MOIRCS DEEP SURVEY. VI. NEAR-INFRARED SPECTROSCOPY OF K-SELECTED STAR-FORMING GALAXIES AT z ∼ 2^*

In order to select star-forming galaxies at z ∼ 2, we apply the BzK color criterion for star-forming galaxies (sBzK; BzK ≡ (z − K) − (B − z)> − 0.2; Daddi et al. 2004) to the K-selected galaxies with K_s < 23 from the MODS catalog, whose limiting magnitude is K_s = 24.9 (5σ, ϕ1.2 arcsec). The BzK diagram of the extracted galaxies is shown in Figure 1. B- and z-band magnitudes are measured on the F435W and F850LP images of the Advanced Camera for Surveys (ACS) on board the Hubble Space Telescope (HST) in the course of the GOODS Treasury Program (ver. 1.0; Giavalisco et al. 2004). In total, 508 galaxies are identified as candidates.

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We chose 37 sBzK galaxies from the candidates for the spectroscopic observation. The coordinates and photometry of the target galaxies are listed in Table 1. Although two of the sample galaxies (MODS12-0125 and MODS41-0194) are not detected in the B band, we include these galaxies because their z − K colors suggest that they are sBzK or passive BzK (pBzK; $BzK > -0.2 \bigcap z-K>2.5$), and their photometric redshifts are consistent with the redshift range of BzK galaxies. We did not select on the basis of spectroscopic redshift, although spectroscopic redshifts were previously known for some galaxies in the present sample (Table 1).

To preferentially select actively star-forming galaxies, we put higher priority on the galaxies identified in the publicly available Spitzer MIPS source catalog (the first delivery of the Spitzer data catalog from the GOODS Legacy Project; R. Chary et al. 2010, in preparation. The flux limit of the catalog (S₂₄ > 80 μJy) corresponds to L_TIR ∼ 4.1 × 10¹¹–3.6 × 10¹² L_☉ at z = 1.4–2.5 using the infrared SED templates by Chary & Elbaz (2001); thus the galaxies in the MIPS source catalog are as luminous as (ultra-)luminous infrared galaxies. MIPS sources in the public catalog are identified with 127 of the sample galaxies (we call them sBzK-MIPS galaxies, hereafter). We selected 18 sBzK-MIPS galaxies. Furthermore, we added 19 sBzK galaxies not listed in the MIPS public catalog to the list of targets (sBzK-non-MIPS galaxies, hereafter).

It is known that the sBzK criterion picks out various types of star-forming galaxies and passively evolving galaxies at 1.4 ⩽ z ⩽ 2.5 (Grazian et al. 2007). In order to check that our sample represents the sBzK population, we compare the B − K color distribution of our target galaxies to that of sBzK-non-MIPS and sBzK-MIPS galaxies with K_s < 23 in the MODS field (Figure 2). The B − K color reflects both the age and the amount of dust of a galaxy at this redshift range. Our sBzK-MIPS sample is reasonably representative of those in the field, according to both the χ²-test (64.65%) and the Kolmorov–Smirnov (KS) test (92.46%). On the other hand, the sBzK-non-MIPS sample is possibly biased toward blue galaxies (χ²-test: 35.35%, KS test: 17.74%). Red sBzK-non-MIPS galaxies are relatively rare, so that the scaled number in the redder bin is too small (≲1) to provide robust statistics given the present sample size.

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We utilize publicly available multi-wavelength data of GOODS-N to exclude possible active galactic nucleus (AGN) which may contaminate the optical star formation emission. The hard X-ray luminosity at the detection limit of the Chandra Deep Field-North 2 Ms Point-Source Catalogs (Alexander et al. 2003) at z ∼ 2 (1 × 10⁴² erg s⁻¹), suggests that the emission lines of the hard X-ray detected sBzK galaxies are powered by an AGN rather than massive stars (Alexander et al. 2002). X-ray sources are identified with 62 of the 508 sBzK galaxies. We exclude these sources from the present sample. Rest-frame infrared colors are often used to identify AGN candidates using Spitzer IRAC bands (3.6 μm, 4.5 μm, 5.8 μm, and 8.0 μm) (e.g., Caputi et al. 2006a; Hayashi et al. 2009). A galaxy with emission dominated by the stellar population has a strong bump in the SED at 1.6 μm (Sawicki 2002). We confirmed that all of the present non-X-ray galaxies have strong 1.6 μm bumps in the SED of the IRAC bands.

The spectroscopic observations for the selected galaxies were performed with MOIRCS (Suzuki et al. 2008) attached to the Subaru Telescope (Iye et al. 2004) during 2007 March 24–27 in the course of the near-infrared spectroscopic observation of X-ray sources (S07A-083, PI: M. Akiyama). We used MOIRCS in MOS mode with the low-resolution grism (HK500), which covers 1.3–2.3 μm wavelengths. The spectral resolution is R ∼ 500 (Δv ∼ 600 km s⁻¹) with 08 slit width. The 37 galaxies were observed with four slit masks as shown in Table 2. One slit mask was used for each night. The total exposure time was 2.5–5 hr for each mask. The slit width was fixed to 08 for targets. Most of the slit lengths were 100 or longer, depending on target positions. To monitor the flux variation due to the atmospheric absorption and photon loss at the slit mask, a bright point source was observed simultaneously on each mask, except for mask CDFN3.

The FWHMs of seeing size in the K_s band during the observations were typically ∼06. According to the atmospheric attenuation archived in the Canada–France–Hawaii Telescope Web site,⁹ the conditions were photometric for the first three nights. For the fourth night, thin cirrus covered the sky. The observation log is summarized in Table 2.

The observation followed the standard procedure for MOS observation with MOIRCS described in Tokoku (2007). The target acquisition was done using 6–7 alignment stars distributed over each slit mask. The telescope was dithered along the slit (we call it an "AB pattern") by 30, to achieve sufficient sky subtraction. The detector was read eight times in non-destructive readout mode to reduce the readout noise down to 16 e⁻. The exposure time was set so that most of the sky emission lines, which were used later for the wavelength calibration, were not saturated. The typical exposure times were 300–900 s. The relative position of the alignment star against the slit mask was checked, and then the telescope pointing was adjusted every hour or two without removing the slit masks from the focal plane. The systematic guide error in 1 hr was measured to be ≲04. At the beginning of each observing night, an A0V type star (HIP53735; K_s = 8.856; Two Micron All Sky Survey point-source catalog; Skrutskie et al. 2006) was observed for flux calibration with the same instrument configuration as the targets.

The data reduction was made, using a semi-automatic reduction script (MOIRCS MOS Data Pipelines; MCSMDP¹⁰), which follows the standard procedure of IRAF¹¹:ONEDSPEC. At first, we make a map of bad pixels; bad pixels are identified as those that show nonlinearity. This includes dead (always zero count) and hot (always high count) pixels. Cosmic rays are detected by a moving block average filter using IRAF:CRAVERAGE. Each input frame for IRAF:CRAVERAGE is first divided by its dithered pair so that cosmic rays stand out behind the sky background. The cosmic-ray mask was combined with the bad pixel mask. The values of bad pixels and cosmic-ray pixels are replaced with values interpolated linearly from the pixels along the slit. Sky emission lines were subtracted using the dithered pair frames (we call the procedure "A − B subtraction"). The dome-flat frames for flat fielding were taken with the same slit mask and grism configuration as used for the observation. For the first-order distortion correction, the database for the imaging mode of MOIRCS was applied to the spectral frames. The tilt of the spectrum against the detector rows was corrected in the cases when it was measurable in the continuum of bright objects.

