To explore the Pop. III contributions to the cosmic star formation history, we ran two simulations on the L10 box, one with all the updates described in Section 2 adopting the fiducial parameters and one without the new physics that we labelled as "NoMini". Given that there are no observational constraints on Pop. III, the choice of a fiducial model is arbitrary. The one adopted in this work has both a low star formation efficiency and a Salpeter-like IMF, which will result in a relatively small global impact of Pop. III star formation compared to Pop. II. The stellar mass function at different redshifts accounts for all, only Pop. III dominated, and only Pop. II dominated galaxies. We classified a galaxy as Pop. III or Pop. II dominated based on which population is brighter in the UV band. The new updates on Meraxes mostly affect the lower end of the SMF with a larger impact at high-redshift.


This reflects the star formation in mini-halos (we are now considering molecular cooling), which is dominant at redshift z >= 15 before the Lyman-Werner background becomes strong enough to photodissociate all the molecular hydrogen. The low-mass peak of the SMF is dominated by Pop. III systems and the more massive one by Pop. II. This bimodality of the SMF could have an impact on the faint end of the luminosity function at redshift z ~ 10. The low mass of Pop. III systems is mainly a consequence of both the shorter lifetimes of Pop. III stars (given their larger mass) and the lower SF efficiency, but also suggests that most of the Pop. III star formation must occur in mini-halos. The halo mass function for Pop. III and Pop. II star-forming halos is shown at several redshifts. The dashed line corresponds to the limit of atomic cooling (virial temperature T_vir = 10^4 K).


Given the small size of the box, we have only a few atomic cooling halos at redshift z ~ 20, so all star-forming systems are mini-halos (and are mostly Pop. III). At lower redshift, the peak of the distribution shifts to higher masses and the impact of Pop. II increases. Finally, Pop. III stars mostly form in mini-halos. This is because the more massive halos are more likely to have already experienced star formation and so will be internally chemically enriched. The total star formation rate density (SFRD) as a function of redshift for both the fiducial and the NoMini model is also analyzed. In the case of the fiducial model, we also show the Pop. III (Pop. II) contribution in the upper (lower) panel. Mini-halos have an appreciable impact on the SFRD only up to redshift z >= 20, while at lower redshift, most star formation occurs in atomic cooling halos.


We see that accounting for Pop. III star formation in mini-halos is crucial during the Cosmic Dawn because Pop. III stars dominate the global star formation history at redshift z >= 20 and are still relevant up to redshift z = 18. The Pop. III SFRD flattens at redshift z ~ 18 and starts to decrease at redshift z ~ 10. The early flattening occurs due to the build-up of the Lyman-Werner background that affects Pop. III star formation in minihalos. The sharp drop at redshift z <= 10 is mostly caused by the photoionizing feedback from reionization that also affects the atomic cooling halos. The feedback from reionization also mildly affects Pop. II. Compared to reionization, the chemical enrichment of the IGM is a much slower process due to the lower velocity of the expansion of the metal bubbles.


As already found in previous works [e.g.][Visbal2020, Yamaguchi2023], the impact of external metal enrichment is subdominant compared to internal metal enrichment. However, external enrichment might still be important for low-mass halos that do not form stars until redshift z ~ 10. The redshift evolution of the average metallicity of the box is shown. As expected, the average metallicity increases monotonically. It crosses the critical value at redshift z ~ 11 and at the end of the simulation is ~ 5 x 10^-3 solar metallicity. The redshift evolution of the IGM metallicity is in very good agreement with [Yamaguchi2023], especially for their "bursty" models, which is the one that most closely resembles the star formation in Meraxes. However, the average metallicity of the IGM does not completely inform us of the average metallicity of the galaxies since, when averaging through the entire box, we are considering voids that have no galaxies and hence zero IGM metallicity.


For this reason, we computed the average metallicity only through cells that have at least one metal bubble (meaning that there must be at least one galaxy that formed stars). The average metallicity, in this case, is much higher, and it is always above the critical value. This suggests that once the first galaxies form, the ejection of metals from Pop. III is quite effective in the nearby regions, allowing a fast Pop. III/II transition in those cells where star formation already occurred. The lower panels instead show the fraction of the IGM that reached the critical metallicity. Considering the entire box, less than 1% of the volume gets enriched above the critical metallicity by redshift z ~ 5. This result is fairly consistent with [Visbal2020] and [Yamaguchi2023] (~ 1% by z = 6).


These small differences are likely due to differences in the star formation models and in the choice of parameters, such as the star formation efficiency. If we focus only on the regions with at least one galaxy that formed stars, we find larger filling factors, and by the end of the simulation, those pixels are all completely enriched. This final result reflects our choice of pixel size that is designed to be similar to the average volume of the metal bubble at redshift z ~ 5. The halo mass function for the externally and internally metal enriched halos at several redshifts is also analyzed. While most halos get internally enriched by their own star formation, at redshift z = 5 halos with virial masses <= 10^7.5 solar masses get their metals mostly from a nearby supernova bubble. This picture reflects the fact that low-mass halos during reionization are not able to form stars because of the LW and photo-ionizing feedback.


Hence, those objects can be chemically enriched only from an external source. In conclusion, when looking at the global evolution of star formation, the effect of the external metal enrichment is quite negligible as it is important only for low-mass halos at low redshift that will hardly form stars due to the radiative feedback effects. We verified this by running a simulation without external metal enrichment. There are no appreciable differences between the two models except at redshift z <= 8 when the dashed line is slightly larger (about 0.1 dex difference). The results in previous sections were obtained adopting quite conservative assumptions for Pop. III stars given the very low SF efficiency and the Salpeter-like IMF. In the following sections, we explore the four main free parameters that regulate Pop. III star formation in Meraxes: the Pop. III star formation efficient, the critical metallicity for Pop.III/II transition, the critical surface density of cold gas for Pop. III star formation to occur and the IMF.


The star formation efficiency determines the conversion of the cold gas into stars and is the free parameter with the largest impact on the Pop. III global SFRD and SMF. This parameter is largely unconstrained, with some simulations supporting very low values (10^-4 - 10^-3, [Skinner2020]) and others suggesting higher values ([Fukushima2020]). We ran two simulations keeping all the Pop. III free parameters unchanged except for the star formation efficiency, which we boosted by one order of magnitude and to a very high value. Pop. III galaxies become more massive as the star formation efficiency increases, erasing the "double peak" feature in the total SMF. For the model with equal Pop. II and Pop. III efficiency, the peak of the Pop. III SMF is still shifted to the left by half dex compared to the Pop. II. This is because, despite having the same star formation efficiency, Pop. III galaxies mostly form in mini-halos.


The star formation efficiency also regulates the Pop. III/II transition as it directly correlates with the SFRD. For a high efficiency, the SFRD is dominated by Pop. III up to redshift z ~ 15 compared to the fiducial model where Pop. II SF becomes larger than Pop. III at redshift z > 20. The Pop. II SFRD does not significantly change between the three models, and so the change in the Pop. III SFRD also affects the total SFRD. All the models converge at redshift z ~ 13 when even with the highest efficiency the total SFRD is entirely dominated by the Pop. II contributions. The critical metallicity defines the metallicity at which there is a change in the IMF. There is currently no consensus on the value of Z_crit, with two competing models that assume the fragmentation driven by either the carbon and oxygen line or the dust cooling. The first class of models determine a Z_crit ~ 10^-2 - 10^-3 solar metallicity while the seconds give a lower value (~ 10^-6 solar metallicity).


Some studies also argue that the value of Z_crit evolves with redshift due to the effect of the CMB ([Chon2022]). Following simulations in the literature ([Schneider2006, Visbal2020]), in this work we adopted an intermediate value Z_crit = 10^-4 solar metallicity as the fiducial value, hereafter we will also show results for the two extreme values of 10^-2 and 10^-6 solar metallicity. The SMF between the fiducial model and Z_crit = 10^-6 solar metallicity does not change. This is because, in the overdense regions where there is star formation, the metallicity becomes larger than 10^-4 solar metallicity very quickly so that changing Z_crit to 10^-6 solar metallicity does not further accelerate the Pop. III/II transition. However, considering a higher value changes the total SMF as we end up with many more Pop. III stars. The high-mass tail of Pop. III SMF extends up to 10^7.5 solar masses at redshift z = 5 when Z_crit = 10^-2 solar metallicity.


The combined effect of Pop. III and Pop. II makes the total SMF computed from the simulation with Z_crit = 10^-2 solar metallicity single peaked as the high-mass peak coming from Pop. II stars are washed out even at redshift z = 5 so that the total SMF peaks at stellar mass ~ 10^4 solar masses at redshift z = 10 - 5 with a high-mass tail that extends up to 10^9 solar masses. The critical metallicity also has a strong impact on the evolution of the SFRD as it affects both the internal and external enrichment. While decreasing the value of Z_crit from the fiducial value only mildly decreases the Pop. III SFRD without altering the total result, choosing Z_crit = 10^-2 solar metallicity strongly changes the SFRD history at redshift z <= 18. The Pop. III SFRD increases by 1-2 orders of magnitude while the Pop. II decreases by the same amount.


Increasing the Pop. III star formation and simultaneously decreasing the Pop. II impacts on the total SFRD. Since Pop. III stars form less efficiently we predict a lower total SFRD compared to the fiducial model. Overall the critical metallicity heavily impacts both the SFRD and SMF during the Cosmic Dawn and the Epoch of Reionization and affects when the universe transitions from being Pop. III dominated Pop. II. In order to trigger the star formation in a galaxy, our model requires the gas density in the disk to be above a certain threshold. We decided to duplicate this free parameter so that we have one for Pop. III and one for Pop. II. While keeping the Pop. II parameter fixed, we explore the extreme case of a Pop. III critical surface density of 0, which is equivalent to assuming that Pop. III stars start to form as soon as the cooling of the gas begins.


This choice results in a larger abundance of low-mass Pop. III systems at high-redshift. However, since the main changes in the Pop. III SF are in low-mass halos, setting the critical surface density to 0 does not significantly change the SFRD history. Overall, this free parameter is the one with the smallest impact on the star formation history, as it only impacts the low-end of the stellar mass function at high-redshift for which there are no observational constraints. The last parameter we explore is the IMF of Pop. III stars. Our fiducial model is quite similar to the Pop. II IMF as we assume a Salpeter IMF that favours the formation of low-mass stars. We consider two log-normal IMFs labelled logA and logE as in [Tumlinson2006]. For these IMFs we have spectral energy distributions (from [Raiter2010]) which will be used in Section 4.


