EMPIRICAL PREDICTIONS FOR (SUB-)MILLIMETER LINE AND CONTINUUM DEEP FIELDS

We start by using the photometric catalog of the Hubble UDF (centered at R.A. = 03^h32^m390, decl. = −27°47'291) described in Coe et al. (2006). This contains aperture-matched, point-spread-function (PSF)-corrected Advanced Camera for Surveys (ACS) BVi'z' and NICMOS3 JH photometry, as well as Bayesian photometric redshifts for all the detected sources, accurate to within 0.04(1 + z_spec) (Coe et al. 2006). The full catalog contains 18, 700 sources, of which a large fraction (8042) are detected at the 10σ level in at least one band. The 10σ limiting AB magnitudes in the B, V, i', z', J, and H are 28.71, 29.13, 29.01, 28.43, 28.30, and 28.22, respectively. Following Coe et al. (2006), to exclude contamination from stars, we exclude sources with i-band stellarity stel ⩾0.8 (about 6% of the sample), leaving us with 17,532 galaxies, with photometric redshifts distributed as plotted in Figure 1(a). For reference, in Figure 1(b), we plot the distribution of the observed ACS V-band magnitude for 16,830 sources detected in that band. We note that the fact that the redshift distribution of our sources peaks at z ≃ 2 and the sudden drop in sources fainter than 30 AB magnitudes in the V band are due to incompleteness. This implies that our predictions may be missing high-redshift, dust-obscured galaxies that are too faint in the optical to be in our sample and may have large (sub-)millimeter fluxes. This is the case of LESSJ0333243.6-274644, the only submillimeter source detected in the UDF as part of the LESS survey (LABOCA observations of the ECDF-S at 870 μm; Weiß et al. 2009), which has no optical counterpart in our optical catalog. Our predictions are therefore lower limits for the possible number of detections at high redshift (z > 2).

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We complement the photometry in the UDF catalog with additional photometry out to the far-infrared. We use 54 galaxies detected in the Herschel/SPIRE bands available in the publicly released HerMES survey (PI: S. Oliver; Oliver et al. 2010) images in GOODS-South, for which we applied the same prior source extraction technique as in Elbaz et al. (2011) for the GOODS-Herschel SPIRE data in GOODS-North. Herschel/PACS images of the GOODS-South are available as part of the GOODS-Herschel program (PI: D. Elbaz). For each of these 250 μm selected galaxies, well-sampled SEDs from the ultraviolet to the far-infrared are available, including photometry in the following bands: U, B, V, i, z, J, K, Spitzer/IRAC 3.6, 4.5, 5.8, and 8.0 μm, Spitzer/IRS at 16 μm, Spitzer/MIPS at 24 μm, Herschel/PACS at 70, 100, and 160 μm, and Herschel/SPIRE at 250, 350, and 500 μm (Elbaz et al. 2011; Magdis et al. 2011). The redshifts of the galaxies in this sub-sample go from z = 0.140 to z = 2.578, with a median of value of 1.0. We use this sub-sample in Section 4.3 to test the reliability of our predictions of the total infrared luminosity and (sub-)millimeter continuum fluxes from observed optical data as described in Section 3.

To test our predictions for a wider field and address the issue of cosmic variance in Section 4.4, we use a wider-area catalog of the CDF-S field which also includes the UDF but is about 10 times larger in area, the FIREWORKS catalog (Wuyts et al. 2008). The FIREWORKS catalog is a K_s-band-selected galaxy catalog which contains multi-wavelength photometry of 6,308 galaxies from the U band to the Spitzer 24 μm band, with a 5σ magnitude limit of 24.3 AB mag in the K_s band (i.e., shallower than the UDF Hubble Space Telescope (HST) catalog described in Section 2.1), over 138 arcmin².

We use the spectral synthesis model of Bruzual & Charlot (2003) to compute the integrated light emitted by stars in galaxies for a wide range of metallicities (distributed between 0.02 and 2 times solar), ages (distributed between 0.1 Gyr and the age of the Universe at each redshift), and star formation histories (parameterized as exponentially declining with a wide range of timescales, and superimposed random bursts of star formation). To account for the attenuation of starlight by dust, we describe the ISM of galaxies using the two-component model of Charlot & Fall (2000): the ambient (diffuse) ISM and the star-forming regions (birth clouds). This prescription accounts for the fact that stars are born in dense molecular clouds, which dissipate typically on a timescale of 10⁷ yr. As a result, the non-ionizing continuum emission from young OB stars and line emission from their surrounding H ii regions is absorbed by dust in these birth clouds and then in the ambient ISM, while the light emitted by stars older than 10⁷ yr propagates only through the diffuse ISM. The main free parameters of this model are the effective V-band optical depth seen by young stars in birth clouds, $\hat{\tau }_{V}$, and the fraction of $\hat{\tau }_{V}$ contributed by dust in the diffuse ISM, μ. Using this model, we compute the attenuated stellar emission of galaxies and the total luminosity absorbed and re-radiated by dust in the birth clouds and the diffuse ISM. We use the model library described in da Cunha et al. (2008), which includes 50,000 attenuated stellar spectra spanning a wide range of star formation histories, metallicities, and dust optical depths.

