HerMES: THE FAR-INFRARED EMISSION FROM DUST-OBSCURED GALAXIES

The 250, 350, and 500 μm far-IR data were obtained using the Herschel/SPIRE (Griffin et al. 2010; Swinyard et al. 2010) as part of HerMES, with an area coverage that completely overlaps with the MIPS observations of the 2 deg² COSMOS field. We use the first data release (DR1) of HerMES maps that were processed using the smap pipeline (Levenson et al. 2010). The reduced maps reach 3σ point-source depths of 8, 10, and 14 mJy, in the 250, 350, and 500 μm channels, respectively, where σ is the combined instrumental and confusion noise. For sources with S₂₅₀ ⩾ 3σ, we use the measured photometry from the HerMES cross-identification catalog. This catalog uses known positions of 24 μm sources as a prior, and estimates SPIRE fluxes via linear inversion methods. Model selections are used to account for and prevent overfitting, and to optimize the 24 μm input. The fitting method is outlined in more detail in Roseboom et al. (2010).

We use deep Subaru Suprime-Cam (Komiyama et al. 2003) aperture-corrected r⁺ photometry supplied by the COSMOS catalog (Capak et al. 2007). The 5σ point-source depth for a 3'' aperture is 26.8 mag.

The near-IR data are from Spitzer observations carried out by the COSMOS Spitzer Survey (S-COSMOS; Sanders et al. 2007) using the Infrared Array Camera (IRAC; Fazio et al. 2004) and MIPS. The IRAC 5σ depths at 3.6, 4.5, 5.6, and 8.0 μm for an aperture-corrected 19 aperture, are 0.50, 0.6, 3, and 5 μJy, respectively. The MIPS 24.0 μm 5σ point-source depth is 80 μJy (Le Floc'h et al. 2009).

We next generate a MIPS 24 μm-selected catalog that combines the Subaru and Spitzer data sets, using a two-step cross-matching process within the 2 deg² of the Subaru deep area in order to find optical counterparts for each source (Fu et al. 2010). Firstly, the 24 μm coordinates are matched to the closest IRAC detection within a 2'' search radius, then the nearest optical counterpart is identified within 1'' of the IRAC position. Finally, sources near bright stars that were within the Subaru/optical and Spitzer/IRAC 3.6 μm coverage were removed from the catalog to avoid contamination. The final 24 μm catalog is ⩾90% complete above S₂₄ ⩾ 80 μJy and contains 28,639 sources with S₂₄ ⩾ 100 μJy.

DOGs are selected in the standard manner, by identifying sources with r⁺ − [24] ⩾ 7.5 (AB mag; $S_{24}/S_{r^{+}}\ge 1000$) and we require [24] ⩽ 18.90 mag (S₂₄ ⩾ 100 μJy) due to the depth of the 24 μm data. Using these criteria, 3077 of the 28,639 (11%) COSMOS 24 μm sources with S₂₄ ⩾ 100 μJy are identified as DOGs (Figure 1). The near-IR SED of each DOG is examined using IRAC photometry (⩾5σ) to classify whether a DOG contains a bump-like feature or resembles a power law. For this study, a "bump" DOG is defined as satisfying one of the following: S_3.6 ⩽ S_4.5 ⩾ S_8.0; S_4.5 ⩽ S_5.8 ⩾ S_8.0; or S_3.6 ⩽ S_4.5 ⩾ S_5.8. Here S_{[3.6, 4.5, 5.8, 8.0]} represent the flux densities in the four IRAC channels. Conversely, we label a DOG as "power-law" if it satisfies S_3.6 ⩽ S_4.5 ⩽ S_5.8 ⩽ S_8.0. Previous studies have interpreted sources that feature a bump in the near-IR SED to be the stellar continuum peak at rest-frame 1.6 μm, tracing stellar emission, and likely star formation dominated (e.g., Yan et al. 2005; Sajina et al. 2007), while a power law is dominated by AGN continuum emission (e.g., Weedman et al. 2006; Donley et al. 2007). Bump DOGs compose 47% of our sample, while power-law DOGs are rarer, totaling 10%. The remaining 43% are not classified due to one of two possibilities: insufficient or low signal-to-noise IRAC data or an SED shape that does not satisfy the above criteria. For the latter case, most of the sources are at z < 2 (median of z = 1.1), such that the rest-frame 1.6 μm stellar continuum peak lies outside the wavelength range of the IRAC channels.