Next, each spectrum was cut out based on a mask design file. For wavelength calibration, OH sky emission lines were identified in each frame using a night-sky spectral atlas (Rousselot et al. 2000). Using the pixel coordinates of the sky emission lines, the sky-subtracted frames were transformed such that the sky lines aligned along the column of the frames and that the wavelength was a linear function of pixel coordinates. Background emission still remained after A − B subtraction due to time variation of OH emission during dithering observations. The residuals were subtracted by fitting with a quadratic equation along the column. Finally, all frames were weighted with the flux of the bright point source and stacked using the weighted average to obtain the final two-dimensional spectra.

The response spectra, including atmospheric transmission and the instrument efficiency, are calculated using the flux standard star to convert the observed count to an absolute flux. The data of the flux standard star are reduced to a two-dimensional spectrum in the same way for galaxy targets and then the continuum is traced with a polynomial to extract a one-dimensional spectrum with IRAF:APALL. We calculate a model spectrum of the A0V flux standard star using SPECTRUM¹² (SPECTRUM uses Castelli & Kurucz 2004 model). The model was scaled to K_s = 8.856 and the spectral resolution of the model was degraded to R ∼ 500 with a Gaussian kernel, so as to give the same hydrogen line width as the observed spectra. Finally, the one-dimensional spectrum of the standard star was divided by the model spectrum to obtain the response spectrum.

Emission lines are detected by visual inspection on the flux-calibrated two-dimensional spectra. The signal-to-noise ratio (S/N) evaluation described later shows that visual inspection corresponds to a larger than ∼3σ detection. The detected emission lines were integrated along the slit direction within an appropriate aperture giving the largest S/N to obtain one-dimensional spectra.

We first consider the strongest emission line as Hα. Taking account of the wavelength range where we observed and the expected redshift range of sBzK galaxies, it is the most plausible candidate for a strong emission line. The identification of Hα is confirmed in all cases when other emission lines, such as [N ii]λλ6548,6583, [O iii]λλ4959,5007, and Hβ are also detected at the expected wavelength. Otherwise, we check the consistency of the redshift of the Hα emission line with that of the SED model fitting using broadband data (see Section 3.4).

Although one of the conventional methods for estimating random error is to obtain the rms of the scattering of background counts at both sides of the object, the slit length for each object is not long enough. The readout noise is suppressed down to 16 e⁻ by non-destructive readout mode, so that the readout noise is about 1/5 of the background noise. Because the readout noise and the photon noise of objects are negligible, we estimated the background noise by assuming that the random error is dominated by sky-background noise, which follows Poisson statistics. In parallel to the reduction of the object spectra, sky-background spectra were reduced without the background subtraction, and the Poisson error of each pixel is estimated from them. We found that the Poisson error is systematically larger than the rms error by a factor of ∼1.5, though the scatter is large. The rms errors can be underestimated, because counts in the two-dimensional spectra correlate with the neighboring pixels due to the sub-pixel shift in distortion correction and wavelength calibration.

To measure the flux and central wavelength of the emission lines, we used SPECFIT (Kriss 1994), which fits multiple emission lines and continuum simultaneously. Gaussian profiles were used for emission lines. The continuum was assumed to be a linear function of wavelength. The flux of [N ii]λ6548 was fixed to be 1/3 of [N ii]λ6583 flux. If Hβ and/or [O iii]λλ4959, 5007 emission lines were detected on the two-dimensional spectra, we measured the flux in the same way. The random errors of the fluxes were estimated from the best-fit parameters of the model fitting based on the sky-background error described above. The emission lines detected with S/N > 3 are listed in Table 1, and the spectra are shown in Figures 3 and 4. The measured flux and the redshift derived from the central wavelength of Hα emission line are shown in Table 3.

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It is difficult to accurately evaluate systematic flux error since part of the flux from the object will be lost by a slit mask; the systematic error due to the flux loss is not negligible if we measure the total flux of emission lines from slit spectroscopy. The flux loss can be caused by a variety of things, for example, a larger size of the object than the slit width, imperfect alignment of the slit mask to the target, and misalignment of the dithering direction along the slit.

First, we estimated the flux loss for a point source. When a Gaussian profile with FWHM = 06, which is the typical seeing during the observations, and a slit width = 08 are assumed, fluxes passing through the slit are 88.4% of the integrated flux; the aperture correction factor is ∼1.13. We applied this factor to the point sources observed. We found that, however, the fluxes of the point sources were still underestimated by up to ∼20%, even if we also correct the effect of the seeing size variation and pointing error, which are measured from alignment stars. This fact indicates that imperfect alignment of dithering and/or short-time variation of the seeing size during the observations can also add to the systematic error.

On the other hand, it is expected that the flux loss for extended sources varies with the size of the sources. We calculated aperture correction factors from the ratio of total flux to that expected to come in through slits, assuming that the regions emitting stellar continuum and Hα had the same spatial distribution. The flux through a slit was measured on the K_s-band image with a rectangle of the same aperture size as for extracting the one-dimensional spectrum and the same position angle of the slit, while the total flux was obtained from MAG_AUTO of SEXTRACTOR. The alignment error of the slit was estimated from the alignment stars on the slit mask. We found the median of the aperture correction factors to be ∼1.9 with little dependence on the size of the objects. The rms scatter of the factor among the objects is ±0.5. We apply the median value to all objects to correct the flux loss. We checked the above aperture correction method by comparing the continuum flux derived from the corrected spectrum with total magnitude, which was measured from K_s- or H-band imaging data according to the redshift of each galaxy. We used seven galaxies in the sample whose stellar continuum around the Hα emission line was detected with sufficient S/N. We applied the aperture correction to the continuum flux, then contrasted the corrected flux with the total magnitude. The comparison shows that both results are consistent and the uncertainty in the aperture correction is within ∼20%.

We need to note that the assumptions on the light distribution of the Hα emission line would also cause systematic errors. The Hα and continuum emitting regions would not always be similar for star-forming galaxies at z ∼ 2. Studies with the integral-field spectrograph show that Hα emission does not trace the stellar continuum in some galaxies (Swinbank et al. 2005, 2006; Kriek et al. 2007). However, this kind of check on our slit-spectroscopy sample is difficult, because both the continuum and emission lines are so faint that the light distribution along the slit is not able to be measured. As a lower limit of the aperture correction, when the Hα emitting region is point-like, the flux becomes smaller by ∼40%.

The flux ratios of rest-optical emission lines such as [N ii]λ6583/Hα, [O iii]λ5007/Hβ, and Hα/Hβ provide us information on physical properties of ionized regions such as the energy sources for ionization ([N ii]/Hα and [O iii]/Hβ; Baldwin et al. 1981; Veilleux & Osterbrock 1987; Kewley et al. 2001; Kauffmann et al. 2003), the amount of dust extinction (Balmer decrement, e.g., Hα/Hβ; Osterbrock 1989), and gas-phase metallicity ([N ii]/Hα; Storchi-Bergmann et al. 1994; Raimann et al. 2000; Denicoló et al. 2002; Pettini & Pagel 2004). However, except for Hα, the emission lines of the present sample are so faint that we cannot evaluate the line ratios of an individual galaxy. Therefore, we stacked the calibrated spectra in the [N ii]–Hα, and [O iii]–Hβ wavelength ranges to obtain high S/N spectra, and discuss the averaged physical properties.

The stacked spectrum $\bar{F_{\lambda _e}}$ was calculated with

$$\begin{equation} \bar{F_{\lambda _e}} = \sum \limits _n \frac{F_{{\rm H}\alpha, 0}}{F_{{\rm H}\alpha, n}} F_{\lambda _e,n}(\lambda _e), \end{equation} \tag{ 1 }$$

where $F_{\lambda _e,n}(\lambda _e)$ and F_Hα,n are the spectrum and the total Hα flux of nth object, respectively. The spectra were re-sampled in the rest-frame wavelength scale (λ_e). The fluxes were scaled with total Hα flux, though the variety of fluxes among the samples are not large. Because our main purpose of stacking is to examine the average physical properties of galaxies from their line ratio, weighted mean (by either inverse square of noise or luminosity) is not applied; the line ratios of weighted mean flux could be biased toward strong emission-line galaxies.

