AGES OF MASSIVE GALAXIES AT 0.5 > z > 2.0 FROM 3D-HST REST-FRAME OPTICAL SPECTROSCOPY

The 3D-HST program (van Dokkum et al. 2011; Brammer et al. 2012) is a 625 arcmin² survey using HST to obtain low-resolution near-infrared spectra for a complete and unbiased sample of thousands of galaxies (Cycles 18 and 19, PI: van Dokkum). It observes the AEGIS, COSMOS, GOODS-S, and UDS fields with the HST/WFC3 G141 grism over 248 orbits, and it incorporates similar, publicly available data in the GOODS-N field (GO: 11600; PI: Weiner). These fields coincide with the area covered by CANDELS (Grogin et al. 2011; Koekemoer et al. 2011) and have a wealth of publicly available imaging data (U band to 24 μm). The 3D-HST photometric catalog is described in Skelton et al. (2014), and it constitutes a fundamental step in interpreting the spectra that often contain only a single emission line, if any, by providing a photometric redshift prior to the redshift fitting.

The WFC3 grism spectra have been extracted with a custom pipeline, described in Momcheva et al. (2015). Redshifts have been measured via the combined photometric and spectroscopic information using a modified version of the EAZY code (Brammer et al. 2008). The precision of redshifts is shown to be $\sigma \left(\tfrac{{dz}}{1+z}\right)=0.3\%$ (Brammer et al. 2012; Momcheva et al. 2015; see Kriek et al. 2015 for a comparison based on the new MOSFIRE redshifts from the MOSDEF survey).

Stellar masses have been determined using the FAST code by Kriek et al. (2009), using Bruzual & Charlot (2003) models and assuming exponentially declining star-formation histories (SFHs), solar metallicity, a Chabrier (2003) initial mass function (IMF), and a Calzetti (2000) dust law.

We separate quiescent galaxies from star-forming galaxies using a color–color technique, specifically rest-frame $U-V$ versus rest-frame $V-J$ (hereafter: UVJ diagram). It has been noted in the past that selecting quiescent galaxies and star-forming galaxies based on a single color is not reliable because heavily reddened star-forming galaxies can be as red as quiescent galaxies (among others, Williams et al. 2009). Adding information from a second color ($V-J$) makes it possible to empirically distinguish between galaxies that are red in $U-V$ because of an old stellar population featuring strong Balmer/D4000 breaks (which are relatively blue in $V-J$) and galaxies that are instead red in $U-V$ because of dust (and therefore are red in $V-J$ too).

The UVJ diagram has been widely used in a variety of high-redshift studies (e.g., Wuyts et al. 2007; Williams et al. 2009; Bell et al. 2012; Gobat et al. 2013). It has been shown to correspond closely to the traditional morphological classes of early-type and late-type galaxies up to at least z ∼ 1 (Patel et al. 2012) and is able to select dead galaxies with low mid-infrared fluxes (Fumagalli et al. 2014).

Effectively, quiescent galaxies are identified with the criteria $(U-V)\gt 0.8\times (V-J)+0.7$, U − V > 1.3, and $V-J\lt 1.5$ (as in Whitaker et al. 2014). The separating lines are chosen with the main criteria that they lie roughly between the two modes of the population seen in Figure 1.

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We select galaxies more massive than $\mathrm{log}({M}_{*}/{M}_{\odot })\gt 10.8$. In order to achieve a sample of high-quality spectra, we exclude spectra contaminated by neighboring objects for more than 10% of their total flux, with a wavelength coverage lower than 80% of the full regime of 1.1–1.7 μm, and with a fraction of bad pixels higher than 10%. All of these quantities are listed in the 3D-HST catalogs. The final sample contains 572 galaxies between redshift 0.5 and 2.0. Figure 1 shows the selection of massive galaxies, divided into star-forming galaxies and quiescent galaxies, in three redshift bins, superposed on the entire population of galaxies from 3D-HST at the same redshift.

