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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


The infalling hot gas is also assumed to follow the singular isothermal sphere (SIS) profile. To ease demonstration in this paper, we term the time-scale of gas being transited from hot reservoir to the star-forming disc as a transition time-scale. We illustrate the cooling prescription in Fig. B1. Most massive haloes are able to create shocks and heat the infalling gas, resulting in hydrostatic equilibrium. In this case, which is termed the hot halo regime, the time-scale of hot gas transitioning to star-forming is determined by the thermal cooling time-scale. However, it is difficult to generate shock heating in less massive systems, leaving little support to prevent gas from infalling onto the central disc, and cooling becomes rapid. In this rapid cooling regime, the prescription assumes the star-forming gas disc is relatively small and such a process happens as free-fall. In making comparisons of the gas reservoir calculated by the SAM and hydrodynamic simulation in [qin2018], we found the transition time-scale equals the dynamical time-scale becomes less accurate when applying the rapid cooling prescription to high-redshift dwarf galaxy modelling.


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


Note that gas within the cooling radius is considered to have reached hydrostatic equilibrium and cool thermally if the cooling radius is less than the virial radius. However, in the case of a large cooling radius, the infalling gas will not be able to form stable shocks or remain in hydrostatic equilibrium. Accordingly, all of the accreted gas directly collapses into the central regions as free-fall. We see that most low-mass galaxies discussed in this work are considered to be in the rapid cooling regime. Next we discuss the gas density profile of galaxies with stellar mass of approximately 10^7 solar masses. We see that, compared to the NOSN_NOZCOOL_NoRe simulation where heating from supernova (and reionization) is not included, galaxies within the same stellar mass range are hosted by larger haloes with more gas particles identified as non-star-forming when the feedback is considered. However, the total gas mass does not change significantly, indicating suppressions of baryonic mass and self-regulation of star formation.


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


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


We investigate the dependence of galaxy clustering at redshift z from approximately 4 to 7 on UV-luminosity and stellar mass. Our sample consists of approximately 10,000 Lyman-break galaxies (LBGs) in the XDF and CANDELS fields. As part of our analysis, the stellar mass versus UV magnitude relation is estimated for the sample, which is found to have a nearly linear slope of the slope of the log10 stellar mass versus UV magnitude relation is approximately 0.44. We subsequently measure the angular correlation function and bias in different stellar mass and luminosity bins. We focus on comparing the clustering dependence on these two properties. While UV-luminosity is only related to recent starbursts of a galaxy, stellar mass reflects the integrated build-up of the whole star formation history, which should make it more tightly correlated with halo mass. Hence, the clustering segregation with stellar mass is expected to be larger than with luminosity. However, our measurements suggest that the segregation with luminosity is larger with approximately 90% confidence (neglecting contributions from systematic errors).


We compare this unexpected result with predictions from the Meraxes semi-analytic galaxy formation model. Interestingly, the model reproduces the observed angular correlation functions, and also suggests stronger clustering segregation with luminosity. The comparison between our observations and the model provides evidence of multiple halo occupation in the small scale clustering. Galaxy clustering provides a probe of the host halo mass of galaxies. The clustering strength is commonly described by the two-point correlation function, which measures the probability of finding galaxy pairs at given spatial separations. [1996MNRAS.282..347M] used the extended Press-Schechter formalism [1991ApJ...379..440B] to show that the ratio between the correlation functions of halos and the underlying matter depends on halo mass. This ratio is known as bias. Since galaxies reside in halos, the bias links galaxy clustering to the mass of their host halos (see [2002PhR...372....1C] for a review).


The dependence of clustering strength on galaxy properties is known as clustering segregation, and reveals the correlation between galaxy properties and halo mass. At high redshifts, clustering segregation is observed for Lyman-break galaxies (LBGs) with UV-luminosity [2006ApJ...642...63L, 2009A&A...498..725H, 2014ApJ...793...17B, 2016ApJ...821..123H, 2018PASJ...70S..11H] and stellar mass [,2017ApJ...841....8I, 2018A&A...612A..42D]. One basic conclusion from these studies is that more luminous and larger stellar mass galaxies are more clustered, and therefore reside in more massive halos. In the context of hierarchical galaxy formation, this correlation between halo and galaxy properties is unsurprising, since halo mass is closely related to the gas reservoir available for star formation and those processes which impact on it. For instance, in the low mass regime, supernova (SN) feedback can effectively suppress star formation [2013MNRAS.428.2741W, 2014MNRAS.445..581H, 2014MNRAS.443.3435D]. Therefore, it is of particular interest to explore which galaxy property is more tightly correlated with the host halo mass.


While UV-luminosity is directly related to the current star formation rate, stellar mass provides integrated information over the star formation history. For this reason, it is expected that, when splitting the same sample by luminosity and stellar mass, clustering segregation with stellar mass should be larger than with UV magnitude. In order to observe the difference between clustering segregation with stellar mass and luminosity, it is essential that the stellar mass is measured from SED fitting including rest-frame optical photometry. Recently, [2016ApJ...821..123H] carried out an analysis of clustering segregation with stellar mass in similar fields to our work. However, the stellar mass used in that study was obtained from a simple conversion of the UV- luminosity to mass using the stellar mass versus UV magnitude relation. Thus, the analysis could not self-consistently infer any difference of the clustering segregation between stellar mass and UV-luminosity. In this paper, we measure stellar masses from SED fitting including rest-frame optical Spitzer/IRAC data, which allows us to measure the clustering segregation in stellar mass at these redshifts for the first time.


Semi-analytic models (SAMs) of galaxy formation are based on halo merger trees provided by N-body simulations, and evolve galaxy properties within these halos using analytic or empirical prescriptions of baryonic physics and feedback processes. Since the theory which links galaxy clustering to dark matter halos also relies on N-body simulations [e.g][1996ApJ...462..563N, 2010ApJ...724..878T], a comparison between observed clustering and that predicted by a SAM tests the link between galaxy properties and halo mass. In this study, we compare our clustering measurements with results from the Meraxes SAM. This model has been shown to be successful in reproducing UV-luminosity functions and stellar mass functions over a wide range of redshifts [2016MNRAS.462..250M, 2016MNRAS.462..235L, 2017MNRAS.472.2009Q]. This somewhat contradicts the above expectation. Examining the clustering dependence beyond this redshift is the main purpose of this paper. This paper is organised as follows. We describe the observational catalogue used in the analysis, and measure the stellar mass versus UV magnitude relation in Section 2.


Methods to measure the angular correlation function (ACF) and galaxy bias are introduced in Section 3, and results are demonstrated in Section 4. We perform the comparison between the observations and predictions from Meraxes in Section 5. Finally, the work is summarised in Section 6. Throughout the paper, unless specified, the cosmology of (h, Omega_m, Omega_b, Omega_Lambda, sigma_8)=(0.678, 0.308, 0.0484, 0.692, 0.815) [2016A&A...594A...1P] is assumed. Magnitudes are in the AB system. The galaxy sample used for our measurements is based on the photometric catalogue from [2015ApJ...803...34B], who selected Lyman break galaxies (LBGs) at z~4-7 based on the Hubble Space Telescope (HST) data in all the CANDELS fields, as well as the very deep XDF and HUDF09 parallel fields.


In particular, our sample is drawn from the XDF [2013ApJS..209....6I], HUDF-091 and HUDF-092 [2011ApJ...737...90B], CANDELS-GN and CANDELS-GS [2011ApJS..197...35G, 2011ApJS..197...36K], ERS [2011ApJS..193...27W], and CANDELS-UDS, CANDELS-COSMOS and CANDELS-EGS [2011ApJS..197...35G, 2011ApJS..197...36K]. These survey regions span an aggregate of ~ 700 arcmin^2 in the sky, and ~ 10,000 LBGs are identified. Photometric redshifts of these sources are estimated using the EAZY code [Brammer08]. For more information on the LBG selection and the photometric redshifts see [2015ApJ...803...34B]. We combine the HST photometry with the large archive of Spitzer/IRAC legacy data available in the CANDELS fields [Ashby13, Ashby15], which includes the ultra-deep IGOODS/IUDF and GREATS surveys [Labbé et al. 2018, in prep.][2015ApJS..221...23L] in the GOODS fields. IRAC photometry is measured in circular apertures after subtracting the contaminating flux of neighboring galaxies using the code mophongo [Labbé et al., 2018, in prep.][Labbe06], which is similar to the code TPHOT [Merlin16].