We selected the objects whose spectra covered the wavelengths of redshifted Hα, [N ii], Hβ, and [O iii]. The emission lines of the objects which had at least Hβ or [O iii]  with S/N >3 were stacked for the measurements of the line ratios of [N ii]/Hα and [O iii]/Hβ. As a result, nine galaxies were used for the stacking. The stacked Hα–[N ii]  and Hβ–[O iii]  spectra are shown in Figure 5. The emission lines of the stacked spectra were fitted with a combination of the same Gaussian profiles and linear continuum measurements used in Section 3.2. The best-fit spectra are also shown in Figure 5. We also made stacked spectra including the objects with neither Hβ nor [O iii]  detection, but the resultant line ratio was not changed.

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To evaluate the effect of galaxies which would happen to have a significantly different line ratio from the others, we used a bootstrap method. We built 1000 bootstrap sets, whose galaxies were randomly selected, allowing duplicate sampling from all galaxies used in the stacking, and stacked the spectra of the galaxies in each set to derive the line ratios. We regard the 16% and 84% quantile of the distribution as 1σ uncertainty in line ratio.

To evaluate the properties of stellar population such as stellar mass, amount of dust extinction (A_V), and age, we use multiband photometric data from the MODS JHK_s images and publicly available images. The publicly available images we use are U(KPNO/MOSAIC; Capak et al. 2004), BViz (ACS/HST; GOODS Treasury Program ver. 1.0; Giavalisco et al. 2004), 3.6 μm, and 4.5 μm (IRAC/Spitzer; DR2 of the Spitzer Legacy Science program for GOODS; M. Dickinson et al. 2010, in preparation). These images are deep enough to detect our galaxies with sufficient S/N and to obtain the properties of stellar population. The flux of the galaxies was obtained with S/N = 8 for >80% of the objects in all bands, except for B (60%) and U (24%) bands. The aperture size for photometry is set to 15, which is approximately 2 × FWHM of the point-spread function (PSF) size in K_s band. Aperture corrections were applied to each band properly (see Kajisawa et al. 2009). The contribution of Hα emission to the flux in K_s or H band, which depends on redshift, was subtracted. The corrections are 0.01–0.34 mag with a median of ∼0.08 mag.

We fitted the photometry data with stellar population SED models. The SED libraries of GALAXEV (Bruzual & Charlot 2003) are used for the fitting. We adopted the libraries with metallicities of 1.0 Z_☉, 0.4 Z_☉, and 0.2 Z_☉ and a Chabrier (2003) initial mass function (IMF). Exponentially declining star formation histories (τ-model) are assumed: SFR ∝e^−t/τ with τ = 0.01, 0.02, 0.05, 0.10, 0.20, 0.50, 1.0, 2.0, and 5.0 Gyr. The starburst attenuation law of Calzetti et al. (2000) was applied to the model with 0.1 steps from A_V = 0.0 to 3.0.

Estimations of the stellar mass are affected by the adopted IMF. Many authors employ a Salpeter (1955) IMF, which is known to overestimate stellar mass due to the steep faint-end slope of the IMF. In addition to the Chabrier (2003) IMF, we fitted the SEDs to the model with the Salpeter (1955) IMF and measured stellar mass for comparison. For our sample, the stellar mass with the Salpeter (1955) IMF is larger by a factor of ∼1.7 than that with the Chabrier (2003) IMF.

Recent evolutionary population synthesis models, for example, Maraston (2005), which take account of contributions of thermal-pulsating asymptotic giant branch stars to SEDs, result in changes in the inferred SED properties. We also fitted the SEDs with the Maraston (2005) model and found that the model made the stellar masses smaller by a factor of ∼1.6.

HYPERZ (Bolzonella et al. 2000) was used to determine best-fit SED models. The redshift for the fitting was fixed to that derived from the Hα emission line. We derived total stellar masses of galaxies, scaling the stellar masses and SEDs with the total magnitude (MAG_AUTO) of the K_s band. The parameter matrix, which gives a χ² value for each grid of SED models, was used to evaluate the uncertainty in the model parameters. The uncertainty is obtained from the probability distributions of the parameters. For example, the probability distribution as a function of stellar mass (P(M_⋆)) was calculated from a minimum χ² value for each stellar mass (χ²(M_⋆)). We determined a 68% confidence level of the distribution as 1σ error of the parameter. The stellar population parameters and the 1σ errors derived from the SED fitting are listed in Table 4. Some examples of the best-fit SEDs and the probability distributions of stellar mass, A_V, and age are shown in Figure 6.

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Hα emission lines are detected from 23 out of the 37 galaxies (62%). The detection rate does not depend on the mid-IR flux. The rate for sBzK-MIPS and sBzK-non-MIPS galaxies are 11/18 (61%) and 12/19 (63%), respectively.

The spectroscopic redshifts are consistent with their photometric redshifts in the MODS catalog (Kajisawa et al. 2009); the median value of the relative errors is <dz > ∼ −0.026 with a scatter of 0.035, where dz ≡ (z_phot − z_Hα)/(1 + z_Hα). The spectroscopic redshifts are newly determined for 10 galaxies. Out of the remaining 13 galaxies with spectroscopic redshifts in the literature (Cohen 2001; Cohen et al. 2000; Dawson et al. 2001; Wirth et al. 2004; Cowie et al. 2004; Treu et al. 2005; Chapman et al. 2005; Reddy et al. 2006a; Barger et al. 2008), two galaxies have slightly different spectroscopic redshift. MODS32-0116 has z_Hα = 1.488 (z_literature = 1.315) and MODS21-2321 has z_Hα = 1.734 (z_literature = 1.879).

The left panel of Figure 7 shows the apparent Hα luminosity and the redshift distribution of the Hα-detected sBzK galaxies. The galaxies lie at redshifts between 1.49 and 2.49 with a median of 2.12. The redshift distribution is consistent with that expected for BzK galaxies (1.4 ≲ z ≲ 2.5; Daddi et al. 2004). The lack of galaxies at 1.8 < z < 2.0 is due to an atmospheric absorption band between H and K bands.

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The luminosity distances of these objects are calculated with their redshifts measured using the Hα emission lines. The fluxes of Hα emission lines (F_Hα) range between 1 and 17 ×10⁻¹⁷ erg s⁻¹ cm⁻², which corresponds to an Hα luminosity (L_Hα) of 0.5–7.0 ×10⁴² erg s⁻¹. F_Hα and L_Hα are listed in Table 3.

Possible reasons for non-detections of emission lines from the remaining 14 galaxies are (1) their Hα emission lines are too faint for the present observations, (2) the redshift of those galaxies are out of the 1.2 ≲ z ≲ 2.5 range, where the grism (HK500) we used covers the Hα wavelength, and (3) the galaxies lie at the redshift where Hα emission line falls on strong sky emission lines or atmospheric absorption bands. Because the photometric redshifts and the errors of the galaxies with emission lines undetected are at 1.2 < z < 2.5, the second possibility is unlikely except for MODS21-2047, whose z_literature is 0.999. In order to check the third possibility, we calculate the detection limit of Hα emission as a function of redshift using the sky-background noise and the response spectrum. A Gaussian profile is assumed for Hα with FWHM = 540 km s⁻¹, which corresponds to the wavelength resolution of the HK500 grism with a 08 slit width. Provided that Hα emitters have a uniform redshift distribution at 1.4 ⩽ z ⩽ 2.5 and an apparent Hα luminosity of 1.8 × 10⁴² erg s⁻¹, which corresponds to the 84% quantiles of the distribution of apparent Hα luminosity, the expected detection rate of the Hα emission line with S/N>4 is 87.7%, which corresponds to ∼32 objects: objects at 1.8 ≲ z ≲ 2.0 are rarely detected. Therefore, for five of the observed samples, Hα emission lines are undetected due to the strong sky emission lines or atmospheric absorption bands, while the remaining nine galaxies have smaller SFR and/or larger attenuation than the galaxies with emission-line detection.

