2.1 Evolutionary equations

We track the time rate of change of gas ( $\dot{m}_{\mathrm{g}}$ ), star formation ( $\dot{m}_{\star}$ ), and supernova-driven wind mass loss ( $\dot{m}_{w}$ ) at time t via a set of coupled differential equations:

(1) \begin{align}\dot{m}_{\mathrm{g}}(t)&=\dot{m}_\mathrm{c,g}(t)-\dot{m}_{\star}(t)-\dot{m}_\mathrm{w}(t), \end{align}

(2) \begin{align} \dot{m}_{\star}(t)&=\unicode{x025B}_\mathrm{sf}\frac{m_\mathrm{g}(t)}{\tau_\mathrm{sf}},\end{align}

(3) \begin{align} \dot{m}_\mathrm{w}(t)&=\unicode{x03B7}_\mathrm{fb}\dot{m}_\star(t-t^\mathrm{d}). \end{align}

It is straightforward to convert between t and redshift z for our adopted cosmology, which is important for evaluating the contribution of the UV background. The various parameters are now described in the following subsections.

2.1.1 Growth of the gas reservoir

Equation (1) tracks the gas reservoir in a galaxy, accounting for the accretion rate of gas from cosmological scales ( $\dot{m}_\mathrm{c,g}$ ), gas that is converted to stars via $\dot{m}_{\star}$ , and gas that is lost via winds ( $\dot{m}_w$ ). As is commonly assumed (e.g. White & Frenk Reference White and Frenk1991), $\dot{m}_{\mathrm{c,g}}$ tracks the accretion rate of the dark matter halo that hosts the galaxy, with $\dot{m}_\mathrm{h}$ representing the halo mass growth rate;

(4) \begin{equation} \dot{m}_\mathrm{c,g}=\unicode{x025B}_\mathrm{in} \left(\frac{\Omega_\mathrm{b}}{\Omega_\mathrm{m}}\right)\dot{m}_\mathrm{h}. \end{equation}

Here the quantities $\Omega_\mathrm{b}$ and $\Omega_\mathrm{m}$ refer to the total baryon and matter densities of the Universe, respectively, and the assumption is that the halo accretes a baryon mass equivalent to the cosmic fraction ( $\Omega_\mathrm{b}/\Omega_\mathrm{m}$ ) per unit dark matter accreted. We also use a pre-factor $0 \leq \unicode{x025B}_\mathrm{in} \leq 1$ , such that $\unicode{x025B}_\mathrm{in}=1$ corresponds to the accretion of the cosmic baryon fraction and $\unicode{x025B}_\mathrm{in}=0$ corresponds to complete suppression of baryon accretion. We discuss this further in the context of UV suppression of cosmological accretion below.

2.1.2 Growth of stellar mass

Equation (2),

\begin{equation} \dot{m}_{\star}=\unicode{x025B}_\mathrm{sf}\frac{m_{g}}{\tau_\mathrm{sf}}, \nonumber \end{equation}

links the star formation rate to the mass of the gas reservoir, $m_\mathrm{g}$ , a star formation timescale, $\tau_\mathrm{sf}$ , and an efficiency factor, $\unicode{x025B}_\mathrm{sf}$ . We assume that $\tau_\mathrm{sf}$ is proportional to the dynamical time,

(5) \begin{equation} \tau_\mathrm{sf} \propto t_\mathrm{dyn}=\frac{R}{\sigma} \end{equation}

where R and $\sigma$ are the characteristic radius and velocity dispersion of the system respectively. If we adopt the dynamical time of the halo as characterising this timescale, we expect that

(6) \begin{equation} t_\mathrm{dyn}=\left(\frac{4}{\Delta_\mathrm{vir}}\right)^{1/2} \frac{1}{H(t)} \simeq \frac{0.15}{H(t)} \end{equation}

where $\Delta_\mathrm{vir} \simeq 178$ is the virial overdensity of the halo and

(7) \begin{equation} H(t)=H_\mathrm{0}\sqrt{\Omega_\mathrm{m}(1+z(t))^3+\Omega_\Lambda}. \end{equation}

is the Hubble parameter at cosmic time, $t\equiv~t(z)$ . We assume that

(8) \begin{equation} \tau_\mathrm{sf} = 0.15 \frac{f_\mathrm{sf}}{H(z)} \end{equation}

and expect that $f_\mathrm{sf} \gtrsim 1$ . We take $\unicode{x025B}_\mathrm{sf}=0.015$ as our fiducial value, following Furlanetto & Mirocha (Reference Furlanetto and Mirocha2022) and motivated by Murray (Reference Murray2011) and Leroy et al. (Reference Leroy2017), but we allow this parameter to vary between $\unicode{x025B}_\mathrm{sf}=0.0015$ (low efficiency) and $\unicode{x025B}_\mathrm{sf}=0.1$ (high efficiency) to gauge its influence on our results.

