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.


We directly compare predictions of dwarf galaxy properties in a semi-analytic model (SAM) with those extracted from a high-resolution hydrodynamic simulation. We focus on galaxies with halo masses of 10^9 < virial mass / solar mass less than or approximately equal to 10^11 at high redshift (redshift z >= 5). We find that, with the modifications previously proposed in [qin2018], including to suppress the halo mass and baryon fraction, as well as to modulate gas cooling and star formation efficiencies, the SAM can reproduce the cosmic evolution of galaxy properties predicted by the hydrodynamic simulation. These include the galaxy stellar mass function, total baryonic mass, star-forming gas mass and star formation rate at redshift z~5-11. However, this agreement is only possible by reducing the star formation threshold relative to that suggested by local observations. Otherwise, too much star-forming gas is trapped in quenched dwarf galaxies.


We further find that dwarf galaxies rapidly build up their star-forming reservoirs in the early universe (redshift z>10), with the relevant time-scale becoming significantly longer towards lower redshifts. This indicates efficient accretion in cold mode in these low-mass objects at high redshift. Note that the improved SAM, which has been calibrated against hydrodynamic simulations, can provide more accurate predictions of high-redshift dwarf galaxy properties that are essential for reionization study. Reionization refers to an important process after the Big Bang, during which the intergalactic medium (IGM) was transiting from neutral hydrogen to its ionized state [Wyithe:2004kb]. According to the observed galaxy sample at high redshift [Bouwens2014, Bouwens2015, Stefanon2016arXiv161109354S, Oesch2016ApJ...819..129O], this process can only happen with ionizing photons from much fainter galaxies taken into account [Robertson2013ApJ...768...71R, duffy2014low, Bouwens:2015hk, Liu2016].


Although there are still some debates on other possible sources that can dominate the high-redshift photon budget such as active galactic nuclei (AGN; [Giallongo2015A&A...578A..83G, Madau2015ApJ...813L...8M, Qin2017c, Hassan2018MNRAS.473..227H]), dwarf galaxies that are beyond our observational capabilities are generally thought to have driven the Epoch of Reionization (EoR). In this context, understanding the formation of these unobserved objects is crucial to studying the EoR and can only be probed at this stage with theoretical simulations. Hydrodynamic simulations evolve dark matter and baryonic particles simultaneously and provide direct insights into the relevant astrophysical process [Vogelsberger2014, Schaye2014, Hopkins2014MNRAS.445..581H, feng2015bluetides]. However, resolving dwarf galaxies within a cosmological volume for reionization studies usually involves more than a few billions of particles, which remains computationally challenging at this stage. A more efficient method is to apply semi-analytic models (SAMs; [croton2006many, Somerville2008, guo2011dwarf, Henriques2015]) to N-body simulations that only consider collisionless particles.


Using the halo properties inherited from the parent simulation, SAMs approximate baryonic physics such as gas accretion, star formation and feedback using simplified scaling relations. These relations are motivated directly from physical processes, or empirically from observational results and more complicated numerical techniques such as hydrodynamic simulations and radiative transfer calculations. The semi-analytic prescriptions that indirectly model galaxy formation introduce free parameters to describe efficiencies which are inevitably accompanied by parameter degeneracies [Mutch2013, Henriques2013]. This can make their predictions sometimes controversial, and potentially disconnected from the true behaviour in the universe. An alternative to validate SAMs in the absence of observations at high redshift is to compare their results against hydrodynamic calculations that start from identical cosmological initial conditions. The goal of this work is to capture emergent behaviours from the hydrodynamic simulations and to improve the parametrised modelling in SAMs as so to replicate these processes.


Under the assumption that hydrodynamic simulations are a more natural description of the astrophysical phenomenon and hence more representative of real galaxies, we can explore the semi-analytic prescriptions for quantities that are, in practice, unobservables and potentially reveal improper assumptions or missing physics in SAMs. [Guo2016] compared the L-galaxies [Cole2000, Bower2006] and GALFORM [Springel2005, Henriques2015] SAMs with the EAGLE hydrodynamic simulations [Schaye2014], and concluded that the models can reproduce the stellar mass function predicted by EAGLE. However, discrepancies were also found in the efficiencies of stellar and AGN feedback as well as the prediction of stellar mass-metallicity and size relations. [Mitchell:2017je] also used EAGLE to assess GALFORM and found the angular momentum as well as the baryon cycling might not be properly traced in the SAM, leading to inaccurate predictions of galaxy sizes.


[Stevens:2017fi], on the other hand, investigated cooling of Milk Way-like galaxies in EAGLE and addressed the necessity of updating the cooling prescription employed in most SAMs (see recent updates of the cooling model in [Hou2018mnras.475..543h, Hou2018arxiv180301923h]). We, in the previous paper [qin2018], also found that the cooling prescription needs revision for more accurate modelling of low-mass galaxies at high redshift and proposed an alternative modification to the current prescription, avoiding the introduction of a new model. [Cote:2017uh] recently extended the comparison from cosmological simulations of smoothed particles to zoom-in simulations of a system with a total mass of approximately 10^9 solar masses, and investigated the difference in dwarf galaxy formation between a SAM and hydrodynamic simulation. They found their SAM was successful in reproducing the hydrodynamic calculation of star formation history but with a prediction of a much narrower distribution of metallicity compared to the hydrodynamic result.


This is the second paper following the work of [qin2018], where we investigated the performance of SAMs when applied to high-redshift dwarf galaxies. We used the Meraxes SAM [Mutch2016a] as an example and focused on gas accretion, cooling and star formation with reionization and supernova feedback isolated. We compared the stellar and gas masses with a high-resolution hydrodynamic simulation from the Smaug suite, and found that, in the SAM, 1) due to the lack of hydrostatic pressure in parent N-body simulations, inheriting halo properties directly from the dark matter halo merger trees overestimates the total mass of haloes hosting dwarf galaxies; 2) the assumption that, in the absence of feedback, haloes consists of a baryonic reservoir with a mass of the cosmic baryon fraction of their total mass is not accurate for dwarf galaxy formation modelling; 3) star formation modelled by consuming the total gas disc in a few dynamical times of that disc cannot capture the evolutionary path of star formation implemented in hydrodynamic simulations; and 4) gas accreted by dwarf galaxies is cold.


Accordingly, we proposed modifications to SAMs, seeking for consistency with hydrodynamic simulations in calculations of the evolution of stellar and gas components of dwarf galaxies. In this work, we include these modifications as well as feedback from reionization and supernovae, and investigate whether the updated SAM can broadly agree with the hydrodynamic calculation of dwarf galaxies in the presence of feedback. We start with a brief review of the Meraxes SAM as well as the modifications proposed in [qin2018] and the Smaug hydrodynamic simulation suite in Section 2. We then present and discuss our comparison results in Section 3. Conclusions are given in Section 4. In this work, we adopt the Chabrier initial mass function (IMF) in the mass range of 0.1-120 solar masses and cosmological parameters from WMAP7.


The Dark-ages Reionization And Galaxy Formation Observable from Numerical Simulation (DRAGONS) project employs N-body simulations, hydrodynamic simulations and SAMs to study reionization and galaxy formation at high redshift (redshift z >= 5). In the previous publication of this series, we used the Meraxes SAM as an example and investigated the semi-analytic modelling prescriptions adopted in the literature. We focussed on comparisons of dwarf galaxy properties calculated by Meraxes with a simplified model that ignores feedback from the Smaug hydrodynamic simulation suite. Based on the comparison, we proposed modifications to the halo properties as well as the cooling and star formation prescriptions. In this work, we include reionization and supernova feedback, and extend the comparison of high-redshift dwarf galaxies modelling with a molecular-hydrogen-based star formation law in the SAM.


We briefly introduce the Meraxes SAM and Smaug hydrodynamic simulation in this section with emphasis on reionization and supernova feedback. Meraxes evolves galaxies with scaling relations capturing baryonic processes. These include gas infall, cooling, star formation, supernova feedback, metal enrichment, stellar mass recycling, reionization, supermassive black hole growth, AGN feedback and mergers. It also calculates the ionization state of the IGM using the 21cmFAST semi-numerical algorithm. Note that, in order to take hydrostatic pressure into account in this work, halo masses inherited from the merger trees that are constructed from collisionless N-body simulations as well as total baryonic masses are updated using the halo mass and baryon fraction modifiers provided in [Qin2017a].


Gas falls into a halo from the IGM and cools through thermal radiation. Within a transition time-scale, this process leads to the formation of a star-forming disc, which is assumed to follow an exponential surface density profile. In previous DRAGONS publications, we follow [croton2006many], form stars by consuming the total star-forming gas, and calculate the star formation rate (SFR) by a formula where the depletion time-scale of total star-forming gas on the disc is proportional to the dynamical time-scale of the host halo. However, this was found to be inconsistent with the implementation in hydrodynamic simulations. In this work, we instead adopt a different equation and use free parameters to directly adjust the evolutionary path of the depletion time-scale. A second star formation prescription will be explored in this work, the detail of which will be presented in Mutch et al. (in prep.).


Note that this prescription is based on the depletion of molecular hydrogen (see [Lagos2011] and references therein) and is considered as a more physically plausible model compared to the total-gas-based star formation prescription. First, we assume stars follow the same distribution as the interstellar medium (ISM) and they are considered as a stellar disc with an exponential surface density profile. We then calculate the pressure of the ISM accounting for both stars and hydrogen through the [Elmegreen1993] approximation. In order to split the disc into molecules and atoms, the observed relation between the ISM pressure and the surface density ratio of neutral to molecular hydrogen is implemented. [Blitz2006] investigated the neutral hydrogen, CO and stellar densities of 14 nearby galaxies and found a power-law relation between the ratio of neutral to molecular hydrogen surface densities and the ISM pressure. With this, we estimate the molecular hydrogen mass and calculate the SFR.