Diagnostic diagrams based on optical emission-line ratios are commonly used to determine the energy source of ionization. Possible sources are massive stars, AGNs, and energetic shocks (Baldwin et al. 1981; Veilleux & Osterbrock 1987). In this paper, we use Hα luminosity as an indicator of SFR, assuming that its energy source is massive stars. We excluded AGN candidates from the sample using multi-wavelength data (see Section 2.1). In addition to these diagnostics, we use optical emission-line ratios to determine the energy sources of the present emission-line objects.

In Figure 8, we plotted [O iii]λ5007/Hβ and [N ii]λ6583/Hα line ratios of the stacked spectra and the galaxy with the four emission lines detected (MODS31-0033). Results for ∼347,000 galaxies from Sloan Digital Sky Survey (SDSS) DR6 (Adelman-McCarthy et al. 2008) and local emission-line galaxies (Moustakas & Kennicutt 2006) are also shown. A classification line empirically determined from the SDSS objects (Kauffmann et al. 2003) and a theoretically determined classification line (Kewley et al. 2001) are also shown. Emission-line galaxies located below and to the left of the lines are expected to have massive stars for their energy sources. The distribution of local SDSS galaxies exhibits the two expected sequences: the populations below (star-forming galaxies) and above (AGNs) the solid line.

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The emission line of the stacked sample places it on the theoretical line, while MODS31-0033 is located to the lower left of the empirical line. Our results are located above the sequence of local star-forming galaxies, closer to the AGN sequence. In fact, a similar offset is seen in star-forming galaxies at z ∼ 2 (Erb et al. 2006a; Kriek et al. 2007). One possible reason for the offset is a higher electron density and a harder ionizing spectrum than for local star-forming galaxies (Erb et al. 2006a).

We note that AGNs can be also identified based on [O iii]–Hα line ratios for objects without an Hβ detection. AGNs typically have [O iii]/Hβ ≳3 on Figure 8. On the other hand, since the Hα/Hβ line ratio must be ∼3 (the Balmer decrement) or larger (if there is a significant dust extinction), the [O iii]–Hα line ratio of an AGN candidate will be larger than 1. Only one object (MODS41-0194) has [O iii]–Hα>1 (1.67). When we exclude MODS41-0194 from the stacking, the stacked line ratios slightly move below the empirical line (Figure 8). We regard this object as a possible candidate of AGN in the later discussion.

In order to evaluate SFR from intrinsic Hα luminosity, the extinction due to interstellar dust needs to be corrected. Moreover, the samples are expected to have large amounts of dust, because about half of the present galaxies are luminous infrared galaxies, in which strong infrared emission is thought to originate from dust thermal emission.

The extinction of emission lines in star-forming regions can be estimated with the reddening of the Balmer decrement through an extinction curve. The intrinsic line ratios of the Balmer emission lines are nearly constant with respect to electron temperature of ionized gas. The amount of reddening is estimated from the ratio of the attenuated-to-intrinsic Balmer line ratio,

$$\begin{equation} E(B-V)_{\rm gas} = \frac{2.5}{k(\lambda _{{\rm H}\alpha }) - k(\lambda _{{\rm H}\beta })} \log \frac{F_i^{\lambda _{{\rm H}\alpha }}/F_i^{\lambda _{{\rm H}\beta }}}{F_o^{\lambda _{{\rm H}\alpha }}/F_o^{\lambda _{{\rm H}\beta }}}, \end{equation} \tag{ 2 }$$

where "i" and "o" denote intrinsic and observed flux, respectively, and k(λ) is an extinction curve. We assume Calzetti et al. (2000) law for k(λ).

The Balmer decrement method can only be applied to the 14 galaxies at z ≳ 1.9 in the present sample, because the Hβ wavelength is not covered by the HK500 grism for galaxies at lower redshift. Both Hα and Hβ are detected in four galaxies (MODS11-0390, MODS12-0255, MODS22-1411, and MODS31-0033), so that E(B − V)_gas is estimated using the above equation. For the rest of the sample, lower limits on the E(B − V)_gas are evaluated using Hβ emission-line upper limits. The resultant E(B − V)_gas values are ≳0.5. We also used the stacked spectrum to measure the average amount of reddening and found it to be E(B − V)_gas = 0.75^+1.19_−0.02.

We estimate the amount of reddening for the ionized gas from the SED fitting and apply these extinction corrections to the Hα luminosities for all of the galaxies. Figure 9 shows the comparison between the amounts of reddening derived from the Balmer decrement and SED fitting. Our result shows that the gas is more attenuated than the stars. From SED fitting, the amount of reddening for stars in the samples used for stacking is found to be E(B − V)_stellar = 0.20^+0.24_−0.08, thus E(B − V)_stellar/E(B − V)_gas ∼ 0.26^+0.42_−0.10. It should be noted that the galaxies which we used to determine the relation suffer relatively less attenuation since we would not be able to detect Hβ emission in the more highly attenuated galaxies. The hatched region in the figure also shows the lower limit of E(B − V)_gas corresponding to the 3σ detection limit of Hβ. The height of the region is calculated from the 68% distribution of the observed Hα fluxes.

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From studies of local starburst galaxies, it is known that ionized gas is more attenuated than stars (Calzetti et al. 2000; Calzetti 2001). The difference is thought to be caused by changes in the geometry of the ionized gas and stars in a typical galaxy. Calzetti (2001) give the relation between the color excess observed for gas and stars:

$$\begin{equation} E(B-V)_{{\rm stellar}} = 0.44 E(B-V)_{{\rm gas}}. \end{equation} \tag{ 3 }$$

This relation is shown with the dashed line in Figure 9. Our stacked result is consistent with the equation within errors. On the other hand, Hβ detected galaxies are also consistent with the case that extinction for gas and stars is equal (defined as "equal extinction," hereafter), though these galaxies would lie at the lower end of reddening as noted above. We, therefore, employ Equation (3) as an averaged nature of dust extinction to correct for the effect of dust extinction on the emission lines and compare these with that calculated assuming equal extinction. We discuss the reliability of the application of these extinction corrections further in Section 5.1.

The right panel of Figure 7 shows the luminosity of the Hα emission lines of the galaxies after the extinction is corrected. The extinction correction factor of our samples ranges from 1.2 to 87, resulting in a wider spread in extinction-corrected Hα luminosities (∼0.6 dex for the Calzetti 2001 relation and ∼0.4 dex for equal extinction) than in the apparent Hα luminosities (∼0.3 dex). This result indicates that proper extinction correction is important to reliably calculate the intrinsic Hα luminosity.

To convert the extinction-corrected Hα luminosity to SFR, we derive a relation between them using stellar population synthesis models, following Kennicutt (1998). We employ GALAXEV models (Bruzual & Charlot 2003) with the Chabrier (2003) IMF and the same parameters as used in Section 3 to calculate ionizing photons (N_Ly). N_Ly is converted to L(Hα) based on a case-B recombination model (Osterbrock 1989). If a constant SFR model and solar metallicity are assumed, the L(Hα)–SFR relation with an age older than 20 Myr is given by

$$\begin{equation} {\rm SFR}({\rm H}\alpha)\; ({M}_\odot \;{\rm yr}^{-1}) = 4.9 \times 10^{-42} L({\rm H}\alpha) \;({\rm erg} \;{\rm s}^{-1}). \end{equation} \tag{ 4 }$$

Using this equation, we derived the SFR with extinction-corrected Hα luminosity. The result is listed in Table 3. The SFR estimate is larger by a factor of ∼1.5, if the Salpeter (1955) IMF is adopted.