The IMF logA is centred at M = 10 solar masses, which enhances the probability of a Pop. III stars ending its life as a core-collapse SN. logE is centred at even a higher mass (M = 60 solar masses), so we expect most of the stars with very short lifetimes to end their lives by collapsing into a black hole without any supernova explosion. Compared to the Salpeter IMF, both log-normal IMFs do not have many low mass stars (mass < 8 solar masses). Decreasing the lifetime of Pop. III stars make the Pop. III SMF shift to lower masses, increasing the separation between the two peaks of the total SMF. The difference between the fiducial model and one adopting the logA IMF is less than one dex. However, in the logE IMF model, most of the stars die in the same snapshot in which they form. This effect strongly prevents the build-up of Pop. III systems.


There are no significant differences in the SFRD from adopting different IMFs, except for the first few snapshots of our simulation where the star formation is mildly reduced in the logE model. This result comes from the stronger feedback typical of the more top-heavy IMF, and it has a stronger impact at redshift z >= 24 when Pop. III star formation is dominating. Unconstrained Pop. III free parameters in our model have an impact on the redshift evolution of the SMF and SFRD. However, our model predicts a consistent number density of Pop. III dominated systems with stellar masses between 10^3 - 10^5 solar masses. Here we will explore the UV luminosity of these systems using the model described in Section 2.6 focusing on the luminosity functions at very high-redshift. Given the same total stellar mass, Pop. III systems have larger luminosities compared to Pop. II galaxies, but shine for shorter times.


We show the evolution in the first 10Myr of the SED of a Pop. III galaxy that forms 10^5 solar masses of Pop. III stars at redshift z = 8 and the absolute UV magnitude at different times since the star formation burst. The black horizontal line corresponds to the magnitude computed assuming a continuous star formation throughout the snapshot. For reference, we also show the magnitude of a Pop. II galaxy forming the same amount of Pop. II stars. Firstly, we notice that Pop. III galaxies with continuous star formation and a log-normal IMF are one magnitude brighter than Pop. II galaxies with the same SFR. Once we focus on the instantaneous star formation model for Pop. III stars, the brightness of the galaxy can be boosted (or reduced) by several magnitudes depending on the time since the burst at which we are observing the system [see also Trussler2023]. Even when we consider a Salpeter IMF, the brightness of a Pop. III galaxy changes ~ 2 mags in ~ 10Myr.


This change is more dramatic for the log-normal IMFs (especially for the logE IMF where in 10 Myr the change in absolute magnitude is approximately 7 mags). The large difference between the instantaneous and continuous star formation models for a single Pop. III galaxy will reflect in the entire population and potentially lead to a higher number density of extremely bright Pop. III galaxies. As a result of instantaneous Pop. III star formation, there are chances to observe these galaxies at a time when their luminosities are higher than what is given by the continuous scenario. When looking at the total luminosity function we see a much significant Pop. III contribution to the number density of galaxies around an absolute magnitude of approximately -16 when assuming a LogE IMF for their star formation. Given their larger brightness, a population of Pop. III galaxies with a top-heavy IMF has been suggested as a possible explanation for the abundance of bright galaxies at redshift z > 10.


Without invoking any exotic physics or a revision of the standard LambdaCDM model of cosmology, other possible explanations are a combination of increased star formation efficiencies at high-redshift and reduced feedback [Qin2023], bursty star formation [Sun2023], cosmic variance [Shen2023] and a modified LambdaCDM power spectrum [Parashari2023, Padmanabhan2023]. Here, we show the predicted UV luminosity functions at several redshifts. Results for different IMFs and different star formation efficiencies are summarized, where all the other Pop. III parameters are taken as in the fiducial model. For all the models considered below, our total luminosity function agrees with early JWST observations at redshift z <= 12 (except for the points at an absolute magnitude < -19 where we do not have galaxies in Meraxes). With our fiducial parameters, both with the Salpeter and with the logA IMF, we find that Pop. III systems are not the brightest galaxies at the redshift considered (absolute magnitude >= -14).


However, when we consider a log-normal IMF with a characteristic mass of 60 solar masses, Pop. III galaxies become significantly brighter and at redshift z = 12 and 16 when some of the brightest systems (absolute magnitude ~ -16) are Pop. III dominated. Even though most of the Pop. III galaxies are still very faint (well below the sensitivity of JWST). This result may indicate that Pop. III dominated galaxies are present at redshift z > 12. This result becomes more robust when considering a Pop. III star formation efficiency equal to the Pop. II efficiency. The bright end of the total and Pop. III dominated galaxy UVLF shifts of 1-2 magnitudes (depending on the IMF) at redshift z = 16 and 12. At redshift z <= 11, there is no or little difference in the total UVLF with different star formation efficiency even though Pop. III dominated galaxies are still ~ 1 mag brighter.


Models with a Salpeter or a log-normal IMF centered at 10 solar masses predict that the bright end of the UVLF at redshift z = 16 is impacted by Pop. III dominated systems while at redshift z <= 12 none or very few of the brightest galaxies are Pop. III dominated. Overall, the abundance of bright galaxies at redshift z >= 12 is better explained by the model with the logE Pop. III IMF and a Pop. III star formation efficiency equal to the Pop. II efficiency. For this model at redshift z >= 12 we find Pop. III dominated galaxies with an absolute magnitude of approximately -18 and at redshift z = 16 all the brightest galaxies (absolute magnitude <= -16) are Pop. III dominated. We compared our results with [Trinca2023], and found a good agreement at redshift z <= 12 and absolute magnitude less than or approximately equal to -16.5 for all the models considered.


The main differences are that at redshift z = 9, we do not have any galaxy with an absolute magnitude of approximately -20, and their model has a steeper evolution predicting many more faint galaxies. At redshift z = 16 only models with high Pop. III star formation efficiency and a log-normal IMF reproduce their results. Even though the total UVLF are similar, their Pop. III contribution is significantly lower, especially when compared to the models with high Pop. III star formation efficiency. The main differences in the model that can explain the different results are (i) the homogeneous feedback and (ii) the Pop. III luminosity calculation. In their model, both the radiative feedback and the chemical enrichment are homogeneous; this might overestimate the suppression of Pop. III systems at redshift z ~ 10-20. Regarding the second point, our model accounts for the fact that a Pop. III galaxies might be observed immediately after they formed stars.


Without accounting for this effect, our Pop. III galaxies would have similar luminosity to those predicted by [Trinca2023]. The main limitation of our results is that, given the small size of the simulation, we intrinsically miss the brightest galaxies. Larger simulation volumes will allow us to be conclusive, but this result suggests that if Pop. III stars are still forming at redshift z >= 10-12, and they have a top-heavy IMF, they do not require increased star formation efficiencies to outshine the Pop. II systems. As Pop. III dominated galaxies become much more rare at redshift z <~ 10, this scenario allows us to increase the abundance of bright galaxies only at redshift z > 10 without boosting the bright end of the UV luminosity function at lower redshift hence we achieve a good agreement with early JWST observations from [Donnan2023, Harikane2023] and [Perez2023].


In this paper, we studied Pop. III star formation in mini-halos with an updated version of the Meraxes semi-analytic model of galaxy formation that includes Lyman-Werner background and the streaming velocities. We also implemented external metal enrichment following the growth of the supernova bubbles according to the Sedov-Taylor model. We computed the bubble size distribution function and found that our results agree with [Trenti2009] (most bubbles are smaller than 150 per h ckpc at z = 6 and have a typical size of 100-200 ckpc at z = 5) hence most halos get self-enriched rather than externally polluted. Only low-mass halos (virial mass <= 10^7.5 solar masses) at redshift z < 10 are more likely to get their metals from the IGM. This is a consequence of the small size of the supernova bubbles that results in a small filling factor (0.1% at z = 12 and less than 1% at z = 5).


The free parameters allow us to explore the global properties of Pop. III dominated galaxies. We ran this model on top of a dark matter-only N-body simulation able to resolve all the mini-halos down to ~ 3 x 10^5 solar masses at redshift z <= 30. We explored the impact on the SMF and SFRD of the main free parameters of our model. All models converge at redshift z <~ 10. However, Pop. III star formation efficiency and IMF lead to differences at high-redshift. Finally, we investigated the SED evolution of a Pop. III dominated galaxy for different IMFs. The shorter lifetime of a Pop. III galaxy motivated us to use an instantaneous star formation model when computing the luminosity function. We computed the total and the Pop. III galaxy UV luminosity functions and compared these to the early JWST results in order to study whether the excess of bright galaxies at high-redshift could be explained by a population of Pop. III dominated galaxies.


Having explored different IMFs and star formation efficiencies, our model predicts, for a log-normal IMF with a characteristic mass of 60 solar masses and a Pop. III star formation efficiency comparable to the Pop. II, most of the brightest galaxies at redshift z >= 12 and all at redshift z = 16 (absolute magnitude = - 18) are Pop. III dominated. Adopting a smaller characteristic mass or a Salpeter IMF, Pop. III dominated systems still have an impact on the bright end of the UVLF, but only at redshift z = 16. In conclusion, this work supports the scenario for which top-heavy Pop. III dominated galaxies might explain the abundance of bright JWST galaxies at redshift z >= 12 without requiring very high star formation efficiencies or extremely weak feedback at high-redshift. In this appendix, we discuss the extent to which our approximation on the external metal enrichment based on the filling factor is appropriate and when it fails.


Since we are avoiding computing all the pairs of distance, there might be some galaxies for which we are not getting a correct enrichment. For this reason, we computed in post-processing the distance between all the pairs of galaxies throughout the entire simulation and we counted how many galaxies in Meraxes are marked as externally metal enriched (pristine) even if they are outside (inside) a metal bubble. We repeated this computation for several grid resolutions: N = 16, 32, 64, 128 and 256. Hereafter, we will call "false pristine galaxies" those galaxies that lie inside a metal bubble (and so should be enriched) but in Meraxes are labelled as pristine and "false enriched galaxies" those galaxies that are not reached by any bubble but in Meraxes are labelled as enriched. At lower resolution, the percentage of the false pristine galaxies increases from 4% at N = 128 to 7% at N = 16.


