We set the specific intensity above the helium ionisation threshold nu_HeII = 54.4 eV to zero, so that there is no reionisation of HeII to HeIII. This is because Meraxes only traces the size of HII bubbles, meaning it does not predict the mean-free path of photons above nu_HeII. This will restrict our temperature model to times earlier than HeII reionisation, thought to complete around z ~ 3 [2008ApJ...682...14F]. We include HeI reionisation, since we expect helium to be singly ionised at the same time as hydrogen [2003ApJ...586..693W]. When constraining the EoR, we also ignore the outputs of our model below z=4, to minimise confusion with the effects of HeII reionisation. In order to produce and test a large number of thermal and ionisation histories, we vary three parameters within Meraxes and the post-processing temperature model. Escape fraction normalisation and redshift-scaling, as well as background ionising spectral slope.


The temperature of the IGM is sensitive to the timing of reionisation, and the timing of reionisation is heavily dependent on the escape fraction of photons from galaxies. We utilise a redshift-dependent, uniform escape fraction for ionising photons, which was shown by [2016MNRAS.462..250M] to allow the model to match electron optical depth and ionising emissivity observations simultaneously. The escape fraction f_esc equals the minimum of [f_5 times ((1+z) divided by 6) to the power of beta, 1.0]. We vary f_5 between 0.03 and 0.12 and beta between 0 and 2.5 to vary the timing and duration of reionisation in the model. These values were chosen to bracket constraints from electron optical depth measurements, producing reionisation histories that finish between redshifts 5 and 10. The spectral shape of the ionising background sets the energy injected per photoionisation, which affects the reionisation temperature and the subsequent cooling rate. We model the stellar spectrum between 13.6 and 54.4 eV as a power law (equation 3), with a slope, alpha, between 0.2 and 5.


The quasar spectral slope in the same frequency range is fixed at 1.57 and the quasar escape fraction is fixed at 1, as in [2017MNRAS.472.2009Q]. The range of spectral slopes considered is both broader and softer than those often used in temperature modelling [2016MNRAS.460.1885U, 2019ApJ...874..154D], which are based on Population II stellar synthesis models. This range was chosen to produce at least one thermal history that is consistent with observations for each reionisation history. We investigate these scenarios when studying the correlations between heat injection, reionisation timing and temperature. When placing constraints on the EoR, however, we restrict alpha to be more consistent with these population synthesis models. For our fiducial model, we start with a 99 per cent ionised (HII and HeII) IGM, with the initial temperature calculated from the UV spectral slope at ionisation and the speed of the ionisation front in Meraxes using fits to radiative transfer simulations, performed by [2019ApJ...874..154D].


If the ionisation front passes through the gas very quickly, the reionisation temperature is decided entirely by the average energy of the ionising photons, the average energy of a photon E_gamma. [2018MNRAS.477.5501K, 1997MNRAS.292...27H], yielding the reionization temperature T_reion is approximately equal to (1 divided by (3 times the Boltzmann constant k_b)) times the average energy of a photon E_gamma. However, [2019ApJ...874..154D] found using one dimensional radiative transfer simulations that the ionisation front can pass through slowly enough for collisional cooling within the hot, partially neutral gas to have a large effect on the reionisation temperaure. It was found that the speed of the ionisation front provided the best estimation for reionisation temperature, as the faster the ionisation front passes through the gas, the less time it spends in the hot, semi-neutral state where collisional cooling is efficient. Ionisation front speeds are calculated in each voxel by finding the distance between ionisation boundaries (where adjacent voxels have different ionisation snapshots) at successive snapshots, and assuming that fronts travel at a constant speed within each 11Myr snapshot.


We take the distance between random points in each voxel, representing our uncertainty at the grid resolution. The reionisation temperature is calculated from the front speeds and the spectral slope of the background, using fits provided by [2019ApJ...874..154D]. The distribution of reionisation temperatures in our box is presented in section 3.2, this approach introduces a correlation between ionising flux amplitude, gas density, and reionisation temperature, which further complicates the picture of patchy reionisation. Since we explore softer spectra than [2019ApJ...874..154D], initial temperatures for models with 3 < alpha <= 5 are given by the lowest of our two upper limits; from 1) the temperature at the speed of the ionisation front with alpha = 3 and 2) the maximum temperature given by the spectral slope in equation 11. This will slightly overestimate initial temperatures for the slower moving fronts, however softer spectra approach their maximum reionisation temperature at slower speeds, so this effect will be small.


The total photo-heating energy, u, is initialised to the mean excess energy of ionising photons, E_excess [2018arXiv180104931P]. Assuming total local absorption of ionising photons, the excess energy E_excess equals the integral from nu_i to infinity of (4 pi J_nu divided by (h_p nu)) times f_i dnu, divided by the integral from nu_i to infinity of (4 pi J_nu divided by (h_p nu)) times h_p(nu - nu_i) times f_i dnu where a factor that accounts for the preferential absorption of different frequencies by different species is f_i equals (n_i sigma_i) divided by the sum over j of n_j sigma_j and j sums over HI, HeI and HeII. Regardless of how long it takes the front to move through the gas, the total amount of energy imparted to the gas will depend only on the background spectrum, assuming the number of recombinations and collisional ionisations that occur as the front passes is small, and that all photons with energies between 1 and 4 Ryd are absorbed within the front.


We note that, in contrast to other works, the reionisation temperature is partly decoupled from the average photon energy. This occurs via collisional excitation cooling described above, from the models in [2019ApJ...874..154D], whereas most previous works assume reionisation occurs very quickly in each region with no cooling in the ionisation front. Using these models lowers Treion without changing the initial u_0, as u_0 only takes into account energy changes via photo-heating, as defined by [2016MNRAS.463.2335N] and [2019ApJ...872..101B]. As a result, we set the initial u_0 to the mean excess energy of the ionising background, described above. Gas on scales below our grid resolution is not homogeneous, and clumping on unresolved scales will increase recombination rates. Increasing the recombination rate will increase temperatures after reionisation by shifting ionisation equilibrium to a more neutral state, allowing more photoionisations to occur, which overcomes the increased cooling rates.


Large scale clumping is taken into account via our spatial grid, of voxel length ~800ckpc. We also expect reionisation to erase much of the small scale clumping due to Jeans smoothing. This still leaves some room for subgrid clumping to affect our results, on scales between our grid resolution and the Jeans length. The increased temperatures due to clumping, while reflective of the total heat inside a voxel, will not represent the wide distribution of temperatures that can exist within the voxel, as it will be dominated by the higher density clumps. In order to compare with higher resolution simulations (see appendix A), and produce a temperature independent of grid resolution, we set the clumping factor to 1. This implies that we are following the gas at the mean density of each voxel throughout the simulation, rather than each voxel as a whole. We note that there is an effect due to the subgrid structure on the background spectrum due to spectral filtering.


This is where intermediary absorbers harden the ionising background by preferentially absorbing lower frequency photons, as the HI ionisation cross-section scales approximately as frequency nu to the power of -3 near the ionisation edge. The amount of hardening is dependent on the structure and clumping of the IGM, as well as the structure or the ionising background. Theoretically this could harden the spectral slope by 3 (see [2009ApJ...703.1416F] appendix D) but measurements of the column density distribution of absorbers suggest a hardening of the spectral slope by approximately 1 [2010ApJ...721.1448S]. However modeling this filtering, as well as the spatially dependent spectrum of the ionising background, is beyond the scope of this work. Two approximations are made when evaluating temperature evolution that should be noted. First, equations 6 and 7 are applied to the Meraxes grids, rather than individual parcels of gas. Second, we calculate temperatures in post-processing, this assumes independence of each voxel with regards to the temperatures of other voxels, and that any gas influx is at the same temperature as the gas within the voxel.


These assumptions can cause some spurious heating or cooling as gas moves through the box, since this will be treated in the same way as structure growth (via the third term of equation 7). Since neighboring voxels tend to have similar ionisation histories and densities, combined with dark matter velocities that are fairly low compared to the voxel sizes, we do not expect this to have a large effect on temperatures. We tested the effect of gas diffusion on the model by running a model where an extra term was added to equation 7 to account for the gas of differing temperature entering the voxel. We keep track of the bulk flow of matter via the Tiamat velocity grids, and conservatively assumed that the temperatures of adjacent regions scaled with the maximum temperature density slope found in our other models, Temperature T is proportional to (1 + delta) to the power of 0.6. Under these assumptions, heating rate differences of order ~ 10% were observed in high density regions, and changes of order ~ 1% were observed at mean density.


Considering that this model overestimates the heating changes due to gas diffusion (nearby voxels are likely to be at similar temperatures shortly after reionisation), we believe that the independence of voxels is a safe approximation for the purposes of this work. The independence of voxels within our model also excludes a treatment of recombination emission, since inhomogeneous recombinations are not yet included in Meraxes. The effects of recombination emission are detailed in [2009ApJ...703.1416F] where they find it can contribute up to 10% of the ionising background for hydrogen. The effect of a 10% increase in the photo-ionisation rate will not greatly affect our results (see Appendix B). However, recombination emission could soften the background spectrum by approximately 1, since the photons from hydrogen recombination will tend to have lower energies than those in the rest of the ionising background. While this would cause a decrease in temperatures, the effect is degenerate with our free parameter for the background spectral slope, so modelling this effect is outside the scope of this work.


We assume that there is negligible heating by HeII reionisation before z=4 [2016MNRAS.460.1885U]. If Helium reionisation is a highly extended process, then we would be underestimating the temperature, and overestimating the IGM cooling rate for any given reionisation history. With our parameterisation, this would bias our models to earlier reionisation scenarios (due to the flatter temperature gradient), with harder spectral slopes (due to the higher temperatures). The reionisation model in Meraxes assumes local absorption of ionising photons. Photons that would redshift below the ionisation threshold in a full radiative transfer model can still ionise in Meraxes. Since Meraxes is tuned to match the ionising emissivity measurements of [2013MNRAS.436.1023B], which were calculated based on radiative transfer models, J_21 could be overestimated by a small amount. This will not have a large effect during Hydrogen reionisation, as the size of HII regions are small enough to make local absorption a good approximation.