In the context of the model described in da Cunha et al. (2008), the energy absorbed by dust is re-radiated at infrared wavelengths through four different components:

• 1.  
  the emission by polycyclic aromatic hydrocarbons (PAHs),
• 2.  
  a hot mid-infrared continuum (with temperature in the range 130–250 K),
• 3.  
  emission by warm dust in thermal equilibrium (with temperature in the range 30–60 K and dust emissivity index β = 1.5),
• 4.  
  emission by cold dust in thermal equilibrium (with temperature in the range 15–25 K and dust emissivity index β = 2).

da Cunha et al. (2008) use a wide library of infrared emission spectra where the temperatures and relative contributions of each component to the total infrared luminosity span a wide range of values. The models in this library are then directly compared to infrared observations of galaxies to constrain each dust emission component. In this paper, since we hardly have any observational constraints on the infrared SED of the galaxies, we do not require such a wide range of models. Therefore, we will adopt a reduced set of dust emission models that reflect the range of infrared SED shapes of local, normal star-forming galaxies.

For this work, our goal is to obtain a range of possible infrared SEDs that are consistent with the observed optical/near-infrared emission in terms of their overall energy balance. Therefore, for simplicity, we fix the values of most free parameters controlling the shape of the infrared SEDs of galaxies to those of three representative infrared SEDs presented in da Cunha et al. (2008): a "standard" model (with equilibrium temperatures of the cold and warm dust components 22 and 48 K, respectively), a "hot" model (25 and 55 K) and a "cold" model (18 and 40 K); the relative contributions of the cold and warm dust components, as well as the PAHs and hot mid-infrared continuum are different for each model and are listed in Table 1 of da Cunha et al. (2008). These three models were calibrated using observed IRAS and ISO infrared fluxes of local star-forming galaxies and span the range of observed infrared colors for this low-redshift sample. The "standard" model reproduces the median infrared colors of local galaxies, the "hot" model is representative of the warmest observed infrared colors, and the "cold" model is representative of the cooler infrared colors. We build a simplified dust emission model library in which we fix the values of the dust temperatures and the contribution by PAHs, hot mid-infrared continuum and warm dust to the total luminosity of birth clouds, as well as the contribution of cold dust to the total dust luminosity of the diffuse ISM, to the values of these representative models, while leaving the fraction of total dust luminosity contributed by the diffuse ISM, f_μ = L^ISM_dust/L_dust, and the total dust luminosity, L_dust as free parameters.

In addition to thermal dust emission, the (sub-)millimeter continuum emission of star-forming galaxies can have a non-negligible radio continuum emission component, especially at the lowest frequencies. This emission is mainly free–free emission from H ii regions and synchrotron radiation from relativistic electrons accelerated in supernova remnants (e.g., Condon 1992). Since the da Cunha et al. (2008) models do not include radio emission, we add a radio continuum component to our model SEDs in order to account for this extra contribution to the (sub-)millimeter continuum. We use the simple prescription described in Dale & Helou (2002), which is based on the observed radio/far-infrared correlation. The radio/far-infrared correlation (e.g., Helou et al. 1985; Condon 1992; Bell 2003) implies that the ratio between the observed far-infrared flux of a galaxy (between 42.5 μm and 122.5 μm) and the radio flux density at 1.4 GHz, q, is constant. We model the radio continuum in star-forming galaxies as a sum of two main components:

• 1.  
  a thermal component, consisting mainly of free–free emission from ionized gas, with spectral shape f^th_ν∝ν^−0.1;
• 2.  
  a non-thermal component, consisting of synchrotron radiation, with spectral shape f^nth_ν∝ν^−0.8.

In normal star-forming galaxies, the contribution of the thermal (free–free) component to the radio continuum at 20 cm is ∼10% (Condon 1992). This allows us to fix the spectral shape of our radio continuum, which we normalize relative to the far-infrared flux of each model in our library using the value found by Yun et al. (2001), q = 2.34. This prescription is based on the assumption that the galaxies fall in the observed radio/far-infrared correlation, and has the advantage of not requiring any extra free parameters to estimate the radio continuum. We also assume that the radio/far-infrared correlation remains constant with redshift (e.g., Sargent et al. 2010).

The libraries of attenuated stellar emission and dust emission are combined by associating models with similar values of f_μ = L^ISM_dust/L_dust (the fraction of total dust luminosity contributed by the diffuse ISM), which are scaled to the same total dust luminosity L_dust. This ensures, for each model, the energy balance between the radiation absorbed and re-emitted by dust in the diffuse ISM and stellar birth clouds.

For each galaxy, we compare the observed optical fluxes in the ACS and NICMOS bands (Section 2) to the predicted fluxes for every model of the stochastic library described above, by computing the χ² goodness of fit for each model. We then build the likelihood distribution of any given physical parameter for the observed galaxy by weighting the value of that parameter in each model by the probability exp (− χ²/2). Our final estimate of the parameter is the median of the likelihood distribution, with an associated confidence interval which is the 16th–84th percentile range of the distribution (this confidence interval is tighter for well-constrained parameters and wider when the parameters are not well constrained by the available observations). In what follows, the values of the physical properties of the galaxies mentioned refer to the median values of the probability density distribution. We also obtained the best-fit model SED for each galaxy, which is the model that minimizes χ².

We use the method described above to fit the observed photometry of each galaxy and produce likelihood distributions of their stellar mass, SFR, dust optical depths, infrared luminosities, dust masses, continuum, and molecular line fluxes in the (sub-)millimeter range.

