The DRAGONS programme was designed specifically to study the Epoch of Reionization (EoR) and the growth of the first galaxies. It consists of two main parts: the Tiamat suite of high resolution N-body simulations, and the Meraxes semi-analytic galaxy formation model. The main Tiamat simulation consists of a (96.9 Mpc)^3 volume with enough mass resolution to resolve dark matter haloes down to approximately three times the atomic cooling mass threshold at z=5. It also possesses a high temporal resolution of ~11 Myr per output snapshot, allowing us to capture the stochastic nature of star formation and quenching during the EoR. Only trees constructed from the main Tiamat N-body simulation are utilized in this work. For more details, including information on the other simulations in the Tiamat suite, please see [Poole2016] and [Angel2016].


The Meraxes semi-analytic model runs atop the merger trees extracted from Tiamat and includes the dominant physical process thought to shape the evolution of high-redshift galaxies, including cooling, star formation, metal enrichment, satellite infall and merger events. The model also features a novel delayed supernova feedback and enrichment scheme and self-consistently incorporates the semi-numerical reionization scheme, 21CMFAST ([Mesinger2007, Mesinger2011]), in order to model reionization and its effects on galaxy evolution both temporally and spatially [Geil2015]. A detailed description of Meraxes and its design can be found in [Mutch2015]. The fiducial Meraxes model calibration (used in this work) was constructed to reproduce the observed evolution of the galaxy stellar mass function from z=5-7 ([Gonzalez2011, Duncan2014, Grazian2014, Song2015]) and to provide a reionization history consistent with the latest Planck optical depth measurements [Planck-Collaboration2015]. The model has also been shown to accurately reproduce the evolution of the observed UV LF from z=5-10 [Liu2015].


The bright end of the z=11.1 UV LF predicted by Meraxes (grey line) along with associated Poisson uncertainties (shaded region). The luminosities of our two analogue galaxies, DR-1 and DR-2, are shown as a green circle and orange square, respectively. No dust extinction has been applied to the model results. The red data point indicates observational estimated value and 1-sigma uncertainty derived from the GN-z11 detection. Also shown are the z=6 and z=8 LFs predicted by the model, along with observational results from [Bouwens2015] which have been dust corrected using the inverse of the methodology outlined in [Liu2015]. The agreement between the model and observations demonstrates the success of our model at reproducing the normalization and evolution of high-z UV LFs. In Fig. 1 we present the bright end of the z=11.1 UV LF predicted by Meraxes (grey line and with shaded Poisson uncertainties). The effects of dust extinction are not included. The orange square and green circle indicate the magnitudes of the two most UV-luminous galaxies in our model whilst the red data point indicates the observationally estimated phi and 1-sigma uncertainty derived from the GN-z11 detection.


As can be seen, these two most luminous Meraxes systems are comparable in luminosity to GN-z11. The properties of the GN-z11 analogues, DR-1 and DR-2, at both z=11.1 and 5. For comparison, the observationally determined values for GN-z11 itself are also shown [Oesch2016]. For reference, we also show in Fig. 1 the z=6 and 8 UV LFs predicted by Meraxes as well as the observational results of [Bouwens2015]. The observations have been corrected for dust extinction using the inverse of the methodology described in [Liu2015]. The agreement between the observational data and our model results indicates that Meraxes correctly predicts both the normalization and evolution of the bright end of the UV LF at lower redshifts, giving us confidence in the z=11.1 result.


Due to the limited volume of the parent Tiamat N-body simulation, the bright end of the LF is poorly constrained for number densities <~10^-5 Mag^-1 Mpc^-3. This precludes us from making any quantitative statements regarding the probability of detecting a galaxy as bright as GN-z11 other than to simply state that we find at least one such system in our ~1x10^6 Mpc^3 volume. However, we note that this is approximately equal to the volume surveyed during the detection of GN-z11 (~1.2x10^6 Mpc^3) and therefore the presence of any galaxies as bright as this observed object in our model is significant. Given the success of Meraxes in reproducing the evolution of the galaxy stellar mass functions [Mutch2015] and observed UV LFs [Liu2015] at z=5-8, the presence of any systems as luminous as GN-z11 in our model is remarkable, opening up the possibility that such galaxies may not be as rare as extrapolations of the observed z~4-8 LFs to higher redshifts may suggest ([see][and references therein]Oesch2016).


Having established that galaxies as UV-luminous as GN-z11 exist in the output of DRAGONS, we selected the two brightest UV magnitude galaxies at redshift z=11.1 for detailed study. The properties of these objects, hence forth referred to as DR-1 and DR-2, show good agreement with the best observational measurements of GN-z11 [Oesch2016], as can be seen in Table 1. This suggests that we can use these model galaxies as analogues with which to investigate the potential formation, evolution and fate of GN-z11.


The observed frame SED of DR-1 (green) and DR-2 (orange) in units of flux density. Lyman-alpha absorption has been included, however, no dust extinction model has been applied. Red points show the observed GN-z11 HST photometric measurements and upper limits. The black thin line indicates the best-fitting SED for GN-z11 [Oesch2016]. Both model analogue galaxies have a spectral shape in good agreement with the GN-z11 measurements. The normalization of both model spectra further show reasonable agreement with GN-z11 photometric measurements due to their comparable UV luminosities. As well as those properties listed in Table 1, DR-1 and DR-2 also possess similar SEDs to GN-z11. In Fig. 2 we plot the observed frame SEDs of DR-1 (green) and DR-2 (orange) in terms of their flux density between 0.2 and 6 micrometers. The red data points show the GN-z11 HST photometric measurements and upper limits, whilst the black thin line indicates the best-fitting SED presented in [Oesch2016].


Lyman-alpha absorption has been included in the model spectra and is manifested as the sharp drop in flux at wavelength <~1.1 micrometers; however, no dust extinction model has been applied. For more details on the methodology used to construct the model SEDs, see [Liu2015]. Both analogue galaxies possess spectra and UV slopes (beta) in close agreement with the GN-z11 observations, supporting the claim that this observed system is relatively dust free [Oesch2016]. The normalization of both model spectra further show reasonable agreement with GN-z11 photometric measurements due to their comparable UV luminosities (cf. Table 1).


The UV magnitude versus stellar mass distribution of Meraxes galaxies at z~11. The thick black line and shaded region indicate the median and 95 pc confidence intervals of the distribution as a function of stellar mass. The top (right) panels indicate the marginalized log-probability distribution of stellar mass (UV magnitude) values. The red point with error bars indicates the position of GN-z11 in this plane, whilst the grey squares show the model analogue galaxies, DR-1 and DR-2. Both analogue galaxies (as well as GN-z11) are rare outliers from the main distribution, but are approximately consistent with an extrapolation of the median relation from lower masses. DR-1 and DR-2 are the two most massive galaxies in the simulation at the redshift at which they were selected (z=11.1) and are hosted by the two most massive subhaloes. They are also rare outliers from the majority of the model galaxy population in terms of their stellar masses, star formation rates and UV luminosities. However, they remain broadly consistent with the mean trends displayed by galaxies at lower luminosities/masses, suggesting that their history is not particularly special or unique.


As an example, in Fig. 3 we present the distribution of all Meraxes z=11.1 galaxies in the UV magnitude versus stellar mass plane. The positions of DR-1 and DR-2 are shown as grey squares, whilst GN-z11 is indicated by the red point with error bars. Although these three objects lie out with the bulk of the main distribution, they are roughly consistent with the median M_1600-M_* trend extrapolated from lower masses.


The detection of such a massive and luminous galaxy at z~11 raises a number of interesting questions. How do such massive systems form so rapidly? Is their extreme brightness merely a transient feature of their evolution brought on by a merger or other significant event? If not, then how can the level of star formation required to produce new, UV luminous stars be maintained? In this section we address these questions using our model analogue galaxies and further explore what the eventual fate of such objects is at lower redshifts.


Property histories for the model analogue galaxies, DR-1 (green) and DR-2 (orange): (a) rest frame intrinsic absolute UV magnitude (1600 Angstrom); (b) star formation rate; (c) stellar mass; (d) subhalo mass; (e) cold gas fraction; (f) half light radius assuming a pure-exponential disc; (g) merger baryonic mass ratios for all galaxy merger events with ratios >0.01 (mergers of lower mass ratio are indicated by tick marks on the x-axis). The red dots with error bars show the best estimates of the properties of GN-z11 [Oesch2016], whilst the vertical dashed line indicates z=11. For comparison, thin grey lines and dots show the histories of the ten next-most-luminous galaxies selected at the same redshift as our analogues (z~11). Fig. 4 shows the full formation histories of DR-1 (green) and DR-2 (orange) for a number of properties from z=18 to 5. The corresponding measured values for GN-z11 are shown as red points with error bars where available.


As with the UV luminosity and stellar masses discussed above, we see close agreement between our model analogues and GN-z11 in terms of star formations rates (panel b) and disc sizes (panel f). The UV luminosities of both DR-1 and DR-2 continually, but extremely rapidly, increase from high redshift (panel a). The formation (or half-mass) redshift of each z=11.1 analogue's dark matter subhalo is z~11.4 and 11.9, respectively. For comparison, theoretical expectations using extended Press-Schechter theory [Bond1991] are for these two haloes to have formed far earlier, at z~12.1, and hence to have grown at a much slower pace [Tacchella2013, Trenti2015]. In our simulation, the host subhalo of DR-1 grows by a factor of approximately 5 in just 65 Myr (panel c) immediately preceding z=11.1, resulting in a growth in stellar mass of a factor of 9 during the same period.


A visual inspection of the evolution of the local environment of DR-1 indicates that this period of rapid growth coincides with infall into a filamentary like structure. During this time, the host subhalo is subjected to multiple minor mergers (of haloes with no stellar mass), as well as significant smooth accretion below the resolution limit of the simulation. It is also immediately apparent from Fig. 4, however, that the extreme UV brightness of DR-1 and DR-2 are not transient features of their evolution; the star formation rates and UV luminosities of both analogue systems remain high or increasing throughout their evolution. For comparison, thin grey lines show the evolution of the 10 next-most-luminous galaxies selected at the same redshift (z=11.1). These luminous counterpart galaxies all possess similar growth histories, with high star formation rates (panel b) driven by continual growth of their host haloes (panel d) and associated accretion of fresh gas. However, their star formation is less sustained than that of DR-1 and 2, with more numerous periods of inactivity.


The distribution of model galaxy UV magnitudes (M_1600) as a function of subhalo mass M_vir at z~11 (top row). The right-hand panel shows a zoom in scatter plot of the highest M_vir (/M_1600) region. Again, both GN-z11 analogue galaxies are outliers of the main population. However, by z=5 (bottom row), this is no longer the case. Whilst these galaxies remain some of the most UV luminous and massive, they have regressed towards the mean trend and there now exists a small number of other, more extreme systems.