The stellar masses derived from the SED fitting are compared with the SFRs calculated using both extinction correction approaches in Figure 10. For the extinction correction using the Calzetti (2001) relation, the SFRs range from 10 to several hundreds M_☉ yr⁻¹. Some massive galaxies (∼10¹¹ M_☉) and a significant fraction of low-mass galaxies show SFR larger than 100 M_☉ yr⁻¹. As a result, there is no clear correlation between SFR and stellar mass. The present sample contains many sBzK-MIPS galaxies, whose SFRs are expected to be high. Although sBzK-MIPS galaxies lie at the massive and high-SFR end of the distribution, there still exist low-mass and high-SFR galaxies among sBzK-non-MIPS galaxies.

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The specific SFR (SSFR), which is the SFR divided by stellar mass, is one of the indicators showing the level of star formation; higher SSFR indicates that a galaxy is accumulating large stellar mass compared to its own stellar mass on a short timescale. In Figure 10, SSFR is depicted by dashed lines. SSFR does not depend on MIPS detection but on stellar mass. There exist actively star-forming galaxies (SSFR>10⁻⁸ yr⁻¹) among the less massive galaxies, while the SSFR of massive galaxies (≳10¹⁰ M_☉) is nearly constant (∼2 × 10⁻⁹ yr⁻¹). It can imply that some low-mass galaxies at z ∼ 2 are actively forming stars.

On the other hand, for the SFRs corrected assuming the equal-extinction case, the SFRs and SSFRs of the high-SSFR galaxies become lower, but they still have higher SSFR when compared to other galaxies. The SSFR of the original high-SSFR galaxies becomes SSFR>3.5 × 10⁻⁹ yr⁻¹. Consequently, there is still no correlation between them. The distribution of SFRs is nearly constant (∼30 M_☉ yr⁻¹).

The possible AGN candidate (MODS41-0194; denoted by a open square) shows lower SFR than other galaxies with similar stellar masses. We note that this object does not change our findings of no correlation between SFR and stellar mass and low-mass galaxies with active star formation.

The inverse of SSFR (SSFR age) gives the estimated age for a galaxy to build up all of its stellar mass with the current SFR. In contrast, mass-weighted age, which is calculated by integrating the ages of stars weighted by SFR at that time, is an indicator of the average age of the stars in a galaxy. The mass-weighted age is given from integration of the star formation history of a galaxy as

$$\begin{equation} \tilde{T}(t) = \frac{\int ^t_0 \phi (t^{\prime }) \times (t-t^{\prime })dt^{\prime }}{\int ^t_0 \phi (t^{\prime }) dt^{\prime }}, \end{equation} \tag{ 5 }$$

where ϕ(t) is the SFR at age t. Comparing the two ages, we can roughly check consistency between the SED fitting result and the SFR estimated from extinction-corrected Hα luminosity. The comparison between the SSFR age for both the cases of extinction correction and mass-weighted age of our sample is shown in Figure 11. For extinction correction with the Calzetti (2001) relation, the SSFR ages agree with the mass-weighted ages to within 1σ error; a galaxy with a higher SSFR has a younger stellar population. This fact suggests that the majority of the stellar mass for the galaxies in our emission-line samples were formed in a recent starburst. On the other hand, for the extinction correction assuming the equal-extinction case, the ages of stars are young compared to the SFR, especially for high-SSFR galaxies. In this case, the galaxies must have experienced much more active star formation in the recent past and terminated the star formation in a very short timescale (≲10 Myr).

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The median value of the reddening derived from the SED fitting is E(B − V)_stellar = 0.42,  0.21 (A_V = 1.7,  0.85) for sBzK-MIPS and sBzK-non-MIPS galaxies, respectively. Both values are relatively larger than those for local galaxies (e.g., <A_V > _median ∼ 0.43 for r-band selected SDSS galaxies; Kauffmann et al. 2003). The extinction corrections are key to the discussion on the luminosities of Hα emission line, and thus the SFRs and SSFRs; larger extinction correction makes extinction-corrected SFR and SSFR larger. In this section, we discuss the reliability of our extinction correction through comparisons of the Hα SFRs with other SFR indicators, and the reason why the higher SSFR galaxies are found by our observation.

5.1.1. UV Luminosity

The UV continuum is used as an SFR indicator, because the stellar continuum at 1500–2800 Å is sensitive to emission from massive stars. We compare the SFRs with those estimated from the UV continuum. We calculate a relation between UV continuum at λ = 1500 Å, L_ν,1500 Å, and SFR as for Equation (4). L_ν,1500 Å is measured from the SED templates of GALAXEV with the same parameters as in Section 4.4 with the B-band filter of ACS/HST redshifted to z = 2. The relation is given by

$$\begin{equation} {\rm SFR(UV)}\; (M_\odot \;{\rm yr}^{-1}) = 7.50 \times 10^{-29} L_{\nu,1500\,\AA}\; ({\rm erg} \;{\rm s}^{-1} \;{\rm Hz}^{-1}). \end{equation} \tag{ 6 }$$

For our samples, L_ν,1500 Å is measured with ϕ15 in B band with an aperture correction obtained from the K_s-band total magnitude. The dust extinction is corrected with that derived from the SED fit.

The comparison between SFR(Hα) and SFR(UV) is shown in Figure 12. For the equal-extinction case, the SFR(Hα) is underestimated for the galaxies with higher SFR(UV). On the other hand, using the Calzetti (2001) relation the SFR(Hα) and SFR(UV) are roughly consistent for a broad range of SFR, but are systematically larger by 0.3 dex for both sBzK-MIPS and sBzK-non-MIPS galaxies.

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We note that the luminosity ratio of the UV continuum to Hα varies with the population of a galaxy. For example, because the UV continuum is contaminated by less massive stars than stars emitting ionizing photon, galaxies populated by younger massive stars are more luminous in Hα. For our template, the ratio is larger by factor ∼3 for age ≲10 Myr and becomes constant for age ≳ 100 Myr. In fact, high-SSFR galaxies have a mass-weighted age ≲ 100 Myr. If a top-heavy IMF is assumed, moreover, the ratio gets larger, so that more ionizing photons are emitted than UV continuum.

5.1.2. MIPS Flux

In order to examine the reliability of the extinction corrections further, we compare the SFR(Hα) estimates with those derived from the thermal infrared luminosities (SFR(IR)), because the infrared luminosity is an SFR indicator almost unaffected by dust extinction. We use the rest-8 μm luminosity (νL_ν,8 μm) derived from the MIPS 24 μm flux as representative of the thermal infrared luminosity. Independently from the public catalog, the 24 μm fluxes of the present galaxies are measured using the Spitzer MIPS image (M. Dickinson et al. 2010, in preparation) with the IRAF/DAOPHOT package (Stetson 1992) for both sBzK-MIPS and sBzK-non-MIPS samples. Some galaxies have 24 μm fluxes severely contaminated by nearby bright 24 μm sources because of the large PSF of MIPS (FWHM ∼ 54). In order to reduce the effect of the contamination, the fluxes are measured by fitting crowded objects simultaneously with the PSF measured from isolated point sources. The objects are centered using the K_s-band light distribution. The photometric error is estimated on a blank-sky region on the image. In order to take into account the residuals of the fitting, the image from which the fitted models are subtracted is measured and the residual is added to the error. Then, νL_ν,8 μm is estimated from the spectroscopic redshift of Hα emission line and the observed 24 μm flux densities with k-corrections derived from the luminosity-dependent SED models of Chary & Elbaz (2001). The measured MIPS-24 μm flux and estimated rest-8 μm luminosity are also listed in Table 3.