In Figure 2, we show an example SED fit and the associated likelihood distributions of some physical parameters: the SFR, stellar mass (M_*), dust luminosity (L_dust), dust mass (M_dust), far-infrared luminosity (L_FIR, defined as the integral of the infrared emission between 42.5 and 122.5 μm), and the continuum flux densities in a number of accessible (sub-)millimeter windows at 345, 230, and 100 GHz (specifically, ALMA bands 7, 6, and 3, respectively, and PdBI/NOEMA band 1, 3, and 4). The ultraviolet to near-infrared part of the SED is the stellar population model that best fits the data. The far-infrared and (sub-)millimeter part of the SED that is plotted corresponds to the model with the best-fit L_dust and f_μ, but all the other parameters controlling the shape of the SED at these wavelengths are randomly drawn from the library of dust emission models. The gray-shaded area shows the range of possible infrared SEDs allowed within the uncertainties in dust luminosity with different dust temperatures and contributions of PAHs, mid-IR continuum and dust in thermal equilibrium reflecting the diversity of possible dust emission models in our model library. The orange points show the (exemplary) median-likelihood estimates of the flux densities at 345, 230, and 100 GHz, where the error bars (16th–84th percentile range) reflect all the different combinations of infrared models that are consistent with the observed optical data.

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We impose a minimum of three photometric bands to define the SED, and we discard galaxies with z > 5 and galaxies for which the fit residuals are larger than 2σ (where σ is the uncertainty associated with the flux measurement) in each band. These criteria allow us to discard the most unreliable SED fits: at very high redshift, our model becomes uncertain and the observations sample only the far-UV emission, making it very difficult to constrain the SED; also, galaxies with very high residuals in a given band may indicate problems with the photometry or wrong photometric redshift, or the presence of an active galactic nucleus (AGN, which is not included in our models). Our final sample used in the remainder of this paper consists of 13, 099 sources with z ⩽ 5.

Stellar masses are constrained to within ±0.35 dex, which reflects uncertainties due to the fact that, for most of our galaxies, observations do not include the rest-frame near-infrared, where the light is dominated by low-mass stars, which constitute the bulk of the stellar mass in galaxies. However, we show in the next section (Section 4.3) that this does not cause any systematic effects on the stellar mass estimates. The SFRs are constrained to within typically ±0.2 dex, due to the fact that the observed SEDs sample the emission by young stars in the rest-frame ultraviolet. The dust luminosities are more uncertain (confidence ranges are typically ±0.45 dex), as expected due to the lack of infrared observations for our sample. The dust luminosity is estimated by our model by calculating the total energy absorbed by dust, taking into account the light emitted by stars and the attenuation by dust. Therefore, by construction, our dust luminosities are consistent with all stellar and dust attenuation parameters (SFR, M_*, μ, $\hat{\tau }_{V}$, f_μ), from an energy-balance perspective (as described in da Cunha et al. 2008 and in Section 3). Even though we have significant uncertainties in the dust luminosity estimates, as expected from our sparse SED sampling, we are still able to predict L_dust well within an order of magnitude (we analyze possible systematic effects in Section 4.3). The dust masses are also estimated by using all the dust emission model templates that are consistent with the statistical estimates on dust luminosities. The confidence range for M_dust is very large (±0.55 dex), and reflects not only the uncertainty in L_dust, but also the large uncertainty in infrared SED shapes and dust temperatures. These dust masses are merely indicative of the range of dust masses that are consistent with the observed SEDs in terms of energy balance, and taking into account a range of possible dust temperatures, 18–25 K for the cold dust (with β = 2), and 40–55 K for the warm dust (with β = 1.5).

The distributions of physical parameters inferred from our SED fits (Figure 3) show that the bulk of our galaxies are low-mass, low SFR, and low dust attenuation ($\hat{\tau }_{V}$ and $\mu \hbox{$\hat{\tau }_{V}$}$) sources, i.e., blue star-forming galaxies (consistent with the finding of a large population of faint blue galaxies in the UDF described in Coe et al. 2006). As expected, galaxies in the highest redshift bin, z ⩾ 2.5, have typically higher stellar masses and SFRs, because only the brightest galaxies are detected. The median dust luminosity of the sources, L_dust increases from log(L_dust/L_⊙) ≃ 8.0 in the lowest redshift bins to log(L_dust/L_⊙) ≃ 9.2 at z ⩾ 2.5. Figures 3(g) and (h) show that this is not necessarily due to an increase in the dust optical depth of galaxies in the highest redshift bin, but rather to the fact that the galaxies detected have larger stellar masses and SFRs, as shown in Figures 3(a) and (b).

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It is important to test how well we can predict the total infrared luminosity of the galaxies in our sample from their observed rest-frame UV/optical SEDs. To do so, we use the sample of 54 UDF galaxies detected in the Herschel/PACS and Herschel/SPIRE bands as part of the GOODS-Herschel program described in Section 2.2. For each of these galaxies, we have well-sampled SEDs from the ultraviolet to the far-infrared, which allows us to test our SED extrapolations. We fit the observed (more complete) SEDs of this sub-sample of galaxies using the same method as described in Section 3.4. First, we include the full observed ultraviolet to far-infrared observations in the SED fits, not only to check that our model can reproduce consistently the SEDs of these galaxies, but also to get the best possible estimates of the stellar masses, SFRs, dust luminosities, and continuum (sub-)millimeter fluxes for these sources. Then, we re-fit the SEDs using only observations between the U band and the K band, to mimic the set of observations available for the majority of galaxies discussed above.