Furthermore, we find that temperature at all redshifts is insensitive to the value of J_21 after the region is reionised, as long as it is large enough to maintain a highly ionised IGM (Appendix B). We only vary the escape fraction (equation 10) to produce different reionisation scenarios, while keeping the same source model. This means that we ignore any degeneracy between our escape fraction parameters and source modelling with regards to the IGM thermal state. In particular the constraints we find due to the scatter in temperatures are likely optimistic, as the patchiness of reionisation will have a significant effect on the range of temperatures observed afterwards. Shock heating of high density regions as they collapse is also not modeled in this work, as we are primarily interested in the diffuse IGM. We only track the increase in temperatures from structure growth on the scale of our grid. This will exclude the ~ 10^5 K gas that exists in hydrodynamic simulations.


However, studies of IGM temperature have so far focused primarily on gas at or below the critical density, excluding shock heated gas in their analyses [2018arXiv181011683O]. Using the temperature evolution model described in section 2, we calculated the thermal and ionisation states of the same 10,000 randomly chosen voxels for 2750 Meraxes realisations, in order to obtain a wide range of density and ionising flux histories per model, as well as a wide range of global reionisation and thermal histories. We have also computed the evolution of one full box, to examine the topology of the temperature field. The model parameters used in the illustrative examples within this section are {f_5, beta, alpha} = {0.08, 1, 2} on the full 128^3 Meraxes box. This run was chosen as all the correlations between reionisation timing, density and temperature of reionisation are clearly shown in its results. However, this is not our highest likelihood model, based on observational data.


Thermal history of 3 ~ 800kpc^3 voxels in Meraxes, showing the effect of density history and ionisation timing on the thermal state. Top panel: Density contrast vs redshift. Middle panel: Temperature vs redshift. Bottom panel: Integrated thermal history, showing the total amount of energy injected by photo-ionisations per unit mass. Temperature and overdensity of a full 100 x 100 x 0.8 Mpc slice of the example model run with parameters {f_5, beta, alpha} = {0.08, 1, 2} at redshifts 4(bottom), 5(middle) and 6(top). An anti-correlation between temperature and density can be observed towards the end of reionisation (z=6), with very hot voids. This state then decays, forming a tighter positive correlation between T and delta. 100 x 100 x 0.8 Mpc redshift of reionisation slice (left) compared to the temperature just after the region ionises (right), showing an inverse correlation between them, with early reionisation corresponding to low reionsiation temperature. As the ionisation fronts speed up in the cosmic voids, the gas is ionised to a hotter temperature.


A few examples of thermal histories output by our model are given in Figure 1. Each voxel will experience a sudden rise in temperature when it is ionised, dominated by photo-ionisation heating. Afterwards, the voxel cools on a cosmological timescale at a rate determined mainly by the ratio of photo-heating and recombination cooling at ionisation equilibrium, as well as adiabatic cooling and inverse compton cooling off the CMB. Density evolution will modulate the temperature, but the previous density history of a voxel will have little effect on its thermal asymptote [2016MNRAS.456...47M]. However, the previous density history will have a substantial effect on the integrated photo-heating, u. Each voxel approches its thermal asymptote within a redshift interval Delta z is approximately equal to 1-3 of its reionisation, creating a distribution in temperatures in the whole box dependent on the inhomogeneous reionisation history. Figure 2 shows a 128^2 voxel slice of the temperature model at selected redshifts between 6 and 4.


It can be seen that shortly after reionisation, at z = 6, the high density regions reach temperatures of T is approximately equal to 2 x 10^4 K, as they have recombination rates high enough to allow continuous photo-heating of the gas. The larger voids also reach similar temperatures, as they reionise last due to the “inside-out" nature of the EoR in our models, and hence have the least time to cool. The coolest regions are the low density regions in close proximity to high density regions, which are reionised early by stars in the nearby dense filament, and cool over time as their recombination rates are not high enough to maintain high levels of photo-heating. Long after reionisation, at z=4, the hotter low density regions cool so that temperature is closely correlated with density. The ionisation topology matches that of previous works modeling inhomogeneous reionisation [2008ApJ...689L..81T, 2012MNRAS.421.1969R, 2018MNRAS.477.5501K, 2014ApJ...788..175L, 2018arXiv181011683O].


Comparing Figure 2 with the left panel of Figure 3, which show the redshifts of reionisation and temperatures within the same region, we see that shortly after reionisation there is a notable anti-correlation between temperature and redshift of reionisation. Over time this correlation diminishes, and the correlation between temperature and density becomes dominant. Reionisation temperature distributions in our grid Top panel: redshift of reionisation distribution. Middle: distribution of reionisation temperatures at the time of their reionisation. Bottom: reionisation temperature distributions for voxels reionising at specific redshifts, showing the increasing reionsation temperature as reionisation progesses. Note that these are the temperatures that voxels reionise to for the subset of voxels that reionise near a specific redshift, not the temperatures of all voxels at the same redshift.


In this section we present the reionisation temperatures in our fiducial model, using fits for the reionisation temperature provided by [2019ApJ...874..154D], to convert ionisation front velocities to reionisation temperatures. Using these fits allows us to include the correlations between temperature, ionisation history and the photon background in more detail; this reduces the number of free parameters we need to include. The temperature to which a region of the IGM is reionised to is inversely correlated with its redshift of reionisation, as shown in Figure 3. This anti-correlation results from the fact that the ionisation fronts speed up as they enter the low-density regions towards the end of the EoR, resulting in higher temperatures because there is less time for collisional cooling to take effect. Differences in simulated reionisation history, and to a lesser extent grid resolution, snapshot cadence, and algorithm to find ionisation front speed result in changes to the distribution.


Figure 4 shows the distribution of reionisation temperatures, compared to redshift of reionisation. As shown in the top panel, reionisation in this model occurs in the redshift range 6 is less than or approximately equal to z is less than or approximately equal to 12. The middle panel shows the full distribution of temperatures to which regions ionise. We find a similar range of reionisation temperatures as [2019ApJ...874..154D], between (15-25) x 10^4 K. The bottom panel splits up this distribution into regions that ionised at different times, showing that the temperature of the ionisation fronts increases as reionisation progresses. Lower spatial resolution, as well as our discrete timestep, likely smooths out the extreme ends of the distribution compared to [2019ApJ...874..154D], and results in a broader distribution of temperatures at any given redshift. However it is difficult to directly compare the distributions due to differences in reionisation history.


The excursion set formalism used to predict reionisation history in Meraxes is likely too crude to predict accurate ionisation front speeds on scales of individual voxels, as the spherical symmetry assumed by 21cmFAST effectively averages over many cells during the later stages of reionisation. However, since we recover a similar range of speeds and reionisation temperatures as [2019ApJ...874..154D], and the reionisation temperature increases as reionisation progresses, we consider this approach to be a valid approximation for testing the statistics of IGM temperature and more accurate than assuming a constant reionisation temperature for all voxels. Temperature density relation for 10,000 voxels in our model at redshifts 4 to 6, showing the large scatter in low density regions at early times, followed by a late time power law, with slope approaching gamma - 1 is approximately equal to 0.5 by redshift 4 for all densities. The solid black line shows mean temperature at binned denisties, and the dotted black lines show the 5th and 95th quantiles in each bin. The colours of each dot show the redshift at which the voxel ionised.


Figure 5 shows the temperature-density relation at various redshifts in our fiducial model. The temperature-density relation (TDR) is a powerful probe of the conditions in the IGM. The shape and scatter in the relation at various densities can reveal information about ionisation history and structure of the IGM during the EoR. A power law fit T=T_0(1+delta)^(gamma-1) is commonly used when characterising the thermal state of the IGM. However, this is only accurate for regions that ionised homogeneously or long before the measurement is taken. Restricting the study of temperature to a power law misses much of the information in the temperature density relation that can be used to constrain a patchy EoR [2018arXiv181011683O, 2019arXiv190704860W]. There is a wide distribution of temperatures in low density regions shortly after reionisation, as the temperature at low density is highly dependent on the redshift of ionisation at this time.


The large scatter lasts long after reionisation itself, halving approximately within a redshift interval Delta z is approximately equal to 1 after reionisation in most of our models. Long after reionisation, regions of all densities cool to their asymptotic temperature, and the temperature density relation is well described by a power law T = T_0(delta + 1)^(gamma - 1), with a slope approaching gamma - 1 is approximately equal to 0.6 by redshift z=4. This is consistent with other inhomogeneous reionisation models and analytic calculations of the thermal asymptote [1997MNRAS.292...27H, 2016MNRAS.456...47M]. High density regions do not show much variance in their temperatures, as their reionisation temperatures tend to be much closer to their early-reionisation asymptote. As a result high density regions settle much more quickly into the late time power-law. Furthermore, dense regions tend to ionise earlier, so any scatter at the high end of the temperature density relation will likely have disappeared by the time of measurement.


In agreement with other inhomogeneous reionisation models [2008ApJ...689L..81T, 2012MNRAS.421.1969R, 2018MNRAS.477.5501K], we find that the large scatter in low density regions shortly after reionisation is highly correlated with the redshift of ionisation of the region (shown in the colours in Figure 5). The size of this scatter, and its correlation with z_r, is what causes the changes in the T-delta slope. The scatter in temperatures can be a powerful probe of the patchiness of reionisation, as it shows differences in reionisation redshift between different regions of the low-density IGM, allowing us to quantify the patchiness of reionisation, as well as estimate how long ago reionisation occurred as the scatter diminishes (see section 4.1). Once every region has reached its thermal asymptote, and the TDR settles into a power law, the thermal memory of reionisation has been lost, as regions that ionised at different times approach the same temperature, based on their density.


[2008ApJ...689L..81T, 2009ApJ...701...94F] and [2012MNRAS.421.1969R] note an inversion of the temperature-density relation, with gamma-1~-0.2, at low densities shortly after reionisation, meaning the lowest density regions are hotter on average than mean density regions at the end of the EoR. This inversion is due to the lowest density regions ionising last, hence having less time to cool, as well as ionising to higher temperatures due to faster ionisation front speeds. In the above model, we find this inversion towards the end of reionisation, lasting until z is approximately equal to 5 as there are many hot low-density regions that have recently ionised, and have yet to cool towards the asymptotic power-law relation. The average slope of the TDR is highly dependent on reionisation history and spectral slope, as they alter the strength of the correlations between density, redshift of reionisation, and temperature. As a result, inversion in the TDR is not observed in all of our models, although the slope will always be at a minimum towards the end of the EoR.


