Within this context, the development of theoretical models that include a self-consistent treatment of the physics of galaxy formation and intergalactic hydrogen will play a key role. Traditional approaches to the study of galaxies and their effects on the IGM utilize either numerical simulation or analytic modelling. The latter allows investigation of average behaviours on large scales but the calculations are inherently linear, meaning that complex feedback processes cannot be addressed. Numerical simulations, on the other hand, include non-linear effects but at the expense of computational cost. To achieve a volume sufficiently large to study ionized structure, a popular and effective approach to simulating reionization is to begin with a collisionless N-body simulation and use a simple prescription to relate halo mass to ionizing luminosity. A radiative transfer method can then be used to calculate the ionization structure on large scales.


In recent years, new hybrid, or semi-numerical models have been developed that combine N-body simulations with analytical methods to enable the calculation of reionization structure in very large volumes with high efficiency. These methods have elucidated the primary features of the ionization structure during reionization, but do not capture the physics of galaxy formation. Therefore, to better understand the physics of galaxy formation, many authors have performed hydrodynamic simulations of galaxy formation which are able to directly model the growth of stellar mass in high-redshift galaxies when coupled with sub-grid models for processes including metal enrichment and feedback. These simulations are able to broadly reproduce the luminosity function of galaxies at high redshift, however, computational expense limits their ability to self-consistently model reionization in volumes large enough to statistically describe the spatial evolution of this process. Instead, a common approach is to impose a simple parametrized model to approximate the average ionizing background as a function of redshift, independent of the properties of the ionizing source population or their spatial distribution.


Recently, hydrodynamical simulations of galaxy formation with coupled radiative transfer have been used to compute the effects of reionization on galaxy formation self-consistently for the first time. However, the extreme computational expense of these simulations limit their size to relatively small volumes and/or few variations on galaxy formation physics or reionization scenarios that can be explored. In addition, the modelling of sub-grid physical processes remains uncertain, requiring systematic studies of the available parameter space in order to draw robust conclusions. Such studies represent an extreme computational challenge which has yet to be overcome. Another approach to the realistic modelling of high redshift galaxies has been through the use of semi-analytic galaxy formation models. While large volumes are available to such models, until now they have not been fully coupled to an accurate description of reionization.


This is in part due to the structure of most existing semi-analytic models, which utilize so-called `vertical' halo merger trees in which galaxies belonging to each tree branch are evolved independently from the rest of the simulation volume. Since galaxies drive the process of reionization, which in turn affects their subsequent evolution, galaxies spatially separated by tens of Mpc cannot be considered and evolved independently as has traditionally been the case. Self-consistently studying reionization instead requires a semi-analytic model designed to run on `horizontal' merger trees where all haloes at each snapshot of the parent N-body simulation are processed simultaneously. Additionally, the reduced dynamical time of dark matter haloes at high redshift requires snapshots with a much higher cadence than is needed to model galaxy formation at lower redshifts.


This is the third paper in a series describing the Dark-ages Reionization And Galaxy Observables from Numerical Simulations (DRAGONS) project, which integrates detailed semi-analytic models constructed specifically to study galaxy formation at high redshift, with semi-numerical models of the galaxy–reionization process interaction. In this work, we introduce Meraxes, the new semi-analytic model of galaxy formation developed for DRAGONS, integrating the 21cmFAST semi-numerical model for ionization structure described in [Mesinger2007]. Meraxes is implemented within the large-volume, high-resolution, and high-cadence Tiamat N-body simulation described in [Poole2016] and [Angel2016]. In subsequent papers we will use Meraxes to carry out a range of studies including the investigation of the high redshift galaxy luminosity function [Liu2015], and the ionization structure of the IGM [Geil2015]. Complimentary, high resolution hydrodynamic simulations are described in [Duffy2014].


Modern semi-analytic galaxy formation models are capable of providing statistically accurate representations of the global properties of galaxies across a broad range of redshifts, and are therefore able to describe the distribution and evolution of the ionizing photons which drive the process of cosmic reionization. These photons generate regions of ionized hydrogen (H II) with characteristic sizes of tens of Mpc during reionization. Thus, in order to take advantage of this information and to self-consistently model the effect of these photons on the growth of galaxies, one must consider the contributions of galaxies separated by similar scales. Traditionally, semi-analytic models have therefore used parametrized descriptions to include the average effect of reionization and the associated photo-suppression of baryonic infall on the growth of galaxies. These parametrizations are typically calibrated using radiative transfer simulations and are provided as a function of redshift and halo mass alone.


Whilst it is computationally efficient to include reionization in this manner, there are a number of important drawbacks. First, the progression of reionization is not self-consistently modified by the growth of the galaxies which are driving it. Therefore it is impossible to investigate how different galaxy physics affect the ionization state of the IGM or to quantify the potential back-reaction on galaxy evolution. Secondly, these simple reionization prescriptions miss the potentially important effects of spatially dependent self-regulation, whereby massive galaxies located at the peaks in the density distribution can reionize their surroundings, delaying or preventing the onset of star formation in nearby lower mass haloes. Our new semi-analytic galaxy formation model, Meraxes, has been written from the ground up to facilitate these modelling requirements. Its key features include the `horizontal' processing of merger trees constructed from a purpose run N-body simulation (Tiamat) and the incorporation of the semi-numerical reionization algorithm, 21cmFAST, as a core component.


When combined, these features allow Meraxes to efficiently couple the growth of galaxies to the process of reionization, both temporally and spatially. It can therefore be used to investigate the potentially complex effects of various reionization models on the properties of high-z galaxies, as well as to test for observational discriminants of different galaxy physics in the distribution and evolution of inter-galactic neutral hydrogen. In order to develop confidence in our newly developed framework, as well as provide a solid foundation for future additions and improvements, our initial implementation of the baryonic physics processes in Meraxes is heavily based on the well-studied L-Galaxies semi-analytic model, in particular the version described in [Croton2006] and extended in [Guo2011]. However, as well as our improved treatment of reionization, the excellent temporal resolution provided to us by the Tiamat merger trees has also necessitated the development of a number of important updates to the treatment of supernova feedback and stellar mass recycling.


The Tiamat collisionless N-body simulation has been designed for the DRAGONS study of high-redshift galaxy formation and the epoch of reionization (EoR). It contains 2160^3 dark matter particles within a 100 Mpc (comoving) periodic box and was run using a modified version of the GADGET-2 N-body code and the latest Planck 2015 cosmology. The volume of Tiamat allows for the investigation of the statistical signatures of reionization and its 21 cm observational signal, whilst the resulting particle mass of 3.89x10^6 solar masses provides the necessary resolution to identify the low-mass sources thought to be driving this process. Furthermore, Tiamat provides high temporal resolution in the form of 100 output snapshots evenly spaced in cosmic time between z=35 and 5, resulting in a cadence of 11.1 Myr per snapshot. This level of temporal resolution is a unique feature of Tiamat which allows our semi-analytic model to accurately simulate the stochastic nature of star formation in a regime where the dynamical time of a typical galactic disc is shorter than the lifetime of the least massive Type II supernova progenitor (~40 Myr).


In addition to the main Tiamat volume, a suite of smaller, higher mass resolution N-body simulations have been run as part of the DRAGONS programme (Paper-I). For this work we make particular use of the TinyTiamat and MediTiamat volumes in order to quantify the effect of resolution on our results. TinyTiamat is the highest resolution simulation of the DRAGONS suite, with a particle mass of 10^5 solar masses in a small box of side length 14.8 Mpc, whilst MediTiamat bridges the resolution gap with the main simulation by providing a particle mass of 1.16x10^6 solar masses in a 33.3 Mpc box. Both simulations maintain the same snapshot cadence as the main Tiamat volume and are described in detail in Paper-I. Halo identification in all simulations used in this work was carried out using the Subfind real-space halo finder down to a minimum mass of 32 particles. The resulting halo catalogues comprise of friends-of-friends (FoF) groups of gravitationally bound particles which themselves are made up of a single mass dominant `central' subhalo along with zero or more sub-dominant `satellite' subhaloes.


