Growth of First Galaxies: Impacts of Star Formation and Stellar Feedback

We use a modified version of the smoothed particle hydrodynamics (SPH) code gadget-3 (Springel 2005) that was previously developed in the Overwhelmingly Large Simulations (OWLS) project (Schaye et al. 2010). This code was extended and modified to include the treatment of population III (Pop III) SF, Lyman–Werner feedback, non-equilibrium primordial chemistry, and dust formation/destruction in the FiBY project (e.g., S. Khochfar & C. Dalla Vecchia et al. 2017, in preparation). The impacts of these new physical processes on galaxy formation have been studied in a series of papers on the FiBY project (e.g., Johnson et al. 2013; Paardekooper et al. 2013; Agarwal & Khochfar 2015; Elliott et al. 2015). In this work, we do not consider the formation of Pop III stars and non-equilibrium primordial chemistry because our focus is on more massive galaxies after the era of first mini halos. We here use the SF model based on the Kennicutt–Schmidt law, the SN feedback, and the equilibrium radiation cooling including metal lines as explained below.

Our simulations follow the dynamics of dark matter and gas from z = 100 to $z\sim 6$. The initial conditions are created by the MUSIC code (Hahn & Abel 2011). We first run a coarse N-body simulation with 128³ particles and identify the dark matter halos using the friends-of-friends (FOF) grouping algorithm. We then choose three halos with ${M}_{{\rm{h}}}=2.4\,\times {10}^{10}\,{h}^{-1}\,{M}_{\odot }$ (Halo-10), $1.6\times {10}^{11}\,{h}^{-1}\,{M}_{\odot }$ (Halo-11), and $7.5\times {10}^{11}\,{h}^{-1}\,{M}_{\odot }$ (Halo-12) at z = 6 for subsequent zoom-in simulations. The entire simulation box size is (20 ${h}^{-1}$ Mpc)³ for Halo-10 and Halo-11, and (100 ${h}^{-1}$ Mpc)³ for Halo-12 in comoving units. For Halo-10 and Halo-11, the effective resolution is 2048³ and the zoom-in region is $\sim {(1\mbox{--}2{h}^{-1}\mathrm{Mpc})}^{3}$ in comoving scale. For Halo-12, the effective resolution is 4096³ and the zoom-in region is ${(6.6{h}^{-1}\mathrm{Mpc})}^{3}$. The relevant simulation parameters are summarized in Table 1. In this work, we set the constant gravitational softening length to $200\,{h}^{-1}\,\mathrm{pc}$ in the comoving scale and the minimum smoothing length to $20\,{h}^{-1}\,\mathrm{pc}$. This corresponds to a resolution of $\lesssim 10\,\mathrm{pc}$ physical at $z\sim 10$, which allows the internal gas structure for galaxies to be resolved.

Table 1.  Parameters of Zoom-in Cosmological Hydrodynamic Simulations

Halo ID	${M}_{{\rm{h}}}\ [{h}^{-1}\,{M}_{\odot }]$	${m}_{\mathrm{DM}}\ [{h}^{-1}\,{M}_{\odot }]$	${m}_{\mathrm{gas}}\ [{h}^{-1}\,{M}_{\odot }]$	${\epsilon }_{\min }\ [{h}^{-1}\,\mathrm{pc}]$	SNe feedback	A	${z}_{\mathrm{re}}$
Halo-10	$2.4\times {10}^{10}$	$6.6\times {10}^{4}$	$1.2\times {10}^{4}$	200	ON	$2.5\times {10}^{-3}$	10
Halo-11	$1.6\times {10}^{11}$	$6.6\times {10}^{4}$	$1.2\times {10}^{4}$	200	ON	$2.5\times {10}^{-3}$	10
Halo-11-lowSF	$1.6\times {10}^{11}$	$6.6\times {10}^{4}$	$1.2\times {10}^{4}$	200	ON	$2.5\times {10}^{-4}$	10
Halo-11-noSN	$1.6\times {10}^{11}$	$6.6\times {10}^{4}$	$1.2\times {10}^{4}$	200	OFF	$2.5\times {10}^{-3}$	10
Halo-11-eUVB	$1.6\times {10}^{11}$	$6.6\times {10}^{4}$	$1.2\times {10}^{4}$	200	ON	$2.5\times {10}^{-3}$	18
Halo-12	$7.5\times {10}^{11}$	$1.1\times {10}^{6}$	$1.8\times {10}^{5}$	200	ON	$2.5\times {10}^{-3}$	10

Note. (1) ${M}_{{\rm{h}}}$ is the halo mass at z = 6. (2) ${m}_{\mathrm{DM}}$ is the dark matter particle mass. (3) ${m}_{\mathrm{gas}}$ is the initial mass of gas particles. (4) ${\epsilon }_{\min }$ is the gravitational softening length on a comoving scale. (5) A is the amplitude factor of the star formation model based on the Kennicutt–Schmidt law. (6) ${z}_{\mathrm{re}}$ is the reionization redshift when the UVB is turned on.

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Figure 1 shows the redshift evolution of the halo mass, the growth rate of halo mass, and the rate of change of the total gas mass. All halos grow as the redshift decreases, although the growth rate depends on the environment. Halo-10 is more massive than Halo-11 at $z\gtrsim 10$; however, Halo-11 grows rapidly and becomes more massive than Halo-10 during $z=6\mbox{--}9$. Halo-12 is always more massive than the other halos. As shown in the middle panel of the figure, the halo growth rate of Halo-10 is higher than that of Halo-11 at $z\gtrsim 10$, but it stalls at lower redshift, whereas Halo-11 rapidly increases its growth rate at $z\lesssim 10$.

