Faint-end of LFs

On the faint-end of the high-$z$ galaxy luminosity function

Abstract

Recent measurements of the Luminosity Function (LF) of galaxies in the Epoch of Reionization (EoR, ) indicate a very steep increase of the number density of low-mass galaxies populating the LF faint-end. However, as star formation in low-mass halos can be easily depressed or even quenched by ionizing radiation, a turnover is expected at some faint UV magnitudes. Using a physically-motivated analytical model, we quantify reionization feedback effects on the LF faint-end shape. We find that if reionization feedback is neglected, the power-law Schechter parameterization characterizing the LF faint-end remains valid up to absolute UV magnitude . If instead radiative feedback is strong enough that quenches star formation in halos with circular velocity smaller than 50 km s, the LF starts to drop at absolute UV magnitude , i.e. slightly below the detection limits of current (unlensed) surveys at . The LFs may rise again at higher absolute UV magnitude, where, as a result of interplay between reionization process and galaxy formation, most of the galaxy light is from relic stars formed before the EoR. We suggest that the galaxy number counts data, particularly in lensed fields, can put strong constraints on reionization feedback. In models with stronger reionization feedback, stars in galaxies with absolute UV magnitude higher than and smaller than are typically older. Hence, the stellar age - UV magnitude relation can be used as an alternative feedback probe.

keywords:
cosmology: observations - galaxies: high-redshift - cosmology: theory - dark ages, reionization, first stars

1 Introduction

Reionization is one of the most important processes during cosmic history. It starts around 1, when the first stars begin to form (see reviews e.g. Abel et al. 2002; Ciardi & Ferrara 2005; Bromm et al. 2009), and it is completed by , when virtually all the intergalactic medium (IGM) is ionized (Becker et al., 2001; Fan et al., 2006; Ouchi et al., 2010; Stark et al., 2010). A deeper understanding of the reionization process relies on observations of its driving sources that are generally believed to be high- normal star-forming galaxies (Salvaterra et al., 2011; Robertson et al., 2013; Lorenzoni et al., 2011; Jaacks et al., 2012; Finkelstein et al., 2012; Robertson et al., 2013, 2015). So far, observations of high- LFs have reached absolute UV magnitude (at ) and redshift up to (Bouwens et al., 2015a; McLure et al., 2013; Oesch et al., 2014; Finkelstein et al., 2015; Bouwens et al., 2015b; Roberts-Borsani et al., 2015) thanks to the deepest HUDF/XDF surveys (Illingworth et al., 2013; Bouwens et al., 2011; Oesch et al., 2013; Ellis et al., 2013) and gravitational lensing magnification (Atek et al., 2015; McLeod et al., 2016; Livermore et al., 2016). Recent observations of high- galaxies are reviewed in Finkelstein (2015).

The observed high- galaxies are mostly the brightest ones among reionization sources. However, in the hierarchical structure formation scenario, massive halos form through a series of mergers of smaller progenitors. Such small halos are more numerous and dominate the collapsed matter fraction budget. Thus, it is natural to expect that more faint galaxies exist, hosting most of the stellar mass and dominating the ionizing photon budget (e.g. Choudhury & Ferrara 2007).

A problem might arise with the above scenario. Sustaining star formation requires continuous cold gas supply, but the available gas content in halos could be limited by various feedback effects, namely supernova feedback and radiative feedback. Both supernova explosions and ionizing radiation inject thermal energy into the interstellar medium, interrupting the gas cooling process. However, the internal feedback due to stars is self-regulated since a decreasing star formation efficiency also reduces the feedback strength. Thus, feedback operates more efficiently when it is driven by ionizing radiation from external sources (nearby galaxies and/or a background). External ionizing photons could completely quench the star formation by either heating the cosmic gas and reducing the efficiency of gas accretion, or by continuously evaporating the gas already contained in halos. These processes are collectively referred to as “reionization feedback” as they occur during the Epoch of Reionization (EoR). The effects mainly depend on the gravitational potential of individual halos, and are particularly evident in smaller galaxies. If reionization feedback is indeed effective, a significant decline in the abundance of galaxies with the lowest luminosities is expected (Wise et al., 2014; O’Shea et al., 2015). It follows that the role played by faint galaxies as reionization sources might be questioned.

The shape of the luminosity function (LF) faint-end during EoR is expected to carry signatures of the star formation modulation imposed by impinging ionizing radiation. However, compared with the wealth of detailed theoretical studies on how external ionizing radiation suppresses star formation in galaxies (see e.g. Sobacchi & Mesinger 2013a, b), there are fewer predictions on the LF of high- galaxies in the full absolute magnitude range, including the faint/low-mass galaxies most sensitive to reionization feedback. This is clearly due to the limited dynamical range that numerical simulations can achieve. Nevertheless, Gnedin (2016) has numerically investigated the LF down to the faint-end. He found evidence for a deviation of the LF from the Schechter function at absolute UV magnitude , although the LF continues to rise up to magnitude . This can be translated into an equivalent sharp cutoff of the Schechter function in the UV magnitude range to to match the required UV emissivity for completing the reionization (Robertson et al., 2015; Mitra et al., 2015).

Observationally, the minimum mass of host halos of high- galaxies has been constrained by Muñoz & Loeb (2011). By assuming that the galaxy stellar mass is gathered in halo growth history through a series of instantaneous star formation bursts triggered by mergers, and comparing the theoretical LFs with deep HST surveys, they derived a minimum mass of at . This minimum mass is consistent with the reionization requirements. Because of the existence of this mass floor, LFs drop gently above a turnover absolute UV magnitude. They also found that the total star formation rate (SFR) in all high- galaxies is only moderately higher than the SFR contained by already observed galaxies, therefore the star formation activity in ultra-faint galaxies must be heavily suppressed by the reionization feedback (however, for a slightly different view see Salvadori & Ferrara 2009).