We measure stellar masses of galaxies based on SED fitting to the HST+Spitzer photometry using ZEBRA+ [Oesch10]. The synthetic template set used here is based on [BC03] with a constant star-formation history, sub-solar metallicities (0.2 Z_sun) and a [2003PASP..115..763C] initial mass function (IMF). Nebular continuum and emission lines are added self-consistently based on the number of ionizing photons emitted by each SED and assuming line ratios relative to H-beta as tabulated by [Anders03]. Dust extinction is applied using the attenuation curve by [Calzetti00]. Following [2015ApJ...803...34B], the absolute magnitudes, M_UV, are computed based on the fluxes in the photometric band that is closest to rest-frame 1600 Angstrom. We first fit the stellar mass versus UV magnitude relation for the LBG sample, which will be used in our clustering analysis to compare stellar mass and luminosity segregation. The form of the relation is assumed to be log10 of the fitted stellar mass M_star equals (d log10 M_star / dM_UV) times (M_UV + 19.5) + log10 of M_star at M_UV = -19.5.


The log-likelihood is then constructed as ln of the likelihood L equals -1/2 times the sum over all LBGs i of [(log10(observed M_star i / fitted M_star))^2 / Delta^2 + ln(2 pi Delta^2)], where the sum is over all LBGs, and Delta is a mass-independent free parameter representing scatter in the stellar mass versus UV magnitude relation. We adopt a Bayesian approach to perform the fit, assume constant priors for all parameters, and apply the emcee MCMC sampler developed by [2013PASP..125..306F]. The resulting stellar mass versus UV magnitude relations are shown in Figure 1. Best-fit parameters are given in Table 1. We find that the stellar mass versus UV magnitude relations are close to linear (M_star is proportional to L) and that the scatter in stellar mass at fixed luminosity is ~ 0.5 dex. Even though our best-fit slopes are slightly shallower, our measurements are consistent with the recent study from [2016ApJ...825....5S].


In this work, every galaxy in our HST sample has an estimate of stellar mass irrespective of the quality of Spitzer data. Low S/N ratios of Spitzer bands could make the stellar masses less precise. To investigate this, in Figure 1, galaxies that have at least one Spitzer band (3.6 um and 4.5 um) with S/N > 3 are shown as yellow empty circles in Figure 1, while the others are shown as small blue dots. No systematic offset is found between them. Since the sample is large enough at z ~ 4, we use multiple stellar mass and luminosity bins for the clustering measurements, and avoid using bins that have no lower bound to reduce possible effects due to low S/N ratios of Spitzer data. The lower bound for the least massive stellar mass bin is shown as red dashed line in the corresponding panel of Figure 1. At all other redshifts, limited by the sample size, we include all galaxies and use two bins to examine clustering segregation with stellar mass and luminosity.


The fraction of LBGs that are included and have at least one Spitzer band with S/N > 3 is 74%, 63%, 53%, and 47% at z ~ 4, 5, 6, and 7 respectively. The advantage of this approach is that the completeness of sample LBGs, which is defined by the selection, is not affected by Spitzer data, and it is important to use the same set of galaxies to compare the clustering segregation between UV-luminosity and stellar mass. We will discuss possible effects on clustering measurements due to uncertainties in stellar mass in Section 4.2. Our approach follows [2014ApJ...793...17B]. We start by determining the angular correlation functions (ACFs), which measure the excess probability of finding galaxy pairs with angular separations between theta and theta + delta theta. The ACF estimator proposed by [1993ApJ...412...64L] is applied, i.e. the observed angular correlation function, omega_obs(theta), is estimated using the Landy-Szalay estimator as (DD(theta) - 2DR(theta) + RR(theta)) / RR(theta).


DD(theta), DR(theta) and RR(theta) are the probability of finding galaxy-galaxy, galaxy-random and random-random pairs respectively. These probabilities are calculated by counting all pairs at separations between theta to theta + delta theta, and normalising by the total number of pairs. Estimates of DR(theta) and RR(theta) require a catalogue of uniformly-distributed random points. This is generated by a random Poisson process. For each field, the random catalogue contains 10,000 points uniformly placed within the corresponding survey regions. We measure the ACFs in logarithmic bins, and estimate errors by bootstrap resampling [1986MNRAS.223P..21L]. We construct bootstrap subsamples by replacing individual galaxies and perform the resampling for ~ 500 times. This approach is also used in [2014ApJ...793...17B] and [2016ApJ...821..123H]. In order to investigate the clustering dependence on both stellar mass and UV magnitude, the total sample is split into subsamples. We choose bins such that they satisfy the stellar mass versus UV magnitude relation at each redshift.


Since the area of each survey region is finite, the observed ACFs are affected by border effects. This is corrected by an additive constant, which is known as the intergral constrain (IC). Following [1999MNRAS.307..703R], we have the true angular correlation function, omega_true(theta), equals the observed angular correlation function, omega_obs(theta), plus the integral constraint, IC, with the integral constraint, IC, equals the double integral of omega_true(theta) over solid angles dOmega_1 and dOmega_2, divided by Omega^2, which equals the sum over i of RR(theta_i) times omega_true(theta_i) divided by the sum over i of RR(theta_i), where omega_true(theta) is the fitting model of the ACF. We assume the ACFs to be a power law, the true angular correlation function, omega_true(theta), equals the amplitude A_omega times (theta divided by 1 arcsecond) to the power of -beta, and construct the log-likelihood using ln of the likelihood L equals -1/2 times the sum over fields and bins i of [(omega_obs(theta_i) - A_omega(theta_i^-beta - IC/A_omega)) / sigma(theta_i)]^2.


This is equivalent to measuring the average ACF of all fields using inverse-variance weighting. Since this approach requires an estimate of the ACF in each individual field, we only include fields that are deep enough such that the mean separation of galaxy pairs is smaller than 100 arcsec. Moreover, the dependence of the IC on the fitting model results in some degeneracy between A_omega and beta [2006ApJ...642...63L], we therefore fix beta = 0.6 following [2006ApJ...642...63L] and [2014ApJ...793...17B]. In addition, since the area of XDF, HUDF-091, and HUDF-092 is only 4.7 arcmin^2 (i.e. one WFC3/IR pointing), the counted number of galaxy pairs in these fields decreases when the angular separation is greater than ~ 140 arcsec. We therefore only include separations smaller than that in the likelihood function. The amplitude of the ACF A_omega could be weakened by contamination of lower redshift sources.


We reduce this effect by removing all LBGs whose best-fit photometric redshift indicates that it might be a low redshift contaminant (photometric redshift z < 2). It is also noted that [2016ApJ...821..123H] used the contamination fraction estimated by [2015ApJ...803...34B] to correct for this effect. They found that the difference is insignificant compared with the statistical error. Thus, no further treatment is employed to correct the effect of contamination. Since linear bins are used in this work, we mainly measure the ACF at large scales, i.e. the 2-halo term. As pointed out by [2013MNRAS.429.2333J], the 2-halo term is mostly sensitive to the halo mass of central galaxies, since, at given stellar mass or luminosity bins, the satellite fraction is small. Real space parameters are obtained by applying the Limber transform to the ACFs.


The real-space correlation function, xi(r), provides three dimensional information on galaxy clustering, which is also approximated by a power law, the real-space correlation function, xi(r), is a power law defined as (r divided by r_0) to the power of -gamma, where r_0 is called the correlation length. In this case, the Limber transform takes the form [1980lssu.book.....P] beta equals gamma - 1, and the amplitude A_omega equals r_0^gamma times the Beta function B(1/2, (gamma-1)/2) times an integral over redshift z involving the redshift distribution N(z) squared, where B(x,y) is the beta function, N(z) is the redshift distribution function of sample galaxies, and H(z) is the Hubble parameter as a function of redshift. The above equations link A_omega and beta to the power law parameters in the real space. N(z) is estimated using the photometric redshifts of each LBG.


We derive the bias using the ratio between the variance of the galaxy and the matter correlation functions smoothed by a top-hat with radius 8 h^-1 Mpc: the bias b equals the galaxy variance sigma_8,g divided by the matter variance sigma_8, where [1980lssu.book.....P] the galaxy variance squared, sigma_8,g^2, equals 72 times (r_0 / 8 h^-1 Mpc)^gamma divided by [(3-gamma)(4-gamma)(6-gamma)2^gamma]. We employ the relation proposed by [2010ApJ...724..878T], which is based on a set of N-body simulations. Figure 2 shows measured ACFs and best-fit power laws for both stellar mass and luminosity subsamples. Results for the angular correlation function amplitude (A_omega), the correlation length (r_0), and bias (b) are summarised in Table 2. All quantities given in the table are the most probable value and the corresponding 1-sigma error.


