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 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. This is consistent with current high redshift observations; when selection effects are accounted for, the observations at high redshift are consistent with no cosmological evolution in these relations [Schulze2014]. Our model predicts no significant evolution in the black hole--host mass relations, with the scatter in the relations decreasing at the highest redshifts. This indicates that there is a connection between the growth of black holes and their host galaxies. Indeed, our model includes joint triggering of star formation and black hole growth during galaxy mergers, and black hole feedback which regulates star formation, meaning that the co-evolution of black holes and galaxies is implicit in our model.


This is not consistent with the scenario proposed by [Peng2007] and [Jahnke2011], for example, where the black hole and galaxy growth is uncorrelated and the relationships are generated naturally within a merger driven galaxy evolution framework, due to a central-limit-like tendency. The median black hole mass to total stellar mass ratio as a function of redshift with galaxies split into black hole mass bins shows that lower mass black holes have lower mass ratios than higher mass black holes. This will lead to a notable selection bias, since when observing the most massive black holes, the measured ratio will be higher than that of the entire population. This is generally expected for any sample selected by black hole mass or luminosity where the scatter in the relation is large [e.g.][Lauer2007]. Finally, we note an interesting effect of changing the parameter controlling the black hole efficiency for converting mass to energy.


For a higher efficiency, the median black hole--stellar mass ratio decreases at redshifts z greater than or approximately equal to 6, instead of remaining constant with redshift. We investigate the cause of this high-redshift decrease in the black hole--host relation by considering the Eddington limit. Increasing the efficiency from 0.06 to 0.2 decreases the Eddington limit. This results in many black holes having Eddington-limited growth at the highest redshifts (redshift z greater than or approximately equal to 6), which is not the case for the lower efficiency model. This causes black holes to grow slower than their host galaxies at high redshifts, resulting in a decreased black hole--stellar mass ratio. Observing the high-redshift black hole--stellar mass relation may therefore probe the Eddington limit and the efficiency of black holes in converting mass to energy.


We consider the cumulative fraction of black hole mass formed through each of the mechanisms in our model: black hole seeding, merger-driven quasar-mode accretion, instability-driven quasar-mode accretion, radio-mode accretion and black hole--black hole coalescence in galaxy mergers. The merger-driven growth mode becomes more important at low redshifts, at both low- and high-black hole masses. On average, instabilities grow the majority of mass in black holes at all redshifts, except for galaxies with black hole mass > 10^9 solar masses at redshift z is approximately equal to 0, whose black hole growth becomes dominated by galaxy mergers. Radio-mode growth slowly increases in significance with redshift, yet still has only contributed to a small proportion of the total black hole mass by redshift z=0, except at the highest masses. Note that we consider growth from disc instabilities that are triggered by earlier galaxy mergers as growth via the instability-driven mode.


We also consider the instantaneous growth fractions of black hole mass formed through each mechanism as a function of redshift. As discussed, the model produces unreliable black hole accretion rates at redshift z<1, and so we only consider these black hole growth rates at redshift z>1. The instability-driven growth mode is the dominant growth mechanism, on average, at all redshifts, regardless of black hole mass. The merger-driven quasar mode and black hole--black hole coalescence mode are sub-dominant at all redshifts. The radio-mode grows more mass at low redshift and in the most massive galaxies, with the percentage of total instantaneous black hole growth from this mode increasing from only 0.1 per cent at redshift z=5 to almost 5 per cent at redshift z is approximately equal to 1. Our finding that mergers are not the dominant mechanism for growing black holes is in agreement with a range of observations.


For example, [Koss2010] find that only 25 per cent of local, moderate luminosity X-ray AGN show signs of mergers, though the fraction is much higher for luminous AGN [Hong2015]. From redshift z from approximately 0.3 to 1.0, [Cisternas2010] find that the vast majority (>85 per cent) of X-ray selected AGN do not show signs of mergers, suggesting that the bulk of their black hole accretion has been triggered by some other mechanism. This is also consistent with the findings of [Georgakakis2009], [Villforth2018], [Schawinski2012], [Mechtley2016], [DelMoro2015] and [Marian2019] for AGN at various redshifts. Our result that disc instabilities cause the majority of black hole growth is also consistent with predictions from other simulations. In the GALFORM semi-analytic model, [Fanidakis2011] found that the growth of black holes is dominated by accretion due to disc instabilities.


Using an updated GALFORM model, [Griffin2019] found that accretion of hot gas dominates the growth of black holes at redshift z<2, with disc-instabilities dominant at higher redshifts. [Hirschmann2012] found that instability-driven black hole growth was required to reproduce AGN downsizing, and that while major mergers are the dominant trigger for luminous AGN, especially at high redshift, disc instabilities cause the majority of black hole growth in moderately luminous Seyfert galaxies at low redshift. [Menci2014] find that in their semi-analytic model disc instabilities can provide enough black hole accretion to reproduce the observed AGN luminosity functions up to redshift z is approximately equal to 4.5, but are not likely to be dominant for the highest luminosity AGN or at the highest redshifts. In contrast, [Shirakata2018] find that the primary trigger of AGN at redshift z less than or equal to 4 in their semi-analytic model is mergers.


The hydrodynamical simulation Horizon-AGN found that only ~35 per cent of black hole mass in local massive galaxies is directly attributable to merging, with the majority of black hole growth instead growing via secular processes [Martin2018]. The Magneticum Pathfinder Simulation also found that merger events are not the dominant fuelling mechanism for black holes in redshift z=0--2, with merger fractions less than 20 per cent, except for very luminous quasars at redshift z is approximately equal to 2 [Steinborn2018]. Finally, we comment on the effect of the efficiency parameters for merger-driven and instability-driven black hole growth in the model. We find the instability-driven efficiency from tuning the model, whereas the merger-driven efficiency is less constrained, with several values producing reasonable model results. Having a merger growth efficiency that is twice, six times or even 18 times larger than the instability-driven growth efficiency may have an effect on the conclusions outlined above.


We find, as expected, that models with larger merger efficiencies result in more merger-driven growth. For a merger efficiency twice the instability efficiency, the instability-driven mode still dominates at redshift z=2, while for a six times larger efficiency, the merger-driven mode begins to dominate at the highest black hole masses. For the model with an 18 times larger merger efficiency, the merger-driven mode contributes even more black hole growth, but is still not the dominant growth mode for black holes with mass between 10^6 and 10^9 solar masses. Thus, while the efficiency parameter for merger-driven growth has some effect on the relative distributions of the instability-driven and merger-driven growth modes, the instability-driven mode is still dominant for the majority of black holes, even if the merger growth efficiency is as much as 18 times larger than the secular growth efficiency.


A popular explanation for the black hole--host correlations is that major mergers drive the growth of both black holes and bulges [e.g.][Haehnelt2000, Croton2006b]. If this were the case, one would expect that black holes would only correlate with galaxy properties directly related to the merger process, such as bulge mass, and not, for example, total stellar mass. [Simmons2017] consider a sample of 101 disc-dominated AGN hosts from the SDSS, which they assume must have a major merger-free history since redshift z is approximately equal to 2. They found that these galaxies lie on the typical black hole mass--total stellar mass relation, but lie offset to the left of the black hole mass--bulge mass relation. This indicates that the substantial and ongoing black hole growth in these merger-free disc galaxies must be due to a process other than major mergers, and that major mergers cannot be the primary mechanism behind the black hole--host correlations.


We plot the black hole mass--total stellar mass and black hole mass--bulge mass relation for disc-dominated and bulge-dominated galaxies at redshift z=0. Our simulated disc galaxies lie on the black hole mass--total stellar mass relation, but lie offset to the left of the black hole mass--bulge mass relation, as they have small bulges relative to their black hole mass. This is consistent with the [Simmons2017] observations, and the results from the Horizon-AGN hydrodynamical simulation [Martin2018]. However, we see a less significant offset, which occurs at lower black hole masses than [Simmons2017] and [Martin2018], since the black holes in our disc-dominated galaxies are less massive in comparison. [Mutlu-Pakdil2017] also find no dependence of the black hole mass--total stellar mass relation on galaxy type in the Illustris hydrodynamical simulation. [Martin2018] suggest that major mergers therefore cannot be primarily responsible for feeding black holes.