Within current detection limits no deviation with respect to the Schechter power-law increase has been reported. However, a sensitivity improvement, as provided by, e.g., the Frontier Fields2, might change the situation dramatically (Yue et al., 2014). In view of these developments it is therefore timely to develop an analytical model of the high- galaxy LF matching the already observed bright part of the LF, and simultaneously predicting the faint-end far below the current detection limit. By comparing the predictions with upcoming observations, we may then characterize reionization feedback effects in detail. This task involves various complications, including the metallicity and stellar evolution, and a mass-dependent star formation efficiency. Moreover, the SFR probably depends on other properties of the halo and its environments as well, in addition to the halo mass, as found by simulations (e.g. Pallottini et al. 2014.)

Pioneering works have derived the high- galaxy properties from the better-known halo properties, using various models to associate the star formation history to the mass assembly history of the host halo (e.g. Choudhury & Ferrara 2007; Trenti et al. 2010; Tacchella et al. 2013; Wyithe et al. 2014; Mason et al. 2015; Behroozi & Silk 2015; Sun & Furlanetto 2016; Mashian et al. 2016). In particular, in a series of works Trenti et al. (2010), Tacchella et al. (2013) and Mason et al. (2015), an analytical method has been developed to calculate the LF of high- galaxies from the dark matter halo mass function. The main assumption in their approach is that the “star formation efficiency” – defined as the ratio of gas converted into stars to the accreted dark matter mass – is a function of halo mass only, i.e. it is independent of redshift. The redshift dependence of the galaxy luminosity is ascribed to its mass assembly history. Therefore, once the star formation efficiency at a given redshift is obtained, it is assumed to hold at any redshifts. Alternatively, to derive the high- galaxy LF from halo mass function, a mean redshift-independent SFR - halo mass relation is used in Mashian et al. (2016). Moreover, Behroozi & Silk (2015) assumed that the specific star formation rate is proportional to the specific halo accretion rate. They found that the stellar mass to halo mass ratio evolves rapidly at .

In this paper3, we extend the Mason et al. (2015) model by including the reionization feedback effect under various assumptions for the feedback strength, then use it to derive the EoR galaxy LF to below current detection limits. The layout is as follows. In Section 2 we introduce the algorithm to compute LFs from halo mass functions and halo star formation histories. We model the influence of reionization feedback on the galaxy abundance self-consistently. In Section 3 we present our predicted LFs extended to galaxies in which star formation is significantly influenced by reionization feedback effects. We also discuss the SFR - stellar mass relations and the mean stellar age - UV magnitude relations there. Conclusions and discussions are given in Section 4.

2 method

2.1 From halos to galaxies

Whether and how a halo contributes to the galaxy LF depends on how efficient it accretes matter and converts accreted matter into stars. Assume that a halo with mass at redshift collects mass at a rate of at cosmic time , and suppose further that an accreted mass element, , later on increases the SFR at cosmic time by , where the star formation efficiency refers to the fraction of accreted mass covered into stars. Following the spirit of Trenti et al. (2010), Tacchella et al. (2013) and Mason et al. (2015), we assume depends only on halo mass , but we introduce an additional mass-independent function to account for the time-dependence. Integrating over all accreted matter elements before , we have the SFR of this halo at cosmic time :

 SFR(Mh,t)=∫ttff(Mh)g(t−t′)dMhdt′dt′, (1)

where is the cosmic time corresponding to the redshift when this halo formed. An accreted gas element is gradually converted into stars in an “extended burst”, for which the time-dependence can be modeled as (Cen & Ostriker, 1992; Gnedin, 1996; Chiu & Ostriker, 2000; Choudhury & Srianand, 2002; Samui et al., 2007):

 g(t−t′)=t−t′κ2t2d(zf)exp[−t−t′κtd(zf)], (2)

where

 td(zf)=√3π32Gρvir(zf) (3)

is the dynamic timescale for a halo with mean density , and is a free parameter that controls the duration of the burst.

Following Tacchella et al. (2013), we only track the halo assembly history back to the half-mass redshift, which is a good approximation as pointed by Mason et al. (2015) (see their Fig. 4): for halos with at , stars formed in progenitors with only contribute 1% - 10% to the total UV luminosity. A further assumption is that the halo mass grows (by accretion and minor mergers) at a constant rate between and , i.e.

 dMhdt′=Mh2(t0−tf), (4)

where is the cosmic time at redshift . Therefore the SFR in this halo can be written as

 SFR(Mh,t) =f(Mh)Mh2(t0−tf)∫ttfg(t−t′)dt′ =SFRM15∫ttfg(t−t′)dt′, (5)

where SFR is the SFR used in Mason et al. (2015). Compared with the SFR, the parameter in the time-dependent factor introduces a new degree of freedom in the SFR prescription. If is a Dirac delta-function, which means that accreted gas forms stars instantaneously, or , or , this factor approaches unity and the SFR reduces to SFR. Nevertheless, can be constrained only from observations (see below).

At redshift , a halo of mass formed at has a luminosity (e.g. Samui et al. 2007):

 L(Mh,z0,zf)=∫t0tfSFR(Mh,t)lν(t0−t)dt, (6)

a stellar mass

 m⋆(Mh,z0,zf)=∫t0tfSFR(Mh,t)m(t0−t)dt, (7)

and a mean stellar age

 t⋆(Mh,z0,zf)=1m⋆∫t0tfSFR(Mh,t)m(t0−t)(t0−t)dt. (8)

Here is the Single Stellar Populations (SSP) SED template, i.e. the luminosity of an instantaneous star formation burst with unit stellar mass at the time after the burst; we always use the luminosity at the wavelength 1600 Å when calculating the absolute UV magnitude. is the mass fraction of surviving stars after the burst. For both and we use the template of Bruzual & Charlot (2003) with Chabrier IMF between 0.1 - 100  and fixed metallicity of .

