The Galaxy Luminosity Function during the Reionization Epoch
The new Wide Field Camera 3/IR observations on the Hubble Ultra-Deep Field started investigating the properties of galaxies during the reionization epoch. To interpret these observations, we present a novel approach inspired by the conditional luminosity function method. We calibrate our model to observations at and assume a non-evolving galaxy luminosity versus halo mass relation. We first compare model predictions against the luminosity function measured at and . We then predict the luminosity function at under the sole assumption of evolution in the underlying dark-matter halo mass function. Our model is consistent with the observed galaxy number counts in the HUDF survey and suggests a possible steepening of the faint-end slope of the luminosity function: compared to at . Although we currently see only the brightest galaxies, a hidden population of lower luminosity objects () might provide of the total reionizing flux. Assuming escape fraction , clumping factor , top heavy-IMF and low metallicity, galaxies below the detection limit produce complete reionization at . For solar metallicity and normal stellar IMF, reionization finishes at , but a smaller is required for an optical depth consistent with the WMAP measurement. Our model highlights that the star formation rate in sub- galaxies has a quasi-linear relation to dark-matter halo mass, suggesting that radiative and mechanical feedback were less effective at than today.
The new Wide Field Camera 3/IR (WFC3) Hubble Ultra Deep Field (HUDF09) observations opened a new window on high-redshift galaxy formation (Oesch et al., 2010a, b; Bouwens et al., 2010a, b; Bunker et al., 2009; McLure et al., 2010; Finkelstein et al., 2009). Yet the sample of galaxies is too small ( z-dropouts in Oesch et al. 2010b, and Y-dropouts in Bouwens et al. 2010a) for a precise determination of the galaxy luminosity function (LF), especially after taking into account the systematic uncertainty introduced by cosmic variance (Trenti & Stiavelli, 2008).
Measuring the galaxy LF is important to assess their contribution to cosmic reionization, which started at , as inferred from the Thomson scattering optical depth in the CMB background (Komatsu et al., 2009). The nature of the reionizing sources is currently debated. Are normal galaxies the agents of reionization, or are other sources responsible, such as Population III stars or Mini-QSOs (Madau et al., 2004; Sokasian et al., 2004; Shull & Venkatesan, 2008)? Within uncertainties, galaxies detected at barely keep the Universe reionized (Stiavelli et al., 2004b; Bunker et al., 2004). The LF evolution established from to (Bouwens et al., 2007) seems to continue into the dark ages, with progressively fewer bright sources (Bolton & Haehnelt, 2007; Bouwens et al., 2008, 2010a; Oesch et al., 2009c).
The exploration of the link between LF and underlying dark-matter halo mass function (MF) helps us understand the processes regulating star formation. This has been studied via the conditional luminosity function (CLF) method locally and at high redshift (Vale & Ostriker, 2004; Cooray & Milosavljević, 2005; Cooray, 2005; Cooray & Ouchi, 2006; Stark et al., 2007; Bouwens et al., 2008; Lee et al., 2009). Key results are: (1) significant redshift evolution of galaxy luminosity versus halo mass, , (Cooray, 2005; Lee et al., 2009); (2) only a fraction of halos appears to host Lyman Break galaxies (LBG) (Stark et al., 2007; Lee et al., 2009); (3) the predicted LF at deviates significantly from Schechter form, missing the sharp drop in density of bright () galaxies (Bouwens et al., 2008). These findings suggest limitations of the current models extrapolated to the highest redshift. In fact, because of the young age of the Universe during the reionization epoch ( corresponds to at ), it becomes difficult to justify rapid evolution of , unless the IMF changes. A low also appears problematic: the halo MF evolves rapidly at : the number density of halos (hosting galaxies) increases by a factor three from to . Hence, implies that the majority of recently formed halos at did not experience significant star formation. Finally, the absence of a well-defined knee in the predicted LF differs from the rarity of observed bright galaxies (see Bouwens et al. 2010a).