In Figure 4(a), we compare the median likelihood of the Herschel/SPIRE 250 μm flux derived from the fits from the U to K band with the actual observed 250 μm flux for each galaxy. We find a small systematic offset of 0.25 dex between the observations and our estimates, in the sense that we tend to underestimate the 250 μm flux of the galaxies on average with our U-band to K-band fits. However, for most of the galaxies, the observed value is still within the confidence range derived from our fits. This effect is likely to be less important for the total dust luminosity and (sub-)millimeter fluxes, as these do not depend as strongly on the exact location of the peak of the infrared SED (i.e., the actual dust temperature spanned by the models). In the next three panels of Figure 4, we compare the median likelihoods of the stellar masses, dust luminosities, and continuum ALMA band 6 fluxes obtained with the two sets of SED fits (U-band to K-band fits versus U-to-SPIRE-fit). Figure 4(b) shows that the stellar masses agree remarkably well between the two fits (with a dispersion around the identity line of 0.13 dex). Not surprisingly, the total dust luminosities and predicted ALMA band 6 fluxes do not agree as well, as shown in Figures 4(c) and (d). The inclusion of infrared data in the fits helps constrain these properties better, as shown by the significantly reduced confidence ranges. In the case of L_dust, this happens because the infrared data allow us to constrain the bolometric dust luminosity by fitting the dust emission itself (as opposed to constraining L_dust from the attenuated spectrum alone); the constraints on SFR are also tightened because we can account for dust-obscured SFR more accurately. The different far-infrared fluxes obtained with PACS and SPIRE help constrain the shape of the dust SED, namely, the dust temperatures and relative contributions of the warm and cold dust components to the total infrared emission. This helps obtaining tighter constraints on the (sub-)millimeter continuum fluxes (namely, the ALMA band 6 flux shown as an example in Figure 4(d)).

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Even though the inclusion of infrared data helps constraining parameters such as the SFR, dust luminosity, and ALMA continuum fluxes, the median-likelihood estimates of these parameters when excluding the infrared data agree very well with the estimates derived from the full SED fits, even if, as expected, the associated confidence ranges are larger. We find very small offsets between the averages of the median-likelihood estimates derived from the two fits: 0.02 dex for SFR, 0.08 dex for L_dust, and 0.07 dex for the predicted ALMA band 6 continuum flux (in the sense of the U-band to K-band fits slightly underestimating the parameters), with a dispersion of ≃ 0.40 dex for all cases. These very small systematic offsets are well within our fit confidence ranges, and show that our approach to predict infrared luminosities and (sub-)millimeter continuum fluxes from modeling UV/optical SEDs is reliable. We note, however, that the difference between 250 μm fluxes and total dust luminosities derived from the fits to the UV/optical data only and those measured with Herschel correlates with the dust optical depth in the galaxies. We tend to underestimate the (sub-)millimeter fluxes/dust luminosity when using only the U-band to K-band fits for galaxies with the highest dust attenuations (which translate into high infrared-to-optical ratios). This is due to the fact that our dust attenuation prior (Section 3; da Cunha et al. 2008) leads to an underestimation of the optical depth for extremely dust-enshrouded, starburst-like sources (such as local ultraluminous infrared galaxy (ULIRGs) or high-redshift submillimeter galaxies (SMGs); see da Cunha et al. 2010). While these galaxies can be a negligible fraction of our sample in number (e.g., Rodighiero et al. 2011; Sargent et al. 2012), they can dominate the bright (sub-)millimeter counts. This is the case for the two GOODS/Herschel sources in our sample with the highest redshift, which are marked in Figure 4 with squares. Due to the flux limit of this sample, at the highest redshifts (z ≃ 2), only very dust-obscured ULIRG-type galaxies were selected. For this particular type of galaxies, the SED models used in Section 3 become limited. However, we expect this kind of galaxies to be rare in our optically selected sample of the UDF, and therefore we do not expect this limitation to greatly affect our results. We also note that our optically selected catalog is also likely to miss completely optically obscured galaxies (e.g., HDF850.1, Walter et al. 2012; GN10, Daddi et al. 2009a; GN20, Daddi et al. 2009b). The very good agreement between parameters derived from fits to the UV/optical SED versus parameters derived from fits to the full SED is consistent with previous results showing that the star formation properties of normal star-forming galaxies up to z ≃ 2 can be reliably derived from UV/optical observations alone (e.g., Daddi et al. 2005, 2007; Reddy et al. 2006). This implies that the ISM of these galaxies is not heavily optically thick, and we can apply our energy-balance technique to interpret the SEDs of most normal star-forming "main-sequence" galaxies.

As a consistency check, we now compare our continuum flux density predictions with previously obtained number counts at (sub-)millimeter wavelengths.

Number counts at 850 μm have been obtained using the SCUBA bolometer array on the James Clerk Maxwell Telescope and LABOCA on APEX by a number of groups over the last decade (e.g., Scott et al. 2002; Smail et al. 2002; Borys et al. 2003; Coppin et al. 2006; Scott et al. 2006; Knudsen et al. 2008; Weiß et al. 2009). These counts can be directly compared to our predictions at 345 GHz (ALMA band 7, PdBI/NOEMA band 4) since this band probes roughly the same wavelength. In Figure 5, we compare our predicted cumulative number counts at 345 GHz with these previous observations. Our predicted number count range to the first order agrees with the observed number counts, but we do not reach the higher fluxes probed by submillimeter observations. The lack of the brightest sources is due to two reasons. First, the field on which we base our predictions is very small (the size of the UDF is only 3.3 × 10⁻³ deg²), i.e., we do not expect the presence of a significant population of bright sources in the field. Second, we are working with an optically detected sample, and the bright submillimeter counts are dominated by optically thick sources that are likely not detected in the optical. For example, LESSJ0333243.6-274644, the only submillimeter source detected in the Hubble UDF as part of the LESS survey (LABOCA observations of the ECDF-S), with a flux density of 6.4 mJy at 870 μm (Weiß et al. 2009), has no optical counterpart in our catalog, presumably because it is an optically thick SMG (cf. Dunlop 2011). We find that the number counts at fluxes ≳ 0.5 mJy are dominated by galaxies with $\mu \hbox{$\hat{\tau }_{V}$}> 1$, i.e., where the ISM is optically thick on average.