This is consistent with our finding that the instability-driven mode is the dominant growth mechanism for black holes. We use the Meraxes semi-analytic model to investigate the evolution of black holes and their relations to their host galaxies. We find the following key predictions of our model: There is minimal statistically-significant evolution in the black hole--bulge and black hole--total stellar mass relations out to high redshifts (redshift z is approximately equal to 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. This indicates that the growth of galaxies, bulges and black holes are all tightly related, even at the highest redshifts. Higher mass black holes have higher black hole--total stellar mass ratios, leading to a significant selection effect in measurements of this ratio when observing only the most massive black holes.


The instability-driven or secular quasar-mode growth is the dominant growth mechanism for black holes at all redshifts. The contribution from merger-driven quasar-mode growth only becomes significant at low redshift for black holes with mass greater than or approximately equal to 10^9 solar masses. Disc-dominated galaxies lie on the black hole--total stellar mass relation, but lie offset from the black hole--bulge mass relation. Our simulation is limited in making predictions for the highest redshift quasars at redshift z=6--7 due to the simulation box size and resolution. In future work we will run Meraxes on larger N-body simulations in order to make predictions for these objects. We calibrate the free parameters in Meraxes to match the observed stellar mass functions at redshift z=0--8 and the [Shankar2009] and [Davis2014] black hole mass function at redshift z=0. The black hole mass functions produced by Tiamat and Tiamat-125-HR are converged at redshift z=2 for black holes with mass > 10^7.1 solar masses [see Marshall2019].


We therefore focus on matching the observed black hole mass functions at masses > 10^7.1 solar masses. While the [Shankar2009] and [Davis2014] relations are different, particularly at black hole mass ~ 10^8.5 solar masses, they are similar relative to the freedom we have in adjusting our model black hole mass function, and so when calibrating we found the most reasonable fit to both. In the model stellar mass function, we also plot the Meraxes stellar mass function produced when AGN feedback is switched off. This shows that AGN feedback has no effect on galaxies in Tiamat at redshift z greater than or equal to 2, but suppresses the growth of the most massive galaxies at lower redshifts as seen in Tiamat-125-HR. Throughout this work, we use the higher resolution Tiamat simulation at redshift z greater than or equal to 2, and Tiamat-125-HR for redshift z<2, where Tiamat is unavailable.


We find that the results discussed in this paper are generally consistent between the two simulations at redshift z is approximately equal to 2. However, one notable result is that the best-fitting black hole--stellar mass relations change rapidly between redshift z=2 (using Tiamat) and z=1 (using Tiamat-125-HR). To verify that this jump is not purely a result of the simulation change, we show the best-fitting relations from redshift z=6--0 using Tiamat-125-HR. The Tiamat-125-HR simulation shows similar results to those found using Tiamat at redshift z greater than or equal to 2, with a slightly milder but still relatively rapid evolution from redshift z=2 to z=1. The qualitative result of the evolution being insignificant relative to the scatter in the relation still holds. Thus, while the change in simulation slightly amplifies the rapid change in the black hole--stellar mass relations from redshift z=2 to z=1, this does not change our conclusions.


We also note that where the black hole mass functions are converged, the black hole--stellar mass relations are in good agreement between the two simulations.


The hyperfine 21-cm transition of neutral hydrogen from the early Universe (z>5) is a sensitive probe of the formation and evolution of the first luminous sources[cite: 1007]. Using the Fisher matrix formalism we explore the complex and degenerate high-dimensional parameter space associated with the high-z sources of this era and forecast quantitative constraints from a future 21-cm power spectrum (21-cm PS) detection[cite: 1008]. This is achieved using Meraxes, a coupled semi-analytic galaxy formation model and reionisation simulation, applied to an N-body halo merger tree with a statistically complete population of all atomically cooled galaxies out to z~20[cite: 1009]. Our mock observation assumes a 21-cm detection spanning z in [5, 24] from a 1000 h mock observation with the forthcoming Square Kilometre Array and is calibrated with respect to ultraviolet luminosity functions (UV LFs) at z in [5, 10], the optical depth of CMB photons to Thompson scattering from Planck, and various constraints on the IGM neutral fraction at z > 5[cite: 1010].


In this work, we focus on the X-ray luminosity, ionising UV photon escape fraction, star formation and supernova feedback of the first galaxies[cite: 1011]. We demonstrate that it is possible to recover 5 of the 8 parameters describing these properties with better than 50 per cent precision using just the 21-cm PS[cite: 1012]. By combining with UV LFs, we are able to improve our forecast, with 5 of the 8 parameters constrained to better than 10 per cent (and all below 50 per cent)[cite: 1013].


Following recombination, the early Universe entered the cosmic Dark Ages characterised by a neutral intergalactic medium (IGM) and the absence of luminous sources[cite: 1014]. The formation of the first stars and galaxies ushered in the era of the Cosmic Dawn[cite: 1015]. The intense ionising ultraviolet (UV) radiation, characteristic of the young and massive stars, as well as X-rays (possibly from high-mass X-ray binaries, e.g. [MesingerXRay]) impacted the thermal and ionisation state of the IGM[cite: 1016]. This period was brought to its conclusion in the Epoch of Reionisation (EoR) when the UV photons ionised the neutral hydrogen rendering it transparent to UV photons[cite: 1017]. Considerable effort has been expended in the last few decades to unravel the complex physics of this period (see [EoRbook] and references therein)[cite: 1018]. The forbidden 21-cm hyperfine transition signal is well-suited for this purpose because of its extreme sensitivity to the formation and evolution of the first stars and galaxies (z <~ 30) as well as the various feedback mechanisms in the early Universe[cite: 1019].


Though the ultimate goal of 21-cm interferometric experiments is to map the tomography of the IGM as a function of frequency (redshift) the first set of observations will be statistical in nature[cite: 1020]. The 21-cm signal is observed as a brightness temperature against a background source (which is almost always assumed to be the cosmic microwave background (CMB) radiation; e.g. [BibleReview, StuartReview, PritchardReview])[cite: 1021]. The 21-cm power spectrum (21-cm PS) quantifies the fluctuations in the brightness temperature of the 21-cm signal across the sky[cite: 1022]. Though a detection has yet to be made, current experiments such as the Murchison Widefield Array (MWA; [MWA_main]), LOw Frequency ARray (LOFAR; [LOFAR_main]), Hydrogen Epoch of Reionization Array (HERA; [HERA_main]) have already begun setting upper limits on the 21-cm PS (see [LOFAR_limits, MWA_limits, HERA_limits])[cite: 1023]. The upcoming Square Kilometre Array (SKA; [SKA_main]) will revolutionise 21-cm EoR cosmology with its unprecedented sensitivity[cite: 1024].


The amplitude and shape of the 21-cm PS are extremely sensitive to the thermal and ionisation state of the IGM [cite: 1025] and hence the astrophysics of early galaxy formation and evolution[cite: 1025, 1025]. It is therefore imperative that realistic and efficient models (see [GnedinMadau] for a recent review) are available to interpret current and upcoming observations[cite: 1026]. Despite significant progress in this direction, there is considerable uncertainty about the properties of the underlying source populations of these models[cite: 1027]. In this paper, we ask: What can we learn about the underlying physical processes driving reionisation from a successful detection of the 21-cm PS? Simulating reionisation requires large volumes (>=200 h^-1 Mpc according to [Iliev2014, Kaur2020] for convergent reionisation and 21-cm statistics)[cite: 1028]. As a result, a number of models have been developed to make large-scale but computationally efficient realisations of the early Universe (e.g. [Battaglia2013, GRIZZLY, Simfast, SCRIPT, 21cmFASTv3])[cite: 1030].


