ISM EXCITATION AND METALLICITY OF STAR-FORMING GALAXIES AT Z ≃ 3.3 FROM NEAR-IR SPECTROSCOPY

Our primary goal of the project is to measure metallicity at z ≳ 3 using strong rest-frame optical emission lines such as [O ii] λλ3726, 3729, ${\rm{H}}\beta$, and [O iii] λλ4959, 5007. At 3 < z < 3.8, [O ii] λλ3726, 3729 are observed in H band and ${\rm{H}}\beta$ and [O iii] λλ4959, 5007 can be accessible in K band.

Our primary sample has been extracted from the zCOSMOS-Deep redshift catalog (Lilly et al. 2007, and in preparation).

We have first selected objects in the redshift range 3 < z_spec < 3.8 that are spectroscopically classified as galaxies (i.e., neither stars nor broad-line AGNs), and the redshifts are reliable with confidence classes (CC) in the range of 2.5 ≤ CC ≤ 4.5, which includes some insecure spectroscopic redshifts but agrees well with photometric redshifts (see Table 1 in Lilly et al. 2009). Then we have given higher priority to those with the expected line positions of [O ii] λ3727, ${\rm{H}}\beta$, and [O iii] λ5007 being more than two MOSFIRE resolution elements (i.e., R ∼ 3600) away from the nearest OH line. Since it is expected that spectroscopic redshifts at z ∼ 3 in the optical spectra have been determined primarily by the presence of $\mathrm{Ly}\alpha$, and a velocity offset in $\mathrm{Ly}\alpha$ relative to the systemic velocity is also expected (Finkelstein et al. 2011; McLinden et al. 2011; Hashimoto et al. 2013; Erb et al. 2014), we gave lower priority to those with some degree of OH contamination instead of fully excluding them from the sample.

The number density of the zCOSMOS-Deep selected objects is ≲5 per MOSFIRE field of view (FOV), so we filled the remaining multiplex with various classes of galaxies in the 2014 run, which will be presented elsewhere. On the other hand, in the 2015 run, star-forming galaxies at 3 < z_phot < 3.8 have been selected from the 30-band COSMOS photometric redshift catalog on UltraVISTA photometry (McCracken et al. 2012; Ilbert et al. 2013). We have further selected objects with predicted ${\rm{H}}\beta$ flux of >5 × 10⁻¹⁸ erg s⁻¹ cm⁻² scaled from the SFR in the catalog to ${\rm{H}}\alpha$ flux by using a conversion of Kennicutt & Evans (2012), assuming an intrinsic ${\rm{H}}\alpha$/${\rm{H}}\beta$ ratio of 2.86, and applying dust extinction by using E(B − V)_star from the catalog with the Calzetti extinction law (Calzetti et al. 2000). Conservatively, we adopt E(B − V)_gas = E(B − V)_star/0.44 extinction in ${\rm{H}}\beta$ following the original recipe of Calzetti et al. (2000), though some studies suggest that the factor appears to be closer to unity at high redshifts (e.g., Kashino et al. 2013; Pannella et al. 2015). A MOSFIRE FOV typically contains ∼25 of these photo-z objects, which is enough to fill the multiplex.

Finally, we observed 54 objects, one of which was observed in both observing runs, with 43 of them having robust spectroscopic identification by one or more emission lines in the MOSFIRE spectra. Table 1 summarizes our sample of 43 objects with detected emission lines.

Table 1.  Properties of the Galaxy Sample and the Observervation Log

ID	R.A.	Decl.	zphot	zzCOSMOS	Run	${t}_{\mathrm{exp},H}$	${t}_{\mathrm{exp},K}$	BAB	KsABa
	(deg)	(deg)				(minutes)	(minutes)	(mag)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
434625	150.09385	2.421970	3.155	3.0717	2014	28	30	25.29	25.38
413136	150.10600	2.436190	2.838	3.1911	2014	28	30	24.91	22.99
413646	150.08025	2.469180	3.132	3.0389	2014	28	30	25.23	23.70
413453	150.13365	2.457430	3.121	3.1892	2014	28	30	25.07	22.24
434585	149.84702	2.373020	3.435	3.3557	2014	28	30	24.66	22.82
434571	149.83784	2.362050	3.142	3.1292	2014	28	30	25.77	24.36
413391	149.78424	2.452890	3.469	3.3626	2014	28	30	24.78	22.44
427122	150.07483	2.032280	2.917	3.0454	2014	28	30	25.14	24.64
434148	150.03551	1.965090	3.144	3.1457	2014	28	30	25.13	23.63
434082	150.35009	1.904380	3.449	3.4750	2014	28	42	24.34	22.65
434126	150.37178	1.947200	3.427	3.2715	2014	28	42	25.35	23.76
434139	150.38895	1.956150	3.108	3.0355	2014	28	42	24.86	23.74
434145	150.40089	1.963460	3.442	3.4208	2014	28	42	25.35	23.05
434242	150.29841	2.075330	3.271	3.2376	2014	28	30	25.74	24.04
406390	150.29878	2.068820	3.149	3.0487	2014	28	30	25.05	23.41
406444	150.33032	2.072270	3.400	3.3060	2014	28	30	24.45	21.71
434227	150.32680	2.053940	3.355	3.2136	2014	28	30	24.71	22.57
191932	150.27605	2.299900	3.414	⋯	2015	56	72	26.03	23.41
434547	150.27003	2.333980	3.261	3.1875	2015	56	72	25.49	23.54
192129	150.30078	2.300540	3.457	3.5002	2015	56	72	24.83	22.89
193914	150.31923	2.306690	3.023	⋯	2015	56	72	24.65	23.43
212863	150.29286	2.367090	3.372	⋯	2015	56	72	26.14	23.69
195044	150.32696	2.310780	3.185	3.1149	2015	56	72	25.14	23.38
208115	150.31325	2.351720	3.661	⋯	2015	56	72	25.71	22.52
200355	150.33103	2.327790	3.476	⋯	2015	56	72	25.87	23.48
214339	150.31607	2.372240	3.392	⋯	2015	56	72	25.94	22.44
411078	150.35142	2.322420	3.106	3.1104	2015	56	72	23.84	22.44
212298	150.34268	2.365390	3.107	3.1024	2015	56	72	25.26	22.79
412808	149.83413	2.416690	3.274	3.3050	2015	72	84	25.05	24.13
223954	149.83188	2.404150	3.436	⋯	2015	72	84	25.62	23.75
220771	149.83628	2.393190	3.473	⋯	2015	72	84	26.63	24.64
219315	149.83952	2.388460	3.240	⋯	2015	72	84	25.72	23.90
434618	149.89213	2.414710	3.289	3.2819	2015	72	84	25.18	23.24
221039	149.86938	2.394510	3.069	⋯	2015	72	84	25.00	23.05
215511	149.84826	2.376170	3.424	⋯	2015	72	84	26.33	23.92
217597	149.86534	2.382770	3.415	⋯	2015	72	84	26.76	24.06
211934	149.84725	2.364040	3.773	⋯	2015	72	84	28.08	24.44
434579	149.86179	2.366931	3.145	3.1889	2015	72	84	26.16	23.87
217753	149.89451	2.383700	3.438	⋯	2015	72	84	26.51	23.06
218783	149.92082	2.387060	3.504	⋯	2015	72	84	25.57	22.80
217090	149.91827	2.381250	3.669	⋯	2015	72	84	26.21	24.41
210037	149.90451	2.357800	3.579	⋯	2015	72	84	27.06	24.33
208681	149.90551	2.353990	3.342	3.2671	2015	72	84	24.20	22.17

Notes. (1) Object ID; (2) R.A.; (3) decl.; (4) photometric redshift from Ilbert et al. (2013); (5) spectroscopic redshift from zCOSMOS-Deep; (6) observing run; (7) exposure time in H band; (8) exposure time in K_s band; (9) B-band total magnitude; (10) K_s-band total magnitude.

^aK_s magnitudes are not corrected for emission lines.

A machine-readable version of the table is available.

Download table as:  DataTypeset image

Observation has been carried out on 2014 January 20–22 and 2015 January 15 using MOSFIRE on Keck I (McLean et al. 2010, 2012). We used 1'' and 07 slit width in 2014 and 2015 runs, respectively, which provides instrumental resolution of R ≃ 2500 and 3600, respectively. We observed in J, H, and K gratings in the 2014 run because there are some lower-redshift objects for which [O ii] λ3727 falls into J band, while only H and K gratings are used in the 2015 run. Following the exposure recommendation, we used 120, 120, and 180 s per exposure in J, H, and K band, respectively. Exposure was done in a sequence of either AB or ABBA dithering with a distance between A and B positions of 25. The observing run and total exposure time for each object in H and K bands are listed in Table 1.

We observed a couple of white dwarfs and A0V stars per night, using them as standard stars for flux calibration.

Data were reduced with the MOSFIRE data reduction pipeline version 1.1⁹ for the science frames and its Longslit branch to make the standard star reduction consistent with the science frames. The pipeline performs flat-fielding, wavelength calibration, sky subtraction, rectification, and co-addition of each exposure and produces rectified 2D spectra with associated noise, as well as exposure maps. One-dimensional (1D) spectra were extracted with a 07–1'' aperture depending on the spatial extent of detected emission lines to maximize the signal-to-noise ratio (S/N). Corresponding 1D noise spectra were also extracted from 2D noise spectra by using the same aperture and summing them up in quadrature. Flux calibration was carried out using the standard star closest in time to the corresponding science exposure. At the same time with the telluric correction, we also carried out absolute flux calibration by scaling the observed standard star spectra to 2MASS magnitudes correcting for the slit loss for a point source.

