STEADILY INCREASING STAR FORMATION RATES IN GALAXIES OBSERVED AT 3 ≲ z ≲ 5 IN THE CANDELS/GOODS-S FIELD

To understand the SFHs of high-redshift star-forming galaxies, we analyze the SEDs of high-redshift (3 ≲ z ≲ 5) LBGs observed in the CANDELS/GOODS-S field. The GOODS is a deep sky survey covering a combined area of ∼320 arcmin² in two fields—the GOODS-N centered on the Hubble Deep Field-North (HDF-N) and the GOODS-S centered on the Chandra Deep Field-South (CDF-S). It carries an extensive wavelength coverage of the panchromatic photometric data from various facilities, including the HST/ACS (Advanced Camera for Surveys) and Spitzer/IRAC (Infrared Array Camera), supplemented with extensive spectroscopic follow-ups (Vanzella et al. 2008, 2009; Popesso et al. 2009). Deep near-infrared (NIR) observation has been carried out in these two GOODS fields with HST/WFC3 (Wide Field Camera 3) as part of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011). The deep and extensive multi-band photometric data of the GOODS/CANDELS provide a unique and useful data set in the study of physical or stellar-population properties of high-redshift galaxies.

In our study, we use the photometric information from observed frame ultraviolet (UV) to mid-infrared (MIR) in fifteen broad bands—U band from the CTIO (Cerro Tololo Inter-American Observatory) and the VLT/VIMOS (VIsible Multi-Object Spectrograph); B₄₃₅, V₆₀₆, i₇₇₅, and z₈₅₀ bands from HST/ACS; F098M, F105W (Y), F125W (J), and F160W (H) from HST/WFC3; Ks bands from VLT/ISAAC (Infrared Spectrometer and Array Camera), and m_3.6 μm, m_4.5 μm, m_5.8 μm, and m_8.0 μm from Spitzer/IRAC. In selecting the high-redshift galaxies based on their color(s) or in analyzing their observed SEDs, it is very important to measure accurately in an unbiased way the colors or the SEDs from the observed photometric data which are typically from the instruments with different spatial resolutions and/or point-spread functions (PSFs). Colors or SEDs with mismatched photometry can lead to unwanted biases in the inferences derived from the analysis of these SEDs or colors. In the case of the GOODS data, the resolution and PSF of the HST/ACS data are ∼003 and ∼01, respectively, while these for Spitzer/IRAC data are as large as ∼06 and ∼2'', each. To construct reliable SEDs of galaxies, the matched photometry among various data from different facilities with different properties is performed using the TFIT (Laidler et al. 2007) photometric software by the CANDELS team. TFIT is a photometric package for matched photometry with the template-fitting technique. It uses the positional as well as the morphological information of high-resolution image (H-band of HST/WFC3 in our case) as prior information to measure the corresponding fluxes at other bands with lower resolutions. Therefore, this package is a very useful tool in measuring colors or SEDs of galaxies from the observational data with different resolutions, especially for faint objects or in a crowded field. The details of the matched multi-band photometric catalogue of the CANDELS/GOODS-S field, which is used in this study, can be found in Guo et al. (2013).

From the CANDELS/GOODS-S multi-band photometric catalogue, we select star-forming galaxies at redshifts z ∼ 3, 4, and 5 using the spectral break at λ ∼ 912 Å in galaxy spectra (i.e., Lyman break) combined with blue color at longer rest-frame UV wavelength. This Lyman break technique (Giavalisco 2002) has been shown to select efficiently high-redshift, star-forming galaxies from optical photometric data sets (e.g., Steidel et al. 2003). We apply the same LBG selection criteria as in L09 and L10. The criteria are as follows:

$$\begin{equation} (U - B_{435}) \ge 0.62 + 0.68 \times (B_{435} - z_{850}) \wedge \end{equation} \tag{ 1 }$$

$$\begin{equation} (U - B_{435}) \ge 1.25 \wedge \end{equation} \tag{ 2 }$$

$$\begin{equation} (B_{435} - z_{850}) \le 1.93 \end{equation} \tag{ 3 }$$

for U dropouts,

$$\begin{equation} (B_{435} - V_{606}) > 1.1 + (V_{606} - z_{850}) \wedge \end{equation} \tag{ 4 }$$

$$\begin{equation} (B_{435} - V_{606}) > 1.1 \wedge \end{equation} \tag{ 5 }$$

$$\begin{equation} (V_{606} - z_{850}) < 1.6 \end{equation} \tag{ 6 }$$

for B dropouts, and

$$\begin{eqnarray} ((V_{606} - i_{775}) &\,{>}\,& 1.4667 + 0.8889 \nonumber\\ &&\times (i_{775} - z_{850})) \vee ((V_{606} - i_{775}) > 2.0) \wedge\quad\ \ \ \end{eqnarray} \tag{ 7 }$$

$$\begin{equation} (V_{606} - i_{775}) > 1.2 \wedge \end{equation} \tag{ 8 }$$

$$\begin{equation} (i_{775} - z_{850}) < 1.3 \end{equation} \tag{ 9 }$$

for V dropouts.

