# The Herschel Multi-tiered Extragalactic Survey: SPIRE-mm photometric redshifts

I. G. Roseboom, R. J. Ivison, T. R. Greve, A. Amblard, V. Arumugam, R.

Auld, H. Aussel, M. Béthermin, A. Blain, J. Bock, et al.

► To cite this version:

I. G. Roseboom, R. J. Ivison, T. R. Greve, A. Amblard, V. Arumugam, et al. The Herschel Multi-tiered Extragalactic Survey: SPIRE-mm photometric redshifts. Monthly Notices of the Royal Astronomical Society, 2012, 419 (4), pp.2758-2773. 10.1111/j.1365-2966.2011.19827.x. hal-01442383

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# The Herschel Multi-tiered Extragalactic Survey: SPIRE-mm photometric redshifts

I. G. Roseboom,<fnref n="1" />,<fnref n="2" />* R. J. Ivison,<fnref n="3" />,<fnref n="2" /> T. R. Greve,<fnref n="4" /> A. Amblard,<fnref n="5" /> V. Arumugam,<fnref n="3" />

R. Auld,<fnref n="6" /> H. Aussel,<fnref n="7" /> M. Bethermin,<fnref n="7" /> A. Blain,<fnref n="8" /> J. Bock,<fnref n="8" />,<fnref n="9" /> A. Boselli,<fnref n="10" /> D. Brisbin,<fnref n="11" />

V. Buat,<fnref n="10" /> D. Burgarella,<fnref n="10" /> N. Castro-Rodríguez,<fnref n="12" />,<fnref n="13" /> A. Cava,<fnref n="12" />,<fnref n="13" /> P. Chanial,<fnref n="7" />

E. Chapin,<fnref n="14" /> S. Chapman,<fnref n="15" /> D. L. Clements,<fnref n="16" /> A. Conley,<fnref n="17" /> L. Conversi,<fnref n="18" />

A. Cooray,<fnref n="5" />,<fnref n="8" /> C. D. Dowell,<fnref n="8" />,<fnref n="9" /> J. S. Dunlop,<fnref n="2" /> E. Dwek,<fnref n="19" /> S. Eales,<fnref n="6" /> D. Elbaz,<fnref n="7" />

D. Farrah,<fnref n="1" /> A. Franceschini,<fnref n="20" /> J. Glenn,<fnref n="21" /> M. Griffin,<fnref n="6" /> M. Halpern,<fnref n="14" />

E. Hatziminaoglou,<fnref n="22" /> E. Ibar,<fnref n="3" /> K. Isaak,<fnref n="6" /> G. Lagache,<fnref n="23" /> L. Levenson,<fnref n="8" />,<fnref n="9" />

N. Lu,<fnref n="8" />,<fnref n="24" /> S. Madden,<fnref n="7" /> B. Maffei,<fnref n="25" /> G. Mainetti,<fnref n="20" /> L. Marchetti,<fnref n="20" /> G. Marsden,<fnref n="14" />

G. Morrison,<fnref n="26" />,<fnref n="27" /> A. M. J. Mortier,<fnref n="16" /> H. T. Nguyen,<fnref n="8" />,<fnref n="9" /> B. O’Halloran,<fnref n="16" /> S. J. Oliver,<fnref n="1" />

A. Omont,<fnref n="28" /> M. J. Page,<fnref n="29" /> P. Panuzzo,<fnref n="7" /> A. Papageorgiou,<fnref n="6" /> C. P. Pearson,³⁰,³¹

I. Pérez-Fournon,<fnref n="12" />,<fnref n="13" /> M. Pohlen,<fnref n="6" /> J. I. Rawlings,<fnref n="29" /> G. Raymond,<fnref n="6" /> D. Rigopoulou,³⁰,³²

D. Rizzo,<fnref n="16" /> G. Rodighiero,<fnref n="20" /> M. Rowan-Robinson,<fnref n="16" /> B. Schulz,<fnref n="8" />,<fnref n="24" /> Douglas Scott,<fnref n="14" />

N. Seymour,<fnref n="29" /> D. L. Shupe,<fnref n="8" />,<fnref n="24" /> A. J. Smith,<fnref n="1" /> J. A. Stevens,³³ M. Symeonidis,<fnref n="29" />

M. Trichas,³⁴ K. E. Tugwell,<fnref n="29" /> M. Vaccari,<fnref n="20" /> I. Valtchanov,<fnref n="18" /> J. D. Vieira,<fnref n="8" /> M. P. Viero,<fnref n="8" />

L. Vigroux,<fnref n="28" /> J. Wardlow,<fnref n="5" /> L. Wang,<fnref n="1" /> G. Wright,<fnref n="3" /> C. K. Xu<fnref n="8" />,<fnref n="24" /> and M. Zemcov<fnref n="8" />,<fnref n="9" />

<footnote n="1">¹Astronomy Centre, Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH</footnote>

<footnote n="2">²Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ</footnote>

<footnote n="3">³UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ</footnote>

<footnote n="4">⁴Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark</footnote>

<footnote n="5">⁵Department of Physics & Astronomy, University of California, Irvine, CA 92697, USA</footnote>

<footnote n="6">⁶Cardiff School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff CF24 3AA</footnote>

<footnote n="7">⁷Laboratoire AIM-Paris-Saclay, CEA/DSM/Irfu - CNRS - Université Paris Diderot, CE-Saclay, pt courrier 131, F-91191 Gif-sur-Yvette, France</footnote>

<footnote n="8">⁸California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA</footnote>

<footnote n="9">⁹Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA</footnote>

<footnote n="10">¹⁰Laboratoire d'Astrophysique de Marseille, OAMP, Université Aix-marseille, CNRS, 38 rue Frédéric Joliot-Curie, 13388 Marseille cedex 13, France</footnote>

<footnote n="11">¹¹Space Science Building, Cornell University, Ithaca, NY 14853-6801, USA</footnote>

<footnote n="12">¹²Instituto de Astrofísica de Canarias (IAC), E-38200 La Laguna, Tenerife, Spain</footnote>

<footnote n="13">¹³Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="14">¹⁴Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada</footnote>

<footnote n="15">¹⁵Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA</footnote>

<footnote n="16">¹⁶Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ</footnote>

<footnote n="17">¹⁷Center for Astrophysics and Space Astronomy, 593 UCB, Boulder, CO 80309-0593, USA</footnote>

<footnote n="18">¹⁸Herschel Science Centre, European Space Astronomy Centre, Villanueva de la Cañada, 28691 Madrid, Spain</footnote>

<footnote n="19">¹⁹Observational Cosmology Lab, Code 665, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA</footnote>

<footnote n="20">²⁰Dipartimento di Astronomia, Università di Padova, vicolo Osservatorio, 3, 35122 Padova, Italy</footnote>

<footnote n="21">²¹Department of Astrophysical and Planetary Sciences, CASA 389-UCB, University of Colorado, Boulder, CO 80309, USA</footnote>

<footnote n="22">²²ESO, Karl-Schwarzschild-Str. 2, 85748 Garching bei München, Germany</footnote>

<footnote n="23">²³Institut d'Astrophysique Spatiale (IAS), bâtiment 121, Université Paris-Sud 11 and CNRS (UMR 8617), 91405 Orsay, France</footnote>

<footnote n="24">²⁴Infrared Processing and Analysis Center, MS 100-22, California Institute of Technology, JPL, Pasadena, CA 91125, USA</footnote>

<footnote n="25">²⁵School of Physics and Astronomy, The University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL</footnote>

<footnote n="26">²⁶Institute for Astronomy, University of Hawaii, Manoa, HI 96822, USA</footnote>

<footnote n="27">²⁷Canada-France-Hawaii Telescope Corp., Kamuela, HI 96743, USA</footnote>

<footnote n="28">²⁸Institut d'Astrophysique de Paris, UMR 7095, CNRS, UPMC Univ. Paris 06, 98bis boulevard Arago, F-75014 Paris, France</footnote>

<footnote n="29">²⁹Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT</footnote>

<footnote>*E-mail: igr@roe.ac.uk</footnote>
Accepted 2011 September 12. Received 2011 September 12; in original form 2011 February 10

## ABSTRACT

We investigate the potential of submm-mm and submm-mm-radio photometric redshifts using a sample of mm-selected sources as seen at 250, 350 and 500 µm by the SPIRE instrument on Herschel. From a sample of 63 previously identified mm sources with reliable radio identifications in the Great Observatories Origins Deep Survey North and Lockman Hole North fields, 46 (73 per cent) are found to have detections in at least one SPIRE band. We explore the observed submm/mm colour evolution with redshift, finding that the colours of mm sources are adequately described by a modified blackbody with constant optical depth $\tau = (v/nu_0)^\beta$, where $\beta = +1.8$ and $v_0 = c/100$ µm. We find a tight correlation between dust temperature and IR luminosity. Using a single model of the dust temperature and IR luminosity relation, we derive photometric redshift estimates for the 46 SPIRE-detected mm sources. Testing against the 22 sources with known spectroscopic or good quality optical/near-IR photometric redshifts, we find submm/mm photometric redshifts offer a redshift accuracy of $|\Delta z|/(1+z) = 0.16$ ($|\Delta z| \rangle = 0.51$). Including constraints from the radio-far-IR correlation, the accuracy is improved to $|\Delta z|/(1+z) = 0.15$ ($|\Delta z| \rangle = 0.45$). We estimate the redshift distribution of mm-selected sources finding a significant excess at $z > 3$ when compared to ~850 µm selected samples.

**Key words:** galaxies: high-redshift – galaxies: statistics – submillimetre: galaxies.

## 1 INTRODUCTION

As is the case in the local Universe, star formation at high redshift takes place in regions rich in molecular gas and dust. Franceschini et al. (1991) and Blain & Longair (1993) thus anticipated that the distant galaxies hosting the majority of this activity would be discovered in the submm/mm waveband, which benefits from a strong, negative *K*-correction for the rest-frame far-infrared (far-IR) portion of their spectral energy distributions (SEDs). Observing at 450 and 850 µm, the Submillimetre Common-User Bolometer Array (SCUBA) submm camera on the James Clerk Maxwell Telescope (Holland et al. 1999) made it possible to detect the first examples of this galaxy population and represented a fundamental turning point in our understanding of galaxy formation in the distant Universe (Smail, Ivison & Blain 1997; Barger et al. 1998; Hughes et al. 1998; Dey et al. 1999; Eales et al. 1999).