To investigate the utility of the 21-cm PS as a probe of galaxy formation, MCMC methods have been utilised to place constraints on the parameterised properties (e.g. UV and X-ray) of population II & III star-forming galaxies (e.g. [CMMC1, CMMC2, Park2019, Yuxiang_tale1, Yuxiang_tale2, Maity_2022, Bevins_2023])[cite: 1031]. In this work, we use Meraxes [Dragons3] - a semi-analytic model (SAM) of galaxy formation (see [SomervilleReview] for a recent review) and evolution self-consistently coupled to a reionisation model - to forecast constraints on astrophysical properties of the early galaxies[cite: 1032]. Unlike semi-numerical models (for example, [21cmFAST], [SCRIPT]), which focus on population-averaged quantities (i.e. these models generally have no galaxies), Meraxes provides a realistic population of galaxies as sources of photons[cite: 1033]. Meraxes incorporates a detailed, physically motivated galaxy formation and evolution model that includes baryonic infall, gas cooling, star formation, supernova feedback, active galactic nuclei feedback, galaxy mergers, etc[cite: 1034].


By efficiently coupling the reionisation of the IGM via a modified version of 21cmFAST [21cmFAST, 21cmFASTv3] and an underlying galaxy population sourced from a dark matter only N-body simulation, Meraxes is thus well-suited to explore the underlying parameter space of the complex astrophysics of this era[cite: 1035]. We deploy Meraxes on a 210 h^-1 Mpc cosmological simulation which resolves all the atomically cooled galaxies down from z~20[cite: 1036]. The large volume and high mass resolution of our simulation make an MCMC analysis using Meraxes prohibitively expensive computationally[cite: 1037]. Using a Fisher matrix analysis, we forecast the constraints on a total of 8 astrophysical parameters in our model that directly control the X-ray luminosity, UV escape fraction, star formation rates and supernova feedback of the galaxies of the high-z Universe[cite: 1039]. Focusing on the upcoming SKA1-low, we forecast constraints from the 21-cm PS before exploring the improvements available when combining information from the UV LFs[cite: 1040].


We use the L210_N4320 dark matter-only N-body simulation of the Genesis suite of simulations (Power et al. in prep)[cite: 1047]. Containing 4320^3 particles in a 210^3 h^-3 Mpc^3 volume, the simulation has a mass resolution of ~5x10^8 h^-1 M_sun[cite: 1048]. L210_N4320 was run with the SWIFT [SWIFT] cosmological code, using the VELOCIraptor [VELOCIRAPTOR] halo identifier and merger trees generated using TreeFrog [TREEFROG][cite: 1049]. The simulation consists of 120 snapshots between redshifts 30 and 5[cite: 1050]. In order to resolve haloes down to the atomic cooling limit at z=20, the L210_N4320 simulation was augmented using the Monte-Carlo algorithm-based code dforest [DARKFOREST] achieving an effective halo mass resolution of ~2 x 10^7 h^-1 M_sun (genpsim of [Balu2022]) [cite: 1050]. dforest achieves this by sampling from a conditional mass function based on the extended Press-Schechter theory [EPS1, EPS2, EPS3] that has been modified to match the halo mass functions from N-body simulations[cite: 1051].


Meraxes [Dragons3] contains detailed and physically motivated prescriptions for galaxy formation and evolution[cite: 1063]. These include, for example, gas infall into dark matter haloes from the IGM followed by its radiative cooling and subsequent star formation, as well as eventual supernova feedback and metal enrichment of the ISM[cite: 1064]. Active galactic Nuclei (AGN) feedback from central black holes of the galaxies was implemented in [Dragons10] and calculations of the galaxies' UV luminosity in [Dragons19][cite: 1065]. In addition, Meraxes also has a coupled treatment of reionisation based on the 21cmFAST semi-numerical code ([21cmFAST, 21cmFASTv3]; see section 2.3 for more details)[cite: 1066]. At every snapshot, the baryonic content of a dark matter halo increases by (1-f_mod)*f_b*M_vir, where f_b is the baryonic fraction and f_mod is the baryon fraction modifier - set by the local IGM ionisation state from the previous snapshot - coupling galaxy growth to the ionisation state of the IGM[cite: 1069].


Star formation in Meraxes follows the disc stability argument of [Kauffmann1996], wherein gas participates in star formation when the cold-gas mass is higher than a critical mass given by Equation 7[cite: 1071]. The new stellar mass, Delta_M_star, formed in the time-step Delta_t is given by Equation 8, where alpha_SF is the star formation efficiency[cite: 1072]. In this work, both Sigma_SF and alpha_SF are free parameters in our astrophysical model[cite: 1073]. The primary impact of SNe feedback is to heat the gas reservoirs of the halo resulting in the transfer of the gas from the cold to the hot gas reservoirs and in extreme cases the removal of the gas from the hot halo[cite: 1077]. Our implementation, modified from [Guo2011] to take advantage of our high cadence merger trees [Dragons19], is based on energy conservation[cite: 1078]. The total stellar mass going SNe at a particular snapshot of the simulation depends on both the current and previous star formation[cite: 1079]. To account for this, we track the stars formed in the present and four previous snapshots[cite: 1080].


Following [Dragons19], the supernova energy coupling efficiency (epsilon) and mass loading factor (eta) are implemented as functions of halo circular velocity and redshift, as shown in Equations 12 and 13[cite: 1085]. These are controlled by two free normalization parameters, epsilon_0 and eta_0, respectively[cite: 1086]. We forecast the fractional uncertainty on both epsilon_0 and eta_0[cite: 1087]. The fraction of ionising photons escaping into the IGM from galaxies plays an important role in regulating the ionisation fraction and morphology[cite: 1088]. Following [Dragons3] we employ an escape fraction (f_esc) prescription for galaxies that is solely redshift dependent ([though see e.g.][Kimm2017, Yeh2023])[cite: 1090]. This results in an f_esc that is skewed towards higher z, motivated by two factors: it is easier for photons to escape the shallower potential wells of high-z galaxies, and early galaxies have less dust attenuation[cite: 1091, 1092, 1093]. The escape fraction is given by Equation 14, and we allow both the normalisation (f_esc,0) and redshift scaling (alpha_esc) to be free parameters[cite: 1094].


The thermal evolution and ionisation state of the IGM follows 21cmFAST [21cmFAST, 21cmFASTv3], modified to take advantage of our realistic galaxy population[cite: 1095]. We divide our simulation into 1024^3 voxels corresponding to a cell dimension of ~0.21 h^-1 Mpc[cite: 1098]. The thermal state of a neutral hydrogen cloud is characterised by the spin temperature T_S. Even though T_S is influenced by both UV and X-ray photons, the latter has a considerably more pronounced impact (e.g. [MesingerXRay]) and is computed via Equation 15[cite: 1101]. We compute the comoving X-ray emissivity as shown in Equation 17, where L'_X/SFR is the specific X-ray luminosity per SFR[cite: 1107]. This is one of the free parameters of our model and has units [erg s^-1 M_sun^-1 yr]. We impose a lower limit E_0, another free parameter, which is motivated by the absorption of low-energy X-rays within the galaxy itself[cite: 1110].


We compute the ionisation (x_HII) grid by employing an excursion-set formalism [Furlanetto2004] by comparing the total number of ionising photons to the combined number of neutral atoms and recombinations within spheres of decreasing radii[cite: 1112]. Grid voxels inside a sphere are deemed to be ionised if Equation 19 is satisfied[cite: 1113]. The number of stellar baryons is inferred from the stellar mass that is dependent on the star formation histories of all star-forming galaxies[cite: 1117]. We compute a baryon modifier, f_mod, for each galaxy based on their local neutral hydrogen value, enabling us to couple a galaxy's growth to its local IGM ionisation state[cite: 1118]. In the next snapshot, the amount of fresh gas accreting onto the galaxy is modulated by f_mod which is calculated following [Sobacchi2013] based on the 'filtering mass'[cite: 1119]. In this manner, by coupling galaxy growth to the local ionisation state as well as the local UV ionising background, Meraxes enables a self-consistent reionisation scenario[cite: 1121].