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Our basic SF prescription is based on the Kennicutt–Schmidt law of local galaxies, and the adopted formulation was developed by Schaye & Dalla Vecchia (2008). This model estimates the star formation rate (SFR) based on the local ISM pressure as follows:

$$\begin{eqnarray}&&{\dot{m}}_{* }={m}_{{\rm{g}}}A{(1{M}_{\odot }{\mathrm{pc}}^{-2})}^{-n}{\left(\displaystyle \frac{\gamma }{G}{f}_{{\rm{g}}}P\right)}^{(n-1)/2},\end{eqnarray} \tag{ 1 }$$

where ${m}_{{\rm{g}}}$ is the mass of the gas particle, $\gamma =5/3$ is the ratio of specific heats, ${f}_{{\rm{g}}}$ is the gas mass fraction in the self-gravitating galactic disk, and P is the total ISM pressure. The free parameters in this SF model are the amplitude A and the power-law index n. These parameters are related to the Kennicutt–Schmidt law,

$$\begin{eqnarray}&&{\dot{{\rm{\Sigma }}}}_{* }=A{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{gas}}}{1{M}_{\odot }{\mathrm{pc}}^{-2}}\right)}^{n}.\end{eqnarray} \tag{ 2 }$$

Local normal star-forming galaxies follow ${A}_{\mathrm{local}}=1.5\,\times {10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ and n = 1.4 for the Saltpeter IMF. Note that, the amplitude should be changed by a factor $1/1.65$ in the case of the Chabrier IMF, i.e., ${A}_{\mathrm{local},\mathrm{Chab}}=2.5\,\times {10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{{\rm{kpc}}}^{-2}$. Schaye et al. (2010) reproduced the observed cosmic star formation rate density (SFRD) using cosmological SPH simulations with this SF model and the parameters $A=2.5\times {10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ and n = 1.4 (see also Schaye et al. 2015). On the other hand, this amplitude can be ∼10 times higher for merging starburst galaxies (Genzel et al. 2010). In addition, Tacconi et al. (2013) suggested that the amplitude increases with redshift.

In this work, we focus on high-redshift galaxies at $z\geqslant 6$, which experience frequent merging processes. Therefore, assuming the Chabrier IMF, we adopt the amplitude factor A, which is 10 times higher than the local normal star-forming galaxies, i.e., ${A}_{\mathrm{fiducial}}=2.5\times {10}^{-3}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ as our fiducial value. The SF timescale with this amplitude factor is $\sim {10}^{8}\,\mathrm{yr}$, which is close to the dynamical time of the halos at $z\sim 10$. As shown in Schaye et al. (2010), the cosmic SFRD is not that sensitive to the value of A due to self-regulation by stellar feedback. We also compare the SF histories of galaxies with the lower value of ${A}_{\mathrm{local},\mathrm{Chab}}$. In this work, we set ${f}_{{\rm{g}}}=1$ and the threshold density ${n}_{{\rm{H}}}=10\,{\mathrm{cm}}^{-3}$ above which SF occurs, similarly to the FiBY project. Above the threshold density, we use an effective equation of state with an effective adiabatic index ${\gamma }_{\mathrm{eff}}=4/3$ with the normalized pressure ${P}_{0}/{k}_{{\rm{B}}}={10}^{2}\,{\mathrm{cm}}^{-3}\,{\rm{K}}$ (see Schaye & Dalla Vecchia 2008 for more details).

The cooling rate is estimated from the assumption of the equilibrium state (collisional or photoionization equilibrium) of each metal species. We allow cooling down to $\sim 100\,{\rm{K}}$, which is close to the CMB temperature at the redshifts we are focusing on. Metal-line cooling is considered for each metal species using a table pre-calculated by the cloudy v07.02 code (Ferland 2000). In this work, we do not follow the formation and dissociation of hydrogen molecules (Johnson et al. 2013). At $z\lesssim 10$, galaxies are being irradiated by the UVB, and it penetrates into the gas with ${n}_{{\rm{H}}}\lt 0.01\,{\mathrm{cm}}^{-3}$, which is the threshold density of the self-shielding derived by Nagamine et al. (2010a) and Yajima et al. (2012b) based on the radiative transfer calculations of the UVB. We switch from the collisional to the photoionization equilibrium cooling table once the UVB ionizes the gas (see Johnson et al. 2013 for details). We use the UVB model of Haardt & Madau (2001) in our simulations.

In this work, we consider SN feedback via the injection of thermal energy into neighboring gas particles as described in Dalla Vecchia & Schaye (2012). The thermal energy is stochastically distributed in the surrounding gas and kept until the gas temperature increases to ${10}^{7.5}\,{\rm{K}}$. The feedback efficiency depends on the local physical properties, e.g., gas density, clumpiness, and metallicity (e.g., Cioffi et al. 1988; Kim & Ostriker 2015). Dalla Vecchia & Schaye (2012) compared the sound-crossing time with the cooling time and derived the following maximum gas density under which the thermal energy is efficiently converted into kinetic energy against radiative cooling loss:

$$\begin{eqnarray}&&{n}_{{\rm{H}}}\sim 100\,{\mathrm{cm}}^{-3}\,\left(\displaystyle \frac{T}{{10}^{7.5}\,{\rm{K}}}\right){\left(\displaystyle \frac{{m}_{{\rm{g}}}}{{10}^{4}{M}_{\odot }}\right)}^{-1/2}.\end{eqnarray} \tag{ 3 }$$

In this work, we treat all star and gas particles as part of one galaxy system inside each halo that is identified by an FOF group finder, and do not distinguish substructures in each halo. Our current focus is on the impact of stellar feedback on the physical properties of galaxies, which sensitively depend on the SFR and gravitational potential of the dark matter halo. We adopt the following cosmological parameters that are consistent with current cosmic microwave background observations: ${{\rm{\Omega }}}_{{\rm{M}}}=0.3$, ${{\rm{\Omega }}}_{{\rm{b}}}=0.045$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.7$, ${n}_{{\rm{s}}}=0.965$, and h = 0.7 (Komatsu et al. 2011; Planck Collaboration et al. 2016).

The SF history contains useful information on the details of galaxy formation and evolutionary processes. Recent observed galaxies at $z\gt 6$ have been detected in the rest-frame UV, and their observed fluxes are directly linked to the SFR if dust extinction effects are negligible (see, however, Yajima et al. 2015b; Cullen et al. 2017).

Since the normalization factor A is treated as a free parameter in our study, we first examine the differences between the three runs of Halo-11, Halo-11-noSN, and Halo-11-lowSF.