Inferred from the stellar lifetime-mass relation and the assumed IMF, we calculate the fraction of the newly formed stars which will have reached the supernova stage at the end of the current time step. These stars recycle their masses to the ISM, and the metals and energy produced by the supernovae provide feedback to the environment. In particular, the metals enhance the cooling rate through the metallicity dependent cooling function while the supernova energy leads to transition of gas between different reservoirs. In practice, supernova energy converts star-forming gas to hot (i.e. non-star-forming gas) and, in the case of strong supernova feedback, further ejects hot gas from the galaxy. In this work, we adopt the [guo2011dwarf] prescriptions to calculate the energy coupled to the surrounding gas. We consider two supernova feedback regimes: 1) contemporaneous feedback; and 2) delayed feedback where long-lived stars formed at earlier times are taken into account.


Therefore, the total energy released in the current snapshot is the sum of the supernova energy from stars formed in the current and previous 4 snapshots. On the other hand, the expected mass of gas that is heated by supernovae depends on the mass loading factor, which is assumed to follow the same form as the coupling efficiency. We first calculate the maximum reheated mass with an upper limit set to be 10 following [Mutch2016a], which is a typical value for high-redshift dwarf starburst galaxies [uhlig2012]. Note that, since the total supernova energy is finite, a galaxy might not be able to heat all the mass estimated from the loading factor. Therefore, we calculate the mass of actually reheated gas. While in the case of an intense supernova event, the energy released by supernovae further unbinds the hot gas which is removed from the galaxy and stored in a reservoir termed the ejected gas. We assume the ejected gas does not contribute to star formation.


In order to model the feedback from reionization, we further inhibit the local baryon fraction of haloes by a factor of f_mod which is defined as 2 to the power of (-M_crit / M_vir), where M_vir is the halo mass and M_crit represents a filtering mass below which haloes are not able to efficiently accrete baryons from the IGM. We calculate the critical mass for each halo following [Sobacchi2013a], where it depends on the local UV background intensity and the redshift at which the surrounding IGM was first ionized, which is determined using the 21cmFAST algorithm. Smaug, a high-resolution hydrodynamic simulation suite, was performed using a modified version of the GADGET-2 N-body/hydrodynamics code, following the same parameter configuration of the OverWhelmingly Large Simulations project (OWLS; [Schaye2010]). The simulations presented in this work start from the same initial conditions generated with the grafic package at redshift z=199 using the Zel'dovich approximation.


Each simulation evolves (2x)512^3 particles including dark matter (and baryons) within a periodic cube of comoving side of 10 per h Mpc. The Plummer-equivalent comoving softening length is 0.2 per h kpc and the particle resolution is 4.7 and 0.9 x 10^5 per h solar masses for dark matter and baryons or 5.7 x 10^5 per h solar masses if only dark matter particles are considered. We summarize the adopted subgrid physics prescriptions in this section. Cooling [Wiersma2009a] consists of both primordial elements and metal emission lines from carbon, nitrogen, oxygen, neon, magnesium, silicon, sulphur, calcium and iron. Star formation [Schaye2008] occurs in the ISM that is assumed to be multiphase. Supernova feedback can be simulated kinetically [DallaVecchia2008] or thermally [DallaVecchia2012]. In order to compare galaxy properties between Meraxes and Smaug, we focus on the hydrodynamic simulation with thermal supernova feedback implemented.


Reionization feedback is implemented as a UV/X-ray background with all gas particles being instantaneously heated to 10^4 K at a given redshift. Although this prescription is numerically achievable and is considered appropriate in the 10 per h Mpc volume of Smaug simulations, it is not an accurate calculation of reionization feedback. Therefore, the semi-analytic prescription of reionization feedback will be compared to two hydrodynamic simulations with reionization redshifts of 9 and 6.5, which bracket the observed CMB and Lyman-alpha forest boundaries of the EoR, and represent the strongest and weakest feedback scenarios, respectively. Simulations utilized in this study is summarized below: (1) DMONLY, a collisionless N-body simulation; (2) NOSN_NOZCOOL, NOSN_NOZCOOL_LateRe and NOSN_NOZCOOL_NoRe, three toy models with cooling in the absence of metal line emission and ignoring supernova feedback; and (3) WTHERM, a complete hydrodynamic simulation including radiative cooling from primordial elements and metals, as well as stellar evolution, thermal supernova feedback and instantaneous photoionization heating from a reionization background at redshift z=9.


In this work, with algorithms described in [qin2018], we 1) include the aforementioned modifications of the semi-analytic properties or prescriptions; 2) build halo merger trees using the DMONLY simulation; 3) match each individual galaxies between Meraxes and Smaug outputs; and 4) identify star-forming and hot gas in Smaug. We present the comparison result between dwarf galaxy properties predicted by the hydrodynamic simulation and SAM in this section. Reionization feedback in the SAM is incorporated by inhibiting the local baryon fraction of haloes using a filtering mass. In this work, we adopt the average filtering mass, which ignores the spatial distribution of the IGM ionization state and depends only on redshift. In order to assess the validity of this feedback prescription, we compare with two Smaug hydrodynamic simulations in which gas particles are instantaneously heated to 10^4 K at different redshifts.


Note that the suppression due to the ionizing background is quantified in the SAM using a baryon fraction modifier, which, in hydrodynamic simulations, can be informed by comparing the baryonic components of galaxies matched between the different reionization feedback runs. We see that, through photoionization heating, reionization plays a significant role in reducing the fraction of baryons, and that the baryon fraction modifier adopted in the SAM is in general agreement with the hydrodynamic result -- the modifier decreases in less massive haloes and towards lower redshifts. In a 10 per h Mpc volume, the UV/X-ray ionizing background adopted in the two hydrodynamic simulations represents the strongest and weakest feedback scenarios that are consistent with the CMB and Lyman-alpha observations. However, since reionization does not affect the gas component before the reionization redshift in these simulations, its feedback on the baryonic reservoirs cannot be captured by the simulations at higher redshifts.


Therefore, the time when reionization feedback becomes important is relatively late compared to the SAM where the onset of reionization is more gradual and realistic on large scales. As a result, the baryon fraction is overestimated in the hydrodynamic simulations at earlier times, which can potentially lead to an overproduction of stellar mass in dwarf galaxies. We next investigate the stellar evolution and feedback in Meraxes and Smaug, starting with a discussion of involved free parameters in the SAM. Cosmological SAMs are usually calibrated against the observed galaxy stellar mass functions where a sufficient sample is available. By doing this, the stellar component is assured to be well modelled in a statistical context, and with more upcoming observations, the parameter space becomes better constrained and missing physics in the SAM might be revealed. However, one of the issues about this calibration strategy is that it cannot guarantee the modelled galaxies are also representative of real galaxies in terms of their unobservable properties.


Taking the gas component as an example, although a handful of radio telescopes are capable of observing the gas component of distant galaxies, the current sample remains small, limiting our understanding of how galaxies accrete baryons and convert their hydrogen into stars in the early universe. In order to illustrate this, we use the total-gas-based star formation model with parameters adopted in [Mutch2016a] as an example. With a short depletion time-scale of the total star-forming gas, the dynamical time-scale for gas transition, and strong supernova feedback, Meraxes was able to reproduce the observed stellar mass function at redshift z=5-7 in [Mutch2016a]. We see that the semi-analytic prediction is in agreement with Smaug above the resolution limit. We next show more detailed galaxy property evolution, including the total baryonic mass, star-forming gas mass and SFR, from the two numerical experiments in two mass ranges.


We see that, although the SAM is in agreement with the hydrodynamic simulation on the stellar mass function at a large range of redshifts, they disagree on the evolutionary path of the gas component. We find that the baryonic mass is about 2-5 times smaller in the SAM compared to the hydrodynamic simulation, suggesting that too much supernova energy has been coupled to the ISM. In addition, the hydrodynamic simulation shows an increasing amount of star-forming gas towards higher redshifts for a given halo mass. This suggests that cooling (or cold-mode accretion) might be more efficient in the early universe. On the other hand, the SAM underestimates the star-forming gas reservoir at higher redshift but predicts a similar SFR. This suggests that the depletion time-scale might be set shorter in the SAM, which happens to result in an agreement with the hydrodynamic result on the stellar mass function.


We recalibrate our chosen parameters in order to simultaneously reproduce the evolutions of the stellar mass function of the hydrodynamic simulation, as well as the following three quantities: total baryonic mass, star-forming gas mass, and SFR. After exploring the parameter space, we identify a set of parameters that lead to a better agreement on the property evolution of galaxies with virial mass > 10^10 solar masses. This model is referred to as SAM_KS_limited. However, in the low-mass range where the virial mass is between 10^9 and 10^10 solar masses, this model fails to reproduce the evolutionary path of the star-forming gas reservoir calculated by the hydrodynamic simulation. During the experiment, we find that the star-forming gas mass at low redshift does not change by incorporating a larger mass loading factor, which is expected to further suppress the star-forming gas mass through supernova heating.


This suggests that the bulk of star-forming gas is stored in quenched galaxies where the star-forming gas mass is less than the critical mass for star formation. According to the total-gas-based star formation prescription, galaxies can only form stars when their gas reservoirs are adequate. This reservoir mass threshold is calculated based on observations at the local Universe. In the current DRAGONS series, we instead have adopted a lower critical mass. This is supported by [Henriques2015], which proposes to reduce the mass threshold of star-forming galaxies to reconcile the issue that previous SAMs have overpredicted the number of quenched galaxies in the low-mass range while these galaxies still possess a significant amount of star-forming gas reservoir. This might explain the evolution of star-forming gas mass of dwarf galaxies predicted by SAM_KS_limited and suggests that the threshold of star formation needs to be further reduced in these low-mass galaxies.