So how can DR-1 and 2 achieve and sustain such a rapid stellar mass growth? High star formation rates should result in large amounts of supernova energy being deposited into the interstellar medium, heating and ejecting cold gas and thus curtailing star formation until this material is replenished. Indeed, this is the dominant mechanism by which star formation is regulated in Meraxes [Mutch2015]. In panel (e) of Fig. 4 we show the cold gas fraction, M_cold / (m_cold + M_*), of our ten luminous comparison galaxies in grey. Many show the expected large drops in cold gas fraction caused by star formation events which deplete gas, both by converting it to stars and ejecting it from the galaxy through supernova feedback. Occasionally these low gas fractions remain for tens of millions of years, but more often gas reserves are quickly replenished by accretion from the IGM. By comparison, our GN-z11 analogues, DR-1 and DR-2, show typically fewer major depletion events.


This is due to the large amounts of cold gas available to these galaxies which results in any one star formation episode removing only a small fraction of the available material. In panel (g) we show the baryonic merger ratios (m_baryon_satellite / m_baryon_central) of all merger events with mass ratios greater than 0.01 for both DR-1 and DR-2, as well as our ten comparison objects. The horizontal grey line indicates a ratio of 1/3, above which we deem the merger to be a major event. The ticks on the lower x-axis indicate merger events with ratios below the range of the plot. Interestingly, we see that neither DR-1 nor DR-2 has experienced a merger with a baryonic mass ratio greater than ~0.1. We further find that none of the luminous comparison galaxies has experienced a major merger prior to z~11.


Although these results refer to galaxy merger events (as opposed to mergers between haloes which may potentially be devoid of stars), we confirm that >90% of the z<11.1 halo mass growth of all plotted objects is driven by smooth accretion of systems below the halo mass resolution limit of our input N-body simulation (~1.4x10^8 h^-1 M_sun). Such quiescent merger histories are an important aspect of the successful growth of GN-z11 analogues in Meraxes. Within our model, and as suggested by hydrodynamical simulations ([e.g.][Cox2008, Powell2013, Kannan2015]), major merger events induce shocks and instabilities which in turn result in efficient star formation events. Such events consume large fractions of the cold gas of both progenitors with the resulting supernova feedback ejecting even more material from the system.


Although both DR-1 and 2 have plentiful and continuous replenishment of cold gas from the IGM, it is likely that numerous major merger events would have curtailed the eventual z~11 stellar mass and would certainly have introduced more variability into the star formation histories of these objects, reducing their duty cycles and limiting their probability of detection.


As well as using the DRAGONS framework to explore the origins of GN-z11 analogues, we can also exploit the realistic galaxy populations it provides to explore the eventual fate of these rare objects. From Fig. 4, we can see that the haloes which DR-1 and 2 occupy continue to grow steadily to z=5. This steady growth brings with it fresh gas, allowing for a slowly increasing star formation rate and UV luminosity. As such, these galaxies remain the most luminous and massive of the twelve objects shown in the figure (selected to be the twelve most UV-luminous objects at z=11.1). However, when compared to the full galaxy population at z=5, our two GN-z11 analogues are no longer the brightest or most massive in the simulation volume. Such a regression of extreme objects towards the mean at later times is an expected feature of structure formation [Trenti2008].


In Fig. 5 we show the absolute UV magnitude versus subhalo mass distribution for the full galaxy population at z=11.1 (top) and z=5 (bottom). The panels on the right-hand side show a zoom in scatter plot of the highest M_vir (/M_1600) regions. As with Fig. 3 above, the thick black line indicates the median of the distribution, with the shaded region showing the 95 pc confidence intervals. Grey squares indicate the positions of DR-1 and DR-2. At z~11 both galaxies lie in the most massive subhaloes and are clear outliers from the bulk of the galaxy population. At z=5, this is no longer the case. Not only have both systems rejoined the tail of the main distribution, but there are several systems which are both more massive and luminous. Our model therefore predicts that although GN-z11 may be a highly biased, rare galaxy in the redshift z~11 Universe, it is likely the progenitor of more common massive galaxies in the post-reionization epoch. For reference, we present the full z=5 properties of DR-1 and DR-2 in Table 1.


The rest-frame intrinsic UV apparent magnitude history of DR-1 (green) and DR-2 (orange) along with the 8-sigma detection threshold (grey horizontal line) corresponding to the example future wide-field JWST/NIRCAM survey presented in [Mason2015]. This survey design corresponds to an 800 hour total exposure time split amongst multiple pointings and bands, covering a total survey area of 4000 arcmin^2. See Section 4 for more details. Our model predicts that future JWST surveys such as this may be able to detect the progenitors of GN-z11 analogues out to z~13-14, corresponding to the early stages of reionization. GN-z11 represents the highest redshift galaxy found to date, and pushes the boundaries of what is technically achievable using HST. JWST will possess the capability to go approximately two magnitudes deeper in the rest frame UV at high redshift, opening up the possibility of detecting more GN-z11 type systems at z~11 [Waters2016], as well as their progenitors at higher redshifts.


In Fig. 6 we present the rest-frame apparent UV magnitude histories of DR-1 and DR-2 from z=5-20 (green and orange lines, respectively). No dust extinction has been included, in agreement with the observed blue UV continuum slope of GN-z11 [Oesch2016]. The grey horizontal line indicates the 8-sigma detection limit for the example wide-field JWST/NIRCAM survey presented in [Mason2015]. This survey design corresponds to a total of 800 hours of exposure time divided up amongst 400 pointings in 5 bands, and covering a total area of ~4000 arcmin^2. Further details can be found in [Mason2015]. The total comoving volume of this example survey is (178.4 Mpc)^3 at z=11.1. This is approximately 6 times the volume of our Tiamat simulation, and ~90% of that of the much larger BlueTides simulation [Feng2015], within which it has been recently shown that approximately 30 systems of similar or brighter luminosity than GN-z11 are found [Waters2016].


Such a survey should therefore provide both a volume and sensitivity to easily detect multiple GN-z11 analogue systems at z~11. Our model further suggests that the progenitors of such systems may remain luminous enough to be detected out to z~13-14, corresponding to the early stages of Reionization when the Universe was less than ~10% ionized ([e.g.][]Greig & Mesinger in prep.[Kuhlen2012, Mutch2015]).


In this work, we identify two GN-z11 analogue systems from the results of the semi-analytic model, Meraxes, created as part of the DRAGONS programme. The presence of these two analogues in a simulated volume approximately equal to that of the observational survey volume supports claims by other authors [Waters2016] that galaxies as luminous as GN-z11 are more common than extrapolations of z~4-8 UV LFs would suggest [Oesch2016]. Using the detailed properties and full formation histories afforded to us by Meraxes, we have investigated the formation, evolution and fate of our two model analogue systems (DR-1 and 2). Our results can be summarized as follows.


Both DR-1 and 2 possess stellar masses, UV luminosities, star formation rates, sizes and SEDs, which are in close agreement with the measured values of GN-z11. These analogues are the most UV-luminous and massive systems in our simulated volume at z=11 and are rare outliers from the bulk of the galaxy population. However, despite their extreme nature, their properties are broadly consistent with extrapolations of the mean trends at lower masses. The extreme UV brightness of our analogues at z=11 is not a transient feature of their histories. Instead both their luminosities and stellar masses increase relatively smoothly, but extremely rapidly, from higher redshifts and continue to increase for the remainder of their evolution to z=5.


The sustained and increasing SFRs of DR-1 and DR-2 are due to rapid growth of their host dark matter haloes, driven predominantly by smooth accretion of objects below the mass resolution limit of our simulation and bringing with it an ample supply of fresh cold gas to fuel star formation. Another important feature of these growth histories is a notable lack of major mergers which would have caused a significant star burst, ejecting a large amount of cold gas and temporarily stalling star formation. Despite being the most extreme and rare outliers of the full galaxy population at z~11, by z=5 neither analogue is the most massive nor UV-luminous system in the simulation. Future, wide-field surveys with JWST will likely be able to identify the progenitors of GN-z11 type galaxies out to z~13-14. The potential ability of JWST to detect GN-z11 progenitors all the way out to z~14 will push the frontiers of galaxy-formation observations to the early phases of cosmic reionization and provide a valuable glimpse of the first galaxies to reionize the Universe on large scales.


This research was supported by the Victorian Life Sciences Computation Initiative (VLSCI), grant ref. UOM0005, on its Peak Computing Facility hosted at the University of Melbourne, an initiative of the Victorian Government, Australia. Part of this work was performed on the gSTAR national facility at Swinburne University of Technology. gSTAR is funded by Swinburne and the Australian Governments Education Investment Fund. AM acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 638809 AIDA). PO and GDI acknowledge the support of NASA grant HST-GO-1387 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. The DRAGONS research programme is funded by the Australian Research Council through the ARC Laureate Fellowship FL110100072 awarded to JSBW.


We investigate high-redshift galaxy sizes using a semi-analytic model constructed for the Dark-ages Reionization And Galaxy-formation Observables from Numerical Simulation project. Our fiducial model, including strong feedback from supernovae and photoionization background, accurately reproduces the evolution of the stellar mass function and UV luminosity function. Using this model, we study the size--luminosity relation of galaxies and find that the effective radius scales with UV luminosity as the effective radius is proportional to UV luminosity to the power of 0.25 at redshift of approximately 5 to 9. We show that recently discovered very luminous galaxies at redshift of approximately 7 [2016arXiv160505325B] and redshift of approximately 11 [2016ApJ...819..129O] lie on our predicted size--luminosity relations. We find that a significant fraction of galaxies at redshift greater than 8 will not be resolved by JWST, but GMT will have the ability to resolve all galaxies in haloes above the atomic cooling limit. We show that our fiducial model successfully reproduces the redshift evolution of average galaxy sizes at redshift greater than 5. We also explore galaxy sizes in models without supernova feedback.


The no-supernova feedback models produce galaxy sizes that are smaller than observations. We therefore confirm that supernova feedback plays an important role in determining the size--luminosity relation of galaxies and its redshift evolution during reionization. The evolution of galaxy size during the Epoch of Reionization (EoR) provides an additional probe for understanding galaxy formation in the early Universe. In the hierarchical structure formation scenario [1978MNRAS.183..341W], dark matter haloes form first, then baryonic gas cools and falls into their potential wells of to form galaxies. Within this scheme, [1980MNRAS.193..189F] studied the formation of galaxy discs. In this model, the spin of a rotationally supported galaxy disc originates from the conservation of angular momentum during the collapse of cooling gas. Further analytic modelling by [1998MNRAS.295..319M] provided a relation between the disc scale length of a galaxy, R_d, and the virial radius of its dark matter halo, R_vir for infinitesimally thin discs with exponential surface density profiles.