Figure 2 presents the maps of projected gas density for Halo-11 with three different models at redshift z = 7. In the case of Halo-11 in our fiducial model, the gas structure shows highly inhomogeneous and clumpy features. No extended galactic gas disk is seen within the central ∼2 kpc of Halo-11. On the other hand, disk-like structures are seen in the Halo-11-noSN and Halo-11-lowSF models. This suggests that either the SF efficiency or the SN feedback has a large impact on the internal gas distribution of galaxies. The SF is rather concentrated in the central nuclear region for the Halo-11-noSN, but the stellar distribution is more extended than the other runs due to the larger amount of stars produced by z = 7. As a result, both SF efficiency and SN feedback affect the stellar distribution as shown in the lower panels of the figure. These effects will be discussed quantitatively in Section 3.7.

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Figure 3 shows the SF histories of the three simulated galaxies. In the case without SN feedback (Halo-11-noSN), the SFR increases smoothly as the halo grows from $\sim 5\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ at $z\sim 10$ to $\sim 51\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ at $z\sim 6$. Note that, however, the SFR somewhat fluctuates even without the feedback due to merging processes or gas consumption by SF. Using the relation ${L}_{\nu ,{UV}}\,=0.7\times {10}^{28}\times (\mathrm{SFR}/{M}_{\odot }\,{\mathrm{yr}}^{-1})\,\mathrm{erg}\ {{\rm{s}}}^{-1}$ (Madau et al. 1998; Kennicutt 1998), we can convert the SFR to AB magnitude at the UV band as follows:

$$\begin{eqnarray}&&{M}_{\mathrm{UV}}=-18.0-2.5{\mathrm{log}}_{10}\,\left(\displaystyle \frac{\mathrm{SFR}}{{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right).\end{eqnarray} \tag{ 4 }$$

For example, based on this equation, the simulated galaxies reach −21.9, −22.3, and −22.2 mag at z = 6 in Halo-11, Halo-11-noSN, and Halo-11-lowSF, respectively. The detection limit of recent galaxy surveys at $z\lesssim 8$ has reached $\sim -18$ mag (e.g., Bouwens et al. 2015). Therefore, if SN feedback is inefficient, galaxies hosted in halos with ${M}_{{\rm{h}}}\gtrsim {10}^{11}\,{M}_{\odot }$ are likely to be detected easily as bright LBGs (e.g., McLure et al. 2013; Oesch et al. 2014; Bouwens et al. 2015). Yajima et al. (2015b) also show that massive galaxies formed in an overdense region are very bright. Note, however, that these authors have suggested that some fraction of UV radiation would be absorbed by interstellar dust (see also Yajima et al. 2014b; Cullen et al. 2017). In this case, galaxies will be bright at rest-frame FIR wavelengths and can be observed by ALMA in the submillimeter wavelengths. In the case of Halo-11-lowSF, the SF continuously occurs, but the SFR at $z\gtrsim 10$ is lower than that of Halo-11-noSN by a factor of a few to 10 due to stellar feedback. The difference in the SFR between Halo-11-lowSF and Halo-11-noSN becomes small as the redshift decreases and is within a factor of 2 at $z\lesssim 7$.

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On the other hand, the SF of Halo-11 proceeds intermittently. At $z\gtrsim 10$, SF is almost completely quenched due to stellar feedback after experiencing an SF event with SFR $\sim 0.1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ in Halo-11. As the SFR decreases, SN feedback ceases to work. This allows the gas to fall down toward the galactic center, which leads to the next starburst. Repeating this cycle causes the oscillation of the SFR. The peak and the minimum of the SFR is different by a factor of $\gtrsim 2$ dex at $z\gtrsim 10$. The mean SFR increases with decreasing redshift as the halo mass increases. Even with the high SF efficiency and SN feedback, the SFR becomes similar to Halo-11-noSN at $z\lesssim 7$. The SFR exceeds $1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ at $z\lesssim 8$ and hence its UV flux can be higher than the detection limits of recent galaxy surveys for $z=6\sim 8$ (e.g., Ouchi et al. 2009; Bouwens et al. 2015). At z = 6, the SFR of Halo-11 reaches $\sim 37\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, whereas that of Halo-11-noSN is $\sim 51\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The Halo-11-lowSF run shows a higher SFR than Halo-11 regardless of the lower SF efficiency. The SFR of Halo-11-lowSF is always between Halo-11 and Halo-11-noSN. This suggests that the SFR is not simply proportional to the SF efficiency when the feedback is considered. We find that the SFR of Halo-11-lowSF is continuous even at $z\gtrsim 10$ unlike Halo-11. The low SF efficiency in the Halo-11-lowSF run prevents the complete evacuation of gas from star-forming regions.

The lower panel of Figure 3 shows the time evolution of gas density within 200 pc (physical) from the center of the halo, which we take as the location of the maximum density in dark matter. The gas density even decreases to ${n}_{{\rm{H}}}\lt {10}^{-3}\,{\mathrm{cm}}^{-3}$ after the peak of the SFR due to SN feedback. For example, at $z\sim 10$, the simulated galaxy has $\mathrm{SFR}\sim 0.1\mbox{--}1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ and the surrounding mean gas density is ${n}_{{\rm{H}}}\sim 10\mbox{--}100\,{\mathrm{cm}}^{-3}$.