Early work on the submm galaxy population was often hindered by small numbers of sources, and significant difficulties in finding multiwavelength counterparts given the relatively low angular resolution (full width at half-maximum, FWHM > 10 arcsec) of ground-based submm/mm facilities. This situation has been much improved by a large increase in the number of submm/mm facilities, larger scale submm surveys [e.g. SHADES, Mortier et al. 2005; LABOCA Extended Chandra Deep Field South Submm Survey (LESS), Weiß et al. 2009], improvements in the quality of overlapping multiwavelength data (in particular radio and mid-IR imaging), as well as advances in cross-matching algorithms (Ivison et al. 2007; Chapin et al. 2009, 2011; Roseboom et al. 2009). As a

result, our understanding of these galaxies has accelerated in recent years (e.g. Alexander et al. 2005; Chapman et al. 2005; Greve et al. 2005; Pope et al. 2008; Swinbank et al. 2008; Tacconi et al. 2008; Menéndez-Delmestre et al. 2009).

The advent of the SPIRE instrument (Griffin et al. 2010) onboard Herschel<fnref n="1" /> (Pilbratt et al. 2010) promises to shed new light on many issues relating to submm galaxies. In photometry mode, SPIRE operates in three submm bands centred on 250, 350 and 500 µm, and thus the large-scale surveys conducted with SPIRE will detect many 10 000 s of submm galaxies (e.g. Eales et al. 2010; Oliver et al., in preparation). SPIRE observations of existing submm/mm survey fields provide the crucial SED information shortward of 500 µm which cannot be routinely accessed from the ground (cf. Kovács et al. 2006), allowing us to characterize their IR luminosity ($L_{IR}$) and dust temperature ($T_D$) with unprecedented accuracy (e.g. Chapman et al. 2010; Magnelli et al. 2010).

Armed with this new information, we can re-evaluate the potential of using submm (and mm) wavelength data alone to estimate the redshifts of submm galaxies. As submm galaxies tend to be both high redshift ($\langle z \rangle \sim 2$; Chapman et al. 2005) and highly dust obscured, they are extremely faint at optical/near-IR wavelengths and hence obtaining redshifts via optical spectroscopy is challenging. Because of this, the prospects for submm/mm photometric

<footnote n="30">³⁰Space Science & Technology Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX</footnote>

<footnote n="31">³¹Institute for Space Imaging Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada</footnote>

<footnote n="32">³²Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH</footnote>

<footnote n="33">³³Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB</footnote>

<footnote n="34">³⁴Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="1">¹ Herschel is a European Space Agency space observatory with science instruments provided by Principal Investigator consortia. It is open for proposals for observing time from the worldwide astronomical community.</footnote>
identification would still pose a serious challenge. The positional uncertainties of both the mm- and SPIRE-detected sources ($\sigma_{pos} \sim 2–3$ arcsec) require the use of a 9–12 arcsec matching radius in order to recover the bulk of the true mm–SPIRE associations. Given the known number density of SPIRE sources at these flux levels ($\sim 7000$ deg$^{-2}$; Glenn et al. 2010), we would expect between 14 and 25 per cent of these associations to be chance alignments.

In this work, we first match the mm-detected source lists to 1.4-GHz sources, using the well-established catalogue based cross-matching techniques of Downes et al. (1986) and then use these 1.4 GHz source positions, along with the source positions of all known 1.4 GHz and 24 µm sources in these fields, to ‘deblend’ the SPIRE maps into their individual contributors and thus provide the best estimate of the SPIRE photometry at the locations of our mm-selected sample.

## 3.1 1.4 GHz identifications of mm-selected sources

In the GOODS-N field, we search for potential 1.4 GHz counterparts in the Morrison et al. (2010) VLA source catalogue to the 41 1.16-mm sources from Penner et al. (2011) using a 10 arcsec search radius. 28 of the 41 sources have at least one potential counterpart within this search radius. To determine the reliability of these matches, we estimate the probability of a chance alignment using the *P*-statistic, as defined by Downes et al. (1986). Defining a reliable match to be the one with a less than 5 per cent probability of being a chance alignment, 32 reliable matches are found for 24 objects, with eight sources having more than one reliable radio counterpart.

For the sources with more than one reliable counterpart, we take the counterpart with the lower *P*-statistic to be the correct identification.

One difficult case for our matching is HDF 850.1 (ID 14 in the catalogue of Penner et al. 2011), the brightest submm source detected in the original 850 µm SCUBA imaging of GOODS-N (Hughes et al. 1998). Our naive cross-matching approach would associate this object with a 56 µJy radio source located 5.3 arcsec away. However, subsequent near-IR and high-resolution Submillimeter Array observations show that the true position is much closer to the original SCUBA position (Dunlop et al. 2004; Cowie et al. 2009), although no distinct 1.4 GHz radio source can be found near this position in the Morrison et al. (2010) image. Here we adopt the Dunlop et al. (2004) position for matching with the SPIRE data.

Other difficult cases are those sources for which the submm emission is later shown, via high-resolution mm-wavelength interferometry, to originate from multiple components. In these cases, the 1.4 GHz identification(s) is often found to lie coincident with one (or more) of these components (e.g. Wang et al. 2011). Amongst our sample are two examples of this, GN20 (G1 in our sample; Daddi et al. 2009a; Carilli et al. 2010, 2011) and GN21 (G19 in our sample; Wang et al. 2011). In both cases, we have simply taken a single radio identification, the one with the lowest *P*-statistic, as being solely responsible for the mm flux density. For G1, the radio identification corresponds to GN20 from Daddi et al. (2009a), while for G19 this corresponds to GOODS 850-13c in the list of Wang et al. (2011). While it may turn out that a large fraction of mm-detected sources do indeed contain multiple components, in both G1 and G19 the single radio identification we have chosen is found to be responsible for $\gtrsim 50$ of the submm/mm flux (Daddi et al. 2009a; Wang et al. 2011). Importantly, our SPIRE photometry pipeline accounts for all 1.4 GHz, mm and 24 µm detected sources, so it should not be affected by this issue.

Encouragingly, for the 28 sources from Penner et al. (2011) which were previously identified in the AzTEC and/or MAMBO surveys, we recover the same identifications made by both Chapin et al. (2009, hereafter C09), for the 25 sources in common, and Greve et al. (2008), for the eight sources in common.

In the Lockmann Hole North field, we make use of the existing matches between the 1.2-mm sources and 1.4 GHz radio catalogue presented in Lindner et al. (2011). 40 of the 41 1.2-mm sources are found to have a 1.4 GHz counterpart within 8 arcsec, with 39 of these found to have less than a 5 per cent probability of being a chance alignment and hence are deemed reliable.

It is worth noting that three of these reliable radio counterparts are not found in the published Owen & Morrison (2008) 1.4 GHz source lists, and come from a deeper extraction performed in the vicinity of the MAMBO sources by Lindner et al. (2011). For these sources, we adopt the quoted 1.4 GHz positions and flux densities from Lindner et al. (2011).

Thus, we are left with a sample of 63 mm-selected sources with high angular resolution 1.4 GHz positions with which to match to the SPIRE data.

Interestingly, there is a significant disparity between the identification rates in the two fields considered here. Both Penner et al. (2011) and Lindner et al. (2011) list 41 mm-detected sources with a similar noise, $1\sigma \sim 0.7$ mJy, although in both cases the noise varies significantly across the field. However, the effective area of the two surveys is quite different. The Penner et al. (2011) combined AzTEC/MAMBO image of GOODS-N covers an area of 0.08 deg$^2$, while the Lindner et al. (2011) MAMBO image of Lockman North covers an area of 0.16 deg$^2$. Thus, it is clear that for the Penner et al. GOODS-N sample to find the same number of objects similar to one-half the area, it must be identifying a fainter population of sources. The differential number counts at these wavelengths are known to be well described by a power law $dN/dS \propto S^\delta$, with an exponent $\delta \sim -3$. So to observe a difference of a factor of 2 in surface density, the GOODS-N sample must be effectively a factor of $\sqrt{2}$ deeper, ignoring the *K*-correction between 1.16 and 1.2 mm.

## 3.2 SPIRE photometric method

The most difficult obstacle in determining accurate SPIRE photometry is the effect of source confusion, i.e. the contributions of numerous faint sources within a single SPIRE resolution element, centred on the target of interest. For this reason, several authors have developed techniques that utilize the positions of sources detected at other wavelengths, usually 24 µm and 1.4 GHz, to disentangle the various contributions from discrete sources to the SPIRE flux in a given beam element (e.g. Roseboom et al. 2010; Chapin et al. 2011). This process is made possible by the high correspondence between the 24-µm and 1.4-GHz populations and those observed at far-IR wavelengths; $>80$ per cent of the cosmic IR background at SPIRE wavelengths can be accounted for by 24-µm sources with $S_{24} > 25$ µJy (Marsden et al. 2009; Pascale et al. 2009), while the strong correlation between the far-IR and radio luminosity is known to hold across a wide range in redshift and luminosity (e.g. Ibar et al. 2008; Ivison et al. 2010).

Here we follow the prescription presented by Roseboom et al. (2010, hereafter R10) to measure the SPIRE fluxes of mm sources in GOODS-N and Lockman Hole North, with a number of modifications. Below, we summarize the R10 methodology, then highlight our departures from it.

