A SIMPLE TECHNIQUE FOR PREDICTING HIGH-REDSHIFT GALAXY EVOLUTION

Galaxy star formation histories are typically parameterized as a function of time, i.e., SFR(t). Yet, physically interpreting SFR(t) is not straightforward because changes in SFR with time come from both the large-scale environment (e.g., the redshift dependence of mass accretion rates) and local properties (e.g., host halo mass and gravitational potential well depth). We therefore consider a more physical parameterization that separates these two effects.

Galaxies at time t_f in a given stellar mass bin will have a well-defined average stellar mass, M_{*, f}, and a well-defined average halo mass, M_{h, f}. The halo mass is determinable through abundance matching/modeling if it is not known through other means (Section 3.2). In this paper we parameterize the average growth of the galaxies in terms of the average growth of their host dark matter halos, i.e., as M_*(M_h(t)). The changing cosmological accretion rate is captured in the average halo mass growth history, M_h(t). The historical efficiency with which the galaxies converted infalling baryons into stars is captured in how M_* depends on M_h. We can write the galaxies' average star formation histories in terms of M_*(M_h) as

$$\begin{equation} {\rm SFR}(M_h(t)) = \frac{dM_\ast }{dt} = \frac{dM_\ast }{dM_h} \frac{dM_h}{dt}. \end{equation} \tag{ 1 }$$

Just as for SFR(t)-parameterized star formation histories, the form of M_*(M_h) will depend on the redshift and stellar mass of the galaxies in question.⁵

As with fitting SFR(t) directly, we have to make an assumption about the functional form of M_*(M_h). Regardless of its true form, we can approximate the recent stellar mass–halo mass history as a power law. This form is especially reasonable for galaxies at z > 4, which are typically not massive enough to experience active galactic nucleus (AGN) feedback and so only experience stellar feedback (supernovae and reionization) over much of their growth histories. We also test this assumption directly in Section 4 for galaxies from z = 4 to z = 8. Hence, we let

$$\begin{equation} M_\ast (M_h) = M_{\ast,f} \left(\frac{M_h}{M_{h,f}}\right)^\alpha \protect, \protect \end{equation} \tag{ 2 }$$

where α is the unknown power-law dependence. We reemphasize that Equation (2) describes the historical evolution of a galaxy population, so α differs from the local slope of the SMHM relation at time t_f if (and only if) the SMHM relation is evolving with redshift.

Combining Equation (2) with Equation (1), the SFR history becomes a simple expression:

$$\begin{equation} {\rm SFR}(t) = \frac{\alpha M_{\ast }}{M_h} \frac{dM_h}{dt}\protect. \protect \end{equation} \tag{ 3 }$$

Letting SSFR(t) be the SSFR ($M_\ast ^{-1} ({dM_\ast }/{dt})$) as a function of time, and letting SMAR(t) be the halo specific mass accretion rate (SMAR) ($M_h^{-1} ({dM_h}/{dt})$), the beautiful symmetry in Equation (3) is apparent:

$$\begin{equation} \frac{{\rm SSFR}(t)}{{\rm SMAR}(t)} = \alpha. \end{equation} \tag{ 4 }$$

That is, the ratio of a galaxy population's average SSFR to its average specific host halo mass accretion rate will be constant, under the weak assumption that the recent historical SMHM relation for the population's progenitors has a power-law form.

If we remove the assumption of a power-law growth history, we note that this ratio still describes the recent growth of the galaxies:

$$\begin{equation} \frac{{\rm SSFR}(t)}{{\rm SMAR}(t)} = \frac{\frac{d\log M_\ast }{dt}}{\frac{d\log M_h}{dt}} = \frac{d\log M_\ast }{d\log M_h}\protect, \protect \end{equation} \tag{ 5 }$$

which is valid as long as SSFR(t) and SMAR(t) are both positive. As shown in Figure (1), Equation (5) is really a geometric statement about the logarithmic SMHM plane. The SSFR sets the galaxies' velocity along the logarithmic stellar mass axis, and the SMAR sets the velocity along the logarithmic halo mass axis. The ratio of these two velocities ("rise over run") corresponds to the slope of the galaxies' trajectory in this plane; this slope will be constant as long as a power-law relationship between the galaxies stellar masses and halo masses holds (Equation (2)).

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Equations (2) and (4) therefore yield a simple way to constrain galaxy progenitors. For this method, two galaxy stellar mass functions (SMFs) at nearby redshifts would suffice for observational constraints. The remaining inputs include host halo masses for the galaxies, which may be found by abundance matching/modeling (Section 3.2); halo mass accretion histories, determined via dark matter simulations (Section 3.3); and galaxy SSFRs, which are set by the growth of the SMF between the two redshifts (Sections 3.1 and 3.2). The ratio of the galaxies' SSFRs to their host halos' specific halo mass accretion rates (Equation (4)) then yields the power-law slope of the progenitors' historical SMHM relation. This can be used to infer basic information about the galaxies' progenitors, including the progenitors' star formation histories, SSFRs, stellar masses, and halo masses. As no observations of the progenitors are required, this method can yield constraints on galaxy populations that are otherwise too faint or at too high of a redshift to be observed directly.

We note in passing that Equation (5) applies for any pair of coupled variables. For example, one may also model the growth of black holes in galaxies using the ratio of specific black hole mass accretion rates to SSFRs. Current research in the coevolution of black holes in galaxies has focused on the correlation between black hole accretion rates and SFRs (see, e.g., Silverman et al. 2008; Aird et al. 2010; Mullaney et al. 2012; Conroy & White 2013; Chen et al. 2013; Hickox et al. 2014). Considering the relationship between the specific rates instead may be an interesting avenue for future research.

A galaxy's stellar mass can change not only because of new star formation (Equation (1)) but also through stellar mass loss (stellar winds and supernovae) and mergers of smaller galaxies. We adopt the stellar mass loss fraction as a function of time from Behroozi et al. (2013c), which is based on the flexible stellar population synthesis (FSPS) model (Conroy et al. 2009; Conroy & Gunn 2010) and the initial mass function (IMF) of Chabrier (2003):

$$\begin{equation} f_\mathrm{loss}(t) = 0.05 \ln \left(1+ \frac{t}{1.4\;\mathrm{Myr}}\right). \end{equation} \tag{ 6 }$$

The logarithmic form results in a very rapid 15% loss over the first 25 Myr, followed by another more gradual loss of 15% over the next 500 Myr. For the high-redshift (z > 4) galaxies we consider, typical SSFRs are on the order of 5–10 × 10⁻⁹ yr⁻¹ (Behroozi et al. 2013c), which leads to characteristic formation timescales of 100–200 Myr. Since these timescales fall in the gradual loss regime, the overall stellar mass loss is quite insensitive to the exact star formation histories and is in the range 15%–20%.

