, 5 and 7 Magorrian relations of the Tiamat result with thick lines compared with the redshift z~0.6 Tiamat-125-HR Magorrian relation. Observations from the local Universe are indicated with different symbols. We see that the model predicts a similar Magorrian relation at redshift z~0.6 compared to the local observations and we find an increasing normalization towards lower redshifts in the mass range of 10^10 solar masses < M_* < 10^12 solar masses. The evolution of the Magorrian relation in our model is due to the black hole and stellar mass evolving with the underlying dark matter halo mass differently. In [Mutch2016], we have shown that the median relation between stellar mass and virial mass does not evolve in our model and can be described by M_* is proportional to M_vir^7/5 in the range of 10^8 solar masses < M_* < 10^11 solar masses, which is supported by a simple analytic model of supernova energy conversation [Wyithe2003].


On the other hand, haloes with a given virial mass host less massive black holes at earlier times in our model (see the right-hand panel of Fig. 1). These result in an increasing normalization of the Magorrian relation towards lower redshifts. From the right-hand panel of Fig. 1, we see that the M_bh - M_vir scaling relation does not get suppressed in massive haloes (at least to M_vir ~ 10^14 solar masses or M_bh ~ 10^9 solar masses). Both black holes and stars grow from the cold gas disc. However, AGN feedback significantly suppresses the cooling flow in massive galaxies [croton2006many], preventing stellar mass from growing. On the other hand, black holes are able to continue accreting until there is enough energy in feedback to overcome the halo potential and unbind the gas [Booth2010]. Because of these, the slope becomes steeper in the Magorrian relation at black hole mass M_bh > 10^8 solar masses.


Fig. 2 presents the galaxy stellar mass functions from our fiducial model for comparison with the available observational data at redshifts 7 to approximately 0.6. The results calculated using the Tiamat and Tiamat-125-HR trees are shown with thick and thin lines, respectively. The shaded regions represent the 1-sigma Poisson uncertainties. We see that the fiducial model is able to reproduce the observed galaxy stellar mass function across the redshift range of redshift z=7-0.6. We also present two additional models in Fig. 2, Mutch2016 and M16BH. Mutch2016 adopts identical parameters as the redshift varying escape fraction for stars model presented in [Mutch2016]. This model is able to reproduce the evolution of the stellar mass function at high redshift (redshift z>5).


The redshift dependence of the escape fraction for stars, f_esc,*,M16, was chosen to simultaneously reproduce the normalization and flat slope of the [McQuinn2011] emissivity measurement at redshift z~5 and the Planck 2015 optical depth measurement ([PlanckCollaboration2015]). Details of the Mutch2016 model at high redshift (redshift z>5) can be found in [Mutch2016]. In this work, we extend the model to lower redshifts and find that, without AGN feedback regulating galaxy formation, the model fails to reproduce the observed stellar mass functions at redshift z<2, especially in larger mass ranges (stellar mass M_* > 10^11 solar masses). However, when AGN feedback is implemented, shown as the M16BH model, the model shows better agreement with observations [croton2006many]. These two models also suggest that radio-mode feedback does not play a significant role in galaxy formation during the EoR, and because reionization is dominated by low-mass galaxies ([Liu2016MNRAS.462..235L], see also Section 5), AGN feedback is expected to have no significant impact on reionization.


With respect to Mutch2016, the fiducial model presented in this work employs a stronger star formation efficiency (alpha_sf) with maximized supernova feedback (alpha_energy and alpha_mass) and more intense radio mode feedback (k_h), in order to gain better agreement with the observed stellar mass function in the intermediate mass range (10^9 solar masses < M_* < 10^11 solar masses) at 1<z<2. Semi-analytic models can predict quasar bolometric luminosity through the modelled central black hole mass. In order to compare with observations, three corrections are usually required: bolometric corrections, duty cycles and obscured fractions. In our model, we calculate the duty cycle self-consistently by assuming Eddington accretion of available gas and a random observation time between snapshots. Black holes that are inactive when observed are not considered in the census.


We convert the total luminosity into a particular band using the bolometric correction of [Hopkins2007]. Finally, we need to take the obscuration due to the presence of a dusty torus surrounding the AGN into account. In practice, each quasar is weighted by 1 - cos(theta/2), where theta represents the opening angle of AGN radiation, during the calculation of the quasar luminosity function. We note that the dependence of theta or the obscured fraction (the ratio of obscured to unobscured AGN) is very complicated. It has been suggested that theta may depend on wavelength, redshift and luminosity [Elvis1994, Hopkins2007]. For simplicity, we only consider constant theta. The opening angle is a free parameter in our model, chosen to reproduce the quasar luminosity function amplitude. In this work, theta=80 deg is adopted. This corresponds to a fraction of visible objects, f_obs, to be ~23.4 per cent, in agreement with [Hopkins2007].


We present the UV 1450 Angstrom luminosity functions of quasars at redshift 6, 5, 4 and 3, and the B-band luminosity functions at redshift z~2, 1.5, 1.3 and 0.6 in Fig. 3, compared to the observational data. The results calculated using the Tiamat and Tiamat-125-HR trees are shown with thick and thin lines, respectively. The shaded regions represent the 95 per cent confidence intervals around the mean using 100000 bootstrap re-samples. Similarly to the black hole mass functions, the Tiamat-125-HR results show a lower number density at high redshift compared to the Tiamat results, and they converge at lower redshifts (redshift z less than or approximately equal to 4). We see that the model shows good agreement with observations across a large redshift range (redshift z~6-0.6).


At high redshift the model is consistent with the samples of bright quasars [willott10, Shen2012, McGreer2013, Jiang2016] while it predicts a lower number density of faint quasars compared to the [Giallongo2015] sample. On the other hand, the model produces a significant number of faint quasars. The turnover (not shown here) of the predicted UV luminosity function is around UV magnitude M_1450 ~ -11, which is much fainter than the observed faintest quasars, UV magnitude M_1450 ~ -18 ([Giallongo2015]). In addition, the observed population of bright quasars (UV magnitude M_1450 < -23) at high redshift is not present in our model due to our limited simulation volume. All of these factors may have an impact on the contribution of AGN to the reionization history, which is discussed in Section 4. The large number of faint AGN identified by [Giallongo2015] has prompted renewed discussion of the contribution of quasars to reionization [Madau2015, Mitra2015, Kulkarni2017].