Before deriving the LF of galaxies from halo mass function, we need to obtain the curve. This is determined by comparing the theoretical mean luminosity - halo mass relations with that from observations. From Eq. (6), halos with the same mass could have different luminosities if they formed at different redshifts. Given the probability distribution of the formation time, , their mean luminosity is

 ¯L(Mh,z0)=∫L(Mh,z0,zf)p(w)dw, (9)

where is related to the formation redshift by

 w=√0.707δc(zf)−δc(z0)√σ2(Mh/2)−σ2(Mh), (10)

with being the critical density contrast for spherical collapse at linearly extrapolated to present time, and being the variance of density fluctuations smoothed on mass scale . In the elliptical collapse scenario the has the simple expression (Giocoli et al., 2007):

 p(w)=2werfc(w/√2), (11)

where erfc is the complementary error function.

On the other hand, a relation between the observed absolute UV magnitude and the halo mass could be constructed by “abundance matching”, i.e. forcing the number density of galaxies with absolute UV magnitude smaller than to match the number density of halos with mass above :

 ∫MUV,obs−∞ΦSch(M′UV,z0)dM′UV=∫∞MhdnhdM′hdM′h, (12)

where is the Schechter parameterization of the LF (Bouwens et al., 2015a). Here the observed absolute UV magnitude corresponds to the dust-attenuated luminosity, which is related to the intrinsic UV magnitude by

 MUV=MUV,obs−A1600, (13)

where is the dust extinction given by (Meurer et al., 1999)

 A1600=4.43+1.99β, (14)

with being the luminosity-dependent UV spectrum slope. must be and is fitted in Bouwens et al. (2014) by a linear form

 β=β0+dβdM0(MUV,obs−M0), (15)

where . The best-fitted parameters of a collection of HST observations given by Bouwens et al. (2014) are used in this work, i.e. and at redshifts , respectively. At intermediate redshifts we linearly interpolate; for higher redshifts we use the linear extrapolation for and fix . At each , by equating with the luminosity of the dust-corrected absolute UV magnitude , we obtain the .

After calibration, we convert the luminosity of the halo into an observed absolute UV magnitude, for which we write its explicit form here for convenience:

 MUV,obs(Mh,z0,zf)= ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1−1.99dβdM0)−1(MUV−M0+4.43+1.99β0)+M0,        (if MUV

where is the absolute UV magnitude of the luminosity given by Eq. (6).

Galaxies with the same luminosity could be hosted by halos with different masses that formed at different redshifts. Among halos of mass , only those formed at a specified redshift, , can host galaxies with observed absolute UV magnitude ; is then obtained by substituting , and into Eq. (16). The LF is then written as

 Φ(MUV,obs,z0)=∫dMhdnhdMhp(wmag)dwmagdMUV,obs, (17)

where is the halo mass function (Sheth & Tormen, 1999; Sheth et al., 2001), and . The integration is performed over the range of for which a solution exists.

The free parameter is determined by comparing the derived LF with the observations from Bouwens et al. (2015a). We use a reduced chi-square to measure the deviation of the predicted from the observations, which is defined as

 χ2red(κ)=1n−1∑i(Φ−Φi)2σ2i, (18)

where is the predicted LFs from Eq. (17) by using the best-fitted Schechter parameterization of Bouwens et al. (2015a) to calibrate the (see Eq. 12), and are observational points of LFs and errors in Bouwens et al. (2015a). The sum is performed on all observational points at redshifts 5, 6, 7 and 8 except for those with upper limits only; is the number of all points used.

The as a function of is shown in Fig. 1. We find that the deviation is small and stable at , implying that observations favor a scenario in which the accreted gas is typically converted into stars on a timescale . We do not find lower limits on the value of . Therefore, we take hereafter. The calibrated at is plotted in Fig. 2.

To derive the uncertainties in , we randomly sample the three Schechter parameters, and , assuming that they all follow a Gaussian distribution with standard deviation equal to the 1 error given in Bouwens et al. (2015a). The region containing 68.3% of the values in each mass bin is plotted in Fig. 2. We do not account for the uncertainties in and . We find that the uncertainties are larger for smaller halos (), because of the weaker constraints on the faint-end slope of LFs. In the following, we will use the calibration at for all calculations.

We now incorporate the reionization feedback effects into the above modeling of galaxy LF. How reionization feedback affects a halo depends on two separate issues: (a) the probability that the halo is located in ionized bubbles, and (b) how fast the gas supply is interrupted, or its gas photo-evaporated. For a given cosmological evolutionary scenario, (a) is determined by the total amount of emitted ionizing photons and the gas recombination rate. The point (b) is instead determined by complex radiation hydrodynamical processes inside galaxies.

Specifically, we assume that the star formation efficiency calibrated by matching the theoretical LF to the Schechter function is valid down to the atomic-cooling halo mass (see Appendix A for a check of the supernova feedback effects), unless in these halos star formation (i) never ignites, or (ii) is totally quenched by the external ionizing radiation. Such events can occur in halos with circular velocity smaller than a threshold , further located in ionized bubbles (see e.g. Choudhury & Srianand 2002). Here depends on both and (Barkana & Loeb, 2001). We note that currently the threshold for quenching star formation by external ionizing radiation is still rather uncertain, and in terms of circular velocity, it varies from km s to km s, depending on the intensity and spectrum of the ionizing flux, the time interval in which the halo is exposed to the radiation, self-shielding effects, and so on (e.g. Thoul & Weinberg 1996; Dijkstra et al. 2004; Kitayama et al. 2000; Gnedin 2000; Sobacchi & Mesinger 2013b).

With star formation being quenched at , analogously to Eq. (6), the emission rate of ionizing photons is

 ˙Nion(Mh,z0,zq,zf)=∫tqtfSFR(Mh,t)˙q(t0−t)dt, (19)

where the rate is taken from Bruzual & Charlot (2003). Taking into account the reionization feedback, we have three possible star formation histories in halos with viral temperature  K (we neglect the contribution from minihalos, as substantial star formation activity in minihalos is unlikely and rather uncertain):

1. A halo with can sustain continuous star formation if it is always located outside ionized bubbles.

2. A halo with that originally formed in a neutral region later became ionized has its star formation quenched4 at (). In this case the star formation activity only happens between and .