Here, ∧ means logical "AND," and ∨ is logical "OR." We also apply the ACS z-band magnitude cut to be z₈₅₀ ⩽ 26.6, which is the S/N = 10 limit for extended sources in the GOODS-S.

From these color selection, 1879, 1066, and 165 U-, B-, and V-dropout candidates are selected. From these color selected candidates we exclude AGN and stellar candidates. AGN candidates are excluded based on: (1) X-ray data from Chandra 4 Ms sample (Xue et al. 2011), (2) radio sample of Padovani et al. (2011), or (3) IR selection using the method of Donley et al. (2012) (provided to the CANDELS team courtesy of Jennifer Donley). Based on these, 34, 23, and 1 objects are excluded as AGN candidates from U-, B-, and V-dropout sample, each. Then, we remove 1, 3, and 1 stellar candidates from each dropout sample by applying the criteria in stellarity parameter (⩾0.9) and in WFC3 H-band magnitude (H_AB ⩽ 24.0).

Among the remaining LBG candidates, 35 (U-dropout), 52 (B-dropout), 15 (V-dropout) galaxies have spectroscopic data, and 3 U-dropout and 3 V-dropout candidate galaxies are excluded from each dropout sample due to their too low redshifts. For the redshift cut, we apply the minimum redshift requirement of z = 1.8 (U dropouts), 2.0 (B dropouts), and 3.0 (V dropouts). These minimum redshift cuts are based on the distribution of photometric redshifts of all LBG candidates. We have determined the minimum redshift for each dropout sample where the distribution shows a minimum or a significantly reduced tail toward low redshift. All 52 spectroscopic B-dropout candidates have redshift greater than 2. The remaining U, B, and V dropouts with spectroscopic redshifts are 32, 52, and 12.

For the LBG candidates without spectroscopic redshift, we use photometric redshift derived by the CANDELS team based on a hierarchical Bayesian approach (T. Dahlen et al. 2013, in preparation). Among 1809, 988 and 148 U, B, and V dropouts with only photometric redshift, we only include the galaxies with photometric redshift in the range of 1.8 ⩽ z_phot ⩽ 4.0 in the case of U-dropout candidates, 2.0 ⩽ z_phot ⩽ 5.0 in the case of B-dropout candidates, and 3.0 ⩽ z_phot ⩽ 6.0 for the V-dropout candidates. Specifically, we exclude 906 (U-dropout), 220 (B-dropout), and 40 (V-dropout) galaxies from our sample for the following analysis based on their photometric redshift. These galaxies which are excluded due to their low photometric redshift have similar dropout and rest-frame UV color with the remaining LBGs, but have, on average, redder color at longer wavelengths. This suggests that the majority of these galaxies are most likely low-redshift galaxies whose 4000 Å breaks mimic the Lyman break of high-redshift LBGs. We cannot completely rule out the possibility that some of these galaxies are actually high-redshift galaxies with redder color, probably due to higher attenuation. However, excluding these galaxies would not significantly affect our conclusion unless these galaxies' star formation properties are clearly different from the remaining LBG sample—for example, deviating largely from the star-forming main sequence. Therefore, we keep our LBG sample conservative and more robust by excluding these potential low-redshift contamination.

Figure 1 shows the redshift distributions of observed LBGs. The red line shows the redshift distribution of U dropouts, and the green line shows the distribution of B dropouts. The blue line is for V dropouts. For galaxies without spectroscopic redshifts, photometric redshifts are used in this figure. The mean redshifts of each dropout samples are 3.0 (U dropouts), 3.8 (B dropouts), and 5.0 (V dropouts). The over-density in this redshift distribution near z ∼ 3.4 is also shown in photometric redshift distribution of total GOODS-S galaxies (Figure 20 in Dahlen et al. 2010).

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In Table 1, we summarize the numbers of galaxies in each dropout sample.