The formation history of subhaloes, in the form of hierarchical merger trees, acts as the raw input to Meraxes and is used to define the positions and growth of galaxies. Many traditional semi-analytic models process such trees in a depth-first (or `vertical') order, whereby small collections of directly interacting dark matter haloes are processed one after the other from high to low redshift and independently of each other. Whilst computationally efficient in terms of minimizing the memory overhead required to process the simulation, the inherent assumption is that haloes (and by extension galaxies) which do not directly interact do not affect each other's evolution. This assumption breaks down when considering the process of reionization during which ionizing photons from galaxies tens of Mpc away can heat the IGM, raising the local Jeans mass and altering the accretion rate of baryons. Meraxes instead processes trees breadth-first (or `horizontally'). In this method all of the haloes in the entire volume are loaded into memory and the associated galaxies evolved for each snapshot sequentially.


We begin by making the standard assumption that as FoF groups grow, any freshly accreted mass, always carries with it the universal baryon fraction, f_b, in the form of pristine primordial gas. However, the fraction of these infalling baryons which will remain bound to the FoF group and participate in galaxy formation may be reduced by a number of factors. In particular, ionizing ultraviolet background (UVB) radiation from both local and external sources can heat the IGM, increasing the local Jeans mass and leading to a non-negligible reduction in the amount of baryons successfully captured by low mass systems. We parametrize this reduction in terms of a baryon fraction modifier, f_mod, which represents the attenuation of the total baryon mass that could have ever been successfully captured by an FoF group in its lifetime. Any baryons which are successfully captured are assumed to be shocked to the virial temperature of the host FoF group and added to a diffuse hydrostatic hot reservoir where they mix with any already present hot gas.


At each time step in the simulation some fraction of the hydrostatic hot reservoir may cool and condense down into the central regions of the group where it can then participate in galaxy formation. In order to calculate the rate at which this occurs we follow the commonly employed methodology outlined in [White1991]. In this model, the cooling time of a quasi-static isothermal hot halo is given by the ratio of the specific thermal energy to cooling rate per unit volume. We assume that the hot gas is shocked to the virial temperature of the FoF group and that it follows a singular isothermal sphere density profile. The cooling model naturally leads to three distinct regimes. When the cooling radius is greater than or equal to the virial radius, any infalling gas will cool so rapidly that there will be no time for a stable shock to form. In this case we assume that the infalling material flows directly into the central regions of the halo over a dynamical time. When the cooling radius is less than the virial radius, the cooling time will be sufficiently long that a quasi-static hot atmosphere will form, and the cooling rate can be calculated from a simple continuity equation.


When the virial temperature is less than or equal to 10^4 K, we set the cooling radius to 0 and no cooling occurs. In the standard model of galaxy formation, haloes with this temperature represent the lowest mass scale for galaxy formation. Below this, the primary mechanism for gas cooling is via molecular hydrogen which is easily photo-dissociated by trace amounts of star formation, making it an inefficient pathway for Pop II star formation. Above this temperature, atomic line cooling provides an efficient mechanism to dissipate energy and remove pressure support. The mass resolution of our input N-body simulation, Tiamat, was chosen such that the minimum halo mass at z=5 is close to the atomic cooling mass threshold of 10^4 K. All material which successfully cools into the central regions of the FoF group is assumed to be deposited directly into the cold gas reservoir of the galaxy hosted by the central halo.


We assume that cooled gas settles into a rotationally supported cold gas disc with an exponential surface density profile. Under the simplifying assumption of full conservation of specific angular momentum, the scale radius of the disc can be approximated from the spin of the host dark matter halo. Based on the well-established observational work of [Kennicutt1998], the star formation rate of local spiral galaxies can be related to the surface density of cold gas above a given threshold. The value of this threshold can be understood in terms of the gravitational instability required to form massive star-forming clouds. If the total amount of cold gas in the disc is greater than the critical mass, the star formation rate is assumed to be a fraction (the star formation efficiency) of the gas mass above the critical value, divided by the disc's dynamical time.


Galaxy mergers can drive strong shocks and turbulence in any participating cold gas, driving this material towards the inner regions of the parent galaxy and resulting in an efficient burst of star formation. We model the fraction of cold gas consumed by such a burst using the prescription introduced by [Somerville2001]. This relation agrees well with the results of numerical simulations of mergers with baryonic mass ratios in the range 0.1--1.0. For merger events where the mass ratio is less than 0.1, we suppress any merger-driven star formation. For simplicity, we assume that all of the stars formed in a merger-driven burst do so within a single snapshot. At z~8, the median dynamical time of a galaxy disc in Meraxes is ~60% of the time between two consecutive snapshots of Tiamat (~11.2 Myr). Hence, this approximation is roughly equivalent to the assumption that the merger-driven burst occurs on a time-scale approximately less than one disc dynamical time for the majority of galaxies.


The radiative and mechanical energy liberated by supernovae can have a profound impact on galaxy evolution, potentially heating significant amounts of gas and even ejecting it from a galaxy or host dark matter halo entirely. This is especially so at high redshift where haloes are on average less massive than in the local Universe, and possess correspondingly shallower potential wells. Many semi-analytic models make the simplifying assumption that all supernova feedback energy is released instantaneously. This approximation is motivated by the reasonable further assumption that the majority of supernova feedback energy is released by massive stars which have short lifetimes. In the cases where the time span between each simulation snapshot is large, the approximation of the instantaneous deposition of all supernova energy into the ISM is valid. However, motivated by the short dynamical time of systems at high redshift, the separation between snapshots in our input simulation is approximately 11.1 Myr. It therefore takes at least three snapshots after a single coeval star formation episode for all stars more massive than 8 solar masses to have gone supernova.


In order to accommodate this matching of time-scales in Meraxes, we have implemented a simple delayed supernova feedback scheme. The basic methodology of our delayed feedback scheme is to calculate the total amount of energy which should be injected into the ISM by a single star formation episode, and to release this energy gradually over time in proportion to the fraction of SN-II which will have occurred. We assume a standard Salpeter initial mass function (IMF) with upper and lower mass limits of 0.1 and 120 solar masses respectively. The number fraction of stars that will end their lives as type II supernovae is then calculated by integrating the IMF. If we further assume that each supernova produced injects a constant 10^51 erg of energy into the ISM then, for a burst of a given mass, the total amount of energy deposited into the ISM is calculated, where a free parameter describes the efficiency with which the supernova energy couples to the surrounding gas.


As is common practice, we model the mass of gas which is reheated by this energy deposition as a mass loading factor. We follow [Guo2013] who adopted parametrizations for the energy coupling efficiency and mass loading factor that scale with the halo's maximum circular velocity. The total amount of supernova energy released by a galaxy at any given snapshot will be dependent on the mass of stars formed both in the current and previous snapshots. We therefore explicitly track the total mass of stars formed in each galaxy for the last 4 snapshots (corresponding to 40 Myr). The eventual fate of the reheated material depends on both its mass and the amount of energy injected. If there is more energy injected into the reheated gas than is required to raise it to the virial temperature of the halo, we assume the gas to be added to the hot halo of the host FoF group. Any excess energy is assumed to then go into ejecting some fraction of the FoF group hot reservoir from the system entirely.


If instead only a fraction of the total reheated mass can be raised to the virial temperature, only that energetically feasible fraction is added to the FoF group hot reservoir, with the rest raining back down on to the galaxy in a galactic fountain. Any gas and metals which are successfully expelled from the system entirely are placed into a separate `ejected' reservoir. Here they are assumed to play no further role in the evolution of the galaxies in the host FoF group until the group falls into a more massive system. At this point, the ejected material is assumed to be re-accreted into the new group and is added to its hot halo component. We implement a simple metal enrichment scheme whereby a fixed yield of metals is released into the ISM per unit mass of stars formed. We assume that these metals are released predominantly by massive stars which end their lives as SN-II and we gradually release them over time as these supernovae occur.