The disc size can be written as: The disc scale length, R_d, is given by the halo spin parameter lambda divided by the square root of 2, multiplied by the ratio of the disc's angular momentum fraction (j_d) to its mass fraction (m_d), all multiplied by the halo's virial radius, R_vir. Here, m_d and j_d are the fraction of mass and angular momentum in the disc relative to the halo and lambda is the spin parameter of the halo. The virial radius of a dark matter halo scales with redshift and virial velocity, V_vir, or virial mass, M_vir, as: The virial radius, R_vir, scales with the virial mass, M_vir, and the Hubble parameter, H(z), or can be expressed as the virial velocity, V_vir, divided by 10 times the Hubble parameter., where H(z) is the Hubble parameter, and the Hubble parameter, H(z), is proportional to (1+z) to the power of 3/2 at high redshifts [1992ARA&A..30..499C]. Therefore, from the first equation, the proportionality of R_d with R_vir predicts that the sizes of discs scale with redshift as (1+z) to the power of -3/2 at fixed circular velocity, or (1+z) to the power of -1 at fixed halo mass.


Caption for Table 1: Observed evolution of galaxy sizes, R_e is proportional to (1+z) to the power of m from literature, where L*_z=3 corresponds to UV magnitude M_UV=-21.0. Observations of Lyman break galaxies (LBGs) show that galaxies are more compact at higher redshift, and that average sizes evolve with redshift as (1+z) to the power of -m with m of approximately 1 to 1.5 [e.g.][2004ApJ...600L.107F, 2004ApJ...611L...1B, 2010ApJ...709L..21O, 2012A&A...547A..51G, 2013ApJ...777..155O, 2015ApJ...804..103K, 2015ApJ...808....6H, 2015ApJS..219...15S]. Semi-analytic models have had considerable success studying the formation and evolution of galaxies in the past two decades [e.g.][1991ApJ...379...52W, 1993MNRAS.264..201K, 2000MNRAS.319..168C, 2006MNRAS.365...11C, 2006MNRAS.370..645B, 2011MNRAS.412.1828L, 2015arXiv150908473L]. Galaxy sizes are important for semi-analytic models since the cold gas is assumed to settle in discs where star formation occurs at a rate depending on the surface density [e.g.][2006MNRAS.365...11C]. Reproducing the evolution of galaxy sizes in the early and dense Universe is therefore important for semi-analytic models of reionization.


On the other hand, feedback mechanisms are already known to play an important role in suppressing star formation in galaxies. Using the observed size evolution and the luminosity function of galaxies, [2011MNRAS.413L..38W] presented a simple model to constrain the feedback mechanism using galaxy sizes: The effective radius, R_e, is proportional to luminosity L raised to the power of 1 divided by 3 times (1+a), and also proportional to (1+z) to the power of -m. Here L is the galaxy luminosity, and a and m are free parameters which can be constrained using both the slope of the galaxy luminosity function and galaxy size evolution. Feedback arising from energy release and momentum outflow could affect the luminosity at fixed disc sizes. Based on the observed relation between size, luminosity and redshift, [2011MNRAS.413L..38W] ruled out the no-supernova feedback model with high confidence, and suggested a supernova feedback model through the transfer of momentum. Here we improve on this analysis using a more realistic semi-analytic model.


Investigation of galaxy sizes using semi-analytic models have previously been made using galaxies in both the local and high redshift Universe [e.g.][2000MNRAS.319..168C, 2009MNRAS.397.1254G, 2010MNRAS.405..948S, 2015MNRAS.447..636X, 2016MNRAS.461..859S, 2016MNRAS.459.4109T]. Our purpose-designed semi-analytic model provides a tool to study galaxy sizes during the EoR. The semi-analytic model, MERAXES [described in][hereafter Paper-III]{2016MNRAS.462..250M}, is a new purpose-built galaxy formation model designed for studying galaxy evolution during the EoR. The MERAXES model is a part of the Dark-ages Reionization And Galaxy-formation Observables from Numerical Simulation (DRAGONS) project. MERAXES includes a temporally and spatially coupled treatment of reionization, and is built upon the high resolution and high snapshot-cadence N-body simulation Tiamat [hereafter Paper-I]{2016MNRAS.459.3025P}. MERAXES successfully reproduces a series of high-redshift galaxy observables including the stellar mass function (Paper-III) and UV luminosity function [hereafter Paper-IV]{2016MNRAS.462..235L}. In this paper, we run simulations to investigate the size--luminosity relation, the size--stellar mass relation and the redshift evolution of galaxy sizes at redshifts between 5 and 10. We aim to use the evolution of galaxy sizes to probe the physics of galaxy formation during the EoR.


The galaxy formation model used in this work is MERAXES (Paper-III). MERAXES is implemented upon dark matter halo merger trees generated from the cosmological N-body simulation Tiamat (Paper-I). Tiamat and MERAXES have special features designed for the study of reionization. The collisionless N-body simulation, Tiamat, was run using a modified version of GADGET-2 [2005MNRAS.364.1105S] and the Planck 2015 cosmology [2015arXiv150201589P]. It includes 2160 cubed particles in a comoving 100 Mpc cube box. The mass of each particle is 2.64 times 10 to the power of 6 h inverse solar masses, which allows us to identify the low mass dark matter haloes close to the hydrogen cooling limit across the redshifts relevant to reionization. Dark matter halo finding was carried out using SUBFIND code [2001MNRAS.328..726S]. This code first identifies dark matter collapsed regions by a friends-of-friends (FoF) algorithm using a link length criterion of 0.2 times of the mean inter-particle separation.


A FoF group typically contains a central halo holding most of the virial mass and a group of lower-mass subhaloes which trace the undigested parts of merger events. Tiamat outputs include 100 snapshots from z=35 to z=5 with a temporal resolution of 11 Myr per snapshot. This high cadence resolves the dynamical time of galaxy discs at high redshift, and is comparable to the lifetime of massive stars. Dark matter halo merger trees constructed from Tiamat are stored and processed in a horizontal form. This allows the semi-analytic model to implement a self-consistent calculation of feedback from reionization on low mass galaxy formation. This is achieved by incorporating the semi-numerical reionization algorithm 21cmFAST [2011MNRAS.411..955M] at each snapshot. MERAXES is a new semi-analytic model based on [2006MNRAS.365...11C] with updated physics for application to redshift greater than 6. It consists of baryonic infall, gas cooling, star formation, stellar mass recycling, metal enrichment, galaxy mergers, gas stripping, and feedback from both supernova and reionization.


To model the formation and evolution of galaxies during the EoR, MERAXES incorporates several improvements in the feedback scheme. Firstly, it considers a delayed supernova feedback mechanism. In an instantaneous feedback scheme, a massive star instantly produces a supernova and so releases energy and mass within the same snapshot that the progenitor star formed. This is appropriate at low redshift, where the stellar lifetime is short compared to the galaxy dynamical time. However, our Tiamat merger trees have a much higher time resolution of approximately 11 Myr, which is shorter than the lifetime of the least massive Type II supernova progenitor stars (e.g., approximately 40 Myr for 8 solar mass stars). Therefore, MERAXES implements a delayed supernova feedback scheme, where a supernova may explode several snapshots after the progenitor star formed. MERAXES also includes feedback from a spatially and temporarily variable ultraviolet background (UVB). The UVB radiation heats the intergalactic medium and reduces baryonic infall within small dark matter haloes suppressing both gas cooling and star formation.


To achieve this, MERAXES integrates the semi-numerical code 21cmFAST [2011MNRAS.411..955M] to construct the reionization structure. We assume a standard [1955ApJ...121..161S] initial mass function (IMF) with stellar mass in the range of stellar mass in the range of 0.1 to 120 solar masses: The initial mass function phi(m_*) is proportional to stellar mass m_* to the power of -2.35. The free parameters in MERAXES were calibrated to replicate the observed stellar mass functions at redshift of approximately 5--7 [2011ApJ...735L..34G, 2014MNRAS.444.2960D, 2015A&A...575A..96G, 2016ApJ...825....5S] and the Planck optical depth to electron scatting measurements [2015arXiv150201589P]. For a more detailed description of MERAXES, see Paper-III. In our semi-analytic model, we adopt the disc scale radius from [1998MNRAS.295..319M] and the standard assumption that the specific angular momentum of the material forming the disc is the same as that of the host halo [1980MNRAS.193..189F].


The spin parameter, lambda, is calculated from the N-body simulation using the definition from [2001MNRAS.321..559B]: The spin parameter lambda is defined as the halo's angular momentum, J_vir, divided by the product of the square root of 2, the virial mass M_vir, the virial velocity V_vir, and the virial radius R_vir., where M_vir and J_vir are the mass and angular momentum enclosed within the virial radius, and V_vir is the circular velocity at R_vir [see][for a discussion of spin parameters for haloes in Tiamat]{2016MNRAS.459.2106A}. The disc scale equation was obtained assuming a simple model in which dark matter haloes have singular spherical isothermal density profiles and the gravitational effects of baryonic discs are neglected. It is therefore important to note that inclusion of gravity from the disc may alter the size and rotation curve of galaxies and modify the dark matter concentration in the inner region of the halo. However, [1998MNRAS.295..319M] showed that a more realistic model with NFW halo profiles [1997ApJ...490..493N] and self-gravitating discs results in only minor modifications to the equation.


Simulations also show that inclusion of the self-gravity of discs will lead to instabilities of gas and stars, which drives disc material towards the centre of galaxies and results in instability-driven star bursts and bulge growth in galaxies. This will have an impact on the distribution of disc sizes [e.g.][2000MNRAS.319..168C, 2006MNRAS.370..645B, 2016MNRAS.461..859S, 2016MNRAS.459.4109T]. Another significant assumption in the model is that the specific angular momentum fraction is 1, since lost angular momentum in the gas component during galaxy assembly would lead to a smaller disc. On the other hand, strong feedback mechanisms which release the energy and angular momentum to the interstellar medium will suppress the formation of small discs. To quantify these effects in semi-analytic models, [2016MNRAS.461.3457G] compared galaxy sizes from semi-analytic models L-galaxies and Galform at redshift less than 2. Galform includes the self-gravity of discs while L-galaxies ignores it.