3. A halo with is massive enough that reionization feedback has no effects on it, no matter whether it is located inside or outside ionized bubbles.

To evaluate reionization feedback we need to know the probability, , that a halo with mass is located in an ionized bubble at redshift . This probability is closely related to the volume filling factor of ionized regions at the corresponding redshift. Once the escape fraction of ionizing photons, , is assigned, we can compute the evolution of the filling factor of ionized regions

 dQHIIdt=fesc˙nionnH−Q2HIIC(z)nH(1+z)3αB, (20)

where is the ionizing photon emissivity, is the comoving total hydrogen number density, is the clumping factor, and is the Case B recombination coefficient. For simplicity we take  K for ionized regions, yielding  cms. We use the form for the clumping factor given by Iliev et al. (2007) normalized to at :

 C(z)=6.8345×exp(−0.1822z+0.003505z2). (21)

Taking into account the three cases of star formation history described above, the ionizing photon emissivity can be written as

 ˙nion(z0)=∫dMhdnhdMh∫dwp(w)∫z0zf˙NionF(Mh,zq,zf)dzq, (22)

where is given by

 F={[1−Pb(Mh,z0)]δ(zq−z0)+dPbdzqvc

in which is the Dirac delta-function. The term corresponds to the case (i), and the terms and identify case (ii) and (iii), respectively. To construct a mapping from to we use a method based on the excursion-set formalism (see Appendix B). Once and are specified, we numerically solve the differential equation (20) from , when it is safe enough to assume and .

Once the ionizing history is determined, the IGM electron scattering optical depth to CMB photons can be derived from

 τ=σTnHc(1+YHe4YH)∫z0QHII(z′)(1+z′)3|dtdz′|dz′, (24)

where cm is the cross-section for Thomson scattering, is the speed of light, is the He mass fraction; the H fraction is ; the singly ionized He fraction is assumed to be equal to the HII fraction.

To gain a physical intuition of the effects of feedback on reionization history, in Fig. 3 we compare obtained for a no-feedback model and a strong feedback ( km s) model, using a fixed . We find that the difference between these two models is modest. Adopting a km s only reduces the Thomson optical depth by and delays the reionization by .

After including the reionization feedback, the luminosity of a galaxy in which star formation is quenched at () is then

 L(Mh,z0,zq,zf)=∫tqtfSFR(Mh,t)lν(t0−t)dt, (25)

where is the quenching time corresponding to redshift . This luminosity can be converted into observed absolute UV magnitude by Eq. (16). The LF including feedback effects is

 Φ(MUV,obs,z0)= (26) ∫dMhdnhdMh[1−Pb(Mh,z0)]p(wmag)I(vc)dwmagdMUV,obs +∫dMhdnhdMh∫z0zfdzqdPbdzqp(wmag,q)I(vc)dwmag,qdMUV,obs +∫dMhdnhdMhp(wmag)[1−I(vc)]dwmagdMUV,obs, (27)

where, in analogy with the no-feedback case, is obtained by substituting , , and into Eq. (16). is a step function: when , and otherwise. The three terms in the above equation correspond to the three cases of star formation history. If the reionization feedback is not accounted for and the above equation reduces to Eq. (17).

3 results

3.1 Reionization history

We first check whether the Thomson optical depth is consistent with observations in this new scenario. In principle, a higher would promote the reionization process and result in a larger . However, it also increases the probability for halos to be located in ionized bubbles, thus reducing the contribution of ionizing photons from small halos.

In Fig. 4 we show the completion redshift of reionization, , and the for different and pairs. In each row, the left, central, and right panels show the minimal, best-fit, and the maximal values allowed by the uncertainty in the star formation efficiency calibration, respectively.

The Thomson optical depth is mainly dependent on , but insensitive to . This is because not all halos with are sterile, and only those that inhabit ionized bubbles lose the ability to produce ionizing photons. In addition, star formation efficiency in small halos is low, as seen from Fig. 2. We find that, taking into account the uncertainties in star formation efficiency calibration, the recent Plank measurements (, Planck Collaboration et al. 2016) rule out both the low escape fraction range (see the right panel for ), and the high escape fraction range (left panel).

Another interesting problem is whether the conclusion that most ionizing photons come from faint galaxies should be revised once reionization feedback is considered. In Fig. 5 we plot the fractional ionizing photon rate produced in galaxies with absolute UV magnitude above , assuming that all galaxies have the same escape fraction of . It is found that with decreasing redshift, a smaller fraction of ionizing photons comes from fainter galaxies, which is a result of the shallower slope of the LF. When the reionization feedback is taken into account, this fraction is further reduced. However, the qualitative trend remains unchanged. In fact, reionization feedback has significant effects only in the late EoR stages or after EoR, while during the early EoR (e.g. ) faint galaxies always emit the majority of ionizing photons. We note that this conclusion is based on the assumption of a fixed escape fraction for all galaxies. If smaller galaxies have larger escape fraction (e.g. Ferrara & Loeb 2013), they could have been even more dominant compared to the results shown in Fig. 5.

3.2 Feedback imprints on the LF

Next we investigate the effect of reionization feedback on the galaxy LF, and how the LF changes with the feedback strength characterized by the threshold and . We start by fixing again , and then predict the resulting LFs at and 8 for models with and 50 km s. These LFs and the corresponding uncertainties are shown in Fig. 6, where we also plot the LFs for a model with corresponding to the atomic-cooling criterion (viral temperature K / km s). This no-feedback model is used as a reference. The observational data from Bouwens et al. (2015a) are also plotted. We find that all LFs match observations in the overlapping magnitude ranges, satisfying our first requirement. In addition, we plot in Fig. 7 the LFs by adopting km s, while equals 0.1 and 0.3 respectively.