A common assumption of many semi-analytic models is the instantaneous recycling approximation (IRA), in which some fixed fraction of the stellar mass formed during each time step is instantaneously recycled back into the ISM. Given our choice of IMF, a recycle fraction of 40% corresponds to all stars more massive than approximately 1 solar mass instantaneously going supernova. However, the lifetime of a 1 solar mass star is close to the current age of the Universe and hence the IRA can only be considered valid for galaxies around z=0. At z>~2, where the majority of galaxies have stellar populations dominated by recent star formation, this approximation becomes invalid and we are forced to consider a more realistic alternative. Our stellar mass recycling prescription is divided into two parts: we assume that the initial stellar mass of all supernovae is returned to the cold gas reservoir of the galaxy and we explicitly track the star formation history of each galaxy for the last 4 snapshots to calculate the recycled mass from older stars.


As haloes inspiral towards more massive systems, tidal forces experienced during repeated pericentric passages can lead to the stripping of loosely bound material from the outer regions. In Meraxes, if the mass of an FoF group drops, then a pro rata fraction of the ejected and/or hot baryonic content of the halo is also removed. This material is taken first from the ejected reservoir, with further mass being removed from the hot halo component if required. No baryons are ever taken from the cold gas or stellar mass reservoirs. Further to these long-range tidal forces, galaxies infalling into groups or clusters are observed to be subjected to a number of dynamical processes which remove gas from the outskirts of the system, including ram-pressure stripping, strangulation, and harassment. We model the combined effects of these processes by assuming that all FoF groups are instantly stripped of their entire hot and ejected gas reservoirs upon infall into a more massive structure, with their combined mass and metals being added to the hot component of the new parent.


Mergers play an important role in the build up of galaxy stellar mass, both through hierarchical mass assembly and induced star formation. This is particularly so at high-z where their prevalence is enhanced. In Meraxes these galaxy merger events are triggered by the merging of the corresponding host dark matter haloes. Following [Croton2006], when a dark matter halo is marked as having merged, we utilize dynamical friction arguments to approximate the time taken for the orbit of the incoming galaxy to decay and the corresponding galaxy--galaxy merger to occur. Galaxy mergers can drive strong shocks and turbulence in any participating cold gas, driving this material towards the inner regions of the parent galaxy and resulting in an efficient burst of star formation. We model the fraction of cold gas consumed by such a burst using the prescription introduced by [Somerville2001]. For merger events where the mass ratio is less than 0.1, we suppress any merger-driven star formation.


A ghost dark matter halo is one which is temporarily unresolved in our input merger trees. This can be due to a number of reasons, but is most commonly a result of a smaller halo passing through or nearby a much more massive structure. The Tiamat merger trees used in this work are carefully constructed to identify these artefacts, resulting in the skipping of a potentially large number of snapshots between haloes and their descendants. In many semi-analytic models, the galaxies hosted by such haloes are simply ignored until their halo is later re-identified. However, we must ensure that we correctly include these objects at all snapshots in order to account for their ionizing photon contribution. Due to the lack of knowledge of the properties of a ghost's host dark matter halo, we are unable to implement many of the physics prescriptions outlined above. We therefore simply allow these galaxies to passively evolve during the time over which they are identified as ghosts, forming no new stars but experiencing the delayed supernova feedback from previously formed generations.


The free parameters of the model were manually calibrated to replicate the observed evolution of the galaxy stellar mass function between redshifts 5 and 7, as well as the integrated free electron Thomson scattering optical depth measurements. The evolution of the stellar mass function has been shown by previous statistical investigations of semi-analytic models to provide a tight constraint on both the star formation efficiency and supernova feedback parameters. By combining this with the Thomson scattering optical observations, we can additionally put constraints on the escape fraction of ionizing photons, and thus all of the free parameters of our model. We make use of the mass functions estimated by [Gonzalez2011], [Duncan2014], [Grazian2014], and [Song2015]. Both [Duncan2014] and [Grazian2014] utilize data collected from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) GOODS South field with stellar masses directly obtained from SED fitting of combined optical and near-infrared space-based observations, and include the effects of both nebular line and continuum emission.


We are able to achieve an excellent match to the normalization, shape, and evolution of the observed mass function across all plotted redshifts. At z=5, where there is the largest divergence between different observational data sets, we chose parameter values which provided a reasonable compromise between each. However, at the low-mass end we have chosen to follow the observations of [Duncan2014] as they use a large data set with stellar masses obtained from SED fitting and provide actual data points rather than a Schechter fit. The quality of the agreement between our model and the observational data gives us faith that our implemented physical prescriptions are both reasonable and applicable at the high redshifts of interest in this work. Although typically producing fewer ionizing photons than their more massive counterparts, low mass galaxies with stellar mass less than the observational limit are expected to contribute a large fraction of the overall ionizing photon budget due to their high number density. For this reason, the low-mass slope of the stellar mass function is of particular importance to reionization.


The median stellar mass--virial mass relation is well described by a power law with a slope of ~1.4. This shows good agreement with simple energy conservation arguments which suggest a slope of ~1.7 for supernova feedback-regulated galaxy growth and a fixed cold gas mass fraction. However, at FoF group masses below approximately 10^9.5 solar masses there is a rapid increase in the spread of stellar mass values. This is due to a combination of supernova and reionization feedback effects, as well as a low star formation efficiency in these small, often diffuse haloes. Interestingly, there is no statistically significant evolution in either the slope or normalization of the relation with redshift. However, the same simple energy conservation arguments which provided a good agreement for the slope of the relation suggest that the normalization should evolve with redshift. Despite this, our model indicates that in order to reproduce the observed evolution of the galaxy stellar mass function over the redshifts considered in this work, the efficiency of galaxy formation and the associated feedback processes must conspire to provide a constant star formation efficieny in haloes of a fixed mass.


Our fiducial model provides an integrated optical depth which is in excellent agreement with the Planck results. An important consideration when assessing the results of any cosmological simulation is the potential loss of stellar mass (and therefore ionizing photon contribution) due to finite mass resolution. This issue is explored in detail for Meraxes and the Tiamat suite of simulations in the appendix. In summary, we find that the halo mass function of the full Tiamat simulation is complete down to approximately 0.5 dex above the atomic cooling mass threshold at z=5, whilst the lower volume but higher mass resolution MediTiamat is fully complete down to this mass limit. Using MediTiamat to quantify the fraction of total stellar mass missing from the full simulation, we find that at z=10 we miss approximately 25% of stellar mass from low mass, unresolved systems. By z=6 this fraction falls to less than 5%.


The ionization structure of a 4 Mpc thick slab produced by Meraxes shows that at early times, ionized bubbles surround the peaks in the density field where the first galaxies form. As reionization progresses the bubbles begin to overlap, and by a neutral fraction of 0.3, a large `lane' of ionized hydrogen, extending the entire length of the simulation volume is formed. The ability to investigate the distribution and evolution of bubble morphologies and the associated observable 21 cm power spectra is a key feature of our Meraxes framework. We find that supernova feedback is the dominant physical mechanism for regulating the growth of galaxy stellar mass both during and immediately prior to reionization. We find that it is only in the absence of supernova feedback that suppression of star formation due to the presence of an ionizing UVB reaches significant levels, and even then, only if star formation is efficient enough so as to be gas supply limited.


We further find that the reionization history of the IGM is similarly insensitive to reionization feedback. This implies that the process of reionization is not self-regulated. However, we note that if the process of reionization is more extended than is predicted by our model (for example, due to inhomogeneous recombinations in the IGM) then reionization feedback may play a more effective role. The fiducial model also predicts that a broad range of halo masses contribute to ionizing photon production during reionization. The reionization feedback is most effective in low mass haloes. However, supernova feedback of the level required to reproduce the observed high-z stellar mass functions dominates the suppression of star formation across all halo masses. The strong halo mass dependence of the stellar mass fraction contrasts the constant value assumed by the majority of reionization structure studies.