[2016MNRAS.461.3457G] showed that self-gravity does not significantly affect the sizes of galaxies with stellar mass less than 10 to the power of 9.5 solar masses. However, for galaxies with stellar mass greater than 10 to the power of 9.5 solar masses, self-gravity of discs in Galform reduces galaxy sizes and results in a decreasing size--mass relation. In this work, which considers the small galaxies that drive reionization, we do not have a large number of galaxies with stellar mass greater than 10 to the power of 9.5 solar masses at redshift greater than 6. Thus, we utilize the simple model of [1998MNRAS.295..319M] in this study, as has been common in semi-analytic models [e.g.][2006MNRAS.365...11C, 2007MNRAS.375....2D]. We see that the disc sizes of galaxies are determined by the properties of dark matter haloes. We assume star formation and feedback processes do not directly modify the disc sizes. On the other hand, the size of the disc does play a fundamental role in the build up of stellar mass.


Based on the observational work of [1998ApJ...498..541K], the global star formation rate of spiral galaxies can be related to the surface density of cold gas above a given threshold. In our model, we adopt a critical surface density for the disc, above which gas cannot maintain stability and will start forming stars. The critical density at a radius r is adopted from [1996MNRAS.281..475K]: The critical surface density Sigma_crit at a radius r is proportional to the virial velocity and inversely proportional to the radius, with units of solar masses per square parsec. where Sigma_norm = 0.2 is a free parameter in MERAXES. Stars are assumed to form within a maximum radius set to 3 times the disc scale length, based on the properties of the Milky Way [2000glg..book.....V]. By integrating the critical density, we obtain the critical mass of the disc. If the mass of cold gas in the disc exceeds this threshold mass the stars will form with a star formation rate given by: The star formation rate is given by the star formation efficiency alpha_SF, multiplied by the difference between the cold gas mass and the critical mass, all divided by the disc's dynamical time.


Galaxy mergers can also trigger a strong burst of star formation. We assume a fraction of the total cold gas of the newly formed system is consumed during such a burst [2001MNRAS.320..504S]. The fraction of cold gas consumed is determined by: The fraction of cold gas consumed in a merger-driven burst, e_burst, is determined by the mass ratio of the merging galaxies raised to the power of gamma_burst, and multiplied by an efficiency factor alpha_burst., where the mass ratio of merging galaxies, and alpha_burst = 0.56 and gamma_burst = 0.7 are chosen to fit the numerical results of [2004ApJ...607L..87C] and [1994ApJ...431L...9M, 1996ApJ...464..641M] for merger mass ratio in the range 0.1 to 1.0 [2006MNRAS.365...11C]. For simplicity, we assume the merger-driven burst occurs within a single snapshot, which is comparable to the disc dynamical time of the majority of galaxies. We do not consider irregular morphologies during galaxy mergers and the sizes of remnants are calculated using the standard disc size equation.


Through the star formation process, disc size affects a number of galaxy properties, including UV luminosities. The size--luminosity relation therefore becomes an important predictor from galaxy formation models. We note that the star forming process is rather complicated. It is not only determined by the galaxy sizes but also by other effects including cooling, mergers and feedback. To study the role of supernova feedback in the build up of the size-luminosity (stellar mass) relation, we also run a simulation with the supernova feedback switched off. This no supernova model cannot reproduce the stellar mass function in detail, but is recalibrated to provide the observed stellar mass density at z = 5 (see Paper-III). In this paper, to compare with observations we present the sizes of model galaxies using the physical effective radius (i.e. half-light radius), R_e, within which half of the galaxy's luminosity originates. Here R_e is estimated using R_e is equal to 1.678 times R_d, where the constant originates from the assumed exponential surface density profile and constant mass-to-light ratio.


Luminosity is the most direct observable of high-redshift galaxies. We calculate the UV luminosities using stellar population synthesis. For each galaxy we obtain the stellar population components by tracking its star formation and merger history. We integrate the stellar populations with model spectral energy distributions (SEDs) calculated using STARBURST99 [1999ApJS..123....3L, 2005ApJ...621..695V, 2010ApJS..189..309L, 2014ApJS..212...14L] with a constant metallicity of Z = 0.05 solar metallicity. We do not include nebular components as they would not affect the UV luminosities of our model galaxies at these redshifts. To obtain the observed luminosities we apply a dust extinction model to each galaxy. We adopt a luminosity dependent dust model [e.g.][2012ApJ...756...14S, 2015ApJ...803...34B] which is based on the IRX-beta relation from [1999ApJ...521...64M] and the observed luminosity-beta relation from [2014ApJ...793..115B]. This dust model is empirical and is calibrated to reproduce the observed properties of galaxies. For more details about the galaxy photometric modeling see Paper-IV.


We first investigate the relationship between the physical size and UV luminosity of model galaxies. Figure 1 caption: Effective radius of galaxies as a function of UV luminosity at redshift of approximately 5 to 10. The colour profile shows the logarithm density of the distribution. The black squares and error bars represent the median and 16th to 84th percentiles of the R_e distribution. The black solid lines are the linear best-fits for galaxies with M_1600 less than -14.5. The pink and orange lines show the observed relations from [2013ApJ...765...68H] and [2015ApJS..219...15S]. The blue and yellow diamonds show the observations at z~7 from [2016arXiv160505325B]. The blue star shows luminous galaxy GN-z11 found by [2016ApJ...819..129O]. For model comparison, the red circles and error bars show the size--luminosity from the model with supernova feedback turned off. The dash-dotted lines represent the minimum measurable effective radii of HST, JWST and GMT.


The figure shows the relation between the effective radius and UV magnitude M_UV for model galaxies at redshift of approximately 5 to 10. We see that at UV magnitudes less than or equal to -14, galaxies with brighter UV luminosity tend to have larger sizes. The effective radius does not significantly change with luminosity for galaxies with luminosities M_UV greater than -14. This is because galaxies fainter than M_UV of approximately -14 are located in the dark matter haloes of the minimum gas cooling mass. This is similar to the turnover at M_UV of approximately -14 in the relation between UV luminosity and the mass of dark haloes found in Paper-IV. We see that at fixed luminosity, the size of galaxies grows from z~10 to 5. For comparison, we show the observed R_e--M_UV relations from [2013ApJ...765...68H] at z~5 and [2015ApJS..219...15S] at z~5 to 8. Our results are in close agreement with the observations.


Recently, [2016ApJ...819..129O] found an unexpectedly luminous galaxy (GN-z11) at z~11, which has M_UV= -22.1 and R_e=0.6 kpc. In [2016MNRAS.463.3556M] we demonstrated that the properties of GN-z11 are in good agreement with the results of our model in terms of stellar mass, star formation rate and UV luminosities. We show the measured size of GN-z11 in the figure and find that it is in agreement with our fitted size--luminosity relation at z~10. The relation between the galaxy size and luminosity is commonly fitted by: The effective radius R_e is equal to a reference radius R_0, multiplied by the ratio of the UV luminosity to a reference luminosity, raised to the power of beta. We set the reference luminosity L_0 = L*_z=3 which corresponds to M_0 = -21 [1999ApJ...519....1S]. This equation can be rewritten as: The logarithm of the effective radius, log10(R_e), is linearly related to the UV magnitude, M_UV, with a slope of -0.4 times beta. We linearly fit this relation for galaxies brighter than M_UV=-14.5 at each redshift.


Caption for Table 2: The best-fitting parameters R_0 and beta for the model galaxies with UV magnitudes M_UV less than -14 at z~5 to 10. We see that the slope of the size-luminosity relation, beta, does not significantly change at z~5 to 9 and has a median value of beta of approximately 0.25 for galaxies with UV luminosity brighter than M_UV of approximately -14. This value agrees with observational studies for both local and high-redshift galaxies. For example, [2000ApJ...545..781D] found beta = 0.253 for local spiral galaxies. [2003MNRAS.343..978S] derived a slope of beta of approximately 0.26 for the late-type galaxies from SDSS. [2007ApJ...671..203C] obtained beta=0.321 from local field and cluster spiral galaxies. [2012A&A...547A..51G] found beta=0.3 to 0.5 for LBGs at z~7, while [2015ApJ...808....6H] derived beta=0.24 using the [2012A&A...547A..51G] data.


In addition, [2013ApJ...765...68H] found beta=0.22 and 0.25 for the galaxies in GOODS and HUDF fields at z~4 and z~5 respectively. Finally [2015ApJS..219...15S] investigated the galaxy effective radius from a large Hubble Space Telescope (HST) sample and obtained beta = 0.27 at z~0 to 8. They also showed that beta does not significantly evolve over this redshift range. Due to limitations in sample volumes and selection biases, observed values of beta often have large uncertainties and vary between studies. For example, observations are generally biased towards galaxies with high surface brightness and are not sensitive to measured properties of fainter, more spatially extended galaxies. Because a model does not suffer from these selection effects and can have a large sample of both bright and faint galaxies, we are able to investigate the true scatter of the size--luminosity relation. The size-luminosity relation fitted to the model predictions is also consistent with the analytic prediction of [2011MNRAS.413L..38W].


In that work they considered a supernova feedback model where supernova-driven winds conserve momentum in the interaction with the galactic gas. The model results in a luminosity scaling of a=1/3 which corresponds to R_e is proportional to L to the power of 0.25. While the model without supernova feedback yields a=0 which corresponds to R_e is proportional to L to the power of 0.33. To study the role of supernova feedback on the build up of galaxy sizes, we show the size-luminosity relation for the no supernova feedback model. The size-luminosity relation for the no supernova feedback model is also flat at M_UV greater than -14. This is because the minimum size is set by the mass scale of efficient cooling in both models. There is no clear difference between the fiducial and no supernova feedback model at M_UV greater than -17, where the accumulated effect from supernova feedback on star formation histories is not significant enough to be observed.


However, at M_UV less than -17, the median size of galaxies from the no supernova feedback model is notably smaller than the fiducial model. In other words, for the same size galaxy, the no supernova feedback model results in a much brighter luminosity. We note that removing supernova feedback allows more stars to form, and so the model has been recalibrated to produce the correct stellar mass density at z=5. The luminosity difference is approximately 2 to 3 mag at z=5 to 7, which is larger than the approximately 1 mag difference at z = 8 to 10. This is also due to the correct galaxy mass only being achieved at z=5 in this recalibrated model. The different size-luminosity relations from these two models arise because the supernova feedback in the fiducial model suppresses star formation resulting in a more gradual star-formation history. In contrast, galaxies without supernova feedback have much burstier star-formation histories and contain more young stellar populations which are UV bright.


The spatial resolution of a telescope with effective diameter D_tel is given by: The spatial resolution, delta l, is given by the Rayleigh criterion, which is 1.22 times the observed wavelength lambda, divided by the telescope diameter D_tel, all multiplied by the angular diameter distance d_A. where delta_theta is the angular resolution determined by the Rayleigh criterion, lambda=1600(1+z) Angstroms is the observed wavelength of UV photons and d_A is the angular diameter distance. In this equation, the observed wavelength is scaled by a factor of (1+z) at fixed intrinsic wavelength, while the angular diameter distance decreases at a similar rate at redshift greater than or equal to 1. Thus the spatial resolution does not rapidly change with redshift. Galaxy sizes are usually measured through light profile fitting [e.g.][2002AJ....124..266P]. As a result, one can trace the galaxy outskirt light, and obtain an effective radius bellow the spatial resolution of the telescope.