Figure 2 (left) shows all the spectra in the sample in observed wavelength sorted by redshift and divided into quiescent galaxies (top) and star-forming galaxies (bottom). Spectra are stacked in 50 redshift bins with a roughly exponential spacing. We show the number of galaxies in each bin in the histograms on the right. We see emission and absorption lines being shifted in the wavelength direction and entering and exiting the observed range at different redshifts. For instance, the Hα line enters the wavelength range of the WFC3 grism at z ∼ 0.7 and exits at z ∼ 1.5. We notice that the subdivision into star-forming galaxies and quiescent galaxies corresponds well to a selection on the presence of emission lines. In the quiescent galaxies sample (Figure 2, left), there are no obvious emission lines visible, while at different redshifts we observe deep absorption bands (Ca ii, Mg, Na, TiO). The star-forming galaxies sample (Figure 2, right) features strong emission lines, such as the already mentioned Hα, and Hβ and [O iii] at higher redshift. Significantly, some absorption bands are also detectable in the star-forming galaxy (SFG) sample.

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This experiment proves the quality of 3D-HST grism redshifts. As a comparison, we show in the right column of Figure 2 the effect of lower-quality redshifts on the stacking procedure, by using photometric redshifts instead of grism redshifts. Even though the photometric redshifts provided in the 3D-HST photometric catalogs (Skelton et al. 2014) reach an excellent absolute deviation from spectroscopic redshifts of just σ = 1%–2% (depending on the field), this level of precision is not good enough to clearly observe the spectral features. Figure 2 shows that the scatter induced by the less accurate photometric redshifts blurs the emission and absorption lines. As an example, [O iii] is very difficult to track in emission-line galaxies if photometric redshifts are used.

This experiment demonstrates that stacking of grism spectra requires the subpercent precision in redshift achieved with the 3D-HST z_grism.

In individual galaxies in our sample, spectral features are often too weak to be used for reliable measurements of stellar population parameters. We therefore achieve the necessary signal-to-noise ratio by stacking spectra in three redshift bins, and in the two populations of star-forming galaxies and quiescent galaxies, as follows. We shift the spectra to rest frame and fit the continuum in each spectrum with a third-order polynomial. In this process we mask regions around known strong emission lines. We normalize the spectra by dividing them by the best-fit polynomial. We next determine the median flux of the normalized rest-frame spectra in a grid of $20\;\mathring{\rm A}$. Errors on the stacks are evaluated via bootstrapping: we perform 100 realizations of each sample by drawing random galaxies from the original sample (repetitions are possible), and we perform the stacking analysis on each resampling. The uncertainty in the flux measurement of each wavelength bin is given by the dispersion of the flux values in the resampled stacks.

The composite spectra are shown in Figure 3 (for quiescent galaxies) and Figure 4 (for star-forming galaxies). Each individual stack is made from the sum of at least 75 galaxies. As the observations cover a constant range of observed wavelength ($11000\;\mathring{\rm A} \lt \lambda \lt 16000\;\mathring{\rm A}$), we probe different rest-frame wavelength regimes at different redshifts. The strongest features are the emission lines of Hα and [O iii](λ = 5007 Å) with a peak strength of 15% over the normalized continuum. The absorption lines have depths of 5% or less. The features are weak due to the limited resolution of the spectra.

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We compare the stacked spectra with predictions from stellar population synthesis models (SPS hereafter). Our goals are to test whether different SPS codes can reproduce the absorption-line properties of high-redshift galaxies and to infer stellar ages for these galaxies. We use models from Bruzual & Charlot (2003, BC03), Conroy & Gunn (2010, Flexible Stellar Population Synthesis, FSPS10), and C. Conroy et al. (2016, in preparation, FSPS-C3K).

We use the most standard settings for each SPS code. The BC03 models are based on the Padova stellar evolution tracks and isochrones (Bertelli et al. 1994); they use the STELIB empirical stellar library (Le Borgne et al. 2003) for wavelengths in the range $3200\;\mathring{\rm A} \lt \lambda \lt 9500\;\mathring{\rm A}$ and the BaSeL library of theoretical spectra elsewhere.

The FSPS10 models are based on a more updated version of the Padova stellar evolution tracks and isochrones (Marigo et al. 2008); they use the MiLeS empirical stellar library for wavelengths in the range $3500\;\mathring{\rm A} \lt \lambda \lt 7500\;\mathring{\rm A}$ and the BaSeL library of theoretical spectra elsewhere.

We have also considered a new, high-resolution theoretical spectral library (FSPS-C3K, C. Conroy et al. 2016, in preparation). This library is based on the Kurucz suite stellar atmosphere and spectral synthesis routes (ATLAS12 and SYNTHE) and the latest set of atomic and molecular line lists. The line lists include both lab and predicted lines, the latter being particularly important for accurately modeling the broadband spectral energy distribution shape. The grid was computed assuming the Asplund (2009) solar abundance scale and a constant microturbulent velocity of 2 km s⁻¹.