The derived νL_ν,8 μm and Hα luminosities are plotted in Figure 13. The Hα luminosities are corrected using both the Calzetti (2001) relation (left) and the equal-extinction case (right) for comparison. The expected relation between νL_ν,8 μm and Hα luminosity is also shown using a solid line in the figure. The relation is derived based on that expected between νL_ν,8 μm and SFR. The νL_ν,8 μm is converted to total infrared luminosity (L_IR) in the wavelength range between 8 and 1000 μm with the luminosity-dependent SED models (Chary & Elbaz 2001), and the relation between the total infrared luminosity and SFR,

$$\begin{eqnarray} {\rm SFR}({\rm IR})\,({M}_\odot \;{\rm yr}^{-1}) &=& 3.0 \times 10^{-44} L_{\rm IR} \; ({\rm erg\;s}^{-1}) \nonumber\\ &=& 1.2 \times 10^{-10} L_{\rm IR}\; (L_\odot), \end{eqnarray} \tag{ 7 }$$

is used. The relation is derived by dividing that in Kennicutt (1998) by 1.5 to convert the Salpeter (1955) IMF to the Chabrier (2003) IMF. As we see in Figure 12, it is suggested that the extinction correction between the stellar continuum and ionized gas depends on the SFR of a galaxy. For higher (lower) SFR, the Hα luminosity corrected with the Calzetti (2001) relation (equal-extinction case) is consistent with νL_ν,8 μm. We divide the present sample into two groups, LIRGs and ULIRGs, at νL_ν,8 μm = 2.8 × 10⁴⁴ erg s⁻¹, which corresponds to L_IR = 10¹² L_☉. For ULIRGs, the systematic offset between SFR(IR) and SFR(Hα) (SFR(IR)/SFR(Hα)) is 0.31 (0.72) dex, with a scatter of 0.47 (0.29) dex for the Calzetti (2001) relation (equal-extinction case); although the Calzetti (2001) relation results better consistency, SFR(Hα) is still underestimated. For LIRGs, the systematic offset is −0.42 (0.02) dex with a scatter of 0.29 (0.43) dex for the Calzetti (2001) relation (equal-extinction case); using the equal-extinction case results in a good agreement between the two SFR indicators.

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We need to note that the SFR estimated from the rest-8 μm luminosity has a large uncertainty. Rest-frame luminosity at ∼8 μm contains a very small fraction of thermal infrared emission, so that the L_IR is estimated by extrapolating the SED to longer wavelengths. The rest-8 μm luminosity could also be affected by polycyclic aromatic hydrocarbons (PAHs) emission lines, whose energy source is dominated by late short-lived B-type stars rather than O-type stars (Peeters et al. 2004). These factors can produce large random error in the estimation of SFR from the rest-8 μm luminosity. For local galaxies, Chary & Elbaz (2001) demonstrated that the 1σ error for the L_IR estimated from rest-6.7 μm was a factor of 2. The uncertainty is consistent with the observed scatter around the solid line in Figure 13.

5.1.3. Radio and Far-Infrared

A correlation between FIR luminosity and radio continuum is known for local galaxies (Condon 1992; Yun et al. 2001). Non-thermal radio continuum is mainly emitted by synchrotron emission from the remnants of supernovae, whose frequency is proportional to the number of massive stars, and thus the SFR. The relation between these luminosities for local galaxies is given by

$$\begin{equation} L_{\rm IR}\,(L_\odot) = 3.5\times 10^{-12}\,L_{\rm 1.4\,GHz}\,({\rm W\,Hz^{-1}}) \end{equation} \tag{ 8 }$$

with about 40% of uncertainty (Yun et al. 2001).

We use a publicly available 1.4 GHz radio catalog and images (Biggs & Ivison 2006), whose 1σ rms noise in the GOODS-N region is 5.8 μJy beam⁻¹ (the beam size = 1.52 × 1.51 arcsec²). The catalog contains sources with a >5σ detection. Although our Hα-emission-line galaxies are not matched with the sources in the catalog, some galaxies are marginally detected (≲3σ), according to our eye-ball check on the 1.4 GHz radio images. Therefore, their 1.4 GHz fluxes are lower than 29 μJy (5σ), and some galaxies possibly have ∼ a few 10 μJy.

If we assume the local radio–IR correlation, we can calculate an expected L_1.4 GHz from SFR using Equations (7) and (8). We assume a spectral index of (α; f_ν ∝ ν^−α) of the non-thermal radio continuum α = 0.8 when we calculate the expected 1.4 GHz flux density. The expected 1.4 GHz flux estimated from the MIPS-24 μm flux exceeds the detection limit for only one emission-line galaxy (MODS12-0125), even if the uncertainty in the estimation of L_IR from rest-8 μm luminosity is considered. This object is an sBzK-MIPS galaxy and has a low SSFR. For this object, the SFR(Hα) is also smaller than SFR(IR), so that only the rest-8 μm luminosity exceeds that at other wavelengths. This infrared excess when compare to the radio luminosity can be explained by a warm dust component, which is thought to be caused by a compact nuclear starburst or dust-enshrouded AGN (Yun et al. 2001).

On the other hand, when we calculate the radio flux from the extinction-corrected Hα luminosity using the Calzetti (2001) relation, the radio emission of five emission-line galaxies (MODS22-1411, MODS21-2612, MODS22-5133, MODS11-0390, and MODS22-2658) exceed the detection limit. The first four galaxies are high-SSFR galaxies. Although most of these galaxies have marginal detection of 1.4 GHz flux, the expected radio fluxes (L_1.4 GHz ∼ 50–150 μJy) are higher than the observed radio flux, even if the uncertainty in the radio–IR correlation is taken into account. On the other hand, for the Hα luminosity calculated assuming the equal-extinction case, none of the galaxies exceed the detection limit. This would suggest that the amounts of extinctions in the ionized gases are overestimated by using the Calzetti (2001) relation, though the uncertainties in their extinctions are large. We also note that, equal extinction in turn underestimates the extinction for ULIRGs (see Figure 13). These Hα-excess galaxies consist of three ULIRGs (MODS11-0390, MODS22-1411, and MODS22-2658).

Thermal emission from dust heated by massive stars (∼30 K) is observed at submillimeter wavelengths. The GOODS-N region is also observed with SCUBA/JCMT (850 μm; Holland et al. 1999), AzTEC/JCMT (1.1 mm; Wilson et al. 2008), and MAMBO/IRAM (1.2 mm; Kreysa et al. 1998) down to ∼2 mJy (e.g., Pope et al. 2005, 2006), ∼1 mJy (Perera et al. 2008; Chapin et al. 2009), and ∼0.7 mJy (Greve et al. 2008), respectively. One of our sample, MODS31-0033, is identified with an AzTEC source, AzGN27, whose 1.4 GHz and 24 μm counterparts are also identified by Chapin et al. (2009), although this object is not detected at both 850 μm and 1.2 mm wavelengths.

Interestingly, the 1.1 mm flux (2.31^+1.16_−1.30 mJy) of MODS31-0033 is consistent with that expected from the 24 μm flux (1.6 mJy), when the Chary & Elbaz (2001) SED templates are assumed, as though the SFR(Hα) is lower than the SFR(IR). None of our samples including MODS31-0033 are detected at 850 μm, in spite of the fact that some of them have SFR(Hα) and/or SFR(IR) exceeding several hundred M_☉ yr⁻¹. It is known that 850 μm selected galaxies seem to have cooler thermal emission than local ULIRGs, which results from the fact that the 850 μm flux is lower than that expected from total IR luminosity (Pope et al. 2006). Therefore, it is possible that 850 μm emission is not detected from all ultra-luminous starburst galaxies at z ∼ 2.