The comparison between the observed R_e and the spatial resolution limits of the Hubble Space Telescope (HST) indicates that values of R_e can be measured which are smaller than the resolution limit of the telescope by roughly a factor of ~ 2 [e.g.][2013ApJ...777..155O, 2015ApJS..219...15S]. In Fig. 1 we show the minimum observable disc size R_min of HST, James Webb Space Telescope (JWST), and the Giant Magellan Telescope (GMT), where we adopt the relation R_min is approximately delta l/2 as discussed above. We see that HST (D_tel=2.4 m) can resolve the R_min of observed galaxies at z~5 to 7, and the structures of typical z>8 galaxies can not be resolved. The larger diameter JWST (D_tel=6.5 m) will resolve the R_min for galaxies brighter than M_UV=(-14, -16, -18) at z=(6, 8, 10). However, with an exposure time of 1 million seconds, JWST will observe galaxies to M_UV=(-15.0, -15.8, -16.3) with signal-to-noise ratio S/N=10 at these redshifts, hence a significant fraction of z>8 galaxies will be still unresolved.


Due to the large mirror size, GMT (D_tel=25 m) will have the ability to resolve all galaxies in haloes above the atomic cooling limit. Figure 2 caption: Size-mass relation of model galaxies at z=5, 6, 8, 10. The colour profile shows the logarithm density of the distribution. The black squares and red circles show the median relation in bins. The error bars represent the median and 16th to 84th percentiles of the intrinsic scatter. The orange diamonds show the observations from [2012ApJ...756L..12M]. The figure shows the relation between the effective radius and stellar mass of galaxies at z~5, 6, 8 and 10 for both fiducial and no supernova feedback models. Observed data from [2012ApJ...756L..12M] are also shown. The model size-mass relation is in good agreement with these observations. We see that for galaxies with stellar masses above 10^6.5 solar masses, more massive galaxies tend to have larger sizes.


The galaxies from the fiducial model have larger sizes than the galaxies from no supernova feedback model at fixed stellar mass. However, the difference in the size-mass relation between the fiducial and no supernova feedback model is much smaller than in the size--luminosity relation. This is expected because we have tuned both models to produce the galaxy stellar mass density. However, star formation histories including supernovae lead to less variable UV luminosities resulting in larger difference seen in the size-luminosity figure. For galaxies with stellar mass less than 10^6.5 solar masses, our two models show similar galaxy sizes due to the inefficient star formation in the minimum cooling mass, as was the case in the size--luminosity relation.


Figure 3 caption: The redshift evolution of the mean effective radius for galaxies in two luminosity ranges. The blue line shows the mean effective radius from the fiducial model and the green line shows the mean effective radius from the model without supernova feedback. The shaded regions show the associated 1-sigma uncertainties of the means. The grey solid lines show the power law fit to our model. For comparison, we show the observed mean sizes from multiple studies. We see that our fiducial model agrees with observations, while the no supernova model significantly underestimates the galaxy sizes. The redshift evolution of galaxy sizes provides another important measurement in addition to the luminosity dependence [e.g.][2004ApJ...600L.107F, 2004ApJ...611L...1B, 2010ApJ...709L..21O, 2012A&A...547A..51G, 2013ApJ...777..155O, 2015ApJ...804..103K, 2015ApJ...808....6H, 2015ApJS..219...15S]. The figure shows the redshift evolution of the effective radius predicted by our model. To compare with observations of size evolution, galaxies were selected using their luminosity in specific ranges. These luminosity ranges correspond to UV magnitudes from -21.0 to -19.7 and from -19.7 to -18.7 respectively.


Both fiducial and no supernova feedback models are shown in the figure. For comparison the observed galaxy sizes from [2004ApJ...611L...1B], [2010ApJ...709L..21O], [2013ApJ...777..155O], [2015ApJ...804..103K], [2015ApJ...808....6H] and [2015ApJS..219...15S] are also shown. We see that the evolution of galaxy sizes from our fiducial model is in good agreement with observations. However, the galaxy sizes in the no supernova feedback model are underestimated at each redshift. For example, sizes at fixed luminosity in the no supernova feedback model are approximately 60 (70) percent of those in the fiducial model at z~5 (10). This corresponds to surface brightness densities which are approximately 3 (2) times larger than the fiducial model prediction. These are distinguishable differences. To investigate the influence of parameter calibration in the no-supernova model, we have also run an uncalibrated no-supernova feedback simulation and find a qualitatively similar result. Therefore, we conclude that the galaxy size evolution provides an additional observable for determining the importance of supernova feedback in early galaxy formation.


Figure 4 caption: Confidence ellipses with delta chi-squared = 1, which projects 1-sigma uncertainties on m and R_e axes. The red, blue and green contours are z > 5 only observations from [2004ApJ...611L...1B], [2010ApJ...709L..21O] and [2013ApJ...777..155O] respectively. The black contours are from all observations shown in the previous figure. Our best-fitting values are shown as black filled circles. We fit the model size evolution at z~5 to 10 using R_e is proportional to (1+z) to the power of -m and find m = 2.00 for the brighter luminosity range and m = 2.02 for the fainter range. The fitted relations are shown as grey solid lines in the previous figure. We also show confidence intervals using the observations. Here we only include the observational data at z>5. We see that the fitted m from our model is comparable to observations. For example, m = 1.64 and m = 1.82 are derived using the combined observations shown in the figure for the two luminosity ranges.


Before concluding, we discuss the applicability of R_d as a measure of galaxy size. In observations, morphologies of LBGs are often found to be irregular and clumpy, sometimes showing multiple components [e.g.][1996ApJ...470..189G, 2006ApJ...652..963R, 2016MNRAS.457..440C, 2016ApJ...821...72S, 2015ApJ...800...39G, 2016arXiv160505325B]. This could be due to two different formation mechanisms: (i) galaxy interactions, such as mergers [e.g.][2006ApJ...636..592L, 2008ApJ...677...37O]; (ii) distributed and clumpy star formation regions within the same collapsing cloud due to instabilities [e.g.][2002ApJ...568..651G, 2007ApJ...656....1L, 2009ApJ...703..785D, 2010ApJ...709L..21O, 2013ApJ...773..153J, 2016ApJ...819L...2B]. Morphological studies at very high redshift are more challenging. [2016ApJ...821...72S] investigated the evolution of clumpy galaxies with large HST samples and found that the clumpy fraction increases from z~0 to z~1 but subsequently decreases from z~1 to 3 to z~8.


On the other hand, high-resolution cosmological simulations show that galaxies at z greater than or equal to 6 are dominated by disc morphologies [e.g.][2011ApJ...731...54P, 2011ApJ...738L..19R, 2015ApJ...808L..17F]. For example, using the large-volume BlueTide simulation, [2015ApJ...808L..17F] found that at z=8 to 10, up to 70 per cent of the galaxy population more massive than 10^10 solar masses are disc galaxies. Detailed measurement of more compact and clumpy galaxies are limited by the angular resolution of instruments, and the origin of observed clumpy morphologies at high-redshift is still under debate. [2016arXiv160505325B] recently published size measurements for a sample of extremely luminous galaxies at z~7. [2016arXiv160505325B] divided the sample into two groups (single and multi-component) according to their morphologies. We see that the size-luminosity relation for the single morphology galaxies is in good agreement with our model while including clumpy morphology galaxies leads to larger sizes. This may suggest that the multi-component galaxies are merging systems [2016arXiv160505325B].


We have used the semi-analytic model MERAXES to study the dependence of galaxy size on UV luminosity, stellar mass and redshift at z~5 to 10. We also studied the effect of supernova feedback on the evolution of galaxy sizes. We show that the rotationally supported disc model generally adopted in semi-analytic models can be used to study the sizes of high-redshift galaxies. Our primary findings are that: The effective radius scales with UV luminosity as R_e is proportional to L to the power of 0.25 for galaxies with luminosity M_UV less than or equal to -14. Galaxies with the same disc size in the no supernova feedback model have brighter UV magnitudes than in the fiducial model. Our fiducial model with strong supernova feedback successfully reproduces the redshift evolution of average galaxy sizes at z>5, which is slightly steeper than z<5 observations. The model with no supernova feedback produces a significantly smaller radius at fixed luminosity than the fiducial model.


The recently identified luminous galaxy GN-z11 at z~11 [2016ApJ...819..129O] lies on our model-fitted size-luminosity relation. The fitted relation is also in agreement with the size measurements of very luminous galaxies containing single components and with individual components of luminous multi-component systems at z~7 [2016arXiv160505325B]. A significant fraction of z>8 galaxies will not be resolved by JWST. However, GMT will have the ability to resolve all galaxies in haloes above the atomic cooling limit. We conclude that galaxy sizes provide an important additional constraint on galaxy formation physics during reionization, and that current observations of galaxy size and evolution reinforce the importance of supernova feedback. These findings are in agreement with results based on the stellar mass function and luminosity function.


This research was supported by the Victorian Life Sciences Computation Initiative (VLSCI), grant ref. UOM0005, on its Peak Computing Facility hosted at the University of Melbourne, an initiative of the Victorian Government, Australia. Part of this work was performed on the gSTAR national facility at Swinburne University of Technology. gSTAR is funded by Swinburne and the Australian Governments Education Investment Fund. This research programme is funded by the Australian Research Council through the ARC Laureate Fellowship FL110100072 awarded to JSBW. AM acknowledges support from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement no. 638809 AIDA).


We investigate how the hydrostatic suppression of baryonic accretion affects the growth rate of dark matter haloes during the Epoch of Reionization. By comparing halo properties in a simplistic hydrodynamic simulation in which gas only cools adiabatically, with its collisionless equivalent, we find that halo growth is slowed as hydrostatic forces prevent gas from collapsing. In our simulations, at the high redshifts relevant for reionization (between redshift of approximately 6 and approximately 11), haloes that host dwarf galaxies (less than or equal to 10^9 solar masses) can be reduced by up to a factor of 2 in mass due to the hydrostatic pressure of baryons. Consequently, the inclusion of baryonic effects reduces the amplitude of the low mass tail of the halo mass function by factors of 2-4. In addition, we find that the fraction of baryons in dark matter haloes hosting dwarf galaxies at high redshift never exceeds approximately 90 per cent of the cosmic baryon fraction. When implementing baryonic processes, including cooling, star formation, supernova feedback and reionization, the suppression effects become more significant with further reductions of approximately 30-60 per cent. Although convergence tests suggest that the suppression may become weaker in higher resolution simulations, this suppressed growth will be important for semi-analytic models of galaxy formation, in which the halo mass inherited from an underlying N-body simulation directly determines galaxy properties. Based on the adiabatic simulation, we provide tables to account for these effects in N-body simulations, and present a modification of the halo mass function along with explanatory analytic calculations.