R10 assume that the SPIRE map can be fitted directly by assuming that the positions of all sources contributing significantly to the
map are known (i.e. previously detected at 24 µm, 1.4 GHz or other wavelengths), and that only the SPIRE flux density of each of these sources is unknown. Algebraically this means the observed SPIRE map, *d*, can be described as a series of point sources with fluxes, *f*, at known positions, *x*. Thus, the observed map can be described as

$$d = P_x f + \delta,$$

where $P_x$ is the point response function (PRF) centred at position *x* and *δ* is some unknown noise term. As discussed in R10, this equation can be easily inverted, using linear algebra, to obtain the best-fitting values for the fluxes. However, one potential pitfall of this approach is that of overfitting. If many of our input sources are intrinsically faint and the source density is high, this approach will give poor results. R10 solve this issue by using model-selection techniques to eliminate those sources that are unnecessary to describe the map. Specifically, the Akaike information criterion (Akaike 1974) is used to filter the input list of sources via backward elimination. This is very time consuming and hence the number of sources that could be fitted simultaneously was limited to ~100–200 in R10.

Here, we replace this model-selection approach with the adaptive Least Absolute Shrinkage and Selection Operator (LASSO; Tibshirani 1996; Zou 2006). In particular, we combine a non-negative application of this technique – the Non-Negative Least Squares path (NNLSpath) algorithm of ter Braak et al. (2010) – with the adaptive weighting scheme suggested by Zou (2006). In Zou (2006), the weights are tunable, based on initial estimates of the parameters; here we use the probability of a chance alignment, $\phi$, as defined in R10. Specifically, $\phi$ is calculated separately for the three samples which make up our input list: radio, 24 µm and mm-detected sources. In cases where a source is detected in more than one of these bands, the lowest value of $\phi$ is adopted. As mm-detected sources have the lowest areal density, these typically have higher weightings, followed by radio sources and then 24 µm.

Accuracy in model selection is the primary advantage. The adaptive LASSO is known to have ‘oracle’ properties<fnref n="3" /> with the correct choice of weightings; however, there is also a significant advantage in computational speed. With this approach, we can solve for large numbers of candidate sources (up to 10 000) in a realistic amount of time.

In addition, we modify the background estimation. Rather than using an unconstrained local estimate of the background, the background is estimated globally, assuming a flat background across the total extent of the map. This approach is preferred as it is more robust against incompleteness in the prior input list, and because there are no significant large-scale correlations in the noise present in our SPIRE maps due to the exceptionally low-frequency ‘knee’ to the 1/f noise (Griffin et al. 2010).

Finally, potential new SPIRE sources, i.e. those that were previously unknown at 24 µm or radio wavelengths, are considered by examining the residual maps after fitting the SPIRE fluxes of known sources.

### 3.3 Estimating the true noise for SPIRE photometry method

Accurately estimating the effect of confusion on an individual source is non-trivial. As the SPIRE maps have a mean intensity

of zero, for blind source catalogues we can expect that the typical confusion noise is given by the width of the distribution of flux densities [$P(D)$] in a PRF-convolved image. For SPIRE, we know these values are 5.8, 6.3 and 6.8 mJy beam⁻¹ at 250, 350 and 500 µm, respectively (Nguyen et al. 2010). However, our source extraction process attempts to find the true zero-point of the map, and simultaneously fits the positions of sources found at other wavelengths, in an attempt to reduce this noise. However, it is difficult to assess for a given source how much we have ‘resolved’ the confusing background, and how much is left as unresolved fluctuations in the map.

One simple way to assess the global improvement found by our method is to use completely synthetic maps and catalogues produced using input mock catalogues known to give reasonable agreement to the known far-IR and submm number counts (e.g. Fernandez-Conde et al. 2008). Testing against simulated SPIRE maps with the same properties as our GOODS-N data (i.e. instrumental noise, number counts and pixel intensity distribution), we find a typical total noise, i.e. confusion and instrumental noise, of $\sigma_{total} = 2.8, 3.9$ and $3.7$ mJy at 250, 350 and 500 µm, respectively. These values compare quite favourably to the SPIRE photometry process described in R10, which gave $\sigma_{total} = 4.9, 5.4$ and $6.5$ mJy at 250, 350 and 500 µm for the same simulations.

While testing the performance of our method on simulated data sets is of interest, we really need an accurate estimate of the true error on the SPIRE photometry of our actual sources. One approach would be to estimate the global confusion noise in our SPIRE photometry by investigating residual intensity fluctuations in the SPIRE maps, after subtracting the contributions from ‘resolved’ sources. This approach is appealing, as we know that the residual maps contain information about sources not included in our prior source list, as well as any artefacts from incorrect astrometry and/or assumptions about the PRF.

However, there is one complication for this approach: variations in sensitivity in the prior source lists across the SPIRE map will mean the ability to resolve confusing sources varies spatially. Thus, we must assess the confusion noise on scales small enough that the prior source list sensitivity is invariant. This is particularly difficult in this work, as the combination of 24 µm and 1.4 GHz prior source catalogues results in large variations across the field. This is due to both the difference in relative sensitivity of 24 µm and 1.4 GHz to SPIRE sources, as well as the tapering of the 1.4 GHz sensitivity towards the edges of its coverage.

We utilize a variant of this approach to estimate the true error for our confused SPIRE sources here. Specifically, we construct PRF-convolved residual maps for each of the SPIRE bands, using our best estimate of the SPIRE photometry at the point of each input source and the known SPIRE beam. Then, at the position of each source we are interested in, we assess the total noise per pixel by measuring the standard deviation of surrounding pixels within a 20-pixel radius (2, 2.8 and 4 arcmin for the SPIRE 250, 350 and 500 µm maps used here).

A further complication arises in converting this noise per pixel into the noise for a point source as we know that the confusing background must be correlated, on small scales by the PRF and on larger scales by the intrinsic clustering of the sources. This means that in calculating the total noise estimate for a given point source, we must take into account the uncorrelated instrumental noise in each pixel and the correlated noise from confusion. For simplicity, we assume that the correlated noise on small scales is dominated by the PRF, and assume that the shape of the PRF gives the covariance between pixels directly, i.e. if we assume the SPIRE photometry is

<footnote n="3">³ In model selection, a method is said to have oracle properties if it can reliably identify which free parameters are truly present in the underlying system, and which are unnecessary or spurious.</footnote>
estimated by

$$S = \frac{\sum_i d_i P_i^2 \sigma_i^2}{\sum_i P_i^2 / \sigma_i^2},$$

where $d_i$ is the intensity in pixel $i$, $P_i$ the PRF for the point source in pixel $i$ and $\sigma_i$ the instrumental noise in pixel $i$, the noise estimate, including contributions from both confusion noise and instrumental noise, can be given by

$$\sigma_{\text{total}}^2 = (\mathbf{P}^T \mathbf{C}^{-1} \mathbf{P})^{-1},$$

where $\mathbf{C}$ is the covariance matrix, the diagonal elements of which are given by $1/\sigma_i^2$, and the off diagonal elements $i, j$ given by $P'_i/\sigma_i/\sigma_j$, where $P'$ is the PRF assuming a point source centred at position $i$.

Using this approach, we find the typical $1\sigma$ total noise for our mm-detected sources is $\sigma_{\text{total}} = 2.2, 2.6$ and $2.5$ mJy in the central part of the GOODS-N field for the 250, 350 and 500 µm bands, respectively, increasing to 3.7, 4.8 and 5.7 mJy at a radius of 20 arcmin. Similarly, in the Lockman North field, the total noise is found to be 3.5, 3.6 and 4.7 mJy at the centre of the field, and 4.4, 4.7, 6 mJy at a radius of 20 arcmin.

It is worth noting that these variations are solely due to the variations in the depth of the prior source list; the SPIRE sensitivity is uniform across both fields within the coverage of the mm observations (Levenson et al. 2010).

# 4 RESULTS

## 4.1 SPIRE detection statistics

We define the SPIRE measurement as a detection if it has >3$\sigma$ significance in the flux density estimate and the goodness of the photometric fit to the map is high (reduced $\chi^2 < 5$, in a 15 × 15 pixel region centred on the source position). The $\chi^2$ cut is necessary to remove cases where bright or unaccounted for sources, or residual non-Gaussian noise (e.g. Galactic cirrus or uncorrected thermal drifts) introduce signal that cannot be accounted for by our input source model.

Table 1 presents our SPIRE photometry for all mm-selected sample in GOODS-N, while Table 2 presents SPIRE photometry for mm sources in the Lockman Hole North field.

In GOODS-N, 17 of the 24 1.16-mm sources with good identifications are detected in at least one SPIRE band, while in Lockman North 30 of the 39 sources with identifications are detected in at least one SPIRE band. These figures give a marginally lower detection rate in GOODS-N (71 per cent) as compared to Lockman North (77 per cent), although this is consistent with the slightly deeper nature of the GOODS-N mm imaging (see Section 3.1).