The emission lines are fit in two steps. In the first step, [O ii] λ3726, [O ii] λ3729, ${\rm{H}}\beta$, [O iii] λ4959, and [O iii] λ5007 are fit simultaneously. We assume that each emission line can be described by a simple Gaussian redshifted by the same amount with a common velocity dispersion σ_vel on a constant continuum component. Therefore, free parameters of the first fitting step are redshift, velocity dispersion, flux of each emission line, and continuum component. The continuum is assumed to be a constant within each band, i.e., [O ii] λ3726 and [O ii] λ3729 have the same continuum in H band and so do ${\rm{H}}\beta$, [O iii] λ4959, and [O iii] λ5007 in K band. In the second step, [Ne iii] λ3869, ${\rm{H}}8$, [Ne iii] λ3969 + ${\rm{H}}\epsilon$, ${\rm{H}}\delta$, and ${\rm{H}}\gamma$ are fit individually by fixing the redshift and line width derived in the first path and leaving the line flux and constant continuum as free parameters. We used MPFIT ¹⁰ (Markwardt 2009) for the fitting. During the procedure, we put constraints on σ_vel and emission-line fluxes to be larger than the instrumental resolution and to be positive, respectively.

To obtain each parameter and the associated uncertainty, we carried out a Monte Carlo simulation by perturbing each pixel value of the spectra with the associated noise multiplied by a normally distributed random number. Mean and standard deviation of the measured distribution of 10³ realizations are adopted for each parameter and its 1σ uncertainty (σ_MC), respectively. The best-fit spectra and observed 1D spectra are shown in Figure 19 in Appendix A. We computed another 1σ error (σ_direct) for each emission-line flux directly from the associated noise spectrum by integrating ±2σ_vel from the line center in quadrature. In the case of F_MC/σ_direct > 3, where F_MC is flux measured by the Monte Carlo simulation, we claim the detection of the emission line and use σ_MC as the corresponding 1σ error. Otherwise, we adopt 3σ_direct as the 3σ upper limit.

Measured ${\rm{H}}\beta$ fluxes were corrected for underlying stellar absorption by assuming EW(${\rm{H}}\beta$) = 2 Å (Nakamura et al. 2004). To compute EW, the continuum flux was estimated from the total K_s-band magnitude. The resulting correction factor is up to ≃20% with a median of 3%.

Table 2 lists measured redshifts, reddening-uncorrected emission-line fluxes, and correction factors applied to stellar ${\rm{H}}\beta$ absorption.

Table 2.  Emission-line Measurements

ID	zMOSFIRE	F([O ii] λ3727)	F([Ne iii] λ3869)	F(${\rm{H}}\beta$)a	F([O iii] λ4959)	F([O iii] λ5007)	fcorr(${\rm{H}}\beta$)	${f}_{\mathrm{em},H}$	${f}_{\mathrm{em},K}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
434625	3.07948	2.41 ± 0.27	<0.88	<1.40	1.70 ± 0.18	5.28 ± 0.20	1.00	0.07	1.00
413136	3.19037	<3.56	<1.21	1.17 ± 0.31	<0.71	1.84 ± 0.31	1.10	0.00	0.08
413646	3.04045	<6.75	<2.31	⋯	<3.61	3.95 ± 0.31	1.00	0.00	0.21
413453	3.18824	<4.84	<3.05	<2.64	2.89 ± 0.31	10.27 ± 0.35	1.00	0.00	0.14
434585	3.36345	2.94 ± 1.23	<3.69	1.56 ± 0.46	<1.14	3.77 ± 0.96	1.09	0.07	0.12
434571	3.12729	<3.57	<4.23	⋯	1.24 ± 0.11	3.03 ± 0.19	1.00	0.00	0.30
413391	3.36544	3.16 ± 0.47	<3.71	0.91 ± 0.26	2.22 ± 0.24	4.84 ± 0.69	1.22	0.04	0.09
427122	3.04157	<2.24	<0.47	0.44 ± 0.13	<0.66	1.90 ± 0.14	1.04	0.00	0.29
434148	3.14780	1.29 ± 0.24	<0.30	<0.51	<0.61	2.26 ± 0.10	1.00	0.04	0.11
434082	3.46860	5.84 ± 0.38	⋯	4.33 ± 0.33	6.97 ± 0.41	20.96 ± 0.51	1.02	0.08	0.49
434126	3.27209	2.90 ± 0.40	<0.87	1.19 ± 0.23	2.14 ± 0.34	7.93 ± 0.36	1.03	0.10	0.50
434139	3.03623	<7.31	<0.79	<1.83	<3.41	6.25 ± 0.42	1.02	0.00	0.35
434145	3.42400	5.15 ± 0.77	<4.47	<3.23	3.65 ± 0.48	11.91 ± 0.48	1.02	0.09	0.35
434242	3.23290	1.44 ± 0.34	<1.44	1.37 ± 0.34	2.16 ± 0.18	6.98 ± 0.33	1.01	0.28	0.59
406390	3.05272	2.16 ± 0.57	<0.56	<2.31	<1.39	4.35 ± 0.48	1.00	0.06	0.18
406444	3.30355	5.15 ± 0.80	1.27 ± 0.27	2.86 ± 0.34	4.27 ± 0.38	9.66 ± 0.73	1.13	0.01	0.10
434227	3.21501	<2.54	<2.40	<2.06	<2.50	5.56 ± 1.05	1.00	0.00	0.10
191932	3.18246	1.50 ± 0.12	<0.52	<0.57	0.92 ± 0.10	2.69 ± 0.11	1.14	0.05	0.11
434547	3.19167	1.33 ± 0.10	<0.23	0.40 ± 0.05	0.90 ± 0.05	2.81 ± 0.07	1.16	0.04	0.14
192129	3.49466	1.82 ± 0.14	⋯	1.25 ± 0.12	1.44 ± 0.14	3.61 ± 0.19	1.10	0.04	0.12
193914	2.96871	1.76 ± 0.28	0.52 ± 0.06	⋯	2.01 ± 0.11	6.45 ± 0.09	1.00	0.04	0.27
212863	3.29183	1.83 ± 0.08	<0.25	1.29 ± 0.11	2.05 ± 0.10	5.46 ± 0.13	1.03	0.25	0.34
195044	3.11091	1.42 ± 0.21	<0.15	0.30 ± 0.11	0.26 ± 0.08	0.76 ± 0.10	1.00	0.03	0.04
208115	3.63568	1.49 ± 0.13	3.70 ± 0.10	6.78 ± 0.16	17.99 ± 0.25	53.21 ± 0.30	1.00	0.06	1.00
200355	3.56796	1.71 ± 0.10	⋯	1.34 ± 0.15	2.94 ± 0.17	7.47 ± 0.27	1.04	0.07	0.37
214339	3.60885	1.59 ± 0.20	<0.43	0.83 ± 0.18	1.34 ± 0.25	3.98 ± 0.31	1.27	0.04	0.08
411078	3.11280	6.07 ± 0.14	1.99 ± 0.07	4.23 ± 0.20	10.26 ± 0.10	29.80 ± 0.13	1.02	0.10	0.55
212298	3.10781	3.97 ± 0.20	0.65 ± 0.10	2.25 ± 0.11	2.54 ± 0.09	7.18 ± 0.24	1.05	0.04	0.21
412808	3.30614	1.48 ± 0.09	0.57 ± 0.06	1.04 ± 0.08	2.25 ± 0.11	7.63 ± 0.13	1.01	0.06	0.67
223954	3.37076	1.72 ± 0.11	<0.61	0.59 ± 0.13	1.30 ± 0.12	3.38 ± 0.14	1.09	0.06	0.21
220771	3.35871	0.77 ± 0.10	0.20 ± 0.04	0.37 ± 0.08	<0.24	1.14 ± 0.10	1.06	0.79	0.18
219315	3.36253	0.96 ± 0.09	0.21 ± 0.05	<0.23	0.75 ± 0.12	1.61 ± 0.10	1.00	0.05	0.10
434618	3.28463	2.45 ± 0.09	<0.26	0.98 ± 0.08	1.72 ± 0.14	5.18 ± 0.09	1.08	0.06	0.21
221039	3.05515	4.67 ± 0.15	0.66 ± 0.05	1.73 ± 0.08	3.00 ± 0.24	8.49 ± 0.10	1.05	0.10	0.29
215511	3.36345	2.44 ± 0.17	0.40 ± 0.08	1.12 ± 0.13	2.32 ± 0.16	5.76 ± 0.20	1.03	0.10	0.43
217597	3.28377	1.69 ± 0.09	0.24 ± 0.04	0.52 ± 0.10	1.39 ± 0.18	3.77 ± 0.12	1.06	0.09	0.31
211934	3.35538	1.32 ± 0.28	<0.09	0.53 ± 0.16	0.43 ± 0.11	1.63 ± 0.16	1.05	0.23	0.22
434579	3.18653	0.82 ± 0.09	<0.34	0.23 ± 0.07	0.54 ± 0.08	1.28 ± 0.09	1.22	0.05	0.09
217753	3.25408	2.38 ± 0.45	<0.54	1.26 ± 0.21	<0.80	1.82 ± 0.18	1.09	0.06	0.08
218783	3.29703	1.91 ± 0.16	<0.30	0.75 ± 0.10	0.87 ± 0.12	2.53 ± 0.22	1.19	0.05	0.07
217090	3.69253	1.69 ± 0.16	⋯	1.65 ± 0.19	3.25 ± 0.21	9.10 ± 0.36	1.00	0.17	1.00
210037	3.69083	2.07 ± 0.43	⋯	<0.61	1.10 ± 0.35	4.10 ± 0.77	1.00	0.48	0.39
208681	3.26734	3.58 ± 0.30	<0.91	2.34 ± 0.16	1.04 ± 0.17	3.31 ± 0.19	1.11	0.05	0.07

Notes. (1) Object ID; (2) spectroscopic redshift measured from MOSFIRE spectra. The associated 1σ error is typically an order of ≲10⁻⁴; (3) [O ii] λ3727 (i.e., [O ii] λ3726 + [O ii] λ3729) flux; (4) [Ne iii] λ3869 flux; (5) ${\rm{H}}\beta$ flux; (6) [O iii] λ4959 flux; (7) [O iii] λ5007 flux; (8) correction factor for ${\rm{H}}\beta$ absorption assuming 2 Å in the equivalent width; (9) and (10) fraction of emission-line contribution in H and K band, respectively. All fluxes are in units of 10⁻¹⁷ erg s⁻¹ cm⁻², and they are not corrected for dust extinction. Quoted upper limits are the 3σ upper limit.