The genpsim simulation from [Balu2022] was calibrated against existing observables including UV LFs from [Bouwens2015] and [Bouwens2021], and stellar mass functions from [Song2016] and [Stefanon2021][cite: 1127]. The reionisation history was calibrated using the Thomson scattering optical depth of free electrons to CMB photons from the [Planck2018][cite: 1128]. Figure 2 shows the dimensionless 21-cm PS from our fiducial model[cite: 1129]. The grey-shaded region represents the sensitivity for a 1000 h observation with the upcoming SKA1-low[cite: 1130]. We use the 'moderate' foreground removal case from [Pober2014] effectively ignoring all the k-modes falling within the 21-cm foreground wedge (setting a k_min=0.16 Mpc^-1)[cite: 1131]. The k_max=1.4 Mpc^-1 is set by a combination of the spatial scales resolved by SKA1-low and scales we trust not to be dominated by the Poisson noise of the sources[cite: 1132].


The sensitivity of a radio interferometer to the 21-cm PS can be divided into two components: thermal noise and sample (cosmic) variance[cite: 1143]. The total noise power is given by adding these two in quadrature[cite: 1144]. We focus on a future observation of the 21-cm PS by the SKA1-low[cite: 1146]. We only include the stations in the 'Central Area', resulting in 296 stations, and calculate the sensitivity using the 21cmSense python package [Pober2013, Pober2014][cite: 1148]. We assume a total of 1080 hours of observation and a sky temperature dominated by galactic synchrotron emission [T_Sky][cite: 1149]. We combine partially coherent baselines to improve power spectrum sensitivity and use the `moderate` foreground removal scenario of 21cmSense wherein we avoid the modes that are contained within the foreground wedge [Datta2010][cite: 1151]. As we ignore the modes that fall within the foreground wedge, we are limited to k_min = 0.16 Mpc^-1 for our analysis[cite: 1152].


In addition to exploring constraints from the 21-cm PS we also consider the improvements achievable with a joint analysis with the UV LFs[cite: 1156]. Here, we adopt the parameterisation for the UV optical depth that depends on the dust-to-gas ratio [Charlot2000] of the galaxies[cite: 1158]. The model differentiates between a short-lived birth cloud of the stars as well as the interstellar medium (ISM) of the galaxy[cite: 1159]. The free parameters of Meraxes have been calibrated [Dragons19, Balu2022] to their fiducial values against infrared excess (IRX)-beta, UV LFs, and stellar mass functions at z>5[cite: 1161]. The blue curves in Fig. 3 are the UV LFs from our calibrated simulation[cite: 1162]. The error bars are determined by multiplying our simulated UV LF data by the corresponding fractional uncertainty on each data point from the [Bouwens2021] observational data[cite: 1163]. In addition to the 21-cm PS, we thus have 6 UV LFs from z in [5, 10][cite: 1164].


To place quantitative constraints on the model parameters we use the Fisher information matrix ([Tegmark1997, Albrecht2009])[cite: 1165]. For any set of observations, the Fisher matrix provides the best possible constraints on the parameters of an assumed model[cite: 1166]. An implicit assumption is that the errors on these parameters are Gaussian and that the observational data points are statistically independent[cite: 1167]. In this limit, by the Cramer-Rao theorem, the covariance matrix of the parameters is given by the inverse of the Fisher matrix[cite: 1168]. Another relevant property of the Fisher matrices is their additive nature enabling one to do joint analyses of a different set of observations by adding the corresponding Fisher matrices together[cite: 1170]. For the present work, we compute the Fisher matrix as shown in Equation 24, where the summation is over all k-modes and z-bins[cite: 1173]. Fisher matrices are thus sensitive to the derivatives of the 21-cm PS, with a larger value indicating increased sensitivity and tighter constraining power[cite: 1174]. Parameters with a similar structure for the derivatives, as a function of k, will be degenerate [Pober2014][cite: 1176].


We obtain tight constraints (<~10 per cent) for the X-ray parameters (luminosity L_X/SFR & minimum X-ray photon energy escaping galaxies E_0), escape fraction normalisation (f_esc,0) and star formation efficiency (alpha_SF) while the SNe ejection efficiency (epsilon_0) is constrained to ~25 per cent[cite: 1204]. On the other hand, the critical mass normalisation (Sigma_SF) and the efficiency of SNe reheating (eta_0) remain relatively unconstrained with ~234 per cent and ~117 per cent fractional 1-sigma uncertainties respectively[cite: 1205]. The relatively poor constraints on Sigma_SF and eta_0 are primarily because these parameters have negligible impact on the 21-cm PS[cite: 1206]. The 21-cm signal is very sensitive to the amount and energy of the X-ray photons in the early Universe[cite: 1208]. The 21-cm PS, therefore, provides tight constraints on the X-ray parameters, E_0 (~11 per cent) and L_X/SFR (~10^-2 per cent), in our model[cite: 1209].


We next add in the Fisher matrix corresponding to the UV LFs from 6 redshifts in [5,10][cite: 1220]. As the UV LFs functions are sensitive to parameters that do not have a pronounced impact on the ionisation morphology, the addition of the UV LFs into the analysis helps improve the overall constraints on the astrophysical model [Park2019][cite: 1221]. Figure 5 shows the constraints from the joint analysis of the 21-cm PS and UV LFs[cite: 1226]. This results in significant improvements in Sigma_SF (from ~234 to ~40 per cent) and eta_0 (from ~117 to ~8 per cent) for the joint case of both the 21-cm PS and the UV LFs as the degeneracies between stellar properties and escape fraction can be weakened[cite: 1230]. This then translates into improvements in other parameters[cite: 1231]. The relatively larger improvement in Sigma_SF compared to alpha_SF following the inclusion of information from the UV LFs stems from the breaking of parameter degeneracies[cite: 1231].


X-rays can contribute to the ionisation of the local IGM of a galaxy leading to photoionisation regulation of the amount of neutral gas available for accretion onto the galaxy and hence subsequent star formation[cite: 1243]. Thus increasing L_X/SFR can be compensated by a decrease in the SNe feedback and/or the ionising UV escape fraction[cite: 1244]. The amount of cold gas available for star formation is influenced by the SNe ejection efficiency which sets the amount of gas removed from the galaxy[cite: 1249]. An increase in f_esc therefore should be accompanied by a decrease in the SNe feedback[cite: 1250]. The strong correlation between alpha_SF and SNe reheat efficiency eta_0 is expected as they are two of the parameters impacting the stellar mass in a galaxy[cite: 1258]. On the other hand, the correlation between alpha_SF and the critical mass normalisation Sigma_SF is surprisingly weak given Equation 8[cite: 1259]. This is because of the relatively small value of Sigma_SF[cite: 1260].


Using the semi-analytic galaxy formation model Meraxes we perform a Fisher matrix analysis to forecast the constraints on physical properties of galaxy formation and reionisation that will be available from future 21-cm PS observations[cite: 1264]. We focused on 8 free parameters in our model that directly impact the X-ray luminosity, UV escape fraction, star formation rate and SNe feedback of the galaxies[cite: 1267]. We constructed a mock observation of the 21-cm PS, focusing on a 1000 hr observation with the forthcoming SKA1-low[cite: 1268]. Using the Fisher matrix formalism, we find that 4 (5) out of the 8 parameters can be constrained to within <~10 (<~50) per cent using the 21-cm PS alone from the EoR[cite: 1270]. Specifically, we forecast tight constraints on our X-ray parameters, star formation efficiency, and the normalisation of the UV escape fraction of the early galaxies[cite: 1271, 1272]. On the other hand, SNe feedback parameters remain largely unconstrained reflecting that the 21-cm PS is relatively insensitive to them[cite: 1273].


The complex astrophysics of the early galaxy formation and evolution is captured in the degeneracies and correlations among the model parameters[cite: 1274]. To improve the overall constraining power of our analysis we added the Fisher matrix corresponding to the UV LFs from redshifts z in [5, 10][cite: 1276]. This results in an improvement in all of our parameter forecasts, most notably the critical mass normalisation Sigma_SF and the SNe reheat efficiency eta_0[cite: 1277]. This is not surprising as these parameters primarily control the stellar mass content of the galaxies to which the UV LFs are very sensitive[cite: 1278]. Incorporating the UV LFs into the analyses results in 5 of our parameters being constrained to <~10 per cent and all 8 of them being to within <~50 per cent[cite: 1279]. Our forecasts illustrate that detailed observations of reionisation with the SKA will be valuable in constraining the astrophysics of the early galaxies[cite: 1280].