This happens since, as the pixel becomes bigger, the assumption of the galaxies randomly distributed inside each pixel is no longer valid as the clustering becomes much more important. However, when we use a very fine grid (N = 256), the mass fraction of false enriched galaxies dramatically increases at redshift z <= 10. This happens because by redshift z = 10, the typical bubble size becomes larger than the pixel volume, and so we lose all the contribution that overflows outside the pixel. The mass percentage of false enriched galaxies instead is only mildly affected by different resolutions (always below 2%). We then compute the absolute value of the difference between the mass percentage of the false pristine and the false enriched galaxies. This quantity tells us if, statistically, we are reproducing the correct global enrichment of the Universe. We can see that having a high-resolution grid significantly improves the quality of our results.


For N = 128, this difference is always below 2%, and at redshift z < 12 is approximately 0%. This tells us that for a high-resolution grid, globally, we are getting the correct enrichment of the Universe as the mass percentage of false pristine and false enriched galaxies cancels out. This agreement is almost perfect during the EoR, while during the Cosmic Dawn we are underestimating the metal enrichment of ~ 1-2%. Finally, we are also showing the sum between the false enriched and pristine galaxies. This quantity instead tells us what is the mass percentage of galaxies for which we are getting a wrong estimation of the enrichment. Increasing the resolution improves our result, which, for N = 128, peaks at ~ 4% during the Cosmic Dawn. In conclusion, our model for N = 128, while reducing the computational cost, still gives an excellent agreement on global enrichment and a very good agreement on the enrichment of single galaxies.


In this case we assumed that the probability p of each galaxy to be enriched is equal to the metal filling factor multiplied by the two-point correlation function. We also note that at the scales of the metal bubbles (< 1 comoving Mpc) we need to use the non linear bias as the linear case would give an underestimation of one order of magnitude ([Dijkstra2008]). The mass percentage of false pristine and false enriched both peak at 4-5% with the percentage of false pristine galaxies decreasing at higher resolution and the percentage of false enriched galaxies increasing at higher resolution. Accounting for the clustering and using a low-resolution grid still allows us to get an excellent agreement on the statistical properties but, compared to the model adopted in this work, it doubles the mass percentage of the galaxies for which we are getting a wrong prediction. This final check further confirms that the model for the external metal enrichment that we are adopting in this work, is the best one.


In this appendix, we discuss how to evaluate the UV luminosity function of Pop. III galaxies assuming instantaneous star formation. Due to the stochasticity involved in the process of star formation, the burst might occur at any time within a snapshot for a galaxy, resulting in different luminosities when observed. We refer to Delta t as the time delay of the star formation burst happening relative to the end of our snapshot, and draw its value from a random uniform distribution between zero and the snapshot duration. One can then evaluate the luminosity of this galaxy, repeat the exercise for all targets after assigning them different Delta t, and calculate their probability distribution as a function of UV magnitude (i.e. the UV luminosity function). To achieve a more efficient computation, we instead sample Delta t in fine steps. Then we calculate the UV magnitudes of each galaxy for given Delta t_i and evaluate the corresponding luminosity function at the condition of fixed Delta t.


Finally, the luminosity function is obtained by summing all conditional probability distributions as the sum over all conditional luminosity functions multiplied by the probability of each time delay. Note that the probability of each conditional luminosity function is simply calculated based on the size of the time step relative to the total snapshot duration.


The highly neutral inter-galactic medium (IGM) during the Epoch of Reionization (EoR) is expected to suppress Lyα emission with damping-wing absorption, causing nearly no Lyα detection from star-forming galaxies at redshift z is approximately 8. However, spectroscopic observations of the 4 brightest galaxies (H-band magnitude is approximately 25) at these redshifts do reveal prominent Lyα line, suggesting locally ionised IGM. In this paper, we explore the Lyα IGM transmission and environment of bright galaxies during the EoR using the Meraxes semi-analytic model. We find brighter galaxies to be less affected by damping-wing absorption as they are effective at ionizing surrounding neutral hydrogen. Specifically, the brightest sources (H-band magnitude is less than or approximately equal to 25.5) lie in the largest ionized regions in our simulation, and have low attenuation of their Lyα from the IGM (optical depth is less than 1). Fainter galaxies (25.5 mag is less than H-band magnitude is less than 27.5 mag) have transmission that depends on UV luminosity, leading to a lower incidence of Lyα detection at fainter magnitudes. This luminosity-dependent attenuation explains why Lyα has only been observed in the brightest galaxies at redshift z is approximately 8.


Follow-up observations have revealed counterparts in the vicinity of these confirmed redshift z is approximately 8 Lyα emitters. The environments of our modelled analogues agree with these observations in the number of nearby galaxies, which is a good indicator of whether Lyα can be detected among fainter galaxies. At the current observational limit, galaxies with greater than or equal to 2 to 5 neighbours within a 2 arcminute by 2 arcminute area are approximately 2 to 3 times more likely to show Lyα emission. JWST will discover an order of magnitude more neighbours, revealing greater than or approximately equal to 50 galaxies in the largest ionizing bubbles and facilitating direct study of reionization morphology. The transmission of Lyman-α (Lyα) photons has played a significant role in observational studies of reionization due to its highly resonant nature in neutral hydrogen, which provides a direct probe of the ionization state of the intergalactic medium (IGM). Toward the end of reionization, the large-scale neutral hydrogen distribution has been studied with the Lyα forest along the line of sight towards bright background quasars.


For instance, by counting the zero-flux pixels on quasar spectra, [McGreer2015MNRAS.447..499M] concludes that the average neutral hydrogen fraction at redshift z~5.9 should be lower than 6 -- 11 per cent. However, individual sight-lines seem to suggest that large chunks (greater than 160 cMpc) of residual neutral hydrogen still exist even at redshift z<6, requiring late reionization and/or that the underlying accountable sources possess unusual ionizing properties. Damping-wing absorption of Lyα emission lines of quasars also facilitates estimates of the neutral hydrogen fraction during the epoch of reionization (EoR). This is done by directly comparing the observed flux to the intrinsic flux inferred from low-redshift analogues using continuum and metal lines. For instance, among some of the 9 quasars discovered at redshift z>7 to date, high-quality spectra have enabled direct measurement of the neutral hydrogen, ranging from a neutral fraction of approximately 10% to 80%.


High-redshift star-forming galaxies that have detectable Lyα emission are referred to as Lyα emitting galaxies (LAEs). The incidence of LAEs has also been used to determine the neutral fraction. This is because Lyα detection during the EoR could provide evidence for whether galaxies sit in large ionized bubbles, so that their Lyα photons are redshifted deep into the damping wing, and are therefore less attenuated by the intergalactic neutral hydrogen. As the Universe becomes more neutral, observations have shown a rapidly decreasing fraction of LAEs among Lyman break galaxies (LBGs) from redshift z=6 to 8. However, despite the increasing neutral fraction in the high-redshift Universe, prominent Lyα lines in some of the brightest sources have been observed even out to redshift z is approximately 9 with a much higher detection rate of Lyα emission than expected.


In particular, the 4 most luminous galaxies (H-band magnitude is approximately 25) in CANDELS have all been spectroscopically confirmed by Keck as LAEs. In contrast, when targeting relatively fainter galaxies, no Lyα emission was detected at redshift z is approximately 8 while the probability of detecting Lyα was found to be less than 2% and less than 15% at redshifts of approximately 7.5 and 7, respectively. Similarly, more recent larger spectroscopic samples of Lyα emission among high-redshift, bright galaxies find the probability of detecting Lyα varying between 20% and 50% at redshifts from 6 to 9. Apart from possibly having higher intrinsic Lyα emission due to recent star formation, these brightest galaxies are likely located in more overdense regions, and may be surrounded by a larger number of star-forming galaxies producing a more transparent IGM.


In this work, we use a semi-analytic galaxy-formation model to explore the Lyα IGM transmission properties and environment of bright galaxies during the EoR. We focus on both the brightest galaxies, which are analogous to those observed at redshift z is approximately 8, as well as relatively fainter galaxies, including those that can be considered promising candidates for the James Webb Space Telescope (JWST). Our semi-analytic model (SAM) follows a number of astrophysical processes of galaxy formation and self-consistently calculates photon-heating feedback from reionzation as well as recombination in ionized regions. The model has been shown to successfully reproduce a large number of observables including galaxy and quasar UV luminosity functions in addition to the inferred EoR history from the comic microwave background (CMB). This paper is organized as follows. We provide a brief review of our model as well as the newly implemented calculation of Lyα damping-wing absorption in Section 2.


We present our results including the strategy for hunting LAEs in Section 3 before concluding in Section 4. In this work, we set the cosmological parameters based on the Planck 2015 results. We use the Meraxes SAM from the DRAGONS program, which is coupled to a parent N-body simulation providing dark matter halo properties from redshift z=35 to 2. With a mass resolution of approximately 4x10^6 solar masses and a box size of 100cMpc, we are able to properly sample low-mass halos down to the atomic cooling threshold and capture massive galaxies of approximately 7x10^11 solar masses at redshift z=8. These allow us to resolve faint galaxies that are responsible for reionization and study galaxies as bright as M_1600 of approximately -22 mag at redshift z=11 or an H-band magnitude of 24.5 mag at redshift z=8.


Meraxes evaluates the accretion and cooling of gas, stellar evolution and feedback, growth of supermassive blackholes and their impact on galaxy formation, as well as other environmental influences such as mergers and photo-heating from reionization. We also consider dust attenuation and forward model the observed galaxy and quasar UV magnitudes. These allow us to adjust our model parameters in order to match the predicted observables with existing measurements during the EoR. In this work, we use the galaxy catalogue generated by the fiducial model presented in [Qin2017a]. We assume all star-forming galaxies during the EoR are effective at producing Lyα photons and their detectability in Lyα is solely determined by neutral hydrogen present along the line of sight. To evaluate the IGM ionization state, Meraxes couples its UV ionizing photon source model with 21cmFAST.


We divide the simulation volume into 512^3 cells having an equal length of 0.195 cMpc. With an excursion-set algorithm, we compare the ionizing photon budget to the number of present neutral hydrogen atoms and estimate the neutral hydrogen fraction in each cell. The model used in this work has a predicted EoR history consistent with the latest Planck results. It is predominately driven by UV ionizing photons emitted by stars in faint galaxies with supermassive blackholes having an insignificant contribution to the overall reionization. The only regions of ionizing background dominated by luminous AGN are in their immediate environment. The ionization field at redshift z=8 has a volume-averaged neutral hydrogen fraction of 0.86. We see reionization (anti-)correlates with underlying densities and brighter galaxies, in general, are located in larger ionized bubbles. We also confirm that, in these star-forming galaxies, the central supermassive black holes have a negligible role in ionizing the surrounding IGM.