The evolution of the instantaneous ionizing emissivity in our fiducial model, which has a constant escape fraction, predicts a high and steeply increasing ionizing emissivity at z=5 that is inconsistent with the observational data from the Lyman-alpha forest. A model with a lower escape fraction, which is only marginally consistent with the Planck optical depth measurements, still fails to reproduce the observational data. However, an evolving escape fraction, with 0.05 <~ f_esc <~ 0.1 at z~6 and a redshift scaling proportional to (1+z)^kappa, where kappa~3, does allow our model to simultaneously satisfy these constraints. The large escape fraction in this model at high-z leads to an early onset of reionization. However, since the escape fraction decreases over time, the speed at which reionization progresses declines with decreasing neutral fraction, resulting in an extended EoR.


By comparing to a simple, homogeneous reionization prescription of the kind traditionally used in semi-analytic models, we find that the inclusion of a self-consistent patchy reionization model can result in significant, environmentally dependent variations in the stellar masses of individual galaxies by factors of 2--3. The distribution in galaxy mass ratios at a fixed stellar mass is largely symmetric, meaning that the total stellar functions produced by these two prescriptions remain in good agreement. However, the different predictions for individual galaxy masses may have important consequences for galaxy clustering statistics and 21cm power spectra owing to the Poisson noise that this scatter adds between the halo and galaxy clustering. This highlights that despite the good agreement in stellar mass functions, there can be important consequences of using a self-consistent, spatially dependent model of reionization that is not fully encoded in a parametrized homogeneous description.


The patchy fiducial model predicts a photoionization suppression which depends on environment with up to an order of magnitude difference in the filtering mass between over- and underdense regions. The trend of increasing filtering mass with increasing over-density is due to the contribution of ionizing flux from nearby galaxies which is not self-consistently included in the homogeneous model. At higher local densities, there is an increase in the average number and mass of nearby sources contributing ionizing photons which in the fiducial, patchy reionization model results in an increased filtering mass value. The inability of parametrized global reionization prescriptions to capture the density-dependent spread in filtering masses, as well as the trend of increasing filtering mass with density, could have important consequences for the galaxy clustering and cross-correlation statistics.


It is important to note that we were only able to calibrate the homogeneous variation through the use of our full fiducial run with self-consistently coupled reionization. Any change to the free model parameters, or underlying cosmological model, would require the mean filtering mass--redshift relationship to be recalculated.


We study dwarf galaxy formation at high redshift (redshift z greater than or equal to 5) using a suite of high-resolution, cosmological hydrodynamic simulations and a semi-analytic model (SAM). We focus on gas accretion, cooling and star formation in this work by isolating the relevant process from reionization and supernova feedback, which will be further discussed in a companion paper. We apply the SAM to halo merger trees constructed from a collisionless N-body simulation sharing identical initial conditions to the hydrodynamic suite, and calibrate the free parameters against the stellar mass function predicted by the hydrodynamic simulations at redshift z=5. By making comparisons of the star formation history and gas components calculated by the two modelling techniques, we find that semi-analytic prescriptions that are commonly adopted in the literature of low-redshift galaxy formation do not accurately represent dwarf galaxy properties in the hydrodynamic simulation at earlier times.


We propose 3 modifications to SAMs that will provide more accurate high-redshift simulations. These include 1) the halo mass and baryon fraction which are overestimated by collisionless N-body simulations; 2) the star formation efficiency which follows a different cosmic evolutionary path from the hydrodynamic simulation; and 3) the cooling rate which is not well defined for dwarf galaxies at high redshift. Accurate semi-analytic modelling of dwarf galaxy formation informed by detailed hydrodynamical modelling will facilitate reliable semi-analytic predictions over the large volumes needed for the study of reionization. About 150 Myr after the Big Bang, the Universe entered the Epoch of Reionization (EoR; [Planck2016A&A...596A.108P]), when the first stars and galaxies were created and started to photoionize the intergalactic neutral hydrogen. Over the past decade, there have been significant advances in the study of galaxies during the EoR and the evolution of the ionized intergalactic medium (IGM).


High-redshift galaxies thought to have sourced reionization have been discovered all the way to redshift z~11 [Oesch2016], when the Universe was only 400 Myr old. However, the sample of these early galaxies remains small with only approximately 1000 candidates at redshift z>6 identified using advanced space-based instruments [Bouwens2014, Bouwens2015]. Achieving a physical understanding between galaxies and the progress of reionization is therefore aided by numerical simulations (see e.g. [Bertschinger1998, Baugh2006, Dolag2008, Somerville2015]). Two of the main numerical approaches to model galaxy formation are hydrodynamic simulations and semi-analytic models (SAMs) applied to dark matter halo merger trees constructed from cosmological N-body simulations. Since they adopt different approximations, their strengths and requirements are diverse. Hydrodynamic simulations include baryons as well as dark matter, and simulate the complex baryonic physics more directly (e.g. [Crain2009, Schaye2010, vogelsberger2014properties, Schaye2014, feng2015bluetides, Pawlik2017]).


However, while galaxies form and evolve in the presence of gravity and hydrodynamical forces in a simulation volume, its large computational expense limits the capability of simultaneously resolving small-scale regions and capturing massive systems. For instance, in order to study reionization and galaxy formation during the EoR, simulations are required to have a volume size of at least 10^6 Mpc^3 [Iliev2014MNRAS.439..725I] and a particle resolution that is able to resolve haloes with masses around 5x10^7 solar masses (i.e. >10 billion particles; [Barkana1999ApJ...523...54B]). While it remains challenging to explore the relevant parameter space in hydrodynamic simulations, SAMs, with their advantages of low expense on computational resources, become the alternative in this case. N-body simulations (e.g. [Navarro1997, springel2005simulating, iliev2008simulating, Boylan_Kolchin2009, Klypin2010, Garrison2017]) solve for the gravitational force on each collisionless dark matter particle but neglect the baryonic physics.


In order to implement the galaxy formation physics of baryons into these simulations, halo merger trees are constructed and based on the properties inherited from the merger trees, SAMs parametrize baryonic processes to evolve galaxies, offering an efficient approach to simulate galaxies in a cosmological context [e.g.][Cole2000, Hatton2003, Baugh2005, croton2006many, DeLucia2007, Somerville2008, guo2011dwarf, Henriques2015]. In general, a SAM first designates haloes as galaxies and endows them with stellar components and several gas reservoirs of varied functionalities. The latter usually include a cold gas disc where star formation occurs and a hot halo of non-star-forming gas. Mass is then manipulated and transferred between these baryonic sectors as a result of varied baryonic processes including accretion, cooling, star formation and feedback from supernovae or active galactic nuclei (AGN). These processes are implemented using scaling functions of astrophysics that are motivated either from first principles, from observations, or from more complicated simulations.


Despite these improvements recently implemented into modern SAMs, some prescriptions are still commonly adopted by many of these models. For instance, in order to initialize the baryonic component of haloes from dark matter only N-body simulations, the universal baryon fraction is applied to every virialized system with a further suppression due to reionization feedback. These baryons are considered as infalling gas and are usually assumed to share the virial temperature of the host halo as a result of experiencing shock--heating. However, we note that the degeneracy between parameters of SAMs, as well as their time and mass dependencies introduced by these simplified baryonic prescriptions are not well understood. In addition, lack of available observational data means that semi-analytic predictions have generally been tested against observables that are massive and in the nearby Universe [croton2006many, guo2011dwarf, Stevens2016]. Their consistency and performance in the low-mass regime or at high redshift remain unclear.


This motivates us to 1) test the performance of SAMs at high redshift, which were originally developed from the literature of low-redshift galaxy formation; and 2) inform more accurate semi-analytic modelling of dwarf galaxies, which will be applied to large-volume simulations to study the EoR in the future. In the absence of detailed observational data, constraints on the properties of SAM galaxies can be supplemented through comparisons against physics-rich hydrodynamic simulations, which are expected to predict galaxies that are more representative than simplified calculations using semi-analytic approaches [e.g. Guo2016, mitchell:2017je, stevens:2017fi, cote:2017uh]. In this work, we take the Meraxes SAM [Mutch2016a] as an example, and apply it to the halo merger trees constructed from a collisionless N-body simulation sharing identical initial conditions to the Smaug suite of hydrodynamic simulations [duffy2014low].