We next calculate the expected gain in stellar mass from mergers. This is a convolution between halo merger rates and the SMHM relation, both of which we take from constraints in Behroozi et al. (2013c). As shown in the left panel of Figure 2, merging halos contribute about 12%–18% of the total stellar mass in most galaxies for 4 < z < 8; very massive galaxies may receive up to a 30% contribution. This small fraction can be understood from the fact that the SMHM ratio declines with halo mass, so most of the incoming stellar mass will come from major mergers (Behroozi et al. 2014b). The aforementioned star formation timescales (∼150 Myr) are shorter than the expected major merger timescales (∼1 per 300 Myr from z = 8 to z = 6; Fakhouri & Ma 2008; Fakhouri et al. 2010; Behroozi et al. 2013c), meaning that mergers contribute a minority of the stellar mass growth.

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As a result, mergers and stellar mass loss nearly cancel each other's effects (Figure 2, left panel), representing only a 5%–10% total correction to α in Equation (4). From z = 8 to z = 4, halos grow by a factor of ∼1 dex, so this correction would result in a 0.1–0.2 dex error in the inferred stellar masses over this range (see also Equation (7)). This is well within typical 0.3 dex systematic errors for stellar masses (Conroy et al. 2009; Behroozi et al. 2010, 2013c). All the same, we correct the stellar mass in Equation (4) to remove stellar mass from mergers and to include stellar mass lost as a result of passive evolution. This process is described in Section 3.2.

Another potential source of error is scatter in halo mass at fixed stellar mass. The average specific halo mass accretion rate is an extremely weak function of halo mass (Figure 2, right panel). For example, a change of ±0.3 dex in halo mass corresponds to a change of ±5.5% in the SMAR for 10¹² M_☉ halos. However, galaxy mass may correlate with a halo's recent accretion rate, which can bias the galaxies' host halo specific accretion rates. We avoid these kinds of selection biases by directly constraining the average stellar mass and the average SFR as a function of halo mass (as in Behroozi et al. 2013c), as discussed in Section 3.2. This allows us to select galaxies at fixed halo mass; the corresponding average specific halo mass accretion rates can then be calculated in an unbiased way from a dark matter simulation (Section 3.3 and Figure 2, right panel).

Finally, the choice of IMF has a minimal effect. The primary effect of switching, e.g., to a Salpeter (1955) IMF would be to identically rescale the SFR and stellar mass, leaving their ratio unchanged (Conroy et al. 2009). Although the stellar mass normalization would be affected, the predicted evolution (from Equation (4)) would remain the same. A secondary effect would be to change the inferred stellar mass loss by 5%–10%, depending on galaxy metallicity (Shimizu & Inoue 2013), but this would not change the ratio of the SFR to the total stellar mass ever formed, which is what is relevant for Equation (4). However, we note the possibility that the IMF may change to be more top-heavy at high redshifts for reasons including lower metallicities, high gas opacity, ultracompact star clusters, tidal shear, or rising cosmic microwave background temperatures (see Scalo 2005; Bonnell et al. 2007 for reviews). Some specific recent examples are given for starbursts in Weidner et al. (2011), for low metallicity at high redshift in Narayanan & Davé (2013), for ultracompact star clusters and low metallicity in Marks et al. (2012), and for disturbed galaxies in Habergham et al. (2010).

SSFRs are difficult to measure directly. This is clearest at low redshifts, where many independent determinations are available; e.g., Figure 3 compares several direct measurements of the z = 1 SSFR from recent literature. These have all been normalized to a Chabrier (2003) IMF; however, significant disagreement is present in both the amplitudes and slopes of the measurements. Behroozi et al. (2013c) found that this level of scatter (±0.3 dex) between different publications' direct measurements is present at all redshifts—even for those now considered nearby (z < 0.5).

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The magnitude of these uncertainties is important because they directly translate into the uncertainty in the power-law slope of the star formation efficiency inferred from Equation (4) (see Section 4). For that reason, it is not appropriate to rely on SSFR measurements from a single source. We instead rely on the meta-analysis technique described in Behroozi et al. (2013c) and Section 3.2, which combines constraints from many different direct SSFR measurements with constraints from the growth rate of the SMF. For our methodology tests (Section 4), we restrict the meta-analysis to data obtained at z ⩽ 5; for our predictions (Section 5), we use the full data for z ⩽ 8.

As shown in Figure 3, this meta-analysis process is largely equivalent to taking the median of the published direct SSFR measurements. However, for completeness, we discuss the results from using single-source direct measurements of the SSFR in Appendix A.

We here distinguish abundance modeling from abundance matching. The latter nonparametrically constrains the SMHM relation by assigning galaxies ranked by decreasing stellar mass to halos ranked by decreasing halo mass within equal volumes. Abundance modeling instead assumes a parametric form (usually informed by abundance matching) for the SMHM relation and uses a Markov Chain Monte Carlo (MCMC) method to constrain the posterior distribution for the SMHM parameters. This approach benefits because observational constraints besides the SMF at a single redshift may be readily incorporated into the MCMC likelihood function. Examples include SMFs at multiple redshifts (Moster et al. 2010, 2013; Behroozi et al. 2010, 2013c; Yang et al. 2012; Béthermin et al. 2012; Wang et al. 2013; Lu et al. 2014), SFRs (Behroozi et al. 2013c), correlation functions (Yang et al. 2012; Leauthaud et al. 2012; Tinker et al. 2013), conditional SMFs (Yang et al. 2012; Lu et al. 2014), and weak-lensing constraints (Leauthaud et al. 2012; Tinker et al. 2013). The ability to include systematic uncertainties and covariances in the likelihood function is also advantageous (Behroozi et al. 2010, 2013c; Yang et al. 2012).