Clustering segregation is observed with both stellar mass and luminosity. From Figure 2, it is clear that the ACFs increase with luminosity at all redshifts. Similar trends are also observed for different stellar mass subsamples. However, the segregation with stellar mass is found to be weaker than with luminosity. To further compare the clustering dependence of these two properties, we plot the measured bias as a function of mean stellar mass and flux in Figure 3. The bias increases with stellar mass, from b = 2.7 to 3.6 at z ~ 4, and b = 4.2 to 4.8 at z ~ 5. The segregation of bias with luminosity is more obvious. At z ~ 4, the bias for the brightest sample is b = 4.1, while that of the faintest sample is smaller at b = 2.5. Similarly, at z ~ 5, the bias has an increase between the faint and bright samples, from b = 3.8 to 5.3.


At z >= 6, the LBG sample is smaller, and uncertainties become much larger. While there is a small increase of the bias with mean UV flux at mean redshift z = 5.9, no significant segregation with stellar mass is detected. For samples at z ~ 7, we find that the bias increases from b = 7.4 to 11.7, and from b = 5.6 to 8.6, for stellar mass and luminosity bins respectively. As a comparison, we also plot the bias estimated by [2016ApJ...821..123H] from their power law fits as a function of mean UV magnitude. We find that the trends of clustering dependence on luminosity are consistent, while the offsets on biases themselves could be due to different methods of computing them. To summarise, we find that both more massive and more luminous galaxies are more highly clustered, implying that they are hosted by more massive dark matter halos.


On the other hand, the comparison of segregation between stellar mass and luminosity disagrees with our prior expectations, especially at mean redshift z = 3.8 and 5.0. In particular, we find that the clustering dependence is larger for luminosity than for stellar mass. In order to quantify this trend, we fit the measured biases of the lightest (faintest) and heaviest (brightest) by straight lines, i.e. the bias b equals the slope alpha_SM times log10(M_star / M_sun) + constant, and the bias b equals -1.1 times the slope alpha_UV times log10(L_UV / L_sun) + constant. If clustering segregation with both properties were the same, one would expect that the ratio of the slopes should recover the slope of the stellar mass versus UV magnitude relation. Therefore, we include a correction factor of -1.1 to equation 9 in order to make alpha_SM and alpha_UV comparable. The value -1.1 is based on the results in Table 1.


We combine the measured biases over all redshifts, assume the same slope but different intercepts for each redshift, and fit these five parameters using the emcee MCMC sampler developed by [2013PASP..125..306F], assuming flat priors. Best-fit results are shown in the left and middle panels of Figure 4. We illustrate the marginalised distributions of the slopes for stellar mass and luminosity subsamples as solid red and blue lines respectively in the bottom left panel of Figure 4. The median and 1-sigma percentiles of the slopes are summarised in the top right corner. We conclude that the slope in the stellar mass case is systematically smaller than in the luminosity case, which indicates larger clustering segregation with luminosity. The significance of this trend can be seen from the bottom right panel of the figure. The probability that the slope alpha_UV is greater than the slope alpha_SM is approximately 90 % (shaded region).


However, UV magnitude only probes recent starbursts of a galaxy, while stellar mass is an integrated quantity, which reflects the whole star formation history. This argument suggests that a tighter relation is expected between stellar and halo mass, and therefore, clustering segregation with stellar mass should be larger. Both observational effects and astrophysical reasons could be responsible for this discrepancy. We investigate this issue further by comparing the results at z ~ 4 with predictions from the SAM Meraxes in the next section. The Meraxes SAM is part of the Dark-ages, Reionization And Galaxy-formation Observables Numerical Simulation (DRAGONS) project. It is designed to self-consistently model galaxy formation and evolution during and after the epoch of the reionisation. Physics implemented in the model is described in [2016MNRAS.462..250M] and [2017MNRAS.472.2009Q].


In the present work, the model is run on the extended Tiamat N-body simulation [2016MNRAS.459.3025P, 2017MNRAS.472.3659P]. The simulation has 2160^3 dark matter particles, with particle mass m_p = 2.64 x 10^6 h^-1 M_sun, and side length 67.8 h^-1 Mpc. The simulation outputs 101 snapshots between z = 35 and z = 5 with time steps separated by ~ 11 Myr, and additional 63 snapshots from z = 5 to z = 1.8. The time steps of these additional snapshots are evenly spaced in dynamical time. We adopt the same parameter configuration as in [2017MNRAS.472.2009Q] for the SAM. All model stellar mass is subtracted by 0.24 dex in order to convert from a [1955ApJ...121..161S] IMF to a [2003PASP..115..763C] IMF.


The computation of photometry of model galaxies is described in [2016MNRAS.462..235L]. Progenitors of each model galaxy are treated a series of single stellar populations (SSPs), and integrated over the whole star formation history. We assume constant metallicity Z = 0.001 for the SSP. Recent hydrodynamical simulations [2016MNRAS.456.2140M, 2016MNRAS.462.3265D] predict a mass-metallicity relation that is close to this value at z ~ 4. [2017MNRAS.470.3006C] point out that intrinsic luminosities only weakly depend on the metallicity. Therefore, this assumption is valid for this work. SSPs templates are generated by starburst99 [1999ApJS..123....3L, 2005ApJ...621..695V, 2010ApJS..189..309L, 2014ApJS..212...14L] assuming the same IMF with meraxes. Modelling of the Lyman absorption due to the intergalactic medium (IGM) is critical for the simulated LBG selection.


We update the transmission curve of the IGM using a recent study from [2014MNRAS.442.1805I], which predicts a more consistent redshift distribution for selected model LBGs. In addition, we also update the dust correction using an empirical model proposed by [2015ApJ...813...21M], which is applicable at z < 4. We follow [2016MNRAS.461..176P, 2017MNRAS.472.1995P] to select model LBGs and calculate the ACFs. This approach mimics the incompleteness of the LBG sample by adding photometric scatter to the magnitudes of each LBG selection band. The level of the scatter is given by the 1-sigma field detection limits. In the present work, we assemble model LBGs according to the flux limits of the deepest field in the our observations, i.e. the XDF. The resulting redshift distribution of model selected LBGs is shown in Figure 5, which agrees with the observed one estimated by photometric redshifts.


In terms of the determination of the ACF, we compute the real-space correlation function across a sequence of snapshots directly from the spatial coordinates of each galaxy, and convert to an ACF by the Limber transform. The readers are referred to [2016MNRAS.461..176P, 2017MNRAS.472.1995P] for a more detailed description. We focus on the comparison between our model and observations at z ~ 4, where the measurements have the smallest errors. We plot predicted ACFs together with measured ACFs in Figure 4. The model ACFs agree very well with observations, and reproduce the observed clustering dependence on both stellar mass and luminosity. In order to check whether the model predicts larger clustering segregation with luminosity as indicated by our observations, we therefore calculate the bias from the model by the bias squared equals the real-space correlation function of galaxies xi(r) divided by the real-space correlation function of dark matter xi_DM(r,z), where xi(r) and xi_DM(r, z) are the real-space correlation functions of galaxies and dark matter.


An average value is taken in the range 5 Mpc <= r <= 10 Mpc. The estimated biases for the lightest (faintest) and heaviest (brightest) samples are shown using star symbols in the left (middle) panel in Figure 4. Subsequently, we derive the slope for these two points for comparison with observational data in Figure 4. Red and blue lines correspond to stellar mass and luminosity bins respectively. We conclude that the measured variation of bias with stellar mass and luminosity is consistent with predictions and that the clustering segregation with luminosity is larger than stellar mass in the model. In other words, the model predictions also contradict our expectation that stellar mass should be more tightly correlated with halo mass. The physical interpretation behind this could be complex, and we defer it to a subsequent paper. The observed clustering dependence on stellar mass could be weakened due to observational uncertainties on the estimations of stellar mass.


The uncertainties, for instance, can be due to the low S/N ratio of Spitzer data. We demonstrate this effect by adding 0.5 dex of Gaussian scatter to the stellar mass of selected model LBGs and remeasure the clustering in stellar mass bins. This level of scatter is also used in abundance matching studies [e.g.][2013MNRAS.428.3121M, 2013ApJ...770...57B]. The recalculated ACFs are shown as green lines in Figure 4. For the most massive bin, the ACF decreases at all scales, while for the other two bins, scatter in stellar mass only affects the small scale correlation functions. We also recalculate the bias and the corresponding slope. The results are demonstrated in Figure 4. In the right panel of Figure 4, the dot dashed line represents the slope in the case where scatter is added to the stellar mass, and shows that scatter reduces the slope relative to that of the original model, from alpha_SM = 1.3 to alpha_SM = 0.7.