In the no-feedback model, the Schechter parameterization (which can be approximated by for ) is valid up to a turnover absolute UV magnitude , above which the number of galaxies drops dramatically. This is because the star formation cannot occur in halos below the atomic-cooling criterion, whose typical luminosity corresponds to the turnover UV magnitude . Above this atomic-cooling mass, the decrease in the star formation efficiency in low mass halos is compensated by the increase in the halo number, as seen from Fig. 2, resulting in a LF with a shallower (albeit still negative) slope with respect to the halo mass function.

The reionization feedback distorts the luminosity - halo mass relation, built from the in Fig. 2 (see detailed discussions in Sec. 3.3), for halos with , resulting in the complex luminosity distributions of faint galaxies hosted by them. We can generally divide the galaxies into two groups: the ones hosted by halos with and those with respectively. In the former halos, the star formation activity has never been interrupted. For halos with , many of them were not able to ignite star formation at all, as they formed in ionized patches; the remaining systems could sustain star formation for some time before being quenched by external ionizing radiation. The latter systems are much fainter than the counterparts that have the same mass assembly history in the reference no-feedback model. As a result the number of galaxies drops dramatically above the new turnover UV magnitude roughly corresponding the mean luminosity of halos with , namely () for (50 km s).

Galaxies in halos with (with ) do not totally disappear. Their number drops down above , but then rises again for higher absolute UV magnitudes. What causes the deficit of galaxies just above ? Why, even under strong feedback conditions, the number of faint galaxies remains large (compared to bright galaxies) instead of gradually dropping to zero? The answer to these questions is simple. For halos with , only those formed before the end of reionization have the chance to form outside ionized bubbles and start their star formation activity. In the model with and km s, the reionization completes at . Halos formed before shine only due to their relic stars, and fade away with time; in halos with , most UV radiation is continuously supplied by newly formed stars. As a result, there is a gap in the LFs corresponding to these two (physically distinct) halo populations, and the gap broadens with time.

The number of faint galaxies starts to deviate from the no-feedback model significantly only in the late stages of reionization (EoR lasts for Myr), because earlier on the time stretch is too short to suppress star formation in most halos. By varying the , one can change the reionization history. For instance, if we decrease to 0.1 (yielding ), as shown in Fig. 7, faint galaxies with could have abundance higher than galaxies as bright as .

For comparison, the LFs from hydrodynamical simulations combined with radiative transfer calculation in Wise et al. (2014) and in Gnedin (2016) are also shown in the corresponding panels for the same (or very similar) redshifts in Fig. 6. We find that at redshift 6 and 7, the Gnedin (2016) LFs are close to our model with km s, however at redshift 8 we predict more faint galaxies above the turnover UV magnitudes than Gnedin (2016). On the other hand, the Wise et al. (2014) LFs are always similar to our no-feedback models.

To get a deeper insight, we plot separately the LFs of halos with and with at different redshifts in Fig. 8, for and 50 km s, respectively. Also shown are the available observational data from Bouwens et al. (2015a). From the figure we clearly see how reionization feedback gradually separates the two components inducing an increasing deviation from the no-feedback LF reference model. It is worthwhile noting that although we just simply adopt a constant circular velocity threshold as a star formation quenching criterion, the LFs have rather complex behaviors at the faintest end. From to , the abundance of galaxies with evolves fast, and these faint galaxies hosted by halos with become EoR relics (see e.g. Ricotti & Gnedin 2005; Salvadori & Ferrara 2009; Benítez-Llambay et al. 2015).

The above discussions are only concerned with the calibrated by using best-fitted Schechter parameters. Considering the uncertainties, predictions on the number of galaxies at the faint-end () are rather uncertain, as shown by shaded regions in each panel of Fig. 6 and Fig. 7. On the other hand, the observed abundance of ultra-faint galaxies can put tight limits on the star formation efficiency of small halos.

3.3 Feedback imprints on galaxy properties

We further investigate the imprints of reionization feedback on galaxies properties. The first is the relation, which critically concerns the galaxy LFs discussed above. As different halo mass assembly histories introduce an intrinsic scatter in this relation, in order to derive the scatter, it is more appropriate to generate Monte Carlo random samples. The samples are generated by using the probability distribution of the (a) halo mass (given by the halo mass function), (b) formation time (from Eq. 11 for a given halo mass , and redshift ), and (c) quenching time (using and ) obtained above. In Fig. 9 we show the results at redshifts 5, 6 ,7 and 8, for the no-feedback reference model, and for and 50 km s. All models have . We use the error bars to represent the intrinsic scatter of the absolute UV magnitudes due to different mass assembly histories of halos in the same mass bin and due to different reionization imprints on them, and shaded regions to represent the full uncertainties considering both the intrinsic scatters and the uncertainties in calibrating the star formation efficiency. The reionization feedback decreases the mean luminosity of halos with at the same time increasing the scatter, imprinting an ankle-knee feature in the relation.

We also extract the SFR - and - relations from the above Monte Carlo samples. The SFR - relation at is shown in Fig. 10, and compared with data from Salmon et al. (2015). Our predicted SFR - relations agree well with observations in the overlapping range. Discrepancies in the SFR - relations of the three models are modest at ; however, in the (50) km s models star formation in all galaxies with has already been quenched before redshift 5. The SFR - relations at higher redshifts have a similar trend, except that the amplitudes increase by about 0.1 dex per redshift.

The analogous stellar age () vs. relation is plotted in Fig. 11. We find that at redshift 5 galaxies with have mean stellar age  Myr. Above this absolute UV magnitude models start to diverge, with stronger feedback models predicting relatively older galaxies. By looking at the - relations it is easier to distinguish models in the range . Therefore, stellar age measurements of galaxies in this range could be used as a probe of reionization feedback strength. At higher redshifts stars are typically younger: for example at galaxies with are about 30 Myr old.