We compute the Lyα optical depth in each cell based on the equation: tau_alpha = L * n_H,0 * x_HI * Delta * (1+z)^2 * sigma_alpha, where L, n_H,0 and Delta are the pixel length, the cosmic mean of the comoving hydrogen number density and its local overdensity in each cell, while sigma_alpha corresponds to the Lyα scattering cross section in neutral hydrogen redwards of the line centre with c and Lambda representing the speed of light and the decay constant for the Lyα resonance. From these two equations, we see that damping-wing absorption increases towards higher redshifts and drops significantly at distances further from the line center. Therefore, we ignore damping-wing absorption at distances longer than our box length (100cMpc) and estimate the total optical depth as the sum of the optical depth in each cell along a sight-line. Here, the optical depth in the i-th cell away from the source along a sight-line where Lyα has shifted to a new wavelength is calculated with the velocity offset of the intrinsic Lyα line and the Hubble parameter at that redshift.


Note that, for each bright galaxy, we randomly select 500 sight-lines and estimate the bubble size and total damping-wing optical depth as well as their associated uncertainties by taking the median and [16, 84] percentiles. There are an increasing number of observed LAEs presenting a velocity offset between the Lyα emission peak and the systemic redshift up to approximately 1000 km/s, as well as a possible correlation showing higher velocity offset with increasing galaxy UV luminosities. This could increase the Lyα transmission in the IGM for brighter galaxies. However, when splitting low-redshift LAEs by their UV magnitudes, we see the opposite with more luminous galaxies showing a lower fraction of LAEs, possibly due to a reduced Lyα escape fraction in deeper gravitational potentials. As current measurements are still inconclusive and limited by the small LAE sample, particularly at higher redshifts, we do not consider the intrinsic line intensity, offset or escape fraction of Lyα.


Since the targets in this work are those mostly sitting in large ionized bubbles, we also expect them to be less affected by the velocity offset. Based on these, we treat the detectability of a LAE governed primarily by its IGM transmission. It is worth noting that fully ionized regions do not contribute to the total damping-wing optical depth. However, to account for resonant absorption by residual neutral hydrogen inside the ionized region, we follow previous work and assume all flux bluer than the circular velocity of the host halo becomes attenuated. This leads to a factor of 2 further reduction to the Lyα transmission so that the transmission T is equal to exp(-tau_total) which equals 0.5 * exp(-tau_D). This results in an optical depth inside ionized regions of approximately 0.69. In this work, we consider galaxies with a total optical depth larger (or lower) than 1 as highly attenuated (or transparent) LAEs subject to damping-wing absorption in the IGM.


The 4 modelled galaxies presented demonstrate some of the characteristic Lyα IGM transmission properties among bright LBGs during the EoR. They also clearly present the real and substantial variations in Lyα detectability within a quite modest range of luminosities (factors of a few). In this section, we use the whole modelled high-redshift galaxy population to quantify how reionization impacts Lyα transmission, and provide guidance for searching for ionized bubbles in the heart of the cosmic reionization epoch. At a given redshift the damping wing optical depth, on average, decreases if the LAE sits inside larger ionized bubbles. Such a transmission-bubble size correlation becomes obvious when the mean density and neutral hydrogen fraction are substituted in the optical depth equation. However, as overdense regions are typically the first to become ionized, we also expect to see a positive correlation between the Lyα transmitted fluxes and large-scale overdensities. This is shown in analysis where we also observe a much larger scatter compared to the transmission-bubble size relation.


When focusing on currently observable galaxies (i.e. H-band magnitude < 27.5), we see that the obscured LAEs are on average located in ionized regions smaller than approximately 7cMpc at redshift z=8 and with an overdensity of less than or approximately equal to 1. The relation between galaxy UV luminosity and Lyα transmission is also explored. In general, brighter galaxies possess higher Lyα transmission as their UV ionizing luminosities are more effective at ionizing the surrounding IGM and therefore creating larger ionized bubbles. However, we also see a bimodal distribution in this relation. After separating galaxies according to their luminosities in each ionized bubble, we find that the two peaks originate from different galaxy populations. Central galaxies, defined as the brightest galaxy in each bubble, dominate the local ionizing photon budget and therefore have lower damping-wing optical depth with increasing luminosities.


However, most galaxies brighter than an H-band magnitude of approximately 25.5 have a similar level of damping-wing absorption and all of them are transparent in Lyα. This is consistent with the 4 observed bright LBGs showing strong Lyα emission and suggests that the brightest, color-selected LBGs are likely to reside in large ionized regions as predicted by generic inside-out models of reionization. On the other hand, fainter satellite galaxies have their ionized bubble size mostly set by the central galaxies. Therefore, we see a horizontal offset towards lower luminosity from the median transmission-luminosity relation of central galaxies to that of the satellites. Comparing between the two galaxy populations, we see that galaxies fainter than an H-band magnitude of approximately 27.5 can only be detected in Lyα when they share the same ionized bubble as another brighter one.


Despite being more than one magnitude fainter than the brightest galaxy in our simulation, another example galaxy possesses a much higher Lyα transmission rate as its high overdensity environment leads to a great number of bright galaxies contributing to the local ionizing photon budget, which pushes the ionizing front much further. Compared to those dominating their UV background alone, galaxies with neighbours in their ionized regions are more likely to reside in high-density regions, and are less obscured by the Lyα damping-wing absorption. To illustrate this in more detail, we select mock (central) galaxies brighter than an H-band magnitude of 27.5 and plot their Lyα transmission fluxes as a function of the number of neighbours. We see a positive correlation between transmission and the number of neighbours -- galaxies with an increasing number of neighbours are more likely to have their surrounding gas ionized to a great distance, and therefore their Lyα emission is less obscured.


It is worth noting that, as transparent LAEs are located in bubbles with a radius of greater than or approximately equal to 7cMpc (larger than the FoV in the mock image), their neighbouring galaxies are also likely to be observable in Lyα. On average, greater than or approximately equal to 4 LAEs can be found in those highly ionized regions. The majority of modelled galaxies with an H-band magnitude of less than or approximately equal to 26 have bright neighbours within a WFC3 FoV. This agrees with preliminary results from an environmental study of 3 confirmed redshift z is approximately 8 LAEs -- up to 6 LBGs brighter than an H-band magnitude of approximately 27.5 were revealed in each of their HST WFC3 images. Finding more of these objects will help understand their properties and impact during the EoR, as well as to infer the ionizing bubble size and to quantify the morphology of reionization. Therefore, in this subsection, we further explore the strategy to search for LAEs during the EoR.


We present the cumulative number density, for all central galaxies with a varying number of bright neighbours as well as for those classified as transparent LAEs. The fraction of modelled LBGs with detectable Lyα emission, p(Lyα), is an informative quantity. Not only can it place constraints on the neutral hydrogen fraction, in a blind survey of Lyα emission among known high-redshift LBGs, p(Lyα) is also an indicator of the detection rate. We find that the success rate for finding LAEs among LBGs regardless of their environment is 100% when targeting the brightest galaxies (i.e. H-band magnitude less than or approximately equal to 25.5). This rate drops significantly towards the fainter end (e.g. 20-30% at H-band magnitude between 26 and 27.5). Considering only LBGs surrounded by bright neighbours can increase the detectability. For instance, at the current observing limit, we find that following up galaxies with at least 2 neighbours can boost the detectability up to 40%.


This number rises to nearly 60% when focusing on galaxies with at least 5 neighbours. However, only less than or approximately equal to 25% of galaxies brighter than an H-band magnitude of 26 have at least 5 neighbours. Therefore, we conclude an optimal survey strategy for finding LAEs is to follow up with spectroscopy all LBGs brighter than an H-band magnitude of 26 while only focusing on fainter ones with at least 2 neighbours. In the coming decade, JWST will allow study of galaxies at magnitudes much fainter than the current limit. To model this we reconsider the detection threshold to be an H-band magnitude of 30, and search for neighbouring galaxies that will be detectable by JWST. For simplicity, we use the same field of view as the mock HST WFC3 survey but assume reduced photometric uncertainties and consider a redshift depth of 0.35. We see the number of neighbours increases by a factor of approximately 10 and that, on average, at least 25 galaxies with an H-band magnitude less than 30 are found in the area surrounding transparent LAEs.


Consequently, we will need to target galaxies with many more neighbours to see an influence of neighbours on the damping-wing optical depth, since having at least 10 neighbours is nearly identical to having at least 0. We find a number of approximately 50 neighbours is optimal in order to reach a detection rate of >50% at an H-band magnitude of less than or approximately equal to 27. At even fainter magnitudes, the Lyα emission from most galaxies will be highly attenuated by damping-wing absorption. In this work, we study Lyα transmission in the IGM by forward-modelling the damping-wing absorption using Meraxes, a coupled galaxy formation and reionization SAM that is consistent with the measured high-redshift LBG luminosity functions and EoR history inferred from Planck. As more luminous galaxies (and their neighbours) are able to ionize surrounding inter-galactic hydrogen to a much larger radius than fainter galaxies, we find the inter-galactic Lyα absorption becomes weaker towards brighter galaxies.


We assume the detectability of Lyα to be determined by the damping-wing optical depth and define LBGs with a total Lyα optical depth less than 1 to be transparent LAEs. Our model shows that transparent LAEs will be found in ionized bubbles that are at least 7cMpc in size at redshift z=8 (where the global neutral hydrogen fraction is predicted to be approximately 86%). In particular, we find that all modelled LBGs with an H-band magnitude of less than or approximately equal to 25.5 are detectable LAEs. This is consistent with the high detection rate of Lyα emission in recent spectroscopic follow-up of the 4 brightest galaxies at redshift z is approximately 8. These results, in addition to the more recent findings of redshift z is approximately 8 LAEs indicate very high LAE fractions among luminous galaxies beyond redshift z is approximately 6.


This is in contrast to measurements of fainter galaxies in deep fields or using less-massive lensed galaxies, which show a significant drop of the probability of detecting Lyα from redshift z=6 to 8. Our simulation suggests that the combination of these measurements at both high and low luminosity provides evidence for massive LAEs being likely to reside in large ionized bubbles. We also find that reionization is more advanced around galaxies in high-density regions compared to those that are isolated, with galaxies having a larger number of bright neighbours being more likely to reside in large ionized bubbles, leading to Lyα emission that is less attenuated by the IGM. This not only provides evidence that overdensity plays an important role in driving reionization, but can also motivate searching for LAEs in high-density environments during reionization.