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All redshifts used in this paper are from COSMOS. Spectroscopic redshifts are used when available (35 sources, 1%, Lilly et al. 2007; J. Kartaltepe et al., in preparation), although virtually all of our DOG sample (2979 sources, 97%) use photometric redshifts. The photometric redshifts are derived from 30 photometric bands (Ilbert et al. 2009), providing σ_{Δz/(1 + z)} = 0.02, for 24 μm sources that lie at z = 1.5–3 and have the same r⁺ mag range as DOGs. The 61 DOGs that are X-ray detected use photometric redshifts that also account for AGN flux variability (Salvato et al. 2009). Also, two sources do not have a redshift estimate and are excluded from our sample. We note that the sharp peak in the redshift distribution at z = 1.95 is due to rounding from the redshift values associated with the bin size used and no spatial correlation is observed.

The redshift distribution of the final sample of 3075 DOGs is shown in Figure 2, with a mean of 〈z〉 = 1.9 ± 0.3. The sample of 90 DOGs in the Boötes field with spectroscopic redshifts from Bussmann et al. (2012), normalized to have an equal peak with our sample, is also shown. The two samples have a consistent mean z of 1.9 ± 0.02 and 2.1 ± 0.5, for our sample and the Bussmann et al. (2012) sample, respectively.

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Using the COSMOS redshifts and Herschel 250, 350, and 500 μm photometry, we fit the far-IR SED and calculate the rest-frame IR luminosity (8–1000 μm) and characteristic dust temperature. We divide the DOGs into two subsamples based on 250 μm detections because this SPIRE channel offers the deepest far-IR observations and the smallest beam size. A DOG is considered to be Herschel-detected if it satisfies S₂₅₀ ⩾ 3σ₂₅₀ (where σ₂₅₀ is the total uncertainty due to the instrumental and confusion noise), and undetected otherwise. Of our DOG sample, 51% are thus Herschel-detected. To calculate the characteristic dust temperature, for each of these we use the available SPIRE flux densities to fit a modified blackbody of the form

$$\begin{equation} S_{\nu } \propto B_{\nu }(T_{{\rm dust}})\nu ^{\beta }, \end{equation} \tag{ 1 }$$

where ν is frequency, β is the dust emissivity, fixed to the typical value of 1.5 (Draine 2003), T_dust is the dust temperature and B_ν is the Planck function, defined as

$$\begin{equation} B_{\nu } = \frac{2h\nu ^3}{c^2}\frac{1}{e^{h\nu /k_{{\rm B}}T_{{\rm dust}}}-1}. \end{equation} \tag{ 2 }$$

Here h is Planck's constant, c is the speed of light, and k_B is Boltzmann's constant. The temperature we calculate is insensitive to and consistent with the reported error bars from varying β slightly. All 250 μm detected sources have measured flux densities at either 350 μm or 500 μm, although it is not required to satisfy the 3σ limit (including confusion) in these wavelengths. For the sources that have low significance detections (⩽3σ) at 350 μm and/or 500 μm, we allow the full range of the uncertainties in flux densities when fitting for their IR luminosities and dust temperatures.

We derive estimates of the IR luminosity by fitting the available SPIRE data to the SED template library of Chary & Elbaz (2001, hereafter CE01). The template with the minimum χ² is chosen for the best fit. The uncertainty in IR luminosity is derived by first producing 1000 mock catalogs for each source that assume a Gaussian distribution centered around the measured SPIRE flux density, with a dispersion equal to the average flux density error. The IR luminosity per source is recalculated 1000 times and the standard deviation of the IR luminosity distribution is the error in our calculation. Examples of the SED template and modified blackbody fitting are shown in Figure 3.