Let us examine in the following paragraphs whether SN feedback can disrupt these dense gaseous clumps in high-z galaxies. Given that the lifetime of massive stars (which causes core-collapse SNe) is less than $10\,\mathrm{Myr}$, the released SN energy results in $E\sim {10}^{55}\,\mathrm{erg}$ for $\mathrm{SFR}=0.1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Here we consider the energy of each SN to be ${10}^{51}\,\mathrm{erg}$, and estimate the number of SN per one solar mass of starsformed to be $\sim {10}^{-2}\times {M}_{\mathrm{star}}/{M}_{\odot }$, where ${M}_{\mathrm{star}}={SFR}\times {t}_{\mathrm{life}}$. The Jeans mass is $\sim {10}^{7}\,{M}_{\odot }$ for the conditions ${n}_{{\rm{H}}}\sim 100\,{\mathrm{cm}}^{-3}$ and $T\sim {10}^{4}\,{\rm{K}}$. The gravitational binding energy of the spherical gas cloud induced by the Jeans instability is $E\sim 2{{GM}}_{\mathrm{Jeans}}^{2}/{L}_{\mathrm{Jeans}}\sim {10}^{53}\,\mathrm{erg}$, which is lower than the released SN energy for $\mathrm{SFR}=0.1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Therefore, if the injected thermal energy is efficiently converted into kinetic energy, star-forming clouds of $\sim {10}^{7}\,{M}_{\odot }$ can be destroyed. Halo-11-noSN and Halo-11-lowSF always retain high-density gas near the center with ${n}_{{\rm{H}}}\geqslant 100\,{\mathrm{cm}}^{-3}$ at $z\lt 10$. This suggests that the disruption of star-forming clouds depends on the SF efficiency.

The slow SF model of Halo-11-lowSF allows the gas to concentrate at the galactic center, resulting in the formation of high-density gas cloud. In the high-density region, the gravitational binding energy is high due to high-density gas clumps populating the small volume. In addition, the SN feedback efficiency becomes weaker due to radiative cooling loss, and the conversion efficiency from thermal to kinetic energy depends on the local gas density as discussed in Section 2. Thus, Halo-11-lowSF cannot evacuate the gas near the galactic center unlike Halo-11. Note that the gas density of Halo-11-lowSF exceeds that of Halo-11-noSN at $z\lesssim 10$. As will be shown in Section 3.5, a large fraction of the gas in Halo-11-noSN is efficiently converted into stars even at $z\gtrsim 6$, resulting in a lower gas mass fraction at the galactic center compared to the Halo-11-lowSF.

Figure 4 presents the redshift evolution of the total gas mass and cold gas mass fraction. Here we define the cold gas as the one with temperature less than ${10}^{4}\,{\rm{K}}$ and hydrogen number density higher than $10\,{\mathrm{cm}}^{-3}$. Total gas mass monotonically increases as the halo grows. The gas mass of the Halo-11 run is lower than that of the other runs at $z\gtrsim 10$, and the difference becomes larger at higher redshift, although they are always within a factor of a few. This suggests that the gas mass evacuated from a halo due to the feedback is insignificant even in the case of Halo-11. On the other hand, as shown in the lower panel of the figure, the cold gas mass shows a large difference among different runs. At high redshift, even if accreted gas experiences virial shock, it can quickly cool down via efficient atomic line cooling and settle into the central galactic disk. Thus, in the case of Halo-11-noSN, more than 10% of the gas always stays in the cold high-density phase. Once stellar feedback becomes effective, the gas can be heated and evacuated from star-forming regions. Halo-11-lowSF can only retain 1%–10% of the total gas in the cold phase. In the case of Halo-11, at $z\gtrsim 10$, most of the gas is evacuated from star-forming regions and becomes low density in the hot phase after the short, active SF. Then, even efficient SF with feedback in Halo-11 cannot erase all cold, high-density gas at $z\lesssim 10$ when the halo mass exceeds $\sim 0.7\times {10}^{10}\,{h}^{-1}\,{M}_{\odot }$. In this phase, although most of the gas is evacuated from the galactic central region as shown in Figure 3, cold sub-clumps that are distributed far from the galactic center keep SF going. This makes the SF history of Halo-11 continuous at $z\lesssim 10$.

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Next, we study the evolution of the first galaxies in halos with different masses. In addition to the three models of Halo-11, we also follow two other halos: one with ${M}_{{\rm{h}}}=2.4\times {10}^{10}\,{M}_{\odot }$ at z = 6 (Halo-10) and another with ${M}_{{\rm{h}}}=0.7\times {10}^{12}\,{M}_{\odot }$ (Halo-12).

Figure 5 shows the SFR and stellar mass of different halos together with recent observational data of spectroscopically confirmed galaxies. We find that a more massive halo (Halo-12) shows a rapid variation in SFR, similarly to Halo-11 at $z\gtrsim 10$. After the SF at z = 12.2, the SN feedback evacuates the gas and quenches the SF for a while until z = 10.5. As halos become more massive toward lower redshift, the deeper gravitational potential well can hold the gas against SN feedback, which allows for more continuous SF at $z\lesssim 10$. We find that Halo-11 and Halo-12 show similar SFRs as the observed LAEs at $z\sim 7$ and the SMG at z = 7.5 discovered by ALMA (Watson et al. 2015). As shown in Figure 1, the halo mass of Halo-11 and Halo-12 is in the range of $\sim 9\times {10}^{10}\mbox{--}4\times {10}^{11}\,{M}_{\odot }$ at $z\sim 7$. These halo masses are consistent with the results of the semi-analytic model of LAEs augmented by radiation transfer calculation (Yajima et al. 2017) and the clustering analysis of LAEs at $z\sim 7$, which indicated a typical host halo mass of $\sim {10}^{11}\,{M}_{\odot }$ (Ouchi et al. 2010).

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Halo-10 has a somewhat higher halo mass than Halo-11 at $z\gtrsim 10$; therefore, Halo-10 has a similar or higher SFR, and a higher ${M}_{\mathrm{star}}$ than Halo-11 at those redshifts. Then, as the redshift decreases, the mass growth of Halo-10 becomes slower than Halo-11, resulting in lower SFRs at $z\lesssim 7$. As a result, the SFR of Halo-10 cannot reach the observed SFR of LAEs. Halo-12 shows high SFRs of $\gtrsim 10\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ at $z\lesssim 8$, which are similar to those of bright LAEs at 7.73 (Oesch et al. 2015).

It is interesting that none of our simulated galaxies reproduce the observed SFR of GN-z11 ($\mathrm{SFR}=24\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$), which is the most distant galaxy observed to date (Oesch et al. 2015). Halo-12 only shows $\mathrm{SFR}\lesssim 1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ at $z\gtrsim 10$ even in its starburst phase. In order to reproduce the high SFR, other conditions may be required, e.g., more massive halos or different halo growth histories.