To overcome these limitations, we present a novel implementation of the CLF model, tailored for application at . Instead of a duty cycle, we adopt another simple assumption: only halos formed within a given time interval host a detectable LBG (Section 2). Section 3 contains the predictions for the LF, compared to WFC3-HUDF09 observations. Section 4 discusses the contribution of galaxies to reionization.
2 An Improved CLF Model for
We adopt a variation on the CLF approach (Vale & Ostriker, 2004; Cooray & Milosavljević, 2005) to construct an empirical relation between the galaxy LF and the halo MF at redshift , close to the reionization epoch and with a well measured LF function. In the standard CLF method, is derived assuming that each dark-matter halo hosts a single galaxy and equating the number of galaxies with luminosity greater than to the number of halos with mass greater than (optionally with ):
In our Improved CLF (ICLF) model, rather than , we modify Equation 1 to include only halos with that have been formed within time interval :
is defined implicitly by:
In the limit and , Equation 4 is equivalent to Equation 1. We adopt but discuss model predictions for . The timescale refers to the global evolution of the halo MF (Equations 2-3) and captures the fraction of halos that experienced a recent change in their mass. This ensemble includes halos that have likely experienced a recent star-formation burst, and are thus more likely to host a UV-bright galaxy. However, for an individual halo star formation is extended over timescales longer than at a lower mass scale. In fact, using our Extended-Press-Schechter modeling (Trenti & Stiavelli, 2008; Trenti et al., 2008), we infer that a , halo had , at ( confidence). This is consistent with the abundant supply of cold gas present at high-redshift (Kereš et al., 2005; Davé et al., 2008), which suggests sustained star formation over several yr. In fact, our ICLF model has an higher at (Figure 1) than the fixed assumed/derived in similar studies (Stark et al., 2007; Lee et al., 2009).
We parametrize the observed LF as a Schechter function:
We calibrate the model to the rest-frame UV parameters measured by Bouwens et al. (2007) for i-dropouts (): (), , with (see also Table 1). In Equations 1 and 4, we do not consider scatter in because we are primarily interested in the insensitive faint end of the relation (Cooray & Milosavljević, 2005). We also neglect multiple halo occupation, motivated by current observational limits. Only halos with are likely to host multiple galaxies (Wechsler et al., 2001). Such halos are rare within a single ACS field of view ( expected in the ACS volume for i-dropouts). Halos hosting multiple galaxies are present in surveys at with a larger volume such as the GOODS field, but their depth is insufficient to detect the fainter (sub-) satellite galaxies. The model of Lee et al. (2009) provides an independent confirmation.
Figure 1 shows the relation at from Equations 1-4. The faintest galaxies () live in halos. The blue-shaded region represents the uncertainty derived by varying the LF parameters within the confidence regions in Figure 3 of Bouwens et al. (2007). We included an additional uncertainty in from cosmic variance ( for the three quasi-independent HUDF05-ACS fields, see Trenti & Stiavelli 2008111Cosmic variance calculator available at http://casa.colorado.edu/~trenti/CosmicVariance.html). is similar for both models. The steep faint-end slope of the observed LF () implies a flattening of compared to the local Universe, where (Cooray & Milosavljević, 2005). For the standard CLF model we derive , while the ICLF model gives . This means that the specific star formation efficiency in small-mass halos depends mildly on halo mass (), compared to the strong suppression () inferred at . This provides clues to the processes that regulate early-time star formation, suggesting a scenario where radiative and supernova feedback were less efficient than today. The UV background decreases at (Haardt & Madau, 1996), likely reducing the impact of photoionization. In addition, halos were more compact, making gas expulsion more difficult.
From the relation, combined with the measure of the total stellar mass in galaxies based on SED fits (Stark et al., 2009; Gonzalez et al., 2010; Labbé et al., 2009), we derive a typical star formation efficiency (6% of gas converted into stars) with a large uncertainty, driven by the measure of the stellar mass (Figure 9 in Stark et al. 2009). Lower-mass halos are slightly less efficient at converting gas into stars. For example, halos with , have , consistent with assumptions in numerical models at (Trenti & Stiavelli, 2009; Trenti et al., 2009). The decrease in with decreasing halo mass is possibly related to suppression of star formation by local photoionization (Cantalupo, 2010).