We next focus on the dwarf galaxies and recalibrate the model without any thresholds of star formation. The result of this model, SAM_KS_unlimited, shows that while SAM_KS_limited with a critical mass of approximately 10^8 solar masses is better at reproducing the hydrodynamically simulated high-mass galaxies, the updated model with no threshold is more consistent with the hydrodynamic result at the low mass range. This indicates high-redshift less massive galaxies, in general, possess lower thresholds of star formation as well. Note that in SAM_KS_unlimited, a rapidly evolving gas transition time-scale is crucial to reproducing the evolution of the star-forming gas mass calculated by the hydrodynamic simulation. However, it also leads to unrealistically rapid changes of the gas transition efficiency in the SAM. We note that, with more intense supernova heating to offset it, additional gas can be allowed to transition from hot to star-forming.


Therefore, incorporating a larger mass loading factor at lower redshifts will decrease the transition time-scale accordingly and allow moderate changes of the transition rate. However, as we will see, supernova heating only plays a secondary role in changing the evolution of star-forming gas of high-redshift dwarf galaxies. In these two SAM_KS models, stars form by consuming the total star-forming gas reservoir. However, due to the unknown time-scale of depleting the total gas, the degeneracy between the processes of cooling and heating exists. [Duffy2017] investigated the molecular hydrogen component of dwarf galaxies using the Smaug simulations and found the depletion time-scale of molecular hydrogen is approximately 0.3 Gyr, independent of the feedback regime. They also discussed the mass and redshift dependencies when applying SAMs with molecular hydrogen-based star formation laws and proposed a redshift-dependent depletion time-scale, the extrapolation of which agrees with the previous findings at the local Universe.


We note that the scaling index of the depletion time-scale was motivated from the KS law with an assumption that galactic discs are self-gravitating and follow exponential surface profiles. The latter might need revising for high-redshift dwarf galaxies. In the early universe, galaxies tend to possess larger velocity dispersions and both mergers and cold-mode accretion are significant. These all indicate that high-redshift galaxies might have thickened discs. In this section, we adopt a less steeply evolving redshift dependency due to thicker discs at high redshift, and then calibrate cooling and supernova feedback efficiencies to reproduce the dwarf galaxy properties from Smaug. We will further discuss the semi-analytic prediction when varying the depletion time-scale. We see that without any degeneracies, the molecular hydrogen-based model can still reproduce the hydrodynamic calculation of the properties of dwarf galaxies as well as the cosmic evolution of the stellar mass function.


Compared to the SAM_KS_unlimited result, SAM_H2 agrees better with Smaug on the evolution of the total baryonic mass of dwarf galaxies, and the calculation of massive galaxies. However, it still overestimates the total baryonic mass and underestimates the mass of star-forming gas and SFR of massive galaxies, suggesting that these galaxies might possess different cooling and supernova feedback efficiencies or shorter depletion time-scale of molecular hydrogen compared to less massive galaxies. The success of our SAM with the molecular hydrogen-based star formation prescription and a fixed molecular hydrogen depletion time-scale is encouraging. It indicates that the accretion-cooling-depletion-heating-and-ejection pathway of gas is still representative for dwarf galaxy formation at high redshift in terms of predicting the gas and stellar properties of the hydrodynamic simulation. We next use the molecular hydrogen-based SAM as an example and illustrate the impact of changing the relevant parameters with comparisons to the fiducial SAM_H2 model.


The gas transition time-scale of the fiducial SAM_H2 model results in a significantly larger value at low redshift. We show this scaling, as well as the result of assigning the transition time-scale with the free-fall time-scale (a common assumption adopted in the literature for the rapid cooling regime), and a time-scale that evolves slower towards higher redshifts compared to the fiducial model. We see that adopting a shorter transition time-scale results in the star-forming gas reservoir receiving more efficient replenishment. In the case of unchanged gas depletion time-scale, SFR increases. Consequently, more energy gets ejected from supernova explosions, leading to more suppressed total baryonic masses given that the energy coupling efficiency and mass loading factor for heating do not change. We see that without changing other parameters, a significantly evolving transition time-scale is required to reproduce the rapidly decreasing star-forming gas mass at lower redshifts as predicted by the hydrodynamic simulation.


We note again that varying the transition time-scale from the dynamical time-scale was proposed to compensate for the overestimated collapse rate attained by assuming SIS profiles for hot gas and the underestimation due to large transition radii between hot and star-forming gas. In the presence of feedback, the star-forming disc shrinks, in particular at lower redshifts. Correspondingly, the overestimation of the transition time-scale becomes insignificant -- gas indeed needs to collapse into the central region to become star-forming gas. Therefore, we need a much longer transition time-scale at low redshift to take into account that hot gas is less dense at the inner regions compared to the SIS profile -- only a small amount of gas can collapse into the centre within the dynamical time-scale. We will investigate below whether the other free parameters can have a significant impact to the star-forming gas reservoir evolution, so that the steep gradient of star-forming gas mass with redshift can be reproduced without the implementation of a rapidly-evolving redshift dependency in the transition time-scale.


The molecular hydrogen depletion time-scale is better constrained than that of the total gas. However, observational results still possess large variance even in the local Universe, from a half to a few Gyr. Therefore, we use the redshift-dependent molecular hydrogen depletion time-scale proposed in [Duffy2017]. For the fiducial model, we use a less steeply evolving redshift dependency due to thicker discs at high redshift. We show the property evolution of using the time-scales proposed by [Duffy2017] as well as a constant value, which is commonly adopted for SAMs in the literature. We see that by increasing the time-scale of converting hydrogen into stars, star formation quenches, leading to weaker supernova ejection and heating. With the current configuration of parameters, we see that a constant 2 Gyr depletion time-scale significantly underestimates star formation at high redshift in agreement with [Duffy2017] and the star-forming gas evolution gradient is not expected to change significantly by varying the redshift dependency.


Supernova explosions increase the thermal energy of the ISM and expel baryons in dwarf galaxies. However, since the relevant region cannot be resolved in cosmological simulations, subgrid physics with free parameters are adopted by both hydrodynamic and semi-analytic modelling approaches. The fraction of supernova energy that contributes to feedback is f_th and epsilon_energy in Smaug and Meraxes, respectively. Since f_th is chosen to be unity, with all the supernova energy being coupled to the ISM, one might expect epsilon_energy=1 as well. However, because SAMs ignore the thermal energy of the star-forming gas and assume the temperature of hot gas does not change during one time step, some energy terms are zero. This means that, despite all supernova energy being coupled to the ISM in the hydrodynamic simulation, only a fraction of it contributes to feedback in the SAM.


How much of the supernova energy is coupled to the ISM and used to heat gas is governed by free parameters describing the mass loading factor in the SAM while in the hydrodynamic simulation, it is the increment of gas temperature that determines the number of gas particles that are affected. This indicates that, in the hydrodynamic simulation, the mean number of instantaneously heated nearby gas particles per stellar baryon is approximately 1.34 [DallaVecchia2012]. This small mass loading factor places the heated gas in the Bremsstrahlung cooling regime, achieving an efficient supernova feedback mechanism through heating. Moreover, due to the increased pressure from the thermal feedback, gas particles within high overdensities tend to move outwards in a wind. As the wind particles travel, they further increase the thermal energy of the nearby gas particles along the path, leading to a much larger effective mass loading factor over a long period of time.


Since the SAM captures the average property over 11Myr, one might expect the SAM mass loading factor to be much larger than 1.34 as well. However, we have also shown that in hydrodynamically simulated dwarf galaxies, gas particles need not be fully virialized to become non-star-forming gas while on the other hand, SAMs ignore the thermal energy of the star-forming gas and assume the non-star-forming hot gas shares the virial temperature of host halo. Without properly tracking the thermal energy of varied gas reservoirs in the SAM, it is challenging to determine the energy-ISM coupling efficiency and mass loading factor. In this work, against the hydrodynamic result of WTHERM, we have calibrated our fiducial SAM. We see that when the mass loading factor is fixed, more coupled energy to the ISM leads to stronger suppression of the total baryonic mass, which subsequently decreases the mass of the star-forming disc and quenches star formation.


From the MaximumSNCoupling model, we see that with all supernova energy used to convert star-forming gas to hot and eject hot gas from the galaxy, the total baryonic mass and star-forming gas become significantly suppressed. On the other hand, when the supernova energy coupling efficiency is fixed, less heating leads to a larger reservoir of star-forming gas and enhanced star formation. Consequently, more supernova energy is coupled to the ISM. With less energy used for heating, more mass in the hot gas reservoir gets ejected. Depending on the increased amount of star-forming gas and stellar mass as well as the decreased hot gas mass, the total baryonic mass varies slightly. In addition, the property evolution does not change significantly between these three models. Therefore, we do not expect that, by changing the heating efficiency of supernovae, the issue of incorporating a rapidly evolving gas transition time-scale can be resolved.


Following [qin2018], we further investigate the semi-analytic modelling prescriptions of galaxy formation that are commonly adopted in the literature. In this work, we include supernova feedback and homogeneous reionization background in both the Meraxes SAM and Smaug high-resolution hydrodynamic simulation, and make comparisons between the stellar and gas reservoirs predicted by these two methods. We focus on galaxies with virial mass between 10^9 and 10^11 solar masses. With the modifications previously proposed in [qin2018] including adjustments to halo masses from the merger trees, suppression of baryon fractions accounting for hydrostatic pressures, and the modulation of time-scales for the transition of gas from hot to star-forming and from star-forming to stars, we find that the current SAM is able to reproduce the hydrodynamic calculation of the cosmic evolution of galaxies with virial mass > 10^10 solar masses at high redshift.