To analyze the SEDs of GOODS-S LBGs and to constrain their physical properties, especially their SFHs, we perform SED-fitting analysis using spectral templates from Bruzual & Charlot (2003, hereafter BC03) stellar population synthesis models. We assume the Calzetti dust attenuation law (Calzetti et al. 2000) for internal dust attenuation, and the Madau (1995) extinction law for intergalactic extinction due to the neutral hydrogen in the intergalactic medium (IGM). We vary metallicity among three allowed values—0.2 Z_☉, 0.4 Z_☉, and 1.0 Z_☉, where Z_☉ is the solar metallicity. For the stellar IMF, we assume the Chabrier (2003) form.

The main focus of this work is to investigate and constrain the SFHs of the observed high-redshift star-forming galaxies selected from their rest-frame UV colors. Therefore, in this SED-fitting analysis, we assume two different forms of SFHs—i.e., the exponentially declining SFHs and the increasing ones.

In the case of exponentially declining SFHs, we vary the star formation timescale parameter, τ—which governs how fast the SFR declines with time—from 0.2 Gyr to τ_max, where τ_max = 2.0, 1.6 and 1.3 Gyr for U, B, and V dropouts, respectively. We vary the parameter t—which is the time since the onset of the star formation—from 50 Myr to t_H, where t_H is the age of the universe at each corresponding redshift.

The increasing SFHs used in this work have the same form as the "delayed SFHs," introduced in L10, with the limited range of τ. The delayed SFHs are parameterized as

$$\begin{equation} \Psi (t,\tau) \varpropto\frac{t}{\tau ^{2}} e^{-t / \tau }, \end{equation} \tag{ 10 }$$

where Ψ(t, τ) is the instantaneous SFR.

This functional form provides the SFHs which initially increase with time and then decline for large t (t > τ) after passing the peak at t = τ. By setting the values of τ to be greater than the age of the universe at specific redshift, we can generate the SFHs in which the SFR keeps increasing until observed at the given redshift. To make a set of galaxy spectral templates with increasing SFHs, we vary the values of τ as τ_min ⩽ τ ⩽ 10.0 Gyr. τ_min values are 2.0 Gyr for U dropouts, 1.5 Gyr for B dropouts, and 1.0 Gyr for V dropouts. This choice of τ range (to limit the range of the allowed SFHs) is motivated by the results of L10, which show that the predicted SFHs of high-redshift (3 ≲ z ≲ 5) star-forming galaxies increase with time from the analysis of SEDs of mock galaxies from the SAMs of galaxy formation. We vary the t within the same range as in the SED fitting with exponentially declining SFHs, i.e., from t = 50 Myr to the age of the universe at given redshift.

In the SED-fitting procedure, we fix the redshift at the given spectroscopic redshift value whenever available, or at the photometric redshift otherwise. In Table 2, we summarize the range of parameters used in SED fitting for each SFH form.

As explained in the previous section, we perform SED fittings with two different assumptions about the SFHs—first with the widely used, exponentially declining SFHs and then with the increasing ones. The main aim of this experiment is to examine what form of SFH is more appropriate for the analysis of the SEDs of high-redshift LBGs—in other words, to see which form of SFH is more representative to the actual SFHs of our observed LBGs.

As our first test on the SFHs of CANDELS/GOODS-S LBGs, we compare the minimum (i.e., best-fit) χ² values from the SED fittings with two different assumptions about the SFHs. The minimum χ² value for each observed galaxy provides the measure of the match between the observed SED and the model galaxy SED templates. Therefore, from this comparison of the minimum χ² values, we can test which form of SFHs produces galaxy SEDs which are closer to the observed SEDs of the CANDELS/GOODS-S LBGs.

In Figure 2, we show the ratio between the minimum χ²s in the SED fitting with the increasing SFHs and the declining SFHs, for U, B, and V dropouts (from left to right panel). From this figure, we can see that the minimum χ² values from the SED fitting with increasing SFHs are smaller than the values from the SED fitting with declining SFHs for the majority of the observed LBGs, especially for V dropouts. This means that the set of synthetic spectral templates with the increasing SFHs, on average, provide a better match to the SEDs of the observed GOODS-S LBGs. In this figure, we can see that the minimum χ² ratios (y-axis) are smaller than 1 for most V-dropout galaxies, while the fraction of galaxies with the ratio ≳ 1 progressively increases with decreasing redshift—i.e., the χ² difference from the SED fitting with different SFH assumptions is more clearly seen at higher redshift. The median values of the χ² ratio are 0.93, 0.88, and 0.86 for U, B, and V dropouts, respectively.

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While the comparison of minimum χ² values with different SFHs favors the rising SFHs, it should be noted that the difference between the two minimum χ² values is not significant for most individual galaxies—the ratio values are clustered near 1 (shown as red dotted horizontal line in each panel). Furthermore, while increasing SFHs appear to be favored on average, we have not explored other possibilities such as varying the dust attenuation law, the IMF, or the IGM attenuation treatment. Therefore, this χ² comparison can only be considered as one of the supporting evidence favoring the rising SFHs, and we will provide further evidences in the following section.