For each observed stack, we perform a least-squares minimization using the three different models to find the best-fit age of the stack. In order to compare the high-resolution models with the low-resolution stacks, we need to downgrade the models to 3D-HST resolution. The dispersion of the G141 grism is $46\;\mathring{\rm A}$ pixel⁻¹ (R ∼ 130 in the raw data) with a raw pixel scale of 0.12 arcsec, sampled with 0.06 arsec pixels. The lack of a slit, combined with the low spectral resolution of the WFC3 grism, implies that emission and absorption "lines" are effectively images of the galaxy at that particular wavelength (see Nelson et al. 2015 for a detailed explanation). In a continuum spectrum, this means that the spectral resolution is different for each galaxy because it is determined by its size. This effect is generally referred to as "morphological broadening" in the slitless spectroscopy literature (e.g., van Dokkum et al. 2011; Whitaker et al. 2013, 2014). We simulate the expected morphological broadening by convolving the high-resolution model spectra with the object morphology in the H_F140W continuum image collapsed in the spatial direction. Model spectra for each galaxy in the sample are continuum-divided and stacked with the same procedure we use for observed spectra. In the fitting of models to data, we allow an additional third-order polynomial continuum component with free parameters.

In summary, for each sample of galaxies, we create mock 3D-HST stacks based on three stellar population models (BC03, FSPS10, FSPS-C3K) with solar metallicity, Chabrier IMF, and three different star-formation histories (single stellar burst, constant star formation, and an exponentially declining model with τ = 1 Gyr), with a spacing of 0.1 Gyr in light-weighted age. Unless specified otherwise, all ages quoted throughout the paper are light-weighted ages. For a single stellar population (SSP), this age is identical to the time elapsed since the beginning (and end) of star formation because all stars have the same age. In other models, this age is some time in between the time elapsed since the onset of star formation and the ages of the youngest stars (see, e.g., van Dokkum et al. 1998).

Our choice of performing the stacking and the fitting analysis on continuum-divided spectra is motivated by the goal of measuring ages of galaxies without being influenced by the slope of the continuum, which is degenerate in dust and age. Aging stellar populations have redder broadband colors, but dust reddening and increasing metallicity have a similar effect. For instance, the difference in g − r color, corresponding to 1 Gyr of passive aging within the BC03 models, can also be caused by 0.5 mag of dust reddening following the Calzetti et al. (2000) dust law or by an increase in metallicity from log(Z) = 0.02 (solar) to log(Z) = 0.05.

In this study, we do not treat galaxies hosting an active galactic nucleus (AGN) separately. We test the influence of AGNs by selecting all sources falling in the InfraRed Array Camera (IRAC) color–color selection presented in Donley et al. (2012). The IRAC-selected AGNs count for less than 5% of each sample of quiescent or star-forming galaxies at different redshifts. The conclusions of the paper do not change when these sources are removed from the stacks.

Figure 5 shows for each of the models (BC03, FSPS10, FSPS-C3K) the best fits to the quiescent galaxy stack for the lower redshift bin (0.5 < z < 1.0). The gray shaded area represents the area around Hα masked in the fitting. With BC03 (Figure 5, red), the best fit is very poor at wavelengths higher than 7500 Å, as shown by the residuals in the lower panel. Moreover, the reduced χ² (χ_red²) of the best fit is high (11.46). Using FSPS10 (green), the best fit also has significant ($\gt 3\%$) residuals at the reddest wavelengths ($\gt 8000\;\mathring{\rm A}$) and around the $\gt 7000\;\mathring{\rm A}$ regime, where the first TiO band lies. The ${\chi }_{\mathrm{red}}^{2}$ value of the best fit is still high (8.5). Finally, using the latest FSPS-C3K models (Figure 5, bottom left), the best fit converges with a lower ${\chi }_{\mathrm{red}}^{2}=5.8$, and residuals are consistently below 2% over the entire wavelength range. We compare residuals from different SPS models (data minus model) in the bottom panel of Figure 5. All best fits have positive residuals in the $7000\;\mathring{\rm A}$ region (up to 4% for FSPS10), underestimating the fluxes at those wavelengths. At wavelengths higher than $8000\;\mathring{\rm A}$, the BC03 models have positive residuals, while FSPS10 and FSPS-C3K have negative ones.