This includes the stellar mass function, total baryonic mass, star-forming gas mass and SFR between redshift z=5-11. However, in less massive galaxies with SFR calculated using the total star-forming gas, we identify a significant amount of star-forming gas stored in quenched galaxies due to the imposed mass threshold of star formation. After reducing the threshold, the SAM successfully mimics the evolution of dwarf galaxies in the hydrodynamic simulation. We also investigate a second star formation prescription, which splits the star-forming gas disc into molecular and atomic hydrogen and forms stars from molecules [Lagos2011]. Fixing the depletion time-scale of molecular hydrogen inferred from a previous study of the Smaug hydrodynamic simulation [Duffy2017], we find that, with only calibrations of the gas transition rate and supernova efficiencies, the SAM can also reproduce the dwarf galaxy properties calculated by the hydrodynamic simulation.


In addition, we find that when reionization and supernova feedback are included, dwarf galaxies tend to accrete a significant amount of star-forming gas at early times (redshift z>10), which quickly becomes suppressed towards lower redshifts. Future work needs to take this into account and incorporate modelling of cold-mode accretion to study dwarf galaxies in the early universe. We show the impact to the semi-analytic calculation, in the presence of reionization and supernova feedback, of incorporating the halo mass and baryon fraction modifiers, which correspond to the slower evolution of haloes and less efficient gas accretion due to hydrostatic pressure. We apply Meraxes with the total-gas-based star formation law and the same parameters adopted in [Mutch2016a] but without the baryon fraction modifier and without any modifiers.


We see that without the halo mass modifier, the halo mass function is overestimated compared to the hydrodynamic result at high redshift, which consequently increases the mass function of gas and stars. In addition, further excluding the baryon fraction modifier increases the amount of gas accreted by the host halo and subsequently causes more stars to form. However, we see that the modifications have an insignificant impact to the stellar mass function in the current observable range, which requires deeper surveys with upcoming space programs such as JWST. [qin2018] shows that in the absence of feedback, the majority of dwarf galaxies in the hydrodynamic simulation accrete gas particles with temperatures around a few of 10^4 K, which is much lower than their halo virial temperatures. This represents a cold-mode accretion of the infalling gas [Keres2005, Keres2009], which in the SAM is currently modelled through the cooling prescription of the rapid cooling regime proposed by [white1991].


The infalling hot gas is also assumed to follow the singular isothermal sphere (SIS) profile. To ease demonstration in this paper, we term the time-scale of gas being transited from hot reservoir to the star-forming disc as a transition time-scale. Most massive haloes are able to create shocks and heat the infalling gas, resulting in hydrostatic equilibrium. In this case, which is termed the hot halo regime, the time-scale of hot gas transitioning to star-forming is determined by the thermal cooling time-scale. However, it is difficult to generate shock heating in less massive systems, leaving little support to prevent gas from infalling onto the central disc, and cooling becomes rapid. In this rapid cooling regime, the prescription assumes the star-forming gas disc is relatively small and such a process happens as free-fall. In making comparisons of the gas reservoir calculated by the SAM and hydrodynamic simulation in [qin2018], we found the transition time-scale equals the dynamical time-scale becomes less accurate when applying the rapid cooling prescription to high-redshift dwarf galaxy modelling.


This is due to the aforementioned two assumptions which lead to over- and under-estimations of the gas transited from hot to star-forming, respectively. 1) Assuming the SIS profile of the accreted mass overestimates the gas density in the inner regions. 2) star-forming gas particles of dwarf galaxies (in the hydrodynamic simulation) possess larger extensions and can be found as far as the virial radius. This means that assuming gas can only transfer from non-star-forming hot gas to star-forming when it reaches the galaxy centre introduces a longer inflow path and hence leads to an overestimated transition time-scale. In this case, the mass of hot gas transitioning to star-forming is underestimated instead. We note that when feedback is included, semi-analytic modelling of dwarf galaxies still suffers from these two factors. First, in order to demonstrate that most high-redshift dwarf galaxies in the SAM are still identified as in the rapid cooling regime when reionization and supernova feedback are included, we calculate the cooling radius, at which the time-scale of thermal cooling is equal to the halo dynamical time in the SAM.


Note that gas within the cooling radius is considered to have reached hydrostatic equilibrium and cool thermally if the cooling radius is less than the virial radius. However, in the case of a large cooling radius, the infalling gas will not be able to form stable shocks or remain in hydrostatic equilibrium. Accordingly, all of the accreted gas directly collapses into the central regions as free-fall. We see that most low-mass galaxies discussed in this work are considered to be in the rapid cooling regime. Next we show the evolution, in terms of the density--temperature phase and spatial distributions of star-forming and non-star-forming gas particles, of the most massive halo in the WTHERM Smaug simulation identified at redshift z=5 as an example, and discuss the gas density profile of galaxies with stellar mass of approximately 10^7 solar masses. We see that, compared to the NOSN_NOZCOOL_NoRe simulation where heating from supernova (and reionization) is not included, galaxies within the same stellar mass range are hosted by larger haloes with more gas particles identified as non-star-forming when the feedback is considered.


However, the total gas mass does not change significantly, indicating suppressions of baryonic mass and self-regulation of star formation. Moreover, although the star-forming regions become relatively smaller in WTHERM, they still possess a large dispersion at high redshift. This can also be observed from the large radius of the maximum rotation of the most massive halo, which suggests the necessity of an enhanced inflow rate between the circum-galactic medium and ISM at earlier times. More accurate semi-analytic modelling of gas accretion should not only distinguish the hot- and cold-mode inflows with gas reaching the star-forming disc on different time-scales, but also account for the larger disc size at higher redshifts. We consider these as a future project with a more complete cooling function implemented. For the purpose of accurately capturing the gas transition time-scale using the current rapid cooling prescription, in [qin2018], we proposed to change the cooling efficiency when galaxies are identified in this regime.


We introduced a maximum cooling factor, which was used to modulate the gas transition time-scale based on the time-scale of free-fall. In this work, we adopt this modification by incorporating a form of the transition time-scale as a function of redshift. However, considering the transition radius between star-forming and non-star-forming gas changes due to feedback, the normalization and scaling index are not expected to possess the same values as adopted in [qin2018]. Therefore, we leave them as free parameters and explore the transition time-scale in this work.


We use the Dark-ages, Reionization And Galaxy-formation Observables from Numerical Simulations (DRAGONS) framework to investigate the effect of galaxy-formation physics on the morphology and statistical signatures of ionized hydrogen (HII) regions during the Epoch of Reioinization (EoR). DRAGONS self-consistently couples a semi-analytic galaxy-formation model with the inhomogeneous ionizing UV background, and can therefore be used to study the dependence of morphology and statistics of reionization on feedback phenomena of the ionizing source galaxy population. Changes in galaxy-formation physics modify the sizes of HII regions and the amplitude and shape of 21-cm power spectra. Of the galaxy physics investigated, we find that supernova feedback plays the most important role in reionization, with HII regions up to approximately 20 per cent smaller and a fractional difference in the amplitude of power spectra of up to approximately 17 per cent at fixed ionized fraction in the absence of this feedback. We compare our galaxy-formation-based reionization models with past calculations that assume constant stellar-to-halo mass ratios and find that with the correct choice of minimum halo mass, such models can mimic the predicted reionization morphology.


Reionization morphology at fixed neutral fraction is therefore not uniquely determined by the details of galaxy formation, but is sensitive to the mass of the haloes hosting the bulk of the ionizing sources. Simple EoR parametrizations are therefore accurate predictors of reionization statistics. However, a complete understanding of reionization using future 21-cm observations will require interpretation with realistic galaxy-formation models, in combination with other observations. Many fundamental questions about the first galaxies remain unanswered despite recent progress in both observations and theoretical studies. Future indirect observations of the first sources of light, through their impact on the surrounding intergalactic medium (IGM), promise to shed new light on some of the important processes involved in their formation and evolution. For this reason, the period during which galaxies reionized the Universe -- the Epoch of Reionization (EoR) -- has become a focus of both theory and observation [see][for reviews][FCK2006, FOB2006, MW2010].


Observations of the cosmic microwave background (CMB) and high-redshift sources (such as quasars, Lyman-break galaxies, Lyman-alpha emitters and gamma-ray bursts) have allowed some constraints to be placed on the timing and duration of the EoR. Without further measurements of the degree of reionization throughout this period, however, its detailed history will remain unknown. Furthermore, the constraints provided by these observations are confined to the global ionization state of the IGM, with limited information on the sources and environmentally-dependent astrophysics at play. The new observational window provided by low-frequency radio experiments will be the first direct probe of neutral hydrogen during reionization. Together, instruments such as LOFAR, MWA, PAPER and GMRT promise to yield statistical measurements of the state and morphology of cosmic hydrogen throughout the EoR through fluctuations in the redshifted 21-cm emission line intensity.


The next generation of instruments, including SKA and HERA, will allow high-resolution tomographic images of the ionized structure around individual regions of ionized hydrogen (HII) to be made [see, e.g.,][MELLEMA2015, WGK2015]. Combined with detailed simulations, these observations should pave the way to a better understanding of the connection between galaxies and reionization. Numerous theoretical methods have been used to study reionization. Analytic modelling [e.g.][SHAPIRO1994, WL2003, FZH2004] can provide insight into some of the processes involved, but cannot include many of the most important physical effects, and do not generate spatial realizations. N-body simulations, post-processed using radiative transfer methods based on ionizing sources placed within dark matter haloes, provide a description of the structure, but not a self-consistent calculation of source properties [see, e.g.,][ILIEV2005, MLZD2007].