Previous studies have shown that there is a positive correlation between stellar masses and SFRs for high-redshift star-forming galaxies (Papovich et al. 2006; Daddi et al. 2007; Elbaz et al. 2007; Noeske et al. 2007; Pannella et al. 2009). This locus of galaxies has been dubbed as the "star-forming main sequence of galaxies." This observed tight correlation indicates that galaxies' mass plays a crucial role in regulating their star formation activity. The investigation of this correlation between galaxies' stellar masses and SFRs and its evolution provides us with an important clue to their formation histories (e.g., Renzini 2009; Sawicki 2012).

Here, we use our LBG sample to explore the SF properties along this star-forming main sequence at z ⩾ 3. Figure 3 shows the distributions of the GOODS-S LBGs as well as the mock LBGs from SAMs of galaxy formation in the stellar-mass–SFR plane. Here, we use the CANDELS/GOODS-S mock galaxy catalogue. This mock catalogue is generated by the CANDELS team combining the Bolshoi N-body simulations (Klypin et al. 2011) and the SAMs of Somerville et al. (2012), which is an updated version of Somerville et al. (2001) and Somerville et al. (2008). This model (and previous versions of it) has been used in various studies of galaxy properties (e.g., Idzi et al. 2004; Fontanot et al. 2009, 2012; Niemi et al. 2012; Porter et al. 2012), providing meaningful insights into the galaxy evolution.

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Our SAM assumes the standard ΛCDM universe and includes the theory of the growth and collapse of fluctuations through gravitational instability. It implements the method of dark matter halo merger tree construction of Somerville & Kolatt (1999). After this construction of dark matter halo merger tree, analytic treatments for various physical processes, including gas cooling, star formation, chemical evolution, mergers of galaxies, and feedback effects from supernova and from AGN (active galactic nuclei), are applied. For the detailed description about these analytic treatments, please refer above references.

The major update done in Somerville et al. (2012) is the treatment of the absorption and re-emission of stellar light by the interstellar medium dust. Briefly, dust extinction is modeled based on Charlot & Fall (2000), considering two components—i.e., dust extinction associated with the birth clouds around young star-forming regions, and extinction due to the diffuse dust in the disc—differently. The wavelength dependence of the dust attenuation is modeled assuming the Calzetti et al. (2000) starburst attenuation law. The absorbed energy is re-emitted in the infrared wavelength range. The model uses dust emission templates to determine the dust emission SEDs. For the detailed explanation about this update in the treatment of dust absorption and emission, please refer Somerville et al. (2012). From this SAM mock catalogue, we select mock U, B, and V dropouts, applying the same color selection criteria which are used to select our observed CANDELS/GOODS-S LBG samples (i.e., Equations (1)–(9)).

For this investigation, we focus on the observed LBGs with their best-fit t values not smaller than 100 Myr. The best-fit t values of t < 100 Myr imply these galaxies might be caught during their star-bursting event or during their very initial phase of formation. By excluding these starburst-candidate (i.e., galaxies with very small best-fit ts), we focus on the stellar-mass–SFR correlations of continuously star-forming galaxies (i.e., star formation main sequence galaxies) because 100 Myr is a longer timescale than the typical dynamical time of high-redshift galaxies (e.g., Wuyts et al. 2011). The fractions of these starburst-candidates are 13% among U dropouts, 15% among B dropouts, and 16% among V dropouts when we fit the observed SEDs with increasing SFHs. The fractions are slightly higher in the case of SED fitting with declining SFHs: 13% for U dropouts, 19% for B dropouts, and 23% for V dropouts. We discuss, in more detail, about these galaxies with the very small best-fit ts in the following section.