In this section we investigate how the distribution of temperatures in the IGM can be used to probe reionisation. We have run a suite of realisations of our model, with different escape fractions and background spectral slopes (see section 2.4), controlling the timing and duration of reionisation, as well as the temperature of reionisation and subsequent cooling rates. In order to simulate earlier and later reionisation scenarios, we vary the escape fraction normalisation and redshift scaling (equation 10) in Meraxes. A higher (lower) escape fraction results in more (less) ionising photons escaping galaxies, and therefore an earlier (later) reionisation, and a colder (hotter) IGM at a fixed redshift, while thermal memory of reionisation still exists. Models with different escape fractions will tend towards the same temperature at late times, as all regions approach their thermal asymptote. The effects of the escape fraction parameters on global redshift of reionisation (defined as when the global neutral fraction is 10 percent) can be seen in Figure 6.


Reionisation history also has an effect on reionisation temperatures, by changing the speed of ionisation fronts in the IGM. Figure 7 shows the temperature at mean density, it's standard deviation, and the energy injected from photoionisations at z=4.5 versus the model's global redshift of reionisation (where the mass-weighted neutral fraction falls below 0.1) for varying spectral slopes. As noted in section 3.3, both the mean and spread of temperature are maximised shortly after the bulk of reionisation occurs. The mean temperature will decrease towards an asymptote dependent on the background spectrum, and the scatter will decrease towards zero. A harder spectral slope will impart more energy to the IGM on average per ionising photon, increasing the ratio of the photoheating rate to the ionisation rate. As a result, the thermal asymptote of each voxel becomes hotter. The reionisation temperature also increases, however this can be limited by collisional cooling if the ionisation front passes through the IGM slowly (see section 3.2 and [2019ApJ...874..154D]).


Changes in temperature due to reionisation timing and those due to background spectrum are difficult to differentiate using temperature measurements alone. However, mean temperature and scatter in temperature have different correlations with the timing of reionisation and background spectrum, and will evolve at different rates over time. As a result, observations of mean and scatter in IGM temperature at multiple redshifts can be used to break the degeneracy between the timing of reionisation and the background spectrum, offering tighter constraints on the redshift of reionisation. The integrated photo-ionisation energy at mean density, u_0, can be used to further tighten constraints. Unlike temperature, an earlier reionisation increases the photo-ionisation energy at mean density, u_0, since residual photo-heating that occurs after reionisation will have been occurring for longer. Because injected energy and temperature have opposite correlations with redshift of reionisation, their observations can be used together to simultaneously constrain the background spectral slope and the timing of reionisation [2016MNRAS.463.2335N, 2019ApJ...872..101B].


In order to explore how temperature observations constrain the EoR, we perform mock observations on our simulation. We create mock observations on the model discussed in section 3 of temperature at mean density T_0, scatter of temperature at mean density sigma(T_0), and injected photo-ionisation energy at mean density u_0 at redshifts 4.5, 5.0 and 5.5, similar to the redshifts of the temperature measurements from [2019ApJ...872..101B], which are 4.2,4.6, and 5.0. T_0 and u_0 are derived from T and u by binning their respective distributions around mean density Delta = 0 +/- 0.1 and averaging. We use 0.125 dex as a 1 sigma measurement error, of similar size to the largest uncertainties given in [2019ApJ...872..101B]. By comparing the mock observations to our series of models, we then estimate the timing of reionisation and background spectral slope, to see how well the input parameters can be recovered.


We estimate the likelihood of each model assuming Gaussian errors, based on the true values from one model. The likelihoods of each of our models, compared to these mock observations from our model (with parameters {f_5, beta, alpha} = {0.08, 1.00, 2.00}, and redshift of reionization is approximately equal to 6.84), are shown in Figure 8. It is important to note that we do not create mock Lyman alpha spectra, due to the low resolution of our simulations. Here, “mock observations" refers to the summary statistics of the IGM thermal state, that we generate from one of our models with some assumed error margin. These statistics correspond to the temperature and energy measurements attained from the analysis of Lyman alpha spectra, for example those published in [2011MNRAS.410.1096B, 2018arXiv180804367W] and [2019ApJ...872..101B].


We recover a strong peak in alpha, and as seen in the contours of the beta versus f_5 panel, the highest likelihood models are those with the same redshift of reionization as the true model. However, we cannot recover our two escape fraction parameters independently with this sample, as our observables are more sensitive to the timing of reionisation than the duration of it. We will require more precise observations, across a wider range of redshifts to begin to distinguish these scenarios using the scatter in temperature. A more extended reionisation will have a wider distribution of temperatures, from a wider distribution of reionisation times. Contours of equal likelihood of our models when compared to each observable separately at redshifts 4.5, 5 and 5.5 with 0.125 dex errors, showing the degeneracies of each thermal statistic in redshift of reionization and alpha. Left panel: mean temperature at mean density. Middle panel: standard deviation of temperature at mean density. Right panel: photo-ionisation energy at mean density.


We next present the constraints from each observable separately. The different correlations of temperature at mean density, its scatter, and injected energy with redshift of reionisation and background spectral slope allow us to constrain these reionisation parameters simultaneously. A similar result was demonstrated by [2019ApJ...872..101B], using the integrated thermal history to break the degeneracy between reionisation timing and initial temperatures. Figure 9 illustrates this for the {f_5, beta, alpha} = {0.08, 1.00, 2.00} model, showing the constraints from mock observations of mean temperature, scatter and photo-heating separately on the redshift of reionisation and background spectrum. While each observable alone is degenerate between the timing of reionisation and the background spectrum, their degeneracies differ in magnitude and direction, allowing us to perform this analysis. We have applied this analysis to each of our models and present the precision achievable for recovery of reionisation redshift and spectral slope in Figure 10.


We can recover the redshift of reionisation in our model within a redshift interval Delta z is approximately equal to 1 and spectral slopes within a slope interval Delta alpha is approximately equal to 2 to 95% confidence, using an error margin of 0.125 dex for temperature and injected energy observations. As discussed in section 4.1, our mock observations with errors corresponding to existing datasets were unable to independently recover escape fraction normalisation and scaling. We now examine a much more optimistic dataset from possible future observations, where temperatures at mean density, scatters, and injected energies are known within a 1 sigma error of 0.05 dex at 9 evenly spaced redshifts between 4 and 6. We show the results for this case in Figure 11 for the parameter set {f_5, beta, alpha} = {0.08, 1.00, 2.00} and Figure 12 for all models. The recovery of parameters is much more precise in this scenario.


With such an extensive dataset we can recover the redshift of reionisation within a redshift interval Delta z is approximately equal to 0.5 and background spectral slope within a slope interval Delta alpha is approximately equal to 0.5 also to 95% confidence. This example is also closer to recovering the escape fraction normalisation and scaling independently. These mock measurements of the scatter in temperatures disfavour reionisation scenarios that are too extended or too sudden, however there is still a significant degeneracy between these two recovered parameters. Considering that this is a very optimistic dataset, recovering the duration of reionisation from its thermal state will likely require a more detailed analysis. In the last section, we demonstrated how IGM thermal state observables constrain the nature of the EoR using mock observations of our suite of models. In this section, we apply this process to recent observations of the thermal and ionisation history of the universe, in order to constrain our model parameters.


We constrain against measurements of temperature at mean density from [2019ApJ...872..101B], and electron optical depth measurements from [2018arXiv180706209P]. Temperature measurements from [2011MNRAS.410.1096B] and [2018arXiv180804367W] were also consdidered, but model likelihoods on these datasets will not be presented in this paper. The former dataset produces similar but looser constraints on reionisation, whereas we are unable to match temperatures from the latter dataset due to the sudden increase in temperatures at z=5. Figure 13 shows the observations utilised, and maximum likelihood models based on each temperature dataset. Electron Optical depth measurements place constraints on the redshift of reionisation, while temperature measurements constrain both the timing of reionisation and the background slope. We restrict our attention to temperature at mean density, T_0, and electron optical depth, tau_e. No measurements currently exist for scatter in temperatures, so we are unable to use this observable for our EoR constraints, as in our mock examples.


Regarding u_0, we are unable to reliably relate the integrated thermal history in the optically thin models that produce this measurement in [2019ApJ...872..101B], with the thermal histories in our patchy reionisation models. A patchy reionisation effectively has a much harder spectrum within ionisation fronts, reducing the effective spectral slope by approximately 3 due to all ionising photons being absorbed within the front. This creates a hotter reionisation, but most importantly, permanently offsets the injected energy by a certain amount compared to an optically thin model, because u_0 is a time-integrated statistic. Since we cannot model how u_0 affects the small scale Lyman alpha power spectrum in our patchy reionisation models, we do not include u_0 in our fiducial constraints. We have ignored temperature measurements at redshifts z<4 to minimise confusion resulting from the beginning of HeII reionisation.


If a substantial amount of HeII ionisation has occurred at the redshift of measurement, this could be confused with a hotter post HI reionisation IGM with a flatter or positive evolution. This would result in a bias towards models where the overall cooling rate is suppressed, either due to a harder ionising spectrum or earlier reionisation, where the gas is closer to its thermal asymptote and would show a flatter evolution. Given the strong degeneracy between escape fraction parameters, we instead show constraints on alpha and the redshift of reionisation, z_reion. Figure 14 shows our constraints on these parameters based on the measurements from [2019ApJ...872..101B] and [2018arXiv180706209P]. We find a strong degeneracy between z_reion and alpha where T_0 alone is considered. Using tau_e to break the degeneracy results in a late reionisation redshift of reionization equals 6.8 with an uncertainty of +0.5 / -0.8 and a soft ionising background alpha equals 2.8 with an uncertainty of +1.2 / -1.0.


In order to match temperature observations at z>4 as well as observations of electron optical depth, our simulations favour a softer UV spectrum than assumed in other works. This may result from tension between the observations, with temperature observations favouring an earlier reionisation than electron optical depth measurements. This is certainly the case when we restrict the background spectral slope to ranges supported by stellar population synthesis models, with 0.5 < alpha < 2.5, given by the red shaded region in figure 14 (see [2016MNRAS.460.1885U] and [2019ApJ...874..154D] for discussions of possible background spectral slopes) which tightens our reionisation timing constraints to redshift of reionization equals 6.9 with an uncertainty of +0.4 / -0.5. However many models without this restriction, with alpha < 2.5 and a slightly earlier reionisation redshift of reionization is approximately equal to 7 are within 1 sigma uncertainty of our results. Since the duration of reionisation also has an effect on temperatures via the ionisation front speed (see section 3.2, [2019ApJ...874..154D]), models with harder spectra require slower reionisation histories in order to agree with both temperature and CMB measurements [2019ApJ...872..101B].