The lower panel of Figure 5 shows the evolution of stellar mass. At first, Halo-10 has a higher ${M}_{\mathrm{star}}$ than Halo-11 at $z\gt 10$. Then, as the halo growth stalls at $z\lt 10$ (see Figure 1), Halo-10 does not increase its ${M}_{\mathrm{star}}$ very much, only reaching ${M}_{\mathrm{star}}\sim 1\times {10}^{9}\,{M}_{\odot }$ at $z\sim 6$. On the other hand, Halo-11 increases its ${M}_{\mathrm{star}}$ smoothly as the redshift decreases, and it reaches ${M}_{\mathrm{star}}\sim 3\times {10}^{9}\,{M}_{\odot }$ at $z\sim 7$, which matches the observed LAEs. The higher SFRs of Halo-12 lead to higher ${M}_{\mathrm{star}}$ than that in Halo-11, achieving ${M}_{\mathrm{star}}\gtrsim 3\times {10}^{9}\,{M}_{\odot }$ at $z\lesssim 8$, which agrees with the bright LAE at z = 7.73. Sometimes, the stellar mass drops suddenly, but this is due to galaxy merger events that spread the stellar components beyond the virial radius. In this work, we only take stars within the virial radius into account, and it sometimes leads to decreasing stellar masses. Note that if the angular resolution of future telescopes will be able to resolve substructures with a scale of the virial radius $\sim 10\,\mathrm{kpc}$, galaxies in such a merger event can be identified separately. The detailed modeling of observational properties of first galaxies will be investigated with radiation transfer calculations in our future work. Again, the ${M}_{\mathrm{star}}$ of Halo-12 is lower than that of GN-z11 by an order of magnitude at $z\sim 11$. The observed high ${M}_{\mathrm{star}}$ indicates that GN-z11 kept high SFRs of $\gtrsim 1\mbox{--}10\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ over $t\gtrsim {10}^{8}\,\mathrm{yr}$, and it is unlikely to be an instantaneously bright galaxy due to a major merger if the observed stellar mass is correct.

We now discuss the relation between the stellar mass and halo mass. Behroozi et al. (2013) derive a relation between the halo and stellar mass based on the abundance matching analysis, and show that the stellar mass increases with halo mass until a specific mass in order to match the faint end of the galaxy stellar mass function. Figure 6 shows the evolution of the SHMR as a function of halo mass for different halos. The SHMR of Halo-11-noSN is higher than that of Behroozi et al. (2013, yellow band) by one to two orders of magnitudes. This means that Halo-11-noSN converts gas into stars too efficiently, which resembles the canonical overcooling problem seen in the hydrodynamic simulations (e.g., Scannapieco et al. 2012). On the other hand, the SHMR of Halo-10, Halo-11, and Halo-12 roughly matches that of Behroozi et al. (2013) at later times. Therefore, we confirm that stellar feedback is the key to regulating SF and reproducing stellar masses. The SHMR of Halo-12 at z = 6 is much higher than Behroozi's result, but the discrepancy can be understood through the following possible reasons: (1) the small sample of simulations—the SHMR derived from the abundance matching method is a statistical mean value. Our small sample might not follow the statistical mean perfectly. (2) The limited sample of observed galaxies at $z\geqslant 6$. Due to the difficulty of estimating the stellar mass of galaxies at $z\geqslant 6$, the estimation of Behroozi et al. (2013) suffers from large uncertainties at $z\gtrsim 6$. (3) Missing other feedback processes: our simulations consider only SN feedback and photoionization heating by UVB, and other processes, such as radiation pressure on dust and AGN feedback, may suppress SF in massive halos.

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The relation between SFR and ${M}_{\mathrm{star}}$ is frequently discussed in the context of SF efficiency, and is often called the main sequence of SF (Daddi et al. 2007). Figure 7 shows the relation between SFR and ${M}_{\mathrm{star}}$ for our simulated halos. We made the smoothed curves by taking the averaged SFR within a time bin of ${\rm{\Delta }}t=47\,\mathrm{Myr}$. The solid lines are the observed main sequence at $z=3.8\mbox{--}5.0$ (Mármol-Queraltó et al. 2016, upper black line) and z = 2.0 (Daddi et al. 2007, lower black line). Our simulated galaxies at $z\geqslant 10$ show higher SFR than the observed main sequence at $z=3.8\mbox{--}5.0$ by a factor of a few, then gradually approaches the observed lines toward $z\sim 6$. This indicates that the SF proceeds efficiently at $z\gt 6$ in simulated galaxies due to higher gas fraction and higher gas density. Note that even Halo-11-noSN shows a similar curve to other runs with feedback. SN feedback suppresses the SFR temporarily (as partially seen in Halo-10), but it does not affect the mean SFR on a timescale ${\rm{\Delta }}t=47\,\mathrm{Myr}$ for Halo-11 and Halo-12. Hence, we suggest that the SF main sequence is not that sensitive to stellar feedback.

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As SF proceeds, the gas mass fraction decreases due to outflow and conversion into stars. Figure 8 shows the gas mass fraction to halo mass normalized by the cosmic mean value ${{\rm{\Omega }}}_{\mathrm{gas}}^{0}\equiv {{\rm{\Omega }}}_{{\rm{b}}}/{{\rm{\Omega }}}_{{\rm{M}}}\approx 0.16$. We find that the gas mass fractions of Halo-10 and Halo-11 significantly decrease due to SN feedback when their halo mass is lower than $\sim {10}^{9}\,{M}_{\odot }$. The minimum reaches around 20% of the cosmic mean. Then, as halos grow and gravitational potential becomes deeper, even the gas kicked by SNe is trapped, i.e., outward velocity is less than the escape velocity of the halo. As a result, the gas mass fraction gradually increases and stays at ∼40% of the cosmic mean when ${M}_{{\rm{h}}}\gtrsim {10}^{10}\,{M}_{\odot }$. On the other hand, the gas mass fraction of Halo-12 is not that low even when ${M}_{{\rm{h}}}\lt {10}^{9}\,{M}_{\odot }$. The redshift when the halo mass reaches ${10}^{9}\,{M}_{\odot }$ is ∼18 for Halo-12, while it is $z\sim 14$ for Halo-11. Hence, the galaxy is more compact and can efficiently trap gas against SN feedback. The gas mass fraction starts to be smaller than the cosmic mean at $z\lesssim 18$ due to the SF. At $z\lesssim 12$, the gas mass of Halo-12 is significantly reduced due to the feedback, and reaches ∼20% of the cosmic mean. Then, as the redshift decreases, the gas mass fraction increases gradually, and becomes similar to that of Halo-11 when ${M}_{{\rm{h}}}\gtrsim {10}^{11}\,{M}_{\odot }$. On the other hand, the gas mass fraction of Halo-11-noSN monotonically decreases due to too efficient SF, resulting in ${{\rm{\Omega }}}_{\mathrm{gas}}/{{\rm{\Omega }}}_{\mathrm{gas}}^{0}=0.3$ at $z\sim 6$.