3.4 Feedback imprints on galaxy counts

We have pointed that our model predicts a Thomson optical depth consistent with the Planck constraints in a wide range. Therefore this indirect observation is not very helpful in discriminating models with different . However, at the faintest magnitudes different models predict substantially different galaxy number counts in a given redshift range from to ,

 N(H160)=∫z2z1r2Φ(MUV,obs,z)drdzdz, (28)

where is the apparent magnitude observed at 1.6 m, corresponding to rest-frame luminosity at  m. This rest-frame luminosity is converted to the absolute UV magnitude at 1600 Å using the as in Eq. (6), but at fixed  Myr for convenience. Hence, number count observations could directly put constraints on , as shown in Fig. 12 for galaxies in the redshift range 5 - 8. Among these ultra-faint galaxies a substantial fraction of them are located in , due to the steeper slope of the LF at compared to at . For example, we check that for the number count of no-feedback model shown in Fig. 12, at about % is in , at this fraction is %. Existing and/or forthcoming galaxy surveys are unlikely to reach the very deep limiting magnitudes required. However, if gravitational lensing (e.g. Yue et al. 2014) can be exploited, it is possible to detect a handful of ultra-faint galaxies, that would allow to put tight constraints on feedback strength. The results of investigations using two Frontier Fields clusters are presented in Castellano et al. (2016).

4 Conclusions

The star formation activity in small halos with shallow gravitational potential well is easily quenched by external ionizing flux from nearby sources and/or an ionizing radiation background. Such quenching effect might play a significant role in shaping the reionization history, when more and more galaxies formed in/entered into ionized bubbles whose size keeps growing throughout the EoR. Thereby the LF of such faint galaxies provides key information on the interplay between the reionization process and its driving sources.

We have investigated the LF of faint galaxies during the EoR by including the above reionization feedback in an analytical model. The model derives the LF from halo mass function by constructing luminosity - halo mass relations from observationally-calibrated star formation efficiency and halo mass assembly history. Reionization feedback effects are included by adopting a constant threshold circular velocity, , below which the star formation of halos located in ionized bubbles is quenched. We computed the LFs for models with different and values, and found that:

• If reionization feedback is neglected, the power-law Schechter parameterization characterizing the faint-end of the LF remains valid up to (corresponding to the atomic-cooling halos mass, see Appendix A for an estimate of supernova explosion effects). Above this absolute UV magnitude the number density of galaxies drops dramatically.

• When feedback is included, small halos () in ionized bubbles fail to collect enough gas to ignite/sustain their star formation. The reionization history, constrained by the Planck electron scattering optical depth, is insensitive to .

• For strong feedback, i.e.  km s, the LF deviates from the Schechter function above , slightly below the detection limit of current surveys of blank fields at (Bouwens et al., 2015a). Hence, we expect that upcoming observations will obtain important constraints on .

• In addition, even for strong feedback, the LF may rise again at luminosities fainter than as a result of the interplay between reionization process and galaxy formation.

• We also pointed out that the - relation might be used as a powerful probe of reionization feedback strength. In models with stronger reionization feedback, stars in galaxies with are typically older. Other constraints on and in our model, can come from galaxy number count data, particularly from those exploiting gravitational lensing magnification.

Our model contains some necessary simplifications and assumptions. The most relevant one is perhaps the use of a constant circular velocity threshold as the criterion for quenching star formation, and its treatment as a free parameter independent of . This is a standard assumption in the literature and is very convenient when performing analytical calculations. Instead, detailed simulations (e.g. Sobacchi & Mesinger 2013b) pointed out that in the presence of reionization feedback, the gas fraction decreases gradually as the halo mass decreases, following a relation , where is a critical halo mass. We believe that this effect would make the LF smoother around the turnover point and may result in more faint galaxies. However, we do not expect significant changes in the basic trend of the galaxy LF found here.

In our work, we take the model parameters that give predictions consistent with the direct observations of the LFs at in the blank fields. There are alternative ways to investigate the reionization feedback and constrain the high- LFs using indirect observations. At intermediate redshifts () the IGM is fully ionized and a global ionizing UV background is in place. Hence the radiative feedback effects should be maximal. The known existence of faint galaxies in these epochs implies the existence of even more fainter galaxies in the EoR.

For example, in Alavi et al. (2014) the observations of galaxies at confirm the validity of the Schechter formula down to absolute UV magnitude . These results show that such faint galaxies support active star formation well after the EoR. As a consequence, it is likely that km s if we incorporate this information into our model.

In addition, the number of ultra-faint satellites in the Local Group also put constraints on the faint-end of the EoR LF once used in combination with their merger tree history, see Weisz et al. (2014); Boylan-Kolchin et al. (2014, 2015). Finally, the rate of high- GRBs is another probe of the abundance of ultra-faint galaxies in the EoR (Trenti et al., 2012). Here, we checked that distinguishing models with different requires very high precision measurements of the star formation rate density (SFRD). Even at the SFRD difference between the no-feedback model and the km s is only about 10%, i.e. much smaller than the current precision of the SFRD derived from GRB observations. Nevertheless, all these alternative techniques nicely complement investigations, as the one presented here, based on direct LFs or number count data.

Acknowledgements

YX is supported by the NSFC grant 11303034, and the Young Researcher Grant of National Astronomical Observatories, Chinese Academy of Sciences.