In Figure 3, we compare the correlation between stellar masses and the SFRs of the observed LBGs to the predictions from SAM. The first two columns are the mass–SFR correlation of GOODS-S LBGs, while the right column shows the same correlation for SAM LBGs. The left column shows this correlation of observed LBGs when the stellar masses and SFRs are estimated through SED fitting with exponentially declining SFHs assumed, while the middle column shows the results for the same observed LBGs, but from the SED fitting with increasing SFHs. In the left and middle columns, contours are drawn from the results of 1000 Monte-Carlo runs of SED fitting. In each Monte-Carlo run, we first scatter the measured flux by the amount of random error. We assume the Gaussian distribution of random error with the measured flux error for each galaxy as the σ in the Gaussian function. We further scatter the flux assuming 5% of systematic error. For the flux uncertainty, we sum the measured flux error and 5% of the measured flux in quadrature. The SFR is the average SFR over past 100 Myr (i.e., the mass of stars formed in the past 100 million year divided by the same time interval) throughout this paper. It is clear from this figure that the stellar masses and the SFRs of GOODS-S LBGs show a tight positive correlation only when these properties are estimated through SED fitting with the rising SFH assumption. When declining SFHs assumed, the SFRs are underestimated, leading to the larger scatter and lower normalization in M_*–SFR relation as shown in the left column of this figure. This underestimation of SFR with declining SFH assumption is consistent with the results of Reddy et al. (2012)—left panel of their Figure 4. The stellar population properties—such as stellar mass, SFR, and age—derived from SED fitting vary depending on the assumed form of SFH because different SFHs result in different mass-to-light ratios (or the relative fraction of stellar masses in old and young stellar populations). Exponentially declining SFHs assume that SFR was always higher in the past, so the resulting mass-to-light ratio is generally higher than the case of increasing SFHs, unless there is huge difference in ages. Therefore, the expected trend when declining SFHs are assumed is that the derived SFR values are smaller if the derived values of stellar mass are similar, or the derived stellar masses are larger if the derived SFRs are similar—resulting in lower values of SSFR in either case, compared to the results with increasing SFHs.

As shown in the right column of this figure, the SAM also predicts a tight, positive correlation between the stellar mass and the SFR of LBGs. If we fit the SEDs of mock LBGs assuming rising SFHs, we recover similarly tight correlations. As shown in L09 and L10, if the SEDs of mock LBGs are fitted with exponentially declining SFHs, the SFR values are significantly underestimated and the amount of underestimation is nearly independent of stellar mass. Therefore, the M_*–SFR correlations would differ significantly from the intrinsic ones with larger scatter and lower normalization (similar behavior shown in the left column of Figure 3). While not conclusive, the similarity between the second and third columns implies that galaxies, on average, have rising SFHs along the star-forming main sequence at z ⩾ 3. In this figure, we can see the relative dearth of massive galaxies in SAM compared to the observation for high-redshift galaxies, as also noticed in Fontanot et al. (2009), Marchesini et al. (2009), and Santini et al. (2012). However, there are only a few galaxies in the high-mass ends, and this discrepancy between model and observation can arise due to the stellar mass errors that can scatter some galaxies to higher masses.

In the case of SED fitting with the rising SFHs, we estimate the best-fit power-law correlations between the stellar masses and the SFRs at different redshifts, parameterized as log (SFR) = a × log (M_*) + b. First, we find the best-fit correlation by varying both of a and b (shown as red line in each panel of the middle column). The best-fit values of (a, b) are (0.77, −6.30) for U dropouts, (0.71, −5.74) for B dropouts, and (0.72, −5.81) for V dropouts. The scatters from these best-fit correlations, measured as the standard deviation of the distances from these correlations, are 0.22, 0.23 and 0.19 dex for U, B, and V dropouts. From this best-fit estimation, we can see that the slope a is close to unity. This means that the SSFR—defined as the SFR per unit stellar mass (SFR/M_*)—at each redshift is nearly constant over the entire range of stellar mass. This strong dependence of galaxies' star formation activity on their stellar mass might be the manifestation of the correlation between the star formation activity and the underling dark matter halo mass.

Next, we estimate the best-fit correlation, by fixing the slope a to be unity (shown as black line in each panel of the middle column). In this case, the best-fit y-intercept (in log–log space) b for each dropout sample is −8.49 (U dropouts), −8.50 (B dropouts), and −8.49 (V dropouts). Our result at z ∼ 3.8 (B dropouts) is in a good agreement with Lee et al. (2011), where the average SFR is derived from rest-frame UV luminosity of stacked photometry for z ∼ 3.7 galaxies.

When investigating the correlation between stellar masses and SFRs of 3 ⩽ z ⩽ 5 LBGs, we exclude the LBGs with small best-fit ts (t < 100 Myr). These small best-fit t values (<100 Myr) can be spurious due to the effects of photometric scatter (e.g., Sawicki 2012). In that case, the dust reddening values (which are subject to the degeneracy with age) as well as the SFR values (which are affected by the estimation of dust reddening) of these galaxies from SED fitting are also unreliable. For this reason, we exclude these galaxies in the analysis of the correlation between stellar masses and SFRs.

These galaxies have, on average, smaller stellar masses and higher SFRs than galaxies with t_best ⩾ 100 Myr at a given stellar mass, thus forming an upper envelope in M_*–SFR correlation. Therefore, if we include these galaxies, the normalization of the correlation would increase and the slope would become slightly flatter than reported in the previous section. However, the main conclusions of this work—including the tight correlation between the stellar masses and SFRs, the slow evolution of this correlation, and the increasing SFHs of LBGs—are not affected.