^aFluxes are not corrected for the underlying stellar absorption. To correct it, one needs to multipy F(${\rm{H}}\beta$) by f_corr(${\rm{H}}\beta$).

A machine-readable version of the table is available.

Download table as:  DataTypeset image

In Figure 1, the MOSFIRE spectroscopic redshifts are compared with those from zCOSMOS-Deep or with photometric redshifts. Agreement between MOSFIRE and zCOSMOS-Deep spectroscopic redshifts is almost perfect with a standard deviation of 0.004 in $({z}_{\mathrm{MOSFIRE}}-{z}_{\mathrm{zCOSMOS}})$. Photometric redshifts also agree well with the spectroscopic ones with no catastrophic failure. The normalized median absolute deviation of $({z}_{\mathrm{MOSFIRE}}-{z}_{\mathrm{phot}})/(1+{z}_{\mathrm{MOSFIRE}})$ is 0.027, which is consistent with what was reported in Ilbert et al. (2013) for high-redshift galaxies. The redshifts of our sample are in the range of 2.97 < z < 3.69, with a median of 3.27.

Zoom In Zoom Out Reset image size

Table 3.  Stellar Mass, Dust Extinction, and Star Formation Rate

ID	log M*	AV (SED)	AV (UV)	log SFR (SED)	log SFR (UV)	log SFR (${\rm{H}}\beta$)
	(M⊙)	(mag)	(mag)	(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)
434625	${9.35}_{-0.47}^{+0.28}$	${0.11}_{-0.11}^{+0.19}$	0.33 ± 0.14	${0.97}_{-0.16}^{+0.29}$	1.09 ± 0.06	<1.40
413136	${10.51}_{-0.04}^{+0.03}$	${0.30}_{-0.07}^{+0.07}$	0.84 ± 0.10	${1.52}_{-0.02}^{+0.02}$	1.86 ± 0.04	1.63 ± 0.12
413646	${9.35}_{-0.13}^{+0.08}$	${0.75}_{-0.12}^{+0.16}$	0.43 ± 0.12	${1.82}_{-0.06}^{+0.37}$	1.32 ± 0.05	⋯
413453	${10.91}_{-0.02}^{+0.02}$	${0.30}_{-0.07}^{+0.07}$	0.83 ± 0.09	${1.46}_{-0.02}^{+0.02}$	1.80 ± 0.04	<1.94
434585	${10.24}_{-0.08}^{+0.08}$	${0.11}_{-0.08}^{+0.08}$	0.38 ± 0.07	${1.56}_{-0.04}^{+0.03}$	1.71 ± 0.03	1.59 ± 0.13
434571	${9.04}_{-0.24}^{+0.15}$	${0.30}_{-0.18}^{+0.17}$	0.08 ± 0.17	${1.17}_{-0.20}^{+0.33}$	0.80 ± 0.08	⋯
413391	${10.17}_{-0.29}^{+0.07}$	${0.53}_{-0.12}^{+0.29}$	0.54 ± 0.08	${1.94}_{-0.10}^{+0.54}$	1.88 ± 0.04	1.48 ± 0.13
427122	${8.93}_{-0.26}^{+0.19}$	${0.00}_{-0.00}^{+0.00}$	0.28 ± 0.20	${0.70}_{-0.04}^{+0.03}$	0.90 ± 0.09	0.88 ± 0.15
434148	${9.70}_{-0.13}^{+0.12}$	${0.10}_{-0.08}^{+0.08}$	0.02 ± 0.14	${1.06}_{-0.02}^{+0.02}$	0.92 ± 0.06	<0.84
434082	${9.75}_{-0.07}^{+0.13}$	${0.03}_{-0.03}^{+0.12}$	0.35 ± 0.07	${1.69}_{-0.24}^{+0.03}$	1.74 ± 0.03	2.03 ± 0.04
434126	${9.13}_{-0.09}^{+0.14}$	${0.48}_{-0.12}^{+0.09}$	0.41 ± 0.12	${1.73}_{-0.25}^{+0.05}$	1.39 ± 0.05	1.44 ± 0.10
434139	${9.18}_{-0.07}^{+0.09}$	${0.44}_{-0.16}^{+0.11}$	0.11 ± 0.12	${1.68}_{-0.23}^{+0.05}$	1.09 ± 0.05	<1.41
434145	${9.71}_{-0.06}^{+0.11}$	${0.69}_{-0.08}^{+0.08}$	0.69 ± 0.11	${2.01}_{-0.07}^{+0.02}$	1.77 ± 0.05	<2.04
434242	${8.78}_{-0.20}^{+0.17}$	${0.04}_{-0.04}^{+0.14}$	0.00 ± 0.22	${0.90}_{-0.24}^{+0.08}$	0.66 ± 0.10	1.30 ± 0.14
406390	${9.61}_{-0.13}^{+0.10}$	${0.66}_{-0.16}^{+0.12}$	0.61 ± 0.11	${1.88}_{-0.19}^{+0.09}$	1.62 ± 0.05	<1.73
406444	${10.87}_{-0.02}^{+0.08}$	${0.10}_{-0.07}^{+0.07}$	0.84 ± 0.05	${1.61}_{-0.02}^{+0.02}$	2.26 ± 0.02	2.07 ± 0.06
434227	${10.14}_{-0.06}^{+0.06}$	${0.42}_{-0.25}^{+0.13}$	0.64 ± 0.11	${1.78}_{-0.45}^{+0.07}$	1.86 ± 0.05	<1.75
191932	${9.93}_{-0.06}^{+0.10}$	${0.64}_{-0.18}^{+0.12}$	0.48 ± 0.16	${1.54}_{-0.24}^{+0.03}$	1.24 ± 0.07	<1.16
434547	${9.62}_{-0.21}^{+0.14}$	${0.65}_{-0.18}^{+0.18}$	0.55 ± 0.12	${1.79}_{-0.21}^{+0.31}$	1.49 ± 0.06	1.05 ± 0.08
192129	${10.33}_{-0.03}^{+0.05}$	${0.00}_{-0.00}^{+0.00}$	0.19 ± 0.10	${1.20}_{-0.04}^{+0.03}$	1.37 ± 0.04	1.46 ± 0.06
193914	${9.32}_{-0.08}^{+0.09}$	${0.63}_{-0.15}^{+0.12}$	0.28 ± 0.11	${1.95}_{-0.24}^{+0.06}$	1.30 ± 0.05	⋯
212863	${9.91}_{-0.13}^{+0.10}$	${0.19}_{-0.13}^{+0.16}$	0.56 ± 0.17	${1.03}_{-0.03}^{+0.19}$	1.31 ± 0.08	1.55 ± 0.08
195044	${9.69}_{-0.13}^{+0.07}$	${0.75}_{-0.11}^{+0.16}$	0.55 ± 0.10	${1.91}_{-0.05}^{+0.28}$	1.56 ± 0.05	0.84 ± 0.16
208115	${10.86}_{-0.02}^{+0.02}$	${0.00}_{-0.00}^{+0.00}$	0.01 ± 0.12	${1.11}_{-0.02}^{+0.02}$	1.12 ± 0.05	2.11 ± 0.05
200355	${9.83}_{-0.12}^{+0.11}$	${0.02}_{-0.02}^{+0.14}$	0.45 ± 0.15	${1.09}_{-0.17}^{+0.07}$	1.35 ± 0.07	1.60 ± 0.08
214339	${10.56}_{-0.02}^{+0.02}$	${0.00}_{-0.00}^{+0.00}$	1.16 ± 0.17	${0.91}_{-0.02}^{+0.02}$	2.19 ± 0.08	1.81 ± 0.12
411078	${9.45}_{-0.02}^{+0.02}$	${0.30}_{-0.07}^{+0.07}$	0.00 ± 0.06	${1.91}_{-0.04}^{+0.02}$	1.38 ± 0.03	1.75 ± 0.03
212298	${10.15}_{-0.08}^{+0.11}$	${0.84}_{-0.16}^{+0.14}$	0.81 ± 0.10	${2.10}_{-0.21}^{+0.04}$	1.83 ± 0.04	1.85 ± 0.04
412808	${8.37}_{-0.12}^{+0.10}$	${0.00}_{-0.00}^{+0.10}$	0.08 ± 0.14	${1.14}_{-0.05}^{+0.40}$	0.99 ± 0.06	1.24 ± 0.07
223954	${9.68}_{-0.13}^{+0.12}$	${0.15}_{-0.13}^{+0.18}$	0.24 ± 0.15	${1.17}_{-0.07}^{+0.18}$	1.16 ± 0.07	1.11 ± 0.11
220771	${9.34}_{-0.21}^{+0.18}$	${0.01}_{-0.01}^{+0.17}$	0.00 ± 0.33	${0.64}_{-0.15}^{+0.12}$	0.56 ± 0.15	0.79 ± 0.16
219315	${9.82}_{-0.09}^{+0.09}$	${0.00}_{-0.00}^{+0.00}$	0.00 ± 0.20	${0.79}_{-0.04}^{+0.03}$	0.80 ± 0.09	<0.56
434618	${10.09}_{-0.09}^{+0.08}$	${0.08}_{-0.08}^{+0.09}$	0.32 ± 0.13	${1.18}_{-0.16}^{+0.04}$	1.28 ± 0.06	1.34 ± 0.06
221039	${9.58}_{-0.04}^{+0.05}$	${0.70}_{-0.07}^{+0.07}$	0.41 ± 0.08	${2.01}_{-0.02}^{+0.02}$	1.51 ± 0.03	1.54 ± 0.04
215511	${9.39}_{-0.18}^{+0.12}$	${0.48}_{-0.21}^{+0.24}$	0.32 ± 0.17	${1.31}_{-0.21}^{+0.29}$	1.03 ± 0.08	1.40 ± 0.08
217597	${9.50}_{-0.22}^{+0.15}$	${0.72}_{-0.23}^{+0.26}$	0.50 ± 0.23	${1.38}_{-0.21}^{+0.30}$	1.05 ± 0.10	1.14 ± 0.13
211934	${9.66}_{-0.24}^{+0.20}$	${0.83}_{-0.23}^{+0.35}$	1.61 ± 0.52	${1.19}_{-0.27}^{+0.32}$	1.82 ± 0.23	1.67 ± 0.24
434579	${10.02}_{-0.10}^{+0.10}$	${0.48}_{-0.15}^{+0.10}$	0.36 ± 0.39	${1.31}_{-0.16}^{+0.03}$	1.05 ± 0.17	0.74 ± 0.20
217753	${10.29}_{-0.07}^{+0.09}$	${0.99}_{-0.15}^{+0.17}$	0.80 ± 0.16	${1.78}_{-0.12}^{+0.19}$	1.56 ± 0.07	1.66 ± 0.09
218783	${10.45}_{-0.15}^{+0.10}$	${0.53}_{-0.09}^{+0.12}$	0.72 ± 0.12	${1.67}_{-0.02}^{+0.13}$	1.75 ± 0.05	1.45 ± 0.07
217090	${8.65}_{-0.28}^{+0.07}$	${0.22}_{-0.22}^{+0.13}$	0.00 ± 0.13	${2.14}_{-1.07}^{+0.41}$	0.95 ± 0.06	1.51 ± 0.07
210037	${8.76}_{-0.11}^{+0.27}$	${0.61}_{-0.20}^{+0.18}$	0.77 ± 0.20	${1.83}_{-0.39}^{+0.74}$	1.50 ± 0.09	<1.43
208681	${10.86}_{-0.02}^{+0.02}$	${0.30}_{-0.07}^{+0.07}$	0.35 ± 0.05	${1.86}_{-0.02}^{+0.02}$	1.75 ± 0.02	1.74 ± 0.04