In recent years, a number of groups have run large volume N-body simulations and used these to investigate the properties of large-scale structure (e.g. Hubble Volume, [Jenkins2001]; Millennium, [springel2005simulating]; CubeP3M, [iliev2008simulating]; Millennium-II, [Boylan_Kolchin2009]; Bolshoi, [Klypin2010]; MICE, [Crocce2010]). In order to connect the observable galaxy population to the halo properties produced by those simulations, semi-analytic models (SAMs) built on dark matter halo merger trees have also been developed (e.g. [Cole2000]; [Hatton2003]; [Baugh2005]; [croton2006many]; [DeLucia2007]; [Somerville2008]; [guo2011dwarf]). SAMs approximate the physics in hydrodynamic simulations (e.g. Illustris, [vogelsberger2014properties]; EAGLE, [Schaye2014]) using analytic descriptions. A crucial difference is that while baryons and dark matter evolve together in hydrodynamic simulations, SAMs explore the properties of galaxies based on the halo properties read from collisionless halo merger trees. This method, therefore, assumes that baryons have little influence on halo properties and that a pure dark matter N-body simulation can provide SAMs with a reliable halo merger tree.


However, recent studies at low redshift have shown that this may not be the case. Both the GIMIC [Sawala2013] and EAGLE [Schaller2014] projects discovered that the mass ratio between haloes extracted from full-hydrodynamic simulations and N-body simulations is less than unity, especially for low-mass haloes. [Sawala2013] found that without baryons, N-body simulations overpredict the halo mass by 30 per cent, 20 per cent and 10 per cent for haloes whose masses are less than or equal to 10^10, approximately 10^11 and approximately 10^12 solar masses, respectively. [Schaller2014] discovered similar differences with an overprediction of approximately 15 per cent for approximately 10^12 solar masses haloes resulting from AGN feedback. The Magneticum project [Bocquet2016] also found that the inclusion of baryons decreases the mass of galaxy clusters (10^12 - 10^15 solar masses) with the effect becoming smaller for larger halo masses. In this work, we analyse the effect of baryons on dark matter halo growth at high redshift (redshift greater than 5) in the mass range relevant for reionization.


Using a suite of high resolution hydrodynamic simulations for comparison with a collisionless N-body simulation, we investigate the baryonic effect, due to a range of galaxy physics, including gas pressure, cooling, star formation, reionization and supernova feedback. We also modify halo mass functions in the collisionless scenario through linear perturbation theory to account for the baryonic effect. This paper is organized as follows. We present simulations in Section 2, and discuss the comparison between full-hydrodynamic simulations and the collisionless case, including mass and baryon fractions. In Section 3.2, we calculate the simulated halo mass functions and provide a modification of halo mass functions from collisionless simulations or analytic calculations. Conclusions are given in Section 5. In this work, we adopt cosmological parameters from WMAP7 (Omega_m, Omega_b, Omega_Lambda, h, sigma_8, n_s = 0.275, 0.0458, 0.725, 0.702, 0.816, 0.968; [Komatsu2011]).


To make a quantitative investigation of baryonic effects, we utilize results from a suite of hydrodynamic simulations (Smaug, [duffy2014low]) performed as a part of the Dark-ages Reionization And Galaxy formation Observables from Numerical Simulations (DRAGONS) project. Each simulation has 512^3 baryonic and 512^3 dark matter particles within a cube of comoving side 10 h^-1 Mpc. This equates to a mass resolution of 4.7(0.9) x 10^5 h^-1 solar masses per dark matter (gas) particle. All simulations have identical initial conditions generated with the grafic package [Bertschinger2001] at z=199 using the Zel'dovich approximation [Zeldovich1970]. The simulations implement different physics and were run with an updated version of the gadget-2 N-body/hydrodynamic code [springel2005cosmological] to redshift z=5. The exceptions are the ADIAB and N-body simulations, which were run to z = 2 and 0, respectively, and will be introduced later. The particle IDs are consistent in the Smaug suite in order to match haloes across different simulations. A brief summary of the simulations is shown as follows, while further details can be found in [duffy2014low] and [Schaye2010].


A collisionless N-body simulation (hereafter N-body) was performed using the same initial conditions from the full simulation but without hydrodynamic forces from the baryonic component. In the ADIAB simulation, gas only cools adiabatically and there is no stellar physics or reionization included. This simple model can be used to investigate the isolated effect on the dark matter halo growth of the hydrostatic suppression of baryonic accretion into the growing potential well. In the set of NOSN_NOSZCOOL simulations, radiative cooling [Wiersma2009] from primordial elements (hydrogen and helium) is turned on and star formation [Schaye2008] is implemented by converting gas particles into collisionless star particles, which represent a single stellar population specified by the Chabrier initial mass function [Chabrier2003]. However, the feedback due to supernova explosion is not included in these models. Moreover, an instantaneous UV/X-ray background [Haardt2001] is switched on at z=9 or 6.5 (NOSN_NOSZCOOL_LateRe) for early and late reionization to ensure the gas is heated to approximately 10^4K when reionization begins [Wiersma2009]. In addition, a NOSN_NOSZCOOL model without reionization is also performed for comparison (NOSN_NOSZCOOL_NoRe).


In the REF simulation, stellar particles explode as Type II supernovae with the feedback coupled by randomly 'kicking' 2 of its neighbours with a velocity of 600 km/s [DallaVecchia2008]. This equates to 40 per cent of the available energy produced by supernova feedback coupled to driving a wind and is termed kinetic feedback. Additionally, radiative cooling from metal elements including carbon, nitrogen, oxygen, neon, magnesium, silicon, sulphur, calcium and iron are pre-tabulated using the public photoionization package cloudy [Ferland1998] and implemented in the simulation. Similarly to the REF simulation, in the REF_EFF model, 3 of the neighbours of a newly formed stellar particle are kicked with a wind velocity of 774.6 km/s. Here, all of the supernova energy is used for kinetic feedback and is used to represent the maximal kinetic feedback from supernovae. An alternative method of modelling supernova feedback is to heat the nearby gas stochastically by increasing the temperature of gas particles, which is termed thermal feedback [DallaVecchia2012]. This feedback is implemented in the WTHERM model, with gas particles heated to 10^7.5 K. We have found that WTHERM disagrees with REF_EFF, indicating that even with the high resolution of Smaug the method of coupling supernova feedback does play a (secondary) role in galaxy formation at our mass scales of interest [duffy2014low].


A summary of the simulations utilized in this study. Structures were identified in all simulations using subfind [Springel2001]. It identifies collapsed regions with a friends-of-friends (fof) algorithm using a standard linking length of b=0.2, then splits them into several self-bound subhaloes according to their local overdensities. In this work, we adopted fof haloes (hereafter haloes for short). However, we note that the differences in the following results between subhaloes and haloes are usually less than 15 per cent, and there are also some offsets between the most massive subhalo within a fof group and their satellites.


First, we match haloes between full-hydrodynamic simulations and the N-body simulation according to their particle IDs. For each halo, if the majority of its particles are located within the corresponding halo of the other simulation, they will be considered as a matched candidate. We only include haloes that are matched bidirectionally to reduce the chance of mismatch. Then we calculate the mass ratio of the matched haloes between the hydrodynamic and the N-body simulations (M_hydro/M_Nbody). In this work, we adopt a spherical top-hat mass for haloes defined as the mass of all the particles (including dark matter, gas and star particles) within a sphere of average density Delta is approximately 18*pi^2 ([Duffy2010], 18*pi^2 for short) times the critical density (we also test our result with the fof halo mass, which is defined as the mass of all the particles linked by the fof halo finder (see Appendix A.1) and find that the difference is less than 15 per cent). We only use haloes with masses higher than 10^7.5 solar masses in the N-body simulation which corresponds to 40 particles [duffy2014low].


Top panels: the evolution of mass ratio, M_hydro/M_Nbody, which is defined as the mass ratio of the matched haloes between hydrodynamic simulations and the corresponding N-body simulation. The spherical top-hat mass, which includes both dark matter and baryonic particles within a sphere of average density equal to 18*pi^2 times the critical density is adopted. The mean values with uncertainties showing 95 per cent confidence intervals around the mean using 100,000 bootstrap re-samples are shown in four mass bins with different colours (based on the top-hat mass in the N-body simulation). In the clockwise direction from the top left panel, these panels represent, respectively, (1) ADIAB, where gas is included and allowed to cool adiabatically compared to the N-body simulation; (2) NOSN_NOZCOOL_NoRe, where radiative cooling from primordial elements and star formation are turned on; (3) NOSN_NOZCOOL and NOSN_NOZCOOL_LateRe (dashed lines), where an instantaneous reionization background is switched on at z=9 or 6.5; (4) REF, where metal cooling and kinetic supernova feedback are implemented; (5) REF_EFF, where a maximal kinetic supernova feedback is adopted; (6) WTHERM, where a maximal thermal supernova feedback is adopted (see more in Table 1). For comparison, the ADIAB result for haloes around 10^9 solar masses is shown as black dotted line in panels (2) to (6). Bottom panels: the ratio of (2) ADIAB to NOSN_NOZCOOL_NoRe, (3) NOSN_NOZCOOL and NOSN_NOZCOOL_LateRe (dashed lines) to NOSN_NOZCOOL_NoRe, (4) REF to NOSN_NOZCOOL, (5) REF_EFF to REF and (6) WTHERM to REF_EFF. Redshifts that reionization background is switched on at are shown with vertical dashed lines. We present the evolution of mass ratios, M_hydro/M_Nbody, in Fig. 1 for different simulations. There are several points to note.


Mass ratios between the ADIAB and N-body simulations are shown in the top left panel of Fig. 1 in four mass bins, based on the N-body simulation. At all redshifts and mass bins, the ratio is less than 1. Since gas can only cool adiabatically in the ADIAB simulation, it is clear that hydrostatic pressure between gas particles keeps baryons from collapsing ([Somerville2002, Simpson2013, Sobacchi2013, Onorbe2015]), which in turn decreases the gravitational potential compared to an N-body simulation and delays the accretion of dark matter as well as baryons. Therefore, the halo mass is smaller at fixed time when baryons are included with the effect being more significant for less massive haloes. At the high redshifts relevant for reionization (between redshift of approximately 6 and approximately 11), haloes that host dwarf galaxies (less than or equal to 10^9 solar masses) are significantly reduced in mass, by up to a factor of 2. In addition, mass ratios rise towards lower redshift, suggesting a decreasing effect of baryons on halo mass.