As GOODS-N and Lockman Hole North are well-established survey fields, a large number of spectroscopic, and good quality photometric, redshifts are available. In GOODS-N, nine sources have spectroscopic redshifts, either from targeted observations of submm galaxies and radio sources by Chapman et al. (2005, and in

<table><caption>**Table 1.** Measured SPIRE photometry for mm sources in GOODS-N. $3\sigma$ upper limits are given for sources which are not detected above this significance in the SPIRE imaging. For the mm photometry, deboosted flux densities are quoted. The $\chi^2$ represents the reduced $\chi^2$ in a 15 × 15 pixel region centred on the source position. ID number refers to the equivalent value from Penner et al. (2011). C09 refers to the ID numbers of sources in common with Chapin et al. (2009). G08 refers to the ID numbers of sources in common with Greve et al. (2008). 1.4 GHz flux density is from the catalogue of Morrison et al. (2010).</caption><thead><tr><th>Name</th><th>C09<sup>a</sup></th><th>G08<sup>b</sup></th><th>RA<br>(°)</th><th>Dec.<br>(°)</th><th>S<sub>250</sub><br>(mJy)</th><th>χ<sup>2</sup><sub>250</sub></th><th>S<sub>350</sub><br>(mJy)</th><th>χ<sup>2</sup><sub>350</sub></th><th>S<sub>500</sub><br>(mJy)</th><th>χ<sup>2</sup><sub>500</sub></th><th>S<sup>c</sup><sub>1.2 mm</sub><br>(mJy)</th><th>S<sub>1.4 GHz</sub><br>(mJy)</th></tr></thead><tbody><tr><td>G1<sup>d</sup></td><td>1</td><td>1</td><td>189.29950</td><td>62.37013</td><td>18.6 ± 2.3</td><td>0.97</td><td>31.7 ± 2.7</td><td>2.16</td><td>24.3 ± 2.7</td><td>1.10</td><td>10.5 ± 0.7</td><td>0.079 ± 0.014</td></tr><tr><td>G2<sup>d</sup></td><td>3</td><td>2</td><td>189.13918</td><td>62.23579</td><td>18.1 ± 2.3</td><td>1.77</td><td>15.2 ± 2.7</td><td>2.11</td><td>&lt;7.8</td><td>1.68</td><td>5.2 ± 0.6</td><td>0.036 ± 0.004</td></tr><tr><td>G3<sup>d</sup></td><td>5</td><td>4</td><td>189.37811</td><td>62.21633</td><td>46.3 ± 2.0</td><td>1.25</td><td>46.5 ± 2.3</td><td>1.75</td><td>32.7 ± 2.3</td><td>1.27</td><td>4.4 ± 0.6</td><td>0.123 ± 0.004</td></tr><tr><td>G4<sup>d</sup></td><td>7</td><td>3</td><td>189.29727</td><td>62.22540</td><td>38.6 ± 2.1</td><td>1.76</td><td>33.0 ± 2.4</td><td>2.62</td><td>23.2 ± 2.3</td><td>1.30</td><td>4.0 ± 0.5</td><td>0.131 ± 0.009</td></tr><tr><td>G5</td><td>2</td><td>13</td><td>189.13301</td><td>62.28749</td><td>&lt;6.9</td><td>1.58</td><td>&lt;8.7</td><td>1.68</td><td>&lt;7.8</td><td>0.88</td><td>4.1 ± 0.6</td><td>0.026 ± 0.005</td></tr><tr><td>G6<sup>d</sup></td><td>6</td><td>5</td><td>189.11346</td><td>62.10158</td><td>9.2 ± 2.4</td><td>1.23</td><td>13.5 ± 3.0</td><td>1.61</td><td>5.2 ± 2.9</td><td>1.14</td><td>4.5 ± 0.7</td><td>0.053 ± 0.012</td></tr><tr><td>G7</td><td>4</td><td>12</td><td>188.95979</td><td>62.17839</td><td>&lt;7.5</td><td>1.53</td><td>&lt;8.1</td><td>2.31</td><td>&lt;8.4</td><td>2.09</td><td>4.6 ± 0.7</td><td>0.028 ± 0.005</td></tr><tr><td>G8<sup>d</sup></td><td>26</td><td>6</td><td>189.30772</td><td>62.30739</td><td>8.2 ± 2.2</td><td>1.91</td><td>8.1 ± 2.8</td><td>3.76</td><td>&lt;7.8</td><td>2.94</td><td>3.1 ± 0.6</td><td>0.652 ± 0.004</td></tr><tr><td>G9<sup>d</sup></td><td>11</td><td>14</td><td>189.14937</td><td>62.11888</td><td>33.9 ± 2.1</td><td>1.45</td><td>20.3 ± 2.7</td><td>1.16</td><td>&lt;8.4</td><td>1.43</td><td>2.9 ± 0.6</td><td>0.046 ± 0.008</td></tr><tr><td>G11<sup>d</sup></td><td>13</td><td>15</td><td>188.97157</td><td>62.22708</td><td>21.0 ± 2.4</td><td>1.25</td><td>&lt;9.3</td><td>4.06</td><td>&lt;8.4</td><td>2.39</td><td>2.6 ± 0.6</td><td>0.045 ± 0.005</td></tr><tr><td>G13</td><td>12</td><td></td><td>189.13604</td><td>62.10586</td><td>&lt;6.9</td><td>1.44</td><td>&lt;8.7</td><td>1.35</td><td>&lt;8.7</td><td>1.25</td><td>2.7 ± 0.7</td><td>0.027 ± 0.005</td></tr><tr><td>G14</td><td>14</td><td>25</td><td>189.21696</td><td>62.20714</td><td>&lt;6.3</td><td>1.05</td><td>&lt;7.5</td><td>1.77</td><td>&lt;7.2</td><td>1.31</td><td>2.4 ± 0.6</td><td>0.016 ± 0.004<sup>e</sup></td></tr><tr><td>G16</td><td></td><td></td><td>189.20117</td><td>62.35201</td><td>&lt;9</td><td>1.00</td><td>&lt;9.3</td><td>1.98</td><td>&lt;8.1</td><td>1.24</td><td>2.5 ± 0.6</td><td>0.030 ± 0.005</td></tr><tr><td>G18<sup>d</sup></td><td>16</td><td></td><td>189.06708</td><td>62.25388</td><td>&lt;7.2</td><td>1.35</td><td>&lt;9.3</td><td>2.95</td><td>14.1 ± 2.6</td><td>1.46</td><td>2.1 ± 0.6</td><td>0.036 ± 0.005</td></tr><tr><td>G19<sup>d</sup></td><td></td><td>29</td><td>189.30020</td><td>62.20341</td><td>11.2 ± 2.1</td><td>1.50</td><td>11.3 ± 2.4</td><td>1.28</td><td>17.3 ± 2.4</td><td>1.09</td><td>2.1 ± 0.6</td><td>0.032 ± 0.004</td></tr><tr><td>G21<sup>d</sup></td><td>9</td><td></td><td>189.40941</td><td>62.29355</td><td>7.1 ± 2.0</td><td>1.10</td><td>12.1 ± 2.6</td><td>1.53</td><td>&lt;8.4</td><td>1.01</td><td>2.1 ± 0.6</td><td>0.026 ± 0.006</td></tr><tr><td>G24</td><td>24</td><td></td><td>189.03583</td><td>62.24325</td><td>&lt;7.5</td><td>1.34</td><td>&lt;9.6</td><td>3.08</td><td>&lt;8.1</td><td>2.36</td><td>2.0 ± 0.6</td><td>0.039 ± 0.004</td></tr><tr><td>G26<sup>d</sup></td><td>15</td><td></td><td>188.94971</td><td>62.25811</td><td>&lt;8.1</td><td>2.33</td><td>&lt;10.2</td><td>4.22</td><td>10.1 ± 2.9</td><td>1.93</td><td>2.1 ± 0.7</td><td>0.124 ± 0.006</td></tr><tr><td>G27<sup>d</sup></td><td>18</td><td></td><td>189.42151</td><td>62.20591</td><td>31.0 ± 2.3</td><td>1.19</td><td>27.9 ± 2.4</td><td>0.99</td><td>30.3 ± 2.5</td><td>0.88</td><td>1.9 ± 0.6</td><td>0.033 ± 0.005</td></tr><tr><td>G29<sup>d</sup></td><td>25</td><td></td><td>189.21550</td><td>62.08417</td><td>12.6 ± 2.9</td><td>1.78</td><td>15.1 ± 3.3</td><td>2.57</td><td>17.7 ± 3.0</td><td>1.43</td><td>2.6 ± 0.9</td><td>0.079 ± 0.006</td></tr><tr><td>G30<sup>d</sup></td><td>17</td><td></td><td>188.92167</td><td>62.24403</td><td>11.7 ± 2.6</td><td>1.59</td><td>&lt;10.2</td><td>3.80</td><td>&lt;9</td><td>2.27</td><td>2.2 ± 0.7</td><td>0.029 ± 0.005</td></tr><tr><td>G31<sup>d</sup></td><td></td><td></td><td>189.32254</td><td>62.13453</td><td>35.1 ± 2.5</td><td>3.20</td><td>32.1 ± 3.0</td><td>1.95</td><td>22.5 ± 2.7</td><td>0.96</td><td>2.0 ± 0.7</td><td>0.050 ± 0.008</td></tr><tr><td>G34</td><td></td><td></td><td>189.14537</td><td>62.32322</td><td>&lt;7.2</td><td>1.36</td><td>&lt;8.7</td><td>2.28</td><td>&lt;10.3</td><td>0.99</td><td>1.8 ± 0.6</td><td>0.085 ± 0.005</td></tr><tr><td>G41<sup>d</sup></td><td></td><td></td><td>189.55217</td><td>62.24859</td><td>25.3 ± 3.2</td><td>2.48</td><td>25.6 ± 3.5</td><td>9.72</td><td>15.3 ± 3.7</td><td>2.97</td><td>1.9 ± 0.7</td><td>0.056 ± 0.006</td></tr></tbody></table>

<sup>a</sup>Typically written with prefix AzGN.

<sup>b</sup>Typically written with prefix GN1200.

<sup>c</sup>Deboosted flux density as described in Penner et al. (2011).

<sup>d</sup>Detected in at least one SPIRE band.