This research was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project #CE170100013[cite: 1283]. Part of this work was performed on the OzSTAR national facility at the Swinburne University of Technology[cite: 1284]. The OzSTAR program partially receives funding from the Astronomy National Collaborative Research Infrastructure Strategy (NCRIS) allocation provided by the Australian Government[cite: 1285]. This research also made use of resources from the National Computational Infrastructure (NCI Australia), another NCRIS-enabled capability supported by the Australian Government[cite: 1286]. This research relies heavily on the python [PYTHON] open source community, in particular, numpy [NUMPY], matplotlib [MATPLOTLIB], scipy [SCIPY], h5py, jupyter [Jupyter], and pandas [Pandas][cite: 1287]. The data underlying this article will be shared on reasonable request to the corresponding author[cite: 1288].


In this section, we show the derivatives of our primary statistics i.e. the 21-cm PS (Fig. A1)[cite: 1289]. These derivatives are computed by perturbing the genpsim simulation about the fiducial model, one parameter at a time[cite: 1290]. These are then used to compute the Fisher matrix (see equation 24)[cite: 1291]. Parameters corresponding to similarly shaped derivatives will be degenerate. Figure A1 shows these derivatives weighted by the noise powers[cite: 1292]. This section shows the analysis using only the UV LFs[cite: 1295]. Figure B1 shows the derivatives of the UV LFs, computed by perturbing the genpsim simulation about the fiducial model[cite: 1296]. We do not vary the X-ray parameters, L_X/SFR and E_0, for the UV LFs analysis as they do not have an impact on the UV LFs[cite: 1297]. In Figure B2 we show the forecasted constraints from the UV LFs Fisher matrix[cite: 1299].


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 [e.g.][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. The outline of the paper is as follows. We give a brief overview of Meraxes in Section 2, and detail the calibration procedure in Section 3. We then investigate the evolution of black holes in the model in Section 4, and conclude in Section 5. Throughout this work, we adopt the [Planck2015] cosmological parameters. 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. For a detailed description of these simulations, see [Poole2016] and [Poole2017].


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. The model outputs a range of properties for each galaxy in the simulation, including the mass of hot gas, cold gas and stars, its star formation rate, and the mass of its central black hole. For a full description of the processes modelled in Meraxes, see [Mutch2016], [Qin2017] (herein Q17) and [Marshall2019] (herein M19).


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. A full description of this model is given in M19, with a brief summary outlined below. 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 will be deposited in the plane of the disc and so it 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. We summarize the model below, however the interested reader is encouraged to refer to Q17 for the full details. 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, suppressing the growth of only the most massive galaxies in Tiamat-125-HR at lower redshifts. 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; this simply adjusts the normalisation of our UV luminosity functions. 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, and so we can reliably use the Tiamat-125-HR simulation at redshift z<2 where Tiamat is unavailable. 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, so unless otherwise specified we only show the model results for one case hereafter. 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. 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.


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. We then consider the relative contributions of the different black hole growth modes, and we consider the black hole--host scaling relations in galaxies of different morphologies. 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: log(M_BH / M_sun) = alpha * log(M / M_sun) + beta, for the total stellar mass and bulge mass at a range of redshifts. We only include galaxies with mass M > 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---including only galaxies with stellar/bulge mass greater than 10^10 or 10^10.5 solar masses reduces the scatter. 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. This is consistent with current high redshift observations; when selection effects are accounted for, the observations at high redshift are consistent with no cosmological evolution in these relations [Schulze2014]. Our model predicts no significant evolution in the black hole--host mass relations, with the scatter in the relations decreasing at the highest redshifts. This indicates that there is a connection between the growth of black holes and their host galaxies. Indeed, our model includes joint triggering of star formation and black hole growth during galaxy mergers, and black hole feedback which regulates star formation, meaning that the co-evolution of black holes and galaxies is implicit in our model.


This is not consistent with the scenario proposed by [Peng2007] and [Jahnke2011], for example, where the black hole and galaxy growth is uncorrelated and the relationships are generated naturally within a merger driven galaxy evolution framework, due to a central-limit-like tendency. The median black hole mass to total stellar mass ratio as a function of redshift with galaxies split into black hole mass bins shows that lower mass black holes have lower mass ratios than higher mass black holes. For example, at high redshifts (redshift z>2), the median ratio for black holes with mass between 10^7 and 10^8 solar masses is higher than those with mass between 10^6 and 10^7 solar masses by ~0.25 dex, with that for black holes with mass > 10^8 solar masses being a further ~0.25 dex higher. This will lead to a notable selection bias, since when observing the most massive black holes, the measured ratio will be higher than that of the entire population.


This is generally expected for any sample selected by black hole mass or luminosity where the scatter in the relation is large [e.g.][Lauer2007]. Finally, we note an interesting effect of changing the parameter controlling the black hole efficiency for converting mass to energy. The median black hole--stellar mass ratio for our best model is compared to a run with a higher efficiency, with all other parameters unchanged. For the higher efficiency, the median black hole--stellar mass ratio decreases at redshifts z greater than or approximately equal to 6, instead of remaining constant with redshift. This effect is not seen by adjusting any of the other black hole parameters we tune in the model. We investigate the cause of this high-redshift decrease in the black hole--host relation by considering the Eddington limit. Increasing the efficiency from 0.06 to 0.2 decreases the Eddington limit.


This results in many black holes having Eddington-limited growth at the highest redshifts (redshift z greater than or approximately equal to 6), which is not the case for the lower efficiency model. This causes black holes to grow slower than their host galaxies at high redshifts, resulting in a decreased black hole--stellar mass ratio. Observing the high-redshift black hole--stellar mass relation may therefore probe the Eddington limit and the efficiency of black holes in converting mass to energy. We consider the cumulative fraction of black hole mass formed through each of the mechanisms in our model: black hole seeding, merger-driven quasar-mode accretion, instability-driven quasar-mode accretion, radio-mode accretion and black hole--black hole coalescence in galaxy mergers. The merger-driven growth mode becomes more important at low redshifts, at both low- and high-black hole masses. On average, instabilities grow the majority of mass in black holes at all redshifts, except for galaxies with black hole mass > 10^9 solar masses at redshift z is approximately equal to 0, whose black hole growth becomes dominated by galaxy mergers.


Radio-mode growth slowly increases in significance with redshift, yet still has only contributed to a small proportion of the total black hole mass by redshift z=0, except at the highest masses; this is discussed in Q17. Note that we consider growth from disc instabilities that are triggered by earlier galaxy mergers as growth via the instability-driven mode, and do not treat them in a more detailed manner as in [IzquierdoVillalba2019], for example. We also consider the instantaneous growth fractions of black hole mass formed through each mechanism as a function of redshift. Here we take the `instantaneous' fraction to be the fraction of growth caused by a mechanism between the specified redshift and the simulation snapshot immediately preceding it. As discussed, the model produces unreliable black hole accretion rates at redshift z<1, and so we only consider these black hole growth rates at redshift z>1.


The instability-driven growth mode is the dominant growth mechanism, on average, at all redshifts, regardless of black hole mass. The merger-driven quasar mode and black hole--black hole coalescence mode are sub-dominant at all redshifts. The radio-mode grows more mass at low redshift and in the most massive galaxies, with the percentage of total instantaneous black hole growth from this mode increasing from only 0.1 per cent at redshift z=5 to almost 5 per cent at redshift z is approximately equal to 1. Our finding that mergers are not the dominant mechanism for growing black holes is in agreement with a range of observations. For example, [Koss2010] find that only 25 per cent of local (redshift z<0.05), moderate luminosity X-ray AGN show signs of mergers, though the fraction is much higher for luminous AGN [Hong2015]. From redshift z from approximately 0.3 to 1.0, [Cisternas2010] find that the vast majority (>85 per cent) of X-ray selected AGN do not show signs of mergers, suggesting that the bulk of their black hole accretion has been triggered by some other mechanism.