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Moving to higher redshifts, the quality of fits with different SPS models is comparable. In the intermediate (1.0 < z < 1.5, Figure 6) and in the high-redshift bin (1.5 < z < 2.0, Figure 7) we examine, all χ_red² values range from 2.8 to 3.3. Residuals in these redshift bins are comparable among different models and tend to be smaller than a few percent. We evaluate residuals corresponding to the Hα line in quiescent galaxies in Section 6.3.

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The stellar ages of quiescent galaxies implied by the best-fit models vary according to the SSP used. In order to evaluate the uncertainty of the age measurement, we bootstrap the sample 100 times and repeat the fitting analysis on the bootstrapped realizations of the stack.

At the lowest-redshift bin (0.5 < z < 1.0, Figure 5), we obtain a stellar age of 3.8 ± 0.6 Gyr with BC03, a younger age (2.4 ± 0.4 Gyr) with FSPS10, and again 4.0 ± 0.2 Gyr with FSPS-C3K (which is the model with the lowest residuals). A similar wide range of age determinations is obtained for the intermediate-redshift bin (Figure 6), ranging from 1.4 ± 0.1 Gyr for FSPS10 to 2.0 ± 0.3 Gyr for FSPS-C3K and to 3.8 ± 0.8 Gyr for BC03.

In the highest-redshift bin (1.5 < z < 2.0, Figure 7), all the age determinations are between 1.2 and 1.4 Gyr. This value is consistent with Whitaker et al. (2013), who studied a sample of galaxies at slightly different masses and redshifts ($10.3\lt \mathrm{log}({M}_{*}/{M}_{\odot })\lt 11.5$, 1.4 < z < 2.2), obtaining an age of 1.25 Gyr computed with the Vakzdekis models. We also agree with Mendel et al. (2015), who investigated the stellar population of 25 massive galaxies with VLT-KMOS, deriving a mean age of ${1.08}_{-0.08}^{0.13}$ Gyr.

Our study relies on the assumption that quiescent galaxies already have a solar metallicity at high redshift. This assumption is supported by the study of Gallazzi et al. (2014), who studied 40 quiescent galaxies at 0.65 < z < 0.75 with IMACS spectra, obtaining a mass–metallicity relation consistent with that at z = 0 from the Sloan Digital Sky Survey (SDSS).

We explore the effect of changing metallicity in Figure 8 on the stack with the highest signal-to-noise ratio. We use the stellar population model that gave the lowest χ_red² with the standard solar metallicity (FSPS-C3K), and we vary the metallicity to twice solar and half solar. The best fit with a Z = 0.5 Z_☉ metallicity has a χ²_red marginally lower than that with a solar metallicity (4.6 versus 5.6), while the Z = 2 Z_☉ case can be excluded by its higher χ²_red = 8.1. We moreover remark that, as seen in Section 4.1, none of the models that we explore properly fit the data at rest-frame wavelengths redder than 7500 Å: at those wavelengths, lines such as TiO and Ca ii could help in constraining metallicities (e.g., Carrera et al. 2007; Conroy et al. 2014), but improved models or higher-resolution spectroscopy are required.

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At higher redshifts, we find that the uncertainty on the ages due to the uncertainty in the metallicity can amount to a factor of two, with the Z = 2 Z_☉ and Z = 0.5 Z_☉ best fits being respectively younger and older than those with a solar metallicity, as expected from the well-known degeneracies between age and metallicity.

This metallicity uncertainty dominates the error budget.

Figure 9 summarizes the best fits to the SFG sample at 0.5 < z < 1.0 obtained with these combinations of models, letting the age t vary. We again mask a 400 Å wide region around Hα in the fit. According to the χ_red² statistics, the models that assume an exponentially declining SFH provide marginally better fits than models that assume a CSF.

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Figures 10 and 11 show the best fits to the SFG samples at 1.0 < z < 1.5 and 1.5 < z < 2.0 obtained with the same combination of models. We mask 400 Å–wide regions around the expected strongest emission lines ([O iii], Hα). In the highest-redshift stack, we see the biggest residual corresponding to the wavelength of the Hγ line ($\lambda =4341\;\mathring{\rm A}$). For these redshifts, the FSPS10 and FSPS-C3K models have a lower best-fit ${\chi }_{\mathrm{red}}^{2}$ than that of BC03. However, the models fit almost equally well.