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The global structure of gas in a galaxy can change dramatically after the starburst and subsequent SN feedback. The standard picture of galaxy formation asserts that the warm gas with $T={10}^{4}\mbox{--}{10}^{5}$ K cools by radiative cooling, falls into the center of the halo, and forms a rotationally supported galactic disk (e.g., Mo et al. 1998). Pawlik et al. (2013) argued that the first galaxies can maintain galactic disks against stellar radiation feedback. However, since the first galaxies are hosted in low-mass halos, SN feedback can significantly affect the global gas structure (e.g., Kimm & Cen 2014). The halo spin parameter of Halo-11 is $\lambda \sim 0.1$ at $z\sim 6$, and if there is no feedback effect, it is likely to have an extended galactic disk with ${R}_{\mathrm{disk}}\sim \lambda {R}_{\mathrm{vir}}$, where λ is the halo spin parameter in the range of $\sim 0.01\mbox{--}0.1$ (Mo et al. 1998).

Here we investigate the impact of the SN feedback on the gas structure of galaxies. Since the current numerical resolution is somewhat poor to fit the galactic disk with an exponential profile, here we roughly measure the aspect ratio of the galactic disk by estimating the disk radius ${r}_{{\rm{d}}}$ and the disk thickness h. The z-axis (normal direction to the disk) is determined by aligning the angular momentum vector of all gas particles within 200 physical pc from the galactic center. The disk radius ${r}_{{\rm{d}}}$ is defined such that the midplane mean density becomes ${n}_{{\rm{H}}}=10\,{\mathrm{cm}}^{-3}$ inside ${r}_{{\rm{d}}}$. The thickness h is the height when the mean density of the cylinder with a radius of $0.2\,{r}_{{\rm{d}}}$ exceeds the same threshold density. Figure 9 shows the aspect ratio of galactic disks, $h/{r}_{{\rm{d}}}$. When the gas is evacuated from the galactic center as shown in Figure 3, $h/{r}_{{\rm{d}}}$ is set to unity for plotting purposes. Note that $h/{r}_{{\rm{d}}}$ can exceed unity when high-density gas is distributed along the face-on viewing angle to the disk. Due to gas expulsion, Halo-11 does not have a disk at $z\gtrsim 10$. Then, at $z\lesssim 10$, a galactic disk forms when gas fills in around the galactic center as shown in Figure 3. However, Halo-11 cannot maintain this disk for a long period due to SN feedback. On the other hand, Halo-11-noSN and Halo-11-lowSF show disky shapes with $h/{r}_{{\rm{d}}}\lesssim 0.5$ for longer periods. In the case of Halo-11-lowSF, a disk forms at $z\gtrsim 10$, but it is destroyed due to SN feedback. Then, as the halo grows, the deeper gravitational potential well allows galaxies to support their galactic gas disks against feedback. Thus, the $h/{r}_{{\rm{d}}}$ of Halo-11-lowSF is always less than ∼0.6 at $z\lesssim 10$. Halo-11-noSN also has low values of $h/{r}_{{\rm{d}}}$ owing to no feedback. Unlike Halo-11-lowSF, the gas clumps can be efficiently converted into stars before they fall into the galactic center, resulting in higher values of $h/{r}_{{\rm{d}}}$ than for Halo-11-lowSF.

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In addition, the evacuation of gas near the galactic center can lead to different stellar distributions between simulations with and without SN feedback. Figure 10 shows the half-mass radius of stars (${R}_{\mathrm{star}}^{\mathrm{half}}$) as a function of redshift. We estimate the stellar mass within that half of the virial radius ${R}_{\mathrm{vir}}({M}_{{\rm{h}}},z)$, and measure ${R}_{\mathrm{star}}^{\mathrm{half}}$ where the stellar mass becomes half of ${M}_{\mathrm{star}}(\lt 0.5{R}_{\mathrm{vir}})$. We first need to define the center of galaxies in order to compute ${R}_{\mathrm{star}}^{\mathrm{half}}$. When galaxies are in major merger processes, galaxies in subhalos can be distributed far from the center of mass of a halo. Considering the large separations of substructures, we define the galactic center as the position of highest density in the dark matter substructure and focus on a substructure with the scale of $0.5{R}_{\mathrm{vir}}$. One might naively expect that ${R}_{\mathrm{star}}^{\mathrm{half}}$ would be small if there is no SN feedback. However, even Halo-11-noSN shows large values of ${R}_{\mathrm{star}}^{\mathrm{half}}$ despite having no feedback. Some peaks of ${R}_{\mathrm{star}}^{\mathrm{half}}$ are caused by subcomponents of star clusters distributed far from the center of galaxy. In the case of Halo-11, ${R}_{\mathrm{star}}^{\mathrm{half}}$ takes large values $\gtrsim 1\,\mathrm{kpc}$, whereas that of Halo-11-noSN steeply decreases after each peak. As shown in Figure 3, the SN feedback evacuates gas from star-forming regions near the galactic center, and suppresses the concentration of stellar distribution. The slow SF of Halo-11-lowSF allows continuous SF at the galactic center. In the case of Halo-11-lowSF, SF in low-density regions far from galactic centers is inefficient, resulting in a lower ${R}_{\mathrm{half}}$ with $\lesssim 0.1\,\mathrm{kpc}$, which corresponds to $\lesssim 1 \%$ of virial radius. These results indicate that the low SF efficiency is more important in making the compact stellar bulge rather than the non-existence of SN feedback. Thus, we suggest that SF efficiency and SN feedback affect the compactness of the stellar distribution, leading to different galaxy growth rates of galactic bulges. The angular resolution of the MIRI imaging of JWST will be able to resolve the stellar distribution with the scale of $\sim 1\,\mathrm{kpc}$ (Bouchet et al. 2015). Therefore, these subgrid models of feedback and SF efficiency can be investigated via comparisons with future observations.