We follow the same technique used in Behroozi et al. (2013c), and we refer readers to that paper for full details. Briefly, we adopt a six-parameter functional form to describe the SMHM relation at a single redshift. This form includes parameters for a characteristic halo mass and stellar mass, a faint-end power-law slope, a massive-end turnoff, a shape for the transition between faint- and massive-end behavior, and the scatter in stellar mass at fixed halo mass. For each parameter, three variables control the evolution at low (z < 0.5), medium (0.5 < z < 2) and high (z > 2), redshifts, giving a total of 18 variables to describe the evolution of the SMHM relation. We additionally include nuisance parameters for systematic effects, including systematic offsets in recovered stellar masses for active and passive galaxies, random errors in stellar masses, incompleteness at high redshifts due to dusty and/or bursty star formation, and the fraction of stellar growth due to in situ star formation versus ex situ mergers.

Every location in this parameter space corresponds to a specific choice for the SMHM relation, SM(M_h, z), as a function of halo mass and redshift. For the halo mass definition, we use the peak historical virial (Bryan & Norman 1998) halo mass along halos' growth trajectories (M_peak), which was found by Reddick et al. (2013) to be the best halo mass proxy for reproducing galaxy clustering and luminosity functions. Each choice of SM(M_h, z) then represents a unique way to assign stellar masses for every halo at every time step in a dark matter simulation. The expected SMF at any redshift is therefore set by the number density of galaxies in dark matter halos; expected SSFRs and CSFRs are set by the growth of galaxies along dark matter merger trees. The closeness of these expectations to observed constraints gives the relative likelihood for the chosen parameter set; an MCMC algorithm then determines the posterior distribution of allowable SMHM relations, as well as the implied SSFRs as a function of halo mass and redshift (required by Sections 2.2 and 3.1). As the dark matter halo merger trees allow full bookkeeping for the amount of stellar mass accreted in mergers and lost through passive stellar evolution, these corrections to galaxy stellar masses can be readily subtracted, as discussed in Section 2.2.

For observational constraints, we use the compilation in Behroozi et al. (2013c). This includes SMFs (Baldry et al. 2008; Pérez-González et al. 2008; Marchesini et al. 2009, 2010; Stark et al. 2009; Mortlock et al. 2011; Lee et al. 2012; Bouwens et al. 2011; Bradley et al. 2012; Moustakas et al. 2013), SSFRs (Yoshida et al. 2006; Salim et al. 2007; Zheng et al. 2007; Smolčić et al. 2009; Shim et al. 2009; Le Borgne et al. 2009; Dunne et al. 2009; van der Burg et al. 2010; Kajisawa et al. 2010; Ly et al. 2011a, 2011b; Rujopakarn et al. 2010; Cucciati et al. 2012; Robotham & Driver 2011; Tadaki et al. 2011; Magnelli et al. 2011; Karim et al. 2011; Bouwens et al. 2012a; Sobral et al. 2013), and CSFRs (Salim et al. 2007; Noeske et al. 2007; Zheng et al. 2007; Daddi et al. 2007; Feulner et al. 2008; Kajisawa et al. 2010; Schaerer & de Barros 2010; Lee et al. 2011; Tadaki et al. 2011; Karim et al. 2011; McLure et al. 2011; González et al. 2014; Reddy et al. 2012; Twite et al. 2012; Whitaker et al. 2012; Salmi et al. 2012; Labbé et al. 2013) from z = 8 to z = 0. For the tests in Section 4, we have also performed a restricted analysis excluding all verification data—i.e., all data at z > 5.

The main output of this model is the posterior distribution for M_*(M_h, z)—i.e., the stellar mass as a function of halo mass and redshift. As averaged SFRs as a function of halo mass and redshift are available from the intermediate steps in the calculation, we can straightforwardly calculate the posterior distribution of SSFRs as a function of redshift and halo mass.

Halo properties are derived primarily from the Bolshoi simulation (Klypin et al. 2011). Bolshoi follows 2048³ (8.6 × 10⁹) dark matter particles in a 250 Mpc h⁻¹ comoving, periodic box from z = 80 to z = 0 using the art code (Kravtsov et al. 1997; Kravtsov & Klypin 1999). Its mass and force resolution (1.9 × 10⁸ M_☉ and 1 kpc h⁻¹, respectively) allow it to resolve dark matter halos down to 10¹⁰ M_☉. The assumed cosmology is a flat, ΛCDM cosmology with parameters Ω_m = 0.27, $\Omega _\Lambda = 0.73$, h = 0.7, σ₈ = 0.82, and n_s = 0.95; these are very close to the WMAP9+BAO+H₀ best-fit values (Bennett et al. 2013). Dark matter halos were found using the Rockstar phase-space temporal halo finder (Behroozi et al. 2013d); merger trees were assembled using the Consistent Trees code (Behroozi et al. 2013e), which reconstructs missing halos in merger trees to preserve gravitational consistency across simulation time steps.

As discussed in Sections 2.1 and 3.2, our approach requires the average growth rate, $\langle \dot{M}_\mathrm{peak}\rangle$, for halos as a function of redshift and mass. Behroozi et al. (2013c) already determined the median mass accretion histories for halos in Bolshoi; we therefore build on this existing fit, which is expressed as a function of the z = 0 halo mass and the scale factor. The full calibration and fitting procedure are discussed in Appendix B. For the halo mass function, we adopt the modified Tinker et al. (2008) form in Behroozi et al. (2013c), which was fit to the Bolshoi halo mass function for 0 < z < 8.

Bolshoi's constraints on halos weaken above z = 8, due to its mass completeness limit. However, extrapolating the Behroozi et al. (2013c) halo mass function fit to higher redshifts compares well to the fit in Watson et al. (2013), which is calibrated to z = 26. We find maximum differences of 15%–25% between the two mass functions from z = 10 to z = 15; these are consistent with systematic biases from the choice of halo finder (Knebe et al. 2011, 2013; Watson et al. 2013). We also have compared to the average mass accretion histories in Wu et al. (2013), finding a maximum difference of 5% in the SMARs at z > 10.

In this section, we check how well Equations (2)–(4) can be used to derive the z > 4 SMHM relation from the z = 4 galaxy SMHM relation, SSFRs, and halo SMARs. Starting at z = 4 is motivated because we approximated galaxies' historical SMHM relationships as pure power laws (Equation (2)). This approximation is most appropriate if the galaxy's history is dominated by a single feedback mode (e.g., stellar feedback) for star formation (see Section 2.1); at z ⩽ 4, AGN-mode or other feedback in massive galaxies results in rapidly declining star formation histories (Behroozi et al. 2013c).⁶ Note that while any starting redshift z ⩾ 4 is reasonable, we choose z = 4 because it gives the longest redshift baseline (Δz = 4 for galaxies up to z = 8) over which we can test the predictions of Equations (2)–(4).