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Star-forming galaxies of these masses are in fact known to follow declining star-formation histories at redshifts lower than 1.5 (e.g., Pacifici et al. 2012). As in the case of quiescent galaxies, different models have qualitatively and quantitatively different residuals. In this case, however, BC03 is the model with the smallest residuals, especially at the longest wavelengths ($\gt 8500\;\mathring{\rm A}$).

The effect of varying metallicity is explored in Figure 12 and compared to that of varying stellar population models. We perform this test on star-forming galaxies at the lowest redshift bin (0.5 < z < 1.0), where the signal-to-noise ratio is high. Figure 12 shows a comparison between residuals from best-fit FSPS-C3K models with solar-metallicity models (Z = 0.190) and models with half of the solar metallicity (Z = 0.096), for a 1 Gyr τ model SFH and a constant SFR. We notice that the difference in ages is smaller than 10% and that residuals do not vary significantly. We conclude that the difference between different SPS models is greater than that obtained by using the same SPS code with different metallicities or SFHs.

For star-forming galaxies at the lowest redshift we examine (0.5 < z < 1.0, Figure 9), the overall best fit (χ_red² = 3.71) is obtained with a young (0.6 Gyr of age) stellar population with the BC03 τ model. We also obtain a young age (0.2 Gyr) when assuming a constant star-formation history for the same stellar population model. The age determinations from FSPS10 and FSPS-C3K indicate instead an older age, from 1 to 5 Gyr (Figure 9). We notice that for star-forming galaxies we cannot exclude any particular age range since the stellar ages inferred from different models vary greatly.

At intermediate redshift (1.0 < z < 1.5, Figure 10), we obtain ages around 2–3 Gyr with different models (the typical error on each inferred age for star-forming galaxies is 1 Gyr). At the highest redshifts (1.5 < z < 2.0, Figure 11), all best fits (with different stellar population models and different star-formation histories) converge to the lowest age value.

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For galaxies with active star formation, the intrinsic strength of features does not vary significantly with time because the spectra are dominated by light from young stars that have been constantly forming. With the current signal-to-noise ratio, we therefore cannot draw any conclusions on the ages of star-forming galaxies at the redshift under consideration.

In order to investigate the origin of the qualitative difference in the best fits described in Section 4, we compare model spectra from different SPS codes. We show model SSPs from the BC03, FSPS10, and FSPS-C3K codes in Figure 13 for different ages. The strengths of the absorption lines vary among models for every age. We notice in particular that at wavelengths higher than 7500 Å different models predict different absorption bands at different wavelengths. We quantify the spread in the models at different wavelengths by computing the mean difference between every possible combination of models at the same age (Figure 13, bottom). This value is lower than 1% at wavelengths between ∼4500 and 6500 Å and is larger otherwise. In particular, the region with wavelengths greater than 8000 Å has a large discrepancy between BC03 on one side and FSPS10 and FSPS-C3K on the other. This explains why determinations of ages from 3D-HST at higher redshifts are more stable between different models than those at lower redshift. For our lowest-redshift sample (0.5 < z < 1.0), we observe the region of the spectrum where discrepancies among models are the largest, while at high redshift we observe rest-frame wavelengths where models are more similar to each other.

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We investigate the evolution of ages of quiescent galaxies in a mass-limited sample. In Figure 14 (left) we show the ages obtained by fitting 3D-HST stacks with different SPS models. Even though the model-dependent spread in ages is large, we observe that quiescent galaxies are younger at higher redshift and that, at each redshift, quiescent galaxies are not maximally old; instead, their age is smaller than half of the age of the universe at the same redshift. We compare to data in a similar mass range by Gallazzi et al. (2014) at z ∼ 0.6 and by Whitaker et al. (2013, who also use spectra from 3D-HST) at 1.4 < z < 2.2, obtaining good agreement. Also, at lower redshifts, Choi et al. (2014) find that quiescent galaxies at 0.2 < z < 0.7 tend to be younger than half of the age of the universe at those redshifts. The age value from the highest redshift mapped by Choi et al. (2014) is lower than the FSPS-C3K determination from this work, but similar to the determination obtained with the FSPS10 code, used by Choi et al. (2014) as well. We also note that the selection criteria and the rest-frame wavelength regime in Choi et al. (2014) are different from those in the present study. We test if the discrepancy between Choi et al. (2014) and our study can be caused by a selection bias by repeating the spectral stacking and the fitting with the same sSFR selection as Choi et al. (2014). We find an age of 1.8 ± 0.6 Gyr, younger than our previous determination, in accordance with Choi et al. (2014).