This effect is most significant for the most massive bins, since the massive end has fewer galaxies. We can use this systematic error of model alpha_SM introduced by adding scatter to the stellar mass to estimate the effect on our measured alpha_SM. This indicates that accounting for uncertainties in stellar mass leads to clustering segregation that is similar for both mass and UV-luminosity, but which is not larger with stellar mass. Hence, although uncertainties in stellar mass can weaken the clustering dependence, the unexpected trend could still result from physical reasons. Another interesting finding from the comparison between observations and the model is the deviation of the ACFs from a power law at small scales (theta < 10 arcsec). In the model, this is due to the 1-halo term, arising from multiple halo occupation, where more than one galaxy resides in the same halo.


To provide evidence of this multiple halo occupation, we calculate the 1-halo and 2-halo terms explicitly from Meraxes by counting galaxy pairs in the same and different FoF groups. These two terms are shown as dashed and dot dashed lines in Figure 4, respectively, demonstrating that the steep increase of the ACFs at small scales is due to multiple halo occupation. Consistency is also found between observations and the model. It can also be seen that the transition of the model ACFs between the 1-halo and 2-halo terms becomes more rapid with decreasing stellar mass. However, there is no such trend in luminosity. This finding may suggest an additional feature of clustering segregation with stellar mass and luminosity, implying different satellite properties for the two cases. However, we caution that the satellite properties of Meraxes have yet to be fully explored and compared with observations, and that achieving realistic recent satellite star formation histories has traitionally been a challenging task for SAMs.


This difference can also be seen from the observed ACFs but with large uncertainties. At the very bright end, the smooth transition between 1-halo and 2-halo terms is observed in [2018PASJ...70S..11H], and explained using non-linear bias [2016MNRAS.463..270J]. Larger surveys with more complete samples might be used to investigate this phenomenon in more details for fainter and less massive galaxies. We have carried out a clustering analysis of LBGs over the range z ~ 4 - 7, with emphasis on the comparison between clustering segregation with stellar mass and luminosity. We also compare our measurements with predictions from the Meraxes SAM. Our findings can be summarised as follows: The observed ACF amplitude and bias generally increase with stellar mass and luminosity over z ~ 4 - 7. The ACFs obtained from the model are consistent with observations, and reproduce clustering segregation with both stellar mass and luminosity.


This suggests that more luminous and massive galaxies are more clustered, and hence hosted by more massive dark matter halos. By combining measurements over all redshifts, a systematic difference is found between clustering segregation with stellar mass and luminosity. In particular, it is observed that clustering strength is more tightly correlated with luminosity. This is in contrast to the expectation that stellar mass should be more tightly correlated with halo mass since stellar mass reflects the whole star formation history, while UV magnitude only corresponds to recent star formation. We find that the model also predicts this surprising result of larger clustering segregation with luminosity. At z ~ 4, the model predicts that the transition between the 1-halo and 2-halo terms of the ACFs is smoother for larger stellar mass galaxies, and that this trend does not appear for samples split by luminosity. Observations show similar behaviour but with large error bars.


This might suggest that samples split by stellar mass and luminosity have quite different satellite properties. Our results extend to higher redshift findings from the recent study at z ~ 3 by [2018A&A...612A..42D], who carried out a clustering analysis of 3236 galaxies discovered in the VIMOS Ultra Deep Survey. They measured the real space correlation functions, and also found a larger difference in the correlation lengths when splitting the sample by luminosity than by stellar mass. This unexpected trend of clustering segregation with stellar mass and luminosity may provide new clues of galaxy formation in the early universe. For instance, some high-redshift galaxies might be formed by a single starburst. In this case, stellar mass and UV-luminosity become similar indicators of the star formation history. Our study also motivates future spectroscopic surveys with the James Webb Space Telescope (JWST), which will provide more complete samples and more accurate stellar mass measurements substantially reducing the systematic errors in clustering studies.


We study the sizes, angular momenta and morphologies of high-redshift galaxies using an update of the Meraxes semi-analytic galaxy evolution model. Our model successfully reproduces a range of observations from redshifts z=0-10. We find that the effective radius of a galaxy disc scales with UV luminosity as R_e is proportional to L_UV^0.33 at z=5-10, and with stellar mass as R_e is proportional to M_*^0.24 at z=5 but with a slope that increases at higher redshifts. Our model predicts that the median galaxy size scales with redshift as R_e is proportional to (1+z)^-m, where m=1.98+/-0.07 for galaxies with (0.3-1)L*_(z=3) and m=2.15+/-0.05 for galaxies with (0.12-0.3)L*_(z=3). We find that the ratio between stellar and halo specific angular momentum is typically less than one and decreases with halo and stellar mass. This relation shows no redshift dependence, while the relation between specific angular momentum and stellar mass decreases by ~0.5 dex from z=7 to z=2.


Our model reproduces the distribution of local galaxy morphologies, with bulges formed predominantly through galaxy mergers for low-mass galaxies, disc-instabilities for galaxies with M_* is approximately 10^10 - 10^11.5 M_sun, and major mergers for the most massive galaxies. At high redshifts, we find galaxy morphologies that are predominantly bulge-dominated.


Size, angular momentum and morphology are three of the most fundamental galaxy properties, and are integral probes for understanding the structure and growth of galaxies. Observing these properties at high redshifts offers an invaluable insight into galaxy formation and evolution in the early Universe. The sizes of high-redshift Lyman-break galaxies have been measured using deep Hubble Space Telescope (HST) fields in a number of studies from z~6-12 ([e.g.][Oesch2010, Mosleh2012, Grazian2012, Ono2013, Huang2013, Holwerda2015, Kawamata2015, Shibuya2015, Kawamata2018]). They find sizes consistent with an evolution of the galaxy effective radius R_e is proportional to (1+z)^-m at fixed luminosity, with measurements of m typically in the range of 1 <~ m <~ 1.5. While observing galaxy morphologies at high-redshift is challenging ([see e.g.][Abraham2001]), observations generally find that high-redshift galaxies are more clumpy and irregular than those at low redshift ([Abraham2001, Papovich2005, Elmegreen2014]).


Several studies find that there is a larger fraction of spheroids at higher redshifts ([e.g.][Bundy2005, Franceschini2006, Ravindranath2006, Lotz2006, Dahlen2007]), though [Ravindranath2006] find that the z~2.5-5 population is not dominated by spheroids but by extended disc-like galaxies and irregulars or merger-like systems. Unfortunately, there are only a few observational studies of galaxy angular momentum at redshifts z>1 ([e.g.][Swinbank2017, Alcorn2018, Okamura2018]), since obtaining large spectroscopic samples to measure kinematics and thus specific angular momenta at high redshifts is difficult. [Okamura2018], for example, estimated the specific angular momentum of z~2, 3 and 4 galaxies from their measured disc sizes, using the analytic model of [Mo1998], and found little redshift evolution of the ratio of stellar to halo specific angular momentum. Because of the difficulty of observing these properties at high redshift, models and simulations play a necessary role for measuring and understanding their evolution in the early Universe.


In the hierarchical structure formation scenario, gas cools in the centres of dark matter haloes to form galaxies. The [Mo1998] analytical model of this process predicts that if galaxies are thin exponential discs with flat rotation curves, and specific angular momentum is conserved, the scale length of a galaxy disc R_s is given by Equation 1. Note that this model is an extension of the [Fall1983] model which assumed R_s = lambda * R_vir / sqrt(2), with J_d=J_H and M_d=M_vir. The virial radius of a dark matter halo is given by Equation 2. Hence, the simple model of [Mo1998] predicts that sizes of discs scale with redshift as (1+z)^-1.5 at fixed circular velocity, or (1+z)^-1 at fixed halo mass. Measurements of m typically lie between these two values ([e.g.][Bouwens2004, Oesch2010, Ono2013, Kawamata2015, Shibuya2015, Laporte2016, Kawamata2018]).


Semi-analytic models often use the [Fall1983] model and assume R_s = lambda * R_vir / sqrt(2) to predict the sizes of galaxies within their simulated dark matter haloes ([e.g.][Croton2006a, Mutch2016, Liu2016]). While simplistic, this model has had success in reproducing the size evolution of galaxies from z~5-9 [Liu2016]. More advanced semi-analytic models improve their disc size estimates by explicitly tracking the angular momentum of galaxy discs ([e.g.][Lagos2009, Guo2011, Benson2012, Stevens2016, Tonini2016, Xie2017, Zoldan2019]). Under the assumption that galaxy discs have a constant velocity profile and an exponential surface density profile, their specific angular momentum is j=2*R_s*V, and so their radii can be estimated as R_s=j/2V. These models generally assume that the velocity of the galaxy disc V is equal to the maximum circular velocity of the surrounding dark matter halo ([e.g.][Guo2011, Tonini2016, Zoldan2019]).