A machine-readable version of the table is available.

Download table as:  DataTypeset image

There are two AGN candidates in the sample that are excluded in the following analysis as nuclear activity is not the main focus of this paper. The object, 413453, has an X-ray counterpart detected in the Chandra-COSMOS survey (CID 3636 in Civano et al. 2011), as well as in the more recent Chandra COSMOS Legacy Survey (Civano et al. 2016; Marchesi et al. 2016). Although the object is not classified as a broad-line AGN in the zCOSMOS-Deep catalog, it actually shows C ivλ1549 emission in the optical spectra. Therefore, the [O iii] and ${\rm{H}}\beta$ emission lines are likely to be contaminated by nuclear activity.

Object 208115 is not detected in X-ray, but it shows very strong [Ne iii] λ3869 emission with [Ne iii] λ3869/[O ii] λ3727 ≃ 2.5 and [O iii] λ4363/${\rm{H}}\gamma$ ≃ 0.7, which appears to be due to AGNs rather than star formation. The median values of [O iii] λ4363/${\rm{H}}\gamma$ for the local SFGs and AGN sample presented by Shirazi & Brinchmann (2012) are 0.13 and 0.39, respectively (M. Shirazi 2016, private communication; see also Francis et al. 1991; Vanden Berk et al. 2001).

In this section, we carry out the SED fitting to the broadband photometry. We started from the PSF-homogenized UltraVISTA photometry catalog (McCracken et al. 2012).¹¹ Since our objects are in a narrow redshift range and have small angular extent (Figure 20 in Appendix B), we adopted uBVrizYJHK aperture magnitudes measured within 2'' apertures. These magnitudes were first corrected for the Galactic extinction based on the calibration of the dust map by Schlafly & Finkbeiner (2011) and an extinction curve by Fitzpatrick (1999) with R_V = 3.1. Then the aperture correction to the total magnitude was made by applying an average difference between aperture and AUTO magnitudes across the used photometric bands. Finally, 0.369 mag was subtracted from B-band magnitude as instructed in the README file of the catalog.¹² Since one of the objects (434579) does not have a counterpart in the UltraVISTA catalog, but was found in the previous CFHT/WIRC K-selected catalog (McCracken et al. 2010), we adopted the photometry from the latter. Optical–near-IR photometry was then matched with Spitzer/IRAC photometry in four channels by the S-COSMOS survey (Sanders et al. 2007) with a search radius of 1 We used 19 aperture flux and corrected it for total flux by using conversion factors listed in the README file of the catalog.

3.3.1. Correction for Emission-line Contributions in Broadband Magnitudes

3.3.2. SED Fitting

SED fitting was carried out for the emission-line-corrected photometry by using the template SED-fitting code ZEBRA+, which is an updated version of the photometric redshift code ZEBRA ¹³ (Feldmann et al. 2006). Redshifts were set to the spectroscopic ones. As templates, we used composite stellar population models generated from the simple stellar population models of Bruzual & Charlot (2003) with a Chabrier initial mass function (IMF; Chabrier 2003). We employed an exponentially declining star formation history (SFH), $\propto \mathrm{exp}(-t/\tau )$, with log τ/year = 8–11 with steps of 0.1 dex. Ages range in log age/year = 6–9.5 with steps of 0.1 dex, where the upper limit of the age is chosen to be an approximate age of the universe at z = 3. Metallicities of 0.2 Z_⊙, 0.4 Z_⊙, and Z_⊙ were used. We also allowed dust extinction with $E(B-V)$ = 0–0.8 mag with steps of 0.05 mag following the Calzetti extinction curve (Calzetti et al. 2000). The median values of stellar mass,¹⁴ SFR, τ, age, A_V, and metallicity and corresponding 68% confidence intervals derived by marginalizing the likelihood distribution were returned as the output. The stellar mass, A_V, and SFR from the SED fitting are shown in Table 3. Figure 22 in Appendix C shows the best-fit template and emission-line-corrected observed photometry.

An exponentially declining SFH may not be the best approximation for SFGs at high redshift (e.g., Renzini 2009; Maraston et al. 2010; Reddy et al. 2012). Although the age derived here is formally meant to be the time elapsed since the onset of star formation, it should actually be regarded as the time interval before the present during which the bulk of stars were formed. Indeed, 85% of the sample show age/τ < 2, which means that the fitting procedure returns a nearly constant SFR. However, in our main analysis of MZR we will use only the stellar mass among the outputs of the SED fitting, as the stellar mass is quite stable against the choice of SFH. For example, employing a constant or delayed-exponential SFHs would change the stellar mass by only 0.1 dex. On the other hand, the SFR and dust extinction will be derived only from the rest-frame UV properties.

Figure 2 compares various outputs from the SED fitting based on the original photometry with those derived from emission-line-corrected photometry, and their distributions. Emission-line-corrected stellar masses (${M}_{\star }^{\mathrm{corr}}$) are generally lower than those from the original photometry (${M}_{\star }^{\mathrm{original}}$), and the effect is more prominent in less massive galaxies. The median difference in stellar mass is $\mathrm{log}{M}_{\star }^{\mathrm{corr}}-\mathrm{log}{M}_{\star }^{\mathrm{original}}=-0.13$ dex. On the other hand, the median differences in SFR, A_V, and age of stellar populations are −0.03, −0.02, and 0.12 dex, respectively.

Zoom In Zoom Out Reset image size

Figure 3 shows the difference in the best-fit parameters as a function of the fraction of emission-line fluxes in the K band. Stellar mass and age show decreasing trends in the difference with increasing emission-line contributions. SFRs are also affected by large emission-line contributions, but the trend appears weaker than those on stellar mass and age. On the other hand, no clear trend can be seen in A_V. This can be understood as longer-wavelength bands are more sensitive to the stellar mass and age, while SFR and dust extinction are primarily captured by the rest-frame UV part of the SED, where emission-line contribution is not important.

Zoom In Zoom Out Reset image size

There are two objects, 434625 and 217090, that have a ∼100% contribution from emission lines in the K band, and they do not seem to follow the general trend of the rest of our sample in Figure 3. These two objects are the only ones without detection both in the K band (after emission-line correction) and in any of the IRAC bands. At z ≃ 3.3, only observing at wavelengths on and beyond the K band can one capture the rest-frame wavelength of >4000 Å, which is essential to properly estimating the mass-to-light ratios of galaxies. This is likely the main reason why the two objects do not follow the trends, especially in stellar mass and age.

All this indicates the importance of a proper assessment of emission-line contribution in near-infrared bands at z > 3 (e.g., Schaerer et al. 2013; Stark et al. 2013). In the following analysis, we adopt ${M}_{\star }^{\mathrm{corr}}$ for the stellar masses of galaxies in our sample.