Transparent LAEs, on average, are located in overdense regions and are found to possess at least 2 (25) neighbouring galaxies brighter than an H-band magnitude of approximately 27.5 (30) within mock images of a 2 arcminute by 2 arcminute field of view. As their neighbours are also within the same ionized bubbles, they are also likely to be observable in Lyα. We find that while nearly 70--80% of redshift z=8 galaxies with an H-band magnitude between 26 and 27.5 experience strong damping-wing absorption, targeting those surrounded by bright neighbours can significantly increase the incidence of Lyα emission. For example, at the current observational limit, 40% galaxies with more than 1 neighbour would be considered detectable LAEs. This number increases to 60% when focusing on galaxies with at least 5 neighbours. Finally, we predict that upcoming JWST observations are likely to reveal a factor of 10 more neighbouring galaxies.


With such a large sample size, we find that targeting galaxies with approximately 50 neighbours will yield a success rate of more than 50% for finding LAEs among LBGs during the EoR. These large samples from JWST will provide insights into the morphology and scale sizes of the ionized regions, and their development and growth with redshift through the EoR, that is just not possible with HST.


We predict the 21-cm global signal and power spectra during the Epoch of Reionisation using the MERAXES semi-analytic galaxy formation and reionisation model, updated to include X-ray heating and thermal evolution of the intergalactic medium. Studying the formation and evolution of galaxies together with the reionisation of cosmic hydrogen using semi-analytic models (such as MERAXES) requires N-body simulations within large volumes and high mass resolutions. For this, we use a simulation of side-length 210 h⁻¹ Mpc with 4320³ particles resolving dark matter haloes to masses of 5×10⁸ h⁻¹ Msun. To reach the mass resolution of atomically cooled galaxies, thought to be the dominant population contributing to reionisation, at z=20 of ~2×10⁷ h⁻¹ Msun, we augment this simulation using the DFOREST Monte-Carlo merger tree algorithm (achieving an effective particle count of ~10¹²). Using this augmented simulation we explore the impact of mass resolution on the predicted reionisation history as well as the impact of X-ray heating on the 21-cm global signal and the 21-cm power spectra.


We also explore the cosmic variance of 21-cm statistics within 70³ h⁻³ Mpc³ sub-volumes. We find that the midpoint of reionisation varies by a change in redshift of approximately 0.8 and that the cosmic variance on the power spectrum is underestimated by a factor of 2-4 at k-values of approximately 0.1-0.4 Mpc⁻¹ due to the non-Gaussian nature of the 21-cm signal. To our knowledge, this work represents the first model of both reionisation and galaxy formation which resolves low-mass atomically cooled galaxies while simultaneously sampling sufficiently large scales necessary for exploring the effects of X-rays in the early Universe. The formation of the first luminous objects during the cosmic dawn resulted in the ionisation of the cosmic neutral hydrogen gas, rendering the intergalactic medium (IGM) transparent to UV photons. This period, termed the Epoch of Reionisation (EoR), constitutes the last major phase change of hydrogen in the Universe and had an impact on subsequent galaxy formation and evolution [BarkanaReview].


A promising probe of this period is the 21-cm hyperfine spin-flip transition of neutral hydrogen which is sensitive to the evolution of the thermal and ionisation states of the IGM [BibleReview]. A number of low-frequency radio telescope arrays are in operation or are planned to detect this signal. Current instruments (MWA, LOFAR, HERA) aim to detect the signal statistically via the 21-cm power spectrum (21-cm PS) [StuartReview]. While a detection has not yet been made, in recent years there has been significant progress in lowering the available upper limits [LOFAR_limits, MWA_limits, HERA_limits]. In addition, the evolution of the all-sky averaged 21-cm global signal (21-cm GS) is being sought with experiments such as EDGES [edges] and SARAS [SARAS3]. In the near future, the Square Kilometre Array (SKA) will provide an unprecedented ability to place observational constraints on the physics of this era by enabling the production of detailed 3-D 21-cm maps showing the distribution and evolution of the cosmic neutral hydrogen [SKAmain].


For interpreting current and future observations it is important that realistic simulations of the early Universe are available and many authors have contributed to this effort [GnedinMadau]. Simulations of the EoR are made challenging by the large range of scales involved. The main drivers controlling the ionisation and thermal states of the neutral hydrogen are respectively the intense UV and X-ray photons from star-forming galaxies [EoRbook]. X-ray photons have mean-free paths of the order of 10s - 100s of Mpc in the high-redshift Universe, while the typical individual ionized hydrogen bubble sizes are ~10 - 15 Mpc [Stu2004, Furlanetto2006]. It has also been shown that simulation volumes of sidelength greater than or equal to 100 h⁻¹ Mpc are needed for convergent reionisation histories [Iliev2014] while greater than or equal to 200 h⁻¹ Mpc are needed for convergent 21-cm power spectra [Kim2016, Kaur2020]. These considerations necessitate simulations capable of resolving structures from a few Mpc in volumes of greater than or equal to 100s Mpc on a side.


At the same time, realistic EoR modelling requires the ability to resolve haloes down to at least the hydrogen cooling limit corresponding to a halo virial temperature of about 10⁴ K and virial mass, which is proportional to the virial temperature over one plus redshift, to the 3/2 power [BarkanaReview]. These so-called atomically cooled haloes provide sites where gas efficiently cools via atomic line transitions to form stars. Thus, to realistically simulate a representative volume of the early Universe, one requires large simulation volumes as well as sufficiently high mass resolutions. Several techniques have been developed to simulate the EoR [GnedinMadau]. Semi-numerical simulations (e.g. [Simfast21, 21cmFAST, Maity2022]) typically associate ionising photon sources with the density peaks of evolved Gaussian random fields. As these models do not require running computationally expensive N-body simulations, they are able to achieve very large volumes [Greig2022c] as well as efficiently explore the available parameter space [CMMC]. Their main drawback is the absence of detailed physics which self-consistently models a realistic galaxy population.


Achieving high resolution in large-volume hydrodynamical simulations is computationally expensive [CROC1, CoDaI, SPHINX, THESAN]. However, the computational overhead associated with hydrodynamical simulations precludes their use in parameter exploration. Semi-analytic models (SAMs) of galaxy formation (e.g. [GalForm, Galacticus, SAGE, SAG, SHARK]) typically take merger trees from comparatively cheaper dark matter-only N-body simulations and evolve key baryonic components which describe the physical processes involved in galaxy formation, growth and evolution using simple but physically motivated prescriptions [SomervilleReview]. Importantly, being based on N-body trees, the galaxies retain their association with the large-scale structure. These galaxy SAMs then provide a realistic galaxy population at a fraction of the cost of full hydrodynamical simulations. Coupling a galaxy SAM with a semi-numerical reionisation code can provide the best of both worlds: large-volume simulations of reionisation with a self-consistent, realistic population of galaxies. In this work, we use MERAXES, developed as part of the DRAGONS program, which couples a galaxy SAM model designed for galaxies in the high-redshift Universe during the EoR with the semi-numerical code 21cmFAST for simulating the reionisation process [Dragons3, RSAGE, Visbal2020, Astraeus].


Additionally, for the first time, we implement the evolution of the neutral hydrogen gas spin temperature into MERAXES, taking into account heating by X-ray photons. We run our updated MERAXES on a new dark matter-only N-body simulation which has a volume of 210³ h⁻³ Mpc³ with 4320³ particles. This is the largest volume on which MERAXES has been deployed (previously 67.8³ h⁻³ Mpc³; [Dragons19]). To achieve sufficient mass resolution (atomic cooling limit at z=20 of ~2×10⁷ h⁻¹ Msun) within our simulations we use DFOREST -- a Monte Carlo algorithm-based code introduced in [DARKFOREST]. This provides a unique dataset modelling both individual galaxy formation and evolution during reionisation in volumes large enough for exploring the effects of X-rays on the 21-cm signal from the cosmic dawn and the EoR. Importantly, this is the first time such a large volume coupled reionisation and galaxy SAM has been performed to study the 21-cm signal into the cosmic dawn. With our large volume, we are able to explore the impact of cosmic variance across the 21-cm statistics.


We use the L210_N4320 (hereafter L210) box of the GENESIS suite of N-body simulations. This simulation is 210 h⁻¹ Mpc on a side and consists of 4320³ dark matter particles of mass m_p = 9.95 × 10⁶ h⁻¹ Msun. The halo mass resolution is ~5 × 10⁸ h⁻¹ Msun based on a minimum of 50 particles. The simulation was evolved from z=99 down to z=5 using the SWIFT code [SWIFT] and the haloes were identified via friends-of-friends by the VELOCIraptor halo-finder [VELOCIRAPTOR]. Halo catalogues are saved over 120 snapshots evenly distributed in dynamical time between redshifts 30 and 5. The merger trees were generated using TREEFROG [TREEFROG]. To increase the mass resolution of the L210 simulation from ~5 × 10⁸ h⁻¹ Msun to the atomic hydrogen cooling limit at z=20 (~2×10⁷ h⁻¹ Msun) we augment it by extending the merger trees to lower mass haloes. This is achieved using DFOREST [DARKFOREST], a Monte-Carlo (MC) based algorithm. We call this new simulation L210_AUG.


DFOREST uses an updated prescription of [Benson2016] for augmenting merger trees and works on what are termed 'simple branches' -- merger tree branches that are composed of a halo and all of its immediate progenitors. To add new haloes to the existing merger trees new simple branches are generated using the algorithm outlined in [Parkinson2008] which employs a conditional mass function, with extra parametrisation derived from the Extended Press Schechter theory [EPS3, EPS2, EPS1]. These new MC branches, by construction, have a higher mass resolution than the N-body trees. Building on the methods employed in [Benson2016], the new branches are used to augment the existing N-body merger tree. For this we first define a mass threshold, M_cut, which serves as a dynamic boundary between the N-body and MC halo populations in the final augmented merger tree. The final augmented merger tree with these MC branches grafted onto it will thus have both N-body as well as MC haloes with the M_cut serving as the barrier separating them.