Predicting the high-redshift SMHM relation is only a fair test if no high-redshift data has been used as a constraint. To that end, we have rerun the abundance modeling framework discussed in Section 3.2 using observational data constraints covering 0 < z ⩽ 5. As noted in Sections 3.1 and 3.2, the inclusion of SMF data at z = 5 allows the growth of the SMF to serve as a constraint on galaxies' SSFRs at z = 4. The resulting best-fit SMHM relation and SSFRs at z = 4 are shown in comparison to the results of (Behroozi et al. 2013c; which used data constraints from 0 < z ⩽ 8) in the top panels of Figure 4. The best-fitting SMHM relation of the restricted analysis is (as expected) slightly different from the best-fitting result of Behroozi et al. (2013c), but it is still well within the observational error bars.

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The quantitative predictions from assuming that galaxies' ratios of SSFRs to halo SMARs remain constant (Equation (4)) over 5 ⩽ z ⩽ 8 are shown in the middle and lower panels of Figure 4. These predictions are extremely consistent with constraints on the evolution of the SMHM relationship to the highest available redshift (z = 8) from Behroozi et al. (2013c). They are also consistent with nonparametric abundance matching of SMFs at those redshifts (Figure 4), within the expected systematic observational errors. While the shape and amplitude of the SMHM relation remain relatively constant, the halo mass corresponding to peak integrated star formation efficiency evolves from ∼10¹² M_☉ at z = 4 to ∼10¹¹ M_☉ at z = 8.

For z = 4 galaxies, the ratio of their SSFRs to their SMARs is fairly close to unity: α ∼ 1–1.5 (Figure 4, top right panel). If the SSFRs and SMARs were equal (α = 1), the relative growth rates of galaxy stellar mass and host halo mass are the same—so that the historical SMHM ratio remains constant. For α slightly greater than unity, this will still be approximately true, so that the historical SMHM ratios will have at most a weak dependence on their historical halo mass. So, the galaxies forming stars at peak efficiency at z > 4 will be the progenitors of the galaxies forming stars at peak efficiency at z = 4. Similarly, the halo mass at which the SMHM ratio reaches any chosen amplitude is also expected to evolve (see also Section 5.2).

The uncertainties in these predictions grow as the redshift baseline increases. At high redshifts, halo growth is very nearly exponential with redshift (Wechsler et al. 2002; McBride et al. 2009; Wu et al. 2013; Behroozi et al. 2013c); we find from the simulations in Section 3.3 that dM_h/dz is 0.25 dex. From Equation (4), the errors in stellar mass predictions then increase as

$$\begin{equation} \Delta M_\ast \approx -0.25 \epsilon \alpha \Delta z {\rm dex,} \end{equation} \tag{ 7 }$$

where α ∼ 1 is the ratio of SSFRs to SMARs (see, e.g., Figure 4), and is the fractional uncertainty in α. The predicted evolution and the fully constrained evolution over z = 4 to z = 8 typically differ by 0.25 dex or less (Figure 4), which implies that is ∼25% or less.

We next calculate how Equation (4) predicts that the SSFR for 10⁹ M_☉ galaxies changes with redshift (Figure 5), given SSFRs and SMARs at z = 4 (Figure 4). At fixed redshift, more massive galaxies have lower SSFRs and larger specific halo mass accretion rates (SMARs) than smaller galaxies (see, e.g., Figure 4). As a result, they would be expected to maintain a smaller SSFR–SMAR ratio over their growth history (Equation (4)). The more massive the galaxy is at z = 4, the higher the redshift when it reached 10⁹ M_☉. Since more massive galaxies have lower SSFR–SMAR ratios (Figure 4), we expect that the SSFR–SMAR ratio at fixed stellar mass will decline with increasing redshift.

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This is exactly what is shown in Figure 5. While the SMAR rises steadily with redshift, the declining SSFR–SMAR ratio leads to an apparent "plateau" (Weinmann et al. 2011), which is extremely consistent with the observed SSFRs at high redshifts.

We also show the expected observed CSFRs in Figure 6, again calculated from galaxy SSFRs at z = 4. To calculate these, we applied the expected SSFRs and the expected SMHM relationship (Section 4.1) to the halo mass function (Behroozi et al. 2013c). We integrated the resulting number density down to an assumed UV luminosity threshold of M_{1500, AB} < −18 (matching recent high-redshift observations; Oesch et al. 2014b), where UV luminosities were computed without including dust as described in Appendix C. As shown in Figure 6, the CSFRs expected from the SSFR to SMAR ratio of z = 4 galaxies are extremely consistent with existing observations from z = 5 to z = 8.

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We note that the uncertainties on SSFRs are not as sensitive to the redshift baseline because the errors are not cumulative. The fractional uncertainty in the SSFR to SMAR ratio α translates directly into the fractional uncertainty in SSFRs. As ≲ 25% (Section 4.1), this suggests that SSFR predictions are extremely robust, as long as the assumption of a power-law historical SMHM relation (Equation (2)) holds. Galaxy SFRs will be the product of the SSFR and stellar mass, which will result in fractional errors of

$$\begin{equation} \frac{\Delta {\rm SFR}}{{\rm SFR}} = \log _{10}(1+\epsilon) - 0.25 \epsilon \alpha \Delta z {\rm dex}\protect. \protect \end{equation} \tag{ 8 }$$

Since the total integrated stellar mass must remain constant, an error in α will result in redistributing when past cosmic star formation happened. The error in Equation (8) thus has a sign change when log₁₀(1 + ) = 0.25αΔz. Noting that log₁₀(1 + ) is well approximated by log₁₀(e) for small , this fractional error becomes zero for redshift baselines of Δz = 4log₁₀(e)α⁻¹, regardless of what value actually takes! For typical values of α (∼1.25), the total additional error from extrapolating the CSFR is minimized for a redshift baseline of Δz = 1.2.

As shown in Figure 5, we expect SSFRs to rise slowly with redshift at fixed galaxy mass. At z = 4 and above, galaxy SSFRs are nearly the same magnitude as host halo SMARs (see, e.g., Figure 4). As argued in Sections 2.1 and 4.1, the ratio of SSFRs to halo SMARs is conserved for progenitors of z > 4 galaxy populations; hence, higher-redshift SSFRs will also be about the same magnitude as the slowly rising halo SMARs. Since the choice of initial redshift is arbitrary, we show the SSFR evolution predicted from assuming constant SSFR-to-SMAR ratios for galaxies starting at z = 5, z = 6, and z = 7 in Figure 5. Consistent with the error analysis in Section 4.2, these predictions are all very similar to each other, as well as to the prediction from the restricted analysis in Section 4.2.