The best-fit ts with t < 100 Myr of these galaxies—if considered as real—imply that they are caught in their very early stage of formation or are experiencing secondary burst events (with certain amount of hidden stellar masses). To see how much stellar mass can be hidden in these galaxies, we fit their SEDs with two components of stellar population.

For both of young and old stellar components, we assume the delayed SFHs given in Equation (10). Because the best-fit ts are smaller than 100 Myr (⩽90 Myr) in the single-component fitting, we allow the values of t_y within 50 Myr ⩽ t_y ⩽ 90 Myr for the young stellar component, and vary the t_o within 500 Myr ⩽ t_o ⩽ t(z) for the old stellar component. Here, t(z) is the age of the universe at given redshift for each galaxy. The parameter τ in Equation (10) is set to be τ = 0.1 Gyr for the young component and 0.5 Gyr for the old component. Because t_y ⩽ 90 Myr, the SFH of the young component is the increasing one with this choice of τ value. We assume that the both components have the same values of metallicity and dust attenuation.

From the results of this two-component fitting, we see how many galaxies can have significant fraction of stellar mass in their old component. Among 110 (U dropouts), 127 (B dropouts), and 19 (V dropouts) galaxies with t < 100 Myr, 18 (16%), 17 (13%), and 3 (16%) galaxies in each dropout sample have their old mass fraction greater than 30%. This implies that majority of these galaxies are in the stages of their initial formation while a small fraction of these galaxies may be experiencing a starburst event with hidden old stellar mass. Definite discrimination between these two possibilities is beyond the scope of this paper.

In previous section, we have shown that the observed GOODS-S LBGs follow a well-defined M_*–SFR correlation with near unity slope in the redshift range of 3 ⩽ z ⩽ 5, when we estimate these quantities with the increasing SFH assumption. This tight correlation is a continued trend from lower redshifts (Daddi et al. 2007; Elbaz et al. 2007; Noeske et al. 2007; Pannella et al. 2009). Renzini (2009) investigated the SFHs of star-forming galaxies from this observed correlation between the stellar masses and the SFRs in the redshift range, z ⩽ 2, based on the results of Pannella et al. (2009). Pannella et al. (2009) found the best-fit evolution of the correlation between the stellar masses and the SFRs of star-forming galaxies (z ≲ 2) as follows,

$$\begin{equation} {\rm SFR} \simeq ({M_{*}}/10^{11}\,M_{\odot })(t/3.4 \times 10^9 {\rm \,yr})^{-2.5}\; ({M_{\odot }}{\rm \,yr}^{-1}). \end{equation} \tag{ 11 }$$

From this evolution of the observed correlation, Renzini (2009) re-parameterized the evolution of the SFR of star-forming galaxies at z ⩽ 2 as,

$$\begin{equation} \frac{{\rm SFR}(t)}{{\rm SFR(2 \,Gyr)}} = 5.66 \times \frac{M(t)}{M({\rm 2 \,Gyr})} \times t^{-2.5}, \end{equation} \tag{ 12 }$$

where t is in billion years. While the normalization of Equation (11) is fixed, we can adjust the normalization, in the case of Equation (12), by changing the values of SFR(2 Gyr) and M_*(2 Gyr).

In this work, we extend the redshift range of the investigation on this correlation between galaxies' stellar masses and SFRs up to z ∼ 5. Based on the results of our work on the evolution of this correlation at redshifts 3 ⩽ z ⩽ 5, we now turn to the analysis of the expected formation histories of our CANDELS/GOODS-S galaxies at this redshift range.

For the LBGs with stellar mass in the range of 10⁹ ⩽ M_*/M_☉ ⩽ 10¹⁰, the mean SSFR is 3.55 ± 2.30 Gyr⁻¹ for U dropouts, and 3.55 ± 2.70 Gyr⁻¹ and 3.92 ± 2.79 Gyr⁻¹ for B and V dropouts, respectively. In Figure 4, we show the mean SSFR values at each redshift as red circles with error bars and the best-fit SSFR evolution as a red solid curve. The error bars show the standard deviation of SSFR. In this figure, we also show the SSFR evolution of model LBGs. Blue squares with error bars show the intrinsic SSFR evolution of our SAM LBGs. Compared to the observed LBGs, our SAM LBGs show steeper SSFR evolution, and higher SSFR at the highest redshift bin (z ∼ 5) by amounts of ∼0.16 dex. Green triangles and error bars show the SSFRs of SAM LBGs, which are derived from the SED fitting with increasing SFHs assumed. The mean SSFR values derived from SED fitting are similar with intrinsic values with slightly larger spread. Here, we do not show how the SSFR values would change if these values are derived from SED fitting with exponentially decaying SFHs. However, it is obvious that the SSFR values would be much smaller if fitted with exponentially declining SFHs considering Figure 3 (left column) as well as the results of L09 and L10. Figure 3 clearly shows that the best-fit SFR values of given stellar mass become much smaller if fitted with declining SFHs. In L09 and L10, we have shown that the SED-derived SFRs of mock LBGs from SAM are much smaller than the intrinsic values if they are derived from SED fitting with declining SFHs.