Meraxes [Mutch2016] is a semi-analytic galaxy formation model designed to study high-redshift galaxy evolution. In [Liu2016] (hereafter L16), Meraxes was used to investigate the sizes of high-redshift galaxies. This model assumed R_s = R_vir * lambda / sqrt(2), included no angular momentum tracking, and assumed that all galaxies were exponential discs for simplicity. In this paper, we use Meraxes to study the evolution of galaxy sizes, angular momentum and morphology. We implement a new method to determine galaxy sizes and velocities, introduce angular momentum tracking, and implement a bulge growth model to allow for morphological studies. This paper is organised as follows. In Section 2 we introduce Meraxes and detail these new model updates... We then make predictions for the high-redshift evolution of galaxy sizes in Section 4, angular momenta in Section 5 and morphologies in Section 6. We conclude in Section 7.


In this section we describe additions to the semi-analytic model Meraxes that allow calculation of the disc scale length, bulge mass, and angular momentum of galaxies in the model. Readers interested in the calibration and results may skip to Section 3. In this work, we apply Meraxes to two collisionless N-body dark matter simulations, Tiamat and Tiamat-125-HR. The Tiamat simulation [Poole2016, Poole2017] has a co-moving box size of (67.8h^-1 Mpc)^3, and contains 2160^3 particles each of mass 2.64x10^6 h^-1 M_sun. The high mass and temporal resolution of Tiamat make it an extremely accurate and ideal simulation for studying galaxy evolution at high-redshifts. Tiamat is used throughout this work unless otherwise specified. To investigate lower redshifts, we run Meraxes on Tiamat-125-HR [Poole2017], a low-redshift compliment to Tiamat, with the same cosmology and snapshot cadence extended to z=0.


Meraxes [Mutch2016] is a semi-analytic model specifically designed to study galaxy evolution during the Epoch of Reionization. Using the properties of the dark matter haloes from Tiamat and Tiamat-125-HR, Meraxes models the baryonic physics involved in galaxy formation and evolution analytically, including prescriptions for physical processes such as gas cooling, star formation, AGN and supernova feedback, and black hole growth. The various free parameters of the model are calibrated to reproduce certain observations, such as the evolution of the galaxy stellar mass function. This work introduces several extensions that have been made to Meraxes: i) we now split galaxies into bulge and disc components, ii) we introduce a new mechanism for black hole growth, iii) we implement a new model for determining galaxy disc sizes, iv) we use a more physical star formation prescription, and v) the angular momenta of the gas and stellar discs are now tracked.


Previously published versions of Meraxes made no attempt to decompose a galaxy into its morphological components, instead assuming that all cold gas and stars in a galaxy are contained in a disc component. Following previous semi-analytic models ([e.g.][Croton2006a, Lucia2006, Guo2011, Menci2014, Tonini2016]), we expand the model to include a second galaxy component---galaxy bulges. These are assumed to contain no gas, and have no angular momentum, for simplicity. Our model for bulge growth is analogous to that of [Tonini2016], with bulges growing during galaxy mergers and disc instabilities. As in [Tonini2016], we split galaxy bulges into two types---merger-driven and instability-driven. We assume that the instability-driven bulge is disc-like, with a mass distribution that is flattened in the disc direction, whereas the merger-driven bulge is spheroidal. These bulges are somewhat comparable to observed classical and 'pseudo' bulges.


When two galaxies merge, we first assume that the gas discs of the primary and secondary galaxies add. Galaxy mergers can induce an efficient burst of star formation. We assume that in a merger with merger ratio gamma = M_2/M_1 >= 0.01 the remnant undergoes a merger-driven starburst. This causes mass M_burst = 0.57 * gamma^0.7 * M_cold,1 to be converted from cold gas into stars [Somerville2001, Mutch2016]. Any gas that is not converted into stars during this burst remains in a gas disc. If the primary is dominated by a discy component, the mass deposition is likely to occur in the plane of the disc, and so these stars are added to the instability-driven bulge. Otherwise, the dominant mass component of the primary is the spheroidal merger-driven bulge, and so we assume that the newly formed stars accumulate in shells around it and are added to the merger-driven bulge.


If a galaxy undergoes a major merger, which we define as a merger with gamma>0.1, all stars and metals from both galaxies are placed into the merger-driven bulge of the remnant—major mergers are assumed to form pure bulge galaxies. In a minor merger (gamma <= 0.1) where the primary is disc-dominated, we assume that the stars from the secondary are added to the primary's disc. This causes a gravitational instability such that an amount of mass equal to the secondary's stellar mass is taken from the disc of the primary and placed into its instability-driven bulge [see][Tonini2016]. In a minor merger where the primary is bulge-dominated, the secondary's mass is simply added to the primary's bulge (either the instability-driven bulge or the merger-driven bulge, depending on which is dominant).


For thin galaxy discs with an exponential surface density and flat rotation curve, a disc is stable if its mass M_disc < M_crit = V_disc^2 * R_s / G [Efstathiou1982, Mo1998]. Thus, we assume that if the galaxy disc accretes enough material such that M_disc > M_crit, then it is no longer in dynamical equilibrium and an amount M_unstable = M_disc - M_crit of this disc mass is unstable. To return the disc to equilibrium, we assume that the unstable stellar mass is transferred from the disc to the instability-driven bulge. We then assume that the disc instability can drive black hole growth. After this black hole growth, the remainder of the unstable gas is consumed in a 100 per cent efficient starburst, with the stars formed added to the instability-driven bulge.


We implement a new density-dependent star formation prescription equivalent to that in [Tonini2016]. This uses the mass of cold gas above the critical density for star formation to determine how many stars are formed. The original Meraxes prescription assumed that the gas in the gas disc was evenly distributed out to a radius of 3*R_s,g. The new model improves this by assuming that the gas in the disc is exponentially distributed. The gas surface density threshold for star formation is Sigma_crit = 10 M_sun/pc^2. For gas discs with an exponential surface density profile, the total mass of gas inside the critical radius is calculated, and we assume that this amount of gas is capable of forming stars. New stars form at a rate of SFR = epsilon * M_crit / t_dyn, where t_dyn is the dynamical time of the disc and epsilon is a free parameter describing the efficiency of star formation.


The simulated properties of dark matter haloes can jump significantly between adjacent snapshots due to errors in the halo-finding process. In previous versions of Meraxes, the scale radius of the disc R_s was approximated from the radius and spin of the host halo. This means that the galaxy properties can also jump significantly between adjacent snapshots due to such errors. Clearly this is unphysical, and decoupling the galaxy properties from the instantaneous halo properties to remove these errors would be ideal. We therefore introduce a new method for determining the scale radius of galaxy discs, in which the disc is evolved incrementally in response to changes that have occurred at each snapshot. In order to do so, we consider any changes to the mass of the disc as a sum (or subtraction) of two discs: the existing disc and a pseudo-disc of material to be added (or removed), and determine the size of the resulting disc using conservation of energy and angular momentum arguments.


This method requires that we also track the vector angular momentum of both the stellar and cold gas discs. We assume that the stellar and gas discs have: i) a constant velocity profile V(r)=V (a flat rotation curve) and ii) an exponential surface density profile, as in [Mo1998]. We consider the sum (or subtraction) of two such discs. If these discs have masses M1 and M2, and velocities V1 and V2, then from conservation of energy, the new velocity is V_new = sqrt((M1*V1^2 + M2*V2^2) / (M1+M2)). Assuming conservation of angular momentum, J1+J2=J_new, the angular momentum of the combined disc is calculated by summing the individual angular momentum vectors. The magnitude of angular momentum is |J| = 2*M_disc*V*R_s. Using these conservation laws, the new disc scale length can be calculated as R_new = |J_new| / (2*V_new*(M1+M2)).


This incremental method is applied to several physical processes. For gas cooling, we assume hot halo gas settles into a disc with the velocity and specific angular momentum of the host halo. For quiescent star formation, new stars form in a disc with the properties of the parent gas disc, and their mass and angular momentum are transferred from the gas to the stellar disc. For starbursts, new stars form in the bulge with zero angular momentum, and the gas disc's radius increases to conserve angular momentum. For minor mergers, we assume the secondary's disc retains its angular momentum and is added vectorially to the primary's. In major mergers, the stellar disc is destroyed and becomes a bulge, while gas discs are combined. For supernovae, gas ejected from the disc is added back to the gas disc, while gas from the bulge is ejected to the halo and re-accretes with the halo's angular momentum. For disc instabilities and AGN feedback, mass is removed from the disc while specific angular momentum is conserved, causing the disc radius to increase.