5.1.4. X-ray

5.1.5. Extinction and Age

5.1.6. Comparison with Other Studies

Our result shows no correlation between stellar mass and SFR. However, Daddi et al. (2007a) demonstrated that BzK-selected star-forming galaxies have a tight correlation within 0.2 dex using SFR derived from UV and mid-IR. We compare our result for SFR and stellar mass with the Daddi et al. (2007a) relation (Figure 14). For the Calzetti (2001) relation, significant fractions of sBzK galaxies (4/18 and 4/19 for sBzK-MIPS and sBzK-non-MIPS, respectively) deviate from the correlation. Massive galaxies (≳10¹⁰ M_☉) in our sample are distributed around the relation given by Daddi et al. (2007a), while some of the less massive galaxies (≲10¹⁰ M_☉) have much higher SFR than that expected from the correlation. Even for the equal-extinction case, which makes the SFRs of the higher SFR galaxies lower, there is still no correlation between stellar mass and SFR like that of Daddi et al. (2007a).

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One of the reasons for the discrepancy could be the different extinction estimates used in Daddi et al. (2007a), who assumed a linear correlation between B − z color and E(B − V) derived from their SED fitting result (Daddi et al. 2004) to estimate extinction-corrected UV luminosities. In fact, if we apply the E(B − V) values estimated from the B − z color, the extinction-corrected SFR(Hα) becomes smaller than the E(B − V) values estimated from our stellar population analysis (Figure 15). Especially when the equal-extinction case is applied, the extinction-corrected SFR(Hα)s are similar to the Daddi et al. (2007a) relation. Although our E(B − V) values correlate with B − z color, the correlation has larger scatter than that of Daddi et al. (2004) (Figure 16). The high-SSFR galaxies have larger E(B − V) values than the galaxies with similar B − z color.

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The Daddi et al. (2007a) relation is also supported by Pannella et al. (2009), who show the averaged properties of extinction from the radio-to-UV luminosity ratio using a large number (∼12, 000) of sBzK galaxies. They also find a strong correlation between the stellar mass and the amount of extinction, and thus the correlation accentuates the Daddi et al. (2007a) relation. In Figure 17, the amount of extinction of our galaxies is shown as a function of the stellar mass. Our results show a larger scatter than the errors of the correlation by Pannella et al. (2009). Although the extinction values of the low-SSFR galaxies correlate with the stellar masses, consistent with that of Pannella et al. (2009), the high-SSFR galaxies have higher extinctions than the low-SSFR galaxies with similar stellar masses. Because our emission-line galaxies would be biased toward Hα bright systems, galaxies that deviate from the average relation shown by Pannella et al. (2009) can be found, which probably results in the larger scatter.

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Daddi et al. (2007a) also show the same correlation using an SFR indicator unrelated to extinction. They estimated the SFR from the sum of the extinction uncorrected UV and mid-IR luminosities. On the other hand, Caputi et al. (2006b) reported that there exist significant fraction of lower mass (∼10¹⁰ M_☉) galaxies with high SSFRs (>10⁻⁸ yr⁻¹) at z ∼ 2 using SFR inferred from the 24 μm flux. Daddi et al. (2007a) argue that the correlation is not accurately recovered with SFRs inferred only from mid-IR, due to a substantial presence of MIPS-excess galaxies, which are excluded from their sample. A MIPS-excess galaxy could harbor a heavily obscured AGN (Daddi et al. 2007b). However, our results on the SFR inferred from Hα do not recover the correlation even if MIPS-excess galaxies are excluded from the sample. We calculate the MIPS excesses of our sample following Daddi et al. (2007a), and found that only a fraction of the high-SSFR galaxies show MIPS excess. In Figure 14, MIPS-excess galaxies are denoted by white dots, which are found to be more typically low-SSFR galaxies rather than high-SSFR galaxies.

Previous studies on the relation between SFR and stellar mass using the SFR derived from the Hα luminosity show a similar trend to ours. The SFRs and stellar masses of BM/BX galaxies (Erb et al. 2006b) and sBzK galaxies (Hayashi et al. 2009) are also plotted on Figure 14. We convert the SFR and stellar mass of these studies to those calibrated with a Chabrier (2003) IMF. In the literature, they applied different extinction corrections to estimate the Hα luminosities from that used to estimate the stellar extinction. The equal-extinction case is applied to the BM/BX galaxies, while the Calzetti (2001) relation is applied to the sBzK galaxies. For comparison, we applied the same extinction correction to the samples in the literature as that for the present sample in each panel. There are no correlations between SFR and stellar mass of the samples in the both panels. However, some high-SSFR galaxies with SSFR >10⁻⁸ yr⁻¹ appear when the Calzetti (2001) relation is applied.

In the studies for BM/BX and sBzK galaxies, validity of the extinction correction is confirmed with the consistency between SFR(Hα) and SFR(UV) (Erb et al. 2006b; Hayashi et al. 2009). We also confirmed the consistency between them in the previous section. The extinction-corrected SFR(Hα) values assuming the equal-extinction case are consistent only for lower SFR galaxies (SFR ≲100 M_☉ yr⁻¹). SFRs calculated assuming the Calzetti (2001) relation are broadly consistent, but the SFR(Hα)s are systematically larger than the SFR(UV)s by 0.3 dex. These are almost consistent with the previous SFR(Hα) works. Although Erb et al. (2006b) applied only the equal-extinction case, galaxies with SFR(UV) >100 M_☉ yr⁻¹ are rare in their sample. On the other hand, Hayashi et al. (2009), who used the Calzetti (2001) relation, also showed that their SFR(Hα) values are systematically larger than SFR(UV) by a factor of 3.

Recently, the relation between Balmer decrement and stellar continuum attenuation has been investigated for galaxies at low-to-medium redshift. Cowie & Barger (2008) studied a spectroscopically confirmed K-selected sample at 0.05 < z < 0.475. Their extinctions from the Balmer decrement and the continuum extinctions are equal on average. On the other hand, Moustakas & Kennicutt (2006) studied nearby galaxies, including starburst galaxies such as infrared-luminous galaxies. Comparison between their extinctions shows large scatter, but the average stellar-to-nebular extinction ratio is consistent with the Calzetti (2001) relation. The dust properties of the samples in these two studies are quite different. For example, the median reddening of Moustakas & Kennicutt (2006) sample is E(B − V) ∼ 0.24, while about half of the galaxies of Cowie & Barger (2008) have E(B − V) < 0.01. The values of reddening of the high-z samples discussed above and our samples are similar to that of Moustakas & Kennicutt (2006), but vary with the sample selection. The median values of reddening are E(B − V) ∼ 0.15,  0.38 for BM/BX galaxies by Erb et al. (2006b) and sBzK galaxies by Hayashi et al. (2009), respectively. This can imply that the Calzetti (2001) relation gives a better estimation of the value of reddening for ionized gas in the active star-forming galaxies at high redshift.

The gas-phase metallicity reflects the past star formation history of a galaxy since the heavy elements are produced by nuclear fusion in stars and fed back into gas. Therefore, it is expected that the gas-phase metallicity of galaxies in the high-z universe is lower than that in the local universe. The metallicities were found to be lower in high-z galaxies than those in local galaxies (e.g., Shapley et al. 2004; Savaglio et al. 2005; Erb et al. 2006a; Maiolino et al. 2008; Hayashi et al. 2009; Pérez-Montero et al. 2009; Lamareille et al. 2009). We measured the gas-phase metallicity of our samples to investigate their star formation histories.