Appendix A Supernova feedback

The star formation efficiency is calibrated by assuming that the Schechter parameterization for LFs always holds down to the atomic-cooling halo mass. This might not be true for small halos in which gas could be totally blown away by supernova explosion, therefore in this section we have a check on it. Halos can sustain the continuous star formation mode if the energy deposited by supernova explosions does not exceed the gravitational binding energy of the halo,

 m⋆ηSNESN<12Mhfgv2esc=Mhfgv2c, (29)

where is the energy released by supernova per stellar mass and (Pallottini et al., 2014) is the fraction that this energy goes into the gas, is the gas fraction and is the escape velocity. In our case , we have

 f(Mh)<∼2fgv2cηSNESN. (30)

If the upper limit at the right hand side is smaller than the calibrated star formation efficiency the halo has to adjust itself to have the new star formation efficiency satisfies the Eq. (30) (Dayal et al., 2014). Simply assuming , and taking from the outputs of Starburst995 (Leitherer et al. 1999; Vázquez & Leitherer 2005; Leitherer et al. 2010, for consistence reason we also replace the in Eq. 6 with the one from Starburst99 in this check), we check that for the Salpeter IMF with the mass ranges of 0.1 - 100 , and , the Eq. (30) always holds for halo mass above the atomic-cooling mass, as long as when the metallicity is , and as long as when the metallicity is . We therefore believe that our assumption is safe enough. This is because in our work for any given metallicity and IMF, we calibrate the corresponding to reproduce the observed LF at . In the IMF models with more massive stars, both the and the are higher. The higher consequently results in a smaller , limiting the total energy released by supernovae , so the host halos could still hold the star formation activity. The effects by Pop III stars are not considered here, as pointed by numerical simulations (Tornatore et al., 2007; Pallottini et al., 2014) Pop III stars are negligible in halos above the atomic-cooling criterion.

Appendix B Pb vs. QHII

We integrate Eq. (4) in Furlanetto et al. (2004a) to calculate the probability that a halo sits in ionized bubbles above a minimum size. This is basically an application of the “bubble model” scenario discussed in Furlanetto et al. (2004b). The minimum bubble size is set by requiring that a target halo has at least one neighbor with within the bubble radius. Namely, we have the two-point halo correlation function:

 ξh(m1,m2,r,z)=ξ(r,z)b(m1,z)b(m2,z), (31)

where is the matter two-point correlation function and is the halo bias (Sheth et al., 2001). We then obtain the number of neighboring halos within radius and above

 N=∫M∗[1+ξh(r,z,m1,Mh)]dndMhdMh; (32)

is then determined by solving the above equation for .

The bubble model uses the cumulative ionizing photons number per collapsed atom, , to calculate , but this number is not explicitly appearing in our algorithm. However, we find that although the reionization history depends on , if we plot the as a function of (), where is the collapse fraction, we actually see limited variations in a large range (see different curves in Fig. 13). Therefore, in this paper we adopt the following approximation: when calculating the we fix and take the value at the time at which . The above algorithm does not take into account the dependence of on the halo formation redshift, . Halos formed earlier are more biased, therefore they might have higher probability to be located in the ionized bubbles. This improvement will be deferred to future work.

Footnotes

1. This is the typical redshift when the 3 fluctuations of the cosmic density field on the molecular hydrogen cooling mass scale collapse to form halos, see Barkana & Loeb (2001). The very early and rare first stars can form from fluctuations at much higher redshifts (Naoz et al., 2006; Trenti & Stiavelli, 2007).
2. http://www.stsci.edu/hst/campaigns/frontier-fields/
3. Throughout the paper, we use the Planck cosmology parameters (Planck Collaboration et al., 2015): =0.308, =0.692, =0.6781, =0.0484, and =0.8149. The transfer function is from Eisenstein & Hu (1998).
4. We warn that our treatment here is simplified, as might be a function of redshift and intensity of the UV radiation field. We neglect this effect here but refer the interested readers to the discussion in Sobacchi & Mesinger (2013b).
5. http://www.stsci.edu/science/starburst99/docs/default.htm