All analysis henceforth is conducted using model galaxies with M_*>10^7 M_sun, the resolution limit of Meraxes when run on the Tiamat simulation [Mutch2016]. To ensure that our conclusions are robust, we therefore only consider galaxies classified as centrals, except in the stellar mass and luminosity functions. We leave a detailed exploration of the effect of including satellite galaxies to future work. We use the same parameter values for both Tiamat and Tiamat-125-HR, and use both simulations to tune the model: Tiamat for matching z>=2 observations and Tiamat-125-HR for z<2. The stellar and black hole mass functions from the two simulations at z=2 are converged above M_*=10^8.6 M_sun and M_BH=10^7.1 M_sun, respectively. We therefore place no significance on the trends observed with Tiamat-125-HR for lower masses.


The free parameters in Meraxes are tuned to match the observed stellar mass functions at z=8-0 and the M_BH - M_bulge relation at z=0. We calibrate to the stellar mass function at a range of redshifts, as this has been shown to provide a tight constraint on both the star formation efficiency and supernova feedback parameters ([see e.g.][Mutch2013, Henriques2013]). We use the M_BH - M_bulge relation to calibrate the black hole growth parameters; we use this relation instead of the black hole mass function as it is a more direct observable, and our model now has the capability of modelling bulge masses. We place an emphasis on matching the canonical [Kormendy2013] observations at high stellar and bulge masses. To produce a median bulge fraction that is in better agreement with local observations at the highest stellar masses, we also modify the definition of a major merger to be one with gamma>0.1, instead of gamma>0.3 as used previously.


Once the model has been tuned, we verify the changes by comparing the output to a range of local observations: the stellar mass-disc size relation, bulge fractions, and angular momentum-mass relations. We present the sizes of model galaxies using the physical effective radius, or half-light radius, R_e. We estimate this as R_e=1.678*R_s, which is the half-light radius for a disc with an exponential surface density profile. Figure 5 shows the relation between the stellar disc effective radius and stellar mass at z=0. The model agrees reasonably well with the [Dutton2010], [Lange2016] and [Lapi2018] observations, with the median matching their relations closely for 10^9 < M_*/M_sun < 10^10.5. At higher masses, our model median jumps to higher radii, though is still reasonably consistent with the observations; this jump is most likely a result of the lower sample size of high-mass galaxies.


To verify that our bulge model is producing galaxies with reasonable morphologies, we consider the median fraction of stellar mass contained within galaxy bulges as a function of total stellar mass at z=0.1. Our model predicts that the majority of stellar mass for low mass galaxies (M_* is approximately 10^9 M_sun) is contained in the bulge component, discs contribute a peak of ~60 per cent of the stellar mass at M_* is approximately 10^9.5 M_sun, and bulges again dominate at larger masses. This dip in the bulge-to-total mass ratio B/T is due to the masses at which the two bulge growth mechanisms are dominant; the merger-driven growth mode peaks for galaxies with M_* is approximately 10^9 M_sun, while the instability-driven growth mode peaks at M_* is approximately 10^11 M_sun. This suggests that high-mass galaxy discs are likely to be unstable and thus form bulges, while it is predominantly only less massive galaxies which form their bulges via mergers.


We compare our results to the SDSS observations of [Thanjavur2016] and GAMA observations of [Moffett2016]. For stellar masses of >~10^11 M_sun the model predicts a bulge fraction in remarkable agreement with the observations. For masses 10^10 < M_* < 10^11 M_sun, our model overpredicts the total bulge fraction by approximately 10 per cent compared to SDSS, but up to 40% compared to GAMA. However, the GAMA analysis does not include pseudobulges, and this mass range is precisely where we predict the instability-driven or 'pseudo' bulge to dominate, which may explain this discrepancy. The trend our model predicts for the B/T ratio is qualitatively consistent with observations such as [Conselice2006], who found that the spiral fraction is largest in galaxies with 10^9 < M_* < 10^11 M_sun. The predictions of our model are also qualitatively very similar to those of the EAGLE simulation [Clauwens2018].


In this section we investigate the success of our model at reproducing the angular momentum-mass relations observed in local galaxies. We consider only disc-dominated galaxies, with bulge-to-total mass ratio B/T<0.3. The correlation between stellar specific angular momentum j_* and total stellar mass M_* (the Fall relation) is claimed to be one of the most fundamental galaxy scaling relations. We show our predicted j_*-M_* relation at z=0 in Figure 8. We also show a range of observations, to which our model shows remarkable agreement. The [Posti2018] observations extend to low stellar masses, where they find that the j_*-M_* relation remains as a single, unbroken power-law, challenging models which predict a flattening at low masses. Our model shows no significant flattening, consistent with the [Posti2018] observations.


We predict a decreasing ratio of stellar-to-halo specific angular momentum (j_*/j_H) with decreasing stellar and halo mass. While our model contains galaxies with a wide range of j_*/j_H values, the distribution has a median lower than one, with j_*/j_H is approximately 0.5 at M_vir is approximately 10^12 M_sun. Our model therefore suggests that the majority of galaxies lose angular momentum as they evolve. Our predictions overlap with the confidence intervals of the [Lapi2018] and [Dutton2012] observations. We make different predictions to previous simulations and semi-analytic models, such as [Pedrosa2015], who find roughly no mass dependence of j_d/j_H, as does the Dark SAGE semi-analytic model [Stevens2016].


Finally, we consider the morphological dependence of the specific angular momentum-mass relation, as observations suggest that mass, angular momentum and bulge-to-total ratio are strongly correlated. We plot the j_*-M_* relation for galaxies in our model, coloured by their bulge-to-total mass ratio B/T, in Figure 10. We see a clear trend of bulge-dominated galaxies lying below the disc-dominated sequence, consistent with the [Fall2018] observations. This trend arises naturally from our assumption that the bulge component has negligible angular momentum. The Illustris [Genel2015] and Magneticum Pathfinder [Teklu2015] simulations also predict a similar correlation between galaxy type and j_*. Overall, our model reproduces the local angular momentum-mass scaling relations well.


We investigate the relationship between the sizes of stellar discs and UV magnitude of galaxies in our model from z=5-10. We find that for all UV magnitudes, brighter galaxies tend to have larger sizes. This is in agreement with the [L16] predictions, however we see a steeper slope at the brightest magnitudes. This is due to the improvements of our new disc size model. In [L16], the galaxy radius was simply related to the halo radius. By tracing the evolution of disc-sizes more physically, our galaxies sizes do not show an artificial flattening at fainter magnitudes. Our model agrees well with the [Grazian2012], [Ono2013], [Huang2013] and [Holwerda2015] observations. At redshifts z=6, 7 and 8, the best-fit relation of [Kawamata2018] agrees well at the brightest magnitudes, but our model shows a flattening in the relation at lower magnitudes that is not seen in those observations.


The relations between the stellar disc effective radius and galaxy stellar mass from z=5-10 are shown in Figure 13. Our model shows a clear trend of increasing disc size with increasing stellar mass. The [Mosleh2012] observations match well with the model predictions at z=5 and 7, while being smaller than our median at z=6. The [Holwerda2015] galaxies also match well with our predictions. We fit the relation R_e = R_0 * (M_*/M_0)^b where M_0 = 10^9 M_sun, as in [Holwerda2015]. The normalization R_0 decreases from z=5 to z=10, while the slope b increases. Our values for b are consistent with some of those derived by [Holwerda2015], though their values vary significantly depending on the observations used.


We show the predictions of our model for the redshift evolution of the stellar disc effective radius in Figure 14. We split our galaxies into two luminosity bins to compare with observations. Our predictions are consistent with these observations, but are somewhat steeper. We fit the relation with the form R_e is proportional to (1+z)^-m. We obtain m=1.98+/-0.07 for galaxies with (0.3-1)L*_(z=3), and m=2.15+/-0.05 for galaxies with (0.12-0.3)L*_(z=3). These are within 3-sigma of the [L16] values. Our predictions are also consistent with the FIRE-2 hydrodynamical simulation [Ma2018], which predicts m=1-2. Our predictions for m are higher than those derived from the individual observational data sets, which all predict 1 <~ m <~ 1.5. However, [L16] show that a combined fit to z>5 observations produces larger values of m that are consistent with our predictions.


We plot the median total stellar specific angular momentum as a function of total stellar mass from z=7 to z=2 in Figure 15. The relation evolves to higher values of j_* for lower redshifts, at a fixed stellar mass. From z=7 to z=2, the relation increases by ~0.5 dex. At lower redshifts, observations show a range of results for the evolution of the j_*-M_* relation. Our model is consistent with the predictions of Dark SAGE [Stevens2016], which finds a weak trend for galaxies of fixed mass to have lower specific angular momentum at higher redshifts. In contrast, the hydrodynamical simulation of [Pedrosa2015] finds that the j_d-M_* relations are statistically unchanged up to z~2. The j_*/j_H - M_vir relation shows no significant redshift evolution.