Predictions for observed CSFRs are shown in Figure 6; as in Section 4.2, we require galaxies to have a threshold UV luminosity of M_{1500, AB} < −18. These SFRs show a smooth decline from z = 4 to z = 15. As with SSFRs, we show predictions from assuming constant SSFR-to-SMAR ratios (taken from Behroozi et al. 2013c) for galaxies starting at z = 5, z = 6, and z = 7, as well as from the restricted analysis in 4.2.

As noted in Section 1, Oesch et al. (2014b) found an apparent sharp steepening in the CSFR evolution beyond z = 8. In our model, we do not find this steepening to occur. Intuitively, this is because galaxy stellar mass doubling times at z = 7 and z = 8 are on the order of 100–200 Myr (see Figure 5). Hence, if galaxy SSFRs at z = 7 and z = 8 are correct, there should still be significant star formation ongoing at z = 9 and z = 10. If the Oesch et al. (2014b) results turn out to be correct, on the other hand, SSFRs at z = 7 and z = 8 would have to be a factor of three higher than current measurements (i.e., beyond existing corrections for nebular emission; Labbé et al. 2013). The tension between the Oesch et al. (2014b) measurements and our predictions is at the 1.5σ–2σ level (Figure 7). We find that the discrepancy is too large to be explained by cosmic variance uncertainties (Trenti & Stiavelli 2008), or by underestimating the effective UV luminosity threshold by a factor of 2.5 (Figure 7). Instead, these results may indicate that the UV–SFR conversion in Oesch et al. (2014b) underestimates the SFR for young stellar populations at lower metallicities (Castellano et al. 2014; Madau & Dickinson 2014). Alternately, the contamination exclusion criteria in Oesch et al. (2014b) may be too restrictive; e.g., removing point sources to exclude contamination from nearby stars may also affect compact galaxies at very high redshifts. We note that high-redshift measurements are rapidly evolving; as this manuscript was being revised, Oesch et al. (2014a) found a doubly imaged z ∼ 10 candidate in the Frontier Fields, which may suggest a somewhat higher CSFR (Figure 7).

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We can also predict the total (observed + unobserved) CSFR; galaxies too faint to be observed at a given redshift can be constrained by their descendants at a later redshift. A UV luminosity threshold of M_{1500, AB} < −18 is expected to miss nearly half of the total SFR at z = 8 and miss two-thirds by z = 12. Long gamma-ray bursts at high redshifts can also constrain the total CSFR. Although significant calibration uncertainties remain for how the long gamma-ray burst rate depends on metallicity and redshift,⁷ our predictions are generally within the 1σ error bars of current constraints (Kistler et al. 2009, 2013). We caution that Kistler et al. (2009, 2013) normalized the ratio of the cosmic gamma-ray burst rate to the CSFR using the CSFR calibration in Hopkins & Beacom (2006). As discussed in Behroozi et al. (2013c), more recent measurements of the CSFR have been systematically lower than the Hopkins & Beacom (2006) calibration by 0.3 dex; this new normalization would lower the GRB-derived SFRs by at least the same factor. As shown in Figure 7, lowering the Kistler et al. (2013) results by 0.3 dex results in excellent agreement with our predictions for the total CSFR.

Given the ratios of galaxy SSFRs (from Behroozi et al. 2013c) to halo SMARs at z = 4, 5, 6, and 7 (from Section 3.3), we have used Equations (2)–(4) to predict SMHM ratios from z = 9.6 to z = 15 (Figure 8). As noted in Section 4.1, typical uncertainties in the stellar mass evolution are ∼0.25 dex for a redshift baseline of Δz = 4 and ∼0.5 dex for a redshift baseline of Δz = 8; these add in quadrature with existing 0.25 dex systematic uncertainties in the stellar mass normalization at z > 4 (Behroozi et al. 2013c; Curtis-Lake et al. 2013). We note that reionization may lead to non-power-law historical SMHM ratios for halo masses below M_h = 10⁹ M_☉ (Gnedin 2000; Noh & McQuinn 2014). However, several authors find that gas may cool in 10⁸ M_☉ halos prior to reionization (Gnedin 2000; Noh & McQuinn 2014; Wise et al. 2014), so that predictions for M_h < 10⁹ M_☉ at very high redshifts (e.g., z > 10) may still be reasonable.

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The median predicted evolution of the SMHM ratio is shown in Figure 9. We find that the redshift evolution seen in the SMHM ratio from z = 4 to z = 8 is expected to continue to very high redshifts. More quantitatively, we can calculate (as a function of redshift) the halo mass at which the stellar mass is 1% of the halo mass. The evolution of this halo mass with redshift is shown in Figure 10. At z < 4, this mass hardly evolves, which suggests that halo mass determines galaxy star formation efficiency at z < 4 (see also Behroozi et al. 2013b). At higher redshifts, this mass evolves continuously from 10^11.5 M_☉ at z = 4 to 10^9.3 M_☉ at z = 15—i.e., over two orders of magnitude across this redshift range. This evolution cannot be explained by appealing to an evolving halo mass definition. For example, it outpaces even that expected for constant v_max (the maximum of $\sqrt{{GM(<R)}/R}$ over the halo profile), which is a direct physical measure of the halo's potential well depth. The evolution at z > 4 may match better with a fixed central density (e.g., density within the central 1 kpc, as shown in Figure 10), which would correspond to a fixed cooling rate. However, galaxy formation is a complex process, and we cannot rule out contributions from other sources.

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5.3.1. Stellar Mass Functions

5.3.2. Cosmology

The recent Planck cosmology results (Planck Collaboration et al. 2013) suggest that Ω_M may be higher and h may be lower than the values used in this paper (see, however, Spergel et al. 2013). We find that the differential comoving volume as a function of redshift (affecting SMFs and CSFRs) decreases by only 6%–7% for z > 4 when we adopt Ω_M = 0.307 and h = 0.68. The increased Ω_M does result in an increase to the high-redshift halo mass function; at fixed cumulative number density, halo masses are ∼0.08 dex larger at z = 4 and ∼0.19 dex larger at z = 15 (Figure 11). This corresponds to an average change in mass accretion rates of 0.01 dex (2%), which we explicitly verify with halo merger trees in Appendix B.