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The IR luminosity (8–1000 μm) is converted to SFR using (Kennicutt 1998)

$$\begin{equation} {\rm SFR} (M_{\odot }\,{\rm yr}^{-1})=1.72 \times 10^{-10}L_{\rm {IR}} (L_{\odot }), \end{equation} \tag{ 3 }$$

which assumes a Salpeter initial mass function (IMF). We note that in our study we assume that UV emission will provide a negligible contribution to the total SFR, as validated by Penner et al. (2012).

To measure the average flux density of the Herschel-undetected DOGs, we bin the sources in redshift and for each bin stack on the SPIRE residual maps. These maps are generated by performing a blind extraction and point-spread function subtraction to prevent contamination of individually detected sources. We use the publicly available idl stacking library from Béthermin et al. (2010) to perform the stacking.²⁴ Each stacked image is converted from the native Jy beam⁻¹ to Jy pixel⁻¹ and aperture photometry with an aperture size equal to 22'', 30'', and 42'' for 250, 350, and 500 μm, respectively, is performed to calculate the flux of the stacked images. These aperture flux densities are consistent with those measured in the central pixel when the stacked map is in units of Jy beam⁻¹.

The observed stacked flux densities are corrected for the boosting from clustering bias by dividing by factors of 1.07, 1.10, and 1.20 at 250, 350, and 500 μm, respectively. The appropriate correction factors vary with clustering strength and are thus population dependent. These values were calculated by Béthermin et al. (2012) for 24 μm sources and are valid for DOGs because the observed correlation lengths, r₀ (a proxy for clustering amplitude), for DOGs (Brodwin et al. 2008) and the parent population of 24 μm sources (Magliocchetti et al. 2008; Starikova et al. 2012) are consistent. Errors in the photometry are calculated from bootstrapping the sources to be stacked. For each redshift bin, the clustering-corrected SPIRE flux densities of Herschel-undetected DOGs are set to equal the median stacked flux densities and the IR luminosity and dust temperature are calculated using the same method as for the Herschel-detected DOGs. The (clustering-corrected) stacked fluxes and errors, and the resulting average IR luminosities and dust temperatures, are shown in Table 1.

We note that the average stacked 250 μm flux density for the Herschel-undetected DOGs is 4.1 ± 0.7 mJy, which is a factor of two lower than the 250 μm catalog detection limit. In Figure 4 we show an example of the median stacked images for 250, 350, and 500 μm from left to right at z = 1.75–2.00 and an example SED using stacked SPIRE flux densities for a Herschel-undetected DOG at z = 1.88 is shown in the right panel of Figure 3. Each image stack is large enough to provide a good estimate for the background noise.

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Figure 5 shows IR luminosities of the Herschel-detected DOGs as a function of redshift. The average IR luminosity for Herschel-detected and undetected DOGs is (2.8 ± 0.3) × 10¹² L_☉ and (6.0 ± 1.0) × 10¹¹ L_☉, respectively. LIRGs (10¹¹ ⩽ L_IR (L_☉) ⩽ 10¹²) comprise 15% of Herschel-detected DOGs and 75% of Herschel-undetected DOGs. ULIRGs (10¹² ⩽ L_IR (L_☉) ⩽ 10¹³) make up 78% of the Herschel-detected and 15% of the Herschel-undetected DOGs.  Hyper-luminous infrared galaxies (HLIRGs; ⩾10¹³ L_☉) are the rarest, totaling 2% of the Herschel-detected DOGs and none of the Herschel-undetected DOGs. Although we note that there is additional uncertainty in the fractional contributions of the undetected sources, due to the use of stacked average fluxes, which minimizes the contribution from extreme sources. Herschel-detected power-law, or AGN-dominated, DOGs have on average L_IR = (4.5 ± 0.5) × 10¹² L_☉, making them more IR-luminous than Herschel-detected bump, star-forming DOGs, which have L_IR = (3.1 ± 0.4) × 10¹² L_☉, which is consistent with the findings of Melbourne et al. (2012). The results for the average IR luminosities and dust temperatures of the different subsamples of DOGs considered for this study are summarized in Table 2. Selection effects are investigated by calculating the IR luminosity of a representative CE01 template, scaled such that S₂₄ = 100 μJy or S₂₅₀ = 8 mJy, as shown in Figure 5. The local maxima in IR luminosity at z ∼ 1.5 in the 24 μm curve is due to the rest-frame 9.7 μm silicate absorption feature. The lack of DOGs below the 24 μm and 250 μm limit at a given redshift leads us to conclude that the apparent trend in Figure 5 that IR luminosity is increasing with redshift is a selection effect.