In Section 3.2.2 we showed that Meraxes can reasonably reproduce the morphological distribution of observed low-redshift galaxies. Here we show the Meraxes predictions for galaxy morphologies at high-redshift. In Figure 16 we show the redshift evolution of the B/T ratio as a function of total stellar mass. As redshift increases, the dip in the B/T ratio becomes shallower and shifts to lower masses. This is a result of an evolution of the masses at which the merger- and instability-driven growth modes are effective. This results in an average B/T that is higher at high redshifts, suggesting that high-redshift galaxies are more likely to be bulge-dominated than those at low redshift. This is consistent with the current observations, which suggest that the spheroidal fraction increases to higher redshifts ([e.g.][Ravindranath2006, Lotz2006, Dahlen2007, Shibuya2015]). This is supported by the IllustrisTNG hydrosimulation [Pillepich2019].


In this paper we introduce updates to the semi-analytic model Meraxes in order to investigate the evolution of sizes, angular momenta and morphologies of galaxies to high redshifts. These updates include a prescription for bulge formation and growth, tracking of angular momentum, and a new way of determining galaxy disc sizes. The model is calibrated to the observed stellar mass functions at z=8-0 and the black hole-bulge mass relation at z=0. At low redshifts, the model reproduces the observed galaxy size-mass relation well. We produce galaxy morphologies that are consistent with observations, with different bulge formation mechanisms dominating at different mass scales. We predict a specific angular momentum-mass relation that is consistent with observations, and shows no flattening at low-masses. We find that the specific angular momentum-mass relation depends on galaxy morphology, with bulge dominated galaxies lying below the disc-dominated sequence. We predict that the ratio between stellar and halo specific angular momentum is typically less than one and decreases with mass.


At high redshifts, our model predicts the following: The size-luminosity relation has a slope of beta is approximately 0.33, with a normalization that decreases with redshift. The size-mass relation has a slope b is approximately 0.24-0.28, which increases with redshift, while the normalization decreases. The median size of a galaxy disc decreases with redshift as R_e is proportional to (1+z)^-m, where m is approximately 2. The specific angular momentum-stellar mass relation evolves to higher values of j_* for lower redshifts, at a fixed stellar mass. The relation between the ratio of stellar and halo specific angular momentum j_*/j_H to stellar mass shows little evolution. Finally, galaxies at high redshifts are predominantly bulge-dominated. Meraxes can now predict a wide-range of relations that can be observed and tested by the next generation of telescopes, such as JWST.


Intergalactic medium temperature is a powerful probe of the epoch of reionisation, as information is retained long after reionisation itself. However, mean temperatures are highly degenerate with the timing of reionisation, with the amount heat injected during the epoch, and with the subsequent cooling rates. We post-process a suite of semi-analytic galaxy formation models to characterise how different thermal statistics of the intergalactic medium can be used to constrain reionisation. Temperature is highly correlated with redshift of reionisation for a period of time after the gas is heated. However as the gas cools, thermal memory of reionisation is lost, and a power-law temperature-density relation is formed, Temperature T equals T_0 times (1 + delta) to the power of (1 - gamma) with gamma approximately equal to 1.5. Constraining our model against observations of electron optical depth and temperature at mean density, we find that reionisation likely finished at redshift of reionization equals 6.8 with an uncertainty of +0.5 / -0.8 with a soft spectral slope of alpha equals 2.8 with an uncertainty of +1.2 / -1.0. By restricting spectral slope to the range [0.5,2.5] motivated by population II synthesis models, reionisation timing is further constrained to redshift of reionization equals 6.9 with an uncertainty of +0.4 / -0.5.


We find that, in the future, the degeneracies between reionisation timing and background spectrum can be broken using the scatter in temperatures and integrated thermal history. Over 13 billion years ago, when the universe was approximately 400,000 years old, it consisted mostly of a neutral atomic gas of hydrogen and helium. Over time the gas began to cool and collapse into the first stars and galaxies. Some of the radiation from these sources was energetic enough to strip electrons from the surrounding atomic gas, ionising it. This period of time, from the birth of the first stars until almost all of the atomic gas in the universe had been reionised, is referred to as the Epoch of Reionisation (EoR). The EoR is the last large-scale cosmic event to be studied in detail, and is of great interest to cosmology as it contains information about the formation processes behind the first galaxies in our universe via their effect on the intergalactic medium (IGM).


There are many unanswered questions concerning the EoR, including details of its structure, duration, and effect on subsequent galaxies. However, observations of the EoR at optical to near infrared wavelengths are made difficult by the absorption of Lyman alpha photons by neutral hydrogen, which is optically thick to this wavelength even at low concentrations. Since all wavelengths below the Lyman alpha line will eventually redshift to that wavelength, measuring the presence of neutral hydrogen using Lyman alpha optical depth probes the tail end of reionisation [2017MNRAS.465.4838G], since the absorption saturates around neutral fractions of neutral hydrogen fraction less than or approximately equal to 10 to the power of -4 [2006ARA&A..44..415F]. While saturation varies between sightlines, the strength of this absorption line prohibits direct observations deeper into the EoR. In addition, we can infer the number of free electrons, and hence ionised gas, along a sightline by measuring the Thomson scattering of CMB photons. However this is an integrated measure that cannot distinguish between reionisation histories of different durations.


The reionisation of the IGM is accompanied by a large increase in temperature, to approximately 2 x 10^4 K, followed by cooling on a cosmological timescale [1994MNRAS.266..343M, 1997MNRAS.292...27H, 2000MNRAS.318..817S, 2003ApJ...596....9H, 2009ApJ...701...94F, 2016MNRAS.460.1885U, 2017ApJ...837..106O, 2018MNRAS.477.5501K, 2018arXiv180104931P, 2018arXiv181011683O, 2018arXiv181201016G, 2019arXiv190704860W]. As a result, the thermal imprint of reionisation will last much longer than reionisation itself. IGM temperature measurements are therefore a potentially powerful way to probe the EoR, as they contain information about the reionisation history of a region that lasts long after it reionises. Modeling the temperature evolution during the EoR allows us to relate various parameters of ionisation history to IGM temperature. Comparing observations of temperature at various redshifts to the model will place constraints on the nature of the EoR and the sources driving it [2002ApJ...567L.103T, 2012MNRAS.421.1969R, 2014ApJ...788..175L, 2017ApJ...847...63O, 2019ApJ...872..101B].


Simulating the thermal history of the IGM is not a new idea, with many authors having studied IGM temperature under various assumptions of the background density and ionisation history. Instantaneous reionisation models [2002ApJ...567L.103T, 2003ApJ...596....9H, 2016MNRAS.460.1885U], radiative transfer [2018arXiv180104931P, 2018MNRAS.477.5501K], and inhomogeneous reionisation simulations [2009ApJ...701...94F, 2019ApJ...874..154D, 2012MNRAS.421.1969R] have all been utilised as a basis for IGM temperature models. Once the density field and ionising background are known, most temperature modelling follows a similar process, where photo-heating, adiabatic cooling due to structure growth and the Hubble flow, along with various cooling processes in the IGM are followed over time. The process used in our model is outlined in 2. This work uses the DRAGONS simulation suite, using the density grids from the N-body simulation Tiamat [2016MNRAS.459.3025P] and the ionising flux grids from the semi-analytic galaxy formation model Meraxes [2016MNRAS.462..250M] to determine temperature evolution over time.


The ionising flux model in Meraxes captures the inhomogeneous nature of reionisation, allowing us to study the entire thermal structure in our simulation. Importantly, Meraxes directly couples high-redshift galaxy formation to hydrogen reionisation, allowing us to constrain the properties of ionising sources using their effect on the IGM. Using the temperature to probe the EoR involves comparing our simulated temperature distribution to observations of thermal history, placing constraints on the timing of reionisation, and the sources driving it. Previous works studying IGM temperature have shown the relationships between the thermal state of the IGM and reionisation. Many of these works connected IGM temperature to statistics of the Lyman alpha forest and offered some measurements and constraints on the nautre of the EoR [2011MNRAS.410.1096B, 2013MNRAS.436.1023B, 2018arXiv180804367W, 2019ApJ...872..101B, 2018arXiv181011683O, 2019arXiv190704860W]. The hydrodynamic and radiative transfer simulations used in this manner have made it possible to make measurements of the IGM temperature, and draw connections between thermal variables and the EoR.