2.1.3 Supernovae-driven wind mass loss

Equation (3),

\begin{equation} \dot{m}_\mathrm{w}=\unicode{x03B7}_\textbf{fb}\dot{m}_\star(t-t^\mathrm{d}),\nonumber \end{equation}

links the rate of mass loss driven by supernovae to $\dot{m}_\star$ at some time $t-t^\mathrm{d}$ , where $t^\mathrm{d}$ is the delayed feedback timescale. If $t^\mathrm{d}=0$ , then this mass loss rate is driven by the instantaneous star formation rate; however, if $t^\mathrm{d}>0$ , then it depends on star formation at some earlier time and corresponds to delayed feedback. We adopt $t^\mathrm{d}$ =0.015 Gyr (15 Myr) as our fiducial value, but have explored values of $0 \leq t^\mathrm{d} \leq 0.03$ Gyr.

We follow Furlanetto et al. (Reference Furlanetto, Mirocha, Mebane and Sun2017) in defining our choice of feedback efficiency parameter; they model momentum-regulated supernova feedback and assume that supernovae accelerate the wind to the escape velocity of the halo. This gives

(9) \begin{equation} \unicode{x03B7}_\textbf{fb}=\unicode{x025B}_\mathrm{fb}\pi_\mathrm{p} \left(\frac{10^{11.5}\,\mathrm{M}_{\odot}}{m_\mathrm{h}}\right)^{1/3} \left(\frac{9}{1+z}\right)^{1/2}, \end{equation}

where $\unicode{x025B}_\mathrm{fb}$ parameterises feedback efficiency and corresponds to the momentum injected by the supernovae driving the wind, and $\pi_{\mathrm{p}}$ is the total amount of momentum from each supernova. We follow Furlanetto & Mirocha (Reference Furlanetto and Mirocha2022) in adopting $\unicode{x025B}_\mathrm{fb}=5$ as our fiducial value, but we also consider values in the range $2\leq \unicode{x025B}_\mathrm{fb}\leq 7$ . Following Furlanetto et al. (Reference Furlanetto, Mirocha, Mebane and Sun2017), we assume $\pi_{\mathrm{p}}$ , to be of the order unity.

2.1.4 UV suppression of cosmological gas accretion

We follow the approach of Kravt

sov & Manwadkar (Reference Kravtsov and Manwadkar2022) by noting that the gas mass within a dark matter halo may be a function of both mass and redshift (e.g. Okamoto, Gao, & Theuns Reference Okamoto, Gao and Theuns2008),

(10) \begin{equation} m_\mathrm{g}(z)=f_\mathrm{b}(m_\mathrm{h},z)m_\mathrm{h}(z). \end{equation}

The quantity $f_\mathrm{b}(m_\mathrm{h},z)$ is related to $\unicode{x025B}_\mathrm{in}$ in Equation (1), as we make clear below. Following Gnedin (Reference Gnedin2000), Okamoto, Gao, & Theuns (Reference Okamoto, Gao and Theuns2008), we write

(11) \begin{equation} f_\mathrm{b}(m_\mathrm{h},z)=\frac{\Omega_\mathrm{b}}{\Omega_\mathrm{m}}s(\unicode{x03BC}_\mathrm{c},\unicode{x03C9}), \end{equation}

where $\Omega_\mathrm{b}/\Omega_\mathrm{m}$ is the cosmic baryon fraction and s(x,y) is defined as:

(12) \begin{equation} s(x,y)= [1+(2^{\frac{y}{3}}-1)x^{-y}]^{-\frac{3}{y}}; \end{equation}

$\unicode{x03BC}_\mathrm{c}=m_\mathrm{h}/M_\mathrm{c}(z)$ , where $M_\mathrm{c}(z)$ is a characteristic mass below which $s(\unicode{x03BC}_\mathrm{c},\unicode{x03C9}) \rightarrow 0$ . Following Kravtsov & Manwadkar (Reference Kravtsov and Manwadkar2022), we adopt