Hydrodynamical simulations that include radiative transfer [see, e.g.,][GNEDIN2000a, PETKOVA2011, PAARD2013, BAUERetal2015] include the effects of feedback and follow structure formation and cosmic reionization self-consistently but do so at great computational expense when performed with the temporal cadence, cosmic volume and resolution required to capture the evolution and relevant spatial scales of reionization. As a compromise, semi-numerical simulations [ZAHN2005, MF2007, GW2008, ALV2009, CHR2009, THOMAS2009, MFC2011] are computationally inexpensive and provide an efficient means of exploring high-dimenisional parameter spaces and very large volumes (and hence rare objects). However, these simulations have not previously included realistic galaxy formation in their formulation. Previous theoretical work constraining properties of the ionizing sources throughout the EoR with 21-cm observations has been carried out using a number of approaches.


[BARKANA2009] fitted properties of the galaxy population during reionization (such as mean halo mass of the ionizing sources) to simulated 21-cm power spectra using analytical modelling. Extensive investigation using radiative transfer simulations run on top of suites of N-body simulations has been performed by [MLZD2007] and [ILIEV2012]. This work investigated whether or not observations will be able to distinguish reionization due to sources within populations of atomically-cooled haloes of different mass ranges, as well as the effects of ionizing source efficiencies and self-regulation. Other work has explored the effect and relative dominance of supernova feedback during reionization [see, e.g.,][KIM2013a], and the ability of observations to constrain galaxy formation through the effect of variation in the mass- and redshift-dependent escape fraction of ionizing photons [KIM2013b].


Using a Markov chain Monte Carlo analysis tool, [GREIG2015] made estimates of astrophysical parameter contraints from simulated observations, including the ionizing efficiency and mean free path of ionizing photons, as well as lower limits on the virial temperature of star-forming haloes. The Dark-ages, Reionization And Galaxy-formation Observables from Numerical Simulations (DRAGONS) project takes a hybrid approach by integrating a semi-numerical calculation of reionization within a semi-analytic model (SAM) of galaxy formation built upon an N-body simulation specifically designed for EoR studies. DRAGONS aims to answer the following questions about galaxy formation and cosmic reionization. What was the nature of the ionizing sources and the processes involved in reionization? What was the relative impact of these processes on cosmic reionization and subsequent galaxy formation? Was reionization a self-regulatory process? Will future observations enable us to distinguish these processes, quantify them, or rule out certain reionization scenarios?


This paper focusses on the morphology and statistical signature of cosmological reionization and is the fifth in a series describing the DRAGONS project. The first paper in the series [[DRAGONS1]] introduces the collisionless N-body simluation, Tiamat, which is used as the basis of this work. The structure of the high-redshift dark matter haloes contained within Tiamat is described in the second paper [[DRAGONS2]]. The third paper [[DRAGONS3]] presents Meraxes, the semi-analytic model of galaxy formation and evolution developed for DRAGONS, while the fourth [[DRAGONS4]] discusses the resulting galaxy luminosity functions. This paper is structured as follows. We begin by describing our simulation and modelling methodologies in Section 2. In particular, Section 2.4 provides a description of the model runs we perform in order to explore the effect of key astrophysical mechanisms included in our simulations.


We then present our results in Section 3, where we compare our model runs using various statistics. We discuss aspects of these results in Section 4 before presenting a summary in Section 5. We include an appendix containing supporting material demonstrating the spatial convergence of our results. All globally-averaged quantities (e.g. neutral fraction) are volume weighted, and distances given in comoving units unless stated otherwise. Our choice of cosmology throughout is the standard spatially-flat Planck LambdaCDM cosmology [PLANCK2015] (h, Omega_m, Omega_b, Omega_Lambda, sigma_8, n_s) = (0.678, 0.308, 0.0484, 0.692, 0.815, 0.968). In this section we summarize our simulation and modelling methodologies, beginning with our collisionless N-body simulation (Tiamat), then our semi-analytic model (Meraxes) and finally how we integrate these with the semi-numerical reionization code 21cmfast [MFC2011].


The collisionless N-body simluation used as the basis of this work is called Tiamat. It consists of 2160^3 particles in a 100~Mpc, periodic, cubed box evolved using GADGET-2 [GADGET22005]. Initial conditions were generated at redshift z = 99 and the simulation was run down to redshift z = 5 providing 100 snapshots of particle data equally spaced in time between redshifts 5--35 (roughly one every 11 Myr). Halo finding was performed using SUBFIND [SUBFIND2001] and merger trees from the resulting halo catalogues were created using the methodology to be presented in Poole~et~al.~(in preparation). A triangular-shaped cloud mass assignment scheme [see, e.g.,][CUI2008] was used to create density grids used by Meraxes/21cmfast. A complete description of Tiamat, resulting halo mass functions and analysis is given in Paper-I.


Semi-analytic models parametrize the physics of galaxy formation to enable fast and accurate realizations of galaxy properties within cosmic volumes. The semi-analytic model of galaxy formation used in this work is called Meraxes, and includes baryonic infall, cooling, star formation, reionization and supernova feedback, metal enrichment, stellar mass recycling, mergers, and ionizing flux from galaxies hosted by dark matter haloes temporarily unresolved in our input merger trees. Calibration was performed so as to agree with observations of the CMB optical depth to electron scattering, and to replicate the observed evolution of the galaxy stellar mass function between redshifts 5--7. A complete description of Meraxes is given in Paper-III. A feature of Meraxes that is important for its application to reionization is that it is run on `horizontal' merger trees, which enable calculation of the evolution of all galaxies at each time-step, so that feedback processes from neighbouring sources can be incorporated self-consistently.


We simulate cosmic reionization by applying the 21cmfast algorithm [details of 21cmfast are described in][MFC2011] within the Meraxes semi-analytic model including the effect of a local/inhomogeneous ionizing UV background (UVB) as described by [SM2013a]. Coupling the galaxy properties modelled by Meraxes with the UVB calculated by 21cmfast provides a self-consistent, UVB-regulated realization of reionization. This is implemented in the following manner: 1. At the end of each snapshot, Meraxes constructs halo mass, stellar mass and star-formation rate grids; 2. 21cmfast's excursion-set filtering algorithm calculates the corresponding ionization state and inhomogeneous UVB-intensity grids; 3. Meraxes keeps track of the redshift at which each voxel was first ionized, and calculates a baryon fraction modifier of each halo which is then used to calculate the effect of the local UVB on the amount of infalling bayonic matter; 4. Meraxes evolves all of the galaxies in the simulated volume forward to the next time-step, and then the process is repeated.


Reionization was simulated on a 512^3 grid, which we find sufficiently resolves behaviour for analysis in this paper (see Appendix A which discusses the spatial convergence of our results). The basic methodology of the 21cmfast algorithm is to utilize an excursion-set approach to identify HII regions and provide a neutral hydrogen fraction grid of the simulation volume. Starting at scales comparable to the mean free path of ionizing photons in an ionized IGM at redshift z ~ 6 and incrementing toward smaller scales, voxels are flagged as being ionized if, in a sphere of radius R, the integrated number of ionizing photons is greater than the number of hydrogen and neutral helium atoms plus the mean number of recombinations. In our prescription this ionization condition can be written in terms of an ionization efficiency factor, xi, and the collapsed fraction of mass, f_coll, in the form of stars within R: xi times f_coll is greater than or equal to 1.


The collapsed fraction is defined as f_coll equals the stellar mass M_star divided by the total mass M_tot. We set the unresolved, sub-cell neutral hydrogen fraction corresponding to the last smoothing scale in our simulations to xi times f_coll, where R_voxel is of the order of the grid resolution of 21cmfast. The ionization efficiency factor depends on the baryon fraction, f_b, mean number of ionizing photons per stellar nucleon, N_gamma_bar, and their escape fraction, f_esc. We set the baryon fraction to its universal value, f_b = Omega_b / Omega_m. The (1-3Y_He/4) factor, where Y_He = 0.24 is the mass fraction of helium, accounts for helium in the ionization budget. Both f_b and Y_He are well constrained, while N_gamma_bar is set by an assumed initial mass function of stars. The value of f_esc is less well constrained for galaxies at high redshift.


We account for recombinations by commencing the reionization excursion-set filtering at a scale comparable to the mean free path of ionizing photons in an ionized IGM. This effectively acts as a `global reionization horizon' [see][for a discussion of this in terms of the framework of 21cmfast][GREIG2015] and slows reionization during its late phase. Alternatively, it is possible to account for homogeneous recombinations by appropriately including a factor of (1 + N_rec_bar) in the equation for the ionization efficiency factor, where N_rec_bar is the globally-averaged number of recombinations. Homogeneous recombinations are, however, degenerate with the ionizing efficiency. The effect of inhomogeneous recombinations has been investigated by a number of authors [see, e.g.,][CHR2009, SM2014, WMPS2015] and is left for future work. In order to simulate the effect of feedback from a UV background on star formation, the 21cmfast algorithm calculates a grid of the local average UVB intensity within a given ionized region.


Following [SM2013a], this is given by the average intensity at 21cm, <J_21>_HII, equals ((1+z)^2 / 4pi) times lambda_mfp times h_P times alpha times f_bias times epsilon, where lambda_mfp is the comoving mean free path of ionizing photons (assumed within the excursion-set algorithm to be equal to the filtering radius, R), h_P is the Planck constant, alpha is the spectral index of the UVB, f_bias is an ionizing emissivity bias factor and epsilon is the ionizing emissivity. We model the background spectrum using a power law with spectral index alpha = 5 [TW1996], which leads to a steep spectrum corresponding to a stellar-driven UVB, effectively ignoring the contribution from harder spectral sources such as quasars. We expect using a much harder spectrum would cause feedback to be more effective [see, e.g.,][TW1996], hence changing our results. We leave this investigation to future work.