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Fairly young ages of quiescent galaxies are expected at z ∼ 1 from the mass build-up of galaxies: a subset of the quiescent galaxies were likely still forming stars at slightly higher redshifts. In addition, quiescent galaxies may grow by merging with lower-mass star-forming galaxies. The processes involved are clearly complex, but a simple estimate of the expected ages of z ∼ 1 quiescent galaxies can be obtained from comparisons of comoving stellar mass densities of quiescent galaxies and star-forming galaxies.

Muzzin et al. (2013) find a stellar mass density of quiescent galaxies at z ∼ 1 of log(${\rho }_{*}$ $[{M}_{\odot }\;{\mathrm{Mpc}}^{-3}])=7.8$ (their Figure 8). This density was reached at a redshift of z ∼ 2.1 for all galaxies (that is, quiescent and star-forming galaxies). If the oldest stars at z ∼ 1 are in the quiescent galaxy population, these stars finished forming at a redshift of z ∼ 2.1. The time difference between z = 2.1 and z = 1 is 2.6 Gyr. This is then the youngest age of this population, while the median age is defined by the redshift at which half of the mass was formed (z ∼ 2.4). The resulting median age is 3 Gyr, similar to what we observe in this study (Figure 14, right). This is strictly an upper limit to the age since z = 1 quiescent galaxies can contain stars that formed later. On the other hand, this argument is strictly valid for the full population of quiescent galaxies, and lower-mass quiescent galaxies might be somewhat younger. It is remarkable that this simple argument gives an age that agrees well with our general results.

6.2.1. Comparison with Photometry

Ages of galaxies can also be inferred from photometry only. In 3D-HST stellar masses, SFRs, ages, and dust extinction are estimated with the FAST code (Kriek et al. 2009), assuming exponentially declining star-formation histories with a minimum e-folding time of ${\mathrm{log}}_{10}(\tau \;{\mathrm{yr}}^{-1})=7$, a minimum age of 40 Myr, $0\lt {A}_{V}\lt 4$ mag, and the Calzetti et al. (2000) dust-attenuation law (see Skelton et al. 2014). The output from the FAST code is an age defined as the time since the onset of star formation, which is not necessarily equivalent to a light-weighted age.

For each galaxy, we therefore compute a light-weighted age (t_lum) following the definition

$$\begin{eqnarray*}&&{t}_{\mathrm{lum}}=\displaystyle \frac{{\displaystyle \sum }_{{\rm{i}}}\mathrm{SFR}({t}_{{\rm{i}}})\times {V}_{\mathrm{SSP}}(t-{t}_{{\rm{i}}})\times (t-{t}_{{\rm{i}}})\times {{\rm{\Delta }}}_{t}}{{\displaystyle \sum }_{{\rm{i}}}\mathrm{SFR}({t}_{{\rm{i}}})\times {V}_{\mathrm{SSP}}(t-{t}_{{\rm{i}}})\times {{\rm{\Delta }}}_{t}}\end{eqnarray*}$$

where

• 1.  
  t is the time of observation (equivalent to the age of FAST);
• 2.  
  $\mathrm{SFR}({t}_{{\rm{i}}})$ is the star-formation rate at time t_i. In the case of the FAST fits, $\mathrm{SFR}({t}_{{\rm{i}}})$ has the functional shape of a τ model $\mathrm{SFR}({t}_{{\rm{i}}})\sim \mathrm{exp}(-{t}_{{\rm{i}}}/\tau )$;
• 3.  
  ${V}_{\mathrm{SSP}}(t-{t}_{{\rm{i}}})$ is the V-band flux of a 1 M_⊙ element formed at t_i and observed at time t;
• 4.  
  Δ_t is the time step we divide the SFH into (we use Δ_t = 50 Myr).

We notice that for an SSP ${t}_{\mathrm{lum}}(t)=t$.

Figure 15 shows the relation between the time from the onset of star formation t and ${t}_{\mathrm{lum}}$ for a range of models: an SSP, a CSF model, and an exponentially declining model with $\tau =1\;\mathrm{Gyr}$. As expected, ${t}_{\mathrm{lum}}$ for an SSP (red line) is well approximated by t. For a CSF (blue line), ${t}_{\mathrm{lum}}$ is always smaller than t, with a increasing difference at later times. This effect is naturally explained by the fact that younger stars are brighter than older stars: V_SSP peaks at 10 Myr and declines afterward (in other words, the mass-to-light ratio $M/{L}_{V}$ increases for older stellar populations; see, among others, Bruzual & Charlot 2003, Figures 1–5). The τ model (purple line) has an intermediate behavior, being similar to the CSF for $t\to 0$ and parallel to the SSP for large t. We note that these relations are well known (see, e.g., Appendix A of van Dokkum et al. 1998).