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A 2% change in halo mass accretion rates is within the existing uncertainties for the mass accretion rate fitting formula that we use (Appendix B); we therefore do not expect this to change predictions for SSFRs or CSFRs. The main effects on the SMHM relation would be systematic increases of 0.08–0.19 dex in halo masses at fixed galaxy mass. For the redshift range from z = 4 to z = 8, the ∼1 dex change in characteristic halo mass (Figure 10) would be reduced by 0.04 dex if we had used the Planck cosmology. Naturally, uniform systematic offsets in halo masses do not affect our conclusions about the evolution of the characteristic halo mass.

As shown in Figures 9 and 10, we find that the evolution of the SMHM relation can be very simply described. At z < 4, the SMHM ratio depends much more on halo mass than on redshift (Behroozi et al. 2013b). At z > 4, however, the characteristic halo mass evolves strongly with redshift. More quantitatively, the halo mass at which M_* is 1% of M_h evolves as

$$\begin{equation} M_{1\%} \sim \left\lbrace \begin{array}{@{}l l}10^{11.5}\, M_{\odot }\quad & {\rm if}\,\; z<4\\ 10^{11.5-0.23(z-4)}\, M_{\odot }\quad & {\rm if }\,z\ge 4\end{array}\!.\right. \end{equation} \tag{ 9 }$$

This evolution at high redshift is close to the average accretion rate of halos (Figure 10), so the SMHM ratio is approximately unchanging as a function of cumulative number density or of peak height, ν.

As discussed in Section 5.3 and as shown in Figure 10, this mass evolution is much larger than can be explained by known systematic biases. We note not only that this mass evolution is consistent with flattening in SSFRs at high redshifts (Weinmann et al. 2011), but also that they are equivalent statements. For a galaxy population with a given stellar mass and redshift, the ratio of the galaxies' SSFRs to their host halos' SMARs is equal to the power-law slopes of their historical SMHM relations (Equations (2)–(4)). If these historical slopes are not completely parallel to the slope of the SMHM relation at the starting redshift, then galaxies will evolve off their original SMHM relationship—that is to say, the SMHM relationship as a whole must evolve with redshift. As a corollary, evolution in the ratio of galaxy SSFRs to their host halo specific halo mass accretion rates (e.g., Figure 5) must be accompanied by evolution in the SMHM relationship: as the ratio changes, the evolutionary trajectories of galaxies will change, and no single SMHM relationship can be parallel to all galaxy trajectories.

Flattening of galaxy SSFRs at high redshifts implies a decrease in the ratio of galaxy SSFRs to their host halo SMARs. In Figure 5, the ratio of galaxy SSFRs to host halo mass accretion rates is consistent (within observational errors) with the slope of the SMHM relation at fixed redshift, at least for z < 4. However, the ratio at higher redshifts is too low to be consistent. From Equation (5), if this ratio falls below the slope of the SMHM relationship at fixed redshift, and if the SMHM relationship has a positive slope, then the SMHM relationship will evolve to higher efficiencies at higher redshifts.

Hence, the growth rate of the SMF and the flattening in observed SSFRs both provide evidence that halos with masses below 10¹² M_☉ formed stars more efficiently at very high redshifts than they did for 0 < z < 4. Part of the reason for this may be a change in the gravitational potential well depth at fixed halo mass (Figure 10, Section 5.2). This is not likely the only explanation, as it would require evolution in the characteristic halo mass over 0 < z < 4 (Figure 10); feedback efficiency at high redshift may also be reduced. However, other factors like increased central density at high redshifts (Figure 10) may decrease cooling times enough to explain the evolution.

Knowing the cosmic star formation history, it is possible to calculate the hydrogen reionization history of the universe. We use the minimal reionization model of Haardt & Madau (2012):

$$\begin{equation} \frac{dQ}{dt} = \frac{\dot{n}_\mathrm{ion}}{n_\mathrm{H+He}} - \frac{Q}{t_\mathrm{rec}}\protect, \protect \end{equation} \tag{ 10 }$$

where Q is the reionized fraction, $\dot{n}_\mathrm{ion}$ is the creation rate density of ionizing photons, n_{H + He} is the number density of hydrogen and helium atoms, and t_rec is the recombination time. In this model, single ionization of helium is assumed to happen at the same time as hydrogen; we assume that the helium mass fraction is Y = 0.24. Ionizing photon production is given by

$$\begin{equation} \dot{n}_\mathrm{ion} = N_\gamma f_\mathrm{esc} {\rm CSFR}\protect, \protect \end{equation} \tag{ 11 }$$

where N_γ is the number of ionizing photons per unit mass of stars, f_esc is the average escape fraction from galaxies, and   is the CSFR. For a low-metallicity Salpeter IMF, N_γ would be $5\times 10^{60}\, M_{\odot }^{-1}$ (Alvarez et al. 2012). Since the Chabrier IMF that we use has fewer low-mass stars but the same number of high-mass (ionizing) stars, we take N_γ equal to $7.65\times 10^{60} \, M_{\odot }^{-1}$; this corresponds to 6400 photons per baryon in stars. While inferences at z < 5 suggest escape fractions less than 10%, the escape fraction at high redshifts is very uncertain (Hayes et al. 2011). Simulations suggest that the escape fractions can be very large (Wise & Cen 2009; Wise et al. 2014); current models typically adopt either a constant f_esc or an f_esc that rises rapidly with redshift (Haardt & Madau 2012). In this paper we use a constant f_esc to simplify the interpretation of our results; we note that this would roughly correspond to the star-formation-weighted average f_esc for models in which f_esc evolves with time. To explore the range of uncertainties, we allow f_esc to vary from 0.1 to 0.6.

The recombination time is given by

$$\begin{equation} t_\mathrm{rec} = (\alpha _B n_\mathrm{H+He} C)^{-1}\protect, \protect \end{equation} \tag{ 12 }$$

where we take α_B to be the case B recombination coefficient at 10,000 K for hydrogen (2.79 × 10⁻⁷⁹ Mpc³ yr⁻¹) and C to be the clumping factor. The determination of C is also uncertain, with estimates ranging from C = 2 to C = 4 at high redshifts (Haardt & Madau 2012); we take C = 3 as our fiducial value and allow C to vary from 2 to 4 for modeling uncertainties.