This is also consistent with the findings of [Georgakakis2009] who claim that a large fraction of AGN at redshift z is approximately equal to 1 are triggered by processes other than major mergers, as do [Villforth2018] at redshift z is approximately equal to 0.9, and [Schawinski2012], [Mechtley2016], [DelMoro2015] and [Marian2019] for AGN at redshift z is approximately equal to 2. Our result that disc instabilities cause the majority of black hole growth is also consistent with predictions from other simulations. In the GALFORM semi-analytic model, [Fanidakis2011] found that the growth of black holes is dominated by accretion due to disc instabilities, with the fraction of mass in black holes produced by disc instabilities more than an order of magnitude larger than that produced by mergers, at all redshifts. Using an updated GALFORM model, [Griffin2019] found that accretion of hot gas dominates the growth of black holes at redshift z<2, with disc-instabilities dominant at higher redshifts.


[Hirschmann2012] found that instability-driven black hole growth was required to reproduce AGN downsizing, and that while major mergers are the dominant trigger for luminous AGN, especially at high redshift, disc instabilities cause the majority of black hole growth in moderately luminous Seyfert galaxies at low redshift. [Menci2014] find that in their semi-analytic model disc instabilities can provide enough black hole accretion to reproduce the observed AGN luminosity functions up to redshift z is approximately equal to 4.5, but are not likely to be dominant for the highest luminosity AGN or at the highest redshifts. In contrast, [Shirakata2018] find that the primary trigger of AGN at redshift z less than or equal to 4 in their semi-analytic model is mergers, while disc instabilities are essential for fuelling moderate luminosity AGN at higher redshifts. The hydrodynamical simulation Horizon-AGN found that only ~35 per cent of black hole mass in local massive galaxies is directly attributable to merging, with the majority of black hole growth instead growing via secular processes [Martin2018].


The Magneticum Pathfinder Simulation also found that merger events are not the dominant fuelling mechanism for black holes in redshift z=0--2, with merger fractions less than 20 per cent, except for very luminous quasars at redshift z is approximately equal to 2 [Steinborn2018]. Finally, we comment on the effect of the efficiency parameters for merger-driven and instability-driven black hole growth in the model. We find the instability-driven efficiency from tuning the model, whereas the merger-driven efficiency is less constrained, with several values producing reasonable model results. Having a merger growth efficiency that is twice, six times or even 18 times larger than the instability-driven growth efficiency may have an effect on the conclusions outlined above. We find, as expected, that models with larger merger efficiencies result in more merger-driven growth. For a merger efficiency twice the instability efficiency, the instability-driven mode still dominates at redshift z=2, while for a six times larger efficiency, the merger-driven mode begins to dominate at the highest black hole masses.


For the model with an 18 times larger merger efficiency, the merger-driven mode contributes even more black hole growth, but is still not the dominant growth mode for black holes with mass between 10^6 and 10^9 solar masses. Thus, while the efficiency parameter for merger-driven growth has some effect on the relative distributions of the instability-driven and merger-driven growth modes, the instability-driven mode is still dominant for the majority of black holes, even if the merger growth efficiency is as much as 18 times larger than the secular growth efficiency. A popular explanation for the black hole--host correlations is that major mergers drive the growth of both black holes and bulges [e.g.][Haehnelt2000, Croton2006b]. If this were the case, one would expect that black holes would only correlate with galaxy properties directly related to the merger process, such as bulge mass, and not, for example, total stellar mass.


[Simmons2017] consider a sample of 101 disc-dominated AGN hosts from the SDSS, which they assume must have a major merger-free history since redshift z is approximately equal to 2. They found that these galaxies lie on the typical black hole mass--total stellar mass relation, but lie offset to the left of the black hole mass--bulge mass relation. This indicates that the substantial and ongoing black hole growth in these merger-free disc galaxies must be due to a process other than major mergers, and that major mergers cannot be the primary mechanism behind the black hole--host correlations. We plot the black hole mass--total stellar mass and black hole mass--bulge mass relation for disc-dominated and bulge-dominated galaxies at redshift z=0. Our simulated disc galaxies lie on the black hole mass--total stellar mass relation, but lie offset to the left of the black hole mass--bulge mass relation, as they have small bulges relative to their black hole mass.


This is consistent with the [Simmons2017] observations, and the results from the Horizon-AGN hydrodynamical simulation [Martin2018]. However, we see a less significant offset, which occurs at lower black hole masses than [Simmons2017] and [Martin2018], since the black holes in our disc-dominated galaxies are less massive in comparison. [Mutlu-Pakdil2017] also find no dependence of the black hole mass--total stellar mass relation on galaxy type in the Illustris hydrodynamical simulation. [Martin2018] suggest that major mergers therefore cannot be primarily responsible for feeding black holes, otherwise major-merger free disc galaxies should have less massive black holes than are observed and simulated. This is consistent with our finding that the instability-driven mode is the dominant growth mechanism for black holes. We use the Meraxes semi-analytic model to investigate the evolution of black holes and their relations to their host galaxies.


We find the following key predictions of our model: There is minimal statistically-significant evolution in the black hole--bulge and black hole--total stellar mass relations out to high redshifts (redshift z is approximately equal to 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. This indicates that the growth of galaxies, bulges and black holes are all tightly related, even at the highest redshifts. Higher mass black holes have higher black hole--total stellar mass ratios, leading to a significant selection effect in measurements of this ratio when observing only the most massive black holes. The instability-driven or secular quasar-mode growth is the dominant growth mechanism for black holes at all redshifts. The contribution from merger-driven quasar-mode growth only becomes significant at low redshift for black holes with mass greater than or approximately equal to 10^9 solar masses.


Disc-dominated galaxies lie on the black hole--total stellar mass relation, but lie offset from the black hole--bulge mass relation. Our simulation is limited in making predictions for the highest redshift quasars at redshift z=6--7 due to the simulation box size and resolution. In future work we will run Meraxes on larger N-body simulations in order to make predictions for these objects. We calibrate the free parameters in Meraxes to match the observed stellar mass functions at redshift z=0--8 and the [Shankar2009] and [Davis2014] black hole mass function at redshift z=0. The black hole mass functions produced by Tiamat and Tiamat-125-HR are converged at redshift z=2 for black holes with mass > 10^7.1 solar masses [see Marshall2019], with Tiamat-125-HR producing more low-mass black holes. We therefore focus on matching the observed black hole mass functions at masses > 10^7.1 solar masses.


While the [Shankar2009] and [Davis2014] relations are different, particularly at black hole mass ~ 10^8.5 solar masses, they are similar relative to the freedom we have in adjusting our model black hole mass function, and so when calibrating we found the most reasonable fit to both. In the model stellar mass function, we also plot the Meraxes stellar mass function produced when AGN feedback is switched off. This shows that AGN feedback has no effect on galaxies in Tiamat at redshift z greater than or equal to 2, but suppresses the growth of the most massive galaxies at lower redshifts as seen in Tiamat-125-HR. Throughout this work, we use the higher resolution Tiamat simulation at redshift z greater than or equal to 2, and Tiamat-125-HR for redshift z<2, where Tiamat is unavailable. We find that the results discussed in this paper are generally consistent between the two simulations at redshift z is approximately equal to 2.


However, one notable result is that the best-fitting black hole--stellar mass relations change rapidly between redshift z=2 (using Tiamat) and z=1 (using Tiamat-125-HR). To verify that this jump is not purely a result of the simulation change, we show the best-fitting relations from redshift z=6--0 using Tiamat-125-HR. The Tiamat-125-HR simulation shows similar results to those found using Tiamat at redshift z greater than or equal to 2, with a slightly milder but still relatively rapid evolution from redshift z=2 to z=1. The qualitative result of the evolution being insignificant relative to the scatter in the relation still holds. Thus, while the change in simulation slightly amplifies the rapid change in the black hole--stellar mass relations from redshift z=2 to z=1, this does not change our conclusions. We also note that where the black hole mass functions are converged, the black hole--stellar mass relations are in good agreement between the two simulations.