As a result constraining reionisation duration will result in tighter constraints on spectral slope. We also note that spectral slopes alpha < 0.5 are ruled out at 2 sigma, as these spectra produce temperatures that are too high compared to observations, even when reionisation occurs relatively early. Using an inhomogeneous reionisation model, we have probed the full distribution of IGM temperatures, and its correlations with structure growth, the ionising background and patchy reionisation. We have begun to show how these correlations can be used to characterise the EoR, placing simultaneous constraints on the timing of reionisation and the background spectrum. We recover thermal behaviours from other inhomogeneous reionisation models [2012MNRAS.421.1969R, 2008ApJ...689L..81T, 2017ApJ...847...63O, 2018MNRAS.477.5501K, 2009ApJ...701...94F, 2018arXiv181011683O], where shortly after reionisation, there is a large scatter in the low-density end of the temperature-density relation, and a strong correlation between temperature and redshift of ionisation.


By redshift z = 4, the temperature correlation shifts towards density, creating a power law temperature density relation, leaving no memory of the redshift of reionisation in the low-density IGM. The large initial scatter in the temperature density relation is highly correlated with the redshift of reionisation of the region, differentiating between regions near the cosmic filaments, which reionise early, and those in large voids, ionising later. Temperature measurements taken while the low-density scatter still exists contain information concerning when a region reionised, and the mean and scatter of multiple measurements can be used to constrain the reionisation history. However interpretations of these measurements will be highly dependent on the nature of the ionising sources, and the structure of the gas in the region. Performing mock observations of the mean and scatter of the temperature density relation at different redshifts illustrates the potential for constraints on the EoR.


Using uncertainties similar to those for recent temperature measurements [2019ApJ...872..101B, 2018arXiv180804367W], we can recover the redshift of reionisation from our mock observations to 1 sigma within a redshift interval Delta z_reion is approximately equal to 0.5 and background spectral slopes within a slope interval Delta alpha is approximately equal to 1. Comparing our suite of models to electron optical depth and temperature measurements, our modeling favours a reionisation history that finishes around redshift of reionization equals 6.8 with an uncertainty of +0.5 / -0.8 and a UV background with a power-law spectral index of alpha equals 2.8 with an uncertainty of +1.2 / -1.0 (1 sigma uncertainties) between 912 A ~and 228 A. If we restrict our models to those with spectral slopes consistent with population II stellar sysnthesis models (0.5 < alpha < 2.5), the redshift of reionisation is restricted further to redshift of reionization equals 6.9 with an uncertainty of +0.4 / -0.5.


Knowledge of the distribution and history of temperatures within the IGM in addition to temperature at mean density will greatly improve our understanding of the EoR. while the timing of reionisation is probed by mean temperature, as discussed in section 4.1, an estimate of the scatter in temperatures between different lines of sight from multiple quasar spectra, or by the large scale features of the Lyman alpha forest [2018arXiv181011683O, 2019arXiv190704860W], would begin to offer information about the duration and patchiness of reionisation. This is because the distribution of temperatures shortly after reionisation is closely related to the distribution of reionisation redshifts within a volume (see Figure 5). This is especially true for low density gas, where the distribution of temperatures is widest. One way to further probe the temperature density relation could involve the cross-correlation of temperature measurements with the galaxy field. In this manner, information could be gained on the correlation of temperatures with density at various scales throughout reionisation.


From the temperature-density plots in this work, we would expect to see the correlation on small scales flattening, or even inverting, shortly after reionisation. This section explores the effects of numerical factors in our model. The grid resolution of Meraxes will be investigated, as well as the parameters used in our DE solver, namely the convergence threshold that we use to determine the timestep throughout the model. For computational reasons, all of model runs were performed using 128^3 grids in Meraxes. However to ensure inhomogeneities below this scale do not greatly affect our results, we examine results from one run on a 256^3 grid. As shown in Figure A1, while the overall clumping factor has increased, the temperatures and photo-heating energy at mean density are largely unaffected. This is due to the fact that we follow the gas at the mean density of each voxel, rather than the voxel as a whole.


This will produce a very similar temperature-density relation regardless of grid resolution; The distribution of densities will change, but the temperature at any particular density will remain the same. We have altered the convergence conditions in our differential equation solver to test for convergence. We change the conversion threshold between the absolute value of the difference between the electron abundance at iteration k+1 and k, |(X tilde)_e,k+1 - (X tilde)_e,k|, is less than 10^-8 and 10^-4. The results of these changes for an individual voxel that ionised at z=10 are presented in Figure A2. The weakest threshold gives a maximum difference in temperatures of is approximately equal to 2 %, and our fiducial threshold of 10^-6 is negligibly different from the strictest thresholds. We are therefore confident that our choice of differential equation solver parameters does not affect our results.


Previous simulations have found little dependence of the long term thermal state on the amplitude of the ionising background, as long as it is strong enough to maintain an ionised IGM [1997MNRAS.292...27H, 2009ApJ...701...94F]. This result is verified in our model, as we can see no change in the long term thermal state when we artificially increase or decrease the amplitude of the ionising background post-reionisation, keeping our other parameters constant. The ionisation rate in a voxel is directly proportional to the amplitude, however the number of ionisations that actually occur is limited by the recombination rate, which is negligibly changed for all highly ionised states. As a result the long term cooling is largely unaffected at most densities. Figure B1 shows the mean temperature at mean density, a third of mean density, and three times mean density, for the three amplitudes tested. It is important to note that the ionising flux amplitude has little effect on temperature only when there is enough ionising flux to maintain the ionised state in a region.


If the amount of ionising flux is low enough such that it is comparable to or lower than the recombination rate, there will be a significant drop in temperature as the gas recombines. Furthermore, the long term temperatures will decrease, due to the ionisation rate falling below the recombination rate for a fully ionised state; this causes the ionisation equilibrium state to shift to a more neutral state, lowering the rate of ionisations and recombinations, and therefore the photo-heating rate near equilibrium, as the cooling rate is now limited by ionisations, rather than recombinations. In our simulation, there is enough ionising flux with the fiducial values from Meraxes to maintain a highly ionised state in the vast majority of the simulated volume, so a significant drop in temperature is observed only for highly dense regions. Voxels at mean density will only show a 5% decrease in temperatures when the ionising flux is decreased by a factor of 10. Differences in temperature will also diminish over time, as the gas cools to its thermal asymptote.


Regions near early galaxies that have their star formation suppressed or move between voxels can also show this behaviour, however in this case the temperature drop is also usually very short-lived, as nearby HII bubbles expand to re-heat the voxel, washing out any memory of previous ionisation events.


We present a new analysis of high-redshift UV observations using a semi-analytic galaxy formation model, and provide self-consistent predictions of the infrared excess (IRX) - β relations and cosmic star formation rate density. We combine the Charlot & Fall dust attenuation model with the MERAXES semi-analytic model, and explore three different parametrisations for the dust optical depths, linked to star formation rate, dust-to-gas ratio and gas column density respectively. A Bayesian approach is employed to statistically calibrate model free parameters including star formation efficiency, mass loading factor, dust optical depths and reddening slope directly against UV luminosity functions and colour-magnitude relations at z ~ 4 - 7. The best-fit models show excellent agreement with the observations. We calculate IRX using energy balance arguments, and find that the large intrinsic scatter in the IRX - β plane correlates with specific star formation rate. Additionally, the difference among the three dust models suggests at least a factor of two systematic uncertainty in the dust-corrected star formation rate when using the Meurer IRX - β relation at z >= 4.


One fundamental question in astronomy is to understand the buildup of stars and galaxies from baryonic matter in the early Universe. During this epoch, observations focus mainly on rest-frame UV properties due to cosmic redshift. These include measurements of UV luminosity functions (LFs) [2010A&A...523A..74V, 2015ApJ...803...34B, 2017ApJ...835..113L, 2018arXiv180707580B, 2018PASJ...70S..10O], and UV continuum slope to UV magnitude relations [2012ApJ...756..164F, 2014ApJ...793..115B, 2014MNRAS.440.3714R], which are also known as the colour-magnitude relations (CMRs). The UV luminosity is a tracer of star formation since most UV photons are emitted by young stars. However, star formation can be heavily obscured by the interstellar dust. One commonly adopted approach to perform dust corrections at high redshifts is to infer the infrared excess (IRX) from the observed UV slopes using a relation calibrated by [1999ApJ...521...64M] [e.g. 2015ApJ...803...34B, 2015ApJ...813...21M, 2016MNRAS.462..235L]. However, the [1999ApJ...521...64M] relation is calibrated against local starburst galaxies, and observations of far infrared data is rather challenging at high redshifts.


Recent observations at z >= 3 show large scatter in the IRX - β relation [2015Natur.522..455C, 2016A&A...587A.122A, 2016ApJ...833...72B, 2017ApJ...845...41B, 2017MNRAS.472..483F, 2018MNRAS.479.4355K]. For instance, the observed IRX by [2016ApJ...833...72B] is much lower than the [1999ApJ...521...64M] relation, while [2018MNRAS.479.4355K] suggest that the IRX - β relation does not evolve with redshift. These observations motivate investigation of the IRX - β at high redshifts from theoretical models. Theoretical studies of dust extinction require intrinsic galaxy properties as input, and one approach is to postprocess the output of a hydrodynamical simulation. This method has been implemented in [2017ApJ...840...15S] and [2018MNRAS.474.1718N] to investigate the origin of the IRX - β relation. At z >= 5, the IRX - β relation has been studied by [2016MNRAS.462.3130M], [2017MNRAS.470.3006C] and [2019MNRAS.487.1844M]. However, their results suggest different extinction curves. [2017MNRAS.470.3006C] pointed out that the reason for the disagreement could be due to systematics associated with different simulations.