(13) \begin{equation} M_\mathrm{c}=1.69\times 10^{10}\frac{\mathrm{exp}(-0.63z)}{1+\mathrm{exp}([z/\unicode{x03B2}]^{\unicode{x03B3}})}\,\mathrm{M}_{\odot}, \end{equation}

where $\unicode{x03B3}=15$ and $\unicode{x03B2}$ is given by:

(14) \begin{equation} \unicode{x03B2}=z_\mathrm{rei} \left[\mathrm{ln} \left(1.82 \times 10^3\mathrm{exp}(-0.63z_\mathrm{rei})-1\right)\right]^{-1/\unicode{x03B3}}; \end{equation}

here $z_\mathrm{rei}$ is the redshift of reionisation. We adopt as our fiducial value $z_\mathrm{rei}=7$ but have checked the sensitivity of our results to this choice.

If we consider the time rate of change of Equations (10) and (11), we see that

\begin{align} \dot{m}_\mathrm{g}(t)=\dot{f}_\mathrm{b}m_\mathrm{h}+f_\mathrm{b}\dot{m}_\mathrm{h}&=\left(f_\mathrm{b}+\dot{f}_\mathrm{b}\frac{m_\mathrm{h}}{\dot{m}_\mathrm{h}}\right)\dot{m}_\mathrm{h}\nonumber\\ &=\left(s+\dot{s}\frac{m_\mathrm{h}}{\dot{m}_\mathrm{h}}\right)\left(\frac{\Omega_\mathrm{b}}{\Omega_\mathrm{m}}\right)\dot{m}_\mathrm{h}\nonumber, \end{align}

which is equivalent to Equation (4), and so we can evaluate $\unicode{x025B}_\mathrm{in}$ , given the form of Equation (12).Footnote b We write,

(15) \begin{align} \unicode{x025B}_\mathrm{in}=&\mathrm{max}\bigg(0,s (\unicode{x03BC}_\mathrm{c},\unicode{x03C9})\bigg[(1+X)- 2\unicode{x025B}(z,\unicode{x03B3})\frac{M_\mathrm{h}}{\dot{M}_\mathrm{h}}X(1+z)H(z)\bigg]\bigg), \end{align}

where

\begin{align*} \unicode{x025B}(z,\unicode{x03B3})&=\frac{0.63}{1+e^{(z/\unicode{x03B2})^{\unicode{x03B3}}}}+\frac{\unicode{x03B3} z^{\unicode{x03B3}-1}}{\unicode{x03B2}^{\unicode{x03B3}}}\frac{e^{(z/\unicode{x03B2})^{\unicode{x03B3}}}}{(1+e^{(z/\unicode{x03B2})^{\unicode{x03B3}}})^2},\\[4pt] X&=\frac{3c_{\unicode{x03C9}} M_{\unicode{x03C9}}}{1+c_{\unicode{x03C9}} M_{\unicode{x03C9}}},\quad c_{\unicode{x03C9}}=2^{\unicode{x03C9}/3}-1,\\[4pt] M_{\unicode{x03C9}}&= \left(\frac{M_\mathrm{c}(z)}{M_\mathrm{h}}\right)^{\unicode{x03C9}},\quad\unicode{x03C9}=2. \end{align*}

This has the effect of suppressing gas accretion at halo masses below the threshold mass $M_\mathrm{c}(z)$ at $z<z_\mathrm{rei}$ , with $M_\mathrm{c}(z)$ increasing with decreasing z (i.e. accretion is suppressed onto progressively higher mass halos). For redshifts $z>z_\mathrm{rei}$ , there is no suppression of gas accretion. We refer the interested reader to Figure 1 of Kravtsov & Manwadkar (Reference Kravtsov and Manwadkar2022) for an illustration of this behaviour.

Figure 1. Mass assembly history of a sample of 100 dark matter halos with a halo mass of $m_\mathrm{h}=10^8\,\mathrm{M}_{\odot}$ at $z=5$ generated using the Parkinson, Cole, & Helly (Reference Parkinson, Cole and Helly2008) Monte Carlo merger tree algorithm. The upper and lower panels show $m_\mathrm{h}$ (in $\mathrm{M}_{\odot}$ ) and $\dot{m}_\mathrm{h}$ (in $\mathrm{M}_{\odot}/{\mathrm{Gyr}}$ ) against cosmic time (in Gyrs; lower horizontal axis) and redshift (upper horizontal axis); red solid and dotted curves indicate the median, 10th, and 90th percentiles of the distributions at a given time.