From we infer a halo mass for galaxies with . This agrees with clustering measurements at and (Overzier et al., 2006).
2.1 Validation of the ICLF Model
To validate the predictions of our ICLF model under the sole assumption of evolution in the underlying dark-matter MF, we apply derived at to at and . The resulting LFs are reported in Table 1 and Figure 2, and compared to those obtained with the standard CLF method. As expected from past studies (Cooray, 2005; Lee et al., 2009), the standard CLF model fails to match the observed LF at primarily because of strong evolution in . The ICLF model predicts instead quasi-constant , because the comoving formation rate per unit time for halos hosting faint () galaxies remains approximately constant between and . The ICLF results are fully consistent with the observed LF, and with the bright-end at . However, the faint-end slope at is underestimated at (predicted versus measured). This clearly indicates that our simple assumption is no longer valid at because becomes too small at the faint end (Figure 1).
3 Luminosity Function Evolution and HUDF09 Dropouts
The current sample of galaxy candidates is too small to provide an independent fit of the LF. For example, Oesch et al. (2010b) measured the faint-end slope assuming a fixed value for and . Here, we apply our ICLF model for a full prediction of the LF. We do not allow evolution of . Significant redshift evolution is present (Figure 2), simply because there is progressively less structure at higher . The decrease in is smaller than predicted by the CLF method. Both models predict a dimming in () and a steepening of the faint-end slope, which becomes close to by . The LF evolution is directly related to evolution of the halo MF shape, which depends exponentially on . As redshift increases, massive halos become rarer, and the relative abundance of smaller mass halos increases.
To have an accurate comparison between predicted and observed number counts, we convolve the LF with the effective volume of the observations as measured with artificial source recovery simulations (Oesch et al., 2010b; Bouwens et al., 2010a). This is crucial because of significant incompleteness at . These numbers are reported in Table 2 and include two additional ICLF models with and . The counts expected from our reference ICLF model agree remarkably well with the number of sources observed in the HUDF: z-dropouts are predicted (with uncertainty including cosmic variance) in agreement with the candidates of Oesch et al. (2010b). The ICLF models with are also consistent with the data at . No LF evolution from predicts 31 sources (rejected at confidence level), while the standard CLF model gives 9.8 sources (rejected at confidence). For Y-dropouts, our reference ICLF model gives sources at (compared to without LF evolution), again fully consistent within with the Y-dropout candidates of Bouwens et al. (2010a).
4 Consequences for Reionization
With our LF model we can investigate the role of galaxies in the reionization of the Universe. Despite large differences in the estimate of reionizing photon production, past studies established that sources below the current detection limit likely provide a significant fraction of the photon budget. The precise number of ionizing photons thus depends on the minimum luminosity chosen for the extrapolation of the LF. Our modeling offers a physically motivated cut-off for the luminosity of the smallest halo capable of forming stars. Theoretical and numerical investigations establish that a halo at irradiated by a UV field comparable to the one required for reionization needs a mass (virial temperature K at ) in order to cool and form stars (Tegmark et al., 1997). For K, the minimum halo mass corresponds to based on the relation at . Galaxies below the HUDF09 magnitude limit contribute of the total luminosity density at integrated to , unless feedback stronger than seen in cosmological simulations (e.g., Ricotti et al., 2008) induces a flattening of the LF below the HUDF09 detection limit.