However these simulations are extremely computationally expensive. In order to compare these variables with a wide range of reionisation histories, a faster model is required. Using the DRAGONS simulation suite, we compare these observations to a wide range of reionisation scenarios in order to statistically constrain the global nature of the EoR. The paper is structured as follows. The DRAGONS simulations will be briefly described and the post-processing temperature model will be laid out in section 2. Overview of the model outputs is given in section 3. Our results, including an investigation of the constraints the IGM thermal history can offer and constraints from current observations, are in sections 4 and 5 respectively, before concluding in section 6. The cosmology utilised throughout this paper is the flat standard LambdaCDM from [2016A&A...594A..13P] with {Omega_m, Omega_b, Omega_Lambda, h, sigma_8, n_s} = {0.308, 0.0484, 0.692, 0.678, 0.815, 0.968}.


Meraxes couples early galaxy formation and reionisation in a spatially and temporally dependent way, tracking gas cooling, star formation, and feedback between galaxies and the IGM amongst other processes (see [2016MNRAS.462..250M] for more details). This paper utilises the 100 cMpc Tiamat simulation box [2016MNRAS.459.3025P] containing 2160^3 particles of mass 2.64 x 10^6 solar masses. We use GBPTREES merger trees [2017MNRAS.472.3659P] from redshifts z=35 to z=2. We use the fiducial parameter balues presented in [2017MNRAS.472.2009Q], apart from the variations listed in section 2.4. Meraxes includes a modified version of the excursion-set algorithm 21cmFAST [2007ApJ...669..663M] to track the progress of inhomogeneous reionisation in the simulated volume. Emissivity within an ionised bubble of radius r in Meraxes is calculated from the star formation rate within the bubble, the star formation rate within the bubble, star formation rate within the bubble(r).


The emissivity is calculated as: the emissivity bar equals the fraction of ionising photons that escape the host galaxy times the number of ionising photons produced per stellar baryon, all divided by (4/3 times pi times r cubed times (1-0.75 times the Helium mass fraction) times the proton mass), multiplied by the star formation rate within the bubble(r), where f_esc is the fraction of ionising photons that escape the host galaxy, N_gamma is the number of ionising photons produced per stellar baryon, and m_p is the proton mass. The specific intensity at the hydrogen ionisation threshold, J_21, within the region is then computed from the emissivity, the specific intensity J bar at the hydrogen ionization threshold equals (1+z) squared divided by 4 pi, times the comoving mean-free path, times the Planck constant, times the assumed spectral power-law slope, times the emissivity bar, where h_p is the Planck constant. The comoving mean-free path, lambda_mfp, is equal to the ionised bubble radius during reionisation, and limited to 30 cMpc throughout the simulation, due to the maximum scale of the excursion-set algorithm. alpha is the assumed spectral power-law slope, a free parameter in our model where a small value corresponds to a harder UV background, such that the specific intensity J at frequency nu equals the specific intensity J bar at the hydrogen ionization threshold times (the frequency nu divided by the ionization threshold of hydrogen) to the power of -alpha, where nu_HI is the ionisation threshold of hydrogen.


Meraxes includes the quasar model detailed in [2017MNRAS.472.2009Q], where radiation from quasars is included when calculating reionisation structure and feedback, using equations analogous to 1 and 2, with a spectral slope of alpha_q = 1.57. Grids of the specific intensity from both galaxies and quasars, as well as density grids, are output from Meraxes. As stated in [2017MNRAS.472.2009Q] quasars have a sub-dominant effect on hydrogen reionisation in our model, due to the low number density of these luminous objects. DRAGONS combines a mass resolution small enough to capture low-mass galaxy formation, a volume large enough to study the structure of reionisation, and an inhomogeneous reionisation model based on galaxy physics. Meraxes can simultaneously reproduce the observed stellar mass function, as well as Thomson scattering optical depth and ionising emissivity measurements with certain parameter choices [2016MNRAS.462..250M].


In this paper we introduce an IGM temperature model in order to better constrain the EoR within DRAGONS. The model is largely based on the semi-numerical approaches of [SRthesis] and [1997MNRAS.292...27H]. Using the ionising background and density grids from the DRAGONS semi-analytic framework, we calculate the temperature and ionisiation of the IGM within the simulated volume. Non-equilibrium photo-ionisation rates, Gamma_i and photo-heating rates, g_i, are calculated post-ionisation from the specific ionising intensity in Meraxes, J_21, assuming an optically thin IGM: the photo-ionization rate, Gamma_i, equals the integral from the ionization threshold nu_i to infinity of (4 pi times the specific intensity J_nu divided by (the Planck constant times nu)) times the frequency-dependent cross-section sigma_i(nu) dnu, the photo-heating rate, g_i, equals the integral from the ionization threshold nu_i to infinity of (4 pi times the specific intensity J_nu divided by nu) times (nu minus nu_i) times the frequency-dependent cross-section sigma_i(nu) dnu, where sigma_i(nu) is the frequency dependent cross-section taken from [1996ApJ...465..487V].


The ionisation state of the IGM is then governed by the following differential equation: the change in the normalized abundance of species i, d(X tilde)_i/dt, equals -Gamma_i times (X tilde)_i plus the sum over species j,k of (X tilde)_j times (X tilde)_k times the recombination rate R_jk, times (the mean baryon density rho_b bar times (1 + delta) divided by the proton mass m_p) for each species i,j,k in {HI,HII,HeI,HeII} where R_jk is the recombination rate of species j and k resulting in i, including recombination and collisional ionisation. (X tilde)_i is defined as (X tilde)_i equals (n_i times m_p) divided by ((1 + delta) times rho_b bar), for the local overdensity, delta, and cosmic mean density, rho_b. The thermal state of the IGM is governed by the balance between photo-heating, adiabatic cooling under the Hubble flow, recombination cooling, and inverse Compton cooling, as well as changes in local overdensity according to [1997MNRAS.292...27H].


The thermal state equation is: the change in temperature dT/dt equals (2 divided by (3 times the Boltzmann constant k_B times the sum over species i of (X tilde)_i)) times (the total photo-heating rate G(t) minus the total cooling rate Lambda(t, (X tilde)_i)) minus 2 times the Hubble parameter H times T plus (2T divided by (3 times (1 + delta))) times d(delta)/dt minus (T divided by the sum over species i of (X tilde)_i) times d(sum over species i of (X tilde)_i)/dt, where G is the total photo-heating rate of all species, G equals the sum over species i of g_i times (X tilde)_i, k_b is the Boltzmann constant, and H is the Hubble parameter. The cooling rate, Lambda, takes into account recombinations, collisions, bremsstrahlung, and inverse Compton cooling. We take the rates for these processes from [2015MNRAS.446.3697L]. Photo-ionisation and heating rates are calculated separately for stellar and quasar sources using the optically thin equations 4 and 5, then added together when solving equations 6 and 7.


Following [SRthesis], the coupled equations 6 and 7 are solved recursively, without assuming ionisation equilibrium, using a first order implicit integration scheme [1997NewA....2..209A, 2007MNRAS.374..493B] until they converge to a solution with the absolute value of the difference between the electron abundance at iteration k and k+1, |(X tilde)_e,k - (X tilde)_e,k+1|, is less than 10^-6 at a given attempt k, where we use the electron abundance as our convergence statistic. To improve the efficiency of our code, we adopt a variable timestep, where the timestep length is doubled for the next timestep each time a solution is found, or halved if a convergent solution cannot be found within 100 attempts. In the same manner, we follow the integrated thermal history via u, the total energy injected into the IGM per unit mass via photo-heating. This can be observed in Lyman alpha power spectra, distinct from temperature [2016MNRAS.463.2335N, 2019ApJ...872..101B] and can be used to simultaneously measure reionisation timing and amount of photo-heating that exists when only considering mean temperature.


The injected energy u is followed by simply integrating the photoheating rate over time. The value of u has been related to the small scale Jeans smoothing of the IGM, as it is dependent on the integrated thermal history throughout the EoR. We cannot calculate the Jeans scale directly in post-processing, but u provides a similar probe into the integrated thermal history throughout the EoR. In order to achieve the computational speeds required to run the model many times, we track the temperature within a 128^3 grid with cell side length approximately 800ckpc of the Tiamat 100 cMpc box; from the redshift of reionisation of each voxel, until z=4. The results for this paper use the same 10,000 (approximately 0.5%) randomly selected voxels in each box, unless otherwise stated, as a sample of the entire volume. As this is a post-processing model, the thermal state of each voxel is treated independently, although their ionising flux intensities and densities are already coupled within Meraxes and Tiamat.