Although our simulations have a relatively small population of large objects, it is likely that the baryonic effect from purely hydrostatic pressure asymptotes to a constant level in massive haloes (approximately 10^9.0 solar masses) at given redshift (e.g. 65 per cent at redshift of approximately 10). The mass ratios between the NOSN_NOZCOOL_NoRe and the N-body simulations are shown in the top middle panel of Fig. 1 (for comparison the bottom sub-panel shows the ratio of NOSN_NOZCOOL_NoRe to the ADIAB result). This comparison demonstrates the effect due to cooling and star formation. The mass ratio becomes higher compared to ADIAB (less than approximately 10 per cent), suggesting that when galaxies are able to cool and remove gas through forming stars, the effect of hydrostatic suppression naturally becomes smaller. Cooling and star formation also show an increasing effect at later time and a complex dependence on halo mass.


Mass ratios between the NOSN_NOZCOOL, NOSN_NOZCOOL_LateRe and N-body simulations are shown in the top right panel of Fig. 1, compared to the NOSN_NOZCOOL_NoRe simulation. When reionization is switched on at z=9 or 6.5, mass ratios decrease dramatically (by up to 30 per cent compared to the model without reionization) because of the heating of the intergalactic medium (IGM) from the UV/X-ray background. Moreover, there is a delay between the onset of reionization and this decrement as the now overpressurized mass can only respond on dynamical time-scales. It is clear that photoionization suppression by reionization only has a significant impact on smaller objects, as seen by comparing the ratio of the bottom sub-panel across different mass bins (there is no discernible effects for haloes >10^9.5 solar masses, in orange). We note that in the simulations with reionization, the IGM cools adiabatically until reionization starts. In reality there may well be an impact from other heating sources such as X-ray binaries, which are not included in our simulations. In Smaug, the UV/X-ray heating is only implemented following reionization through the assumption of a [Haardt2001] UV/X-ray background.


Since the growth rate is affected by the photoionization/heating of the IGM, the physics of stellar feedback is expected to have an impact as well [Governato2009, Trujillo-Gomez2014]. The bottom right panel of Fig. 1 shows the effect of supernovae feedback on the halo growth rate (less than approximately 20 per cent). The top and bottom sub-panels show the result from the REF simulation and its ratio to the NOSN_NOZCOOL simulation, respectively. We see that supernova feedback has an increasing impact on the halo mass at lower redshift and larger objects. In addition, we expect that for much larger haloes (greater than or equal to 10^11 solar masses), supernova feedback will have less influence, while AGNs become the dominant heating source [Somerville2015]. When supernova feedback becomes stronger, the mass ratio is further suppressed (less than or equal to 10 per cent, see the REF_EFF result in the bottom middle panel of Fig. 1).


In the left bottom panel of Fig. 1, the REF_EFF and WTHERM simulations show slightly different mass ratios (less than 5 per cent) although they both have a strong supernovae feedback mechanism coupling 100 per cent of available supernovae energy. However, the supernovae feedback in the REF_EFF simulation is implemented kinetically [DallaVecchia2008], so that it uses the supernovae energy to drive winds and expel gas particles from galaxies. On the other hand, the thermal feedback in the WTHERM simulation stochastically heats the neighbouring gas particles and increases the temperature of heated gas by a certain value [DallaVecchia2012]. In practice, WTHERM removes more baryons for haloes less massive than 10^10 solar masses, and consequently reduces the mass ratio by a greater amount.


Mass ratio versus halo mass for all subhaloes. The red triangles and dashed line are from the data and fitting function in [Sawala2013] at z=6, while the grey points indicate the result from our REF simulation with 256^3 particles at z=5. The square-solid line shows the median value in each mass bin. Halo masses in our result are rescaled to have a consistent cosmology. The circle-solid line indicates the result from the REF simulation with 512^3 particles. [Sawala2013] use the GIMIC [Crain2009] simulations, which assume instantaneous reionization at z is approximately 9 [Haardt2001], and include star formation [Schaye2008], metal cooling [Wiersma2009] and kinetic supernovae feedback [DallaVecchia2008]. Their particle masses are 9.05 and 1.98x10^6 solar masses per dark matter and baryon particle, respectively. This value lies between our REF simulations with 256^3 (closer) and 512^3 particles (see the convergence test in Appendix B).


We compare our results to theirs at redshift approximately 6 in Fig. 2. In order to have a consistent cosmology during the comparison [Angulo2010], we rescale the halo masses of our simulations, multiplying by (Omega_m' / Omega_m) * (H'^2 / H^2) * s^3, where H, H' are the Hubble constants in the two cosmologies, Omega_m' = 0.25 and s=0.83. The mass ratio of all subhaloes at redshift approximately 5 are in excellent agreement with the result at z=6 from [Sawala2013]. This result gives us confidence in our quantitative results for lower masses at higher redshifts.


Top panels: the evolution of baryon fraction, f_b, which is defined as the mass ratio of the baryonic particles to all particles including baryons and dark matter within a sphere of average density equal to 18*pi^2 times the critical density. The mean values with uncertainties showing 95 per cent confidence intervals around the mean using 100,000 bootstrap re-samples are shown in four mass bins with different colours (based on the top-hat mass in the N-body simulation). In the clockwise direction from the top left panel, these panels represent, respectively (1) ADIAB, where gas is included and allowed to cool adiabatically compared to the N-body simulation; (2) NOSN_NOZCOOL_NoRe, where radiative cooling from primordial elements and star formation are turned on; (3) NOSN_NOZCOOL and NOSN_NOZCOOL_LateRe (dashed lines), where an instantaneous reionization background is switched on at z=9 or 6.5; (4) REF, where metal cooling and kinetic supernova feedback are implemented; (5) REF_EFF, where a maximal kinetic supernova feedback is adopted; (6) WTHERM, where a maximal thermal supernova feedback is adopted (see more in Table 1). For comparison, the ADIAB result for haloes around 10^9 solar masses is shown as black dotted line in panels (2) to (6). Bottom panels: the ratio of (2) ADIAB to NOSN_NOZCOOL_NoRe, (3) NOSN_NOZCOOL and NOSN_NOZCOOL_LateRe (dashed lines) to NOSN_NOZCOOL_NoRe, (4) REF to NOSN_NOZCOOL, (5) REF_EFF to REF and (6) WTHERM to REF_EFF. Redshifts that reionization background is switched on at are shown with vertical dashed lines. The baryonic effect on halo mass shown in the previous section represents a combined impact on collapse of dark matter and baryons. The change in the dark matter component is essentially a consequence of the change in gravitational potential caused by baryons. Therefore, we expect a more significant impact on the baryonic component. In this section, we discuss the fraction of baryons present in a dark matter halo.


In Fig. 3 baryon fractions, f_b, are shown in the same mass bins (binned by the halo mass in the N-body simulation) for different hydrodynamic simulations. The baryon fraction of a halo is calculated through the mass ratio of the baryonic particles to all particles within a sphere of average density equal to 18*pi^2 times the critical density. The different behaviour of the baryon fraction using the fof mass is shown and discussed in Appendix A.1. We see that the fraction of baryons in dark matter haloes hosting dwarf galaxies at high redshift never exceeds approximately 90 per cent of the cosmic mean, Omega_b/Omega_m, in the presence of hydrostatic pressure (in agreement with [Crain2007] but here even with no cooling or feedback, see the top left panel of Fig. 3); the baryon fraction f_b also becomes larger for massive haloes, suggesting that more massive haloes have a deep enough potential well to overcome the hydrostatic pressure of the baryons [Gnedin2000, Hoeft2006, Okamoto2008, Noh2014]. However, unlike the mass ratio, the baryon fraction depends weakly on time, indicating that the growth rate of baryons is close to the dark matter component within the virial radius (at least, see Appendix A.1).


Top panels: halo mass functions at redshifts from 13 to 2 (clockwise direction from the top left panel). Different simulations are shown using different colours. Bottom panels: ratios of halo mass functions from the full-hydrodynamic simulations to the N-body simulation. Cooling and star formation help haloes retain more baryons, so that haloes have approximately 20 per cent more baryons in the NOSN_NOZCOOL_NoRe simulations compared to ADIAB. The ratio of baryon fractions between NOSN_NOZCOOL_NoRe and ADIAB reaches its maximum of approximately 1.25 when the halo mass is around 10^8.5 solar masses, with a decrement for larger objects, illustrating that the combination of cooling and star formation has a non-monotonic response to halo mass. In contrast to ADIAB, the NOSN_NOZCOOL_NoRe simulation allows star formation, which decreases the heating due to virial shocks. Because there is no feedback to prevent gas from cooling and forming stars, star formation can consume plenty of gas and make a significant difference to the strength of shock heating. This decrement of shock heating also helps haloes retain more baryons. However, this decrement of shock heating has less effect to the mass of more massive haloes. This is because their gravitational potentials are strong enough to retain the ejecting particles or reincorporate the ejected gas regardless of this runaway star formation.


Reionization plays a significant role in reducing the baryon fraction [Somerville2002, Simpson2013, Sobacchi2013, Onorbe2015]. In the top right panel of Fig. 3, baryon fractions decrease rapidly when reionization starts. Moreover, reionization is able to remove the majority of baryons (by up to 90 per cent) in the haloes hosting dwarf galaxies, with the suppression becoming smaller in massive objects; supernova feedback also has a noticeable impact on baryon fractions ([Governato2009, Trujillo-Gomez2014], see the bottom right panel of Fig. 3). The ratio between REF and NOSN_NOZCOOL shows an increased effect in more massive haloes before reionization, indicating that supernova feedback also regulates galaxy formation in haloes with 10^8-10^10 solar masses. When supernova feedback becomes stronger, baryon fractions get further suppressed (by up to 40 per cent, see the bottom left and middle panels of Fig. 3); These calculations of mass ratio and baryon fractions for ADIAB and WTHERM simulations can be included into SAMs, in order to account for the loss of baryons due to hydrostatic pressure alone in the former and in the latter with additional feedback induced gas removal. This will be discussed further in Section 4.1. We also provide a simple analytic model to illustrate this baryonic effect in Appendix C.


In this section, we discuss the effect of baryons on the dark matter halo mass function. The previous sections demonstrated a suppression of halo mass due to the inclusion of baryons. We therefore expect an impact on the halo mass function. We first present the halo mass functions from the hydrodynamic simulations and their ratios to the N-body simulation at z=13, 9, 5 and 2 in Fig. 4 (clockwise direction from the top left panel). We see that baryons suppress the halo mass function [Sawala2013,Vogelsberger2014,Velliscig2014,Schaller2014,Bocquet2016]. The effect is dramatic at high redshift, with the number density reduced by as much as a half at redshift approximately 13, but converges at later times. This suggests that the halo mass function extracted from N-body simulations is biased significantly at high redshift. The halo mass function is further suppressed when more complete physics is considered [Velliscig2014], with the effect varying at different stages.