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As an additional test on our number count predictions, we turn to a wider field covered by the LESS survey. To do so, we expand our analysis to the FIREWORKS photometric catalog (Wuyts et al. 2008) on the CDF-S, described in Section 2.2. FIREWORKS covers an area that is about 10 times the area of the UDF and about 10 times smaller than the full ECDF-S. The photometric catalog is much shallower than that for the UDF. For the area covered by FIREWORKS, we estimate between 6 and 23 sources to have 870 μm fluxes above 4.7 mJy (the flux limit of the LESS catalog). For comparison, Weiß et al. (2009) find 10 sources over the same area. Our prediction is thus broadly consistent with the LESS measurements.

We can also compare our predictions with measurements of the integrated EBL in the submillimeter. Using our median-likelihood estimate of the 345 GHz flux density for each galaxy in the Hubble UDF, we obtain an integrated continuum 870 μm flux density of 45.6 Jy deg⁻². If we add the contribution from LESSJ0333243.6-274644 (which is not part of our sample but is detected in the LESS survey with a 6.4 mJy flux), we obtain an EBL value of 47.5 Jy deg⁻². This value is fully consistent with measurements of the EBL from COBE observations 44 ± 15 Jy deg⁻² (Puget et al. 1996; Fixsen et al. 1998). We note that the use of fixed SED templates to derive the (sub-)millimeter continuum fluxes of the galaxies would lead to EBL values that are inconsistent with the observed value (see Appendix A). In Table 1, we list our estimates of the EBL in different (sub-)millimeter bands.

The estimated EBL at 870 μm using the FIREWORKS catalog over the CDF-S area is 36.3 Jy deg⁻², broadly consistent with our estimate based on the UDF area only (Table 1). This is lower than the COBE observed value quoted above, but it is still consistent with the observations, since the FIREWORKS catalog does not reach very deep, and therefore it is likely to miss the large number of faint galaxies that make up for a significant fraction of the EBL.

In this section, we present our general predictions for the continuum flux densities of our galaxies. In Table 6, we make our continuum predictions in all relevant (sub-)millimeter bands available for all the galaxies in our sample.

As an example, in Figure 6, we plot continuum map of the UDF at 230 GHz (ALMA band 6, PdBI/NOEMA band 3) using our predictions (right-hand panels). Such images can be directly compared to future deep fields performed with ALMA or other facilities. We note that this figure is based solely on our optically based predictions, and so they are missing the only known bright (sub-)millimeter source in the UDF: LESSJ0333243.6-274644. The top panels of Figure 6 show our full sample, and we then divide the sample in two redshift bins (z < 1; middle panels) and (z ⩾ 1; bottom panels). This illustrates the differences between galaxy detections as a function of redshift between the optical and the (sub-)millimeter, in particular that we expect galaxies to be relatively brighter in the (sub-)millimeter at high redshift thanks to the negative k-correction. Therefore, we will be able to detect "normal" galaxies out to higher redshifts in the (sub-)millimeter with new facilities, as we discuss in more detail in Section 5.

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In Figure 7, we plot the distribution of the predicted continuum flux densities of all the galaxies in our sample in all current and future ALMA, JVLA, and IRAM PdBI/NOEMA bands from 38 GHz to 870 GHz. We plot the expected number of galaxies per flux bin in the total UDF area in each band. The distribution of fluxes peaks at higher fluxes in the highest frequency ALMA bands, because, even taking into account k-correction effects, these bands sample the emission from the galaxies closer to the peak of the dust SED. In Section 5, we discuss the feasibility of performing a blind survey of the UDF with ALMA at full operation, and use these predicted fluxes, combined with the projected ALMA sensitivities, to obtain an estimate of the expected number of continuum detections.

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ALMA and JVLA will detect CO and [C ii] line emission from high-redshift galaxies, which will allow us to determine redshifts, molecular gas reservoirs, and dynamical masses (e.g., Solomon & Vanden Bout 2005; Daddi et al. 2010b; Genzel et al. 2010; Tacconi et al. 2010; Walter et al. 2011). The rest-frame frequency, ν_rest, of the CO lines corresponding to J → J − 1 transitions from J = 1 to J = 7 and of the [C ii] line are given in Table 5. The observed frequency of each line varies with redshift as ν_obs = ν_rest(1 + z)⁻¹. In Figure 14 (Appendix B), we plot the observed frequency of the seven first CO transitions, CO(1–0) (i.e., J = 1) to CO(7–6) (i.e., J = 7) and of the [C ii] line as a function of redshift, with the frequency ranges covered by each ALMA, PdBI/NOEMA, and JVLA band shaded in gray and green. In Table 5 (Appendix B), we explicitly list the redshift ranges where the CO lines and [C ii] are observable in each band. It is clear that the lowest frequency bands, such as JVLA K, Ka, and Q will be crucial to probe the low J CO transitions, in particular the CO(1–0) line, in z > 1 galaxies. All the other PdBI/NOEMA and ALMA bands will potentially detect higher J CO transitions at different redshifts, depending on the excitation state of the gas in galaxies. The highest frequency ALMA bands will not only sample high-J CO lines at low redshifts and the continuum dust emission nearest to its peak, as mentioned in the previous section, but also the [C ii] line out to high redshifts.