We therefore further discuss the low number density in the faint end predicted by the model compared with the G15 data. When constructing an observed AGN luminosity function, the sample is typically identified spectroscopically or via colour-colour selections if spectra are not available, following which contamination from host galaxies is removed. However, this may not be the case for faint AGN. In particular, the G15 AGN sample is selected using X-ray activity and no AGN-galaxy separation is possible. Thus, the total UV luminosity may have a large fraction of stellar light, suggesting that the G15 sample may be potentially impacted by stellar light contamination. Despite the recent claim that only 12 of the 22 reported X-ray detections in G15 are high-redshift AGN [Parsa2017], this conjecture is supported by [Ricci2017], who used X-ray observations as a proxy and derived the quasar UV luminosity function down to much fainter ranges.


They showed that the luminosity function is in agreement with UV/optical observations (e.g. [Croom2009, Glikman2010, Masters2012, Palanque-Delabrouille13]), and have much lower amplitudes than the G15 results, and that the high number density of faint AGN in G15 can be explained by the contribution from the luminosity of host galaxy with UV magnitude M_1450 ~ -20. In order to account for this, we calculate the galaxy UV luminosity by integrating model SEDs based on the modelled star formation history [Liu2016MNRAS.462..235L] and add the stellar light to mimic the observed total UV luminosity (AGN+galaxy). We present the resulting luminosity function of faint objects in Fig. 3 with dashed lines. We see that including stellar light can significantly increase the number density of faint AGN inferred at high redshift, by up to approximately 1 order of magnitude. The fitting functions provided by [Giallongo2015] as shown with thin black dashed lines in Fig. 3 are more consistent with the AGN+galaxy luminosity function.


If this is the case, the estimated emissivity at high redshift based on [Giallongo2015] is likely overestimated. Motivated by the G15 sample, which suggests a numerous population of faint AGN at redshift z=4-6 ([Giallongo2015]), [Madau2015] extrapolated the emissivity calculated from [Giallongo2015] to higher redshifts and assessed a model of reionization, in which quasars are the dominant ionizing sources. They found that due to the high escape fraction of ionizing photons produced by those luminous objects, quasars are able to ionize the neutral hydrogen by redshift z~5.7 if the high emissivity of quasars derived at redshift z~5 continues to higher redshifts. Later, [Mitra2015] revisited the model with a revised extrapolation and also found that quasars have a significant role during the EoR. However, their analysis still prefers models with a non-zero escape fraction of approximately 12 per cent from galaxies.


It is worth noting that with different formalism, [Manti2017] fit the observed quasar UV luminosity function at redshift z=0.5-6.5, including the G15 sample. They recalculated the emissivity by integrating the luminosity function and confirmed a large number of ionizing photons from quasars at high redshift using the Schechter luminosity function. However, the fitting result using a double power law presents a rapidly decreasing emissivity at redshift z>6, suggesting that the extrapolated high-redshift quasar emissivity is strongly dependent on the assumed shape of the quasar luminosity function. Taking advantage of the Meraxes semi-analytic model with 21cmFAST [Mesinger2011], we investigate the contribution of quasars to reionization, within a frame work that accounts for black hole growth and feedback on star formation. The semi-numerical reionization code 21cmFAST [Mesinger2011] uses an excursion set formalism to identify HII bubbles in which the cumulative number of ionizing photons is more than the number of absorbing atoms.


The reionization condition is: N_star * N_gamma,star * f_esc,star + N_q * N_gamma,q * f_obs * f_esc,q >= (1 + N_rec_bar) * N_HI, where N_star and N_q are the numbers of baryons in stars and quasars, N_gamma,star ~ 4000 [Loeb2001] and N_gamma,q are the mean numbers of ionizing photons produced per baryon incorporated into the stellar or quasar components. The parameters f_esc,star and f_esc,q are the escape fractions of ionizing photons produced by stars and quasars. f_obs ~ 0.234 represents the observable fraction due to obscuration. N_HI is the cumulative number of atoms being ionized and N_rec_bar is the mean number of recombinations per baryon. Inhomogeneous recombinations are ignored, which can have a large impact [Sobacchi2014]. In this work, N_rec_bar is set to be 0 as suggested by the high-redshift Lyman alpha forest in the IGM [Bolton2007, McQuinn2011].


When the local volume around a galaxy is ionized, the UV background provides an extra heating mechanism, which modifies the baryonic fraction of the host halo. Following [Sobacchi2013], when the virial mass, M_vir is smaller than a filtering mass, which can be calculated through M_filt = 2.8x10^9 solar masses * J_21^0.17 * ((1+z)/10)^-2.1 * [1 - ((1+z)/(1+z_ion))^2]^2.5, the suppression of gas becomes significant. Here, z_ion is the redshift when the local volume is first ionized. J_21 represents the intensity of the local UV background. This can be calculated through an equation involving the mean-free path of ionizing photons, which is approximated by the HII bubble radius, R. The parameters alpha_star=5.0 [Loeb2001] and alpha_q=1.57 ([Giallongo2015]) are the spectral indexes for a stellar-driven and a quasar-driven spectra in the UV band. m_dot_star and m_dot_q are the growth rates of stellar mass and black hole mass, respectively.


Fig. 4 shows the instantaneous ionizing emissivity averaged over the entire Tiamat simulation volume from different models as a function of redshift... From the Lyman alpha opacity, several measurements of the total emissivity (AGN and stars) at high redshift have been estimated [Bolton2007, McQuinn2011, Becker2013]. There are relatively large uncertainties in these measurements. In this work, we compare our models using the most recent data from [Becker2013]. The top panel of Fig. 4 shows that the fiducial model agrees with the [Becker2013] data and the bottom shows the ratio between ionizing photons from black holes and stars in the fiducial model, suggesting that during the EoR, quasars are subdominant in our model. The evolutions of the mass-weighted global neutral hydrogen fraction and the integrated Thomson scattering optical depth are shown in Fig. 5.