For any reionization history, we can also calculate an optical depth τ for Thomson scattering, given by

$$\begin{equation} \tau = \sigma _T \int _0^{t_\mathrm{now}} (Q n_\mathrm{H+He} + Q_\mathrm{He\,\scriptsize{III}} n_\mathrm{He})c \mathrm{d} t\protect, \protect \end{equation} \tag{ 13 }$$

where σ_T is the Thomson cross section of an electron (6.986 × 10⁻⁷⁴ Mpc²), Q_He iii is the fraction of helium that has been doubly ionized, and n_He is the number density of helium. Since the double ionization of helium happens much later than single ionization, it changes τ by only ∼0.001; we therefore fix the redshift of double ionization to z = 3.5.

The largest uncertainty for τ is the escape fraction, followed by uncertainties in the normalization of the CSFR (Figure 12), followed by subdominant uncertainties in the clumping factor. If the CSFR at z > 4 were currently underestimated by a factor of two, an average escape fraction of 0.2 would be consistent with the most recent Wilkinson Microwave Anisotropy Probe (WMAP) results at the 1σ level. However, if current high-redshift CSFRs are correct, f_esc would need to average at least 0.5 during reionization. These escape fractions agree with other models that match the optical depth (e.g., Haardt & Madau 2012; Alvarez et al. 2012; Bouwens et al. 2012b; Robertson et al. 2013) and are plausible for certain types of galaxies (Wardlow et al. 2014; Dijkstra et al. 2014), but they are higher than typically seen in low-redshift observations (Hayes et al. 2011).

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We also show the expected reionization history of the universe in Figure 13 for f_esc = 0.5. As with the optical depth, the normalization of the CSFR is a substantial uncertainty. For this escape fraction, 68% confidence limits on the redshift of half-reionization are 7 < z < 10. Absent independent constraints, the escape fraction is completely degenerate with the normalization of the cosmic star formation history in these models. This makes it difficult to use τ or the reionization history as a constraint on galaxy formation (or vice versa).

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As noted in Alvarez et al. (2012) and Wise et al. (2014), an escape fraction that varies strongly with halo mass would result in an escape fraction that varies strongly with redshift (as assumed in Haardt & Madau 2012). This is because the median halo mass for star formation (i.e., the halo mass below which 50% of star formation occurs) is a strong function of redshift (Figure 14). From Figures 9 and 14, we expect that the galaxies that would reionize the universe at z > 8 are typically below 10⁸ M_☉ and their host halos are typically below 10¹⁰ M_☉. While escape fractions for these low-mass galaxies can be observed locally, we note that the nature of star formation in their high-redshift counterparts will likely be very different, as evidenced by the very different SMHM ratios (Figure 9).

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If low-mass halos continue to become more efficient at z > 8 (Figure 9), the JWST should be able to see many M_* > 10⁸ M_☉ galaxies out to at least z = 15 (Figure 15; see also Appendix C for luminosity functions). By comparison, if the SMHM relation were fixed at z = 8 and did not evolve to z = 15, the expected number density of galaxies at z = 15 would drop by over two orders of magnitude (Figure 15). We expect that the JWST will rule out at least one of these two scenarios very shortly after launch. Owing to the rapid buildup of the halo mass function from z = 15 to z = 8, the number density of galaxies observed with the JWST will be very constraining for galaxy formation models.

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We find that the expected faint-end slopes for the SMFs are −1.85 to −1.95 from z = 9 to z = 15. These are steep, suggesting that deeper JWST pointings will result in many more observed galaxies. However, none of the slopes are significantly steeper than the faint-end slopes we expect at z = 8 (−1.8), suggesting that the evolution in faint-end slopes observed over 4 < z < 8 (Bouwens et al. 2012b) does not continue to higher redshifts.

We note that Equation (4) also constrains the maximum attainable SSFRs for galaxy populations as a function of redshift. From Equation (4), we have

$$\begin{equation} {\rm SSFR} = \alpha {\rm SMAR}\protect. \protect \end{equation} \tag{ 14 }$$

Hence, maximum SSFR would require maximizing the specific halo mass accretion rates (SMARs) and the power-law slope (α) of the historical SMHM relation. The maximum value of α is limited by available mechanisms to regulate star formation. For halos that are just large enough to accrete reionized gas (∼10⁹ M_☉; Gnedin 2000) and form their first stars, the SSFRs could be effectively infinite. At larger masses (e.g., >10¹⁰ M_☉), the most important mechanisms become stellar winds and supernovae from massive stars (Agertz et al. 2013). Typical values of α for 10¹⁰ M_☉ and larger halos range from α ≈ 2 at z = 0 to α ≈ 1 at z > 4. It is difficult to imagine that α could be much higher than two because the energy and momentum requirements for leaving a halo both scale as a low power of the halo mass. For example, the energy needed to escape from a halo scales as halo mass to the two-thirds power, and the escape velocity scales as halo mass to the one-third power. That said, we consider values from α = 1 to α = 4 to cover any exotic future feedback mechanisms.

Note that all variation from galaxy physics is contained in the slope α, and the range α = 1 to α = 4 constitutes only a ±0.3 dex uncertainty in the maximum SSFRs for halos larger than 10¹⁰ M_☉. The uncertainty in specific halo mass accretion rates is negligible in comparison, as those are directly measurable from dark matter simulations (Appendix B). Hence, the maximum average SSFRs for a mass-selected galaxy population are limited to be within a factor of one to four of the SMARs at all redshifts, as shown in the top panel of Figure 16.

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These constraints are consistent with existing measurements of galaxy SSFRs (Figure 16). They also place strong constraints on what typical stellar populations will look like at higher redshifts. For example, at z = 10, averaged SSFRs are limited to no more than 10^−7.3 yr⁻¹. If a survey returned higher average SSFRs for galaxies in a chosen stellar mass bin, it would most likely indicate that the survey was biased toward selecting highly star-forming galaxies, or that the analysis pipeline was giving biased stellar masses or SFRs.⁸

Redshift evolution in the SMHM relation at z > 4 can strongly affect the stellar mass for satellite galaxies. For example, the scatter in satellite stellar masses at fixed peak halo mass could be very large, depending on the distribution of satellite formation times. This is especially true for the satellite halo mass range expected to host the brightest dwarf satellites of the Milky Way (∼10⁹–10¹⁰ M_☉). Boylan-Kolchin et al. (2012) identified several Milky Way satellites that were much too bright compared to the z = 0 SMHM relation. Conversely, Boylan-Kolchin et al. (2012) found that not all of the most massive satellite halos of the Milky Way could host the brightest satellite galaxies (i.e., the "too big to fail" problem). While several papers have shown that the observations may be explained by a combination of cyclic feedback and tidal interactions with the disk of the Milky Way (Governato et al. 2012; Zolotov et al. 2012; Teyssier et al. 2013; Pontzen & Governato 2014), it remains possible that some of the satellites were formed and accreted very early in the history of the universe. In this case, Figures 9 and 10 would suggest that these early-formed satellites should have a higher stellar mass than later-formed satellites at fixed circular velocity.