Semi-analytic models are another popular approach for studying galaxy formation [e.g. 2011MNRAS.413..101G, 2015MNRAS.453.4337S, 2016ApJS..222...22C, 2016MNRAS.462.3854L, 2018MNRAS.479....2C, 2018MNRAS.481.3573L, 2019arXiv190101906C]. Semi-analytic models solve a system of differential equations that govern the mass accretion and transition of several key baryonic components of galaxies such as gas and stellar mass. Their construction is relatively simple, and hence they are computationally efficient. These models also introduce several free parameters to describe the unknown efficiency or strength of certain physics processes. These parameters bring flexibility, and allow the exploration of different galaxy formation scenarios, which is very useful for identifying which galaxy processes regulate certain observations. This work utilises MERAXES to predict intrinsic galaxy properties, and combines it with a simple and flexible dust attenuation model. The dust optical depths are calculated empirically using relevant galaxy properties. By taking full advantage of the fast computational speed, we carry out a Bayesian analysis on all the model free parameters, and use UV LFs and CMRs as constraints, which are the most fundamental observables at high redshift.


The MERAXES semi-analytic model is the backbone of the present work. It extends the models of [2006MNRAS.365...11C] and [2011MNRAS.413..101G] to high redshifts, and is modified to run on high cadence halo merge trees with a delayed supernova feedback scheme. It also implements gas infall, radiative cooling, star formation, supernova feedback, metal enrichment, and reionisation feedback. The active galactic nuclei (AGN) feedback of the model is later introduced by [2017MNRAS.472.2009Q]. This work also applies several updates to the model, aiming to improve the predicted gas phase metallicity. We utilise the halo merger trees of the Tiamat N-body simulation [2016MNRAS.459.3025P, 2017MNRAS.472.3659P]. The simulation contains 2160³ particles in a (67.8 h⁻¹)^3 Mpc³ box, with mass resolution m_p = 2.64 × 10⁶ h⁻¹ Msun. Halos and friends-of-friends groups are identified using SUBFIND [2001MNRAS.328..726S]. The high cadence of the simulation is critical to this study since UV magnitudes are sensitive to starbursts in the recent 100 Myr.


Since this work requires evaluating the model many times, and does not focus on ionising structures, we adopt homogeneous reionisation feedback [2000ApJ...542..535G] instead of using 21cmfast [2007ApJ...669..663M]. Both approaches are described in [2016MNRAS.462..250M] and found to have almost the same predictions on global galaxy properties such as the stellar mass function up to z ≈ 5. However, the homogeneous prescription is more computationally efficient. Our model assumes that gas undergoes shock heating and forms a quasi-static hot halo when it is accreted by the host dark matter halo. The gas can cool and form a cold disk, which then becomes fuel for star formation. Following the disk stability argument of [1996MNRAS.281..475K], our model assumes that gas can only form stars when its mass is greater than a critical mass, which is a function of the halo's maximum circular velocity V_max and the disk scale radius r_disk. The mass of new formed stars is then proportional to the star formation efficiency α_SF and the excess gas mass, divided by the disk's dynamical time.


We update the supernova feedback model with a different treatment of supernova energy, and a different parametrisation of mass loading factor and energy coupling efficiency. Our original model is a modified version of [2011MNRAS.413..101G], taking into account the high cadence of our halo merger trees. The effect of supernova feedback is to transfer the gas in the cold disk to the hot halo. The amount of mass that is reheated is determined by a two-case equation: it is estimated by the mass loading factor argument, and reduced if the energy injected by supernova is smaller than the underlying energy increase of the hot halo. Moreover, if the injected energy is greater than what is required for reheating, materials can be further ejected from the hot halo. The ejected mass is subtracted from the hot gas and put into a separated component. The injected supernova energy plays an important role in this model, calculated as an integral over the star formation history weighted by the energy release rate of a simple stellar population.


While our original model uses an analytic fit to estimate the energy release rate from supernovae, this work generates it using STARBURST99 [1999ApJS..123....3L, 2005ApJ...621..695V, 2010ApJS..189..309L, 2014ApJS..212...14L] with metallicity dependence, assuming a Kroupa IMF [2002Sci...295...82K]. This treatment provides more reasonable and self-consistent estimates of the supernova energy. To evaluate the integral, MERAXES tracks the mass of new formed stars and their metals in four previous snapshots. The mass loading factor η and energy coupling efficiency ϵ are given by a broken power law based on the FIRE simulations [2014MNRAS.445..581H, 2015MNRAS.454.2691M]. This form is primarily motivated by its impact on metallicity, as previous studies found that explicit redshift-dependent models can lead to evolution of the mass metallicity relation [2016MNRAS.461.1760H, 2018MNRAS.481..954C]. We set the redshift dependence of reheating α_reheat = 2 and assume no redshift dependence on the energy coupling efficiency (α_eject = 0), leaving η₀ and ϵ₀ as free parameters.


We also apply STARBURST99 to the mass recycling and metal enrichment. The mass of materials produced by type-II supernova and released into the ISM is obtained by an integral over the star formation history, weighted by the stellar yield. This quantity depends on the IMF and varies with different elements. We generate the table of yields using STARBURST99, including metallicity dependence and assuming a Kroupa IMF [2002Sci...295...82K]. This age-dependent mass recycling scheme is more realistic than the commonly adopted constant recycling fraction, particularly at high redshift. The ejected gas can be transferred back to the hot gas halo via reincorporation. We employ the model proposed by [2013MNRAS.431.3373H] where the reincorporation timescale is inversely proportional to the virial mass of the halo. We also force this timescale to be smaller than the halo dynamic time. This model provides a better fit of stellar mass functions against observations at z <= 3 [2013MNRAS.431.3373H]. Reincorporation is more efficient in our model relative to [2013MNRAS.431.3373H] at high redshifts, particularly for low mass halos.


We implement the dust model proposed by [2000ApJ...539..718C]. This model's transmission function accounts for the relative stars-dust geometry of different stellar populations. Photons from young stars are absorbed by an additional component from the surrounding molecular cloud, which is assumed to have a lifetime of 10 Myr [2000ApJ...539..718C, 2008MNRAS.388.1595D]. Older stars' light is only absorbed by diffuse ISM dust. The attenuation is described by optical depths for the birth cloud and ISM, which vary between galaxies. We explore three different parametrisations: M-SFR, M-DTG, and M-GCD, linked to star formation rate (SFR), dust-to-gas (DTG) ratio, and gas column density (GCD) respectively. These properties are indirectly related to dust; for example, one dust production channel from supernova ejecta is proportional to the SFR [2018PhR...780....1D]. Since M-DTG and M-GCD primarily depend on gas density, they are expected to yield similar results. The dependence of dust optical depths on SFR is motivated by observations that more UV luminous galaxies have redder UV continuum slopes [2012ApJ...756..164F, 2014ApJ...793..115B, 2014MNRAS.440.3714R].


The M-SFR model parameterizes the dust optical depth as a function of SFR, redshift, and wavelength, with five free parameters. For all three parametrisations, we include an exponential redshift dependence factor to fit the model against multiple redshifts, motivated by the work of [2019MNRAS.483.2983Y]. In the literature, dust optical depths are often linked to gas column density, converted using the DTG ratio [2007MNRAS.375....2D, 2011MNRAS.413..101G, 2012MNRAS.423.1992S, 2019MNRAS.483.2983Y]. The M-DTG model expresses optical depths as a function of cold gas metallicity, mass, and disk radius, with five free parameters. We also propose the M-GCD model, which is independent of metallicity. In MERAXES, we assume metals are first fully mixed with cold gas, but in reality, this may take time. Thus, the M-GCD model parameterizes optical depths as a function of cold gas mass and disk radius, with a power law scaling on cold gas mass instead of metallicity. The computation of galaxy SEDs follows standard stellar population synthesis, integrating the star formation history, SSP luminosity, and the ISM transmission function.


We generate the SSP luminosity using STARBURST99 [1999ApJS..123....3L, 2005ApJ...621..695V, 2010ApJS..189..309L, 2014ApJS..212...14L], assuming a metallicity range from Z = 0.001 to Z = 0.040 and a Kroupa IMF [2002Sci...295...82K]. Nebular continuum emissions are also added. To compute UV magnitudes, we apply a tophat filter centred at 1600 Å with a width of 100 Å. UV slopes are obtained by a linear fit using five of the ten windows proposed by [1994ApJ...429..582C] for computational speed. This treatment introduces negligible errors. We also make a numeric approximation to accelerate the SED evaluation by first computing the intrinsic luminosity in necessary filters and then applying the dust transmission using the central wavelength. This approximation has a negligible effect on the results. An essential part of this work is to determine the free parameters in both the galaxy formation and dust attenuation models. We carry out a Bayesian analysis on these parameters, using observed UV LFs and CMRs at z ~ 4 - 7 as constraints.


A key goal of a Bayesian analysis is to estimate the posterior distribution of model parameters. The MCMC method has been applied in several studies [2013MNRAS.431.3373H, 2013MNRAS.428.2001M, 2015MNRAS.451.2663H], but it has several drawbacks, including difficulties in determining convergence and handling multimodal parameter spaces [doi:10.1080/01621459.1996.10476956, Neal1996]. In this work, we use the multimodal nested sampling introduced by [2009MNRAS.398.1601F] to achieve higher sampling efficiency and more stable results. This algorithm is a competitive alternative to MCMC methods. The nested sampling was designed to evaluate the Bayesian evidence [2004AIPC..735..395S], but its output can also be used to estimate posterior distributions. No burn-in phase is required, and the stopping criterion is based on an estimated error of the Bayesian evidence. The algorithm's efficiency is improved by using existing sample points to approximate iso-likelihood surfaces as hyper ellipsoids and includes a special treatment for multimodal problems by using a clustering algorithm to detect and split multiple modes [2007MNRAS.378.1365S].


The Bayesian posterior distribution is comprised of the likelihood and prior distributions. We construct the log-likelihood as a chi-squared sum over LF and CMR data points. Observational data of LFs and CMRs are taken from [2015ApJ...803...34B] and [2014ApJ...793..115B] respectively. We convert the Hubble constant from h = 0.7 to h = 0.678 for consistency. Due to the limited size of the simulation box, the model is unable to probe the full range of the LFs and CMRs. Therefore, we drop bins if the expected number of galaxies is less than five for the LF and twenty for the CMR. We focus on four galaxy formation parameters: star formation efficiency, critical mass normalisation, mass loading factor, and supernova energy coupling efficiency, with uniform logarithmic priors. Each of the three dust models has five free parameters with uniform linear priors. The prior ranges are chosen based on experiments to improve sampling efficiency. We utilise a modified version of the open-source Python package nestle, coupled with the MERAXES Python interface MHysa. We set the number of active points to 300 for the sampler.