It shows that the fiducial model has a reasonable reionization history, with the mean global neutral hydrogen fraction decreasing from 90 per cent at redshift z~10 to 0 by redshift z~6 and a Thomson scattering optical depth in agreement with the latest Planck limits ([PlanckCollaboration2016]). There are two additional models in the top panel of Fig. 4 as well as in Fig. 5: StellarReion and QuasarReion. The ionizing source in the StellarReion model is only stars, with an evolving escape fraction and zero escape fraction for quasars, while quasars are the only reionization contributor in the QuasarReion model, with an escape fraction of 1 for quasars and 0 for stars. Note that only changing the feedback from reionization has little impact on the stellar mass function, the quasar luminosity function or the Magorrian relation. By preventing gas infall, reionization only affects less massive objects in our model.


In Fig. 4, we see that the emissivity of quasars grows rapidly in the QuasarReion model by a factor of 10 from redshift z~7-5. However, if quasars are the only reionization contributor, even with an escape fraction of 1 for quasars, the number of ionizing photons cannot reach the [Becker2013] data. Moreover, due to the deficiency in the photon budget at high redshift, quasars can only start reionization at redshift z~6 resulting in an end at redshift z~3. Together with the predicted optical depth, our model rules out the quasar-only reionization scenario. One may recalibrate the model with a more efficient black hole growth rate at high redshift in order to match the G15 luminosity function and the estimated emissivity. [Mitra2015] also suggest that if G15 emissivity is correct, quasar-only reionization is possible and it results in a small value of tau_e due to the rapid evolution of the Lyman-limit systems.


However, simultaneously matching the model with the G15 faint AGN luminosity function and the other observations of bright systems at high redshift is difficult. For example, comparing to observations at redshift z~6, our models produce a flatter luminosity function, which is more consistent with the bright quasar sample. This suggests that a mass-dependent black hole growth efficiency would be required, in order to steepen the luminosity function and produce more small quasars. In the following sections, we explore the relative contribution of quasars to reionization based only on the presented black hole growth model. Comparing the neutral hydrogen fraction and the optical depth between the fiducial and StellarReion models also suggests that quasars do not have a significant role in reionization in this model. Their contribution helps finish reionization earlier by a redshift interval Delta z less than or approximately equal to 0.1 and decreases the optical depth by less than 10 per cent.


Our models are calibrated against the black hole - galaxy scaling relation and quasar luminosity function, in order to reproduce a realistic AGN catalogue for the study of the contribution of quasars to hydrogen reionization. However, it has recently been suggested that the black hole sample used to derive scaling relations is likely different from the entire population, leading to a selection bias [Bernardi2007]. For instance, using Monte Carlo simulations [Shankar2016] recovered the intrinsic scaling relation assuming the selection bias comes from unresolved black holes and showed that such a bias can lead to factors of >= 3 and ~50-100 higher normalizations of the M_bh - sigma_* and M_bh - M_* relations, respectively. We note that a Magorrian relation with a smaller normalization can be achieved in our model using a smaller black hole growth efficiency.


In this case, the reconstructed black hole population becomes less massive, more efficient AGN feedback and a larger fraction of observable AGN in the UV band are required to simultaneously reproduce the observed stellar mass function and quasar luminosity function. With the model calibrated against the quasar luminosity function, the black hole - galaxy scaling relation is coupled with the observable AGN fraction - a lower Magorrian relation requires a larger fraction of observable AGN. We note that the total emissivity of quasars is integrated from the luminosity function. Therefore, scaling relations do not have a significant impact on reionization in this work. Noting the difficulty of observationally determining the fraction of obscured AGN and the large uncertainties in the black hole - galaxy scaling relation, in this section we use a range of models which predict similar Magorrian relations as shown in Fig. 1 and explore the contribution of quasars to reionization.


We have shown that with an opening angle of 80 deg, the model is able to reproduce the observed quasar luminosity function from redshift z~6-0.6. However, at high redshift, the model predicts significant stellar contribution to UV flux in the G15 sample, and consequently less ionizing photons from AGN. Based on this, we find that quasars do not have a significant role during EoR. In the top panel of Fig. 5, we show the estimated total emissivity from [Giallongo2015] with the model proposed by [Haardt2012, Madau2015] and [Mitra2015]. The modelled emissivity from QuasarReion is shown for comparison. Our quasar reionization-only model (QuasarReion) predicts lower emissivities compared to these two estimations, with only a third of the G15 value at redshift z~6. Consequently, in disagreement with [Madau2015] and [Mitra2015] we conclude that quasars cannot be the dominate sources during reionization.


We could increase the emissivity by excluding the obscuration from dust (setting theta=180, shown as QuasarReion_nodust in Fig. 5), which gives a closer emissivity compared to the G15 estimation and in agreement with the model proposed by [Mitra2015]. In this model, quasars have a more significant role during the EoR and can reionize the IGM alone by redshift z~4.5. However, this model overestimates the number density of bright and low-redshift AGN, leading to an incorrect evolution of the quasar luminosity function. A lower fraction of observable AGN, f_obs, towards brighter luminosities and lower redshifts is required to solve this conflict. However, observations suggest the opposite trend in optical, infrared and X-ray bands [Hopkins2007] and more constraints are required to clearly establish a f_obs - z relation. In this section, we explore possible combinations of stars and quasars that could result in an overall photon budget at redshift z>5 consistent with the observed optical depth and ionizing flux at redshift z~2-5.


Noting this requirement, in this section we assume constant escape fractions both for simplicity and to ease interpretation. Motivated by the recent claim that the escape fraction of low-luminosity AGN is possibly less than unity at high redshift [Micheva2016], we run Meraxes with different combinations of the escape fraction for stars, f_esc,star, and the escape fraction for quasars, f_esc,q, without any changes to the other parameters. In the top panel of Fig. 6, the left-hand panel shows the Thomson scattering optical depth, tau_e. For comparison, shaded regions are shown corresponding to the best fit and 1-sigma range of the [PlanckCollaboration2016] measurements. Based on the [Becker2013] data at redshift z~2-5 and the [PlanckCollaboration2016] measurement, the top right two panels show the 68, 90 and 99 per cent confidence contours on each parameter of the best fit via the standard minimum-chi-squared technique. The 2D histogram shows the distribution of the ratios of quasar emissivity to stellar emissivity at redshift z~6.