DFOREST determines the positions and assigns velocities to the newly added MC haloes. We apply the non-linear halo bias prescription from [Ahn2015] on the input dark matter density field from L210 to generate a halo density field. This is normalised and used as a one-dimensional probability distribution from which the MC haloes are assigned their positions by random sampling. We performed a convergence test to determine the resolution providing the best performance and use 512³ cells for our calculations. This method is used to assign positions to every new MC that is not a progenitor of another MC halo. The evolution of the MC haloes' position with time is based on their peculiar velocity field, using the linear continuity equation where the divergence of the peculiar velocity field is related to the difference in the halo density field at two different times. The halo density field uses the linear halo bias from [Tinker2010]. We run a number of tests to ensure that the MC haloes are introduced without compromising the accuracy of the underlying L210 simulation.


We use the MERAXES [Dragons3] SAM which was specifically designed to study the interplay and feedback between galaxy formation and evolution, and reionisation. Since its introduction MERAXES has undergone several updates including AGN feedback [Dragons10] and updates to supernova feedback, recycling and chemical enrichment of the ISM, and reincorporation of the ejected gas [Dragons19]. MERAXES includes detailed, physically motivated prescriptions for processes including baryonic infall, radiative cooling, star formation, supernova feedback, mass recycling, metal enrichment of the interstellar medium (ISM), and reincorporation of gas. At each snapshot, the baryonic content of a dark matter halo increases up to a fraction of the virial mass in the form of pristine primordial gas. This fraction is determined by the universal baryon fraction and a baryon fraction modifier which couples the feedback of reionisation to galaxy formation. This newly acquired baryonic gas is deposited into a shock-heated hot-gas reservoir. A fraction of this hot gas cools radiatively to a much colder gas cloud, which then participates in star formation following the [Kauffmann1996] model.


To model reionisation and investigate the role of photoionisation feedback on the high-z galaxies, MERAXES includes a modified version of 21cmFAST [21cmFAST, 21cmFASTv3]. In MERAXES the density and velocity fields come directly from the N-body simulations while the stellar mass grids are computed realistically by MERAXES making use of the full galaxy properties. We extend the reionisation calculations of MERAXES to additionally follow the evolution of the spin temperature, T_S, of neutral hydrogen by incorporating the heating and ionisation of the IGM by X-rays following the same approach taken within 21cmFAST. The ionisation state of the IGM is determined directly from the stellar mass grids following the excursion-set formalism [Furlanetto2004]. A simulation cell is flagged as ionised if the number of stellar baryons multiplied by the average number of ionising photons per baryon and the escape fraction is greater than or equal to the total number of baryons in the same volume, accounting for recombinations and secondary ionisations by X-ray photons. We account for recombinations inside Lyman limit systems via a sub-grid prescription [Sobacchi2014].


The 21-cm signal depends upon the spin temperature T_S, which is sensitive to the thermal state of the IGM influenced by X-ray photons. Its inverse is a weighted average of the inverse temperatures of the CMB, colour, and gas kinetic temperatures, with weights determined by the Wouthuysen-Field (WF) coupling constant [Wouthuysen1952, Field1958] and the collisional coupling coefficient. Following 21cmFAST, we compute the gas kinetic temperature T_K and the ionised fraction. The thermal evolution of the gas incorporates contributions from Compton heating, adiabatic cooling/heating, and changes in internal energy. These calculations depend on the angle-averaged specific X-ray intensity, which is computed by integrating the comoving X-ray specific emissivity back along the light-cone. We relate the X-ray emissivity to the star formation rate density (SFRD) computed directly from MERAXES. This differs from 21cmFAST, where SFRD is calculated from the density field and collapse fraction. The X-ray emission is characterized by three free parameters: the soft-band X-ray luminosity per SFR (L(X<2keV)/SFR), a threshold energy E₀, and a power-law index α_X.


The 21-cm brightness temperature field depends on the ionization, density, velocity, and spin temperature fields. Below redshift z~25, the signal reflects three broad periods [PritchardReview]: 1. WF coupling (Lyman-alpha pumping), which drives the signal into absorption. 2. X-ray heating, where the IGM is heated and the signal shows an emission feature. 3. Reionisation, where the signal goes to zero. Most studies focus on the post-heating regime assuming the spin temperature is much greater than the CMB temperature, an assumption that breaks down in the Dark Ages and Epoch of Heating. The main drivers of heating are X-ray photons [Furlanetto2006, McQuinn2012]. Large-scale simulations with low mass resolution are unable to simulate the effects of X-rays since the build-up of the stellar mass is delayed. We use MERAXES combined with our augmented N-body simulations for calculations of the full brightness temperature field including contributions from heating, the spin temperature, recombinations, and peculiar velocities. This allows for a more complete picture of the early Universe's thermal history.


SAMs contain many free parameters that require calibration against existing observations. MERAXES involves two sets of calibrations: one for galaxy formation parameters and another for reionisation calculations. We calibrate the L210_AUG simulation by varying a subset of MERAXES input parameters related to star formation efficiency, critical mass, mass loading, supernova energy coupling, and dust scaling. We use the same parameter values for all of our simulations. Following [Dragons19], we calibrate against observed luminosity functions (LFs) and stellar mass functions (SMFs) at z~5-8, using data from [Bouwens2015, Bouwens2021] and [Song2016, Stefanon2021]. We re-calibrated the dust parameter which governs how dust optical depth scales with cold gas metallicity, as our larger simulation volume allowed us to extend calibrations to brighter LFs than were previously available. The second calibration set is for reionisation. The photon budget is influenced by the escape fraction (f_esc), which is poorly constrained. We use a prescription where f_esc evolves with redshift, skewed towards higher redshifts where smaller galaxies with shallower potentials allow more photons to escape.


We tune the escape fraction parameters (normalization and redshift scaling) so that our reionisation history matches measured constraints on the IGM neutral fractions and the integrated optical depth of CMB photons (τ_e) from scattering off free electrons from [Planck2018]. In this section, we demonstrate the full reionisation model from our L210_AUG simulation. We focus on comparing the impact of the missing low-mass haloes from the L210 simulation to illustrate the importance of mass resolution. We note that our model does not include smaller galaxies in molecularly cooled 'mini-haloes', which likely contain PopIII stars, so the appearance of features in this work will be delayed relative to simulations which include them [Yuxiang2020, Yuxiang2021, EmanueleMaria_pop3_1]. The missing small halos in L210 delay the build-up of cosmic stellar mass, and consequently any physical property dependent on it. The L210_AUG simulation, however, includes a realistic galaxy population capable of producing the whole X-ray background, allowing us to present the first large-scale simulation of the thermal and ionisation history incorporating realistic galaxy formation down to the atomic cooling limit.


The L210_AUG box starts to reionise much earlier than L210 owing to the introduction of low-mass galaxies found only in Monte-Carlo haloes. However, the fiducial L210_AUG and L210 simulations both finish reionisation at approximately the same redshift. There are two main reasons for this. Firstly, towards the end of the EoR, reionisation is primarily maintained by larger mass haloes (which are accurately simulated across both simulations) while the lower mass galaxies are more relevant at earlier times. Secondly, the impact of inhomogeneous recombination is different in the two simulations. Since small galaxies, which initiate reionisation in L210_AUG are short-lived, the cosmic gas recombines until sufficiently big galaxies have had time to form and produce enough ionising photons to complete reionisation. To check this, we ran two additional simulations where we turned off inhomogeneous recombinations. Turning off this process results in the augmented simulation reionising much earlier than the un-augmented one, as expected. Ionization field slices show that at a fixed neutral fraction, the L210_AUG simulation has smaller ionized regions, as they are driven by lower stellar mass galaxies at an earlier cosmic time.


The 21-cm global signal for L210_AUG and L210 shows that the augmented simulation has a similar but broader absorption feature, occurring earlier in redshift. This highlights the importance of including low-mass haloes. By including them, the Lyman-alpha and X-ray backgrounds build up at earlier times. The Lyman-alpha background couples the spin temperature to the much lower gas kinetic temperature, resulting in the broader and earlier absorption. The delayed but sudden formation of sources in L210 results in a comparatively rapid build-up of stellar mass and radiation backgrounds. The spherically-averaged 3-D 21-cm power spectrum at fixed spatial scales (k~0.1 and 1 Mpc⁻¹) shows a qualitatively similar evolution for both simulations, but the delayed formation of stellar mass in L210 causes considerable differences in the timing of the peaks. The Lyman-alpha-coupling peak in L210 is delayed by a redshift difference of about 3 relative to L210_AUG. Below z~7, when X-rays have initiated heating and reionisation is underway, the power in both simulations becomes similar.


Even though the large-scale 21-cm power is expected to have three peaks corresponding to Lyman-alpha-pumping, X-ray heating, and reionisation epochs [Pritchard2007, MesingerXRay], we observe only two peaks. The Epoch of Heating peak, expected at z~12 for L210_AUG, is masked by the Lyman-alpha peak, merging into one broad peak due to the timing and build-up of the backgrounds. On the other hand, the small-scale (k~1 Mpc⁻¹) power is characterised by two peaks corresponding to a combined Lyman-alpha-pumping and heating, and an EoR peak [Yuxiang2020]. On small scales, the impact of the Epoch of Heating is harder to disentangle as it primarily impacts larger scales due to the larger mean free path of X-ray photons. The large volume of our simulations enables the exploration of the effects of X-rays on the EoR morphology with a full source population. We vary the X-ray luminosity per SFR, keeping other parameters fixed, considering values from 3.16 × 10³⁸ to 3.16 × 10⁴² erg s⁻¹ Msun⁻¹ yr to encompass a plausible range [Fialkov2017, GreigCMMC2].


Recent results from HERA have ruled out many 'cold reionisation' models corresponding to low X-ray luminosity [HERA_limits], making our lowest luminosity model an unlikely scenario, though useful for developing intuition. The ionisation photon budget is dominated by UV photons, with X-rays contributing at most 10-15 per cent in extreme models [MesingerXRay]. We find that X-rays can hasten reionisation, in agreement with other studies. The 21-cm brightness temperature light cone slices show a delayed but rapid evolution for the lower-resolution L210 simulation. For the low-luminosity model, the signal remains in absorption across our full redshift range, whereas for the high-luminosity model it is mostly in emission. As shown by the global signal, X-ray photons heat the cosmic gas. Lower X-ray luminosity results in colder gas and a stronger absorption dip. Higher X-ray luminosity leads to a hotter IGM, causing the signal to go into emission at z less than or equal to 17 and a merging of the Lyman-alpha-pumping and heating epochs.