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For each quiescent galaxy in the sample, we infer ${t}_{\mathrm{lum}}$ from the best fit to its photometry, and we find the average value in the three redshift bins $0.5\lt z\lt 1.0$, 1.0 < z < 1.5, and 1.5 < z < 2.0. Figure 14 (left) shows how the quantity compares to the ages measured from the spectra (Section 4) with different sets of models. For each redshift bin, ages derived from photometry tend to be comparable to the lowest values obtained with the spectral fitting.

6.2.2. Comparison with Fundamental Plane Studies

At redshifts lower than 1.5, we can quantify the Hα emission¹⁰ in quiescent galaxies from the residuals to the best fits (Figures 5 and 6). We subtract the best-fit model with the FSPS-C3K SPS from the stacks and fit residuals with a Gaussian centered at the Hα wavelength ($\lambda =6563\;\mathring{\rm A}$). Since the stacks are continuum-subtracted, this is effectively a direct measurement of EW(Hα+[N ii]). At the lowest redshifts ($0.5\lt z\lt 1.0$), we do not obtain a significant detection, with $\mathrm{EW}({\rm{H}}\alpha +[{\rm{N}}\;{\rm{II}}])=0.5\pm 0.3\;\mathring{\rm A}$, while at $1.0\lt z\lt 1.5$ we robustly detect the emission line, measuring EW(Hα+[N ii]) = $5.5\pm 0.8\;\mathring{\rm A}$. In the same mass/redshift regime, typical star-forming galaxies have EW(Hα+[N ii]) $\sim \;60\;\mathring{\rm A}$ (Fumagalli et al. 2012). This shows that, assuming no dust absorption, the Hα emission in quiescent galaxies is quenched by a factor of ∼10.

To estimate the Hα fluxes, we multiply the EW(Hα+[N ii]) by the median continuum flux of galaxies in the stack, and we assume a 0.25 ratio for [N ii]/(Hα+[N ii]). We finally estimate SFR(Hα) with the Kennicutt (1998) relation, obtaining that quiescent galaxies have $\mathrm{SFR}({\rm{H}}\alpha )=0.46\pm 0.06\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$ at $1.0\lt z\lt 1.5$ and $0.10\pm 0.05\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$ at $0.5\lt z\lt 1.0$, assuming no dust absorption.

In Fumagalli et al. (2014), we reported for quiescent galaxies higher SFRs inferred from mid-infrared emissions, up to $3.7\pm 0.7\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$ at $1.1\lt z\lt 1.5$. Since the sample in Fumagalli et al. (2014) has a slightly different selection ($1.1\lt z\lt 1.5$, log $({M}^{*}/\;{M}_{\odot })\gt$ 10.3), we test if the discrepancy between Hα and IR-inferred SFRs can be caused by a selection bias by repeating the spectral stacking and the Hα measurement in the same redshift/mass window as in Fumagalli et al. (2014). We obtain SFR(Hα) $=\;0.32\pm 0.07\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$, which would require a significant dust extinction for the Hα line (${A}_{{\rm{H}}\alpha }\;\sim$ 2.6) to be reconciled with the SFR(IR) measurement. This extinction value is similar to the A(Hα) measured for star-forming galaxies of similar masses at both redshift z = 1.5 (Sobral et al. 2012; Kashino et al. 2013; Price et al. 2014; Fumagalli et al. 2015) and in the local universe (Garn & Best 2010) with no significant redshift dependence.

A variety of studies (Fumagalli et al. 2014; Hayward et al. 2014; Utomo et al. 2014) suggest that SFRs inferred from IR are overestimated because of the contribution of dust heating by old stars or thermally pulsing asymptotic giant branch (TP-AGB) stars to the mid-infrared fluxes. SFRs measured from Hα are instead contaminated by potential AGN or LINER emission and are affected by dust extinction. Our combined multiwavelength findings agree, however, in indicating that SFRs of quiescent galaxies are very low, they are negligible in comparison to those of star-forming galaxies at the same redshift, and they are potentially consistent with zero.