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Figure 11 shows the gas density–temperature phase diagram of Halo-11 and Halo-11-noSN. Due to the balance between Lyα cooling and radiative heating from UVB, a large fraction of gas results in a neutral warm state with $T\sim {10}^{4}\,{\rm{K}}$ in both runs. At ${n}_{{\rm{H}}}\gtrsim 1\,{\mathrm{cm}}^{-3}$, the gas temperature gradually decreases to $\lt {10}^{4}\,{\rm{K}}$ as density increases due to metal cooling. SN feedback suppresses the formation of high-density gas with ${n}_{{\rm{H}}}\gtrsim {10}^{3}\,{\mathrm{cm}}^{-3}$, but some fraction of gas can reach ${n}_{{\rm{H}}}\sim {10}^{3}\mbox{--}{10}^{4}\,{\mathrm{cm}}^{-3}$. As shown in Equation (3), SN feedback in this high-density region becomes inefficient due to radiative cooling.

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The apparent difference between Halo-11 and Halo-11noSN is the low-density, high-temperature region. Halo-11 shows some gas distribution at ${n}_{{\rm{H}}}\sim {10}^{-3}\mbox{--}{10}^{-2}\,{\mathrm{cm}}^{-3}$ and $T\sim {10}^{5}\mbox{--}{10}^{6}\,{\rm{K}}$. The low-density, high-temperature gas is shock-heated by SN feedback, and the temperature corresponds to the relative velocity of $\sim 100\,\mathrm{km}\,{{\rm{s}}}^{-1}$ between the pre- and post-shock regions. For a low-metallicity gas, the cooling rate at $T\sim {10}^{6}\,{\rm{K}}$ is low, hence the shock-heated gas remains hot for ${t}_{\mathrm{cool}}\sim \tfrac{{k}_{{\rm{B}}}T}{{n}_{{\rm{H}}}{\rm{\Lambda }}}\sim 2\times {10}^{7}\,\mathrm{yr}$. Note that this timescale is shorter than the typical dynamical timescale of halos. Therefore, the hot gas can cool down and join star-forming regions as a warm medium if its outflowing velocity is less than the escape velocity of the halo. In Halo-11noSN, there is no such low-density, high-temperature gas due to the lack of SN feedback. A small fraction of gas that is heated due to gravitational shocks exists at ${n}_{{\rm{H}}}\sim {10}^{-2}\,{\mathrm{cm}}^{-3}$ and $T\sim {10}^{5}\,\mbox{--}{10}^{6}\,{\rm{K}}$ . The gas cools down to a warm state after ${t}_{\mathrm{cool}}\lt {10}^{7}\,\mathrm{yr}$. The lower bounds are seen at high density ${n}_{{\rm{H}}}\gtrsim {10}^{2}\,{\mathrm{cm}}^{-3}$. This is due to our effective EOS as explained in Section 2 (see also Johnson et al. 2013; Schaye et al. 2015). The slope of 4/3 prevents spurious fragmentation due to the finite resolution (Robertson & Kravtsov 2008; Dalla Vecchia & Schaye 2012).

It is well-known that cosmological N-body simulations often nearly universally show cusps in the DM density profile at galactic centers (e.g., Fukushige & Makino 1997; Navarro et al. 1997; Moore et al. 1999; Jing & Suto 2000). However, some observations of local galaxies indicated a core-like structure of DM near the galactic center (e.g., Gentile et al. 2004). In order to solve the discrepancy, impacts of baryonic processes have been discussed (Governato et al. 2010, 2012; Ogiya & Mori 2014; Chan et al. 2015; Oñorbe et al. 2015). In particular, Pontzen & Governato (2012) showed that the rapid evacuation of gas from a star-forming region near the galactic center results in the heating of DM and the formation of a DM core. Davis et al. (2014) showed that the DM profiles of low-mass dwarf galaxies in the early universe become shallower due to SN feedback. As shown in the previous subsection, our simulations shows rapid gas evacuation due to strong SN feedback. Therefore, the rapid change of baryonic distribution at the galactic center could also change the DM distribution.

Figure 12 shows the redshift evolution of the power-law index of the dark matter density profile between $r=30\mbox{--}300\,\mathrm{pc}$ in physical units. The lower panel of the figure represents the differences in power-law indices between different runs. We find that the DM profile of Halo-11 periodically changes, ranging from ${\alpha }_{\mathrm{DM}}\sim -1$ (cusp) to ∼0 (core-like). After starbursts, DM profiles develop cores, but go back to cusps after a short time. The difference in ${\alpha }_{\mathrm{DM}}$ between Halo-11 and Halo-11-noSN frequently exceeds unity. In the case of Halo-11-lowSF and Halo-11-noSN, the DM profile also changes with time, but it is always less than $\sim -1$. Therefore, only the efficient SF model with SN feedback can produce cored profiles of DM. This oscillation of the DM profile was also shown in previous works (e.g., Pontzen & Governato 2012; Davis et al. 2014). They claimed that the the oscillation of the DM profile has the cumulative effect of heating up the DM and keeps the core structure. On the other hand, Zhu et al. (2016) found that DM cores were not seen in the dwarf galaxies in their moving mesh cosmological simulations. Our current simulations also do not show a clear difference between simulations with and without SN feedback at $z\sim 6$. Thus, we argue that the heating effect on DM up to $z\sim 6$ is not enough to make lasting cores.