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Recently, it has been suggested that the IRAC-bands (channels 1 and 2) might be affected at redshift z ≳ 4 due to the effects of nebular emission lines, such as Hα or [O iii], resulting in overestimation of stellar mass and underestimation of SSFR (de Barros et al. 2012; González et al. 2014; Stark et al. 2013). With the inclusion of the effect of nebular lines, González et al. (2014) found modest increases in the SSFR at z ∼ 4 and 5, while the effect is larger at z ∼ 6. This results were acquired when they assumed their maximal emission line model where the equivalent width (EW) of Hα increases with redshift as

$$\begin{equation} {\rm EW}(z) \sim 15.8 \times (1+z)^{1.52} \,{\mathring{\rm{A}}}. \end{equation} \tag{ 13 }$$

Stark et al. (2013) estimated the median excess of 0.27 magnitude at IRAC 3.6 μm for 3.8 < z < 5.0 galaxies, which corresponds to EW of 360–450 Å. From this, they found the average reduction of stellar masses by factors of 1.1 and 1.3 at z ∼ 4 and z ∼ 5, respectively.

We try to estimate and correct the effects of nebular emission lines in estimating SSFRs of our LBGs. First, we repeat the SED-fitting (with rising SFHs) correcting the flux values of the bands which can be affected by the existence of emission lines adopting the results of González et al. (2014; their maximal emission model). We get higher values of mean SSFRs—4.08 ± 3.24 Gyr⁻¹ and 5.17 ± 3.88 Gyr⁻¹ for B and V dropouts, respectively, for the galaxies with 10⁹ ⩽ M_*/M_☉ ⩽ 10¹⁰. We show these values as well as the best-fit SSFR evolution as open black circles and black solid curve, respectively in Figure 4. Then, we repeat the SED fitting again adopting the results of Stark et al. (2013). We show the corresponding results as open purple circles and purple curve in Figure 4. The derived mean SSFRs are 4.59 ± 3.66 Gyr⁻¹ for B dropouts and 5.70 ± 4.13 Gyr⁻¹ for V dropouts. In either case, we get a steeper slope for SSFR evolution which shows good agreement with the model prediction as well as similar SSFR values with the SAM LBGs at z ∼ 5.

If we parameterize the SSFR (with Stark et al. (2013) nebular correction) as SSFR = c × t^r, the best-fit evolution of SSFR (i.e., purple curve in Figure 4) is given as

$$\begin{equation} \log {\rm SSFR} = -8.18 - 0.9 \log t, \end{equation} \tag{ 14 }$$

where t is in Gyr.

From this best fit, the evolution of SSFR within the redshift range of 3 ⩽ z ⩽ 5 is given as

$$\begin{equation} {\rm SFR} = 66.1 \times (M_{*}/10^{10}\,M_{\odot }) (t/1.0 \times 10^{9} \,{\rm yr})^{-0.9}. \end{equation} \tag{ 15 }$$

Equation (15) has a smaller power in the time-dependent term than Equation (11) because the time evolution of SSFR at 3 ⩽ z ⩽ 5 is much slower than at z ⩽ 2. From this best-fit SSFR evolution, we can connect the time evolution of SFR and the time evolution of the stellar mass as

$$\begin{equation} \frac{{\rm SFR}(t)}{{\rm SFR}(z=5)} = (1.43 \times 10^8) \frac{M_{*}(t)}{M_{*}(z=5)} t^{-0.9}, \end{equation} \tag{ 16 }$$

where t is in year.

The growth of stellar mass can be obtained by integrating Equation (16) as follows (by setting SFR = dM_*/dt),

$$\begin{equation} \frac{M_{*}(t)}{M_{*}(z=5)} = \exp (-65.69) \times \exp (8.15 t^{0.1}). \end{equation} \tag{ 17 }$$

In the previous section, we analyze the history of star formation activity and of stellar mass assembly within the redshift range, 3 ≲ z ≲ 5. In this section, we compare our results with previous works in literature.