References

1. Abel T., Bryan G. L., Norman M. L., 2002, Science, 295, 93
2. Alavi A. et al., 2014, ApJ, 780, 143
3. Atek H. et al., 2015, ApJ, 800, 18
4. Barkana R., Loeb A., 2001, Phys. Rep., 349, 125
5. Becker R. H. et al., 2001, AJ, 122, 2850
6. Behroozi P. S., Silk J., 2015, ApJ, 799, 32
7. Benítez-Llambay A., Navarro J. F., Abadi M. G., Gottlöber S., Yepes G., Hoffman Y., Steinmetz M., 2015, MNRAS, 450, 4207
8. Bouwens R. J. et al., 2011, ApJ, 737, 90
9. Bouwens R. J. et al., 2014, ApJ, 793, 115
10. Bouwens R. J. et al., 2015a, ApJ, 803, 34
11. Bouwens R. J. et al., 2015b, ArXiv e-prints, 1506.01035
12. Boylan-Kolchin M., Bullock J. S., Garrison-Kimmel S., 2014, MNRAS, 443, L44
13. Boylan-Kolchin M., Weisz D. R., Johnson B. D., Bullock J. S., Conroy C., Fitts A., 2015, MNRAS, 453, 1503
14. Bromm V., Yoshida N., Hernquist L., McKee C. F., 2009, Nature, 459, 49
15. Bruzual G., Charlot S., 2003, MNRAS, 344, 1000
16. Castellano M. et al., 2016, ApJ, 823, L40
17. Cen R., Ostriker J. P., 1992, ApJ, 399, L113
18. Chiu W. A., Ostriker J. P., 2000, ApJ, 534, 507
19. Choudhury T. R., Ferrara A., 2007, MNRAS, 380, L6
20. Choudhury T. R., Srianand R., 2002, MNRAS, 336, L27
21. Ciardi B., Ferrara A., 2005, Space Science Reviews, 116, 625
22. Dayal P., Ferrara A., Dunlop J. S., Pacucci F., 2014, MNRAS, 445, 2545
23. Dijkstra M., Haiman Z., Rees M. J., Weinberg D. H., 2004, ApJ, 601, 666
24. Eisenstein D. J., Hu W., 1998, ApJ, 496, 605
25. Ellis R. S. et al., 2013, ApJ, 763, L7
26. Fan X., Carilli C. L., Keating B., 2006, ARA&A, 44, 415
27. Ferrara A., Loeb A., 2013, MNRAS, 431, 2826
28. Finkelstein S. L., 2015, ArXiv e-prints, 1511.05558
29. Finkelstein S. L. et al., 2012, ApJ, 758, 93
30. Finkelstein S. L. et al., 2015, ApJ, 810, 71
31. Furlanetto S. R., Hernquist L., Zaldarriaga M., 2004a, MNRAS, 354, 695
32. Furlanetto S. R., Zaldarriaga M., Hernquist L., 2004b, ApJ, 613, 1
33. Giocoli C., Moreno J., Sheth R. K., Tormen G., 2007, MNRAS, 376, 977
34. Gnedin N. Y., 1996, ApJ, 456, 1
35. Gnedin N. Y., 2000, ApJ, 542, 535
36. Gnedin N. Y., 2016, ArXiv e-prints, 1603.07729
37. Iliev I. T., Mellema G., Shapiro P. R., Pen U.-L., 2007, MNRAS, 376, 534
38. Illingworth G. D. et al., 2013, ApJS, 209, 6
39. Jaacks J., Choi J.-H., Nagamine K., Thompson R., Varghese S., 2012, MNRAS, 420, 1606
40. Kitayama T., Tajiri Y., Umemura M., Susa H., Ikeuchi S., 2000, MNRAS, 315, L1
41. Leitherer C., Ortiz Otálvaro P. A., Bresolin F., Kudritzki R.-P., Lo Faro B., Pauldrach A. W. A., Pettini M., Rix S. A., 2010, ApJS, 189, 309
42. Leitherer C. et al., 1999, ApJS, 123, 3
43. Livermore R. C., Finkelstein S. L., Lotz J. M., 2016, ArXiv e-prints, 1604.06799
44. Lorenzoni S., Bunker A. J., Wilkins S. M., Stanway E. R., Jarvis M. J., Caruana J., 2011, MNRAS, 414, 1455
45. Mashian N., Oesch P. A., Loeb A., 2016, MNRAS, 455, 2101
46. Mason C., Trenti M., Treu T., 2015, ArXiv e-prints, 1508.01204
47. McLeod D. J., McLure R. J., Dunlop J. S., 2016, ArXiv e-prints, 1602.05199
48. McLure R. J. et al., 2013, MNRAS, 432, 2696
49. Meurer G. R., Heckman T. M., Calzetti D., 1999, ApJ, 521, 64
50. Mitra S., Choudhury T. R., Ferrara A., 2015, MNRAS, 454, L76
51. Muñoz J. A., Loeb A., 2011, ApJ, 729, 99
52. Naoz S., Noter S., Barkana R., 2006, MNRAS, 373, L98
53. Oesch P. A. et al., 2013, ApJ, 773, 75
54. Oesch P. A. et al., 2014, ApJ, 786, 108
55. O’Shea B. W., Wise J. H., Xu H., Norman M. L., 2015, ApJ, 807, L12
56. Ouchi M. et al., 2010, ApJ, 723, 869
57. Pallottini A., Ferrara A., Gallerani S., Salvadori S., D’Odorico V., 2014, MNRAS, 440, 2498
58. Planck Collaboration et al., 2016, ArXiv e-prints, 1605.03507
59. Planck Collaboration et al., 2015, ArXiv e-prints, 1502.01589
60. Ricotti M., Gnedin N. Y., 2005, ApJ, 629, 259
61. Roberts-Borsani G. W. et al., 2015, ArXiv e-prints, 1506.00854
62. Robertson B. E., Ellis R. S., Furlanetto S. R., Dunlop J. S., 2015, ApJ, 802, L19
63. Robertson B. E. et al., 2013, ApJ, 768, 71
64. Salmon B. et al., 2015, ApJ, 799, 183
65. Salvadori S., Ferrara A., 2009, MNRAS, 395, L6
66. Salvaterra R., Ferrara A., Dayal P., 2011, MNRAS, 414, 847
67. Samui S., Srianand R., Subramanian K., 2007, MNRAS, 377, 285
68. Sheth R. K., Mo H. J., Tormen G., 2001, MNRAS, 323, 1
69. Sheth R. K., Tormen G., 1999, MNRAS, 308, 119
70. Sobacchi E., Mesinger A., 2013a, MNRAS, 432, 3340
71. Sobacchi E., Mesinger A., 2013b, MNRAS, 432, 51
72. Stark D. P., Ellis R. S., Chiu K., Ouchi M., Bunker A., 2010, MNRAS, 408, 1628
73. Sun G., Furlanetto S. R., 2016, MNRAS, 460, 417
74. Tacchella S., Trenti M., Carollo C. M., 2013, ApJ, 768, L37
75. Thoul A. A., Weinberg D. H., 1996, ApJ, 465, 608
76. Tornatore L., Ferrara A., Schneider R., 2007, MNRAS, 382, 945
77. Trenti M., Perna R., Levesque E. M., Shull J. M., Stocke J. T., 2012, ApJ, 749, L38
78. Trenti M., Stiavelli M., 2007, ApJ, 667, 38
79. Trenti M., Stiavelli M., Bouwens R. J., Oesch P., Shull J. M., Illingworth G. D., Bradley L. D., Carollo C. M., 2010, ApJ, 714, L202
80. Vázquez G. A., Leitherer C., 2005, ApJ, 621, 695
81. Weisz D. R., Johnson B. D., Conroy C., 2014, ApJ, 794, L3
82. Wise J. H., Demchenko V. G., Halicek M. T., Norman M. L., Turk M. J., Abel T., Smith B. D., 2014, MNRAS, 442, 2560
83. Wyithe J. S. B., Loeb A., Oesch P. A., 2014, MNRAS, 439, 1326
84. Yue B., Ferrara A., Vanzella E., Salvaterra R., 2014, MNRAS, 443, L20