In this section, we attempt to predict the CO and [C ii] line fluxes for the galaxies in our UDF sample using empirical relations that relate line luminosities with the infrared luminosity of the galaxies, for which we have a statistical estimate from our SED fits (Section 4.1).

4.7.1. CO Emission

The CO line luminosity of galaxies depends on various factors, such as the gas heating by starbursts, AGNs, and the cosmic microwave background at high redshifts (e.g., Combes et al. 1999; Obreschkow et al. 2009b; E. da Cunha et al., in preparation), as well as the clumpiness and metallicity of the gas (e.g., Obreschkow et al. 2009b). In this section, for simplicity, we predict the CO line luminosity of the galaxies in our sample using simple, empirically calibrated prescriptions. It has been found for a wide range of galaxy types, both in the local and high-redshift Universe, that the CO line luminosity of star-forming galaxies correlates with their infrared luminosity (e.g., Solomon & Vanden Bout 2005; Genzel et al. 2010; Daddi et al. 2010a).

The following relation between CO line luminosity and far-infrared luminosity was derived by Daddi et al. (2010a) for BzK galaxies, i.e., gas-rich star-forming disks at high redshifts:

$$\begin{equation} \log (\hbox{$L_\mathrm{IR}$}) = 1.13 \log (\hbox{$L^{\prime }_\mathrm{CO}$}) + 0.53 \,, \end{equation} \tag{ 1 }$$

where L_IR is the infrared luminosity (integrated between 8 and 1000 μm) in L_☉ and L'_CO is the CO(1–0) line luminosity in K km s⁻¹ pc². We obtain an estimate of L'_CO using this equation and the statistical estimate on L_IR obtained for the SED fits of our galaxies; the resulting distribution of L'_CO for the whole sample is plotted in the top left-hand panel of Figure 8. We chose this empirical calibration between L_IR and L'_CO because our physical parameter estimates in Section 4.2 indicate that most of these galaxies would be comparable to normal, "main-sequence" star-forming disks, with typical infrared luminosities L_IR ≲ 10¹¹ L_⊙. We note that Equation (1) is similar to the relation found by Genzel et al. (2010) for isolated, star-forming galaxies out to z = 2. Other calibrations of this relation have been derived which include more extreme galaxies such as starbursts, mergers, and AGNs (e.g., Solomon & Vanden Bout 2005; Riechers et al. 2006). When including these extreme galaxies, the relation between infrared and CO line luminosity becomes steeper, e.g., Solomon & Vanden Bout (2005) find log(L_FIR) = 1.7log(L'_CO) − 5.0. Using this steeper relation for the infrared luminosity range of our galaxies (L_dust ≲ 10¹¹ L_☉) would result in CO line luminosities over one order of magnitude higher than those predicted using Equation (1) for the L_IR range of the galaxies in our sample. We discuss the implications of using these different assumptions for the predicted number of CO line detections in Section 5.

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From L'_CO (computed using Equation (1)), we then compute the corresponding flux of the CO(1–0) line, S^CO(1–0)_ν, using (e.g., Solomon & Vanden Bout 2005):

$$\begin{equation} \hbox{$L^{\prime }_\mathrm{CO}$}= 3.25\times 10^7 S_\nu ^\mathrm{CO(1\hbox{--}0)} \, \Delta v \, \nu ^{-2}_\mathrm{obs} \, D_L^2 (1+z)^{-3}, \end{equation} \tag{ 2 }$$

where S^CO(1–0)_ν is the flux density in Jy, Δv is the line width in km s⁻¹ (the velocity-integrated flux of the line is I_CO(1–0) = S^CO(1–0)_ν Δv), ν_obs is the observed frequency of the line in GHz, and D_L is the luminosity distance in Mpc. We assume a typical line width of 300 km s⁻¹, consistent with typical line widths measured in high-redshift star-forming galaxies (e.g., Daddi et al. 2010a; Genzel et al. 2010; Tacconi et al. 2010). In the top right-hand panel of Figure 8, we plot the distribution of the velocity-integrated flux of the CO(1–0) line for all the galaxies in our sample computed using Equation (2).

The fluxes of higher transition CO lines depend highly on the excitation of the CO gas in galaxies. Different physical conditions in the gas produce different CO spectral line energy distributions (SLEDs; e.g., Weiss et al. 2007), which translate into different ratios between the CO(1–0) line and the higher J lines. To compute the predicted flux of the CO(2–1), CO(3–2), CO(4–3), CO(5–4), CO(6–5), and CO(7–6) lines, we assume two possible CO SLEDs from Weiss et al. (2007): the Milky Way CO SLED and the M 82 center CO SLED. These two cases correspond to very low and high excitation of the gas, respectively, and should bracket a large realistic range of possible physical conditions in star-forming galaxies. In the six bottom panels of Figure 8, we show the distribution of expected velocity-integrated CO line fluxes for the galaxies in our sample, and compare the predictions for these two excitation scenarios. For each CO line, these two extreme excitations should bracket the full range of line fluxes expected for our sample of star-forming galaxies (as supported by observations of multiple CO lines in a wide range of systems from local quiescent galaxies to high-redshift QSOs; see, e.g., Weiss et al. 2007). The difference between the predictions of CO line fluxes using these two CO SLEDs increases with increasing J: the higher J CO lines are stronger in the case of the M 82 center SLED, which corresponds to a higher CO excitation. In Table 7, we provide the predicted CO line fluxes for all the galaxies in our sample (including both CO excitation scenarios).