The gas-phase metallicity is estimated from the [N ii]λ6583–Hα line ratio (N2 index), defined by

$$\begin{equation} {\rm N2} \equiv \log {\lbrace [\mathrm{N_{II}}]\lambda 6583/\mathrm{H\alpha }\rbrace }, \end{equation} \tag{ 9 }$$

which correlates with oxygen abundance (Storchi-Bergmann et al. 1994). The relation between the N2 index and gas-phase oxygen abundance is given by

$$\begin{equation} 12+\log {(\mathrm{O/H})} = 8.90 + 0.57 \times {\rm N2} \end{equation} \tag{ 10 }$$

(Pettini & Pagel 2004). [N ii] and Hα are so close to each other in wavelength that the ratio is not affected by reddening. Therefore, the N2 index has been used to estimate the oxygen abundances of distant galaxies (e.g., Shapley et al. 2004; Erb et al. 2006a; Hayashi et al. 2009). We note that the N2 index saturates at ∼1 Z_☉ (12 + log(O/H) = 8.66; Asplund et al. 2004).

In order to gain a sufficiently high S/N to reliably measure the line flux ratio, we stacked the [N ii]–Hα emission lines. We divided the present sample into three groups based on stellar masses at 9 × 10⁹ M_☉ and 2.5 × 10¹⁰ M_☉. In addition, we also divide the sample into two groups based on an SSFR of 10⁻⁸ yr⁻¹. The stacking, measurement of flux, and estimates of error are performed in the same manner as described in Section 3.3. The results are plotted in Figure 18 as a function of the median stellar mass of each group. For comparison, ∼39, 500 local objects from SDSS DR4,¹³ local Lyman Break Analog (LBA) galaxies (Overzier et al. 2010), BM/BX galaxies (Erb et al. 2006a), and sBzK galaxies (Hayashi et al. 2009) are also plotted with the same calibration.

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The mass–metallicity properties of our high-SSFR galaxies are possibly different from that of rest-UV-selected galaxies. The mass–metallicity relation of the two largest mass bins is consistent with that of BM/BX galaxies described by Erb et al. (2006a), while the smallest mass bin has higher metallicity than the similar mass bins of Erb et al. (2006a); our sample is closer to the local galaxies. The discrepancy in the smallest mass bin could be due to the galaxies with high SSFR. The high-SSFR group has higher metallicity, close to that of the local galaxies with similar stellar mass. As shown in Figure 10, high-SSFR galaxies dominate our sample in the smallest mass bin, while most of the galaxies in Erb et al. (2006a) have SSFR smaller than 10⁻⁸ yr⁻¹.

The K-band-selected samples would provide many galaxies with high SSFR and metallicity. The existence of higher metallicity galaxies in the K-selected sample is also supported by the observations of sBzK galaxies by Hayashi et al. (2009), as shown in Figure 18. On the other hand, the local LBAs, which are local analogs of high-z UV-selected galaxies, tend to have a lower mass–metallicity relation. This would suggest that the sample by Erb et al. (2006a) is biased toward lower metallicities due to the rest-UV selection. The results indicate that in order to reveal the unbiased cosmic evolution of mass–metallicity relation, we need to evaluate metallicities of a larger sample of K-band-selected (i.e., mass-selected) galaxies.

It is surprising that the high-SSFR galaxies have already acquired metallicities as high as local galaxies at z ∼ 2. The age of the stellar population of the high-SSFR galaxies is young, so that the metal enrichment must have occurred in a short timescale. The N2 index is known to be sensitive to a hard ionizing radiation field from an AGN or strong shock excitation and ionization parameter (Kewley & Dopita 2002). In fact, our stacked [O iii]λ5007/Hβ and [N ii]λ6583/Hα line ratios show evidence for a harder ionizing spectrum. If the metallicity is really high, the result is in conflict with the observations of local galaxies. These galaxies cannot keep their stellar masses and metallicities after the observed epoch, because the studies for local galaxies demonstrate that galaxies in a similar mass range formed a significant fraction of their stars in the low-redshift universe (Heavens et al. 2004), suggesting that these local galaxies must have increased their metallicity recently. It is possible that such high-SSFR galaxies evolved into larger stellar mass galaxies with low star-forming efficiency in the local universe, depleting the gas and/or merging with higher mass galaxies.

We estimate the contribution of the observed galaxies to the cosmic SFR density at z ∼ 2 with the SFR(Hα)s, dividing the sample into two groups with SSFR >10⁻⁸ yr⁻¹ and <10⁻⁸ yr⁻¹ for the extinction-corrected SFRs with the Calzetti (2001) relation. Since we observed only a limited number of sBzK galaxies, we need to assume that the SFR and redshift distributions of the other galaxies are the same as the observed galaxies; this is not unreasonable since we randomly selected the present sample from the sBzK-MIPS and sBzK-non-MIPS population. First, we further divide the sample based on the MIPS detection in the public catalog. Second, we calculated the total SFR in a certain group; we multiply the total SFR in the group by the ratio of the number of the entire galaxies in the group to that of the number of galaxies observed. In the calculation of total SFR, we assumed that the SFR of the galaxies without Hα detection is negligible; we set those SFRs to zero. Then, the total SFRs of galaxies with and without MIPS detection are summed. Third, we divide the total SFR by the expected detection rate of Hα emission line (87.7%; see Section 4.1) to recover the Hα emission lines not detected due to strong sky emission lines or atmospheric absorption bands. Finally, we divide the total SFRs by comoving volume of the survey region for 1.4 < z < 2.5.

The resultant total cosmic SFR densities of the sBzK galaxies are 0.136^+0.020_−0.028 and 0.029^+0.002_−0.002 M_☉ yr⁻¹ Mpc⁻³ for extinction correction calculated using the Calzetti (2001) relation and the equal-extinction case, respectively. As we see in the previous discussions, the relation between extinction estimated from the stellar continuum and the emission lines probably depends on the intrinsic luminosities and thus the SFRs. To examine this effect, when we apply extinction correction with the Calzetti (2001) relation for the sBzK-MIPS galaxies and equal extinction for the sBzK-non-MIPS galaxies (differential correction), the result is 0.089^+0.008_−0.021. In any case of extinction correction, high-SSFR galaxies have a large contribution to the SFR density. Although these galaxies account for only ∼20% of all of the galaxies, they contribute to ∼75% of the SFR density for the extinction correction with the Calzetti (2001) relation and the differential extinction. Even for the equal-extinction case, they contribute to no less than ∼50% of the SFR density.

We compare our SFR density at z ∼ 2 with previous works (e.g., Hopkins 2004; Hopkins & Beacom 2006, and references therein; Wang et al. 2006; Caputi et al. 2007). If we use the same calibration of IMF as that of these studies, the SFR densities are 0.203, 0.044, and 0.134 M_☉ yr⁻¹ Mpc⁻³ for the Calzetti (2001) relation, the equal-extinction case, and the differential correction, respectively. The values compiled by Hopkins (2004) and Hopkins & Beacom (2006) are 0.05–0.38 M_☉ yr⁻¹ Mpc⁻³, which are consistent with both those with the Calzetti (2001) relation and differential extinction. Recent studies on the SFR density at z ∼ 2 using IR luminosities (850 μm; Wang et al. 2006; 24 μm; Caputi et al. 2007) show slightly lower SFR densities (0.08–0.09 M_☉ yr⁻¹ Mpc⁻³), which are still consistent with the differential extinction correction. This would also suggest that the relation between the dust properties of the stellar continuum and the emission lines is different depending on the intrinsic SFR. The SFR density using the equal-extinction case is lower than that of any study. The lower SFR density implies that applying the equal-extinction case to all galaxies underestimates SFR(Hα).