For the three different dust models, we obtain 5,000 - 6,000 sample points from the nested sampling algorithm. The point with the highest posterior distribution value is chosen as the best-fit result. The three models all fit the observational data extremely well. All parameters are well constrained for the M-DTG and M-GCD models, while the mass loading factor η₀ is less constrained for the M-SFR model. An interesting finding is that the derived galaxy formation parameters preferred by these three dust models are quite different. M-DTG and M-GCD suggest similar mass loading and supernova energy coupling efficiency, but M-DTG shows evidence of a more active star formation scenario. M-SFR requires much smaller supernova energy coupling efficiency. The variation among the posterior distributions implies that these parameters fit the data in a complex way and the constraints depend on the assumptions used to model the dust attenuation. We find similar correlations among the parameters of the supernova feedback, dust relation scaling, and reddening slope. The supernova energy coupling efficiency is positively and inversely correlated with γ and the reddening slope n, respectively.


The intrinsic LFs of the best-fit M-SFR is roughly a factor of two higher than for the other two models. The supernova coupling efficiency of the best-fit M-SFR model is much smaller, which is likely the main reason for the difference in the intrinsic LFs. The number density at a fixed UV magnitude decreases with an increasing mass loading factor η₀. While higher energy coupling efficiency ϵ₀ decreases the LFs, the effect is more significant at the bright end, as it allows more gas to be reheated in galaxies hosted by more massive halos. The shape of both the LFs and CMRs are quite sensitive to the dust parameter γ. The effective UV optical depth is a function of γ and the optical depth normalisations. The effective optical depth should be degenerate with the intrinsic UV LFs, which are primarily controlled by the supernova feedback parameters. The observed UV continuum slope β depends on the reddening curve. The cold gas is more metal enriched in the best-fit M-DTG than in M-GCD. The normalisation of the critical mass Σ_SF is the primary driver for this difference.


By simultaneously fitting our models to the observed UV LFs and CMRs, we can estimate the infrared luminosity F_IR and the infrared excess (IRX) using energy balance arguments. The resulting IRX - β relations for galaxies with stellar mass greater than 10⁸ Msun are shown in Figure
ef{plot_IRX}. Our results cover the observations from [2018MNRAS.479.4355K] and [2017ApJ...845...41B]. The [1999ApJ...521...64M] relation is frequently used to correct dust extinction at high redshifts [2014MNRAS.444.2960D, 2015ApJ...803...34B, 2015ApJ...813...21M, 2016MNRAS.462..235L, 2018PASJ...70S..11H]. The best-fit M-SFR predicts higher IRX than the [1999ApJ...521...64M] relation at a fixed β, while the other two models suggest lower IRX. This implies that a direct application of the local relation at high redshifts may lead to systematic errors. The best-fit models have quite different reddening slopes 'n': M-SFR has the shallowest slope, while M-DTG and M-GCD have much steeper slopes. This difference is directly reflected on the IRX - β plane.


Similar disagreements in the reddening slope can be found in other studies. For example, [2017MNRAS.470.3006C] suggest n = -0.55, which is more consistent with our best-fit M-SFR model. On the other hand, [2016MNRAS.462.3130M] reproduce observed LFs and CMRs using a steeper, SMC-like extinction curve, similar to our best-fit M-DTG and M-GCD models. Since all our models can well reproduce observed LFs and CMRs, we cannot draw any firm conclusions on the reddening slope and treat this as a systematic uncertainty. Observations at z >= 3 show considerable scatter in the IRX - β plane [2015Natur.522..455C, 2016A&A...587A.122A, 2016ApJ...833...72B, 2017ApJ...845...41B, 2017MNRAS.472..483F, 2018MNRAS.479.4355K], which might be explained by the large intrinsic scatter in our predicted relations. Low IRX galaxies vanish in the best-fit M-SFR model due to the nature of our star formation prescription. In the M-SFR model, the dust optical depths of galaxies with zero SFR are also zero, resulting in the disappearance of the IRX. This unrealistic feature shows the limitations of this model.


The IRX - β relations at z ~ 5 as functions of stellar mass and sSFR show that massive galaxies form a tight correlation between IRX and β in the high IRX and red β regions. The trend that more massive galaxies have higher IRX is also observed by [2016A&A...587A.122A] and [2017MNRAS.472..483F]. However, we also find several larger stellar mass galaxies which have lower IRX and redder β, which might explain some of the outliers detected by [2017ApJ...845...41B]. The scatter of the IRX - β relation is tightly correlated with sSFR. At a fixed IRX, redder galaxies typically have lower sSFR, consistent with other theoretical studies [2017MNRAS.472.2315P, 2017ApJ...840...15S, 2018MNRAS.474.1718N, 2019arXiv190101747C]. Dust corrections are typically required for the conversion between UV luminosity and SFR. The [1999ApJ...521...64M] relation is widely used, though it is calibrated against local galaxies. The dust extinction predicted by our models is rather different from this relation. We directly present the predicted SFRs, and the difference among the three models allows us to estimate the systematic uncertainties.


We compare our predicted cosmic SFRD with [2015ApJ...803...34B], whose estimations are based on the CMRs of [2014ApJ...793..115B] and the [1999ApJ...521...64M] relation. The comparison quantifies the systematic errors of using the [1999ApJ...521...64M] relation. Our models suggest bluer intrinsic UV continuum slopes than the one used in [1999ApJ...521...64M]. The best-fit results of M-DTG and M-GCD have shallower IRX - β relations. Thus, compared with the [1999ApJ...521...64M] relation, the dust extinction in these two models is stronger for bluer galaxies but weaker for redder galaxies. On the other hand, the dust attenuation is stronger for all galaxies in the best-fit M-SFR. The cosmic SFRD of the best-fit M-DTG and M-GCD are consistent with those of [2015ApJ...803...34B], while the results of the best-fit M-SFR is roughly a factor of two higher. We also compare our results with [2018MNRAS.475.2891D], whose dust-corrected SFRs are derived from the energy balance SED-fitting code MAGPHYS [2008MNRAS.388.1595D]. Better consistency is found between their measurements and our best-fit models of M-DTG and M-GCD.


This work investigates the IRX - β relation and cosmic SFRD at z ~ 4 - 7 by combining the MERAXES semi-analytic galaxy formation model [2016MNRAS.462..250M, 2017MNRAS.472.2009Q] and the [2000ApJ...539..718C] dust attenuation model. The supernova feedback model of MERAXES is updated using results from previous studies [2015MNRAS.454.2691M, 2016MNRAS.461.1760H, 2018MNRAS.479....2C]. We introduce three different parametrisations of the dust optical depths (M-SFR, M-DTG, M-GCD). We adopt a Bayesian approach, calibrating these parameters against the UV LFs of [2015ApJ...803...34B] and CMRs of [2014ApJ...793..115B] at z ~ 4 - 7. The posterior distribution is estimated using multimodal nested sampling [2009MNRAS.398.1601F]. We find that these observations can be fit extremely well by all three dust models. However, the preferred parameter ranges are quite different among the models. Our analysis indicates that the combination of the LFs and CMRs can put strong constraints on a given dust attenuation model. The differences in our results are due to the different assumptions of the dust models.


Using energy balance arguments, we estimate the IRX for each model galaxy. We find that the predicted IRX - β relations are quite different from the [1999ApJ...521...64M] relation, and contain large intrinsic scatter, which might explain the current discrepancy among several high redshift observations [2015Natur.522..455C, 2016A&A...587A.122A, 2016ApJ...833...72B, 2017ApJ...845...41B, 2017MNRAS.472..483F, 2018MNRAS.479.4355K]. We also confirm the correlation between the intrinsic scatter and sSFR, consistent with other theoretical studies [2017MNRAS.472.2315P, 2017ApJ...840...15S, 2018MNRAS.474.1718N, 2019arXiv190101747C]. Secondly, we present model predictions for the cosmic SFRD, and compare these with observations [2015ApJ...803...34B, 2018MNRAS.475.2891D]. The difference among the three dust models implies at least a factor of two systematic uncertainty in the observed SFRD when corrected using the Meurer IRX - β relation. This work has simultaneously constrained the free parameters of a semi-analytic model and dust parameters using UV properties. This approach is particularly useful for studies at high redshifts where UV properties are the most robust observables. This work could be further improved by explicitly modelling the dust evolution.


We implemented Population III (Pop. III) star formation in mini-halos within the Meraxes semi-analytic galaxy formation and reionisation model, run on top of a N-body simulation with a side length of 10 per h comoving Megaparsecs with 2048^3 particles resolving all dark matter halos down to the mini-halos (approximately 10^5 solar masses). Our modelling includes the chemical evolution of the IGM, with metals released through supernova-driven bubbles that expand according to the Sedov-Taylor model. We found that SN-driven metal bubbles are generally small, with radii typically of 150 comoving kiloparsecs at redshift z = 6. Hence, the majority of the first galaxies are likely enriched by their own star formation. However, as reionization progresses, the feedback effects from the UV background become more pronounced, leading to a halt in star formation in low-mass galaxies, after which external chemical enrichment becomes more relevant. We explore the sensitivity of the star formation rate density and stellar mass functions on the unknown values of free parameters.


We also discuss the observability of Pop. III dominated systems with JWST, finding that the inclusion of Pop. III galaxies can have a significant effect on the total UV luminosity function at redshift z = 12 - 16. Our results support the idea that the excess of bright galaxies detected with JWST might be explained by the presence of bright top-heavy Pop. III dominated galaxies without requiring an increased star formation efficiency. The first episodes of star formation likely occurred at redshift 30-40 inside low-mass (10^5-7 solar masses) halos with a pristine chemical composition (metal-free or extremely metal-poor). These environments were responsible for the chemical enrichment of the intergalactic medium (IGM) leading to the Pop. III/Pop II transition at redshift 15-10. However, due to the low mass and the high-redshift, the study of Pop. III star formation in mini-halos is challenging both in terms of observations and simulations [Klessen2023].