To predict the number of expected CO line detections given a certain flux limit in various (sub-)millimeter bands (ALMA, JVLA, or PdBI/NOEMA), we can build the expected distribution of CO line fluxes in each band based on the predictions described above. First, for each frequency band listed in Table 5, we retain galaxies for which the redshift falls in a range where one of the CO lines can be observed (these ranges are listed in Table 5, see also Figure 14). Then, we compute the expected line fluxes, S^line_ν in Jy, for each galaxy, by assuming a typical line width of Δv = 300 km s⁻¹ (using Equation (2) to the get flux of the CO(1–0) line and the CO SLEDs to get the higher J lines). In Figure 9, as examples, we plot the distribution of line fluxes from CO(1–0) to CO(7–6) if one were to fully cover the bands from 201 to 275 GHz (ALMA band 6, PdBI/NOEMA band 3), 80–116 GHz (ALMA band 3, PdBI/NOEMA band 1), and 18–50 GHz (ALMA band 1/JVLA bands K, Ka, and Q). We show the CO line fluxes corresponding to two gas excitation scenarios (i.e., CO SLEDs): Milky Way type (left-hand panels) and M82-type (right-hand panels). For example, in the frequency range 80–116 GHz, we plot the distribution of CO(1–0) line fluxes only of galaxies with redshifts z ⩽ 0.44, for which the CO(1–0) line would be redshifted to that frequency band (Table 5). Similarly, for the distribution of CO(2–1) line fluxes in that band, we include only galaxies with 0.99 ⩽ z ⩽ 1.88, and so on until the CO(7–6) line. For clarity, the distributions plotted in Figure 9 are cumulative: the darkest-colored histogram shows the number of galaxies per line flux bin for the CO(1–0) line, the next, lighter histogram shows the number of galaxies per line flux bin for the CO(1–0) and CO(2–1) line, the next histogram adds the number of galaxies per line flux bin for the CO(3–2) line, etc. That is, the lightest-colored histogram shows the total number of galaxies per line flux bin when including all CO lines from J = 1 to J = 7. In general, Figure 9 shows that the number of galaxies in the highest flux bins is largest if the CO SLED is M82-like, as expected, so the number of detections given a certain flux limit greatly depends on the CO excitation (as discussed in Section 5.2).

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4.7.2. [C ii] Emission

The [C ii] line at 158 μm is the main cooling line of the ISM in galaxies, and it arises mainly from photo-dissociation regions—at the interface between the ionized gas and the neutral and molecular gas—which are typically associated with star-forming regions. This line is therefore one of the main far-infrared tracers of star formation in galaxies (Stacey et al. 1991, 2010; Boselli et al. 2002; de Looze et al. 2011). Since this is the brightest far-infrared line, it will be readily detected in deep observations with ALMA, particularly using the highest frequency bands.

We rely on previous observational studies of the ratio of [C ii] line to far-infrared luminosity (L_[C II]/L_FIR) of galaxies to estimate the [C ii] line luminosity for each galaxy in our sample. The ratio L_[C II]/L_FIR of normal star-forming galaxies varies typically between 1% and 0.1%, and has been shown to anti-correlate with dust heating intensity (e.g., Brauher et al. 2008). Here, for simplicity, and considering the relatively large error bars on our L_FIR estimates, we adopt a constant ratio of log(L_[C II]/L_FIR) = −2.5, which corresponds to the average value for normal star-forming galaxies with low far-infrared luminosities L_IR ≲ 10¹¹ L_⊙ (i.e., similar to the galaxies in our sample) and average dust heating (Boselli et al. 2002; Brauher et al. 2008; Graciá-Carpio et al. 2011; Cox et al. 2011). In Figure 10(a), we plot the distribution of the expected velocity-integrated flux of [C ii], $I_{\mathrm{[C\,\scriptsize{II}]}}=S_\nu ^{\mathrm{[C\,\scriptsize{II}]}} \, \Delta v$ for all the galaxies in our sample, computed using Equation (2):

$$\begin{equation} \hbox{$L_{\mathrm{[C\,\scriptsize{II}]}}$}= 1.04\times 10^{-3} S_\nu ^{\mathrm{[C\,\scriptsize{II}]}} \, \Delta v \, \nu _\mathrm{rest} \, D_L^2 (1+z)^{-1}, \end{equation} \tag{ 3 }$$

where L_[C II] is the [C ii] line luminosity in L_☉, $S_\nu ^{\mathrm{[C\,\scriptsize{II}]}}$ is the line flux in Jy, Δv is the velocity dispersion in km s⁻¹, D_L is the luminosity distance, and ν_rest = 1900.54 GHz. At z < 5, the observed frequency of the [C ii] line falls in the highest frequency ALMA bands, namely, bands 7, 8, 9, and 10 (Figure 14). In Figure 10(b), we plot the distribution of expected detections in these bands as a function of [C ii] line flux (computed assuming Δv = 300 km s⁻¹). In Table 7, we provide the predicted [C ii] line fluxes for all the galaxies in our sample.