We calibrate the SAM against the Smaug hydrodynamic simulations and make direct comparisons of the calculated galaxy properties using the two modelling techniques. This paper focuses on modelling of gas accretion, cooling and star formation, the implementation of which is common among many modern SAMs [croton2006many, Somerville2008, guo2011dwarf, Henriques2015]. The effect of feedback on star formation is excluded and will be presented in a companion paper. We begin with a brief introduction of the N-body/hydrodynamic simulations and SAM utilized in this work in Section 2. Then we discuss the consequence of applying SAMs directly to collisionless N-body simulations and propose modifications accordingly in Section 3. In Section 4, we proceed with the proposed modifications and calibrate the SAM to reproduce the hydrodynamic result. We investigate galaxy properties in detail and make comparisons between the hydrodynamic simulation and the SAM results. We conclude in Section 5.


In this work, we adopt cosmological parameters from WMAP7 (Omega_m, Omega_b, Omega_Lambda, h, sigma_8, n_s = 0.275, 0.0458, 0.725, 0.702, 0.816, 0.968; [Komatsu2011]) in all simulations. The Dark-ages Reionization And Galaxy Formation Observable from Numerical Simulation (DRAGONS) project employs N-body simulations [Poole2016], hydrodynamic simulations [duffy2014low, Qin2017a] and SAMs [Mutch2016a, Qin2017c] to study reionization and galaxy formation at high redshift (redshift z greater than or equal to 5, [Geil2016, Geil2017, Liu2016, Liu2017, Mutch2016b, Park2017, Duffy2017, Qin2017b]). The resulting galaxy catalogues from both the hydrodynamic simulation and SAM with feedback included are in good agreement with observations including the stellar mass and galaxy UV luminosity functions across cosmic time.


In this work, we focus on high-redshift modelling (redshift z greater than or equal to 5) of currently unobservable dwarf galaxies (virial mass M_vir less than or approximately equal to 10^10.5 solar masses) which are believed to be the dominant source of ionizing photons driving reionization [duffy2014low, Liu2016]. Using simplified models, we isolate prescriptions of gas accretion, cooling and star formation from reionization and supernova feedback, which will be further discussed in a companion paper. In the following two subsections, we summarize the DRAGONS SAM and hydrodynamic simulation, named Meraxes and Smaug, respectively. Meraxes is based on the Munich SAM [croton2006many, guo2011dwarf] and optimized to study galaxy formation at high redshift. Based on halo properties read from merger trees, it evolves galaxies using simplified prescriptions, which calculate baryonic infall, cooling, star formation, supernova feedback, metal enrichment, stellar mass recycling, AGN feedback and mergers.


In addition, Meraxes incorporates 21cmFAST [Mesinger2010], a semi-numerical approach to evolve ionization fields, for the purpose of investigating reionization feedback and exploring the IGM state during the EoR. We refer the interested reader to [Mutch2016a] and [Qin2017c] for a detailed description of Meraxes and provide a brief review of the relevant modelling prescriptions in this section. To connect N-body simulations of dark matter with baryonic physics, the first implementation is the accretion of gas. Following convention, a universal baryon fraction with suppression due to reionization heating is applied and each halo group accretes gas from the IGM. The change in hot gas mass Delta m_hot equals f_mod*f_b*M_vir - sum over N_halo of m_baryon, where m_baryon is the total baryonic mass including hot gas, cold gas and stellar mass. In the case of Delta m_hot > 0, all of the mass is assumed to be accreted by the central halo.


Gas falling into a halo is stored in the hot gas reservoir, which is assumed to share the halo virial temperature, T_vir, due to shock heating. It then cools within a certain time-scale. Assuming the hot gas follows a singular isothermal sphere (SIS) profile, we calculate the cooling radius, r_cool, at which the cooling time is equal to the halo dynamical time. Based on the ratio of the cooling radius to the virial radius, we calculate the cooling rate and consider two scenarios: static hot halo (r_cool < R_vir) and rapid cooling (r_cool >= R_vir). Therefore, the cooling rate can be calculated by the cooling rate dot m_cool equals (m_hot / t_dyn) times the minimum of (1, r_cool / R_vir). Cooling leads to the build-up of a cold gas reservoir, which is assumed to form a rotationally supported disc, where stars form, with an exponential surface density profile.


When the cold gas mass, m_cold, exceeds a critical value, m_crit, we use a phenomenological relation suggested by observations [kennicutt1998global] to calculate the star formation rate (SFR). The star formation rate dot m_star equals alpha_sf times (max(0, m_cold - m_crit)) / t_dyn,disc, where t_dyn,disc is the dynamical time of the cold gas disc inferred by the halo spin parameter [bullock2001apj...555..240b]. A free parameter, alpha_sf, is introduced for the purpose of adjusting the star formation efficiency. We note that, in order to be consistent with the hydrodynamic simulations, the newly formed stars are assumed to follow an initial mass function in the mass range of 0.1-120 solar masses with the form from [chabrier2003galactic]. Smaug is a series of high-resolution hydrodynamic simulations which were run with a modified GADGET-2 N-body/hydrodynamics code [springel2005cosmological]. These models follow the OverWhelmingly Large Simulations project (OWLS; [Schaye2010]).


These models have been shown to successfully reproduce observed galaxy properties at low redshift including the cosmic star formation history [Schaye2010] and the stellar mass function at redshift z<2 [Haas2013]. In addition, [Katsianis2017] recently showed that the EAGLE simulations, an extension of OWLS, are also in agreement with the observed SFR function across cosmic time (redshift z from approximately 0 to 8). These achievements motivate us to make use of the hydrodynamic simulations and to investigate the validity of semi-analytic prescriptions at high redshift where observational constraints are unavailable. In this work, the simulation from Smaug comprises 512^3 baryon and 512^3 dark matter particles within a cube of comoving side of 10 per h Mpc. This equates to a mass resolution of 4.7 (0.9) x 10^5 per h solar masses per dark matter (gas) particle.


The time interval between two outputs is around 11 Myr, and the initial conditions were generated with the grafic package [Bertschinger2001] at redshift z=199 using the Zel'dovich approximation [Zeldovich1970]. We summarize the subgrid physics adopted in Smaug in terms of cooling and star formation for comparison with the semi-analytic prescriptions. Cooling [Wiersma2009a] consists of only primordial elements and is pre-tabulated using the cloudy package [ferland1998cloudy]. Star formation [Schaye2008] occurs by stochastically converting gas particles to star particles in the ISM. When the gas density exceeds a critical value, the gas particle is assumed to be multiphase ISM with pressure proportional to density to the power of 4/3. The gas depletion time, t_g, in the hydrodynamic simulation is calculated based on the observed KS ([kennicutt1998global]) star formation law, and the SFR is given by the gas mass divided by the depletion time.


The two simulations used in this work are (1) DMONLY, a collisionless N-body simulation performed using the same initial conditions from the full simulation but neglecting hydrodynamic forces from the baryonic component. (2) NOSN_NOZCOOL_NoRe, a toy model with cooling in the absence of metal line emissions and reionization heating. It ignores feedback from exploding supernovae and heating due to reionization, and will be used to compare with the SAM for the purpose of investigating gas accretion, cooling and star formation without impacts from any feedback. The top left panel of Fig. 2 presents an example of the gas density distribution at redshift z=5 in the NOSN_NOZCOOL_NoRe Smaug simulation. In the SAM, cold gas is considered as potentially star-formation gas. In order to facilitate direct comparisons of the gas reservoir with the semi-analytic calculation, we specify, for each galaxy in the hydrodynamic simulation, a star-formation gas component.


This reservoir comprises all potential star forming particles of the galaxy, making it comparable to the assumption adopted in the SAM. As mentioned before, a gas particle located in a cold dense region is considered as multiphase and can potentially form stars. More specifically in Smaug, this requires the particle to satisfy a combination of four criteria [Schaye2010]: A) a high physical density; B) a high comoving density; C) a high physical density and a low temperature; D) a low temperature. Hot gas in the SAM represents a reservoir where the temperature is as high as the halo virial temperature as a result of shock--heating during gas accretion. Therefore, in order to define a hot gas component for a galaxy in hydrodynamic simulations, which can be compared with the SAM, we first exclude all potential star forming particles from the galaxy.