2.2 Modelling dark matter halo growth

We have used the algorithm of Parkinson, Cole, & Helly (Reference Parkinson, Cole and Helly2008) to generate Monte Carlo dark matter halo merger trees based on Extended Press-Schechter theory (e.g. Bond et al. Reference Bond, Cole, Efstathiou and Kaiser1991; Lacey & Cole Reference Lacey and Cole1993). This has been calibrated to provide predicted halo masses as a function of cosmic time that are consistent with merger trees derived from the Millennium Simulation (cf. Springel et al. Reference Springel2005). For each of the 11 halo mass bins equally spaced (linearly) in the interval $10^6\,\mathrm{M}_{\odot} \leq m_\mathrm{h} \leq 10^{11}\,\mathrm{M}_{\odot}$ , we sample 100 realisations following mass assembly histories in the redshift range $5 \leq z \leq 25$ equally spaced in the logarithm of the expansion factor, $a=1/(1+z)$ . This provides us with $m_\mathrm{h}$ as a function of z; we calculate $\dot{m}_\mathrm{h}$ as a function of t(z) by constructing a smooth spline interpolant, which we differentiate numerically.

In Fig. 1, we show the mass assembly histories of a sample of dark matter halos that have masses of $m_\mathrm{h}=10^8\,\mathrm{M}_{\odot}$ at $z=5$ . Each individual halo is represented by a light grey curve; the median, 10th, and 90th percentiles are indicated by the red solid and dotted curves. The upper panel shows the variation of $m_\mathrm{h}$ with cosmic time (lower horizontal axis) and redshift (upper horizontal axis), starting at $z=25$ , while the lower panel shows the corresponding values of $\dot{m}_\mathrm{h}$ . This shows that our Monte Carlo merger trees produce a diversity of assembly histories and that halos that have the same value of $m_\mathrm{h}$ at $z=5$ can differ by up to an order of magnitude in $m_\mathrm{h}$ and $\dot{m}_\mathrm{h}$ at earlier times. This allows us to explore how such variation in $\dot{m}_\mathrm{h}$ and the corresponding influence on $\dot{m}_\mathrm{c,g}$ and $\dot{m}_\star$ affect the halos’ baryon mass content.

Figure 2. Baryon mass assembly history of an example halo with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . The upper and middle panels show $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs); solid (dotted) curves correspond to instantaneous star formation and feedback without (with) UV suppression of accretion (instantaneous + No UV, instantaneous + UV), while dashed (dot-dashed) curves correspond to bursty star formation from delayed feedback without (with) UV suppression of accretion (delayed + No UV, delayed + UV). For comparison, we show also the growth of $m_\mathrm{h}$ with cosmic time in the lower panel.

3.1 Bursty star formation & UV suppression of accretion

We begin by considering the growth of gas mass (upper panel), stellar mass (middle panel), and halo mass (lower panel; for comparison) in Fig. 2, focusing on an example halo with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ , drawn from the sample of 100 halos in this mass bin. We choose this halo mass bin because it marks the transition between low-mass halos that can be efficiently quenched via delayed feedback alone and high-mass halos that continue to accrete and form stars after the onset of UV suppression; we show examples of low- and high-mass systems’ assembly histories in Appendix 1. To illustrate the effects of bursty star formation and UV suppression on the growth of gas and stellar mass as a function of cosmic time, we consider the cases of instantaneous star formation and feedback and no UV suppression of accretion (hereafter instantaneous + No UV; solid curves); instantaneous star formation feedback and UV suppression of gas accretion with $z_\mathrm{rei}=7$ (hereafter instantaneous + UV; dotted curves); bursty star formation from delayed feedback and no UV suppression of accretion (hereafter delayed + No UV; dashed curves); and bursty star formation and feedback and UV suppression of accretion with $z_\mathrm{rei}=7$ (hereafter delayed + UV; dot-dashed curves). We adopt our fiducial parameters of $\unicode{x025B}_\mathrm{sf}=0.015$ and $\unicode{x025B}_\mathrm{fb}=5$ .