Like the global signal, the shape and amplitude of the 21-cm power spectrum (PS) are strongly affected by X-ray luminosity, meaning accurate PS measurements have great constraining power [HERA_limits]. At large scales, the low X-ray luminosity model has the highest power for most epochs due to large temperature contrasts from the cold IGM, and the heating and reionisation peaks merge. The fiducial simulation shows the three expected peaks, with power peaking at z~16. The high luminosity model has less power during all epochs due to reduced temperature contrast. On small scales, the high luminosity model has clearly differentiated Lyman-alpha and heating peaks, with the heating peak merging with the reionisation peak due to the extended Epoch of Heating. Measurement of any statistical signal from a finite volume of the Universe introduces an inherent uncertainty termed cosmic variance. We explore the cosmic variance of the 21-cm signal by dividing each augmented simulation into 27 equal sub-volumes of side 70 h⁻¹ Mpc. Each sub-volume is larger than typical ionised regions and comparable to many state-of-the-art simulations [CROC3, BLUETIDES, SpringelTNG, CODA, THESAN, Dragons19].


We find the range in redshift for reionisation histories among sub-volumes at a neutral fraction of approximately 0.5 is a change in redshift of about 0.8. We see the same trend among the simulations, except features in the high-luminosity simulation occur earlier. For the 21-cm global signal, the fractional change in brightness temperature is higher than in the neutral fraction. During the Epoch of Heating, the scatter in brightness temperature is driven by variations in spin temperature, while during reionisation the scatter in the neutral fraction dominates. The scatter in the 21-cm power spectrum increases for decreasing k-value (larger scales) and decreasing redshift. This is due to sample variance, since there are fewer modes at large scales to average over. At low redshifts, most of the 21-cm emission comes from sparse, isolated neutral patches, leading to considerable scatter. The power spectrum quantifies the variance in amplitudes of a random field. A purely Gaussian-random field is fully specified by its power spectrum [PeeblesLSS]. However, higher-order statistics are required to capture information for non-Gaussian fields.


The 21-cm field is non-Gaussian, especially on small scales and during the final stages of the EoR. Initially, it traces the underlying matter-density field which is Gaussian on large scales. However, once the complex 3D morphology of the radiation fields (ionisation, X-ray or Lyman-alpha) impacts the signal, the statistics deviate from Gaussianity [StuartReview]. Hence the cosmic variance of the 21-cm power spectrum will be larger than the Poisson sampling error. We explore the impact of non-Gaussianity on this uncertainty. The full error-covariance matrix of the PS includes a trispectrum component arising from non-Gaussianity [Mondal2016]. Studies often assume the field is Gaussian and ignore this term. Any deviation from the Gaussian expectation measured from our 27 sub-volumes must occur as a result of non-Gaussianity. We compute the ratio of the measured cosmic variance to the theoretical Gaussian expectation. For a Gaussian field, this ratio is unity. We find similar trends among all simulations, though we caution we may over-predict variance due to our sub-volume size being slightly smaller than needed for convergence.


The ratio of measured-to-theoretical variance increases from small to large k-values, implying large scales are more Gaussian. Our results agree qualitatively with [Mondal2015, Mondal2016], who show that non-Gaussianity grows as reionisation progresses. During early stages (80% neutral), non-Gaussianity's contribution to variance is comparable to the Gaussian term for k between 0.1 and 0.4 Mpc⁻¹. As reionisation progresses (30% neutral), this ratio becomes 2-4 for the same scales. Our model's transition to non-Gaussianity appears to occur earlier than in [Mondal2016], likely due to detailed physics including spin temperature fluctuations. Our results show that assuming the 21-cm field is Gaussian underestimates cosmic variance in ~100 Mpc boxes by a factor of ~2 on scales relevant to current and upcoming telescopes. This paper introduced an updated MERAXES model including X-ray heating and thermal evolution. We utilized a new, large-volume N-body simulation (L=210 h⁻¹ Mpc) and performed Monte-Carlo augmentation to resolve all atomically cooled haloes out to z=20, creating a unique dataset for exploring galaxy formation physics and its impact on the EoR.


We found that including Monte-Carlo haloes significantly impacts the build-up of stellar mass and consequently reionisation, which commences earlier and is more gradual. Lyman-alpha-coupling and X-ray heating also occur earlier in the higher-resolution simulation. The timing and duration of the 21-cm power spectrum peaks are also different. These results underscore the need for both large volume and sufficient mass resolution for EoR simulations. The large volume and implementation of thermal evolution in MERAXES enables exploration of X-ray luminosity's impact on heating. In agreement with semi-numerical studies [MesingerXRay, GreigCMMC2], we show that while X-rays' impact on reionisation history is minimal, their effect on both the 21-cm global signal and power spectrum is appreciable. Observations of the 21-cm PS will thus provide constraints on the X-ray properties of early sources. We explored the scatter in reionisation history and the 21-cm global signal within 27 sub-volumes of side 70 h⁻¹ Mpc. As previously described in [Mondal2016], we find that non-Gaussianity contributes significantly to the variance of the 21-cm PS on all scales, increasing towards smaller scales.


This work is the first study of the error-variance of the 21-cm power spectrum at high redshifts in a model that also includes both a model of galaxy formation and spin temperature fluctuations. We find that the assumption of Gaussianity for the 21-cm field results in underestimating the cosmic variance of the 21-cm power spectrum by a factor of greater than or equal to 2 for the scales relevant for the SKA (k~0.1-0.5 Mpc⁻¹).


Correlations between black holes and their host galaxies provide insight into what drives black hole--host co-evolution. We use the Meraxes semi-analytic model to investigate the growth of black holes and their host galaxies from high redshift to the present day. Our modelling finds no significant evolution in the black hole--bulge and black hole--total stellar mass relations out to a redshift of 8. The black hole--total stellar mass relation has similar but slightly larger scatter than the black hole--bulge relation, with the scatter in both decreasing with increasing redshift. In our modelling the growth of galaxies, bulges and black holes are all tightly related, even at the highest redshifts. We find that black hole growth is dominated by instability-driven or secular quasar-mode growth and not by merger-driven growth at all redshifts. Our model also predicts that disc-dominated galaxies lie on the black hole--total stellar mass relation, but lie offset from the black hole--bulge mass relation, in agreement with recent observations and hydrodynamical simulations.


Extensive low-redshift studies reveal a complex interplay between galaxies and the supermassive black holes that reside at their centres, with clear correlations observed between black hole mass and host bulge mass, total stellar mass, velocity dispersion and luminosity [e.g.][Magorrian1998, Gebhardt2000, Merritt2001, Tremaine2002, Marconi2003, Haring2004, Bentz2009, Kormendy2013, Reines2015]; see the review by [Heckman2014]. These tight correlations suggest a co-evolution between galaxies and supermassive black holes, which may be causal, due to feedback from the active galactic nucleus [AGN; e.g.][Silk1998, Matteo2005, Bower2006, Ciotti2010] or the efficiency with which the galaxy can fuel the black hole [e.g.][Hopkins2010, Cen2015, AnglesAlcazar2017], or coincidental, simply due to mergers causing both black hole and galaxy growth [e.g.][Haehnelt2000, Croton2006b, Peng2007, Gaskell2011, Jahnke2011]. To understand what drives black hole--host co-evolution, it is necessary to study how these correlations change with redshift.


Observing high-redshift black hole--host correlations is fraught with difficulties. Host galaxies are hard to detect since they are often completely outshined by the AGN light, particularly in the rest-frame optical where common stellar mass estimators can be used [e.g.][Zibetti2009, Taylor2011]. Subtracting the quasar light has resulted in host detections out to redshift z is approximately equal to 2 [Jahnke2009, Mechtley2016], but is yet to be successful for detecting the highest redshift quasars at redshift z is approximately equal to 6 [Mechtley2012]. For these quasars, host masses are often estimated using the widths of observed submillimeter and millimeter emission lines, such as the [CII] 158 micron and CO (6--5) lines [e.g.][Wang2013]. However, dynamical masses determined from emission line widths are highly dependent on the assumptions made, such as the gas-disc geometries and inclination angles [e.g.][Valiante2014]. In fact, inclination angle assumptions can change the determined black hole mass to bulge mass ratio measurements by roughly 3 orders of magnitude [Wang2013].


In addition, the emission regions may not trace the spatial distribution of the stellar component of the galaxy, meaning that these dynamical masses may not be representative of the total stellar mass [Narayanan2009]. Determining the black hole masses of high-z quasars is also difficult, with emission-line based estimators relying on calibrations at low redshift. Where these observations are unavailable, Eddington accretion rates are instead often assumed to estimate the black hole mass [as in e.g.][Wang2013, Willott2017], which also leads to large uncertainties. High-redshift studies of the black hole--host mass relations are thus very uncertain. With this in mind, high redshift observations find black holes that are more massive than expected by the local relation, where the canonical black hole--bulge mass ratio is 10 to the power of (-2.31 +/- 0.05) for a bulge mass of 10^11 solar masses [Kormendy2013].


For example, ALMA observations of five redshift z is approximately equal to 6 quasar hosts show black hole to dynamical mass ratios ranging from 10 to the power of -1.9 to 10 to the power of -1.5 [Wang2013]. Similar studies at redshift z is approximately equal to 4--7 [e.g.][Maiolino2007, Riechers2008, Venemans2012] also give estimates for individual quasars of a black hole mass to dynamical mass ratio greater than or approximately equal to 10 to the power of -2, which is significantly larger than the local value if dynamical masses and bulge masses are assumed to be roughly equivalent. This suggests a faster evolution of the first supermassive black holes relative to their host galaxies [Valiante2014], which could potentially be a result of super-Eddington accretion [Volonteri2015]. The high observed black hole mass to dynamical mass ratio relation at high redshift could, however, be a result of selection effects [Lauer2007, Schulze2011, Schulze2014, DeGraf2015, Willott2017].


[Willott2017] suggest that since only the most massive z>6 black holes are observed, if the relation has a wide dispersion then one would expect to see a higher value due to the Lauer bias [Lauer2007]: since the luminosity function falls off rapidly at high masses, the most massive black holes occur more often as outliers in galaxies of smaller masses than as typical black holes in the most massive galaxies. Indeed, [Willott2017] found that black holes with mass less than 10^9 solar masses at redshift z>6 fall below the black hole mass--dynamical mass relation for low redshift galaxies, in contrast to the opposite being true for higher mass black holes. Similarly, [Schulze2014] claim that selection effects are the reason for the observed evolution of the black hole mass--bulge mass relation; on applying a fitting method to correct for selection effects, they find no statistical evidence for a cosmological evolution in the black hole mass--bulge mass relation.