We computed the UV spectral slope ${\beta }_{\mathrm{UV}}$ defined as ${f}_{\lambda }\propto {\lambda }^{{\beta }_{\mathrm{UV}}}$ by fitting a linear function to the observed broadband magnitudes from the r to the J band, which cover the rest-frame wavelength range 1400 ≲ λ ≲ 2800 Å at z = 3.27. We used the rest-frame wavelengths corresponding to the Galaxy Evolution Explorer (GALEX) far-UV (FUV; λ_c ≃ 1530 Å) and near-UV (NUV; λ_c ≃ 2300 Å) bands to derive β_UV photometrically, following the recipe used in Pannella et al. (2015). For all objects in the sample, this choice of passbands encloses the rest-frame UV wavelengths with which UV slopes are determined. Following Nordon et al. (2013), magnitude errors are further weighted by $1+| {\lambda }_{\mathrm{filter}}-{\lambda }_{\mathrm{FUV}}| /{\lambda }_{\mathrm{FUV}}$, where λ_filter and λ_FUV are the rest-frame central wavelengths of each photometric band and that of GALEX FUV. By using the best-fit relation, ${\beta }_{\mathrm{UV}}$ was derived as

$$\begin{eqnarray}&&{\beta }_{\mathrm{UV}}=\displaystyle \frac{\mathrm{log}({f}_{\lambda ,\mathrm{FUV}}/{f}_{\lambda ,\mathrm{NUV}})}{\mathrm{log}({\lambda }_{\mathrm{FUV}}/{\lambda }_{\mathrm{NUV}})}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&=\;-0.4\displaystyle \frac{({m}_{\mathrm{FUV}}-{m}_{\mathrm{NUV}})}{\mathrm{log}({\lambda }_{\mathrm{FUV}}/{\lambda }_{\mathrm{NUV}})}-2,\end{eqnarray} \tag{ 2 }$$

where m_FUV and m_NUV are the AB magnitudes in the GALEX FUV and NUV bands, respectively (Nordon et al. 2013; Pannella et al. 2015). Then assuming the Calzetti extinction law (Calzetti et al. 2000; Pannella et al. 2015), ${\beta }_{\mathrm{UV}}$ was converted to dust attenuation at λ = 1530 Å with

$$\begin{eqnarray}&&{A}_{\mathrm{UV}}=4.85+2.31{\beta }_{\mathrm{UV}},\end{eqnarray} \tag{ 3 }$$

which assumes an intrinsic (un-extincted) slope ${\beta }_{\mathrm{UV}}$(A_V = 0) = −2.10 (Calzetti et al. 2000).

In Figure 4, the values of A_V from UV and SED fitting are compared, having converted A_UV to A_V by assuming the Calzetti extinction law as shown in Table 3. There appears to be no tight correlation between the two estimates, perhaps because dust extinction degenerates with other parameters such as SFH and age in broadband SED fitting (e.g., Kodama et al. 1999; Michałowski et al. 2014). Two outliers, 214339 and 211934, as indicated in Figure 4, turn out to be the faintest galaxies in our sample. Indeed, the broadband SED of these objects shown in Section C appears to be quite noisy, which makes a robust estimate of ${\beta }_{\mathrm{UV}}$ difficult.

Zoom In Zoom Out Reset image size

Figure 5 shows A_UV versus stellar mass. There is a trend with more massive galaxies being more dust attenuated. The sample of z ≃ 3.4 SFGs from Troncoso et al. (2014) is also shown, with on average a higher dust extinction compared to our sample. Note that in their study extinction was derived from broadband SED fitting, while ours was based on ${\beta }_{\mathrm{UV}}$ slope. Troncoso et al. (2014) specifically selected Lyman break galaxies (LBGs) with redshifts from near-IR spectroscopy, while we selected objects partly based on the availability of spectroscopic redshifts, which automatically puts a limit on B_AB = 25 mag for those galaxies culled from the zCOSMOS-Deep sample. Moreover, targets were selected based on the predicted ${\rm{H}}\beta$ flux of >5 × 10⁻¹⁸ erg s⁻¹ cm⁻². Though this flux limit does not seem to be too high, the combination of these two criteria could result in selecting bluer, less dust-attenuated objects compared to the standard LBG selection employed by Troncoso et al. (2014).

Zoom In Zoom Out Reset image size

Another noticeable feature of Figure 4 is that there is an appreciable number of objects (∼20%) showing A_V close or equal to zero, which seems somewhat at odds with the relatively high SFR of galaxies in our sample. The adopted calibration of the ${\beta }_{\mathrm{UV}}$–A_UV relation in Equation (3) was established at z = 0, but the relation could be different at high redshift for a number of reasons, such as different intrinsic β slope owing to stellar population properties, IMF, SFH (e.g., Reddy et al. 2010; Wilkins et al. 2011), or a different extinction curve than the one assumed here (e.g., Reddy et al. 2015). Castellano et al. (2014) investigated the UV dust extinction for a sample of LBGs at z ≃ 3, obtaining an intrinsic slope of ${\beta }_{\mathrm{UV}}$ = −2.67, which is significantly steeper than widely used values of ${\beta }_{\mathrm{UV}}$ = −2.10 (Calzetti et al. 2000) and ${\beta }_{\mathrm{UV}}$ = −2.23 (Meurer et al. 1999). Indeed, for our galaxies with A_V close or equal to zero the measured ${\beta }_{\mathrm{UV}}$ is close to or slightly steeper than −2.10, either because of measurement errors or because the intrinsic slope is actually steeper than −2.10.

In this section, we derive SFR based on UV and Hβ luminosities. The resulting SFRs are listed in Table 3.

3.5.1. UV Luminosity

We adopt the calibration of the UV luminosity-based SFR from Daddi et al. (2004) after converting from Salpeter IMF to Chabrier IMF by applying an offset of 0.23 dex:

$$\begin{eqnarray}&&{\mathrm{SFR}}_{\mathrm{UV}}({M}_{\odot }\;{\mathrm{yr}}^{-1})=6.65\times {10}^{-29}{L}_{\mathrm{UV},\mathrm{corr}}\ (\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{Hz}}^{-1}),\end{eqnarray} \tag{ 4 }$$

where L_UV,corr is extinction-corrected UV luminosity derived using

$$\begin{eqnarray}&&{L}_{\mathrm{UV},\mathrm{corr}}={L}_{\mathrm{UV},\mathrm{obs}}\times {10}^{0.4{A}_{\mathrm{UV}}}.\end{eqnarray} \tag{ 5 }$$

Here the UV luminosity is measured at rest-frame 1500 Å.

In the left panel of Figure 6, we compare the UV-based SFRs with those from SED fitting, showing that the two measurements agree well with each other.

Zoom In Zoom Out Reset image size

Figure 7 shows SFR as a function of stellar mass for z ≃ 3.3 SFGs. Our sample covers about 2.5 dex in stellar mass from $\mathrm{log}{M}_{\star }/{M}_{\odot }\lesssim 8.5$ to ≃11, essentially following the MS at z = 3.27 derived by Speagle et al. (2014), which also traces the parent sample of z ≃ 3.3 SFGs shown with gray scale. We also derived the best-fit relation between stellar mass and SFR for our sample, finding

$$\begin{eqnarray}&&\mathrm{log}\;\mathrm{SFR}/({M}_{\odot }\;{\mathrm{yr}}^{-1})\\ &&\quad =(1.52\pm 0.05)+(0.49\pm 0.08)\times (\mathrm{log}M/{M}_{\odot }-10),\end{eqnarray} \tag{ 6 }$$

shown with the solid line. Because of the sample selection based either on the availability of spectroscopic redshift or on the predicted ${\rm{H}}\beta$ flux, our sample is likely to be biased against especially low mass and low SFR galaxies. We believe that this is the main reason why we obtained a flatter MS slope compared to that of the parent sample and that of the Speagle et al. (2014) MS. We will use Equation (6) only to separate the sample into bins of M_⋆ and sSFR to create composite spectra (see Section 3.6).

Zoom In Zoom Out Reset image size

For comparison, overplotted in Figure 7 is the sample of z ≃ 3.4 SFGs from Troncoso et al. (2014). Their sample shows an even flatter slope than ours. Again, this could be due to their sample selection based on the LBG technique, which prefers galaxies with blue SEDs and shows a flat SFR–stellar mass relation (e.g., Erb et al. 2006b). Note that their SFRs are based on ${\rm{H}}\beta$ luminosity with extinction correction assuming an extra extinction in the nebular component, i.e., $E{(B-V)}_{\mathrm{neb}}=E{(B-V)}_{\mathrm{star}}/0.44$, while UV-based SFRs are used for our sample. Thus, both our best-fit and Troncoso et al. (2014) distributions appear to be flatter than the canonical MS (Speagle et al. 2014), which can result from a bias favoring low-mass SFGs with above average sSFR and disfavoring high-mass galaxies with high dust extinction. Relatively low dust extinction is indeed obtained for our sample in the analysis above. In particular, we might miss high-metallicity objects owing to this possible bias, if the correlation between metallicity and dust extinction claimed at an intermediate redshift z ≃ 1.6 (Zahid et al. 2014) still holds at z ∼ 3.3.

3.5.2. Hβ Luminosity

We converted ${\rm{H}}\beta$ luminosities to the ${\rm{H}}\alpha$ luminosities, assuming an intrinsic ${\rm{H}}\alpha$/${\rm{H}}\beta$ ratio of 2.86 (Case B recombination with T = 10⁴ K and n_e = 10² cm⁻³; Osterbrock & Ferland 2006). Then the SFR based on ${\rm{H}}\alpha$ was computed following Kennicutt & Evans (2012),

$$\begin{eqnarray}&&\mathrm{log}\;{\mathrm{SFR}}_{{\rm{H}}\alpha }({M}_{\odot }\;{\mathrm{yr}}^{-1})=\mathrm{log}{L}_{{\rm{H}}\alpha }(\mathrm{erg}\;{{\rm{s}}}^{-1})-41.27.\end{eqnarray} \tag{ 7 }$$

For the objects in which ${\rm{H}}\beta$ is not detected with more than 3σ, we adopted 3σ upper limits.