In the instantaneous + No UV case, we find that both $m_\mathrm{g}$ and $m_\star$ show an initial steep rise at early times ( $\leq$ $0.5$ Gyr), and continue to grow but at a slower at later times (up to 2 Gyr). There are noticeable increases in $m_\mathrm{g}$ at approximately 0.5 and 0.6 Gyr; these correlate with increases in $m_\mathrm{h}$ evident in the lower panel, which translate into increases in $m_\mathrm{g}$ given our assumption that $\dot{m}_\mathrm{c,g}$ tracks $\dot{m}_\mathrm{h}$ . In contrast, for instantaneous + UV, we find that $m_\mathrm{g}$ and $m_\star$ show a similar initial steep rise at early times but the effect of the UV background shutting off cosmological accretion is evident – $m_\mathrm{g}$ peaks after 0.75 Gyr and then declines with increasing time, while $m_\star$ plateaus. Because $m_\mathrm{g}$ is linked to $\dot{m}_\mathrm{h}$ and the magnitude of UV suppression is both halo mass and redshift-dependent, we see that $m_\mathrm{g}$ shows an initial sharp decline but plateaus after 1.5 Gyr. For the delayed + No UV case, we see the characteristic strong oscillations in $m_\mathrm{g}$ – and to a lesser extent in $m_\star$ – evident during the initial 0.5 Gyr, but as the halo grows, these oscillations damp away, and both quantities are indistinguishable from the case with instantaneous star formation and feedback. Similarly, for the delayed + UV case, we see these oscillations in $m_\mathrm{g}$ and $m_\star$ repeated.

3.2 Halo mass assembly history

In Fig. 3, we show the variation in $m_\mathrm{g}$ and $m_\star$ with cosmic time for a sample of 100 halos, each with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . We now focus on models with the UV suppression of accretion of gas from the IGM. The dotted (dot-dashed) curves correspond to the median values of $m_\mathrm{g}$ (upper panel) and $m_\star$ (lower) for the instantaneous + UV (delayed +UV) case, while the coloured bands indicate the range of the 10th and 90th percentiles. This shows how variations in the assembly history of the underlying dark matter halo, whose growth rate $\dot{m}_\mathrm{h}$ governs the growth rate of the gas mass and consequently the stellar mass. The variations introduced by the diversity of halo assembly histories correspond to approximately $0.5-1$ dex in both $m_\mathrm{g}$ and $m_\star$ . The variations in $m_\mathrm{g}$ are larger for the delayed + UV case, as we might expect, while we see that the variations in $m_\star$ are similar in magnitude for both the instantaneous + UV and delayed + UV cases ( $\sim$ $0.5$ dex at early times, $\sim$ 1 dex at later times).

Figure 3. Influence of halo mass assembly history: Baryon mass assembly history of a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . The upper and lower panels show the median values of $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs); dotted curves correspond to the instantaneous + UV case, while dot-dashed curves correspond to the delayed + UV case. The coloured bands indicate the range of the 10th and 90th percentiles.

3.3 Efficiency of star formation, $\unicode{x025B}_\mathrm{sf}$ , and feedback, $\unicode{x025B}_\mathrm{fb}$

We now examine how our choices of star formation and feedback efficiency affect the growth of gas mass (upper panel) and stellar mass (lower panel) in Figs. 4 and 5. As in Fig. 3, we focus on a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ , but for cases of low and high star formation and feedback efficiencies, $\unicode{x025B}_\mathrm{sf}$ and $\unicode{x025B}_\mathrm{fb}$ , respectively. We keep $z_\mathrm{rei}=7$ fixed and look at the cases of low (high) efficiency in the upper (lower) panels, with reference curves corresponding to our fiducial values plotted in light grey ( $\unicode{x025B}_\mathrm{sf}$ =0.015 and $\unicode{x025B}_\mathrm{fb}$ =5).

Figure 4. Influence of star formation efficiency: Baryon mass assembly history of a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . The upper and lower panels show the median values of $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs) for $\unicode{x025B}_\mathrm{sf}$ =0.0015 (inefficient) and $\unicode{x025B}_\mathrm{sf}$ =0.1 (efficient) respectively. Dotted curves correspond to the instantaneous + UV case, while dot-dashed curves correspond to the delayed + UV case. The coloured bands indicate the range of the 10th and 90th percentiles. The greyed bands and curves correspond to the counterpart cases with the fiducial value of $\unicode{x025B}_\mathrm{sf}$ =0.015.