We see that a lower escape fraction of ionizing photons from stars, f_esc,star requires a higher contribution from quasars, in order to reach the observational constraint. This also returns a higher ratio of quasar emissivity to stellar emissivity. However, because there is not a significant number of quasars at high redshift, changing the escape fraction for quasars, f_esc,q, has little impact to the optical depth. Through the best fitting contours, we see that if the escape fraction of ionizing photons from stars is only a few percent (<5 per cent; [Ciardullo2014, Matthee2016b]), the model requires an escape fraction for quasars, f_esc,q, of approximately 1.0. Although the escape fraction of ionizing photons depends on the local environment, many theoretical and observational works suggest an evolving or mass-dependent escape fraction with a decreasing average value at lower redshifts or in more massive galaxies [Kuhlen2012, Haardt2012, Paardekooper2013, Kimm2014, Wise2014, Bauer2015, Price2016].


Therefore, as discussed in [Mutch2016], in order to simultaneously match the normalization and flat slope of the observed ionizing emissivity at redshift z less than or approximately equal to 6, and the Planck tau_e measurements, a redshift-dependent escape fraction for galaxies was proposed: the escape fraction for stars as a function of redshift, f_esc,*(z), equals the minimum of [f_esc,* at z=5 times ((1+z)/6)^beta, 1]. We have shown two models with evolving escape fractions (fiducial and StellarReion) in Section 4. In this section, we further explore the possible evolution of the escape fraction by running the semi-analytic model with different combinations of the escape fraction for stars at z=5 and beta. Note that all of the ionizing photons from quasars are included in this section (the escape fraction for quasars f_esc,q = 1) in order to investigate the evolution of the stellar escape fraction with the contribution of quasars to reionization.


The bottom panels of Fig. 6 show the optical depth and the best-fitting confidence limits as functions of the normalization of the escape fraction for stars as a function of redshift, f_esc,*(z), the escape fraction for stars at z=5, and the scaling, beta, when the escape fraction for quasars f_esc,q = 1. We see that because a larger scaling suppresses the escape fraction at lower redshifts, which results in a lower emissivity at redshift z less than or approximately equal to 5, a larger normalization is required. In addition, a larger beta gives less ionizing photons at redshift z~8, which slows the process of reionization and consequently increases the optical depth. When the escape fraction for stars at z=5 reaches 0, the model becomes quasar-dominated, returning a low tau_e of around a half of the [PlanckCollaboration2016] measurement (see Fig. 5). We see that when including the contribution from quasars, the model prefers a combination of an escape fraction for stars at z=5 of approximately 6 per cent and beta of approximately 0.5.


This corresponds to an escape fraction for stars, f_esc,*, of approximately 6.5 per cent at redshift z=6 with a ratio between the emissivities of quasars and stars of approximately 0.12. In addition to the possibility that quasars do not have a very high escape fraction [Barkana2000], we note that the very faintest quasars predicted in the model are not observed. The recent detection by [Giallongo2015] only reaches to UV magnitude M_1450 ~ -18, while our fiducial models predict a significant population of faint quasars down to UV magnitude M_1450 ~ -11. Whether those undetected quasars are able to contribute a significant amount of ionizing photons is still unknown. For instance, they might be buried in the dust with a large obscuration fraction [Hopkins2007]. The critical mass above which quasars can contribute ionizing photons, coupled with the previously discussed escape fraction, represents limiting cases of a mass-dependent escape fraction for quasars.


In the top panel of Fig. 7, we present the cumulative fraction of ionizing photons as a function of black hole mass (or the corresponding UV magnitude M_1450 during the Eddington state as shown in the top axis) assuming an escape fraction for quasars f_esc,q = 1 from the fiducial model using the Tiamat simulation. We see that quasars fainter than UV magnitude M_1450 = -18 contribute approximately 80 per cent of the total emissivity at redshift z~7 with a decreasing contribution towards lower redshifts (10 per cent at redshift z~3, the end of the EoR in the QuasarReion model). This suggests that the number of fainter quasars becomes relatively smaller at later times, which can also be observed from the slope of the predicted quasar luminosity function becoming flatter from redshift z~7-3 (see Fig. 3). However, at redshifts higher than redshift z~6, the total emissivity from quasars is low.


For example, the total emissivity at redshift z~7 is five times lower than redshift z~5, suggesting that faint quasars below current observational limits provide only a small contribution to reionization. In addition, the AGN light curve adopted in this work, which assumes that black holes are either accreting with epsilon=1 or stay quiescent (epsilon=0) depending on the amount of accretion mass, has been shown to underestimate the number density of faint AGN at low redshift [Bonoli2009]. For instance, allowing epsilon to decrease progressively when the accretion disc has been mostly consumed predicts more faint AGN with M_B ~ -16 by a factor of 2 at redshift z~0.1. However, the impact becomes insignificant at brighter ranges and higher redshifts. Due to the small contribution of ionizing photons from faint quasars, AGN light curves are therefore not expected to have a significant impact on our conclusions regarding reionization.


On the other hand, due to the limited simulation volume the brightest quasars at high redshift (redshift z>=4) in our model only reach UV magnitude M_1450 ~ -23, above which the contribution of ionizing photons is not considered. However, the G15 emissivity accounts for bright quasars up to UV magnitude M_1450 = -28, 100 times brighter than the brightest quasar in our model. In order to estimate the emissivity of the missing bright quasars, we integrate the fitting functions provided by [Giallongo2015] with a magnitude interval of -28 < M_1450 < -23. We find that the total emissivity at high redshift increases by less than 1 per cent with the inclusion of the ionizing photons from these quasars. Therefore, the conclusion that quasars do not have a significant role during the EoR is not affected by the volume size. However, with the flattening luminosity function at lower redshifts (redshift z<4), bright quasars become more important and their contribution to reionization is not ignorable.