In the original recipe by Calzetti et al. (2000), calibrated with local UV-luminous starbursts, nebular emission lines are attenuated more than stellar light following $E{(B-V)}_{\mathrm{neb}}=E{(B-V)}_{\mathrm{star}}/0.44$. However, there have been various claims in recent years indicating that at high redshift these two components may suffer similar attenuations (e.g., Erb et al. 2006b; Kashino et al. 2013; Pannella et al. 2015; Puglisi et al. 2016). Therefore, we assumed the same amount of attenuation for the stellar light as for emission lines, which actually provides a good agreement between UV-based and ${\rm{H}}\beta$-based SFRs as shown in the left panel of Figure 6.

In addition to the relation between nebular emission lines and stellar continuum extinction, there is an uncertainty in the choice of the extinction curve for nebular emission lines. Related to the original derivation of the Calzetti et al. (2000) relation, the Milky Way extinction curve (Cardelli et al. 1989) or Small Magellanic Cloud (SMC) extinction curve (Gordon et al. 2003) is often preferred (e.g., Steidel et al. 2014). In the estimate of ${\rm{H}}\beta$-based SFRs we use the Calzetti et al. (2000) extinction curve. Note that for our range of A_V, the change between Calzetti et al. (2000) and Cardelli et al. (1989) extinction curves is very minor: at most 2% for the ${\rm{H}}\beta$ flux and 0.03 dex for log([O iii]/[O ii] λ3727).

In the next sections we will compare various properties of the ionized gas with global properties of galaxies, such as stellar mass, SFR, and sSFR. Besides carrying out such a comparison for individual galaxies, we will also compare average properties. For this purpose, we created composite spectra in bins of stellar mass and SFR. The sample, excluding the two AGN candidates, was split into three bins in stellar mass, log M/M_⊙ < 9.5, 9.5 < log M/M_⊙ < 10.0, and log M/M_⊙ > 10.0, and two bins in SFR, above and below the best-fit MS shown as the dashed line in Figure 7, i.e., Δ_MS < 0 and Δ_MS > 0, where Δ_MS ≡ sSFR/sSFR_best-fit.

First, spectra of each object in both J and H band were normalized by total [O iii] λ5007 flux and registered by a linear interpolation to the rest-frame wavelength grid of the 0.25 Å interval, which is slightly finer than the spectral resolution for the highest-redshift object of our sample. We also normalized the corresponding noise spectra by total [O iii] λ5007 flux and registered to the identical rest-frame wavelength grid but interpolated in quadrature. Then composite spectra were constructed by taking an average at each wavelength pixel weighted by the inverse variance. We constructed the associated noise spectra via the standard error propagation from the individual noise spectra.

Emission-line fluxes are measured by fitting a Gaussian to each emission line by assuming a common velocity shift and a velocity dispersion. We fit simultaneously [O ii] λλ3726, 3729, ${\rm{H}}\beta$, [O iii] λλ4959, 5007, and [Ne iii] λ3869, with continuum described by a second-order polynomial. We computed the flux values and their 1σ errors by means of a Monte Carlo simulation with 10³ realizations.

The resulting stacked spectra are shown in Figure 8. We adopt median stellar mass, SFR, and A_V whenever they are used in the subsequent analysis (see Table 4).

Zoom In Zoom Out Reset image size

Table 4.  Properties of Stacked Spectra

Nobj	log M⋆	log SFR	AV	log q	log ne	$12+\mathrm{log}({\rm{O}}/{\rm{H}})$
	(M⋆/M⊙)	(M⊙ yr−1)	(mag)	(cm s−1)	(cm−3)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Above the Best-fit MS
8	9.26	0.97	0.20	${7.77}_{-0.01}^{+0.01}$	${1.92}_{-0.31}^{+0.21}$	${8.15}_{-0.10}^{+0.09}$
6	9.82	1.20	0.34	${7.86}_{-0.01}^{+0.01}$	${2.51}_{-0.10}^{+0.09}$	${8.04}_{-0.12}^{+0.11}$
5	10.29	1.37	0.35	${7.69}_{-0.01}^{+0.01}$	${2.86}_{-0.06}^{+0.05}$	${8.38}_{-0.09}^{+0.08}$
Below the Best-fit MS
7	9.13	1.32	0.28	${7.95}_{-0.00}^{+0.01}$	${2.45}_{-0.06}^{+0.06}$	${7.97}_{-0.10}^{+0.10}$
7	9.66	1.62	0.55	${7.76}_{-0.01}^{+0.01}$	${2.01}_{-0.18}^{+0.15}$	${8.15}_{-0.10}^{+0.09}$
8	10.34	1.86	0.76	${7.74}_{-0.01}^{+0.01}$	${1.92}_{-0.41}^{+0.25}$	${8.41}_{-0.09}^{+0.08}$

Note. (1) Number of objects in the bin; (2) median stellar mass; (3) median SFR; (4) median A_V; (5) ionization parameter; (6) electron density; (7) gas-phase oxygen abundance.

Download table as:  ASCIITypeset image

The primary indicator for gas-phase metallicity, $12+\mathrm{log}({\rm{O}}/{\rm{H}})$, in this study is R₂₃ ≡ ([O ii] λ3726 + [O ii] λ3729 + [O iii] λ4959 + [O iii] λ5007)/${\rm{H}}\beta$ (Pagel et al. 1979; Kobulnicky & Phillips 2003). For the metallicity measurement, we adopt the empirical calibration by Maiolino et al. (2008), in which the low-metallicity regime ($12+\mathrm{log}({\rm{O}}/{\rm{H}})$ ≲ 8.3) is directly calibrated by the electron temperature method. At this metallicity scale, the theoretical calibration has been known to have difficulties in reproducing the observed emission-line ratios (e.g., Kewley & Dopita 2002). This appears to still exist in the recent photoionization models by Dopita et al. (2013), as shown in Figure 9, i.e., calibration lines by Maiolino et al. (2008) show higher ratios at low metallicity than those from Dopita et al. (2013). The line ratios from the stacked spectra (Figure 8) are shown as hexagon symbols in Figure 9, clearly showing that at low metallicities the photoionization models by Dopita et al. (2013) cannot account for all five line ratios simultaneously.

Zoom In Zoom Out Reset image size

For our metallicity estimates, following Maiolino et al. (2008), we used the five extinction-corrected line ratios shown in Figure 9, namely, R₂₃, [O iii] λ5007/${\rm{H}}\beta$, [O iii] λ5007/[O ii] λλ3726, 3729, [O ii] λλ3726, 3729/${\rm{H}}\beta$, and [Ne iii] λ3869/[O ii] λλ3726, 3729, for the metallicity estimate. For the stacked spectra, we corrected for dust extinction with the median A_V derived from the ${\beta }_{\mathrm{UV}}$ slope.

For the metallicity analysis of individual galaxies we removed seven objects without 3σ detection in [O ii] λ3726 + [O ii] λ3729 and the two AGN candidates (one of two is also [O ii] λ3727 nondetection). This leaves us with 35 objects. In the case that either [O iii] λ4959 or [O iii] λ5007 is undetected at the 3σ level, the other [O iii] flux is complemented by assuming an intrinsic line ratio of [O iii] λ5007/[O iii] λ4959 = 3. For eight objects without >3σ ${\rm{H}}\beta$ detection, we used the ${\rm{H}}\beta$ flux estimated from the UV-based SFR, given the relatively tight correlation between the SFRs from the two estimators as shown in Figure 6. [Ne iii] λ3869 is detected for 10 out of the 35 objects.

The gas-phase oxygen metallicity was then derived with the maximum likelihood method, first computing

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i}\;\displaystyle \frac{{(\mathrm{log}{I}_{i,{\rm{M}}08}-\mathrm{log}{I}_{i,\mathrm{obs}})}^{2}}{{\sigma }_{i,\mathrm{obs}}^{2}+{\sigma }_{i,\mathrm{rms}}^{2}},\end{eqnarray} \tag{ 8 }$$

where ${I}_{i,{\rm{M}}08}$ and ${I}_{i,\mathrm{obs}}$ are the ith line ratio from Maiolino et al. (2008) calibration at a given $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ and the one from observed spectra. Further, ${\sigma }_{i,\mathrm{obs}}$ and ${\sigma }_{i,\mathrm{rms}}$ are the errors in the observed line ratio and intrinsic scatter measured as in Jones et al. (2015) for z = 0.8 galaxies. Then, we translated χ² to the likelihood distribution using ${ \mathcal L }\propto \mathrm{exp}(-{\chi }^{2}/2)$. The metallicity and its confidence interval are then defined as the median and the 16th–84th percentiles of the probability distribution, respectively.

Note that the Maiolino et al. (2008) calibration adopted in this study implicitly assumes the ionization parameter at a given metallicity to be independent of redshift, as it is derived for local SFGs. Recent studies of gas-phase metallicity of high-redshift SFGs show an offset in the BPT diagram with higher [O iii]/${\rm{H}}\beta$ ratio at a given [N ii]/${\rm{H}}\alpha$ ratio, which can be ascribed either to an elevated ionization parameters in high-redshift galaxies (e.g., Kewley et al. 2013) or to an enhanced N/O abundance ratio at a fixed O/H ratio (e.g., Cullen et al. 2014; Masters et al. 2014; Nakajima & Ouchi 2014; Yabe et al. 2015; Sanders et al. 2016). Moreover, Sanders et al. (2016) have argued that there is no significant change in ionization parameter at a fixed metallicity from z ∼ 0 to z ∼ 2.3. Unfortunately, the [N ii] λ6583/${\rm{H}}\alpha$ ratio that is used to break the degeneracy of metallicity estimates from R₂₃ cannot be obtained from the ground for the redshift of our galaxies, but will become possible with the James Webb Space Telescope. Maier et al. (2015) suggest that the [O iii] λ5007/Hβ ratio can be used to break the degeneracy, with galaxies with log [O iii] λ5007/${\rm{H}}\beta$ > 0.26 having $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ < 8.6 on the Kewley & Dopita (2002) metallicity scale. Indeed, our sample with $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ < 8.6 always has log [O iii] λ5007/${\rm{H}}\beta$ > 0.26, confirming the Maier et al. (2015) result.