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Figure 6 shows dust temperature as a function of IR luminosity for DOGs, color-coded by redshift. The average characteristic dust temperature is T_dust = (34 ± 7) K and (37 ± 5) K for Herschel-detected and undetected DOGs, respectively. Herschel-detected power-law DOGs and bump DOGs have average T_dust = (37 ± 6) K and (35 ± 7) K, respectively, which is consistent with each other. We investigate sample selection effects in the T_dust–L_IR plane by considering both the IR luminosity of fixed temperature modified blackbody SEDs (Equation (1)) and the CE01 templates at z = 0.5 and z = 2.0 for S₂₅₀ = 8 mJy (Figure 6). The temperatures of the CE01 SEDs are calculated by fitting the template 250, 350, and 500 μm flux densities with a modified blackbody as in our data. Figure 6 also confirms that the 250 μm flux density limit biases against lower luminosity sources and the luminosity limit is a function of redshift (see also Figure 5). Furthermore, the apparent correlation between T_dust and z is in fact caused by a combination of the redshift-dependent L_IR selection limit and the correlation between L_IR and T_dust (e.g., Symeonidis et al. 2013; Hwang et al. 2010). The selection limits at z = 0.5 on Figure 6 also show that at low redshift (z ≲ 1) there is a bias against the warmer sources, which results in an apparent difference in the dust temperature distribution of Herschel-detected DOGs compared to the observed relationship locally and at higher redshifts (e.g., Chapman et al. 2004; Symeonidis et al. 2013; Hwang et al. 2010). The 24 μm flux density limit produces a similar effect as the 250 μm limit.

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The mean stacked IR luminosities and dust temperatures, per redshift bin, of the Herschel-undetected DOG population are displayed in Figure 6 and are less sensitive to these selection biases. The lowest redshift bin (bin 1, z ⩽ 1.5) is offset relative to the other redshift bins because it covers a wide redshift interval in which the IR luminosity limit has a steep slope (Figure 5). The dearth of sources at high luminosities and low dust temperatures is not a selection effect as these sources would have been detected by our data. This is consistent with Symeonidis et al. (2013), who found that cold cirrus-dominated SEDs (Rowan-Robinson et al. 2010) are rare in the most IR luminous galaxies.

The results shown in Figure 6 suggest that z ∼ 2 DOGs span a wider range of dust temperatures than z ∼ 2 SMGs (by which we mean 850 μm or 1 mm selected sources) due to the different selection effects associated with each galaxy population. SMGs are biased toward detecting cold-dust dominated sources (T_dust ∼ 30–40 K) because hot sources are missed by submillimeter surveys (Chapman et al. 2004; Casey et al. 2009; Chapin et al. 2009). The discovery of optically faint radio galaxies (OFRGs; Chapman et al. 2004; Casey et al. 2009; Magnelli et al. 2010) at z ∼ 2, which have similar stellar masses, radio luminosities, and UV spectra as SMGs, but have T_dust ∼ 40–60 K, demonstrate this, while we also note that the radio-detection limit is biased against the coldest sources (e.g., Wardlow et al. 2011). DOGs are more insensitive to these selection biases and thus show a wider range of temperatures at z ∼ 2. Magdis et al. (2010) found similar results when investigating the characteristic dust temperatures for IRAC peakers and showed that mid-IR selected sources bridge the gap in temperature ranges between OFRGs and SMGs. We note that the 250 μm selected sources suffer from the same selection biases as our Herschel-detected DOGs but shifted to higher luminosities due to their shallower 250 μm detection limit.