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Davis et al. (2014) showed the dependence of SF efficiency on the heating of DM. They showed that $\mathrm{SFR}\gtrsim 0.2\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ was required to change the DM profile by SN feedback. Our simulations indicate that the SF efficiency can be an important factor in the heating of DM in the current SF and SN feedback models. The slow SF model tends to form stars in higher density regions, resulting in lower feedback efficiency owing to more efficient radiative cooling loss. As a result, although the SFR of Halo-11-lowSFR is higher than that of Halo-11 as shown in Figure 3, the DM density profile does not change significantly with SF.

Note that the current analysis is only one example. Furthermore, our simulations focus on high-redshift galaxies that are violently evolving via frequent merger events, whereas the observational data for the core profile are for well-relaxed, low-mass galaxies. The relation between stellar feedback and DM density profile should be investigated using a larger galaxy sample evolved to low redshifts.

SF in first galaxies can also be affected by external UVB (e.g., Susa & Umemura 2004; Okamoto et al. 2008). In our fiducial model, we have turned on the UVB at z = 10 to mimic the effect of cosmic reionization. The photoionization by the UVB suppresses the cooling rate of primordial gas at $T\sim {10}^{4}\mbox{--}{10}^{5}$ K. Recent observations of the cosmic microwave background have indicated that the cosmic reionization took place at $z\sim 9\mbox{--}11$ (Komatsu et al. 2011; Planck Collaboration et al. 2016). Cosmic reionization proceeds with a patchy ionization structure as shown in the work using cosmological simulations (e.g., Iliev et al. 2014). By $z\sim 6$, cosmic reionization is completed once the H ii bubbles completely overlap. This patchy reionization scenario suggests that the external UVB sensitively depends on the environment. If galaxies reside in high-density regions where the ionization of IGM proceeds earlier, they are likely to be affected by the external UV flux at higher redshift. Here we investigate the impact of different reionization redshifts, when the UVB starts to irradiate, on the SF history of galaxies, as one of the environmental effects. Using the same initial condition as Halo-11, we carry out an additional simulation with the reionization redshift ${z}_{\mathrm{re}}=18$ (Halo-11-eUVB).

Figure 13 shows the SFR and stellar mass assembly histories. A clear difference is seen in the SFR at $z\gt 10$. The SFR of both simulations proceeds intermittently due to the SN feedback as explained in Section 3. However, the quenching time of SF is longer in the case of Halo-11-eUVB. Since the mean gas density in halos at $z\gt 6$ is ${n}_{{\rm{H}}}\gtrsim {10}^{-2}\,{\mathrm{cm}}^{-3}$, the interstellar medium in high-redshift galaxies is easily self-shielded from UVB. However, once the interstellar gas expands due to SN feedback, the decreased gas density allows the UVB to penetrate and ionize the gas. As a result, the photoionization heating decreases the gas density further and suppresses SF. This makes the no-SF phase longer in Halo-11-eUVB at $z\gt 10$. At $z\leqslant 10$, the UVB is turned on in both simulations. The different SF history due to UVB continues until $z\sim 8$, and then the SFR converges at $z\lesssim 8$. In addition, the gas photoionized by the UVB with $T\sim {10}^{4}\,{\rm{K}}$ can be bound in the gravitational potential of halos at $z\lesssim 10$. Thus, the impact of UVB on SF history can be secondary compared to the SN feedback at lower redshifts.

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Unlike the SFR history, the stellar mass assembly history under different UVB models does not show clear differences. The low SFR phase does not contribute to the stellar mass assembly. The high SFR phase does not depend on the UVB models very much, because the star-forming clouds are completely shielded from the UVB. As a result, the difference in the stellar mass is always within 20%. Thus, we suggest that the environmental effect of UVB on galaxy formation is not that strong in our galaxy sample. Note that, however, the large dispersion of SFR can change the shape of the faint-end slope of the UV luminosity function. Hence, combining UV luminosity and the stellar mass function allows us to understand the SF history and feedback efficiency. The current work focuses on a small region with a high resolution; hence, we cannot discuss the statistical feature of the first galaxies. We will investigate the statistical feature using a larger galaxy sample and multiwavelength radiative transfer calculations in our future work.

Our galaxy sample is limited due to the computational costs of the simulations of massive halos with high resolution. Therefore, there is a possibility that the SF behavior shown above is not general, just due to the formation site, i.e., environmental effects. In order to study the general behavior of SF in first galaxies, we need to investigate larger galaxy samples statistically. As a first step, here we study the SF history of another halo. We find another halo in the whole box of ${(20{h}^{-1}\mathrm{Mpc})}^{3}$ of which the halo mass assembly history is similar to Halo-11, and we call this halo as "Halo-11b." As a test, we carry out a zoom-in simulation targeting Halo-11b with the same mass resolution of SPH and dark matter particles with the Halo-11 run, and compare the results.

Figure 14 presents the SFR, halo mass assembly history, and time evolution of gas mass in halos. Both halos show similar halo growth histories. The halo mass of Halo-11b is somewhat larger than that of Halo-11 at $z\gt 8$. Then, Halo-11b slowly grows, and the halo mass becomes lower than that of Halo-11 at $z\lesssim 8$. Finally, the halo mass of Halo-11b reaches $1.1\times {10}^{11}\,{M}_{\odot }$ at z = 6. The behavior of the SF history of Halo-11b is similar to that of Halo-11. The SF intermittently proceeds at $z\gtrsim 10$ due to the stellar feedback, then goes smoothly at lower redshift $z\lesssim 10$ as the halo mass grows. Therefore, we furthermore support the intermittent SF history as the common feature in the evolution of first galaxies. The peaks of SFR of Halo-11b at the bursty phases are higher than those of Halo-11. As shown in the bottom panel of the figure, the gas accretion rate onto Halo-11b is higher than that onto Halo-11 at $z\gtrsim 8$. This higher gas accretion rate can cause starbursts with higher SFR and longer quenching time. Therefore, our results suggest that instantaneous SFR can vary significantly even if the halo mass is almost the same, but the overall SF history is roughly similar.

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