To evaluate the likelihood that galaxies ionize the universe, we resort to the widely used conversion of luminosity density in star formation rate (SFR; Equation 2 in Madau et al. 1998). We compare this SFR (Figure 3) to the critical rate (Madau et al., 1999) required for reionization,
where is the escape fraction of ionizing photons and the hydrogen clumping factor. Both Equation 2 in Madau et al. (1998) and Equation 6 above depend on explicit assumptions on the stellar IMF (Salpeter 1955 in ) and metallicity (Solar). Equation 6 has a large uncertainty, including the IMF-dependent efficiency of Lyman-Continuum photon production. We adopt , consistent with the very blue UV slope of small galaxies (Bouwens et al., 2010b), and (Bolton & Haehnelt, 2007; Pawlik et al., 2009). Figure 3 shows that galaxies with produce enough photons to reionize the Universe.
To obtain the evolution of the reionization fraction, , we follow a complementary approach (see Equation 9 and its derivation in Stiavelli et al. 2004a). We adopt an IGM temperature , include the effect of Helium () and assume conservatively , , and no ionizing flux at . The production rate of ionizing photons is obtained from the LF by assuming different SEDs for the sources. Figure 3 shows that if the LF is integrated to , reionization by is achieved for any SED, including the unlikely scenario with Salpeter IMF and (consistent with the analysis). However, in this case, the optical depth to reionization is underestimated compared to the WMAP5 measurement because of the sharp drop of at ( versus [Komatsu et al. 2009]). Metal-poor and top-heavy SEDs alleviate this problem, as they achieve complete reionization at , predicting . Alternatively, is needed. Without a steeper faint-end, , the ionizing flux is reduced by , requiring a corresponding decrease in for a constant . Without the contribution from sources below the HUDF detection limit (i.e., LF integration to ), only a top-heavy and metal-poor SED can reionize the universe by , but that model predicts .
In this Letter we construct a model for the evolution of the galaxy luminosity function at based on a modification of the CLF method. We derive the relation between galaxy luminosity and dark-matter halo mass at , assuming a one-to-one correspondence between observed galaxies and halos that formed in a period . Using fixed at , we derive the expected LFs between and , assuming only evolution of the underlying dark-matter MF. The LF is consistent with observations, but our model is less accurate at lower redshift because it underestimates the faint-end slope. At , we predict a moderate decrease of , a possible steepening of the faint-end slope, and continued evolution of toward lower values (Figure 2 and Table 1). At all epochs, our predicted LF is well fitted by a Schechter function with a prominent “knee”.
The predicted number counts for the HUDF09-WFC3 field are a good match to the dropouts observed at and (Table 2). Overall our ICLF model is consistent with the observed galaxy LF from to with no evolution in . DM halos assembly can explain LF evolution at without invoking a change in the properties of LBG star formation. This is in agreement with the constant specific star formation rate inferred at (Gonzalez et al., 2010).
Our model provides evidence for a reduced impact of feedback in low-mass halos. In fact, we derive a star formation efficiency weakly dependent on halo mass (), compared to the strong quenching of star formation derived at (; Cooray & Milosavljević 2005), providing a testable prediction for cosmological simulations. The strong suppression of star formation in halos suggested by Bouché et al. (2010) and Maiolino et al. (2008) contrasts with measured for halos with (Section 2). Such strong feedback would also imply that galaxies appear incapable of sustaining reionization. In fact, with a steep LF, sources below the HUDF-WFC3 detection limit may contribute of the ionizing flux, sufficient for full reionization if . A metal-poor and top-heavy IMF, or smaller , are required to complete reionization at for consistency with . While our extrapolation is physically motivated to , it extends for magnitudes. Deeper observations are thus crucial to verify that the LF faint-end remains steep.
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|CLF model||ICLF model||Observations|
|-1.99||-2.22bbAsymptotic faint-end slope is|
Note. – Best-fit Schechter parameters for LFs in Figure 2 (second to fourth column: CLF model; fifth to seventh column: ICLF; last three columns: Bouwens et al. 2007 measurements). Fit holds for . Relative residuals are .
Note. – Observed (Oesch et al., 2010b; Bouwens et al., 2010a) and predicted number counts for galaxies at (z-dropouts) and (Y-dropouts) in the HUDF09 field for different (I)CLF models. Predictions include convolution with effective HUDF09 volume as function of source magnitude. uncertainty includes cosmic variance.