We plot the temperature distribution of non-star-forming gas particles of two haloes in the NOSN_NOZCOOL_NoRe Smaug simulation in the top panels of Fig. 1. In the more massive halo, three populations can be observed, corresponding to, from right to left, (1) infall hot gas; (2) infall gas that has been through cooling, and the cooling rate decreases significantly when temperature drops to a few 10^4 K [Wiersma2009a]; and (3) cold gas which is about to reach the multiphase ISM disc and become dense enough to form stars. However, we can only identify the group of less dense (non-star-forming) cold gas in the less massive halo. Based on the density--temperature phase diagram of the non-star-forming gas particles of a galaxy, the hot gas mass and temperature of the galaxy can be determined. In practice, we use the KMeans clustering algorithm. We consider two features (density and temperature) and separate the non-star-forming gas particles into two groups.


Hydrostatic pressure keeps gas from collapsing into shallow potential wells of low-mass haloes, which consequently decreases the gravitational potential within these haloes and slows their growths compared to a collisionless universe. This has been illustrated using comparisons between hydrodynamic simulations and N-body simulations of dark matter only [Sawala2013, Schaller2014, Velliscig2014], which show that the inclusion of baryons significantly reduces halo masses. We investigated the same baryonic effect in dwarf galaxies at high redshift (redshift z>5) in [Qin2017a]. We found that the reduction of mass can be up to a factor of 2 and that the fraction of baryons in haloes with masses between 10^7 and 10^9 solar masses, which host dwarf galaxies, never exceeds 90 per cent of the cosmic mean during reionization. Thus, applying SAMs to halo merger trees generated from a collisionless N-body simulation and assuming the universal baryon fraction overestimates halo and baryon masses of dwarf galaxies at high redshift.


We proposed two modifiers using a simplified hydrodynamic simulation in [Qin2017a] -- ADIAB, where gas only cools adiabatically with no stellar or galactic physics involved. In this work, which aims to inform a better semi-analytic description of high-redshift dwarf galaxies, we make use of these two modifiers, which are shown in the top panels of Fig. 2 as functions of halo mass at redshift z=13-5. In practice, the halo mass modifier is included when halo properties are read from the merger trees and the baryon fraction modifier is embedded in f_mod. However, we also point out the fact that these modifiers do not offer a consistent modification accounting for all aspects of astrophysical processes. In [Qin2017a], these have been shown to provide additional approximately 10 per cent alterations to halo mass due to radiative cooling, star formation and supernova feedback. Reionization, on the other hand, provides a more dramatic suppression.


In order to demonstrate the effect of incorporating the modifiers within semi-analytic galaxy formation modelling, we apply Meraxes with different implementations. We first apply the SAM with both the halo mass and baryon fraction modifiers implemented and calibrate the free parameters in the NOSN_NOZCOOL_NoRe regime where feedback is not included (SAM_HB). Next with all free parameters remaining the same, we then apply the SAM with only the halo mass modifier included (SAM_H) and without any modifiers (SAM). We see that compared to the hydrodynamic simulations, the collisionless N-body simulation as well as the uncorrected SAM overestimate the virial mass function, especially at higher redshifts. However, the halo mass modifier results in a SAM virial mass function that is in better agreement with the hydrodynamic calculation. This leads to a reduction as much as a factor of 2 in of the halo number density at redshift z~12 for a given mass, and since the growth of the system becomes suppressed, it harbours fewer baryons.


The first row of Fig. 3 presents the stellar mass function of matched galaxies between SAM_HB and Smaug at redshift z=13-5. We see that the two redshift z=5 mass functions are in agreement at stellar mass M_* > 10^7 solar masses, below which the models are limited by resolution and cooling mechanism. This indicates that the two modelled galaxy catalogues comprise similar stellar components at redshift z=5 in a cosmological context. However, the results diverge towards higher redshifts -- SAM_HB produces more massive galaxies than Smaug. This highlights the issue that stellar build-up proceeds faster in the SAM. As a consequence of the high-redshift modelling, the SAM produces more ionizing photons at earlier times compared to the hydrodynamic calculation, overestimating the contribution of high-redshift dwarf galaxies to reionization [Liu2016].


We investigate the properties of individual galaxies in the two models, and show the comparisons of the total baryon mass, stellar mass and SFR predicted by the hydrodynamic simulation and SAM in the bottom three rows of Fig. 3. We see that the overall baryon mass is underestimated by the SAM, with the offset becoming larger at the low-mass end. This is because the NOSN_NOZCOOL_NoRe Smaug simulation predicts larger halo masses and baryon fractions than ADIAB due to cooling and star formation [Qin2017a]. Therefore, modifying the halo mass and baryon fraction using the ADIAB Smaug simulation overestimates the baryonic effect when feedback is not included. In addition, we see that both stellar mass and SFR are similar between the two models at redshift z=5, which are, however, underestimated in the SAM result when the stellar mass approaches 10^6 solar masses or SFR reaches 10^-2 solar masses per year. This causes an underestimation of the number of ionizing photons at the high redshifts.


We next discuss the involved prescriptions that might induce such discrepancies in the history of star formation. In the NOSN_NOZCOOL_NoRe case when feedback is not implemented, there are only two processes that might cause the different stellar growth histories between SAMs and hydrodynamic simulations: 1) star formation efficiency; and 2) gas fuelling and cooling efficiencies. We note again that since star formation is not resolved in cosmological simulations, both the hydrodynamic simulation and SAM start from an empirical relation between SFR and density -- the KS law [kennicutt1998global], which proposes that a galaxy is able to form stars when its surface density exceeds a critical value, and the surface SFR density can be estimated by a power law. We have shown that cooling is not well modelled in the SAM when the virial temperature is lower than the atomic cooling threshold, which leads to significant underestimations of the number of low stellar mass and SFR objects in Fig. 3.


In the hydrodynamic simulation, local 3D densities can be calculated through smoothing particle mass. Smaug converts the critical surface density to three dimensions assuming a self-gravitating disc. When the local pressure of a gas particle with a relatively low temperature (<10^5 K) exceeds the critical value, the particle is considered as a potential star forming region. This infers a SFR of dot m_star,hydro is proportional to (1+z)^1.1. On the other hand, the average SFR of a galaxy in the SAM is calculated using equation (6). For a given halo mass and with m_cold proportional to M_vir, the redshift dependency of the SAM SFR is approximately dot m_star,SAM is proportional to (1+z)^1.5. Comparing these equations, we find that although both numerical calculations of star formation are derived from the KS law, they possess different evolutionary histories for a given halo mass. This is essentially due to the different assumption of gas depletion time-scale adopted in the two modelling approaches.


In order to be consistent with the hydrodynamic simulation, a suppressed star formation efficiency of alpha_sf proportional to (1+z)^(-m) towards higher redshifts is required in the SAM. At higher redshifts, galaxies have larger velocity dispersions, indicating increased turbulence and thickened discs [Newman2012, Price2015]. Meanwhile, simulations suggest that mergers which happen frequently at high redshift [Poole2016] can also thicken discs [Moster2010, Moster2012]. Therefore, the assumption of a self-gravitating disc might not be valid at high redshift. If we assume an SIS profile for star-forming gas, the density scales as (1+z)^3, which leads to a larger value of m~0.7. Furthermore, during the experiment we find that, in the SAM, galaxies are more likely to have an insufficient cold gas reservoir at lower redshifts. This requires a more suppressed star formation efficiency at higher redshift, and in practice, m=1.3 is adopted in this work.