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For each ALMA band, given the area of the primary beam, we compute how many effective pointings are needed to cover the whole area of the UDF (Table 2). The number of pointings needed to cover the UDF increases from band 1 to band 10, from only 3 pointings to over 1000 pointings. In this back-of-the-envelope calculation we take the integration time per pointing in each band as the total 500 hr divided by the required number of pointings, and then use the ALMA Sensitivity Calculator (ASC)¹³ to compute the continuum sensitivity that can be reached in each integration time, assuming a 8 GHz bandwidth and the default weather conditions. In Table 2, we list the sensitivities and the expected number of 3σ continuum detections in each band for the whole field, based on our continuum predictions of Section 4.6. The resulting number of predicted continuum detections is a combination of the intrinsic flux of each galaxy, the k-correction for each galaxy at each redshift, and the changing integration time per pointing due to varying beam size; for reference, 1 hr integration time corresponds to an rms of 4.81, 12.6, and 483 μJy at 38 GHz (ALMA 1), 230 GHz (ALMA 6), and 870 GHz (ALMA 10), respectively. Table 2 shows that the number of expected continuum detections is maximum for band 6 at 230 GHz (601 detections), with a large number of predicted detections also in band 3 at 100 GHz (221), band 4 at 144 GHz (363), and band 7 at 345 GHz (522). In Figures 1 and 11, we compare the distributions of the properties of the 601 galaxies that would be detected at the 3σ level in the ALMA band 6 to those of the original sample. We note that the redshift distribution of these sources is relatively flat, thanks to the negative k-correction in the submillimeter. Also not surprisingly, these figures show that we expect to detect only the most massive, highly star-forming, and dusty galaxies in our sample. These galaxies are still over an order of magnitude star-forming and dusty than classic SMGs detected in blind "pre-ALMA" (sub-)millimeter surveys, which typically have SFRs ≳ 100 M_☉ yr⁻¹ and dust luminosities ≳ 10¹² L_☉. The median detected galaxy has a stellar mass of 3 × 10⁹ M_☉, an SFR of ∼5 M_☉ yr⁻¹, dust luminosity of 4 × 10¹⁰ L_☉, dust mass of 5 × 10⁷ M_☉, and V-band effective optical depth in stellar birth clouds of 1.6. This implies that the typical detected galaxy would be about 100 times more star-forming, more massive (in terms of stellar content), and more dusty, and about 10 times more obscured in the optical than the median of the whole UDF sample.

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We now discuss the possibility of detecting CO and [C ii] lines in the UDF using frequency scans in all ALMA bands. We compute the expected line sensitivities for a total on-source time of 500 hr on the UDF as in Section 5.1, but taking into account the time needed to scan in band in frequency space. The frequency interval covered in each frequency setting is Δν = 8 GHz. Therefore, the total number of settings required to scan a given ALMA band is the total frequency range of the band divided by Δν (we note that the exact setup will depend on the sideband separations in the various bands). The total time spent in each frequency setting is then the integration time per pointing divided by the required number of settings in each band. We assume the same mosaicking scheme of the UDF and use the same integration time per pointing in each band as for the continuum (Section 5.1). The total number of frequency settings, time per setting, and resulting line sensitivity σ_line (computed using the ASC and assuming a bandwidth of 300 km s⁻¹ in order to resolve the lines in velocity space) are listed in Table 2. In Table 3, we show the predicted number of 5σ line detections in each band given the line sensitivities σ_line of Table 2, assuming both the Milky Way and the M 82 CO SLED, as in Section 4.7.1. It is clear that with the integration times per frequency setting of Table 2 we predict a relatively low number of line detections, compared with the predicted number of continuum detections. In bands 2–6, we predict a minimum of about 20 CO line detections in each band, when assuming the Milky Way CO SLED; around 100 detections are predicted if the CO is more excited. To get more line detections, one would need to go deeper, which likely implies a compromise with the area of the survey. For example, to reach an rms of 10 μJy in band 6 (approximately 8 times deeper than reached in the "default" survey setup considered so far), which would yield at least between 200 and 800 detections over the whole UDF, one would need over 400 hr of effective integration time for each of the 81 pointings.

As mentioned in Section 4.7.1, if we use the Solomon & Vanden Bout (2005) calibration to convert the infrared luminosities of our galaxies into CO line luminosities, we would obtain intrinsically brighter CO lines by about one order of magnitude. This would make the number of predicted CO line detections in the observational setup discussed here much higher than the numbers listed in Table 3. In the bands with most predicted detections, ALMA 2 to ALMA 6, the minimum number of detections (when assuming the Milky Way SLED) would increase from on average 22 to 116 in each band, and the maximum number of detections (when assuming the M 82 SLED) would increase from on average 50 to 242 in each band.

We use the same method to estimate the number of expected [C ii] line detections in the highest frequency ALMA bands, based on our estimates of [C ii] line fluxes from Section 4.7.2. We predict a total of 15 [C ii] 5σ detections using the integration times per pointing and per frequency setting in each band listed in Table 2 (Table 4). In bands 8–10, these numbers represent a low fraction (≲ 0.1%) of the total number of galaxies in the observable redshift range because the large number of required pointings to cover the whole UDF implies a very short integration time per pointing and hence a relatively high rms. The minimum dust luminosities and SFRs of the galaxies that can be detected in [C ii] using this observational setup are 5 × 10¹¹ L_☉ and 10 M_☉ yr⁻¹, at the very high end of the distribution for the whole sample (e.g., Figure 11). With band 7, the number of detections is higher and because the rms is smaller thanks to a higher integration time, allowing us to go deeper and detect [C ii] emission from galaxies with dust luminosities as low as 1.4 × 10¹¹ L_☉.