The bias factor in the UV background equation accounts for the higher than average ionizing emissivity at halo locations due to their clustering. We use a value of f_bias = 2 based on discussion in [MD2008]. The ionizing emissivity is the number of ionizing photons emitted into the IGM per unit time, per unit comoving volume. This is calculated for each voxel using the grid of gross stellar mass, M_star,gross, through epsilon equals f_esc times N_gamma_bar times (M_star,gross / m_b_bar) times (1 / t_H(z)) times (1 / V_R), where the gross stellar mass has been averaged over a sphere of radius R with volume V_R, t_H(z) is the Hubble time which acts as the average star-formation time-scale, and m_b_bar is the mean mass of each baryon. A UV background can lead to negative feedback on cosmic reionization in various ways. Its presence reduces baryonic infall by way of heating the IGM, therefore affecting its gas cooling properties.


It can also photo-evaporate gas from shallow potential wells surrounding small galaxies. Both of these mechanisms lead to the quenching of star formation. We parametrize this effect in Meraxes using a spatially- and temporally-dependent baryon fraction modifier, 0 <= f_mod <= 1, which acts to reduce the mass fraction of baryons contained in freshly accreted matter from its universal value, f_b, to f_mod times f_b. Following the work of [SM2013b], we calculate f_mod using f_mod = 2^(-M_filt / M_vir), where M_vir is the halo mass and M_filt is a `filtering' mass, defined to be the total halo mass at which the baryon fraction is half the universal value. Together with the snapshot redshift, z, and the redshift at which the voxel was first ionized, z_ion, the local average UVB intensity within a given ionized region is used to calculate the filtering mass. The free parameters were fit by [SM2013b] to one-dimensional collapse simulations.


The ionizing efficiency of very high-redshift galaxies is poorly constrained by observations of galaxy UV luminosity functions and our knowledge of the escape fraction of ionizing photons and the abundance of faint galaxies is poor. Furthermore, simulations show a strong mass dependency and anisotropy of the escape fraction [e.g.][PAAR2015], both of which can strongly impact the topology of reionization. This lack of observational constraint and expected complexity allows for a wide variety of plausible reionization scenarios [examples of recent work in this area include studies by, e.g.,][KFG2012, KIM2013b, PAAR2015]. As shown in Paper-III, while we are able to match high-redshift galaxy stellar mass functions and satisfy constraints imposed by CMB measurements using an escape fraction which does not evolve with redshift, we are unable to also match the normalisation and slope of the observed ionizing emissivity at redshift z <= 6 without using a redshift-dependent escape fraction.


Furthermore, Lyman-alpha emitter studies suggest that reionization ends at redshift z ~ 6 [see, e.g.,][DIJKSTRA2014, CPHB2015]. An evolving escape fraction prolongs reionization and delays its completion, hence simultaneously satisfying these constraints. Similarly, inhomogeneous recombinations [a la][SM2014] are expected to prolong the late stages of reionization. However, for the purpose of comparing models presented in this work, we have kept our prescription for the escape fraction as simple as possible by assuming it to be spatially and temporally invariant. This section describes the set of models used in this work. Models that ignore the effects of supernova feeback have been recalibrated so as to provide the same total redshift z = 5 stellar mass density as the fiducial model. This has been achieved by using a lower star-formation efficiency parameter, alpha_SF.


In order to facilitate direct comparison between our models, we `tune' the ionization efficiency factor of each by varying the escape fraction of ionizing photons so as to match the globally-averaged neutral fraction of our fiducial model at redshift z is approximately equal to 8.4 (the mean neutral hydrogen fraction is approximately 0.68). This results in the models having very similar reionization histories. Therefore, with each model having a similar global neutral fraction and the same underlying density field at each snapshot/redshift, we can sensibly compare models by way of their 21-cm power spectra (which are sensitive to both neutral fraction and density). We choose to match at a mean neutral hydrogen fraction of approximately 0.7 as our cosmic volumes at this stage of reionization provide a fair sample of different HII region sizes and forms. Fiducial model (F): We couple both the spatial and temporal evolution of the reionization structure and ionizing field to the growth of the source galaxy population through the incorporation of feedback by an inhomogeneous UVB.


Supernova feedback is included and a spatially-homogeneous, redshift-independent escape fraction of ionizing photons of f_esc = 0.2 is used. This model is identical to the fiducial model in Paper-III and is discussed at length in that paper and Paper-IV. No SN feedback (NoSNeFB): This is a recalibrated version of the fiducial model which ignores the effect of feedback from supernovae. A constant f_esc = 0.239 is used. No feedback (NoFB): This is a recalibrated version of the fiducial model which ignores the effect of feedback from both reionization and supernovae. A constant f_esc = 0.2328 is used. Constant stellar-to-halo mass ratio (CSHR): This model ignores the galaxy properties calculated by Meraxes and is therefore decoupled from the effects of any form of feedback. Stellar mass is calculated assuming a constant ratio between stellar mass and the virial mass of each Friends-of-Friends (FoF) halo group using M_star / M_vir = 0.055.


This is based on the value we find in the high-mass regime of a simulation with no feedback from either reionization or supernovae. This model (along with the following mass-cut models) is included so as to facilitate comparison of our fiducial model simulation with previously published semi-numerical simulations and models that do not include realistic galaxy formation and feedback effects. A constant f_esc = 0.01547 is used. Constant stellar-to-halo mass ratio with 10^9 M_sun mass cut (CSHR.Mcut.9): This is the same as the CSHR model but only includes galaxies whose FoF virial mass is M_vir >= 10^9 M_sun. A constant f_esc = 0.0312 is used. Constant stellar-to-halo mass ratio with 10^10 M_sun mass cut (CSHR.Mcut.10): This is the same as the CSHR model but only includes galaxies whose FoF virial mass is M_vir >= 10^10 M_sun. A constant f_esc = 0.1302 is used.


Figure 1 shows the evolution of the globally-averaged neutral fraction, of each model (top panel) and their difference with respect to the fiducial model (middle panel) as a function of redshift. Reionization for our fiducial model occurs over a period of 411~Myr (period between the simulation volume being 1--99~per~cent ionized) and is 99~per~cent ionized at redshift z is approximately equal to 6.9. The data point in Figure 1 indicates the neutral fraction at which the models have been matched. This figure provides a comparison between the duration and rates of reionization for each model. Each model follows a similar reionization profile, the greatest difference being for the NoFB model. The CSHR model reionizes least rapidly, owing to the more complete distribution of ionizing source masses for this model. The bottom panel in Figure 1 shows the cumulative CMB optical depth to Thomson scattering as a function of integrated redshift for our reionization model simulations.


These results have been calculated in the same manner as in Paper-III. The horizontal dashed line and surrounding shaded region mark the latest Planck observations and +/-1-sigma uncertainty [PLANCK2015]. The total CMB optical depth to electron scattering of our fiducial model falls within this observational constraint. While the other models have had their reionization histories matched to the fiducial model, small differences in their reionization profiles result in a small variation in their optical depths. All results, however, fall well within the constraints provided. The spatial progression of reionization can be seen in Figure 2 which shows slices of the neutral gas density for our fiducial model at selected redshifts. The underlying dark matter density contrast is shown in fully ionized regions (`bubbles'). In this section we focus on the morphology of reionization, and compare the effect of different ionizing source populations and feedback mechanisms.


The corresponding results for 21-cm power-spectra statistics are discussed in Section 3.3. Reionization on intergalactic scales is expected to proceed in an `inside-out' manner whereby more-over-dense regions (e.g. sites of the first-formed galaxies) reionize before less-over-dense regions [see, e.g.,][ILIEV2006, MLZD2007, MFC2011, BAUERetal2015]. The spatial correlation between the higher-density cosmic web and regions of ionized hydrogen is clear in the neutral gas density slices of our fiducial model shown in Figure 2. The effect of feedback and different ionizing source populations can be seen in Figure 3, which shows slices through the ionization field of our models at three different global neutral fractions. In order to help explain differences and similarities between the ionization fields of our models, we refer to the plots in Figure 4 which show the following ionization properties of their source populations. In the top panel we have calculated the median fraction of mass in the form of stars at redshift z is approximately equal to 8.4 as a function of FoF virial mass, weighted by the escape fraction of each model.


This quantity acts as a proxy for the average ionizing luminosity of sources as a function of their host halo virial mass but includes no information about the size of their populations. In the bottom panel we show the probability distributions of the integrated ionizing photon contribution up to redshift z is approximately equal to 8.4 as a function of FoF virial mass for each model. As these histograms have been weighted by the gross stellar mass in each halo mass bin, the distributions include the effect of source population. The ionization field for the NoSNeFB model has a slightly larger population of small (1--3~Mpc) ionized regions and its overlapping regions tend to be smaller than those of the fiducial model. As evident from the top panel of Figure 4, including the star-formation suppressing effect of SNe feedback and tuning the escape fraction so as to match the models' reionization states effectively lowers the specific luminosity of sources hosted by medium-mass haloes, but increases it for the very lowest- and highest-mass sources.


Due to their relative populations, these medium-mass sources are the dominant drivers of reionization in the NoSNeFB model up to this period of reionization. As these haloes are less biased than more-massive haloes, they tend to populate the less-dense regions of the simulation volume away from the clustered sources central to the overlapping ionized regions. Therefore, without the feedback effects of supernovae, more small isolated ionized regions will exist and the overlapping regions will be smaller at fixed ionized fraction. This is in agreement with previous work investigating SNe feedback [see, e.g.,][KIM2013a], although the effects are significantly weaker in our results. We attribute this difference to Tiamat having an order of magnitude higher temporal resolution than the simulation used by [KIM2013a]. This resolution is required to resolve the dynamical time at redshift z > 6. The ionization field for the NoFB model is almost identical to that of the NoSNeFB model.