The ionization parameter, q, is defined as $q\equiv {Q}_{{{\rm{H}}}^{0}}/4\pi {R}_{s}^{2}n$, where ${Q}_{{{\rm{H}}}^{0}}$ is the flux of ionizing photons above the Lyman limit, R_s is the Strömgren radius, and n is the local number density of hydrogen atoms (e.g., Kewley & Dopita 2002). The ionization parameter can be derived through a q-sensitive line ratio, O₃₂ ≡ [O iii] λλ4959, 5007/[O ii] λλ3726, 3729 (e.g., McGaugh 1991). Here we adopt a metallicity-dependent [O iii]/[O ii]–q relation of Kewley & Dopita (2002) parameterized by Kobulnicky & Kewley (2004) as

$$\begin{eqnarray}\mathrm{log}q & = & \{32.81-1.153{y}^{2}\\ & & +\;[12+\mathrm{log}({\rm{O}}/{\rm{H}})]\\ & & \times \;(-3.396-0.025y+0.1444{y}^{2})\}\\ & & \times \;\{4.63-0.3119y-0.163{y}^{2}\\ & & +\;[12+\mathrm{log}({\rm{O}}/{\rm{H}})]\\ & & \times \;{(-0.48+0.0271y+0.02037{y}^{2})\}}^{-1},\end{eqnarray} \tag{ 9 }$$

where y = log O₃₂. The results are reported in Table 5.

Note that the ionization parameters derived in this way are not fully self-consistent, as we used the Maiolino et al. (2008) calibration for gas-phase metallicity (see Section 4.1), which implicitly assumed ionization parameters of normal star-forming galaxies in the local universe. Note also that in this and the following analysis all emission-line fluxes have been corrected for dust extinction using E(B − V)_star derived based on ${\beta }_{\mathrm{UV}}$ slope by assuming the Calzetti extinction curve and $E{(B-V)}_{\mathrm{neb}}=E{(B-V)}_{\mathrm{star}}$.

The relatively high spectral resolution of MOSFIRE allows us to resolve the [O ii] λλ3726, 3729 doublet as seen in individual spectra. The ratio of the two [O ii] lines is sensitive to electron density (Osterbrock & Ferland 2006), and we used the PyNeb ¹⁵ package (Luridiana et al. 2015) to compute the electron density n_e of the line-emitting regions of our galaxies. We assumed an electron temperature of T_e = 10⁴ K. At low to intermediate redshift, T_e is indeed observed to be ∼(1–2) × 10⁴ K via the direct measurements of the [O iii] λ4363 line (e.g., Izotov et al. 2006; Nagao et al. 2006; Andrews & Martini 2013; Ly et al. 2014; Jones et al. 2015). Assuming a higher T_e, e.g., 2 × 10⁴ K, the resulting n_e will become systematically higher by ∼0.15 dex.

When one of the [O ii] lines is not detected at the 3σ level, either upper or lower limits of n_e are derived from the 3σ flux limit of the line. In the case that both of the [O ii] lines are detected, we have carried out a Monte Carlo simulation with 500 realizations by perturbing the measured line ratios with the associated 1σ uncertainties. The median and 16th and 84th percentiles of the resulting distribution have been taken as n_e and 1σ confidence interval, respectively.

Figure 10 shows the relation between R₂₃ and O₃₂. The local, z ≃ 0 galaxy sample shown in the background was selected from the OSSY catalog (Oh et al. 2011) as star-forming based on the BPT diagram (Baldwin et al. 1991; Kewley et al. 2006) by requiring all four emission lines (${\rm{H}}\beta$, [O iii] λ5007, ${\rm{H}}\alpha$, and [N ii] λ6583), as well as [O ii] λ3727 with S/N > 3. The higher line ratios of z ∼ 3.3 galaxies relative to the local SFGs indicate that our galaxies have a higher ionization parameter, on average (Nakajima & Ouchi 2014; Shirazi et al. 2014). Our z = 3.3 galaxies lie along the tail of the local distribution and extend it to higher log R₂₃ and log O₃₂ values, typical of SFGs at z = 2–3 (e.g., Henry et al. 2013; Nakajima & Ouchi 2014; Sanders et al. 2016). Locally, the tail of the distribution consists of metal-poor galaxies, typically with $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ ≲ 8.5. Since the majority of our sample also shows $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ ≲ 8.5 (See Section 4.1), we argue that the ionization parameter could be similar at a given metallicity in both low- and high-redshift galaxies, consistent with Sanders et al. (2016).

Zoom In Zoom Out Reset image size

Figure 11 compares various galaxy physical quantities with the ionization parameter. The strong correlation between log q and $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ is trivial, as both $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ and log q strongly depend on [O iii]/[O ii]. Among the other parameters shown in Figure 11, we do not find significant correlations between any of them, with the exception of the SFR–log q plot. The Spearman's correlation coefficient between SFR and log q is −0.35, corresponding to a two-sided p-value of 0.04, when considering objects without an upper or lower limit in log q. A similar but more significant correlation between SFR and O₃₂ is found by Sanders et al. (2016) for z ∼ 2.3 SFGs, with a two-sided p-value of 0.002. This correlation can be understood as a result of a correlation between SFR and metallicity and an anticorrelation between ionization parameter and metallicity (Pérez-Montero 2014; but see Dors et al. 2011). Nakajima & Ouchi (2014) and Sanders et al. (2016) found correlations of O₃₂ with stellar mass and sSFR, as well as with SFR. While we do not see such correlations for individual objects, stacked data points show some hints of a similar correlation for log q with stellar mass and sSFR.

Zoom In Zoom Out Reset image size

In Figure 12 we also compare various physical parameters with the electron density measured from the line ratio of the [O ii] λλ3726, 3729 doublet. The measured electron densities of ∼100–1000 cm⁻³ are about one order of magnitude higher than those of typical SFGs at z = 0, roughly consistent with those reported for other high-redshift galaxies (e.g., Masters et al. 2014; Shirazi et al. 2014; Shimakawa et al. 2015; Sanders et al. 2016). In Figure 12 there is no indication of strong correlations in any of the parameters with the electron density, which is also confirmed by the very low Spearman's rank correlation coefficients, indicating less than 1σ significance, for individual objects and for stacked points.

Zoom In Zoom Out Reset image size

Shimakawa et al. (2015) measured the electron density of a sample of ${\rm{H}}\alpha$ narrowband-selected SFGs at z = 2.5 through the resolved [O ii] λ3727 doublet. They found a positive correlation with a 4σ significance between n_e and sSFR. However, from panel (c) of Figure 12, the correlation between sSFR and n_e does not seem to be strong in our sample. If at all, there is a hint for an opposite trend with higher-sSFR galaxies with lower n_e. Based on a larger sample at z ∼ 2.3, Sanders et al. (2016) also do not find any correlation of electron density with stellar mass, SFR, or sSFR.

The [O ii] λ3727 emission line luminosity has been used as an indicator of SFR (e.g., Kennicutt 1998), though it depends not only on SFR but also on metallicity and ionized gas properties. Figure 13 shows the relation between the [O ii] luminosity and the UV-based SFR. Here, both quantities were extinction corrected using UV-based estimates and assuming $E{(B-V)}_{\mathrm{neb}}=E{(B-V)}_{\mathrm{star}}$ as before. The [O ii] luminosities of our sample do not seem to follow the calibration of SFR([O ii]) by Kewley et al. (2004), shown as the dashed line in Figure 13. We derived the best-fit calibration of [O ii] λ3727 SFR for our z ≳ 3 MS galaxies as

$$\begin{eqnarray}&&\mathrm{log}\;{\mathrm{SFR}}_{[{\rm{O}}{\rm{II}}]}({M}_{\odot }\;{\mathrm{yr}}^{-1})=\mathrm{log}\;{L}_{[{\rm{O}}{\rm{II}}]}(\mathrm{erg}\;{{\rm{s}}}^{-1})-41.17,\end{eqnarray} \tag{ 10 }$$

having fixed the slope to unity. Our best fit gives 0.22 dex lower SFRs compared to the Kewley et al. (2004) calibration. If the extinction toward the H ii region were higher, like the original Calzetti law, i.e., $E{(B-V)}_{\mathrm{gas}}=E{(B-V)}_{\mathrm{star}}/0.44$, then the discrepancy would become more prominent.

Zoom In Zoom Out Reset image size

The elevated [O ii] luminosity relative to the local relation could be due to a change in the physical condition of star-forming regions as discussed above, i.e., higher ionization parameter and electron density in high-redshift galaxies.

Figure 14 shows the MZR for our galaxies, both individually and for the stacked spectra, in comparison with the MZRs at lower redshifts. The MZRs at z = 0.07, 0.7, and 2.2 are from Tremonti et al. (2004), Savaglio et al. (2005), and Erb et al. (2006a), respectively, converted to the same metallicity calibration, and parameterized by Maiolino et al. (2008). We also plot the MZR at z ≃ 3.4 from Troncoso et al. (2014) and that at z = 0 taken from Mannucci et al. (2010). The majority of our sample follows the previously defined MZR at z ≃ 3.4, i.e., our MZR offsets by ≃0.7 dex and ≃0.3 dex from those at z ≃ 0 and z ≃ 2, respectively. There are, however, a few objects showing higher metallicity, by up to 0.3–0.4 dex compared to the stack points. Owing to a small sample size especially at M_⋆ > 10^10.5 M_⊙, we do not attempt to constrain the turnover mass of MZR here. Instead, we carried out a linear regression and found the best-fit MZR for the individual objects in our sample as

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})\\ &&\quad =\;(8.36\pm 0.03)+(0.31+0.05)\times (\mathrm{log}{M}_{\star }/{M}_{\odot }-10),\end{eqnarray} \tag{ 11 }$$

as shown in the left panel of Figure 14 with a dashed line.