At high redshift (redshift greater than or equal to 9), stellar physics has little impact on the halo mass function (less than approximately 5 per cent). However, at later times, the effects of cooling and supernovae feedback become noticeable across the halo mass range of interest (10^8-10^10 solar masses) for reionization (redshift approximately 11 to 6). After the Epoch of Reionization, the number densities of haloes with masses between 10^7.5 and 10^9 solar masses get further suppressed by the global UV/X-ray background. This reduction by photoionization is enhanced for simulations with stronger supernova feedback as noted by [Pawlik2009].


By connecting collapsed haloes with their initial density field, the Press-Schechter formalism (PS, [Press1974, Bond1991, Lacey1993]) provides an approximation to the halo mass function. It assumes the initial density fluctuations were Gaussian, and can be implemented using the standard spherical collapse model or the ellipsoidal model (SMT, [Sheth2001]). The PS formalism describes the halo mass function as Equation 1, where sigma is the mass variance. The halo mass function has also been further updated with new parameters and functional forms to better match N-body simulations. For instance, [Tinker2010] performed a large set of collisionless N-body simulations with the flat LambdaCDM cosmology and the spherical overdensity halo finder. They tested several overdensity thresholds and provided fitting functions of the halo mass function as Equation 4, where coefficients alpha, beta, eta and gamma are functions of mass and redshift (see more examples in [Reed2007, Tinker2008, Tinker2010, Watson2013], or the summary in [Murray2013]).


In this section, rather than parametrizing the halo mass function from the full-hydrodynamic simulation, we instead develop a simple method to account for the hydrostatic suppression of baryonic accretion, and use this to modify the halo mass function found in the theoretical collisionless scenario described above. We take the ADIAB simulation as an example. However, the same method can be employed for the other simulations with more complete physics regimes although they do not show significant differences from ADIAB in the halo mass function at the mass and redshift ranges tested (see Fig. 4). First, we match fof haloes in the ADIAB and N-body simulations at a given redshift, z. In the N-body simulation, we then search for the redshift z' when M_Nbody(z') = M_hydro(z) by following the most massive progenitor in the halo merger tree. We calculate the ratio of (1+z') to (1+z) and plot its median value as a function of mass (M_adiab, the virial halo mass in the ADIAB simulation) in Fig. 5 for z=7, 9 and 12. The 2D histogram illustrates the distribution of these quantities at z=7. We see that the ratios of (1+z') to (1+z) are nearly constant for 10^7.5 - 10^10 solar masses haloes and very weakly dependent on redshift in the early universe.


2D histograms of (1+z')/(1+z) versus M_adiab at z =7. The solid, dashed and dash-dotted lines indicate the median values at z=7, 9 and 12, respectively. Note that the lines overlap, which suggests a very week evolution of (1+z')/(1+z).


Top panels: halo mass functions at redshift from 13 to 2 (clockwise direction from the top left panel). The filled and empty circles are from the ADIAB and N-body simulations while the dashed and solid lines indicate the original and modified SMT halo mass functions [Sheth2001], respectively. The error bar indicates the 1-sigma Poisson uncertainties. Bottom panels: the circles represent the ratios of halo mass functions from ADIAB to N-body while the solid lines indicate the ratios of modified SMT halo mass functions to the original collisionless halo mass functions. In the collisionless scenario, when the initial overdensity reaches the critical value, delta_c = 1.686*D^-1(z'), haloes collapse at z' with masses equal to M. However, when baryons are included, because of the hydrostatic suppression, haloes with the same initial overdensities suffer delayed formation and attain the same mass only at later times, z < z' as shown in Fig. 5. As a result, haloes with the same overdensity, 1.686*D^-1(z'), will collapse at redshift z in the presence of baryons. We use the median value of (1+z')/(1+z) to modify the collisionless halo mass function at z by replacing delta_c from 1.686*D^-1(z) to 1.686*D^-1(z').


We present the modified SMT halo mass functions compared with the original collisionless halo mass function and their ratios in Fig. 6 for z=13, 9, 5, 2 (clockwise direction from the top left panel). Since the halo mass functions in Fig. 6 are calibrated against the ADIAB simulation, we show the simulated halo mass functions from ADIAB (filled circles) for comparison with the N-body simulation (empty circles) in the top panels. In the bottom panels of Fig. 6, ratios between halo mass functions from ADIAB and N-body simulations are shown with circles. Again, we see that the number densities of haloes are significantly reduced with the inclusion of baryons and the offsets between scenarios with and without baryons vanish at the high mass end and towards lower redshift.


This method provides an accurate modification of the collisionless halo mass function. In the bottom panels of Fig. 6, the difference between lines and circles is less than 0.1 for haloes larger than 10^8 solar masses but starts to increase at the low mass end when approaching the resolution limit. This significant effect has been missed in previous work, which mainly focused on lower redshift where this effect is subdominant. However, this modification of the halo mass function is a very important consideration for galaxy formation modelling at high redshift during the Epoch of Reionization.


Mass ratio at redshift approximately 5 for the ADIAB simulation. Tables of mass ratios, baryon fractions and redshift ratios for the ADIAB and WTHERM simulations are available in a machine-readable form in the online journal and in the HDF5 format on the DRAGONS website: http://dragons.ph.unimelb.edu.au/resources/smaug.html. z: redshift; log10(mass_lower): lower limit of the mass bin in logarithm. The unit of mass is solar masses; log10(mass_upper): upper limit of the mass bin in logarithm; mass_mean: average mass in the unit of solar masses; mass_errl: lower bond of the uncertainty in mass_mean. The uncertainty is the 95 per cent confidence interval around the mean using 100,000 bootstrap re-samples; mass_erru: upper bond of uncertainty in mass_mean; ratio_mean: average mass ratio; ratio_errl: lower bond of the uncertainty in ratio_mean; ratio_erru: upper bond of the uncertainty in ratio_mean;


Tabulated values of mass ratios (see Table 2), baryon fractions and redshift ratios are provided online as functions of halo mass in the collisionless simulation and redshift for the ADIAB and WTHERM simulations, which are shown in the left panels of Figs 1, 3 and 5. We note that the provided tables are based on our simulations [duffy2014low] with 512^3 particles within 10h^-1 comoving Mpc boxes running with an updated version of the gadget-2 code. The quantities are based on top hat properties. However, different codes, box sizes, property definitions (see Appendix A.1) or resolutions (see Appendix B) might have different quantitative results. The WTHERM result can be used to evaluate the baryon fraction and the baryonic effect on halo mass within a more complicated physics paradigm, including radiative cooling, stellar evolution and reionization.


The simplistic ADIAB simulation's result can be incorporated into SAMs, in order to account for the baryonic effect on halo mass from the hydrostatic pressure in the presence of gas and improve the connection between halo merger trees and galaxy formation compared to use of pure dark matter collisionless simulations. We note that if one intends to use the result of the WTHERM simulation, which includes additional astrophysical processes, the baryonic effect on predicted galaxy properties will be double-counted. The first time is from the feedback on baryons in the hydrodynamic simulation while the second is due to calculations of feedback in the SAM. Therefore, one should use the ADIAB simulation to modify the halo mass first, then calculate the galaxy properties. If one wants to compare the properties between galaxies and their host haloes, the simulation including all astrophysical effects can then be used after calculating the galaxy properties because SAMs usually do not alter the halo properties.


Whilst we advocate this approach to approximate the baryonic effect on halo growth in SAMs, we note that modifying halo mass using ADIAB is not self consistent. This is because the additional baryonic physics also has an impact on the halo mass and consequently affects galaxy properties, especially at lower redshifts (see Figs 1, 3 and 4). However, most of these baryonic processes such as radiative cooling, supernovae feedback, reionization heating are implemented in SAMs for the purpose of gaining sensible galaxy properties, which can also be calibrated through the related free parameters representing efficiencies.


Through hydrodynamic simulations, we have found that simulated halo masses during the reionization era are significantly reduced when baryons are included. For example, in a simulation where gas only cools adiabatically and no stellar physics is involved, the mass reduction is approximately 45 per cent for 10^8 solar masses haloes, and approximately 30 per cent for 10^9 - 10^10 solar masses haloes at z = 10 compared to a collisionless N-body simulation. We note that this result is not converged (see Appendix B) for haloes with masses less than 10^9.5 solar masses. Therefore, the suppression is weaker in higher resolution simulations. We have also found that the suppressed growth of dark matter haloes becomes more dramatic at lower masses and at higher redshifts. The size of this effect depends on the physics of feedback and star formation as well. When supernovae feedback or reionization is implemented, halo masses are further suppressed because these act to remove gas or prevent its accretion.


In addition, we have found that while the mass ratios between hydrodynamic simulations and N-body simulations increase with time (before reionization starts), the baryon fractions of haloes usually have no dependence on time. The reduction of halo mass in the presence of baryons has important implications for the halo mass function during reionization. We found that simulated mass function amplitudes are reduced by factors of 2-4 in hydrodynamic models compared to the N-body simulation for low-mass galaxies during reionization. Motivated by this, we have developed a methodology to modify the collisionless halo mass function, in order to take baryons into account, which is calibrated against hydrodynamic simulations. The modifications of halo mass and baryon fraction will have important consequences for SAMs that utilize the halo merger tree constructed from N-body simulations, particularly for simulations of low-mass systems at high redshift. Merger trees generated from N-body simulations should therefore be modified to account for dark matter growth in the presence of baryons. To aid this we have provided tables online for the modified halo mass and baryon fraction, which can be utilized in SAMs to more accurately describe the growth of galaxies during and after reionization.


We would like to thank the anonymous referees for providing helpful suggestions that improves the paper substantially. This research was supported by the Victorian Life Sciences Computation Initiative (VLSCI), grant ref. UOM0005, on its Peak Computing Facility hosted at the University of Melbourne, an initiative of the Victorian Government, Australia. Part of this work was performed on the gSTAR national facility at Swinburne University of Technology. gSTAR is funded by Swinburne and the Australian Governments Education Investment Fund. This research programme is funded by the Australian Research Council through the ARC Laureate Fellowship FL110100072 awarded to JSBW. This work was supported by the Flagship Allocation Scheme of the NCI National Facility at the ANU, generous allocations of time through the iVEC Partner Share and Australian Supercomputer Time Allocation Committee. AM acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant No. 638809 -- AIDA).