Even with the launch of JWST, we do not have confirmed observations of Pop III stars and only a few potential candidates [Welch2022, Maiolino2023]. As pointed out by [Trussler2023], a direct detection of a Pop. III galaxies will be extremely challenging as, in the best-case scenario, hundreds of hours of integration time are needed in order to detect an unlensed Pop. III system with 5-sigma confidence. A complementary tool to direct observation could be the redshifted 21cm global signal whose depth might be determined by the radiation emitted from Pop. III stars and early accreting black holes during the Cosmic Dawn [among the most recent works see Mebane2018, Mirocha2018, Mebane2020, Munoz2022, GesseyJones2022, Magg2022, Ventura2023, Hegde2023]. So far, there is no detection confirmed of the 21cm redshifted line, however, a large number of facilities aiming to the observation of the 21cm global signal and power spectrum are already operating or becoming operative in this decade.


There are a number of Pop. III star formation simulations at different scales. Hydrodynamical simulations that follow the cooling of gas in a single pristine or very metal-poor cloud suggest that the inefficient cooling due to the lack of metals might favour an initial mass function (IMF) that is shifted to larger masses [e.g.][Bromm1999, Hirano2014, Stacy2016, Chon2021]; however, there is no general consensus (e.g. [Wollenberg2020, Jaura2022, Prole2022] predict IMF shifted to lower masses). Simulations at larger scales (>=1 cMpc) are instead used to follow the global evolution of Pop. III stars with redshift and their impact on the cosmic metal enrichment and reionization. In the last decade, many of these simulations have been performed both with hydrodynamical and semi-analytical codes. At the same time we also need to consider a volume large enough in order to have a statistically significant sample of the Universe. However, to satisfy both requirements of large volumes and high-resolution is impossible and thus all the semi-analytical and hydrodynamical simulations have to make compromise in one of the two directions.


In this work, we chose to use the semi-analytical model of galaxy formation Meraxes ([Mutch2016], M16 hereafter) within a simulation that allows us to resolve all the mini-halos down to a few 10^5 solar masses in a simulation of L = 10 Mpc per h. As so, in this paper we ran Meraxes on top of a N-body simulation of L = 10 Mpc. The size of this simulation is not large enough to be representative of the Universe (we will miss the most massive galaxies). However, the scales are large enough to investigate the impact of the main physical processes on the Pop. III star formation. We incorporated a number of new physical processes that are relevant to Pop. III star formation in mini-halos including: (i) molecular hydrogen cooling functions for mini-halos, (ii) baryon-dark matter streaming velocities, (iii) photo-dissociation of H2 molecules from the Lyman-Werner background and (iv) the chemical evolution of the intergalactic medium.


This latter effect has been implemented assuming that metals are released through supernova explosions and within a bubble that expands accordingly to the Sedov-Taylor model (the approach is very similar to the analytical calculation shown in [FL2003]). We found that these bubbles are generally small and that they roughly agree with the previous estimate by [Trenti2009], with typical radii of 150 ckpc at redshift z = 6. The implementation of both internal and external metal enrichment allows us to understand whether a galaxy will form Pop. III or Pop. II stars and thus quantify the impact of Pop. III galaxies on the total luminosity function at redshift z >= 5. We assumed that the Pop. III star formation occurs in an instantaneous burst at a random Delta t from the end of the snapshot. Considering an instantaneous rather than a continuous star formation makes some of the Pop. III dominated systems significantly brighter, hence at high-redshift the brightest systems are likely to host Pop. III stars.


The semi-analytical model Meraxes was first developed by [Mutch2016] (hereafter M16), [Qin2017, Qiu2019] in order to study galaxy formation and growth through the Epoch of Reionization. Despite the variety of physical processes included in Meraxes assumes that all the galaxies form in a previously chemically enriched Universe and inside atomic cooling halos. Such an approximation does not allow us to study the first episodes of star formation that mostly occurred in mini-halos when the Universe did not have any metals. The version of Meraxes presented in this work allows us to compute the physics of the first episodes of star formation from the initial molecular cooling of the gas to the external metal enrichment from the supernova feedback. As in the previous work, Meraxes is coupled to the reionization so that all the radiative backgrounds are computed in a self-consistent way from the galaxy properties. This has been done by implementing a modified version of 21cmFAST ([Mesinger2011]) that includes the local ionizing UV background from [Sobacchi2014] and the X-ray heating ([Balu2023]).


In this version, we also included the Lyman-Werner background. The updates to Meraxes for Pop. III necessitate high mass and spatial resolution. For this purpose, we introduce L10_N2048 (hereafter L10) from the Genesis suite of dark matter only N-body simulations. L10 is a periodic cubical simulation of side 10 per h Mpc and consists of 2048^3 dark matter particles of mass m_p = 9.935 x 10^3 per h solar masses resulting in a halo mass resolution of ~ 3.18 x 10^5 per h solar masses (based on a minimum of 32 particles per halo). The simulation, run using the SWIFT [Schaller2018] cosmological code, evolves these dark matter particles from redshift z = 99 down to z = 5. The halos were identified using the friends-of-friends phase space halo-finder VELOCIraptor [VR] and the merger trees were constructed using TREEFROG [treefrog].


We note that the mass resolution achieved in this simulation allows us to resolve all the mini-halos below redshift z ~ 30 and is the highest resolution on which Meraxes has been run. The output trees of the N-body simulation used in this work are available on Zenodo at [balu2024]. The first process in order to enable star formation is the cooling of the gas. For dark matter halos with virial temperature T_vir >= 10^4 K, the main coolant is the atomic hydrogen, while in mini-halos (10^3 K <= T_vir < 10^4 K), the cooling occurs via roto-vibrational transitions of molecular hydrogen [e.g.][Tegmark1997]. For the details on how the cooling of the gas is implemented in Meraxes, we refer the reader to M16; in this section, we only highlight the main differences between the atomic and the molecular cooling. As in M16, we compute the ratio of the specific thermal energy to the cooling rate per unit volume: t_cool(r) = (1.5 * mu_bar * m_p * k * T) / (rho_hot(r) * Lambda(T, f_H2)).


M16 included only atomic cooling halos. Thus they set the mean molecular weight assuming a fully ionized gas and the cooling function from [Sutherland1993]. For mini-halos we instead set the mean molecular weight for a fully neutral gas and implement the molecular hydrogen cooling functions from [Galli1998]. This choice is valid as long as we assume that the cooling inside mini-halos occurs only due to H2. This may not be valid as, if a mini-halo is chemically enriched with metals by a nearby halo, metals are much more effective in the cooling of the gas [e.g.][Nebrin2023]. However, this enrichment is almost ineffective at redshift z > 10. The molecular cooling function depends on the gas density of the halo and on the molecular hydrogen fraction f_H2. For the latter we assumed a fixed value of 0.1%, which is consistent with results from [Nebrin2023] for halos of virial temperature of approximately 5x10^3 K at redshift z = 15-20.


As a first approximation, gas in a mini-halo can start to cool down the gas once it reaches the virial temperature of approximately 10^3 K. This requirement would correspond to a minimum virial mass of M_min,H2 = 2.5 x 10^5 * (26 / (1+z)) solar masses [Visbal2015]. However, there are a number of effects that can decrease the amount of molecular hydrogen present in the halo, reducing the cooling efficiency and ultimately increasing the minimum virial mass of a halo capable of cooling. A non-radiative process that can delay the gas cooling in very low-mass halos is the streaming velocity between baryons and dark matter ([Tseliakhovich2010]). This effect is a consequence of the different decoupling of baryons and dark matter particles from photons that results in a velocity difference between the two species. The presence of a relative motion between baryons and dark matter particles will make it harder for baryons to fall into the potential wells of dark matter halos, delaying the accretion and, hence, the cooling of the gas in mini-halos.


The main outcome of this physical process is to delay the very first episodes of star formation around redshift z ~ 40. Throughout this work, we implemented the effect of the streaming velocities as per other semi-analytical models based on a fitting function found by [Fialkov2012], which is calibrated to reproduce the results of the hydrodynamical simulations of [Greif2011] and [Stacy2011]: V_cool,H2 = (a^2 + (b*v_bc)^2)^0.5, where a = 3.714 km/s and b = 4.015 km/s. This equation provides the minimum circular velocity that a halo needs to have in order to have enough H2 to cool down the gas. This can be easily converted into a virial mass. In this work we fixed the streaming velocity v_bc(z) = sigma_bc(z) and we assumed this value for the entire box. Once the first mini-halos are forming stars, the effect of the streaming velocities becomes widely subdominant compared to the photo-dissociation of the Lyman-Werner background.


A self-consistent model of star formation in mini-halos must consider the photo-dissociation of H2 from UV photons in the Lyman-Werner (LW) band ([11.2 - 13.6] eV). LW photons destroy H2 and thus prevent mini-halos gas from cooling ([Haiman2006]). We implement this effect by changing the minimum mass for molecular cooling using the fitting from [Visbal2015]: M_crit,MC = M_cool,H2 * [1 + 22.87 * J_LW^0.47]. This critical mass only applies to minihalos which are below the atomic cooling halo mass threshold. J_LW is the LW flux that reaches the minihalo. LW photons have a mean free path of ~ 100Mpc, so each minihalo will be affected even by distant galaxies that formed at higher redshift. Thus, we modelled the LW background by integrating contributions across the cosmic history. As for all the radiative backgrounds in Meraxes, we follow an excursion-set formalism ([Furlanetto2004]) which counts the number of photons in a certain band in spheres of radius R.


At each of these locations, we compute the LW emissivity using the spectral energy distributions from [Barkana2005]. We assume that LW photons are absorbed only at resonant frequencies. Under these approximations, we can compute the LW emissivity smoothed over R at redshift z and location x for sources emitted at redshift z' as an integral over the star formation rate density for both Pop. III and Pop. II stars. This sum accounts for the resonances in the Lyman series. Given the large mean free path of LW photons, we must also account for distant galaxies at higher redshift z' > z with the redshifted spectrum. The sum of these two effects causes the peculiar shape of the LW spectrum. Following [Barkana2005] we account for all the Lyman resonances with n <= 23. Finally, we need to convert the emissivity into a flux following ([Qin2020]).


The approach described above differs from the one in 21cmFAST only for the computation of the SFRD. In [Qin2020], the SFRD is estimated from the density field and the collapsed fraction, while in this work, it is computed directly from Meraxes, which tracks the formation of each galaxy and its entire star formation history. Together with the LW background, the main radiative feedback that suppresses star formation is the UV photo-ionization. We used the same prescriptions as in M16, which consists of reducing the baryon content through a baryon fraction modifier that stops the gas infall. In difference from the LW feedback, the UV photo-ionization is relevant only during reionization (redshift z <= 10), and it also affects atomic cooling halos as massive as ~ 10^9.5 solar masses. However, given that the volume of our simulation (L = 10 cMpc) is much smaller than the mean-free path of LW photons, each halo is losing the contribution of the most distant sources, thus neglecting the self-shielding should partially counterbalance the loss of the LW background from distant galaxies.