<sup>e</sup>1.4 GHz flux density from Dunlop et al. (2004).
<table><caption>**Table 2.** Measured SPIRE photometry for mm sources in Lockman Hole North. $3\sigma$ upper limits are given for sources which are not detected above this significance in the SPIRE imaging. For the mm photometry, deboosted flux densities are quoted. The $\chi^2$ represents the reduced $\chi^2$ in a 15 × 15 pixel region centred on the source position. ID number refers to the equivalent value from Lindner et al. (2011). 1.4 GHz flux density is from the catalogue of Owen & Morrison (2008), as presented by Lindner et al. (2011).</caption><thead><tr><th>ID</th><th>RA<br>(°)</th><th>Dec.<br>(°)</th><th>S<sub>250</sub><br>(mJy)</th><th>χ<sup>2</sup><sub>250</sub></th><th>S<sub>350</sub><br>(mJy)</th><th>χ<sup>2</sup><sub>350</sub></th><th>S<sub>500</sub><br>(mJy)</th><th>χ<sup>2</sup><sub>500</sub></th><th>S<sub>1.2 mm (deboosted)</sub><br>(mJy)</th><th>S<sub>1.4 GHz</sub><br>(mJy)</th></tr></thead><tbody><tr><td>L1<sup>a</sup></td><td>161.75083</td><td>59.01878</td><td>80.1 ± 3.4</td><td>0.92</td><td>63.2 ± 3.8</td><td>0.85</td><td>40.6 ± 4.4</td><td>0.89</td><td>3.5<sup>+0.6</sup><sub>-0.6</sub></td><td>0.279 ± 0.015</td></tr><tr><td>L2<sup>a</sup></td><td>161.61192</td><td>59.09578</td><td>&lt;11.7</td><td>1.07</td><td>12.5 ± 4.0</td><td>0.99</td><td>&lt;13.8</td><td>1.08</td><td>3.8<sup>+0.7</sup><sub>-0.7</sub></td><td>0.036 ± 0.006</td></tr><tr><td>L3</td><td>161.63112</td><td>58.84889</td><td>&lt;10.8</td><td>0.84</td><td>&lt;11.1</td><td>0.92</td><td>&lt;13.2</td><td>0.87</td><td>3.8<sup>+0.8</sup><sub>-0.7</sub></td><td>0.459 ± 0.025</td></tr><tr><td>L4</td><td>161.53050</td><td>58.90389</td><td>&lt;10.8</td><td>0.83</td><td>&lt;11.4</td><td>1.03</td><td>&lt;14.1</td><td>0.79</td><td>2.7<sup>+0.5</sup><sub>-0.5</sub></td><td>0.030 ± 0.004</td></tr><tr><td>L5<sup>a</sup></td><td>161.85592</td><td>59.06014</td><td>32.8 ± 3.9</td><td>1.21</td><td>44.5 ± 4.1</td><td>1.08</td><td>35.6 ± 4.6</td><td>0.92</td><td>4.0<sup>+0.8</sup><sub>-0.9</sub></td><td>0.051 ± 0.006</td></tr><tr><td>L6<sup>a</sup></td><td>161.66112</td><td>58.93681</td><td>52.4 ± 3.6</td><td>0.91</td><td>51.5 ± 3.6</td><td>0.75</td><td>17.5 ± 4.4</td><td>2.17</td><td>2.3<sup>+0.4</sup><sub>-0.4</sub></td><td>0.160 ± 0.010</td></tr><tr><td>L7<sup>a</sup></td><td>161.75054</td><td>58.91150</td><td>21.0 ± 3.6</td><td>0.84</td><td>18.6 ± 3.5</td><td>0.97</td><td>31.7 ± 4.4</td><td>0.78</td><td>2.3<sup>+0.4</sup><sub>-0.5</sub></td><td>0.042 ± 0.005</td></tr><tr><td>L8<sup>a</sup></td><td>161.63779</td><td>58.86642</td><td>30.0 ± 3.6</td><td>0.99</td><td>29.8 ± 3.7</td><td>0.94</td><td>16.5 ± 4.4</td><td>1.25</td><td>2.7<sup>+0.6</sup><sub>-0.6</sub></td><td>0.098 ± 0.010</td></tr><tr><td>L9</td><td>161.77042</td><td>58.83556</td><td>&lt;10.8</td><td>0.96</td><td>&lt;11.1</td><td>0.91</td><td>&lt;13.8</td><td>1.21</td><td>3.8<sup>+1.0</sup><sub>-0.9</sub></td><td>0.023 ± 0.005</td></tr><tr><td>L10<sup>a</sup></td><td>161.59604</td><td>58.99300</td><td>13.3 ± 3.5</td><td>0.72</td><td>&lt;14.1</td><td>0.94</td><td>19.8 ± 4.4</td><td>0.91</td><td>2.4<sup>+0.5</sup><sub>-0.5</sub></td><td>0.078 ± 0.007</td></tr><tr><td>L11<sup>a</sup></td><td>161.48721</td><td>58.88856</td><td>58.2 ± 3.7</td><td>1.02</td><td>56.6 ± 3.8</td><td>1.26</td><td>18.8 ± 4.6</td><td>1.44</td><td>2.7<sup>+0.6</sup><sub>-0.6</sub></td><td>0.315 ± 0.019</td></tr><tr><td>L12<sup>a</sup></td><td>161.19829</td><td>59.00997</td><td>25.3 ± 3.7</td><td>0.84</td><td>36.0 ± 3.9</td><td>1.11</td><td>24.4 ± 4.7</td><td>0.78</td><td>2.7<sup>+0.6</sup><sub>-0.7</sub></td><td>0.273 ± 0.019</td></tr><tr><td>L13<sup>a</sup></td><td>161.53646</td><td>58.97458</td><td>12.1 ± 3.6</td><td>0.98</td><td>17.4 ± 3.6</td><td>0.80</td><td>&lt;14.1</td><td>0.82</td><td>2.1<sup>+0.5</sup><sub>-0.5</sub></td><td>0.198 ± 0.010</td></tr><tr><td>L14<sup>a</sup></td><td>161.64937</td><td>59.13014</td><td>39.0 ± 3.9</td><td>0.99</td><td>46.2 ± 4.0</td><td>1.07</td><td>33.9 ± 4.6</td><td>1.65</td><td>3.0<sup>+0.9</sup><sub>-0.9</sub></td><td>0.097 ± 0.006</td></tr><tr><td>L15<sup>a</sup></td><td>161.86654</td><td>58.87058</td><td>55.8 ± 3.9</td><td>1.01</td><td>69.1 ± 3.6</td><td>0.95</td><td>49.7 ± 4.8</td><td>0.80</td><td>3.0<sup>+0.9</sup><sub>-1.0</sub></td><td>0.180 ± 0.018</td></tr><tr><td>L16<sup>a</sup></td><td>161.83633</td><td>58.86472</td><td>&lt;11.7</td><td>1.04</td><td>11.2 ± 3.6</td><td>0.90</td><td>&lt;14.1</td><td>0.95</td><td>2.7<sup>+0.8</sup><sub>-0.8</sub></td><td>0.048 ± 0.007</td></tr><tr><td>L17<sup>a</sup></td><td>161.54250</td><td>59.04508</td><td>10.4 ± 3.7</td><td>1.15</td><td>14.6 ± 3.9</td><td>1.26</td><td>&lt;13.5</td><td>0.84</td><td>2.0<sup>+0.5</sup><sub>-0.5</sub></td><td>0.027 ± 0.003</td></tr><tr><td>L18<sup>a</sup></td><td>161.73021</td><td>58.83444</td><td>39.9 ± 3.6</td><td>0.81</td><td>37.8 ± 3.7</td><td>0.91</td><td>25.4 ± 4.6</td><td>0.95</td><td>2.9<sup>+1.1</sup><sub>-1.1</sub></td><td>0.105 ± 0.008</td></tr><tr><td>L19</td><td>161.25833</td><td>59.06758</td><td>&lt;11.4</td><td>0.91</td><td>&lt;12.3</td><td>1.23</td><td>&lt;14.7</td><td>1.45</td><td>2.0<sup>+0.6</sup><sub>-0.6</sub></td><td>0.029 ± 0.004</td></tr><tr><td>L21<sup>a</sup></td><td>161.37575</td><td>59.11008</td><td>&lt;11.7</td><td>1.02</td><td>12.0 ± 4</td><td>1.11</td><td>16.0 ± 4.9</td><td>1.03</td><td>1.8<sup>+0.5</sup><sub>-0.5</sub></td><td>0.036 ± 0.004</td></tr><tr><td>L22<sup>a</sup></td><td>161.51763</td><td>59.08039</td><td>32.7 ± 3.6</td><td>1.06</td><td>23.7 ± 3.9</td><td>1.01</td><td>&lt;14.1</td><td>1.00</td><td>2.0<sup>+0.6</sup><sub>-0.6</sub></td><td>0.166 ± 0.010</td></tr><tr><td>L23</td><td>161.67129</td><td>58.89050</td><td>&lt;11.1</td><td>0.97</td><td>&lt;10.8</td><td>0.92</td><td>&lt;13.2</td><td>0.72</td><td>1.7<sup>+0.5</sup><sub>-0.5</sub></td><td>0.032 ± 0.005</td></tr><tr><td>L24<sup>a</sup></td><td>161.25279</td><td>59.12600</td><td>11.0 ± 3.6</td><td>0.92</td><td>&lt;12.0</td><td>0.76</td><td>17.6 ± 4.6</td><td>0.79</td><td>1.8<sup>+0.6</sup><sub>-0.5</sub></td><td>0.068 ± 0.009</td></tr><tr><td>L25</td><td>161.41679</td><td>59.06333</td><td>&lt;12.0</td><td>1.25</td><td>&lt;12.6</td><td>1.66</td><td>&lt;15.0</td><td>6.59</td><td>1.7<sup>+0.6</sup><sub>-0.6</sub></td><td>0.024 ± 0.007</td></tr><tr><td>L27<sup>a</sup></td><td>161.76000</td><td>58.85086</td><td>17.4 ± 3.5</td><td>0.91</td><td>19.5 ± 3.6</td><td>0.71</td><td>&lt;13.8</td><td>1.16</td><td>2.0<sup>+0.8</sup><sub>-0.9</sub></td><td>0.078 ± 0.014</td></tr><tr><td>L28</td><td>161.58708</td><td>58.90944</td><td>&lt;10.5</td><td>0.74</td><td>&lt;11.1</td><td>1.25</td><td>&lt;13.5</td><td>1.04</td><td>1.5<sup>+0.5</sup><sub>-0.5</sub></td><td>0.025 ± 0.006</td></tr><tr><td>L29<sup>a</sup></td><td>161.48129</td><td>59.15453</td><td>140.5 ± 3.9</td><td>2.47</td><td>64.4 ± 3.9</td><td>0.83</td><td>29.1 ± 4.8</td><td>1.46</td><td>1.8<sup>+0.8</sup><sub>-0.8</sub></td><td>0.307 ± 0.043</td></tr><tr><td>L30<sup>a</sup></td><td>161.29329</td><td>59.06864</td><td>&lt;11.7</td><td>1.02</td><td>13.3 ± 4.1</td><td>1.28</td><td>&lt;15.0</td><td>1.59</td><td>1.5<sup>+0.6</sup><sub>-0.6</sub></td><td>0.074 ± 0.007</td></tr><tr><td>L31<sup>a</sup></td><td>161.60379</td><td>58.89644</td><td>&lt;10.5</td><td>1.04</td><td>20.8 ± 3.7</td><td>1.08</td><td>13.1 ± 4.3</td><td>0.89</td><td>1.5<sup>+0.6</sup><sub>-0.5</sub></td><td>0.043 ± 0.004</td></tr><tr><td>L32<sup>a</sup></td><td>161.41579</td><td>58.90692</td><td>18.6 ± 3.8</td><td>1.07</td><td>21.7 ± 3.9</td><td>0.98</td><td>17.6 ± 5.2</td><td>1.08</td><td>1.7<sup>+0.8</sup><sub>-0.7</sub></td><td>0.047 ± 0.004</td></tr><tr><td>L33<sup>a</sup></td><td>161.39608</td><td>58.84714</td><td>28.3 ± 3.7</td><td>1.19</td><td>32.5 ± 3.8</td><td>0.96</td><td>26.5 ± 5.1</td><td>1.13</td><td>2.7<sup>+1.2</sup><sub>-1.2</sub></td><td>0.071 ± 0.010</td></tr><tr><td>L34<sup>a</sup></td><td>161.22346</td><td>58.97819</td><td>&lt;10.8</td><td>0.94</td><td>13.3 ± 3.8</td><td>0.91</td><td>&lt;14.1</td><td>0.74</td><td>1.5<sup>+0.7</sup><sub>-0.7</sub></td><td>0.041 ± 0.008</td></tr><tr><td>L35<sup>a</sup></td><td>161.82546</td><td>58.92383</td><td>19.7 ± 3.7</td><td>0.95</td><td>24.4 ± 3.7</td><td>0.97</td><td>18.2 ± 4.5</td><td>1.14</td><td>1.5<sup>+0.7</sup><sub>-0.7</sub></td><td>0.060 ± 0.006</td></tr><tr><td>L36</td><td>161.53375</td><td>59.12889</td><td>&lt;11.7</td><td>1.40</td><td>&lt;11.7</td><td>0.98</td><td>&lt;14.1</td><td>1.13</td><td>1.7<sup>+1.1</sup><sub>-0.9</sub></td><td>0.016 ± 0.003</td></tr><tr><td>L37<sup>a</sup></td><td>161.54575</td><td>58.87914</td><td>52.0 ± 3.6</td><td>1.01</td><td>39.9 ± 3.8</td><td>1.09</td><td>16.8 ± 4.5</td><td>0.95</td><td>1.5<sup>+0.8</sup><sub>-0.7</sub></td><td>0.160 ± 0.011</td></tr><tr><td>L38</td><td>161.18704</td><td>59.13836</td><td>&lt;12.0</td><td>0.86</td><td>&lt;11.7</td><td>0.81</td><td>&lt;14.4</td><td>4.57</td><td>1.5<sup>+0.9</sup><sub>-0.9</sub></td><td>0.068 ± 0.016</td></tr><tr><td>L39<sup>a</sup></td><td>161.55012</td><td>59.04261</td><td>&lt;14.1</td><td>1.12</td><td>&lt;11.4</td><td>1.22</td><td>23.3 ± 4.5</td><td>0.81</td><td>1.4<sup>+0.7</sup><sub>-0.6</sub></td><td>0.034 ± 0.003</td></tr><tr><td>L40<sup>a</sup></td><td>161.74304</td><td>59.10931</td><td>20.4 ± 3.7</td><td>1.09</td><td>12.2 ± 4.0</td><td>0.89</td><td>&lt;14.1</td><td>1.05</td><td>1.5<sup>+1.2</sup><sub>-1.1</sub></td><td>0.057 ± 0.005</td></tr><tr><td>L41<sup>a</sup></td><td>161.50129</td><td>58.91794</td><td>12.1 ± 3.7</td><td>0.86</td><td>11.6 ± 3.8</td><td>0.87</td><td>&lt;14.1</td><td>1.19</td><td>1.4<sup>+0.7</sup><sub>-0.6</sub></td><td>0.072 ± 0.012</td></tr></tbody></table>