We compute the IR luminosity function of DOGs using the 1/V_max method (Schmidt 1978), defined as

$$\begin{equation} \Phi (L)\Delta L = \displaystyle \sum _{i}\frac{1}{V_{{\rm max},\,i}}, \end{equation} \tag{ 4 }$$

where V_max is the maximum comoving volume of the ith source such that it would be detected and included in the sample. We consider the peak of the redshift distribution using only DOGs at z = 1.5–2.5. For the Herschel-detected DOGs we use two flux limits to determine V_max: S₂₄ = 100 μJy and S₂₅₀ = 8 mJy. These are the two detection limits of the survey. For the Herschel-undetected sample, the 24 μm flux limit alone was used to calculate V_max, as all redshift bins are detected in the stacks. We then calculate each Herschel-undetected DOG's IR luminosity and contribution to the luminosity function using its redshift and the relevant stacked flux densities. The uncertainties are from Poisson statistics and binning errors, where the binning errors are calculated by generating the IR luminosity function 1000 times from IR luminosities calculated from artificial SPIRE flux densities described in Section 3.1 and taking the standard deviation per IR luminosity bin. The DOG IR luminosity function at z ∼ 2 is presented in Figure 7 and Table 3. The faint end of the IR luminosity function for Herschel-detected and undetected DOGs is coadded, which affects the lowest luminosity bin for Herschel-detected DOGs the most, showing a 0.20 dex increase.

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For comparison, the DOG IR luminosity function for Herschel-detected DOGs and all DOGs, calculated by extrapolating the IR luminosity from S₂₄ using CE01 templates, is also shown in Figure 7. We find that the IR luminosities using this method are overestimated by a median factor of 1.8, consistent with the previous studies of 24 μm selected galaxies at z ∼ 2 (Houck et al. 2005; Yan et al. 2007; Daddi et al. 2007; Papovich et al. 2007; Pope et al. 2008; Nordon et al. 2010, 2012; Elbaz et al. 2010, 2011; Magnelli et al. 2011), and affects both the shape and normalization of the IR luminosity function.

We compare the number densities of DOGs to the parent population of sources with S₂₄ ⩾ 100 μJy (Figure 7). The luminosity function of 24 μm sources is calculated self-consistently using SPIRE data, including stacking on the Herschel-undetected galaxies. There are 5932 sources in COSMOS with S₂₄ ⩾ 100 μJy at z = 1.5–2.5, of which 32% are DOGs. Figure 7 shows that DOGs have a smaller overall normalization in their luminosity function (since they are fewer in number) and their relative contribution to the 24 μm number density increases with luminosity in agreement with the IR luminosity distribution of DOGs compared to 24 μm sources in Penner et al. (2012).

Figure 7 also shows the z = 1–3 SMG luminosity function by combining the z = 1–2 and z = 2–3 redshift bins from Wardlow et al. (2011) for comparison of the distinct, but overlapping DOG and SMG (Pope et al. 2008) populations. DOGs are more common than SMGs at z ∼ 2, which is reflected in the higher normalization in the luminosity function, although in the HLIRG regime, SMGs dominate. This is consistent with the picture in which DOGs represent an evolutionary stage toward the end of the peak of SFR, observed as the SMG phase (Narayanan et al. 2010), they must have lower IR luminosities and SFRs on average. In this case, the relative scaling of the two luminosity functions indicates that the DOG phase is longer lived than the SMG phase. As is shown in Figure 8, the total star formation rate density (ρ_SFR) provided by the two populations are approximately even despite the number and intensity of the sources.