In Fig. 3, the semi-analytic result using the redshift-dependent star formation efficiency (SAM_HBS) is presented using dashed lines. We see that suppressing star formation at higher redshifts results in a better agreement of the stellar mass function, stellar mass and SFR with the hydrodynamic calculations. We define hot gas fraction as the mass ratio of the hot gas to the non-star-forming gas, and show its correlation with halo mass in Smaug at redshift z=13-5 in the left-hand panel of Fig. 4. We highlight galaxies above the atomic cooling threshold with red circles and we see that these galaxies in general comprise little or no hot gas. We see that the majority of galaxies at high redshift do not possess any hot gas, and their non-star-forming gas particles are identified as one group -- the cold non-star-forming gas.


However, due to the lack of molecular cooling, galaxies with virial mass M_vir less than or approximately equal to 10^8 solar masses do comprise two components. In the right-hand panel of Fig. 4, we show the 2D histograms of the hot (red) and cold (blue) non-star-forming gas temperatures as functions of halo mass in Smaug. We see that for galaxies with virial mass less than or approximately equal to 10^8 solar masses, their hot and cold non-star-forming gas particles possess median temperatures >10^3 K and <10^3 K, respectively. This broad distribution on the density--temperature phase diagram is from adiabatic cooling of the gas as the universe expands, which is also indicated by the decreasing median temperature at lower redshifts of these less massive objects. Gas in the IGM falls into the gravitational potential of a halo and fuels star formation in the host galaxy.


While gas particles cover a broad range of temperatures from approximately 10^3 - 10^8 K [Wiersma2009b] in the hydrodynamic simulation, (hot) gas initially accreted by a galaxy in the SAM is assumed to be shock-heated to the virial temperature. In order to test the validity of this simplified gas infall prescription for modelling of dwarf galaxies, we compare the temperature of non-star-forming gas in the hydrodynamic simulation to the halo virial temperature. In the right-hand panel of Fig. 4, we highlight galaxies with virial mass > 10^9 solar masses in Smaug with circles and show the correlation between virial temperature and halo mass with green solid lines. We see that fewer than 10 galaxies with virial mass > 10^9 solar masses in the Smaug simulation comprise hot gas at redshift z=5, and their hot gas temperatures are close to the virial temperatures, suggesting that theses galaxies have been heated through shocks.


However, the non-star-forming gas of most galaxies in Smaug is considered as cold with temperature T less than or approximately equal to 10^4 K, which is much lower than the virial temperature and becomes more common at higher redshifts. This is consistent with the prediction of EAGLE simulations [Schaye2014], where the mass ratio of hot to all baryons is less than 1 per cent on average for haloes around 10^10 solar masses [Correa2017], with the fraction decreasing towards higher redshifts and in less massive haloes. We see that in the SAM, cold and hot gas, where stars can and cannot form, are defined geometrically. They represent a star forming disc and the outer region, following an exponential and an SIS profile, respectively. However, we show that full virialization of infall gas might be problematic for high-redshift modelling of dwarf galaxy formation. In these less massive systems, virial shocks might not form and the infall gas could then only reach the virial temperature when it arrives at the star forming disc [Birnboim2003, Cattaneo2017].


Instead, these cold and less dense non-star-forming gas particles of dwarf galaxies in Smaug represent a cold accretion mode [Keres2005, Keres2009, benson2011, Correa2017]. Therefore, in the SAM, instead of being fully virialized, the infall gas of dwarf galaxies should be separated into hot and cold components, which reach the disc and contribute to star formation on different time-scales. In addition, since the non-star-forming gas particles of the majority of high-redshift dwarf galaxies are not hot in the hydrodynamic simulation, we advocate that the terminology of hot gas becomes misleading in the redshift and mass ranges discussed in this work. The gas infall and cooling prescription adopted for the SAM distinguishes two regimes -- a static hot halo and rapid cooling gas. In the second regime, gas cools onto the disc at the dynamical time-scale, which has the properties of cold accretion if one combines the process of gas infall and rapid cooling.


Therefore, altering the rapid cooling rate is an alternative to incorporating cold accretion directly. The relative importance of the two cooling modes is determined by the ratio of the cooling radius to the virial radius, which is shown in the left panel of Fig. 6. We see that galaxies with stellar mass < 10^9 solar masses are mostly in the rapid cooling regime. The determination of static hot halo and rapid cooling regimes has been discussed in [croton2006many]. Comparing with hydrodynamic simulations, they found virial mass ~ 10^11 solar masses separates the two regimes, which is approximately independent with redshift up to redshift z~6 (see also [Correa2017]). This is in agreement with our finding of stellar mass ~ 10^9 solar masses. The balance of instantaneous cooling rate between the two numerical approaches can be inferred from the evolution of the mass ratio of star forming to non-star-forming gas. This is shown in the right panel of Fig. 6.


We see that while the SAM_HBS result is in agreement with the hydrodynamic calculation at low redshift, the ratio of star-forming to non-star-forming gas in the SAM is underestimated at higher redshifts. We have shown that in the SAM, dwarf galaxies at high redshift are identified as being in the rapid cooling regime. Therefore, in order to understand the different transition rates between non-star-forming reservoir and star-formation gas in Meraxes and Smaug, we next discuss the assumed SIS profile of hot gas reservoir in the SAM. We present the median radial profile of the non-star-forming gas of galaxies with stellar mass = 10^(7 +/- 0.5) solar masses in Smaug at redshift z=13-5 in Fig. 7. We see that the density of non-star-forming gas drops at the inner region when compared to an SIS mass profile. The non-star-forming gas particles 1) have a large dispersion up to greater than or approximately equal to 2 times the virial radius; and 2) are less concentrated at higher redshifts.


The SAM assumes the SIS profile for hot gas. Therefore, in the rapid cooling regime, the cooling mass during one time step can be calculated. We see that the SIS profile significantly overestimates the non-star-forming gas density in the inner region compared to the simulation result. Therefore, with the assumption that gas can only transform from non-star-forming to star forming gas when collapsing into the centre, the SIS profile leads to an overestimation of the inward collapse rate. However, due to the large radial region of star-formation gas observed in Fig. 7, this assumption might not be accurate. Gas particles in dwarf galaxies at redshift z>9 can be triggered as star forming regions as far as the virial radius. The combination of corresponding effects between the overestimated collapse rate from the assumed SIS profile and the underestimated transition radius of gas reservoir is the primary cause of the discrepancy in the ratio of star-forming to non-star-forming gas between Meraxes and Smaug.


Whilst adopting the correct gas profile and finding the accurate transition radius of gas reservoir can help explain the underestimation of the ratio of star-forming to non-star-forming gas in the SAM at high redshift, an analytic solution is not well defined. The large dispersion in the gas profile is mainly due to frequent mergers at these redshifts, which can be affected by numerical configurations, as well as by physics implementations including feedback. In order to take the underestimated cooling rate of dwarf galaxies at high redshift into account, we propose an alteration to the cooling equation and allow the free parameter, kappa_cool representing the maximum cooling factor to exceed unity at high redshift. We show the result of the ratio of star-forming to non-star-forming gas calculated with kappa_cool = min(5, (1+z)/6) in Fig. 6, and we see that with higher cooling rates, the semi-analytic calculation becomes more consistent with the hydrodynamic result at high redshift.


However, the stellar mass function becomes higher at redshift z=13 compared to the Smaug result (see Fig. 3), suggesting the necessity of stronger suppression of the star formation efficiency due to the relatively larger disc mass at higher redshifts in the SAM. Lastly, we point out that without a self-consistent radiative transfer calculation, atomic cooling and gas temperature might not be properly simulated in the hydrodynamic simulation either, which alters the importance of thermal radiation cooling. For instance, with local sources of ionizing radiation neglected, the cooling rate is potentially overestimated [Schaye2014], and since self-shielding cannot be properly captured in our hydrodynamic simulations, the cooling rate might be underestimated in dense regions [McQuinn2011]. Moreover, supernova feedback and reionization are expected to regulate galaxy formation through altering the gas component, which will be the topic of a forthcoming paper.