Figure 5. Influence of feedback efficiency: Baryon mass assembly history of a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . The upper and lower panels show the median values of $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs) for $\unicode{x025B}_\mathrm{fb}$ =2 (inefficient) and $\unicode{x025B}_\mathrm{fb}$ =7 (efficient) respectively. Dotted curves correspond to the instantaneous + UV case, while dot-dashed curves correspond to the delayed + UV case. The coloured bands indicate the range of the 10th and 90th percentiles. The greyed bands and curves correspond to the counterpart cases with the fiducial value of $\unicode{x025B}_\mathrm{fb}$ =5.

As before, we focus on the cases of instantaneous + UV (dotted curves) and delayed + UV (dot-dashed curves). By reducing (increasing) the star formation efficiency, we expect the number of stars to be formed over a fixed interval to be lower (higher) than in the fiducial case; however, the coupling of star formation to feedback means that a higher star formation efficiency should boost the strength of the feedback, which, in the case of delayed feedback, should lead to stronger oscillations in the ${m}_\mathrm{g}$ and ${m}_\star$ (see, e.g. Furlanetto & Mirocha Reference Furlanetto and Mirocha2022). We anticipate that stronger (weaker) feedback efficiency should impact $\dot{m}_\mathrm{c,g}$ and consequently $m_\mathrm{g}$ , but it’s likely the that the difference with respect to the fiducial run will be more pronounced for weaker feedback – it may evacuate the halo of gas more quickly, but the accretion rate onto the halo is unchanged.

In Fig. 4, we fix feedback efficiency at $\unicode{x025B}_\mathrm{fb}$ =5 and look at star formation efficiencies of $\unicode{x025B}_\mathrm{sf}$ =0.0015 (low efficiency, upper panel) and $\unicode{x025B}_\mathrm{sf}$ =0.1 (high efficiency, lower panel). As expected, in the case of higher star formation efficiency, we see evidence for enhanced star formation efficiency driving enhanced feedback. Consider the instantaneous + UV case. At a given time, we see that $m_\mathrm{g}$ is smaller by approximately 0.5 dex compared to the fiducial run up to the point at which the effect of UV suppression kicks in, following which $m_\mathrm{g}$ rapidly declines as its converted to stars; the evolution of $m_\star$ is relatively unchanged from the fiducial case. In contrast, the delayed + UV case shows some interesting changes compared to the fiducial case; the initial evolution of $m_\mathrm{g}$ is similar but from approximately 0.2 Gyr it experiences 3 longer periods in which $m_\mathrm{g}\rightarrow 0$ , while we also note that the typical system forms more stars at a given time, and $m_\star$ is $\sim 1$ dex higher after 1.2 Gyr. For the low star formation efficiency case, we see that the differences between the two models are dramatically reduced. In particular, the strong oscillations apparent in the fiducial delayed + UV case are very weak – only a dip in $m_\mathrm{g}$ is evident at 0.2 Gyr – and the evolution in $m_\mathrm{g}$ and $m_\star$ are very similar, with the delayed + UV case having higher values of $m_\mathrm{g}$ by $\sim$ $0.1$ dex and lower values of $m_\star$ by $\sim$ $0.1$ dex, when compared to the instantaneous + UV case.

In Fig. 5, we fix star formation efficiency at $\unicode{x025B}_\mathrm{sf}$ =0.015 and look at feedback efficiencies of $\unicode{x025B}_\mathrm{fb}$ =2 (low efficiency, upper panel) and $\unicode{x025B}_\mathrm{fb}$ =7 (high efficiency, lower panel). As anticipated, stronger feedback acts to accelerate when the median halo loses its gas mass (i.e. $m_\mathrm{g}\rightarrow 0$ ), especially following the onset of UV suppression, and the spread in times when this occurs is larger than in the fiducial model (0.3 Gyr compared to 0.2 Gyr in the fiducial case); however, it has little effect on the evolution of the stellar mass. We find that weaker feedback has a dramatic effect on the evolution of $m_\mathrm{g}$ in the delayed +UV model; the initial oscillations seen in the fiducial case are present, but after 0.3 Gyrs $m_\mathrm{g}$ grows up to the onset of UV suppression, before showing a slow decline; the total gas mass in this model is larger than in the instantaneous feedback case after the initial oscillations have damped away. The behaviour of the instantaneous +UV models in both the strong and weak feedback cases track that in the fiducial case, with the key difference being in the rate of decline of $m_\mathrm{g}$ after the onset of UV suppression – the halo retains more (less) gas mass at a given time when feedback is strong (weak).