preparation) or from the many magnitude-limited redshift surveys of the field which have been conducted (see Barger, Cowie & Wang 2008, for a recent compilation of spectroscopic redshifts in GOODS-N). In the Lockman Hole North, spectroscopic redshifts are available for four sources, mostly from mid-IR spectroscopy of 24 µm selected sources (Fiolet et al. 2010) or WIYN spectroscopy of radio sources (Owen & Morrison 2009). Photometric redshifts are available for an additional nine mm sources in Lockman Hole

North from the catalogue of Strazzullo et al. (2010). Here we only consider photometric redshifts which have a quality flag of ‘AA’, i.e. the estimated photo-z error is less than 0.2(1 + z), from Strazzullo et al. (2010). In total, good quality redshifts are available for 22 of our parent sample of 63 mm sources. These redshifts are quoted in Tables 1 and 2. All of the sources with known spectroscopic and/or reliable optical/near-IR photometric redshifts are detected by SPIRE. This is not particularly surprising, as those sources with

<footnote><sup>a</sup>Detected in at least one SPIRE band.</footnote>
redshift information from the optical/near-IR wavelength range tend to be more intrinsically luminous and/or at slightly lower redshift (Wardlow et al. 2011).

To estimate the reliability of our detections, we test random positions in the SPIRE maps. Test positions are required to be >3 arcsec from other sources in our input list, and within the region covered by our 24-µm and 1.4-GHz catalogue. This is the typical minimum separation of sources in our input list, due to the resolutions at 24 µm and 1.4 GHz. If we impose the same selection criteria on the fluxes recovered for these random positions, 2.7 per cent are detected in any SPIRE band at a significance of >3 σ.

Taking this as a guide, if our parent sample of 63 mm sources were completely spurious, then we would expect to recover SPIRE detections in any band for two sources.

## 4.2 Submm-mm colours

For the use of submm/mm photometry, either alone or in combination with mid-IR or radio data, to be useful as a redshift estimator, there must be a reasonably strong, and unique, matching between submm/mm colour evolution and redshift.

In Fig. 1, we compare the SPIRE-mm colours to the expected evolution for a modified blackbody parametrized by a single dust temperature, $T_D$, and optical depth $\tau = (v/v_0)^\beta$, i.e.

$$S_v \propto \left\{ 1 - \exp \left[ - \left( \frac{v}{v_0} \right)^\beta \right] \right\} B_v(T_D),$$

where $B_v$ is the Planck function for a single dust temperature $T_D$. In order to give a better representation to the mid-IR SED, we replace the blackbody SED on the Wien side of the SED with a power law, $S_v \propto v^\alpha$, where $\alpha$ is fixed to -2 (Blain et al. 2003; Conley et al. 2011). To ensure a smooth SED, the power law is used at frequencies where dlog S/dlog v ≤ $\alpha$. As we have a limited number of photometric bands to constrain this model, we fix the dust opacity and optical depth to values of $\beta = +1.8$ and $v_0 = 100$ µm, respectively. We adopt this definition of a modified blackbody throughout this paper.

The sources with known spectroscopic or reliable optical/near-IR photometric redshifts from Tables 3 and 4 are coloured coded by redshift.

While the errors on the SPIRE and mm photometry are large, the modified blackbody model can sufficiently explain the observed submm-mm colours of the majority of SPIRE-mm sources. This is qualitatively in agreement with recent results from Kovács et al. (2010), although they warn that more sophisticated models (in particular, one which takes account of the range of dust temperatures likely to occur within a star-forming galaxy) are needed to match shorter wavelength data (i.e. less than rest frame 30 µm). A modified blackbody redshift and temperature are completely degenerate at $z < 4$; thus, changing the assumed dust temperature will change the predicted submm-mm colour as a function of redshift, but it will not change the position of the model track in colour-colour space. Thus, if the dust temperature were known, the redshift could be inferred directly, or vice versa.

It can be seen that the highest redshift objects are found to have the ‘reddest’ submm/mm colours, i.e. large $S_{mm}/S_{SPIRE}$ or $S_{500}/S_{350}$ flux ratios, supporting claims that identifying red, or ‘cold’, SPIRE/mm sources is an effective way of isolating very high redshift submm galaxies (e.g. Pope & Chary 2010).

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Submm-mm colour evolution of mm-selected sources: $S_{500}/S_{350}$ versus $S_{350}/S_{250}$ (top), $S_{1.2mm}/S_{350}$ versus $S_{350}/S_{250}$ (middle) and $S_{1.2mm}/S_{500}$ versus $S_{500}/S_{350}$ (bottom). Only sources with >3σ detections in each of the relevant SPIRE bands are shown in each panel. The track taken by a modified blackbody, with $T_D = 40$ K and optical depth $\tau = (v/v_0)^\beta$, where $\beta = +1.8$ and $v_0 = c/100$ µm, is also shown. Sources located in the GOODS-N field are shown in squares, those in Lockman Hole North are shown as triangles. Symbols are colour coded by redshift, with filled symbols designating those sources with spectroscopic redshifts and open symbols those with reliable [$\delta z < 0.2(1+z)$] optical/near-IR photometric redshifts. Black open symbols represent sources with no redshift estimate. The submm-mm colours are reasonably well described by our simple modified blackbody model.</figcaption></figure>

## 4.3 IR luminosity to dust temperature relation for SPIRE-mm sources

As a significant fraction of SPIRE-mm sources have good redshift information (22 of 46 SPIRE-detected mm sources), we can investigate the IR luminosity to dust temperature relation directly for our sample of SPIRE-mm sources.