While most assumptions adopted for semi-analytic galaxy formation models, including the gas reservoirs, are in good agreement with hydrodynamic simulations for Milky Way size objects [Guo2016, Stevens2016b], they become less accurate for less massive galaxies at high redshift. In this work, we propose modifications to SAMs based on the comparison of dwarf galaxy properties calculated by the Meraxes SAM and the Smaug high-resolution hydrodynamic simulation. We focus on gas accretion, cooling and star formation at redshift z>=5, and consider scenarios in the absence of reionization and supernova feedback. We summarize the modifications below: The parent cosmological simulation of a SAM usually includes only collisionless particles, where baryonic physics is neglected. In making comparisons between N-body and hydrodynamic simulations starting from identical initial conditions, we previously showed ([Qin2017a]) that dwarf galaxy host halo masses are significantly overestimated when hydrostatic pressure is not considered, and that the fraction of baryons accreted by dwarf galaxies cannot reach the level of cosmic mean.


While inclusion of halo masses directly from N-body dark matter only simulation and assumption of a universal baryon fraction are standard features of SAMs, the impact of these assumptions becomes significant at high redshift and small scale. We find that the star formation prescription in the SAM that is based on the consumption of a cold gas reservoir does not represent the evolutionary path followed by gas in the hydrodynamic simulation, while both of the two modelling approaches start from the observational relation between surface density and SFR [kennicutt1998global]. We find that this results from variation in the calculated depletion time-scale of the gas reservoir. We address this by modifying the efficiency that modulates star formation in the SAM with a redshift dependency. In this work, we adopt alpha_sf(z) = 0.05 * [(1+z)/6]^-1.3, which allows the model to follow the NOSN_NOZCOOL_NoRe hydrodynamic simulation in the Smaug suite.


The majority of dwarf galaxies at high redshift are in the rapid cooling regime, where the infalling gas cannot form stable shocks or remain in hydrostatic equilibrium. This represents a cold accretion mode, which is not well modelled with the assumption that the hot gas reservoir follows the SIS profile and falls into the centre within the dynamical time. We find, in the hydrodynamic simulation, that gas in high-redshift dwarf galaxies can form stars as far as the virial radius and that the SIS profile overestimates the density in the inner regions of these low-mass objects. In order to take this into account, we propose of modulation of the cooling prescription with a redshift-dependent collapse rate where kappa_cool(z) = min[5, (1+z)/6] leads to an agreement with the hydrodynamic calculation on the cold gas mass evolution. Furthermore, we point out that the terminology of hot gas and cold gas becomes misleading when applying SAMs to the formation of high-redshift dwarf galaxies.


The hot gas representing non-star-forming gas within a galaxy does not experience full virialization and possess a median temperature that is much lower than the virial temperature of the host halo. In a future paper (Qin et al. in prep.), we will discuss star formation in the presence of feedback from reionization and supernovae. We will compare the SAM calculation with the most complete hydrodynamic simulation in the Smaug suite that considers reionization as an instantaneous heating background [Haardt2001] and distributes supernova energy in thermal form [DallaVecchia2012]. We will also consider an additional star formation prescription to the one shown in this work, in which molecular gas directly drives the star formation history [Lagos2011]. The goal of these two papers will be more complete and faithful recreation of hydrodynamic simulations at high redshift, which will further leverage the observations that we can make in this early history of our Universe.


In this work, we make direct comparisons of the semi-analytic and hydrodynamic galaxy properties calculated by Meraxes and Smaug, respectively. We connect galaxies produced by Meraxes to their host haloes in the N-body simulation, and then match galaxies in Meraxes with the corresponding hydrodynamic simulation. The same algorithm of matching haloes between snapshots is adopted to match between simulations. Dark matter (and baryon) particles evolve (co-evolve) in Smaug, and friend-of-friend (or fof) groups and subgroups (or haloes), to which particles belong are determined using a standard fof halo finder and subgroup finder ([subfind], [springel2008aquarius]) in post-processing. Then by tracking halo particles in consecutive snapshots, each halo can be linked to its progenitors and form a merger tree. This process is applied to all haloes at one snapshot, in order to horizontally construct the merger trees, which allows the SAM to evolve all galaxies at each snapshot and calculate reionization feedback self-consistently.


We next give a brief overview of the gbptrees algorithm used to build dark matter halo merger trees. For each halo, subfind sorts particles according to their kinetic and potential energies, with, in general, more bounded particles ranking higher. For each candidate, a pseudo-radial moment is defined. We define the goodness of a matching candidate as the difference between the pseudo-radial moment at m=-1 and m=0. The two halos are considered as a good match when this difference is > -0.2 [poole2017mnras.472.3659p]. Amongst all good matches, the best match is identified by maximizing the statistic of the pseudo-radial moment at m=-1 during the process of scanning for good matches forwards and backwards over 16 snapshots. A consequence of this algorithm is that matching is performed by tracking halo cores instead of the majority of halo particles. This central weighting algorithm allows us to build more realistic halo merger trees.


Meraxes reports the Most Bounded Particle (MBP) of the host halo for each galaxy, which bridges SAM galaxies and dark matter haloes in DMONLY. In order to further connect haloes in the collisionless N-body simulation with simulations including baryonic physics, we match haloes between simulations using the same matching strategy described in Appendix A.2. Instead of matching 33 snapshots in one simulation, we match haloes from two simulations (e.g. NOSN_NOZCOOL_NoRe and DMONLY) using their dark matter particles at each snapshot. This is achievable because all the Smaug simulations including DMONLY start from the identical cosmological initial conditions and their dark matter particles possess the same IDs over different simulations. However, with information from only two snapshots, mismatches and pathologies occur in particular when reaching the resolution limit. Therefore, final matched sample is limited to haloes which are matched bidirectionally.


Fig. A2 illustrates how galaxies are finally matched between Meraxes and Smaug with this two-step manner that MBPs connect galaxies from Meraxes to their host haloes which are matched with Smaug galaxies through dark matter particles. In this work, only central galaxies with virial masses exceeding the resolution threshold of 10^7.5 solar masses in the SAM results are considered, and in order to minimize the impact from central-satellites switching mentioned in Appendix A.2, satellite galaxies identified in the SAM are excluded from the final sample as well. In the NOSN_NOZCOOL_NoRe Smaug simulation, there are 195562 galaxies, in which 129043 galaxies (~ 66 per cent) are matched with Meraxes. In this work, we use a machine learning algorithm to spilt non-star-forming gas particles into hot and cold components. The two groups are considered as one when the offset between their median temperatures is small.


Fig. A3 shows the fraction of hot gas in the non-star-forming gas at redshift z=5 with different choices of the temperature offset threshold Delta_T,crit, and we see that a smaller Delta_T,crit introduces more galaxies with a hot gas fraction > 0. However, this only affects galaxies with inefficient cooling (i.e. below the atomic cooling thresholds) while the hot gas fraction of more massive galaxies remain zero or small. On the other hand, when non-star-forming gas particles of a galaxy are considered as one group, which is common for galaxies with efficient cooling, whether they are determined as hot or cold is based on their median temperature. We vary the minimum temperature of particles being identified as hot (T_crit) from 10^5 K to 10^4 K and we find that the hot gas fraction is insensitive to T_crit. Based on these, we note that the choice of temperature thresholds does not have a significant impact to our conclusion that high-redshift dwarf galaxies accrete gas in cold mode.


A numerical experiment evolves a galaxy by calculating physics processes in a predefined order. For instance, Meraxes in sequence calculates reionization, baryonic infall, cooling, star formation and the impact of supernova feedback. Thanks to the exquisite time step of calculation of hydrodynamic simulations, their outputs can be considered as instantaneous galaxy properties at a given redshift. However, this is not the case for the SAM used in this work, which records properties that either have or have not been processed or are average values during one time step. In the right panel of Fig. 6, we show the ratio of star-forming to non-star-forming gas, which is the mass ratio of cold to hot gas in the SAM. Three moments of calculation can be adopted: (1) after star formation; (2) before star formation and after cooling; (3) before cooling and after baryon infall. We note that case (1) is adopted in the main context, and we show the difference in Fig. A4. We see that because of the high cadence (~11Myr) of the SAM, the choice of calculation does not have a significant impact to our results.


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.