3.4 Delayed feedback timescale, $t^\mathrm{d}$

In Fig. 6 we assess how our choice of the delayed feedback timescale, $t^\mathrm{d}$ , influences the growth of $m_\mathrm{g}$ and $m_\star$ with cosmic time. We focus on a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ , with $z_\mathrm{rei}=7$ fixed and our fiducial values of star formation and feedback efficiency of $\unicode{x025B}_\mathrm{sf}$ =0.015 and $\unicode{x025B}_\mathrm{fb}$ =5. According to our formulation, the instantaneous star formation and feedback model corresponds to $t^\mathrm{d}$ =0, while our fiducial delayed feedback model assumes $t^\mathrm{d}$ =0.015 Gyr; we consider two further cases, with $t^\mathrm{d}$ =0.0075 Gyr (upper panel) and 0.03 Gyr (lower panel).

Figure 6. Influence of delayed feedback timescale: Baryon mass assembly history of a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . The upper and lower panels show the median values of $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs) for $t^\mathrm{d}$ =0.0075 Gyr and $t^\mathrm{d}$ =0.03 Gyr respectively. Dotted curves correspond to the instantaneous + UV case, while dot-dashed curves correspond to the delayed + UV case. The coloured bands indicate the range of the 10th and 90th percentiles. The greyed bands and curves correspond to the counterpart cases with the fiducial value of $t^\mathrm{d}$ =0.015 Gyr.

Compared to our fiducial model with $t^\mathrm{d}$ =0.015 Gyr, we see that a shorter (longer) time delay increases (decreases) the number of initial oscillations in both $m_\mathrm{g}$ and $m_\star$ , and the peak median masses are similar. There are two noticeable differences – in the lengths of the intervals when $m_\mathrm{g}\rightarrow 0$ at early times, with $t^\mathrm{d}$ =0.03 Gyr experiencing longer intervals when $m_\mathrm{g}\simeq 0$ ; and in the evolution of $m_\mathrm{g}$ after the onset of UV suppression of gas accretion, such that $m_\mathrm{g}$ plateaus close to its peak value in the model with the long time delay through to 1.2 Gyr, whereas $m_\mathrm{g}\rightarrow 0$ after the onset of UV suppression as in the fiducial case, albeit with a delay of $\sim$ $0.1$ Gyr. These trends highlight the complex interplay between gas accretion, star formation, and delayed feedback – especially the manner in which gas mass is retained at later times when $t^\mathrm{d}$ =0.03 Gyr.

3.5 Redshift of reionisation, $z_\mathrm{rei}$

In Fig. 7, we assess how our choice of the redshift of reionisation, $z_\mathrm{rei}$ , influences the growth of $m_\mathrm{g}$ and $m_\star$ with cosmic time. Again we focus on a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ , with our fiducial values of star formation and feedback efficiency of $\unicode{x025B}_\mathrm{sf}$ =0.015, $\unicode{x025B}_\mathrm{fb}$ =5 fixed. We look at the case of $z_\mathrm{rei}$ =10, which corresponds to the time when the Universes was approximately 60% of its age at $z=$ 7 (our fiducial redshift of reionisation). The trends are qualitatively similar to those we see when $z_\mathrm{rei}$ =7; the peak values of $m_\mathrm{g}$ and $m_\star$ are lower, and the rate at which $m_\mathrm{g}\rightarrow 0$ is more rapid in the case of instantaneous feedback with UV suppression, as we would expect because of the shorter dynamical times in these systems.

Figure 7. Influence of redshift of reionisation: Baryon mass assembly history of a sample of 100 halos with $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . We show the median values of $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs) for $z_\mathrm{rei}$ =10. Dotted curves correspond to the instantaneous + UV case, while dot-dashed curves correspond to the delayed + UV case. The coloured bands indicate the range of the 10th and 90th percentiles. The greyed bands and curves correspond to the counterpart cases with the fiducial value of $z_\mathrm{rei}$ =7.

3.6 Baryon mass content – variation with halo mass

We have explored how local factors – star formation and feedback efficiencies, burstiness of star formation, variation in dark matter halo assembly histories – and global factors – the onset of UV suppression arising from cosmological reionisation – influence the time evolution of the gas and stellar masses of halos. We focused on a halo mass scale – $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ – that we argued is transitional, being sufficiently massive enough not to experience early truncation of gas accretion and star formation, but not overly massive that it’s unaffected by delayed feedback and UV suppression of gas accretion from the IGM. We have this information for halos spanning a mass range $10^6\,\mathrm{M}_\odot\leq m_\mathrm{h} \leq 10^{11}\,\mathrm{M}_\odot$ at that epoch, as we show in Appendix 1.