For each source with a spectroscopic or optical/near-IR photometric redshift (excluding the very low z source L29), we fit a modified blackbody to the SPIRE and MAMBO photometry (and AzTEC and SCUBA where available), assuming a dust emissivity of $\beta = +1.8$, the redshift and the functional form for a modified blackbody given in Section 4.2. In the Lockman Hole North field,
<table><caption>**Table 3.** Radio–far-IR photometric redshifts, far-IR luminosities and dust temperatures for mm-detected sources in the GOODS-N field. Spectroscopic redshifts from the literature are quoted where available. Far-IR luminosities (8–1000 µm) and dust temperatures are presented for sources with good redshift information.</caption><caption>GOODS-N</caption><thead><tr><th>ID</th><th>Redshift</th><th>Radio/far-IR photo-z</th><th>L<sub>IR</sub>(L<sub>☉</sub>)</th><th>T<sub>D</sub>(K)</th></tr></thead><tbody><tr><td>G1<sup>a</sup></td><td>4.05<sup>b</sup></td><td>5.00<sup>+0.00</sup> c<sub>-0.4</sub></td><td>13.2 ± 0.1</td><td>39 ± 1</td></tr><tr><td>G2<sup>a</sup></td><td>4.05<sup>d</sup></td><td>4.20<sup>+0.08</sup> c<sub>-0.21</sub></td><td>13.1 ± 0.2</td><td>43 ± 2</td></tr><tr><td>G3<sup>a</sup></td><td>2.98<sup>e</sup></td><td>2.63<sup>+0.04</sup> c<sub>-0.06</sub></td><td>13.2 ± 0.1</td><td>50 ± 2</td></tr><tr><td>G4<sup>a</sup></td><td>1.996<sup>f</sup></td><td>2.13<sup>+0.00</sup> c<sub>-0.17</sub></td><td>12.7 ± 0.2</td><td>36 ± 2</td></tr><tr><td>G5</td><td></td><td>4.5<sup>g</sup></td><td></td><td></td></tr><tr><td>G6<sup>a</sup></td><td></td><td>4.07<sup>+0.32</sup> c<sub>-0.36</sub></td><td></td><td></td></tr><tr><td>G7</td><td></td><td>4.6<sup>g</sup></td><td></td><td></td></tr><tr><td>G8<sup>a</sup></td><td></td><td>3.7<sup>+0.4</sup> h<sub>-0.2</sub></td><td></td><td></td></tr><tr><td>G9</td><td>0.95<sup>i</sup></td><td>0.91<sup>+0.13</sup> c<sub>-0.15</sub></td><td></td><td></td></tr><tr><td>G11<sup>a</sup></td><td>2.098<sup>j</sup></td><td>0.92<sup>+0.22</sup> c<sub>-0.41</sub></td><td>12.8 ± 1.3</td><td>73 ± 36</td></tr><tr><td>G13</td><td></td><td>3.95<sup>g</sup></td><td></td><td></td></tr><tr><td>G14</td><td></td><td>4.1 ± 0.5<sup>j</sup></td><td></td><td></td></tr><tr><td>G16</td><td></td><td>3.7<sup>g</sup></td><td></td><td></td></tr><tr><td>G18<sup>a</sup></td><td>2.58<sup>f</sup></td><td>2.56<sup>+0.18</sup> c<sub>-0.27</sub></td><td>12.5 ± 0.6</td><td>35 ± 8</td></tr><tr><td>G19<sup>a</sup></td><td>2.91<sup>f</sup></td><td>2.91<sup>+0.24</sup> c<sub>-0.23</sub></td><td>12.6 ± 0.3</td><td>36 ± 3</td></tr><tr><td>G21<sup>a</sup></td><td>3.19<sup>f</sup></td><td>2.82<sup>+0.34</sup> c<sub>-0.27</sub></td><td>12.6 ± 0.4</td><td>38 ± 4</td></tr><tr><td>G24</td><td></td><td>3.1<sup>g</sup></td><td></td><td></td></tr><tr><td>G26<sup>a</sup></td><td></td><td>0.24<sup>+0.05</sup> c<sub>-0.09</sub></td><td></td><td></td></tr><tr><td>G27<sup>a</sup></td><td></td><td>2.43<sup>+0.04</sup> c<sub>-0.18</sub></td><td></td><td></td></tr><tr><td>G29<sup>a</sup></td><td></td><td>2.81<sup>+0.26</sup> c<sub>-0.31</sub></td><td></td><td></td></tr><tr><td>G30<sup>a</sup></td><td></td><td>2.41<sup>+0.42</sup> c<sub>-0.26</sub></td><td></td><td></td></tr><tr><td>G31<sup>a</sup></td><td></td><td>0.25<sup>+0.03</sup> c<sub>-0.11</sub></td><td></td><td></td></tr><tr><td>G34</td><td></td><td>2.2<sup>g</sup></td><td></td><td></td></tr><tr><td>G41<sup>a</sup></td><td></td><td>1.87<sup>+0.17</sup> c<sub>-0.21</sub></td><td></td><td></td></tr></tbody></table>

<sup>a</sup>Detected in at least one SPIRE band.

<sup>b</sup>Spectroscopic redshift from Daddi et al. (2009a).

<sup>c</sup>Redshift estimate from simultaneous fitting of L<sub>IR</sub>–T<sub>D</sub> and L<sub>IR</sub>–L<sub>1.4 GHz</sub> relations.

<sup>d</sup>Spectroscopic redshift from Daddi et al. (2009b).

<sup>e</sup>Spectroscopic redshift from Chapman et al. (in preparation).

<sup>f</sup>Spectroscopic redshift from Chapman et al. (2005).

<sup>g</sup>Redshift estimate from mm–1.4 GHz spectral slope.

<sup>h</sup>Redshift estimate from fitting of L<sub>IR</sub>–T<sub>D</sub> as radio flux boosted by AGN activity.

<sup>i</sup>Spectroscopic redshift from Barger et al. (2008).

<sup>j</sup>Radio–far-IR photo-z estimate from Dunlop et al. (2004).

PACS 100 and/or 160 µm photometry is available from the HerMES survey for six sources and is included in the SED fits. In GOODS-N, SCUBA 850 µm photometry is available for five sources from the catalogue of Pope et al. (2005) and is included in the SED fits. IR luminosities are calculated for each source by integrating the modified blackbody fit in the range 8–1000 µm. IR luminosities and dust temperatures for sources with good redshift information in GOODS-N and Lockman Hole North are presented in Tables 3 and 4, respectively.

Fig. 2 shows the best-fitting modified blackbody IR luminosity versus dust temperature for our sample of SPIRE-mm sources. One source, G11, is omitted as our modified blackbody model is not well constrained by the observed data (see Table 3). Also shown is the local relationship for IRAS 1.2 Jy sources from Chapman