We note that including these objects will bring forward reionization in the QuasarReion model, but have no impact to the fiducial model. Our choice of black hole seed mass, 1000 per h solar masses lies between the light seed (approximately 10^2 - 10^3 solar masses) from a remnant Pop III star and the massive seed (approximately 10^3 - 10^5 solar masses) from the direct collapse of a gas cloud at early times [Greene2012]. The massive seed mass is frequently used to initialize massive haloes in hydrodynamic simulations [Springel2005, Vogelsberger2014, Schaye2014, Feng2016] while the approximately 10^3 solar masses seeds are also often adopted in semi-analytic models [Somerville2008, Bonoli2009]. We note that this seed mass assumption only affects the black hole mass at early times and in the least massive galaxies. The main conclusions of this work are not significantly affected by this assumption.


For example, with exactly the same adopted parameters but 10 times larger seed mass, the properties such as the black hole mass function, the Magorrian relation and the UV luminosity function are changed by less than 5 per cent in massive galaxies (stellar mass M_* > 10^9 solar masses). On the other hand, the model predicts a significant number of less massive black holes with masses approximately 10^5 solar masses, which is more than approximately 1 order of magnitude larger than the [Shankar2009] sample. However, this has negligible impact on the total instantaneous emissivity and consequently, the reionization history does not change significantly. Fig. 8 presents the evolution of emissivity, neutral hydrogen fraction and optical depth for the fiducial and QuasarReion models with larger black hole seed masses of 10^4 per h solar masses. Compared to the original models, we see that the quasar emissivity increases with a larger seed mass while the stellar emissivity decreases due to the stronger feedback from black holes.


However, the changes are negligible, resulting in a small perturbation to the reionization history and optical depth. We have updated the Meraxes semi-analytic model ([Mutch2016]) with a detailed prescription of black hole evolution as part of the Dark-ages Reionization And Galaxy formation Observables from Numerical Simulations (DRAGONS) project to study the role of AGN in reionization and galaxy formation at high redshift. The model is calibrated against the observed stellar mass function (redshift z~7-0.6), black hole mass function (redshift z less than or approximately equal to 0.5), quasar luminosity function (redshift z~6-0.6), ionizing emissivity (redshift z~5-2) and the Thomson scattering optical depth. The model is in agreement with the observed Magorrian relation at low redshift (redshift z<0.5) and predicts a decreasing black hole mass towards higher redshifts at a fixed stellar mass.


An opening angle of 80 deg, which corresponds to an un-obscured fraction of approximately 23.4 per cent, allows the model to reproduce the observed quasar luminosity function across a large redshift range (redshift z~6-0.6). Our model suggests that the radiation observed from recently discovered faint AGN at high redshift [Giallongo2015] may include a significant fraction of UV flux from stars. Previous direct estimates of quasar contributions to reionization based on these observations [Madau2015, Mitra2015] therefore result in an overestimate of the emissivity of quasars by a factor of 3 at redshift z~6. When we include the contribution of AGN to reionization, we find that quasars do not dominate the ionizing photon budget at redshift z>6. In a quasar-only reionization model, where the escape fractions of ionizing photons are 1 and 0 for quasars and stars, respectively, we find that reionization happens very late, redshift z~3, with a Thomson scattering optical depth of only half of the [PlanckCollaboration2016] measurement ([PlanckCollaboration2016]).


However, at low redshift, quasars are able to provide a large number of ionizing photons. With quasars contributing all of their ionizing photons (the escape fraction for quasars f_esc,q = 1), our model prefers a redshift-dependent escape fraction for stars, having the form of the escape fraction for stars as a function of redshift, f_esc,*(z), equals the minimum of [0.06 times ((1+z)/6)^0.5, 1]. This corresponds to quasars contributing 10 per cent of the total ionizing photons at redshift z~6. During one time step, for a black hole with a given initial mass of M_BH, its bolometric luminosity at Eddington rate can be calculated through the right hand of equation (17). Since the bolometric correction [Hopkins2007] adopted in this work is dependent on the bolometric luminosity, the UV magnitude, M_1450 of the quasar changes during its accretion, so does the emissivity.


In our model, because the accretion mass is always smaller than the black hole mass (Delta M_BH < M_BH, see Fig. A1), for the sake of calculation speed, we estimate the mean number of ionizing photons produced per black hole, N_gamma,q with the bolometric luminosity at the beginning of accretion. We calculate N_gamma,q as follows: 1. We calculate the UV magnitude, M_1450 using equations (19)-(22). 2. We calculate the UV flux with M_1450 in units of erg per second per Hz... 3. We calculate the flux at Lyman limit following [Giallongo2015]... 4. We calculate the instantaneous emissivity by the rate of ionizing photons dot N_ion is defined as the integral from nu_912 to infinity of F_912 * (nu/nu_912)^-alpha_q * dnu/(h*nu), which equals F_912 / (h*alpha_q). 5. The duration of accreting mass, Delta M_BH can be calculated... Therefore, the total number of ionizing photons emitted is the rate of ionizing photons dot N_ion times t_acc and the mean number of ionizing photons produced per black hole is N_gamma,q which is approximately equal to (dot N_ion at t=t_acc/2 * t_acc) / ((1-eta) * Delta M_BH / m_p).


We note that during the accretion, with an exponential increase of black hole mass, the AGN bolometric luminosity, L_bol increases exponentially. Since the B-band bolometric correction, k_B, decreases with increasing luminosity following a double power law, the rate of ionizing photons dot N_ion is a convex function of time. Therefore, the approximation in the equation for N_gamma,q underestimates the number of ionizing photons produced by black hole. In order to test whether this has a significant impact on our conclusion, we rerun the QuasarReion model assuming a constant bolometric correction with k_B(t) approximately equal to k_B at t=t_acc. Eliminating the complex dependence of time from k_B, N_gamma,q can be analytically calculated by integrating the AGN light curve. However, we note that since k_B(t) <= k_B at t=t_acc, this approximation overestimates N_gamma,q. Fig. A2 presents the evolution of emissivity, neutral hydrogen fraction and optical depth for different QuasarReion models assuming constant dot N_ion (QuasarReion) and k_B (QuasarReion_kB), respectively.