We implemented Population III (Pop. III) star formation in mini-halos within the Meraxes semi-analytic galaxy formation and reionisation model, run on top of a N-body simulation with a side length of 10 per h comoving Megaparsecs with 2048^3 particles resolving all dark matter halos down to the mini-halos (approximately 10^5 solar masses). Our modelling includes the chemical evolution of the IGM, with metals released through supernova-driven bubbles that expand according to the Sedov-Taylor model. We found that SN-driven metal bubbles are generally small, with radii typically of 150 comoving kiloparsecs at redshift z = 6. Hence, the majority of the first galaxies are likely enriched by their own star formation. However, as reionization progresses, the feedback effects from the UV background become more pronounced, leading to a halt in star formation in low-mass galaxies, after which external chemical enrichment becomes more relevant. We explore the sensitivity of the star formation rate density and stellar mass functions on the unknown values of free parameters.


We also discuss the observability of Pop. III dominated systems with JWST, finding that the inclusion of Pop. III galaxies can have a significant effect on the total UV luminosity function at redshift z = 12 - 16. Our results support the idea that the excess of bright galaxies detected with JWST might be explained by the presence of bright top-heavy Pop. III dominated galaxies without requiring an increased star formation efficiency. The first episodes of star formation likely occurred at redshift 30-40 inside low-mass (10^5-7 solar masses) halos with a pristine chemical composition (metal-free or extremely metal-poor). These environments were responsible for the chemical enrichment of the intergalactic medium (IGM) leading to the Pop. III/Pop II transition at redshift 15-10. However, due to the low mass and the high-redshift, the study of Pop. III star formation in mini-halos is challenging both in terms of observations and simulations [Klessen2023].


Even with the launch of JWST, we do not have confirmed observations of Pop III stars and only a few potential candidates [Welch2022, Maiolino2023]. As pointed out by [Trussler2023], a direct detection of a Pop. III galaxies will be extremely challenging as, in the best-case scenario, hundreds of hours of integration time are needed in order to detect an unlensed Pop. III system with 5-sigma confidence. A complementary tool to direct observation could be the redshifted 21cm global signal whose depth might be determined by the radiation emitted from Pop. III stars and early accreting black holes during the Cosmic Dawn [among the most recent works see Mebane2018, Mirocha2018, Mebane2020, Munoz2022, GesseyJones2022, Magg2022, Ventura2023, Hegde2023]. So far, there is no detection confirmed of the 21cm redshifted line, however, a large number of facilities aiming to the observation of the 21cm global signal and power spectrum are already operating or becoming operative in this decade.


There are a number of Pop. III star formation simulations at different scales. Hydrodynamical simulations that follow the cooling of gas in a single pristine or very metal-poor cloud suggest that the inefficient cooling due to the lack of metals might favour an initial mass function (IMF) that is shifted to larger masses [e.g.][Bromm1999, Hirano2014, Stacy2016, Chon2021]; however, there is no general consensus (e.g. [Wollenberg2020, Jaura2022, Prole2022] predict IMF shifted to lower masses). Simulations at larger scales (>=1 cMpc) are instead used to follow the global evolution of Pop. III stars with redshift and their impact on the cosmic metal enrichment and reionization. In the last decade, many of these simulations have been performed both with hydrodynamical and semi-analytical codes. At the same time we also need to consider a volume large enough in order to have a statistically significant sample of the Universe. However, to satisfy both requirements of large volumes and high-resolution is impossible and thus all the semi-analytical and hydrodynamical simulations have to make compromise in one of the two directions.


In this work, we chose to use the semi-analytical model of galaxy formation Meraxes ([Mutch2016], M16 hereafter) within a simulation that allows us to resolve all the mini-halos down to a few 10^5 solar masses in a simulation of L = 10 Mpc per h. As so, in this paper we ran Meraxes on top of a N-body simulation of L = 10 Mpc. The size of this simulation is not large enough to be representative of the Universe (we will miss the most massive galaxies). However, the scales are large enough to investigate the impact of the main physical processes on the Pop. III star formation. We incorporated a number of new physical processes that are relevant to Pop. III star formation in mini-halos including: (i) molecular hydrogen cooling functions for mini-halos, (ii) baryon-dark matter streaming velocities, (iii) photo-dissociation of H2 molecules from the Lyman-Werner background and (iv) the chemical evolution of the intergalactic medium.


This latter effect has been implemented assuming that metals are released through supernova explosions and within a bubble that expands accordingly to the Sedov-Taylor model (the approach is very similar to the analytical calculation shown in [FL2003]). We found that these bubbles are generally small and that they roughly agree with the previous estimate by [Trenti2009], with typical radii of 150 ckpc at redshift z = 6. The implementation of both internal and external metal enrichment allows us to understand whether a galaxy will form Pop. III or Pop. II stars and thus quantify the impact of Pop. III galaxies on the total luminosity function at redshift z >= 5. We assumed that the Pop. III star formation occurs in an instantaneous burst at a random Delta t from the end of the snapshot. Considering an instantaneous rather than a continuous star formation makes some of the Pop. III dominated systems significantly brighter, hence at high-redshift the brightest systems are likely to host Pop. III stars.


The semi-analytical model Meraxes was first developed by [Mutch2016] (hereafter M16), [Qin2017, Qiu2019] in order to study galaxy formation and growth through the Epoch of Reionization. Despite the variety of physical processes included in Meraxes assumes that all the galaxies form in a previously chemically enriched Universe and inside atomic cooling halos. Such an approximation does not allow us to study the first episodes of star formation that mostly occurred in mini-halos when the Universe did not have any metals. The version of Meraxes presented in this work allows us to compute the physics of the first episodes of star formation from the initial molecular cooling of the gas to the external metal enrichment from the supernova feedback. As in the previous work, Meraxes is coupled to the reionization so that all the radiative backgrounds are computed in a self-consistent way from the galaxy properties. This has been done by implementing a modified version of 21cmFAST ([Mesinger2011]) that includes the local ionizing UV background from [Sobacchi2014] and the X-ray heating ([Balu2023]).


In this version, we also included the Lyman-Werner background. The updates to Meraxes for Pop. III necessitate high mass and spatial resolution. For this purpose, we introduce L10_N2048 (hereafter L10) from the Genesis suite of dark matter only N-body simulations. L10 is a periodic cubical simulation of side 10 per h Mpc and consists of 2048^3 dark matter particles of mass m_p = 9.935 x 10^3 per h solar masses resulting in a halo mass resolution of ~ 3.18 x 10^5 per h solar masses (based on a minimum of 32 particles per halo). The simulation, run using the SWIFT [Schaller2018] cosmological code, evolves these dark matter particles from redshift z = 99 down to z = 5. The halos were identified using the friends-of-friends phase space halo-finder VELOCIraptor [VR] and the merger trees were constructed using TREEFROG [treefrog].


We note that the mass resolution achieved in this simulation allows us to resolve all the mini-halos below redshift z ~ 30 and is the highest resolution on which Meraxes has been run. The output trees of the N-body simulation used in this work are available on Zenodo at [balu2024]. The first process in order to enable star formation is the cooling of the gas. For dark matter halos with virial temperature T_vir >= 10^4 K, the main coolant is the atomic hydrogen, while in mini-halos (10^3 K <= T_vir < 10^4 K), the cooling occurs via roto-vibrational transitions of molecular hydrogen [e.g.][Tegmark1997]. For the details on how the cooling of the gas is implemented in Meraxes, we refer the reader to M16; in this section, we only highlight the main differences between the atomic and the molecular cooling. As in M16, we compute the ratio of the specific thermal energy to the cooling rate per unit volume: t_cool(r) = (1.5 * mu_bar * m_p * k * T) / (rho_hot(r) * Lambda(T, f_H2)).