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To calculate the contribution of DOGs with S₂₄ ⩾ 100 μJy to the ρ_SFR of the universe at z ∼ 2, we integrate the IR luminosity function and use Equation (3). Figure 8 shows DOGs compared to other z ∼ 2 galaxy populations. The total uncertainty in ρ_SFR is calculated from the quadrature sum of individual SFR uncertainties and the standard deviation of ρ_SFR from the mock catalogs discussed in Section 3.1. Horizontal error bars represent the considered redshift interval. The value of ρ_SFR for DOGs at z = 1.5–2.5 is (3.2 ± 0.5) × 10⁻² M_☉ yr⁻¹ Mpc⁻³ which contributes to 12%–29% of the overall ρ_SFR at z = 2 calculated from UV and IR data shown by Hopkins & Beacom (2006) and Burgarella et al. (2013). When comparing to z = 1.5–2.5 sources with S₂₄ ⩾ 100 μJy, DOGs contribute 33% to the 24 μm ρ_SFR.

The Herschel-undetected and power-law sources provide non-dominant contributions to the total ρ_SFR of DOGs. The Herschel-undetected DOGs contribute 18% and power-law DOGs contribute just 9%. We note that even though power-law DOGs are thought to be dominated by AGN emission in the IRAC bands, their far-IR emission is still likely dominated by star formation, as is the case for other far-IR luminous samples containing AGNs (Elbaz et al. 2010; Lutz et al. 2008, 2010). Indeed, even studies of the most active AGNs have revealed that SED fits for Herschel-detected AGNs always required a starburst component in order to appear bright in the far-IR (Hatziminaoglou et al. 2010). As an attempt to quantify this claim, we use a simplified method to calculate an upper limit on the AGN contribution to the IR luminosity and SFR in power-law DOGs and hence the contamination of ρ_SFR by AGNs. We begin by scaling the AGN SEDs from Kirkpatrick et al. (2012) to the 24 μm flux density of each power-law DOG and calculate the luminosity from the warm dust component. Then, by assuming that the warm dust component is entirely AGN-dominated and the cold-dust component is entirely star formation dominated, we can subtract the warm IR luminosity from the CE01 IR luminosity to calculate the residual contribution from star formation. We find that power-law DOGs each have a maximum average contribution of 70% to the IR luminosity, which could contaminate ρ_SFR by ∼0.2 × 10⁻² M_☉ yr⁻¹ Mpc⁻³, which is only 6% of the total DOG ρ_SFR. In addition, we also estimate the dispersion of AGN contribution by normalizing quasar SED templates from Elvis et al. (1994), Richards et al. (2006), Polletta et al. (2007), and Dai et al. (2012) to the average power-law DOG 24 μm flux density at z = 1.5–2.5 and assume that the SEDs have no emission associated with star formation. Under this assumption, the average AGN contributions to the individual galaxies' IR luminosities range from 5% to 65%, depending on the SED, which corresponds to 0.005% to 6% contribution to the total DOG ρ_SFR.

We note that Pope et al. (2008) also examined bump (star-forming) DOGs at z ∼ 2 down to S₂₄ = 100 μJy and they calculated ρ_SFR ∼ 1 × 10⁻² M_☉ yr⁻¹ Mpc⁻³, under the assumption that the average DOG has an SFR of 200 M_☉yr⁻¹. This value is lower than the bump ρ_SFR = 1.9 ± 0.3 × 10⁻² M_☉ yr⁻¹ Mpc⁻³ that we measure. However, it is difficult to determine whether these two values are significantly different because Pope et al. (2008) do not provide an error on their measurement. We use their reported fractional error on the average IR luminosity ((1.1 ± 0.7) × 10¹²L_☉) to estimate that the minimum error on their ρ_SFR is ∼0.6 × 10⁻² M_☉ yr⁻¹ Mpc⁻³, in addition to the contribution from the counting error from their 62 sources (compared to our 1137 bump sources at z = 1.5–2.5). We also note that the selection criteria for the two studies are slightly different and if we were to use the bump DOG selection scheme in Pope et al.  (2008; S_3.6 < S_4.5 > S_5.8 and S_4.5 > S_8.0, or S_4.5 < S_5.8 > S_8.0 and S_3.6 > S_8.0) we would identify 100 fewer bump DOGs (9% of our sample of bump DOGs are at z = 1.5–2.5). We conclude that the two results are consistent but since our study uses a larger sample and employs a combination of direct observations and redshift-binned stacking to determine our IR luminosities, we consider this measurement more accurate.