Since the time interval between two snapshots is much smaller than the Eddington accretion time-scale (t_Edd ~ 450 Myr), the black hole mass increment is still within the linear regime and therefore we see that the impact from the calculation of N_gamma,q is not significant.


Correlations between black holes and their host galaxies provide insight into what drives black hole--host co-evolution. We use the Meraxes semi-analytic model to investigate the growth of black holes and their host galaxies from high redshift to the present day. Our modelling finds no significant evolution in the black hole--bulge and black hole--total stellar mass relations out to a redshift of 8. The black hole--total stellar mass relation has similar but slightly larger scatter than the black hole--bulge relation, with the scatter in both decreasing with increasing redshift. In our modelling the growth of galaxies, bulges and black holes are all tightly related, even at the highest redshifts. We find that black hole growth is dominated by instability-driven or secular quasar-mode growth and not by merger-driven growth at all redshifts. Our model also predicts that disc-dominated galaxies lie on the black hole--total stellar mass relation, but lie offset from the black hole--bulge mass relation, in agreement with recent observations and hydrodynamical simulations.


Extensive low-redshift studies reveal a complex interplay between galaxies and the supermassive black holes that reside at their centres, with clear correlations observed between black hole mass and host bulge mass, total stellar mass, velocity dispersion and luminosity [Magorrian1998, Gebhardt2000, Merritt2001, Tremaine2002, Marconi2003, Haring2004, Bentz2009, Kormendy2013, Reines2015]; see the review by [Heckman2014]. These tight correlations suggest a co-evolution between galaxies and supermassive black holes, which may be causal, due to feedback from the active galactic nucleus [AGN; e.g.][Silk1998, Matteo2005, Bower2006, Ciotti2010] or the efficiency with which the galaxy can fuel the black hole [e.g.][Hopkins2010, Cen2015, AnglesAlcazar2017], or coincidental, simply due to mergers causing both black hole and galaxy growth [e.g.][Haehnelt2000, Croton2006b, Peng2007, Gaskell2011, Jahnke2011]. To understand what drives black hole--host co-evolution, it is necessary to study how these correlations change with redshift.


Observing high-redshift black hole--host correlations is fraught with difficulties. Host galaxies are hard to detect since they are often completely outshined by the AGN light, particularly in the rest-frame optical where common stellar mass estimators can be used [e.g.][Zibetti2009, Taylor2011]. Subtracting the quasar light has resulted in host detections out to redshift z is approximately equal to 2 [Jahnke2009, Mechtley2016], but is yet to be successful for detecting the highest redshift quasars at redshift z is approximately equal to 6 [Mechtley2012]. For these quasars, host masses are often estimated using the widths of observed submillimeter and millimeter emission lines, such as the [CII] 158 micron and CO (6--5) lines [e.g.][Wang2013]. However, dynamical masses determined from emission line widths are highly dependent on the assumptions made, such as the gas-disc geometries and inclination angles [e.g.][Valiante2014]. In fact, inclination angle assumptions can change the determined black hole mass to bulge mass ratio measurements by roughly 3 orders of magnitude [Wang2013].


In addition, the emission regions may not trace the spatial distribution of the stellar component of the galaxy, meaning that these dynamical masses may not be representative of the total stellar mass [Narayanan2009]. Determining the black hole masses of high-z quasars is also difficult, with emission-line based estimators relying on calibrations at low redshift. Where these observations are unavailable, Eddington accretion rates are instead often assumed to estimate the black hole mass [as in e.g.][Wang2013, Willott2017], which also leads to large uncertainties. High-redshift studies of the black hole--host mass relations are thus very uncertain. With this in mind, high redshift observations find black holes that are more massive than expected by the local relation, where the canonical black hole--bulge mass ratio is 10 to the power of (-2.31 +/- 0.05) for a bulge mass of 10^11 solar masses [Kormendy2013].


For example, ALMA observations of five redshift z is approximately equal to 6 quasar hosts show black hole to dynamical mass ratios ranging from 10 to the power of -1.9 to 10 to the power of -1.5 [Wang2013]. Similar studies at redshift z is approximately equal to 4--7 [e.g.][Maiolino2007, Riechers2008, Venemans2012] also give estimates for individual quasars of a black hole mass to dynamical mass ratio greater than or approximately equal to 10 to the power of -2, which is significantly larger than the local value if dynamical masses and bulge masses are assumed to be roughly equivalent. This suggests a faster evolution of the first supermassive black holes relative to their host galaxies [Valiante2014], which could potentially be a result of super-Eddington accretion [Volonteri2015]. The high observed black hole mass to dynamical mass ratio relation at high redshift could, however, be a result of selection effects [Lauer2007, Schulze2011, Schulze2014, DeGraf2015, Willott2017].


[Willott2017] suggest that since only the most massive z>6 black holes are observed, if the relation has a wide dispersion then one would expect to see a higher value due to the Lauer bias [Lauer2007]: since the luminosity function falls off rapidly at high masses, the most massive black holes occur more often as outliers in galaxies of smaller masses than as typical black holes in the most massive galaxies. Indeed, [Willott2017] found that black holes with mass less than 10^9 solar masses at redshift z>6 fall below the black hole mass--dynamical mass relation for low redshift galaxies, in contrast to the opposite being true for higher mass black holes. Similarly, [Schulze2014] claim that selection effects are the reason for the observed evolution of the black hole mass--bulge mass relation; on applying a fitting method to correct for selection effects, they find no statistical evidence for a cosmological evolution in the black hole mass--bulge mass relation.


A lack of evolution in the black hole--host relations is consistent with the findings of cosmological hydrodynamical simulations such as Horizon-AGN [Volonteri2016], which observes very little evolution in the black hole mass--stellar mass relation from redshift z=0 to 5, and BlueTides [Huang2018], which finds a black hole mass--stellar mass relation at redshift z=8 that is consistent with the local [Kormendy2013] relation. [DeGraf2015], on the other hand, found that the relation evolves slightly for redshift z greater than or equal to 1 for the highest mass black holes, with a steeper slope at the high-mass end at higher redshifts, making selection effects important. The more statistical study of [Schindler2016] found that the ratio of the black hole to stellar mass density is constant within the uncertainties from redshift z=0 to 5, with a slight decrease in the ratio at redshifts between 3 and 5; this is also consistent with no cosmological evolution in the black hole mass--stellar mass relation.