In Fig. 8 we quantify how $m_\mathrm{g}$ and $m_\star$ vary with $m_\mathrm{h}$ , looking at three epochs – z=10, 7, and 5 (from right to left). Because we are interested in relative trends, we plot the quantity $m_\star/(m_\star+m_\mathrm{g})$ against $m_\mathrm{h}$ at that epoch. As the system becomes gas depleted, we expect that $m_\star/(m_\star+m_\mathrm{g})\rightarrow 1$ because $m_\mathrm{g}\rightarrow 0$ , which we expect at lower halo masses and should move to a higher mass scale following the onset of the UV suppression of gas accretion from the IGM.

Figure 8. Baryon mass assembly history of a $m_\mathrm{h}=10^7\,\mathrm{M}_{\odot}$ at $z=5$ . The upper and lower panels show $m_\mathrm{g}$ and $m_\star$ (in $\mathrm{M}_{\odot}$ ) against cosmic time (in Gyrs); solid (dotted) curves correspond to instantaneous star formation and feedback without (with) UV suppression of accretion, while dashed (dot-dashed) curves correspond to bursty star formation without (with) UV suppression of accretion.

At $z=10$ , we see that this ratio shows little variation across the mass range, with the median $m_\star/(m_\star+m_\mathrm{g})\simeq 0.05$ in the instantaneous + UV case, compared to $m_\star/(m_\star+m_\mathrm{g})\simeq 0.03$ in the delayed + UV case. There is a weak trend for $m_\star/(m_\star+m_\mathrm{g})$ to decrease (increase) with increasing mass in the instantaneous (delayed) + UV case. Interestingly, we see an enhancement $m_\star/(m_\star+m_\mathrm{g})$ in the delayed + UV case in the halo mass range $10^7 \mathrm{M}_\odot \leq m_\mathrm{h} \leq 10^8 \mathrm{M}_\odot$ – this reflects the transitional nature of this halo mass range and how halo mass assembly histories interact with burstiness in star formation and delayed feedback to produce a range in gas mass.

By $z=7$ , we now see a marked variation across the mass range. At higher masses, the median $m_\star/(m_\star+m_\mathrm{g})\simeq 0.05$ in the instantaneous + UV case, unchanged from $z=10$ ; in contrast, $m_\star/(m_\star+m_\mathrm{g})\simeq 0.02$ in the delayed + UV case, which has is lower than at $z=10$ . The ratio increases smoothly towards lower masses in the instantaneous + UV case, whereas the ratio is flat down to $m_\mathrm{h} \simeq 10^7\,\mathrm{M}_\odot$ before sharply turning up to $m_\star/(m_\star+m_\mathrm{g})\simeq 1$ . At $z=5$ , the marked variation with mass has strengthened. At higher masses, the median $m_\star/(m_\star+m_\mathrm{g})\simeq 0.05$ in the instantaneous + UV case is unchanged from z=10, whereas $m_\star/(m_\star+m_\mathrm{g})\simeq 0.01$ in the delayed + UV case, which is marginally lower than at $z=7$ . We now see a gradual rise in $m_\star/(m_\star+m_\mathrm{g})$ towards lower $m_\mathrm{h}$ becoming evident at $m_\mathrm{h} \simeq 10^{10}\,\mathrm{M}_\odot$ ; by $m_\star/(m_\star+m_\mathrm{g})$ =1 by $m_\mathrm{h} \simeq 10^7\,\mathrm{M}_\odot$ in both the instantaneous + UV and delayed + UV cases.

At the three epochs we have considered, we see a general trend for $m_\star/(m_\star+m_\mathrm{g})$ to be approximately a factor of 3–5 larger in the instantaneous + UV case compared to the delayed + UV case. Depending on epoch and mass scale, this can mean both a relative enhancement in stellar mass or a relative deficit in gas mass in the instantaneous + UV model. However, the trend is for the lower mass galaxies to have higher stellar masses in the delayed + UV case, while – in the context of this particular physical model – higher mass galaxies can retain their gas reservoirs. Physically, we would expect seed supermassive black hole growth and associated feedback to reduce gas mass in these high-mass galaxies. We will revisit this in a future paper.