M16 included only atomic cooling halos. Thus they set the mean molecular weight assuming a fully ionized gas and the cooling function from [Sutherland1993]. For mini-halos we instead set the mean molecular weight for a fully neutral gas and implement the molecular hydrogen cooling functions from [Galli1998]. This choice is valid as long as we assume that the cooling inside mini-halos occurs only due to H2. This may not be valid as, if a mini-halo is chemically enriched with metals by a nearby halo, metals are much more effective in the cooling of the gas [e.g.][Nebrin2023]. However, this enrichment is almost ineffective at redshift z > 10. The molecular cooling function depends on the gas density of the halo and on the molecular hydrogen fraction f_H2. For the latter we assumed a fixed value of 0.1%, which is consistent with results from [Nebrin2023] for halos of virial temperature of approximately 5x10^3 K at redshift z = 15-20.


As a first approximation, gas in a mini-halo can start to cool down the gas once it reaches the virial temperature of approximately 10^3 K. This requirement would correspond to a minimum virial mass of M_min,H2 = 2.5 x 10^5 * (26 / (1+z)) solar masses [Visbal2015]. However, there are a number of effects that can decrease the amount of molecular hydrogen present in the halo, reducing the cooling efficiency and ultimately increasing the minimum virial mass of a halo capable of cooling. A non-radiative process that can delay the gas cooling in very low-mass halos is the streaming velocity between baryons and dark matter ([Tseliakhovich2010]). This effect is a consequence of the different decoupling of baryons and dark matter particles from photons that results in a velocity difference between the two species. The presence of a relative motion between baryons and dark matter particles will make it harder for baryons to fall into the potential wells of dark matter halos, delaying the accretion and, hence, the cooling of the gas in mini-halos.


The main outcome of this physical process is to delay the very first episodes of star formation around redshift z ~ 40. Throughout this work, we implemented the effect of the streaming velocities as per other semi-analytical models based on a fitting function found by [Fialkov2012], which is calibrated to reproduce the results of the hydrodynamical simulations of [Greif2011] and [Stacy2011]: V_cool,H2 = (a^2 + (b*v_bc)^2)^0.5, where a = 3.714 km/s and b = 4.015 km/s. This equation provides the minimum circular velocity that a halo needs to have in order to have enough H2 to cool down the gas. This can be easily converted into a virial mass. In this work we fixed the streaming velocity v_bc(z) = sigma_bc(z) and we assumed this value for the entire box. Once the first mini-halos are forming stars, the effect of the streaming velocities becomes widely subdominant compared to the photo-dissociation of the Lyman-Werner background.


A self-consistent model of star formation in mini-halos must consider the photo-dissociation of H2 from UV photons in the Lyman-Werner (LW) band ([11.2 - 13.6] eV). LW photons destroy H2 and thus prevent mini-halos gas from cooling ([Haiman2006]). We implement this effect by changing the minimum mass for molecular cooling using the fitting from [Visbal2015]: M_crit,MC = M_cool,H2 * [1 + 22.87 * J_LW^0.47]. This critical mass only applies to minihalos which are below the atomic cooling halo mass threshold. J_LW is the LW flux that reaches the minihalo. LW photons have a mean free path of ~ 100Mpc, so each minihalo will be affected even by distant galaxies that formed at higher redshift. Thus, we modelled the LW background by integrating contributions across the cosmic history. As for all the radiative backgrounds in Meraxes, we follow an excursion-set formalism ([Furlanetto2004]) which counts the number of photons in a certain band in spheres of radius R.


At each of these locations, we compute the LW emissivity using the spectral energy distributions from [Barkana2005]. We assume that LW photons are absorbed only at resonant frequencies. Under these approximations, we can compute the LW emissivity smoothed over R at redshift z and location x for sources emitted at redshift z' as an integral over the star formation rate density for both Pop. III and Pop. II stars. This sum accounts for the resonances in the Lyman series. Given the large mean free path of LW photons, we must also account for distant galaxies at higher redshift z' > z with the redshifted spectrum. The sum of these two effects causes the peculiar shape of the LW spectrum. Following [Barkana2005] we account for all the Lyman resonances with n <= 23. Finally, we need to convert the emissivity into a flux following ([Qin2020]).


The approach described above differs from the one in 21cmFAST only for the computation of the SFRD. In [Qin2020], the SFRD is estimated from the density field and the collapsed fraction, while in this work, it is computed directly from Meraxes, which tracks the formation of each galaxy and its entire star formation history. Together with the LW background, the main radiative feedback that suppresses star formation is the UV photo-ionization. We used the same prescriptions as in M16, which consists of reducing the baryon content through a baryon fraction modifier that stops the gas infall. In difference from the LW feedback, the UV photo-ionization is relevant only during reionization (redshift z <= 10), and it also affects atomic cooling halos as massive as ~ 10^9.5 solar masses. However, given that the volume of our simulation (L = 10 cMpc) is much smaller than the mean-free path of LW photons, each halo is losing the contribution of the most distant sources, thus neglecting the self-shielding should partially counterbalance the loss of the LW background from distant galaxies.


Once stars start to form in the Universe, they also explode as supernovae, releasing metals. Most of these will stay inside the same galaxy, contributing to the chemical enrichment of the galaxy itself. This process is commonly known as "genetic" enrichment. However, some of the metals escape their parent galaxy. In this case, they will pollute the nearby IGM, and if later a galaxy forms in a region where the IGM was enriched, the new galaxy will be pre-enriched with metals. This latter mechanism is referred to as "external" metal enrichment [e.g.][Pallottini2014, Smith2015, Hartwig2018, Visbal2020, Yamaguchi2023]. Keeping track of the metallicity evolution of the Universe is crucial in order to put constraints on when the Pop. III/II transition occurred. We account for both processes, and we are able to follow the metallicity evolution of the IGM. Firstly, we choose a critical metallicity Z_crit = 10^-4 solar metallicity as the threshold value below which a galaxy will form Pop. III stars.


All new galaxies, unless externally polluted, will accrete pristine gas (without any metals) onto the hot gas reservoir. This gas, once it cools, will provide the reservoir for the star formation. Hence, a galaxy that is not externally polluted will always form Pop. III stars for the first time. At each snapshot, we compute the metallicity of the cold gas reservoir from the amount of metals released by earlier supernovae and if this is higher than Z_crit, the galaxy will form Pop. II stars, otherwise it will form Pop. III. While internal enrichment is the main mechanism that drives the Pop.III/II transition, some galaxies can be externally enriched through supernova winds originating in a nearby galaxy. Once several supernovae in a galaxy go off, they will form a "super-bubble" that will expand outside the galaxy escaping the binding energy of the dark matter halo.


We followed the expansion in time of this "metal bubble" using the analytic approximation in [Dijkstra2014]: the bubble radius r_bubble(t) equals (Delta E_SN / (m_p * n_gas))^0.2 * t^0.4. All quantities that appear in this equation are computed in Meraxes. Delta E_SN is the total supernova energy released at a certain snapshot. n_gas is the number density of the gas to which the bubble has expanded, and t is the time since the explosion occurred. We assume that all the supernova events will occur in the middle of the snapshot. Note that since Meraxes accounts for both contemporaneous and delayed supernova feedback, each galaxy has several bubbles associated with the same star formation episode. However, we consider only the largest of these bubbles so that each galaxy has only one associated bubble. Having calculated the bubble size, we can predict if a nearby galaxy will accrete pristine or enriched gas.


In order to reduce the computational cost, we avoid computing the distance between all the pairs of galaxies and instead use a grid-based approach. We build a high-resolution grid with 128^3 cells. For each cell, we compute the average metallicity Z_IGM,i as the ratio between the sum of the metals ejected by all galaxies inside the cell and the total gas in the cell. We only let the galaxies with a bubble radius r_bubble >= 3 * R_vir contribute. For each cell, we compute the volume fraction filled with metals (the metal filling factor), summing the volume of the largest bubble surrounding each galaxy within the pixel and dividing by the cell volume. At the beginning of each snapshot, we assign the probability p for external metal enrichment to each galaxy inside the cell. This probability will be given by the metal filling factor. We assign a random number m between 0 and 1 to each newly formed galaxy, and when m <= p is satisfied, we label that galaxy as externally enriched.