In this work we explore the evolution of the black hole--host relations with the Meraxes semi-analytic model [Mutch2016]. Meraxes is designed specifically to study galaxy formation and evolution at high redshifts, making it ideal for studying the evolution of black holes and their host galaxies. In this work we use Meraxes, a semi-analytic model designed to study galaxy evolution at high redshifts [Mutch2016]. Using the properties of dark matter halos from an N-body simulation, Meraxes analytically models the physics involved in galaxy formation and evolution. We run Meraxes on the collisionless N-body simulations Tiamat and Tiamat-125-HR [Poole2016, Poole2017]. Tiamat is ideal for studying high redshifts, with a high mass and temporal resolution. Tiamat runs from redshift z=35 to z=1.8, with a box size of (67.8 per h Mpc)^3, 2160^3 particles of mass 2.64x10^6 per h solar masses, and a high cadence of 11.1 Myr per output snapshot at redshift z>5.


Tiamat-125-HR is a low-redshift counterpart to Tiamat, running from redshift z=35 to z=0 with the same temporal resolution, but with a lower mass resolution (1080^3 particles of mass 1.33x10^8 per h solar masses) and larger box size of (125 per h Mpc)^3, more suited for low-redshift studies. Throughout this work, we use the higher resolution Tiamat at high-redshifts, and Tiamat-125-HR for redshift z<2, unless otherwise specified. Meraxes assumes that galaxies reside in the centre of dark matter haloes produced by the N-body simulation. Using the properties of these haloes, Meraxes analytically models the baryonic physics involved in galaxy formation and evolution, such as gas cooling, star formation, black hole growth, and supernova and black hole feedback. These analytical prescriptions involve a range of free parameters, which must be calibrated using observations such as the stellar mass function.


In Meraxes, stars in galaxies reside in three components: an exponential disc, a spheroidal merger-driven bulge and a disc-like instability-driven bulge. Bulges grow through both galaxy-galaxy mergers and disc-instabilities. In Meraxes, we assume that galaxy mergers with merger ratio greater than 0.01 trigger a burst of star formation, by causing shocks and turbulence in the cold gas of the parent galaxy. The galaxy will also accumulate the mass of the secondary galaxy. We assume that the dominant mass component of the primary galaxy will regulate where these stars produced by the burst and the secondary's mass will be deposited. If the primary is dominated by a discy component, the mass is added to the instability-driven bulge. Otherwise, we assume that the new stars will accumulate in shells around the spheroidal merger-driven bulge. In major mergers, where the merger ratio is greater than 0.1 or 0.3, we assume that the stellar disc and instability-driven bulges are destroyed, with all stars placed into the merger-driven bulge.


In our model we assume that the galaxy discs are thin, with an exponential surface density and flat rotation curve. Such discs become unstable if the disc mass is greater than the disc velocity squared times the scale radius divided by the gravitational constant, which equals the critical mass [Efstathiou1982, Mo1998]. Here, we take the disc mass as the combined mass of both gas and stars in the disc, and the disc velocity and scale radius as the mass-weighted velocity and scale radius of the stellar and gas discs. If such a disc instability occurs, Meraxes returns the disc to stability by transferring the unstable mass of stars from the disc to the instability-driven bulge. The Meraxes black hole model was introduced in Q17, and updated to include instability-driven growth in M19. In Meraxes, black holes are seeded in every newly-formed galaxy, with a seed mass of 10^4 solar masses. Black holes then grow by accretion of both hot and cold gas, through the radio- and quasar modes, respectively.


We also assume that black holes grow in galaxy mergers, with the black holes in each galaxy merging together. Black holes accrete hot gas from the static hot gas reservoir around the galaxy, at a fraction of the Bondi-Hoyle accretion rate. We consider this fraction a free parameter, which adjusts the efficiency of radio-mode black hole growth [Croton2016]. This accretion is limited by the amount of hot gas in the reservoir and the Eddington limit. A fraction of this accretion mass is radiated away and so during one snapshot, black holes grow through the radio-mode by the remaining mass. We include the effects of radio-mode AGN feedback by assuming that a fraction of the radiated energy is coupled to the surrounding gas, adiabatically heating a mass which is subtracted from the cooling flow, regulating the accretion of new gas onto the black hole [Croton2006a, Croton2016]. This AGN feedback has no significant effect on the results of Tiamat at redshift z greater than or equal to 2.


Black holes accrete cold gas from the galaxy, when triggered by either a galaxy-galaxy merger or a disc instability. During such an event, the black hole mass grows by a certain amount, where the virial velocity and a free parameter adjust the growth efficiency. For merger-triggered growth, we take the efficiency parameter to be proportional to the merger ratio. For instability driven growth, we consider two separate free parameters. During the quasar mode, black holes are assumed to accrete at the Eddington rate, and thus the mass accreted by the black hole during one simulation snapshot is limited. This can result in the mass being accreted over multiple simulation snapshots. We incorporate quasar-mode AGN feedback by considering the energy injected into the gas during a simulation time-step. We assume that this energy generates a wind that heats the cold disc gas and transfers it to the hot gas reservoir, depleting the supply of cold gas available for the black hole to accrete. If sufficient energy is injected by the quasar, this wind can also eject the hot gas.


We calculate the bolometric luminosities of each black hole in the model following the Q17 method, which assumes Eddington luminosity for all accreting black holes, and self-consistently calculates the duty cycle. We consider the luminosities from both the quasar- and radio-modes of accretion. At high-redshifts the contribution from the radio-mode is negligible. At the lowest redshifts (redshift z less than or equal to 2), the radio-mode becomes a more significant growth mechanism for the most massive black holes, and so their luminosities are enhanced slightly by the addition of the radio-mode luminosity. We convert from bolometric to B-band luminosities using the [Hopkins2007] bolometric correction, and then assume a continuum slope of 0.44 to convert to UV luminosities. We also account for obscuration due to quasar orientation, by scaling the UV luminosity function by a factor related to the opening angle of quasar radiation. In our model we assume a constant opening angle, for simplicity, which is a free parameter in our model.