Zoom In Zoom Out Reset image size

In Figure 14 we plot ranges between minimum and maximum stellar masses in each bin as error bars for the stack points. The metallicities from the stacked spectra appear to be lower than the average or median of the individual measurements at a given stellar mass bin. This may be due to the way they were employed for stacking: we normalized each spectrum by the total [O iii] λ5007 flux and then stacked with weights proportional to the inverse variance. This procedure gives more weights for [O iii] bright objects, which tend to have lower metallicities. Indeed, as shown in Figure 23 in Appendix D, objects with higher S/N in [O iii] tend to have lower metallicity. The same trend is also seen in Troncoso et al. (2014) for SFGs at z ≃ 3.4. Their best-fit MZR for the average of individual measurements shows higher metallicity than that of the measurement on the stacked spectra (dot-dashed and dotted lines in the left panel of Figure 14, respectively). We also tried stacking with the [O iii] normalization but without weighing, and measuring metallicity in the same way, finding about 0.1 dex higher $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ compared with those presented above. However, our conclusion does not depend on the choice of the stacking method since our analysis on the metallicity is based mainly on individual objects.

Figure 15 shows that there appears to be no correlation between $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ and SFR relative to the distances from the best-fit MZR and MS within the error bars for both stacked and individual points, respectively. A similar behavior has been reported for z ∼ 2 SFGs (Steidel et al. 2014; Wuyts et al. 2014; Sanders et al. 2015), while Zahid et al. (2014) found an anticorrelation between SFR and metallicity in a sample of SFGs at z ∼ 1.6 (see also Yabe et al. 2015). This suggests that the role of SFR as a second parameter in the MZR may be less important at z ≃ 3.3 compared to that in the local universe (e.g., Mannucci et al. 2010; Andrews & Martini 2013). We shall return to this issue in Section 5.4.

Zoom In Zoom Out Reset image size

The redshift evolution of metallicity at a stellar mass of 10¹⁰ M_⊙ is illustrated in the right panel of Figure 16. For the comparison, data points at different redshifts are taken from Tremonti et al. (2004) for z = 0.07, Savaglio et al. (2005) for z = 0.72, and Erb et al. (2006a) for z = 2.2 after being corrected to the same metallicity calibration used here by Maiolino et al. (2008), Troncoso et al. (2014) for z = 3.4, Zahid et al. (2014) for z = 1.55, Henry et al. (2013) for z = 1.7, Cullen et al. (2014) for z = 2.2, Steidel et al. (2014) for z = 2.3, and Sanders et al. (2015) for z = 2.3. Note that all these data are converted to the metallicity calibration of Maiolino et al. (2008), when needed.

Zoom In Zoom Out Reset image size

Mannucci et al. (2010) found that in the local universe SFGs lie close to a 3D surface in the space with SFR, stellar mass, and metallicity as coordinates (see also Lara-López et al. 2010). They also found that SFGs lie close to the same surface at least to z ≃ 2.5, suggesting the existence of the so-called FMR, and showed that a projection of the surface over the plane with coordinates $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ and ${\mu }_{0.32}\equiv \mathrm{log}M/{M}_{\odot }-0.32\mathrm{log}\;\mathrm{SFR}/({M}_{\odot }\;{\mathrm{yr}}^{-1})$ is able to minimize the scatter about the surface itself. Figure 17 shows such a 2D plane, where most of the objects in our sample are offset from the locally defined FMR by ≃0.3 dex.

Zoom In Zoom Out Reset image size

Comparing the left panel in Figures 14 and 17, it is not obvious whether in the case of our galaxies using μ_0.32 as a coordinate reduces the scatter in metallicity in the MZR, as it does locally. We left the α in ${\mu }_{\alpha }\equiv \mathrm{log}({M}_{\star })-\alpha \mathrm{log}(\mathrm{SFR})$ as a free parameter and computed the standard deviation around the best-fit linear regression in the $12+\mathrm{log}({\rm{O}}/{\rm{H}})$–μ_α relation for our sample. The scatter as a function of α is shown in Figure 18. Mannucci et al. (2010) claimed that at z = 0 with α = 0.32 the scatter decreases to ≃0.02 dex compared to ≃0.06 dex in the case of the simple MZR (i.e., α = 0). In contrast, varying α does not seem to reduce the observed scatter around the best-fit line for our z = 3.3 galaxies, but it remains essentially constant around 0.15 dex. We should note that the typical uncertainties of metallicity from strong lines are in general comparable to the scatter in the MZR, in this study and in the literature (e.g., Marino et al. 2013; Steidel et al. 2014), which may explain why by varying α the scatter does not change. Indeed, the number of objects in our sample is still too small to overcome the dominance of the measurement error and to draw firm conclusions on the intrinsic scatter of our z ∼ 3.3 MZR. To push down the sampling error, orders of magnitude larger samples would be required, e.g., ≳10⁵ objects are used in the study of Mannucci et al. (2010) for the local galaxies, whereas z ≳ 2 samples include only ≳10^1–2 objects.

Zoom In Zoom Out Reset image size

Because of the redshift evolution in both the MS and metallicity, our sample spans a parameter space that is not well covered by the local galaxy population (Maier et al. 2014). Therefore, an extrapolation of the $Z({M}_{\star },\mathrm{SFR})$ relation is inevitable when trying to compare the local relation with the high-redshift one. A more physically motivated relation among the three parameters, $Z({M}_{\star },\mathrm{SFR})$, is proposed by Lilly et al. (2013), which indeed can reproduce the FMR at z = 0 and at z ≃ 2.3 with a sensible choice of parameters. We then use the $Z({M}_{\star },\mathrm{SFR})$ relation in Equation (40) in Lilly et al. (2013):

$$\begin{eqnarray}&&{Z}_{\mathrm{eq}}({M}_{\star },\mathrm{SFR})={Z}_{0}\\ &&\phantom{\rule{1.em}{0ex}}+\;\displaystyle \frac{y}{1+\lambda {(1-R)}^{-1}+{\varepsilon }^{-1}\left\{(1+\beta -b)\mathrm{SFR}/{M}_{\star }-{(1-R)}^{-1}\displaystyle \frac{1.2}{t}\right\}},\end{eqnarray} \tag{ 12 }$$

where Z_eq is the equilibrium metallicity, Z₀ is the metallicity of the incoming gas, y is the chemical yield, λ is the mass-loading factor (≡ outflow rate/SFR), R is the fraction of mass return due to stellar evolution, ε is the star formation efficiency (SFE ≡ SFR/M_gas), β is the MS slope defined as $\mathrm{SFR}\propto {M}_{\star }^{1+\beta }$, and t is the age of the universe in units of Gyr. In Lilly et al. (2013) λ and ε are parameterized as follows:

$$\begin{eqnarray}&&\lambda ={\lambda }_{10}{m}_{10}^{a}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\varepsilon ={\varepsilon }_{10}{m}_{10}^{b},\end{eqnarray} \tag{ 14 }$$

where m₁₀ is the stellar mass in units of 10¹⁰ M_⊙.

Following Lilly et al. (2013), we fix the ε and λ parameters to the values that reproduce the local FMR of Mannucci et al. (2010), and then we use Equation (12) to derive the MZR at z = 3.3 predicted by this regulator model. The result is shown in Figure 14, with the orange shaded area encompassing the cases between Z₀/y = 0 and Z₀/y = 0.1.

Observations suggest that the SFE may be higher at higher redshift (e.g., Daddi et al. 2010; Genzel et al. 2010). We also show with the green shaded area in Figure 14 the model predictions in the case for $\varepsilon \propto (1+z)$. These two cases virtually enclose our z ≃ 3.3 sample, while the majority of them are consistent with the prediction with nonevolving ε.

In Section 5.4 we have shown that the scatter of the MZR does not change appreciably by introducing the SFR as a second parameter. Together with the observed data, Figure 15 shows the corresponding relations as predicted by the regulator models, as well as those observationally defined and extrapolated for the MS at z = 3.3. In general, compared to our data, a steeper dependence is predicted for the metallicity offset from the MZR as a function of the SFR offset from the MS. Although this apparent mismatch may be due to large errors and small sample size, it may give some insight into the functional form of the SFH of the galaxies studied here. Indeed, the dependence of metallicity on SFR would disappear if galaxies were evolving at constant SFR, i.e., $d\mathrm{SFR}/{dt}\simeq 0$ (see Appendix E for details).

In Figure 16 we compare the redshift evolution of the MZR at M_⋆ = 10¹⁰ M_⊙ with the model predictions described above. The evolving $Z({M}_{\star },\mathrm{SFR})$ relation assuming ε ∝ (1 + z) traces the observed trend up to z ≃ 2.5, with an exception for the data point of Cullen et al. (2014). As mentioned above, our sample appears to prefer a nonevolving star formation efficiency, as do the data by Maier et al. (2014) at z ∼ 2.3. However, the observational scatter of ≳0.1 dex for the high-redshift measurements may hamper us from distinguishing models at better than the ≃2.5σ level. On the other hand, the departure in metallicity from the original z = 0 FMR by Mannucci et al. (2010), as seen in Figure 17, can be due to the extrapolation of the local FMR into a parameter space that is basically unpopulated at z = 0. Perhaps a more suitable γ would lie in between the two cases above, 0 and 1. For instance, a recent study by Genzel et al. (2015) derived γ = 0.34 ± 0.15, based on a combined analysis of CO and dust scaling relations at 0 ≲ z ≲ 3. Of course, the other parameters of the regulator model and their mass dependence may also evolve with redshift, hence being different at z ≃ 3.3, but the current sample size is too small to explore the entire parameter space.