Using the stellar masses derived in Ilbert et al. (2010, corrected to assume a Salpeter IMF by adding +0.24 dex and to be consistent with our SFR calculations) and our derived SFRs using Herschel data, we investigate where DOGs lie in the star formation rate–stellar mass (SFR–M_*) plane. Disk galaxies with a steady star formation mode are observed to form a tight correlation in their SFRs as a function of stellar mass, defining a "main sequence" (Daddi et al. 2007; Elbaz et al. 2011). Outliers above this relation are thought to be merger-driven starburst galaxies (Rodighiero et al. 2011 and references therein). In the top panel of Figure 9 we show the SFRs and stellar masses for Herschel-detected DOGs, considering only those at z = 1.5–2.5 to minimize the effects of redshift evolution. Average error bars are plotted for SFRs and the uncertainties in stellar mass are fixed to 0.5 dex, which covers the systematic offset range due to the choice of extinction laws and stellar population synthesis models.

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Figure 9 shows that power-law DOGs and bump DOGs cover the same ranges in stellar mass and star formation rate in the SFR–M_* plane, as expected if the far-IR is star formation dominated. Our findings are also consistent with previous studies that investigated the similarities in properties of far-IR SEDs of Herschel-selected star-forming galaxies and AGN (Mullaney et al. 2012).

The IR main sequence from Elbaz et al. (2011) for Herschel-selected star-forming galaxies at z = 2 is also shown in Figure 9. DOGs have a significant amount of scatter about this relation, with 46% within a factor of two of the main sequence, 24% above it and consistent with starbursts, and 31% below it in the more quiescent regime.

The bottom panel of Figure 9 shows the specific SFR (sSFR = SFR/M_*) as a function of stellar mass. The sSFR quantifies the weighted SFR and stellar mass, with its inverse giving the mass-doubling time for the current episode of star formation activity. An apparent negative correlation in which lower mass z = 2 DOGs exhibit higher sSFRs than their higher mass counterparts is observed; however, this is largely a selection effect due to the flux limit of our sample. On the top panel of Figure 9 we use the minimum IR luminosity at z = 2 from our sample to represent a minimum detectable SFR limit, shown as the horizontal line. We convert this to sSFR for a range of masses and this is shown as the diagonal line in the bottom panel. The logarithmic inverse age of the universe in Gyr at z = 2 is ≈−9.5 (dashed line in Figure 9) and most DOGs have sSFRs larger than this, indicating that the observed phase of star formation could be responsible for their total observed stellar mass.

Finally, we use the known redshift distribution and the sSFRs of DOGs to compare their volume densities to their proposed progenitors, SMGs. The volume density of observed DOGs with S₂₄ ⩾ 100 μJy at z = 1.5–2.5 is 8 × 10⁻⁵ Mpc⁻³. Using the median DOG sSFR to estimate the characteristic lifetime of the DOG phase to be approximately 1 Gyr, we can correct this density for the burst duty cycle to derive a volume density for the progenitors to be approximately 10⁻⁴ Mpc⁻³. This is consistent to the volume density for SMGs at z = 1.5–3 with S₈₇₀ > 4 mJy derived from Wardlow et al. (2011), which assumes the lifetime of the SMG phase to be 100 Myr, 10 times shorter than for DOGs. In this scenario, DOGs would have the same descendants as z ∼ 2 SMGs, which are likely to be 2–3L* early-type galaxies (Wardlow et al. 2011; Hickox et al. 2012).