In M19 we calibrated the free parameters in Meraxes to match the observed stellar mass functions at redshift z=0--8, and the black hole--bulge mass relation at redshift z=0. Using this model, we find that the black hole mass function and quasar luminosity functions are much larger than predicted by the observations. In addition, we note that [Shankar2016] find significant selection biases in the black hole--bulge mass relation---a topic of recent debate [see e.g.][Kormendy2019]. Due to the M19 predictions and this potential bias, we assume that the [Shankar2009] redshift z=0 black hole mass function is a less biased indicator of the local black hole population, and retune the model here to better reproduce the black hole observations. Note that we use the same parameter values for Tiamat and Tiamat-125-HR, and use both simulations to tune the model: Tiamat for matching redshift z greater than or equal to 2 observations and Tiamat-125-HR for redshift z<2.


We find that our results from the two simulations are generally consistent at redshift z is approximately equal to 2, with broad qualitative agreement at higher redshifts. We calibrate the free parameters in the model to match the observed stellar mass functions at redshift z=0--8, the [Shankar2009] and [Davis2014] black hole mass function at redshift z=0, and the quasar X-ray luminosity functions from redshift z=5 to 2. Since [Shankar2016] find that the observed black hole--bulge mass relation is biased to high black hole masses, we also require our model to not over-predict this relation, however we do not otherwise tune to it. We note that our best models produce black hole--host mass relations lower than the observations, consistent with the expectations of [Shankar2009], and have steeper slopes. We find that these criteria are met by a range of free parameter values for the merger-driven black hole growth efficiency, and the definition of a major merger.


We note that all of these parameter sets produce very similar results. As a further check of the black hole population, we plot the black hole accretion rate density as a function of redshift for models with these different merger-driven black hole growth efficiencies. We find that the models with lower efficiencies give black hole accretion histories in approximate agreement with the observations. The larger efficiencies overproduce measurements of the black hole accretion rate density [e.g.][Delvecchio2014]. The opening angle of AGN radiation, theta, adjusts the normalization of the UV luminosity function. We tune this to match the observations, finding a preferred theta of 70 degrees, corresponding to an observable fraction of UV quasars of 18 per cent. We show the quasar X-ray luminosity functions at redshift z=5--0, with X-ray luminosities calculated using the [Hopkins2007] bolometric to X-ray correction.


At redshift z=2 the model and the observations agree remarkably well. At redshift z>2 the model over-predicts the observed quasar X-ray luminosity function at intermediate luminosities, by up to ~0.7 dex at redshift z=4, while at redshift z<2 the model under-predicts the luminosity function at these luminosities. Our model shows better agreement with the observations than previous versions of Meraxes. While the observations show a slight increase in the X-ray quasar luminosity functions from redshift z=4 to 2, the model predicts a slight decrease. In fact, we cannot find a combination of black hole parameters that results in a redshift evolution that matches that of the observed X-ray quasar luminosity function at redshift z>2. However, the key quantity of black hole accretion rate density is predicted by the model to peak at redshift z=2 as observed.


In addition to published uncertainties in the observations, it may also be the case that at higher redshifts X-ray AGN are more likely to be obscured, which is consistent with evidence from a range of X-ray observations [Treister2006, Vito2014, Buchner2015]. Thus we argue that the inability of our model to match the redshift evolution of the X-ray quasar luminosity function may not represent a significant concern. We show the quasar UV luminosity functions at redshift z=5--0. We find that, as with the X-ray luminosity function, the UV luminosity function decreases from redshift z=5 to 0, though it agrees well with observations at redshift z>2. At redshift z<2, however, we note that the faint-end of the UV luminosity function becomes flat, and by redshift z<1 there is a significant disagreement with the observations, with the model producing too many luminous quasars.


As seen in analysis, the black hole accretion rate density becomes significantly higher than the observations at redshift z<1, consistent with the quasar luminosities being overestimated at these redshifts. This excess black hole accretion is most likely a result of the model missing important physics required for modelling low-redshift galaxy evolution, particularly in the quenching of massive galaxies, or due to the simplifications assumed in the model such as a constant black hole accretion efficiency. However, as the overall accretion rate density at these redshifts is low, this will not have a significant impact on the black hole mass, an integrated quantity. Thus, while the redshift z<1 black hole accretion rates are overestimated, the black hole mass function and black hole--host mass relations are reliable at low redshifts. Indeed, we find that assuming a lower Eddington ratio significantly improves the match between the model and observed UV luminosity functions at redshift z<1.


However, this causes the model to no longer match the observations at higher redshifts. Thus, some evolving Eddington ratio is necessary for Meraxes to accurately reproduce the redshift z<1 quasar UV luminosity function. We now use the model described in Sections 2 and 3 to explore black hole growth. We investigate the redshift evolution of the black hole--host scaling relations. To investigate the redshift evolution of the black hole--bulge and black hole--total stellar mass relations we first perform linear least squares fits to the relations at a range of redshifts. We only include galaxies with mass > 10^9.5 solar masses in our fits, so that they are not biased by the large number of low-mass galaxies. Both relations have a slope and normalization that increase with redshift from redshift z=0 to 2, with much weaker evolution for redshift z>2. Relative to the scatter in the relations, we see minimal evolution in both the black hole--bulge and black hole--total stellar mass relations from redshift z=0 to 6.


This lack of evolution in the black hole--host mass relations is consistent with the findings of cosmological hydrodynamical simulations such as Horizon-AGN [Volonteri2016] and BlueTides [Huang2018]. We find that our black hole--total stellar mass relation has similar but slightly larger scatter than the black hole--bulge relation, with the scatter in both decreasing with increasing redshift. While the black hole mass has a slightly stronger relationship with the bulge stellar mass, the black hole and total stellar mass are still tightly correlated. The scatter in the relations is slightly larger than the 0.28 dex observed by [Kormendy2013] locally. However, they are very consistent with those from the BlueTides simulation at high redshift. The scatter decreases with increasing stellar mass. The median black hole mass to total stellar mass ratio as a function of redshift for galaxies with black hole mass > 10^6 solar masses shows no statistically-significant evolution out to redshift z is approximately equal to 8.