This is due to the dominance of supernova feedback over reionization feedback in suppressing star formation as discussed in Paper-III. The ionization field for the CSHR model has a larger population of small ionized regions, including very small (< 1~Mpc) regions, and its overlapping regions tend to be smaller than those of the fiducial model. Galaxies with low-mass FoF hosts in the Meraxes models contain, on average, less stellar mass than that expected using a constant stellar-to-halo mass ratio prescription. This is due to both the galaxy-formation modelling, and star-formation suppressing effects of reionization and (predominantly) supernova feedback. As evident in the bottom panel of Figure 4, sources hosted by these low-mass haloes in the CSHR model dominate reionization. This leads to a larger population of small isolated ionized regions, with a greater number of very small regions. Despite the relatively large shift in the host halo mass scale of sources which dominate reionization between the Meraxes-based models and the CSHR model, the resulting ionization fields are remarkably similar.


The ionization field of the CSHR.Mcut.9 model bears a striking resemblence to those of the no-feedback and CSHR models, but contains fewer very small isolated ionized regions. The mass cut imposed on this model removes the ionizing contribution of sources hosted by the lowest-mass haloes present in the fully-populated CSHR model but, as evident in the bottom panel of Figure 4, leaves reionization to be dominated by sources hosted by haloes in the same medium-mass range as the no-feedback models (hence their similarity). Ionized regions in the CSHR.Mcut.10 model are larger, more clustered and more spherical than those of the other models. On average, ionizing sources of the CSHR.Mcut.10 model are more luminous than those in all of our other models. Since only high-mass ionizing sources have been included in this model and these sources are more biased, they tend to cluster within the densest environments, forming large overlapping ionized regions.


In order to quantify these differences in the real-space morphology of reionization we calculate the ionized region (or `bubble') size distributions for each snapshot using the Monte Carlo method described in [MF2007]. In this method, an ionized voxel is randomly selected and its distance from an ionization phase transition in a randomly chosen direction is recorded. This is repeated 10^7 times to form a probability distribution function of region size. This methodology provides an approximate measure of the mean free path of ionizing photons inside ionized regions and has been extensively used as a proxy for bubble radius in other work [e.g.][FO2005, MLZD2007, MF2007]. The top panel of Figure 5 shows the mean size of ionized regions, as a function of global neutral fraction for each model. The bottom panel shows the ratio of the resulting mean scale of ionized regions relative to the fiducial model. This figure quantitatively supports the qualitative results illustrated in Figure 3.


We show the probability distribution of ionized region size for our models at a globally-averaged neutral fraction of approximately 0.68 in Figure 6. Relative to the fiducial model, we find the following comparisons of average `bubble' size: i) without SNe feedback, regions are approximately 19 per cent smaller; ii) without SNe or reionization feedback, regions are approximately 18 per cent smaller; iii) using a constant stellar-to-halo mass relationship results in regions that are approximately 22 per cent smaller, iv) using a constant stellar-to-halo mass relationship, but including only sources with haloes of mass greater than 10^9 M_sun, regions are approximately 17 per cent smaller, and; v) using a constant stellar-to-halo mass relationship but including only sources with haloes of mass greater than 10^10 M_sun leads to regions that are approximately 19 per cent larger.


We calculate the spatially-dependent 21-cm differential brightness temperature, delta T_b, between hydrogen gas and the CMB along the line of sight. For redshift z >> 1, the evolution of delta T_b can be written as delta T_b is approximately equal to 27 * x_HI * (1 + delta) * ((1+z)/10 * 0.15 / (Omega_m * h^2))^0.5 * (Omega_b * h^2 / 0.023) mK, where delta is the local dark matter overdensity and x_HI is the local neutral fraction of the gas. By using this formulation we ignore redshift-space distortions and assume the spin temperature of the cosmic gas, T_s, to be much higher than the CMB temperature, T_gamma. As a result, this signal is seen in emission. The latter assumption is valid in the post-heating regime where X-ray heating and the Lyman-alpha background act to decouple the 21-cm transition from the CMB [see, e.g., the discussion by][FOB2006].


Caution must be taken, however, if claiming the spin temperature can be neglected during the early stages of reionization since the effect of X-rays on the 21-cm signal depends on the timing of the X-ray heating epoch. As discussed in [MFS2013], an overlap of these epochs can lead to 10--100 times more power at wavenumber k is approximately equal to 0.1 Mpc^-1 at a mean neutral fraction >~ 0.9 than predicted assuming T_s >> T_gamma. At a mean neutral fraction of approximately 0.7, this assumption is found to overpredict 21-cm power by a factor of 1--2. Therefore, while we show results for redshift z <~ 15 only in this work, these effects must be kept in mind. They should, however, not affect our model-relative results as we compare the power spectra of our models in terms of their power ratio at the single global neutral fraction of approximately 0.68.


The spherically-averaged dimensionless 21-cm power spectrum is a measure of the relative power of fluctuations in the 21-cm brightness temperature field as a function of scale: the dimensionless power spectrum Delta^2_21(k,z) equals (k^3 / (2*pi^2*V)) times the angle-averaged power of the Fourier transform of the brightness temperature contrast field. Here V is the volume of the simulation and the brightness temperature contrast field is calculated using delta_21 = (delta T_b / mean delta T_b) - 1. A dimensional form of the 21-cm power spectrum (with units of mK^2) can be obtained through the mean brightness temperature squared times the dimensionless power spectrum. Both can be calculated from observational data but the dimensional power spectrum has the advantage of including the effect of the global ionization state on the overall power. Simulations show that the variance of the brightness temperature field reaches a maximum approximately midway through the reionization process [see, e.g.,][LIDZ2008, ICHIKAWA2010, WP2014].


These maxima are also present in the dimensional 21-cm power spectra but not in the dimensionless power spectra. This feature may enable the observational determination of the redshift at which the IGM is approximately half ionized. The evolution in the dimensional 21-cm power spectra for our fiducial model is shown in Figure 7. The redshifts and global neutral fractions shown correspond to those of the maps in Figure 2. At early times, when the simulation volume is fully neutral, the 21-cm power spectrum reflects that of the underlying density field. Soon after the first regions ionize, the 21-cm power spectrum drops briefly. This is due to what [LIDZ2008] describe as an `equilibration phase', when over-dense and under-dense regions have similar brightness temperatures due to the `inside-out' nature of reionization.


As reionization progresses, ionized regions grow resulting in an increase in large-scale power (small k) and a supression of small-scale power. Power on scales corresponding to wavenumber k ~ 0.1--1 Mpc^-1 tends to a maximum and the slope of the power spectrum flattens when reionization approaches its midpoint at a mean neutral fraction of approximately 0.5 (which occurs at redshift z is approximately equal to 8 for our fiducial model). Past this point power on all scales falls until there is no signal. The shaded region in Figure 7 corresponds to wavenumbers k > 0.7 k_Ny, where k_Ny is the Nyquist frequency of our simulation. Due to aliasing effects in our simulation, power for modes in this region are unreliable [see, e.g.,][CUI2008]. However, based on fiducial instrumental specifications, current- and future-generation radio interferometers will be most sensitive to modes with k ~ 0.1--1 Mpc^-1 [see, e.g.,][LIU2014, PRITCHARD2015], therefore these aliasing effects do not affect our predictions for observable differentiation between models.


Figure 8 shows the dimensional 21-cm power spectra for our models at redshift z is approximately equal to 8.4 where they have been matched, together with their ratios with respect to the fiducial model. The metric we use to compare the power spectra of our models is the maximum fractional difference in dimensional 21-cm power with respect to the fiducial model in the wavenumber interval k = 0.2--1 Mpc^-1. Using this metric, our results show there is a difference of up to: i) approximately 17 per cent without SNe feedback; ii) approximately 15 per cent without SNe or reionization feedback; iii) approximately 18 per cent using a constant stellar-to-halo mass relationship; iv) approximately 15 per cent using a constant stellar-to-halo mass relationship including only sources with haloes of mass greater than 10^9 M_sun, and; v) approximately 69 per cent using a constant stellar-to-halo mass relationship including only sources with haloes of mass greater than 10^10 M_sun.


The evolution of spatial fluctuations in the 21-cm signal has been shown to be sensitive to the nature of the sources driving reionization and to the size of their populations [see, e.g.,][MLZD2007]. Observing how the 21-cm power spectrum evolves may therefore be used to probe the clustering characteristics and bias of the sources responsible for reionization. In order to explore this we show the evolution of our models' dimensional power spectra for specific wavenumbers in Figure 9. It can be seen that while each model exhibits similar evolutionary behaviour on large scales, there are noticable differences on smaller scales. In particular, not all models' 21-cm power spectra pass through a local minimum during the first half of reionization. These minima are a feature of models with relatively weakly biased ionizing sources. The CSHR and CSHR.Mcut.10 models provide useful examples to help explain this.


The CSHR model includes ionization due to sources associated with haloes of any mass resolved by Tiamat, in contrast with the CSHR.Mcut.10 model which has fewer sources hosted by smaller-to-medium-mass haloes. During the early stages of reionization, ionized regions form around these `low-mass sources' in the CSHR model, reducing 21-cm power on scales comparable to the source separation. This leads to persistant minima for wavenumber k <~ 1 Mpc^-1. However, when only `high-mass sources' are included in the CSHR.Mcut.10 mass-cut model, ionized regions only form about the most clustered, highly biased haloes, leaving 21-cm power on other scales. This leads to the absence of a local minimum in the power spectrum evolution for this model on scales k >~ 0.4 Mpc^-1. This behaviour has been investigated previously [see, e.g.,][MLZD2007, LIDZ2008] using models that assume ionizing source luminosities that are proportional to their host halo mass and implementing different source-luminosity relationships.