Once stars start to form in the Universe, they also explode as supernovae, releasing metals. Most of these will stay inside the same galaxy, contributing to the chemical enrichment of the galaxy itself. This process is commonly known as "genetic" enrichment. However, some of the metals escape their parent galaxy. In this case, they will pollute the nearby IGM, and if later a galaxy forms in a region where the IGM was enriched, the new galaxy will be pre-enriched with metals. This latter mechanism is referred to as "external" metal enrichment [e.g.][Pallottini2014, Smith2015, Hartwig2018, Visbal2020, Yamaguchi2023]. Keeping track of the metallicity evolution of the Universe is crucial in order to put constraints on when the Pop. III/II transition occurred. We account for both processes, and we are able to follow the metallicity evolution of the IGM. Firstly, we choose a critical metallicity Z_crit = 10^-4 solar metallicity as the threshold value below which a galaxy will form Pop. III stars.


All new galaxies, unless externally polluted, will accrete pristine gas (without any metals) onto the hot gas reservoir. This gas, once it cools, will provide the reservoir for the star formation. Hence, a galaxy that is not externally polluted will always form Pop. III stars for the first time. At each snapshot, we compute the metallicity of the cold gas reservoir from the amount of metals released by earlier supernovae and if this is higher than Z_crit, the galaxy will form Pop. II stars, otherwise it will form Pop. III. While internal enrichment is the main mechanism that drives the Pop.III/II transition, some galaxies can be externally enriched through supernova winds originating in a nearby galaxy. Once several supernovae in a galaxy go off, they will form a "super-bubble" that will expand outside the galaxy escaping the binding energy of the dark matter halo.


We followed the expansion in time of this "metal bubble" using the analytic approximation in [Dijkstra2014]: the bubble radius r_bubble(t) equals (Delta E_SN / (m_p * n_gas))^0.2 * t^0.4. All quantities that appear in this equation are computed in Meraxes. Delta E_SN is the total supernova energy released at a certain snapshot. n_gas is the number density of the gas to which the bubble has expanded, and t is the time since the explosion occurred. We assume that all the supernova events will occur in the middle of the snapshot. Note that since Meraxes accounts for both contemporaneous and delayed supernova feedback, each galaxy has several bubbles associated with the same star formation episode. However, we consider only the largest of these bubbles so that each galaxy has only one associated bubble. Having calculated the bubble size, we can predict if a nearby galaxy will accrete pristine or enriched gas.


In order to reduce the computational cost, we avoid computing the distance between all the pairs of galaxies and instead use a grid-based approach. We build a high-resolution grid with 128^3 cells. For each cell, we compute the average metallicity Z_IGM,i as the ratio between the sum of the metals ejected by all galaxies inside the cell and the total gas in the cell. We only let the galaxies with a bubble radius r_bubble >= 3 * R_vir contribute. For each cell, we compute the volume fraction filled with metals (the metal filling factor), summing the volume of the largest bubble surrounding each galaxy within the pixel and dividing by the cell volume. At the beginning of each snapshot, we assign the probability p for external metal enrichment to each galaxy inside the cell. This probability will be given by the metal filling factor. We assign a random number m between 0 and 1 to each newly formed galaxy, and when m <= p is satisfied, we label that galaxy as externally enriched.


Furthermore, when a galaxy experiences a star formation episode, we enforce the probability p = 1 (in this latter case, we know that this galaxy will be inside its own metal bubble and thus cannot accrete pristine gas). With this latter condition and internal enrichment, we effectively stop Pop. III star formation inside a galaxy after the first supernova episode. The main limitation of this technique is that we are not accounting for the overlap of the bubbles and thus are overestimating the metal-filling factor and underestimating the metallicity of those galaxies that are polluted from more than one galaxy. However, given the small size of the bubbles, this is not a major factor, especially prior to reionization. The model shows the bubble growth over time from a typical radius of ~ 30 ckpc at redshift z = 20 to ~ 150 ckpc at redshift z = 5. These values are consistent with the results shown in [Trenti2009].


A galaxy that already formed stars or a galaxy that formed in a region enriched with metals so that its metallicity is larger than Z_crit = 10^-4 solar metallicity will form Pop. II stars. We adopted the same prescription and parameters for Pop. II star formation as described in M16 and [Qiu2019]. The uncertainty around the properties of Pop. III stars motivated us to include Pop. III star formation in a flexible way so that it is easy and fast to investigate the impact of Pop. III stars changing a few parameters. In particular, once a galaxy reaches a mass of cold gas larger than a critical value, this galaxy will convert the cold gas into stars according to the star formation efficiency. Since we are now considering two different stellar populations, we adopted two different free parameters, both for the star formation efficiency alpha_SF,III and for the critical surface density Sigma_crit,III.


Recent full hydrodynamical simulations that follow the collapse of a pristine gas cloud until a Pop. III star is formed [e.g.][Hirano2014, Stacy2016, Chon2021], suggest that the Pop. III IMF is shifted to larger masses as a result of less fragmentation due to inefficient cooling. Within this work, we adopted the IMFs from [Raiter2010], while for Pop. II stars, we assumed a [Kroupa2001] IMF. For the fiducial model, we chose a Salpeter IMF with a mass between 1 and 500 solar masses. The choice of the IMF is crucial as it determines many properties of the stellar population, including what fraction of stars will explode as supernovae and, hence, the amount of energy and metals injected into the IGM. Since the focus of this work is on the first galaxies formed during the Cosmic Dawn, we do not explore parameters describing the UV ionizing and X-ray radiation (e.g. escape fraction), and we will take the same fiducial values as in the previous works.


The fate of a zero-metallicity star is quite uncertain. Here, we adopt a simplified picture of the final fate of a Pop. III star depends only on its initial mass ([Heger2002]). For masses below 8 solar masses, there will be no SN event. If stellar mass is in [8,40] solar masses it will explode as a core-collapse SN (CCSN) leaving a remnant, if stellar mass is in [140,260] solar masses there will be a pair-instability SN (PISN) leaving no remnant and if stellar mass is in [40,140] solar masses or stellar mass > 260 solar masses stars collapse directly into a black hole (BH) with negligible feedback. While we use the technique of [Qiu2019] for Pop. II stars with precomputed tables assuming a Kroupa IMF ([Kroupa2001]), for Pop. III supernova feedback we estimated the amount of SN energy and metal yields with an analytic calculation as in M16.


The total energy provided by Pop. III supernovae explosions at the snapshot j is the sum of delayed contributions from core-collapse supernovae and contemporaneous contributions from pair-instability supernovae. These are computed by integrating the chosen IMF over the correct mass limits. Assuming that all-star formation occurred in the middle of the snapshot, the mass limits for CCSN are computed from the lifetimes for Pop. III stars using [Schaerer2002] assuming no mass loss and zero metallicity. We highlight that a zero metallicity star with a mass between 140 and 260 solar masses has a lifetime smaller than the time separation between two consecutive snapshots in meraxes; hence, it will explode as a PISN in the same snapshot in which it forms. For CCSN, instead, we keep track of the star formation history over the last 17 snapshots, which correspond to >= 40 Myr, after which all stars with mass >= 8 solar masses will already be exploded.


We also update the amount of ejected gas and metals from Pop. III stars. These are taken from [Heger2010] assuming non-mixing and a supernova explosion of 1.2 x 10^51 erg. In all the previous Meraxes works, all the stellar mass locked up in remnants (neutron stars and BHs) was neglected as it was recycled into the gas mass budget of the galaxy. We decided to drop this approximation for Pop III stars as they are likely to leave more massive remnants. Firstly, we need to consider the BHs that formed after a "failed SN scenario" typical of a star with an initial mass between 40 and 140 solar masses and greater than 260 solar masses. Finally, we need to account for the BH remnants that are left after a CCSN. Meraxes, does not evolve the remnants (via accretion), as the main focus of this work is Pop. III stars. However, the impact of the first accreting BHs on the formation of the supermassive black holes might be important even at high-redshift ([Ventura2023]).


In order to investigate the observability of the first Pop. III galaxies we implement spectral energy distributions (SED) for Pop. III stars. We use SEDs from [Raiter2010] that have been computed for the IMFs listed in Table 2 assuming that star formation occurs in an instantaneous burst. These SEDs also include the nebular continuum emission and the UV ionizing properties. To compute the luminosity of galaxies at a specific wavelength, we used the model introduced by [Qiu2019] (see also [Mutch2023]), and we extended it to Pop. III galaxies. Having the Pop. III SEDs, we can compute the luminosity of a Pop. III galaxy at time t as an integral over its star formation history, modulated by the luminosity of a stellar population per unit mass and a dust transmission function. For the latter term, we refer the reader to [Qiu2019] while the luminosity per unit mass for each Pop. III IMF has been taken from [Raiter2010].


This calculation is nearly identical to [Qiu2019], with the difference being that we do not have the metallicity dependence because Pop. III stars SEDs are defined with a zero metallicity. In this framework, we assume that the star formation occurs continuously throughout the snapshot. While this is a good approximation for Pop. II stars, Pop. III star formation is expected to occur in a single burst because of the feedback from Pop. III stars are likely to prevent continuous star formation. For this reason, we expanded the calculation of the luminosity of Pop. III galaxies assuming that new stars form instantaneously at a specific time. As we will discuss in Section 4, instantaneous (instead of continuous) Pop. III star formation has an impact on the estimated luminosity function of galaxies hosting Pop. III stars. This is because Pop. III stars have lifetimes that can be shorter than the duration of the snapshot.


Hence, when we consider continuous star formation, we average the star formation over the entire snapshot, leading to lower luminosity with a higher duty cycle. When we instead consider Pop. III stars to form in a single burst, if this burst will occur toward the end of the snapshot the galaxy will appear much brighter when it is observed in that snapshot. In order to account for instantaneous star formation, we assumed that a Pop. III star formation episode in a galaxy can occur at random Delta t within the snapshot duration prior to the end of the snapshot. For the detailed evaluation of the UV luminosity function accounting for the stochasticity in the time at which the burst of star formation occurs in different galaxies, we refer the reader to Appendix B.