A lack of evolution in the black hole--host relations is consistent with the findings of cosmological hydrodynamical simulations such as Horizon-AGN [Volonteri2016], which observes very little evolution in the black hole mass--stellar mass relation from redshift z=0 to 5, and BlueTides [Huang2018], which finds a black hole mass--stellar mass relation at redshift z=8 that is consistent with the local [Kormendy2013] relation. [DeGraf2015], on the other hand, found that the relation evolves slightly for redshift z greater than or equal to 1 for the highest mass black holes, with a steeper slope at the high-mass end at higher redshifts, making selection effects important. The more statistical study of [Schindler2016] found that the ratio of the black hole to stellar mass density is constant within the uncertainties from redshift z=0 to 5, with a slight decrease in the ratio at redshifts between 3 and 5; this is also consistent with no cosmological evolution in the black hole mass--stellar mass relation.


In this work we explore the evolution of the black hole--host relations with the Meraxes semi-analytic model [Mutch2016]. Meraxes is designed specifically to study galaxy formation and evolution at high redshifts, making it ideal for studying the evolution of black holes and their host galaxies. In this work we use Meraxes, a semi-analytic model designed to study galaxy evolution at high redshifts [Mutch2016]. Using the properties of dark matter halos from an N-body simulation, Meraxes analytically models the physics involved in galaxy formation and evolution. We run Meraxes on the collisionless N-body simulations Tiamat and Tiamat-125-HR [Poole2016, Poole2017]. Tiamat is ideal for studying high redshifts, with a high mass and temporal resolution. Tiamat runs from redshift z=35 to z=1.8, with a box size of (67.8 per h Mpc)^3, 2160^3 particles of mass 2.64x10^6 per h solar masses, and a high cadence of 11.1 Myr per output snapshot at redshift z>5.


Tiamat-125-HR is a low-redshift counterpart to Tiamat, running from redshift z=35 to z=0 with the same temporal resolution, but with a lower mass resolution (1080^3 particles of mass 1.33x10^8 per h solar masses) and larger box size of (125 per h Mpc)^3, more suited for low-redshift studies. Throughout this work, we use the higher resolution Tiamat at high-redshifts, and Tiamat-125-HR for redshift z<2, unless otherwise specified. Meraxes assumes that galaxies reside in the centre of dark matter haloes produced by the N-body simulation. Using the properties of these haloes, Meraxes analytically models the baryonic physics involved in galaxy formation and evolution, such as gas cooling, star formation, black hole growth, and supernova and black hole feedback. These analytical prescriptions involve a range of free parameters, which must be calibrated using observations such as the stellar mass function.


In Meraxes, stars in galaxies reside in three components: an exponential disc, a spheroidal merger-driven bulge and a disc-like instability-driven bulge. Bulges grow through both galaxy-galaxy mergers and disc-instabilities. In Meraxes, we assume that galaxy mergers with merger ratio greater than 0.01 trigger a burst of star formation, by causing shocks and turbulence in the cold gas of the parent galaxy. The galaxy will also accumulate the mass of the secondary galaxy. We assume that the dominant mass component of the primary galaxy will regulate where these stars produced by the burst and the secondary's mass will be deposited. If the primary is dominated by a discy component, the mass is added to the instability-driven bulge. Otherwise, we assume that the new stars will accumulate in shells around the spheroidal merger-driven bulge. In major mergers, where the merger ratio is greater than 0.1 or 0.3, we assume that the stellar disc and instability-driven bulges are destroyed, with all stars placed into the merger-driven bulge.


In our model we assume that the galaxy discs are thin, with an exponential surface density and flat rotation curve. Such discs become unstable if the disc mass is greater than the disc velocity squared times the scale radius divided by the gravitational constant, which equals the critical mass [Efstathiou1982, Mo1998]. Here, we take the disc mass as the combined mass of both gas and stars in the disc, and the disc velocity and scale radius as the mass-weighted velocity and scale radius of the stellar and gas discs. If such a disc instability occurs, Meraxes returns the disc to stability by transferring the unstable mass of stars from the disc to the instability-driven bulge. The Meraxes black hole model was introduced in Q17, and updated to include instability-driven growth in M19. In Meraxes, black holes are seeded in every newly-formed galaxy, with a seed mass of 10^4 solar masses. Black holes then grow by accretion of both hot and cold gas, through the radio- and quasar modes, respectively.


We also assume that black holes grow in galaxy mergers, with the black holes in each galaxy merging together. Black holes accrete hot gas from the static hot gas reservoir around the galaxy, at a fraction of the Bondi-Hoyle accretion rate. We consider this fraction a free parameter, which adjusts the efficiency of radio-mode black hole growth [Croton2016]. This accretion is limited by the amount of hot gas in the reservoir and the Eddington limit. A fraction of this accretion mass is radiated away and so during one snapshot, black holes grow through the radio-mode by the remaining mass. We include the effects of radio-mode AGN feedback by assuming that a fraction of the radiated energy is coupled to the surrounding gas, adiabatically heating a mass which is subtracted from the cooling flow, regulating the accretion of new gas onto the black hole [Croton2006a, Croton2016]. This AGN feedback has no significant effect on the results of Tiamat at redshift z greater than or equal to 2.


Black holes accrete cold gas from the galaxy, when triggered by either a galaxy-galaxy merger or a disc instability. During such an event, the black hole mass grows by a certain amount, where the virial velocity and a free parameter adjust the growth efficiency. For merger-triggered growth, we take the efficiency parameter to be proportional to the merger ratio. For instability driven growth, we consider two separate free parameters. During the quasar mode, black holes are assumed to accrete at the Eddington rate, and thus the mass accreted by the black hole during one simulation snapshot is limited. This can result in the mass being accreted over multiple simulation snapshots. We incorporate quasar-mode AGN feedback by considering the energy injected into the gas during a simulation time-step. We assume that this energy generates a wind that heats the cold disc gas and transfers it to the hot gas reservoir, depleting the supply of cold gas available for the black hole to accrete. If sufficient energy is injected by the quasar, this wind can also eject the hot gas.


We calculate the bolometric luminosities of each black hole in the model following the Q17 method, which assumes Eddington luminosity for all accreting black holes, and self-consistently calculates the duty cycle. We consider the luminosities from both the quasar- and radio-modes of accretion. At high-redshifts the contribution from the radio-mode is negligible. At the lowest redshifts (redshift z less than or equal to 2), the radio-mode becomes a more significant growth mechanism for the most massive black holes, and so their luminosities are enhanced slightly by the addition of the radio-mode luminosity. We convert from bolometric to B-band luminosities using the [Hopkins2007] bolometric correction, and then assume a continuum slope of 0.44 to convert to UV luminosities. We also account for obscuration due to quasar orientation, by scaling the UV luminosity function by a factor related to the opening angle of quasar radiation. In our model we assume a constant opening angle, for simplicity, which is a free parameter in our model.


In M19 we calibrated the free parameters in Meraxes to match the observed stellar mass functions at redshift z=0--8, and the black hole--bulge mass relation at redshift z=0. Using this model, we find that the black hole mass function and quasar luminosity functions are much larger than predicted by the observations. In addition, we note that [Shankar2016] find significant selection biases in the black hole--bulge mass relation---a topic of recent debate [see e.g.][Kormendy2019]. Due to the M19 predictions and this potential bias, we assume that the [Shankar2009] redshift z=0 black hole mass function is a less biased indicator of the local black hole population, and retune the model here to better reproduce the black hole observations. Note that we use the same parameter values for Tiamat and Tiamat-125-HR, and use both simulations to tune the model: Tiamat for matching redshift z greater than or equal to 2 observations and Tiamat-125-HR for redshift z<2.


We find that our results from the two simulations are generally consistent at redshift z is approximately equal to 2, with broad qualitative agreement at higher redshifts. We calibrate the free parameters in the model to match the observed stellar mass functions at redshift z=0--8, the [Shankar2009] and [Davis2014] black hole mass function at redshift z=0, and the quasar X-ray luminosity functions from redshift z=5 to 2. Since [Shankar2016] find that the observed black hole--bulge mass relation is biased to high black hole masses, we also require our model to not over-predict this relation, however we do not otherwise tune to it. We note that our best models produce black hole--host mass relations lower than the observations, consistent with the expectations of [Shankar2009], and have steeper slopes. We find that these criteria are met by a range of free parameter values for the merger-driven black hole growth efficiency, and the definition of a major merger.


We note that all of these parameter sets produce very similar results. As a further check of the black hole population, we plot the black hole accretion rate density as a function of redshift for models with these different merger-driven black hole growth efficiencies. We find that the models with lower efficiencies give black hole accretion histories in approximate agreement with the observations. The larger efficiencies overproduce measurements of the black hole accretion rate density [e.g.][Delvecchio2014]. The opening angle of AGN radiation, theta, adjusts the normalization of the UV luminosity function. We tune this to match the observations, finding a preferred theta of 70 degrees, corresponding to an observable fraction of UV quasars of 18 per cent. We show the quasar X-ray luminosity functions at redshift z=5--0, with X-ray luminosities calculated using the [Hopkins2007] bolometric to X-ray correction.


At redshift z=2 the model and the observations agree remarkably well. At redshift z>2 the model over-predicts the observed quasar X-ray luminosity function at intermediate luminosities, by up to ~0.7 dex at redshift z=4, while at redshift z<2 the model under-predicts the luminosity function at these luminosities. Our model shows better agreement with the observations than previous versions of Meraxes. While the observations show a slight increase in the X-ray quasar luminosity functions from redshift z=4 to 2, the model predicts a slight decrease. In fact, we cannot find a combination of black hole parameters that results in a redshift evolution that matches that of the observed X-ray quasar luminosity function at redshift z>2. However, the key quantity of black hole accretion rate density is predicted by the model to peak at redshift z=2 as observed.


In addition to published uncertainties in the observations, it may also be the case that at higher redshifts X-ray AGN are more likely to be obscured, which is consistent with evidence from a range of X-ray observations [Treister2006, Vito2014, Buchner2015]. Thus we argue that the inability of our model to match the redshift evolution of the X-ray quasar luminosity function may not represent a significant concern. We show the quasar UV luminosity functions at redshift z=5--0. We find that, as with the X-ray luminosity function, the UV luminosity function decreases from redshift z=5 to 0, though it agrees well with observations at redshift z>2. At redshift z<2, however, we note that the faint-end of the UV luminosity function becomes flat, and by redshift z<1 there is a significant disagreement with the observations, with the model producing too many luminous quasars.