However, these models do not capture the complex interplay between galaxies and the ionization state of the IGM through the effects of feedback. We find that on smaller scales, semi-analytic modelling of galaxies and feedback reduces the signature of the equilibration phase that is seen in constant stellar-to-halo mass ratio models. In addition to the 21-cm power spectrum amplitude, the evolutionary behaviour of its slope also serves as a diagnostic of reionization. Since the global neutral fraction is not directly measurable, we follow the work of [LIDZ2008] and [KIM2013a, KIM2013b] by exploring the joint evolution of the 21-cm power spectrum amplitude and its slope at selected scales. Figure 10 shows loci of the power spectrum gradient versus amplitude for our models. As was shown in Figure 9 for the amplitude, we find that the models exhibit similar evolutionary behaviour during the first half of the reionization process. As before, the main exception is the CSHR.Mcut.10 model.


We find a smaller level of difference between models with and without supernova feedback than described in [KIM2013a, KIM2013b]. This is most likely due to the high temporal resolution of our simulation as mentioned in Section 3.2. As mentioned in Section 3.2, there is a remarkable similarity between the ionization fields of the no-feedback and CSHR models despite the relatively large difference in the host halo mass scale of sources which dominate their reionization. This motivated the implementation of the CSHR mass-cut models, of which, CSHR.Mcut.9 is the most similar to the no-feedback models. As explained, this follows from the fact that their dominant ionizing sources are hosted by haloes in the same mass range. This is in contrast to the CSHR.Mcut.10 model which has a significantly different ionization field owing to the bias of its ionizing sources.


While this may not be surprising, since none of these models include feedback effects, what is surprising is the similarity between either the CSHR or CSHR.Mcut.9 model and the fiducial model (which does include feedback effects). This suggests the existance of a mass cut for the CSHR model which best replicates the morphology and 21-cm power spectra results of the fiducial model. Considering the added complexity of implementing a SAM-based prescription to simulate reionization, such `simple' semi-numerical prescriptions have their utility. While reionization morphology at fixed neutral fraction is not very sensitive to the details of galaxy formation, it is sensitive to the mass of the haloes which host the bulk of the ionizing sources. Simple EoR parametrizations may therefore be used to predict reionization morphology, and from that infer the dominant ionizing source population. This utility, however, has its limitations.


By not including realistic galaxy-formation physics in their formultion, these models cannot be used to investigate the effect of reionization on galaxy formation. Therefore, unlocking the details of galaxy formation requires interpretation with SAMs, in combination with other observations. This work has investigated the dependence of the morphology and statistics of HII regions on galaxy-formation physics during the Epoch of Reioinization using the Dark-ages, Reionization And Galaxy-formation Observables from Numerical Simulations (DRAGONS). DRAGONS includes the galaxy properties modelled by the semi-analytic model Meraxes and includes calculation of the inhomogeneous ionizing UV background using the 21cmfast algorithm, providing a self-consistent realization of reionization. This allows us to explore the effect of environment on galaxy formation, subsequent reionization of the IGM, and its observable signatures.


We use this coupled model to make updated cosmic 21-cm signal simulations, and predictions for low-frequency 21-cm experiments. We have presented results for a range of models that capture important physical mechanisms such as reionization and supernova feedback. We have demonstrated that the morphology and statistics of reionization are sensitive to both the ionizing source population and to feedback effects. We have shown how galaxy formation can modify the observable morphology of reionization by looking at the sizes of ionized regions and 21-cm power spectra. Of the galaxy physics we have investigated, we find that supernova feedback plays the most important role in reionzation, and that in the absence of this feedback, HII regions are up to approximately 20 per cent smaller, while the fractional difference in amplitude of power spectra is up to approximately 17 per cent at fixed ionized fraction. We have compared our SAM-based reionization models with past calculations that assume constant halo mass-to-luminosity ratios.


We find that the correct choice of minimum halo mass in these models leads to a reionization morphology that mimics that of a realistic galaxy-formation model. Therefore, reionization morphology at fixed neutral fraction is not uniquely predicted by the details of galaxy-formation physics. However, morphology is sensitive to the mass of the haloes which host the bulk of the reionizing sources. Simple EoR parametrizations therefore have utility for predicting the cosmic 21-cm signal, however, a better understanding of galaxy-formation physics using future 21-cm observations requires interpretation including a model of galaxy formation, in combination with other observations. In future work we will include a number of improvements to our modelling, including redshift and mass dependency of the escape fraction of ionizing photons, the effect of gas temperature and X-rays, a Markov chain Monte Carlo exploration of the model parameter space, and inhomogenous recombinations.


In this appendix we demonstrate the spatial convergence of our simulations through refinement of the grids used by 21cmfast. This is done for both the reionization history and 21-cm power spectra of our fiducial model. Figure A1 shows the globally-averaged neutral fraction, as a function of redshift, z, for our fiducial model using a 128^3, 256^3 and 512^3 21cmfast grid and the differences between them. We find that our final refinement results in convergence to within +/- 0.01 neutral fraction at all redshifts. In order to show spatial convergence of the power spectrum, Figure A2 shows the dimensional 21-cm power spectra for our fiducial model at a mean neutral fraction of approximately 0.68 using a 128^3, 256^3 and 512^3 21cmfast grid and their ratios with respect to the 512^3 simulation. The vertical dotted line shows the scale at which current- and future-generation radio interferometers are most sensitive. We find that successive refinement results in convergence to within 3 per cent on this scale.


Using Hubble data, including new grism spectra, Oesch et al. recently identified GN-z11, an M_UV=-21.1 galaxy at z=11.1 (just 400 Myr after the big bang). With an estimated stellar mass of ~10^9 solar masses, this galaxy is surprisingly bright and massive, raising questions as to how such an extreme object could form so early in the Universe. Using Meraxes, a semi-analytic galaxy-formation model developed as part of the Dark-ages Reionization And Galaxy-formation Observables from Numerical Simulations (DRAGONS) programme, we investigate the potential formation mechanisms and eventual fate of GN-z11. The volume of our simulation is comparable to that of the discovery observations and possesses two analogue galaxies of similar luminosity to this remarkably bright system. Existing in the two most massive subhaloes at z=11.1 (M_vir=1.4x10^11 solar masses and 6.7x10^10 solar masses), our model analogues show excellent agreement with all available observationally derived properties of GN-z11. Although they are relatively rare outliers from the full galaxy population at high-z, they are no longer the most massive or brightest systems by z=5.


Furthermore, we find that both objects possess relatively smooth, but extremely rapid mass growth histories with consistently high star formation rates and UV luminosities at z>11, indicating that their brightness is not a transient, merger-driven feature. Our model results suggest that future wide-field surveys with the James Webb Space Telescope may be able to detect the progenitors of GN-z11 analogues out to z~13-14, pushing the frontiers of galaxy-formation observations to the early phases of cosmic reionization and providing a valuable glimpse of the first galaxies to reionize the Universe on large scales.


Using Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) grism spectroscopy, [Oesch2016] recently identified the most distant galaxy known to date (GN-z11). The spectrum, combined with photometric data from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) survey, placed the system at z=11.09 (+0.08 / -0.12), with lower redshift spectral energy distribution (SED) solutions excluded with high confidence. At M_UV = -22.1 +/- 0.2, GN-z11 is approximately three times more luminous than L* at z=7-8, and is therefore extremely bright for such a high-z object. Extrapolations of the z=4-8 ultraviolet luminosity function (UV LF) suggest that such objects should be rare with fewer than ~0.06 such systems expected in the total volume surveyed (~1.2x10^6 Mpc^3; [e.g.][Bouwens2015, Finkelstein2015]).


LF evolution models based on hierarchical assembly ([e.g.][Trenti2010, Mason2015]), the extrapolation of abundance matching results to higher luminosities/redshifts ([e.g.][Trac2015, Mashian2016]), and cosmological hydrodynamic simulations such as BlueTides ([Waters2016, Waters2016b]), all similarly point to GN-z11 being an exceptional and rare system. This begs the question of what the formation mechanisms might be for such a galaxy, as well as what its descendants are at lower redshifts? In this work we use the semi-analytic galaxy-formation model Meraxes [Mutch2015], developed for the Dark-ages Reionization And Galaxy-formation Observables from Numerical Simulations (DRAGONS) programme ([Angel2016, Geil2015, Liu2015]), in order to investigate whether galaxy-formation models of this type predict the existence of such extreme systems as GN-z11 and, if so, what their properties, origins and potential fates are.


The DRAGONS parent N-body simulation (Tiamat; [Poole2016]) possesses a comparable volume to that of the discovery observations, yet Meraxes predicts two z=11.1 galaxies of comparable UV magnitude to GN-z11, suggesting that such luminous galaxies may be a more common outcome of early galaxy formation than previously thought. This paper is laid out as follows. In Section 2, we give a brief overview of the DRAGONS framework, including both the Tiamat N-body simulation and Meraxes semi-analytic model. In Section 3, we discuss the presence and properties of luminous GN-z11 analogues in our model galaxy population before going on to investigate the origin and fate of these objects from z=5-20. In Section 4, we discuss the prospects for detecting the progenitors of GN-z11 analogues with the forthcoming James Webb Space Telescope (JWST) and provide forecasts for the maximum redshift out to which these may be observed. We close with a summary of our findings in Section 5. The Tiamat simulation was run with a standard, spatially flat LambdaCDM cosmology, utilizing the latest [Planck-Collaboration2015] parameters. However, for consistency with [Oesch2016] we present all results with h=0.7 throughout.