We briefly note that abundance matching methods typically match galaxies to halos using the SMHM relation at a single redshift. For satellites, this creates some consistency issues because it is often not clear whether it is better to use the SMHM relation at the time of the satellite's accretion or peak halo mass/peak maximum circular velocity (Behroozi et al. 2010; Yang et al. 2012; Moster et al. 2013). At z < 2, clustering and conditional luminosity functions indicate that satellite galaxy masses are best set by using the SMHM relation at a single redshift (Reddick et al. 2013; Watson & Conroy 2013); physically, this indicates continued star formation in a fraction of satellites after accretion (Wetzel et al. 2013). However, strong evolution in the SMHM relation at high redshifts suggests that abundance matching using the SMHM relationship at the time of accretion or peak halo properties will better capture satellite galaxy masses at z > 4.

Several previous models, such as Trenti et al. (2010) and Wyithe et al. (2014), have assumed that high-redshift star formation occurs in bursty episodes. Under this interpretation, many galaxies at z > 4 are unobserved because they are temporarily not forming stars. This assumption is in part based on rest-frame UV angular clustering measurements at z > 4, such as Lee et al. (2009). We note, however, that modeling angular clustering measurements can be extremely challenging. The Lee et al. (2009) conclusions are largely based on the discrepancy between the one-halo autocorrelation term of their model and the observed data. However, the one-halo term is dominated by satellites, which tend to have lower SFRs than central galaxies. As a result, the Lee et al. (2009) results largely constrain the duty cycle of satellites; if only 30% of satellites are star-forming at z = 4, this would still allow 90% of all galaxies to be star-forming, since the satellites make up a small fraction at these redshifts. Additional issues generic to all angular correlation functions are appropriately propagating uncertainties in the redshift distribution (which are difficult to determine for faint galaxies with UV-only detections) and understanding how the sample selection function (e.g., color–color cuts) affects the redshift-dependent average bias of the selected galaxies. We note that Conroy et al. (2006) finds no duty cycle to be necessary when matching angular correlation functions at z = 4 and z = 5, further suggesting that it is very difficult to use angular correlation functions as robust constraints on galaxy duty cycles.

Separately, Wyithe et al. (2014) argued for lower duty cycles so that the growth of the stellar mass density matches the CSFR; however, previous discrepancies have largely been resolved (Reddy & Steidel 2009; Bernardi et al. 2010; Moster et al. 2013; Behroozi et al. 2013c). Also, the concern in Wyithe et al. (2014) that z > 4 galaxies cannot have formed stars at their observed rates for an entire Hubble time is resolved by strong evidence for individual galaxies having rising SFRs (Papovich et al. 2011; Behroozi et al. 2013a, 2013c; Moster et al. 2013; Lee et al. 2014), as well as the overall rise in the total CSFR with time.

Because of problems with angular correlation functions, duty cycles at z > 4 may not be fully determined until future deep rest-frame optical observations (Wyithe et al. 2014). In our approach, we chose not to assume low duty cycles due to several heuristic arguments. Most importantly, UV light continues to be emitted by stellar populations up to 100 Myr old (Conroy et al. 2009; Castellano et al. 2014; Madau & Dickinson 2014). At z = 8, this is comparable to specific halo mass accretion rates—so even if a major starburst expels all the gas from a galaxy, an equal amount of replacement gas will have reaccreted by the time the first stellar population fades in the UV. At z = 4, halo SMARs are somewhat lower ((2–3) × 10⁻⁹ yr⁻¹), but the typical halo still grows by 20%–25% in 100 Myr. These high accretion rates make it difficult for stellar feedback to completely quench star formation on long timescales. Second, high quenched fractions at z = 4 are inconsistent with K-band-selected SMFs at z < 4 because they would imply a discontinuous jump between Lyman-break galaxy (LBG)-selected (at z > 4) and K-selected (at z < 4) SMFs. Instead, the growth of LBG-selected SMFs into K-selected mass functions is extremely consistent with measured SSFRs (Behroozi et al. 2013c; Moster et al. 2013).⁹ Third, the quenched fractions measured in K-selected samples at 2 < z < 4 are very low (Brammer et al. 2011; Muzzin et al. 2013) except at the very highest stellar masses, which is inconsistent with a large quenched fraction at z > 4—unless the quenched galaxies at z > 4 all return to being star-forming at z < 4. Fourth, regardless of the feedback prescription, the observational duty cycle of galaxies above 10⁹ M_☉ is unity in hydrodynamical simulations (G. Snyder & P. Hopkins, private communication; Jaacks et al. 2012b; Wise et al. 2014).

Despite the differences with our approach, both Trenti et al. (2010) and Wyithe et al. (2014) also find that galaxy star formation must be more efficient at high redshifts—in their language, that galaxies have higher duty cycles. Their conclusion is driven by the same evidence as ours, namely, that if the same SMHM or star formation–halo mass relationship for z = 4 halos were applied to z = 8 halos, the resulting stellar mass, SFR, and luminosity functions would be an order of magnitude less than observational constraints (Section 5.3.1). However, we note that the Trenti et al. (2010) extension to high redshifts (z > 8) is very different than ours because their duty cycle is nearly 100% at z = 8 and has no more room to grow. Hence, the Trenti et al. (2010) predictions are very similar to the case in Figure 15, where the SMHM relationship is assumed to be fixed for z > 8. As noted in Section 6.3, the JWST should be able to very quickly distinguish these two scenarios.

We note finally that Tacchella et al. (2013) took a very different approach, which was to model halo star formation histories as a combination of a burst and a longer-duration star formation mode. While this model provides an excellent match to luminosity functions, star formation histories of halos are not self-consistent across redshifts (S. Tacchella 2012, private communication). For example, the star formation history of z = 5 halos at z > 6 in the model is not consistent with the star formation histories of z = 6 halos at z > 6; so if applied to merger trees, the model would require retroactive modification of galaxy star formation histories. It is nonetheless still interesting mathematically to consider the resulting redshift evolution; the Tacchella et al. (2013) model also does not predict any break in the slope of the CSFR for z > 8.