Papovich et al. (2011) have derived the time evolution of the mean SFR at redshifts, 3 ⩽ z ⩽ 8 by analyzing the high-redshift galaxy samples with constant comoving number density. From this study, they show that the average SFR of high-redshift galaxies increases as a power law with decreasing redshift (with best-fit power α = 1.7).

By combining Equations (16) and (17), we estimate the time evolution of SFR (i.e., the SFH) in the range of 3 ⩽ z ⩽ 5. By parameterizing the evolution of SFR as SFR(t)/SFR(z = 5) = (t/r)^α, we find the best-fit power α = 5.85 with the timescale parameter r = 1.16 Gyr. From this, we can confirm that the mean SFH of our GOODS-S LBGs steadily increases with time from z ≃ 5 to z ≃ 3 during about 1 billion years, which is in qualitative agreement with Papovich et al. (2011). Our derived best-fit power is steeper than the value from Papovich et al. (2011). When we try to find the power using only the data points at z = 3.1, 3.8, and 5.0 from Papovich et al. (2011), we find that the best-fit power is α = 2.9.

Here, we derive the SFH from the best-fit estimation of SSFR evolution. The SFR(t) can also be derived more easily by finding the best-fit correlation between the mean SFR and the cosmic time, t, at each redshift bin. However, the mean SFR values can vary depending on the stellar mass range of sampled galaxies because of the positive correlation between stellar masses and SFRs. The mean SSFR values are less subject to this problem because SFRs are tightly correlated with stellar masses with the slope near unity. Therefore, SFH—i.e., SFR(t)—derived from the evolution of SSFR is more robust, independent of the sampled stellar-mass range.

Next, we compare our best-fit time evolution of SSFR with the one derived by Pannella et al. (2009) (or Renzini 2009). In Figure 5, the blue curve shows the time evolution of SSFR for M_* ∼ 3 × 10¹⁰ M_☉ star-forming galaxies derived by Pannella et al. (2009)—i.e., Equation (11)—up to z = 3. Because Equation (11) is derived from the observational data up to z ⩽ 2.4, we show the SSFR(t) as the solid curve up to z ⩽ 2.4 and as the dotted curve within the redshift range 2.4 ⩽ z ⩽ 3. The time evolution of SSFR of our LBG sample at 3 ⩽ z ⩽ 5 is shown as the red curve in the same figure. The solid and dashed red curves show the SSFR evolution when we adopt the nebular emission correction of Stark et al. (2013) and of González et al. (2014), respectively. The dotted red curve is for the case when we do not apply any correction for the nebular emission. It is clear that the SSFR(t)s of Pannella et al. (2009) and of ours show significant discrepancy at z ∼ 3—SSFR(z = 3) from Pannella et al. (2009) is much larger than our best-fit SSFR(z = 3).

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For comparison, we show the mean SSFR value of z ∼ 4 LBGs (B dropouts) from Daddi et al. (2009) as the black circle, which shows a relatively good agreement with our results. The mean SSFR of z ∼ 4 submillimeter galaxies (SMGs) from the same authors is also shown in this figure as the red square. It should be noted that our estimation of SSFRs at z ⩾ 4 is higher than previous results in literature. Purple diamond and blue triangles show the SSFR values between z ∼ 4–7 from González et al. (2010). The SSFRs at z ∼ 4, 5, and 6 are estimated by authors from the results of Stark et al. (2009). Besides the lower values of SSFR, previous results also show a plateau between z ∼ 2–7, which is contradictory to the theoretical prediction from SAMs. Our results, on the other hand, indicate that the SSFR keeps increasing above z ∼ 2, which is in good agreement with the prediction of our SAM, which is shown as the green curve. Our improved agreement between observation and model prediction is encouraging because our result is based on (1) the consistent estimation of SFRs and stellar masses of our LBGs from the same SED fitting instead of estimating these quantities separately, and (2) appropriate assumption about the SFHs of high-redshift LBGs—i.e., increasing SFHs.

Recently, Bouwens et al. (2012) reported higher dust extinction at 4 ≲ z ≲ 7 based on their new measurement of the UV-continuum slope, β. This leads to higher values of SSFR in this redshift range, shown as the black triangles (Figure 5). Even though the values are higher, the SSFR values still show little evolution with redshift.

As mentioned above, de Barros et al. (2012), González et al. (2014), and Stark et al. (2013) have included the effects of nebular emission in the estimation of SSFR through SED fitting, and have reported higher SSFR values than previous results (for example González et al. 2010), as well as increasing SSFRs with redshift in qualitative agreement with our results or the prediction from our SAM as shown in Figure 5.