<table><caption>**Table 4.** Radio–far-IR photometric redshifts, far-IR luminosities and dust temperatures for mm-detected sources in the Lockman Hole North field. Spectroscopic or reliable [$\delta z < 0.2(1+z)$] optical/near-IR photometric redshifts from the literature are quoted where available. Far-IR luminosities (8–1000 µm) and dust temperatures are presented for sources with good redshift information.</caption><caption>Lockman Hole North</caption><thead><tr><th>ID</th><th>Redshift</th><th>Radio/far-IR photo-z</th><th>L<sub>IR</sub>(L<sub>☉</sub>)</th><th>T<sub>D</sub>(K)</th></tr></thead><tbody><tr><td>L1<sup>a</sup></td><td>2.562<sup>b</sup></td><td>2.06<sup>+0.09</sup> c<sub>-0.18</sub></td><td>13.3 ± 0.2</td><td>54 ± 3</td></tr><tr><td>L2<sup>a</sup></td><td></td><td>4.09<sup>+0.27</sup> c<sub>-0.28</sub></td><td></td><td></td></tr><tr><td>L3</td><td></td><td>1.8<sup>d</sup></td><td></td><td></td></tr><tr><td>L4</td><td></td><td>4.4<sup>d</sup></td><td></td><td></td></tr><tr><td>L5<sup>a</sup></td><td>3 ± 0.42<sup>e</sup></td><td>3.30<sup>+0.07</sup> c<sub>-0.29</sub></td><td>13.1 ± 0.2</td><td>43 ± 2</td></tr><tr><td>L6<sup>a</sup></td><td>2.03<sup>f</sup></td><td>1.81<sup>+0.06</sup> c<sub>-0.17</sub></td><td>12.9 ± 0.2</td><td>43 ± 2</td></tr><tr><td>L7<sup>a</sup></td><td></td><td>2.74<sup>+0.11</sup> c<sub>-0.20</sub></td><td></td><td></td></tr><tr><td>L8<sup>a</sup></td><td></td><td>2.07<sup>+0.20</sup> c<sub>-0.23</sub></td><td></td><td></td></tr><tr><td>L9</td><td></td><td>3.9<sup>d</sup></td><td></td><td></td></tr><tr><td>L10<sup>a</sup></td><td></td><td>3.01<sup>+0.28</sup> c<sub>-0.21</sub></td><td></td><td></td></tr><tr><td>L11<sup>a</sup></td><td>1.95<sup>f</sup></td><td>1.71<sup>+0.18</sup> c<sub>-0.13</sub></td><td>12.9 ± 0.2</td><td>41 ± 3</td></tr><tr><td>L12<sup>a</sup></td><td>2.2 ± 0.2<sup>e</sup></td><td>2.64<sup>+0.08</sup> c<sub>-0.23</sub></td><td>12.7 ± 0.2</td><td>33 ± 2</td></tr><tr><td>L13<sup>a</sup></td><td></td><td>0.32<sup>+0.18</sup> c<sub>-0.21</sub></td><td></td><td></td></tr><tr><td>L14<sup>a</sup></td><td>2.3 ± 0.5<sup>e</sup></td><td>2.81<sup>+0.16</sup> c<sub>-0.18</sub></td><td>12.9 ± 0.2</td><td>37 ± 2</td></tr><tr><td>L15<sup>a</sup></td><td>2.8 ± 0.2<sup>e</sup></td><td>3.16<sup>+0.09</sup> c<sub>-0.16</sub></td><td>13.3 ± 0.2</td><td>45 ± 2</td></tr><tr><td>L16<sup>a</sup></td><td></td><td>3.24<sup>+0.51</sup> c<sub>-0.33</sub></td><td></td><td></td></tr><tr><td>L17<sup>a</sup></td><td></td><td>2.65<sup>+0.29</sup> c<sub>-0.19</sub></td><td></td><td></td></tr><tr><td>L18<sup>a</sup></td><td></td><td>2.15<sup>+0.11</sup> c<sub>-0.24</sub></td><td></td><td></td></tr><tr><td>L19</td><td></td><td>4.1<sup>d</sup></td><td></td><td></td></tr><tr><td>L21<sup>a</sup></td><td></td><td>2.52<sup>+0.31</sup> c<sub>-0.20</sub></td><td></td><td></td></tr><tr><td>L22<sup>a</sup></td><td>1.4 ± 0.4<sup>e</sup></td><td>0.38<sup>+0.14</sup> c<sub>-0.28</sub></td><td>12.3 ± 0.5</td><td>32 ± 4</td></tr><tr><td>L23</td><td></td><td>3.6<sup>d</sup></td><td></td><td></td></tr><tr><td>L24<sup>a</sup></td><td>3.2 ± 0.4<sup>e</sup></td><td>2.52<sup>+0.43</sup> c<sub>-0.20</sub></td><td>12.7 ± 0.5</td><td>40 ± 5</td></tr><tr><td>L25</td><td></td><td>3.5<sup>d</sup></td><td></td><td></td></tr><tr><td>L27<sup>a</sup></td><td></td><td>1.62<sup>+0.45</sup> c<sub>-0.27</sub></td><td></td><td></td></tr><tr><td>L28</td><td></td><td>3.8<sup>d</sup></td><td></td><td></td></tr><tr><td>L29<sup>a</sup></td><td>0.004<sup>g</sup></td><td>0.38<sup>+0.28</sup> c<sub>-0.33</sub></td><td>8.0 ± 0.3</td><td>22 ± 2</td></tr><tr><td>L30<sup>a</sup></td><td></td><td>0.71<sup>+0.20</sup> c<sub>-0.41</sub></td><td></td><td></td></tr><tr><td>L31<sup>a</sup></td><td>2.9 ± 0.3<sup>e</sup></td><td>2.44<sup>+0.21</sup> c<sub>-0.24</sub></td><td>12.7 ± 0.4</td><td>40 ± 4</td></tr><tr><td>L32<sup>a</sup></td><td>1.3 ± 0.2<sup>e</sup></td><td>2.21<sup>+0.25</sup> c<sub>-0.27</sub></td><td>12.0 ± 0.4</td><td>24 ± 2</td></tr><tr><td>L33<sup>a</sup></td><td></td><td>2.63<sup>+0.16</sup> c<sub>-0.34</sub></td><td></td><td></td></tr><tr><td>L34<sup>a</sup></td><td></td><td>2.46<sup>+0.50</sup> c<sub>-0.32</sub></td><td></td><td></td></tr><tr><td>L35<sup>a</sup></td><td></td><td>2.14<sup>+0.18</sup> c<sub>-0.24</sub></td><td></td><td></td></tr><tr><td>L36</td><td></td><td>4.5<sup>d</sup></td><td></td><td></td></tr><tr><td>L37<sup>a</sup></td><td>1.7 ± 0.2<sup>e</sup></td><td>1.17<sup>+0.18</sup> c<sub>-0.14</sub></td><td></td><td></td></tr><tr><td>L38</td><td></td><td>3.6<sup>d</sup></td><td></td><td></td></tr><tr><td>L39<sup>a</sup></td><td></td><td>2.59<sup>+0.31</sup> c<sub>-0.21</sub></td><td></td><td></td></tr><tr><td>L40<sup>a</sup></td><td></td><td>0.78<sup>+0.13</sup> c<sub>-0.46</sub></td><td></td><td></td></tr><tr><td>L41<sup>a</sup></td><td></td><td>1.49<sup>+1.01</sup> c<sub>-0.87</sub></td><td></td><td></td></tr></tbody></table>

<sup>a</sup>Detected in at least one SPIRE band.

<sup>b</sup>Spectroscopic redshift from Polletta et al. (2006).

<sup>c</sup>Redshift estimate from simultaneous fitting of L<sub>IR</sub>–T<sub>D</sub> and L<sub>IR</sub>–L<sub>1.4 GHz</sub> relations.

<sup>d</sup>Redshift estimate from mm–radio spectral slope as presented by Lindner et al. (2011).

<sup>e</sup>Photometric redshift from Strazzullo et al. (2010).

<sup>f</sup>Spectroscopic redshift from Fiolet et al. (2010).

<sup>g</sup>Spectroscopic redshift from Owen & Morrison (2009).
<figure><img src="image_3.png" /><figcaption>**Figure 2.** IR luminosity versus dust temperature for those SPIRE-mm sources with good redshift information. Source G11 is omitted as a modified blackbody model is not well constrained by the observed data. Symbols are colour coded by redshift, with GOODS-N sources denoted by squares and Lockman Hole North sources denoted by triangles. The solid line is the local $L_{IR}-T_D$ relation from the 1.2Jy IRAS sample as determined by Chapman et al. (2003), where we have converted from $S_{100}/S_{60}$ to $T_D$ assuming a modified blackbody with optical depth $\tau = (\nu/v_0)^\beta$, with $\beta = +1.8$ and $v_0 = c/100$ µm. The dashed line shows a modification to this relation given by fitting equation (1) directly to our SPIRE-mm sources. The best-fitting parameters are $\gamma = 0.193$, $\delta = -0.03$ and $C_* = 0.3$. $L_*$ is held fixed at $5 \times 10^{10}$.</figcaption></figure>

et al. (2003), after converting from IRAS $S_{100}/S_{60}$ colour to $T_D$ via our definition of a modified blackbody. It is clear that the SPIRE-mm sources form a tight sequence in $L_{IR}-T_D$ space. However, we can see that our sample is systematically biased towards colder dust temperatures when compared to the local $L_{IR}-T_D$ relationship (Chapman et al. 2003). This bias towards colder dust temperatures for submm/mm-selected samples is now well established (Chapman et al. 2010; Magdis et al. 2010; Magnelli et al. 2010).

The bias towards colder dust temperatures causes the $L_{IR}-T_D$ relation to be much tighter than for an unbiased sample. To parametrize the observed $L_{IR}-T_D$ relation for mm-selected sources, we fit the functional form given by Chapman et al. (2003) to our mm-selected sample. Specifically, Chapman et al. (2003) describe the mean IRAS colour via

$$ \log C_0 = \log C_* + \delta \log \left( 1 + \frac{L_{\text{star}}}{L_{\text{TIR}}} \right) + \gamma \log \left( 1 + \frac{L_{\text{TIR}}}{L_{\text{star}}} \right), \quad (1) $$

where $C_* = 0.45$, $\delta = -0.02$, $L_{\text{star}} = 5 \times 10^{10} L_{\odot}$ and $\gamma = 0.16$, while $L_{\text{TIR}}$ is the integrated IR luminosity from 3–1100 µm and $C_0$ is the ratio of the 60-to-100 µm IRAS flux density. We fit equation (1) to our observed $L_{IR}-T_D$ relation of our SPIRE-mm sources, adopting a conversion between IRAS 60-to-100 µm flux ratio and $T_D$ given by a modified blackbody with $\beta = +1.8$. We leave $L_*$ fixed at the Chapman et al. (2003) value as we do not have enough low-luminosity sources to constrain this parameter. Our best fit is given by $C_* = -0.3$, $\delta = -0.03$ and $\gamma = 0.193$. The track taken by our best-fitting $L_{IR}-T_D$ relation is shown in Fig. 2.

# 5 DISCUSSION

## 5.1 Submm photometric redshifts

As shown in Section 4.2, there is a reasonable correspondence between submm/mm colour and redshift, which is likely a by-product of our SPIRE-mm sources demonstrating a strong relationship between dust temperature and luminosity, as seen in Fig. 2. Thus, if we can assume a relationship between dust temperature and IR luminosity, we can break the temperature-redshift degeneracy and produce photometric redshifts which rely on submm/mm data alone.

We explore the potential of such an approach here by making use of the 22 SPIRE-mm sources with good redshift information as a test set. We assume that the $L_{IR}-T_D$ relationship for SPIRE-mm sources is given by our best fit of equation (1) as seen in Fig. 2. Additional photometry from PACS and SCUBA is not considered in the photometric redshift estimation as these are only available for a small subset of the sample; we want a fair assessment of the ability of SPIRE and mm-wavelength data alone to estimate the redshift.

Fig. 4 compares our photometric redshift estimates to the known redshifts for these sources. The mean photometric redshift accuracy is found to be $|\Delta z|(1+z) = 0.16$ with the mean offset ($|\Delta z| = 0.51$). All of our SPIRE-mm sources are detected in the radio at 1.4 GHz, so we can hope to improve on our photometric redshift estimates by also requiring that the sources lie on the radio-to-far-IR correlation (van der Kruit 1971). As our SPIRE photometry is close to the peak of the far-IR emission at $z \sim 1-3$, it is not sufficient to assume a spectral index between the SPIRE/mm bands and 1.4 GHz as is typically done for submm-radio photo-z (e.g. Carilli & Yun 1999). Here we assume the ratio of $L_{IR}$ to $L_{1.4 GHz}$ is given by a constant value $q_{IR} = \log_{10}[(L_{IR}/3.75 \times 10^{12} W)/(L_{1.4 GHz}/W Hz^{-1})] = 2.4$ (Ivison et al. 2010). We also assume that the radio SED is adequately described by a power law, $S_v \propto v^\alpha$, with $\alpha = -0.75$ (Ibar et al. 2010). Fig. 3 shows $q$ as a function of redshift for the 22 SPIRE-mm sources with good redshifts. Good agreement is seen between our sample and the median (and standard deviation) of $q$ for SPIRE sources as measured by Ivison et al. (2010).

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Radio-to-far-IR correlation as a function of redshift as probed by the $q_{IR}$ parameter, where $q_{IR} = \log_{10}[(L_{IR}/3.75 \times 10^{12} W)/(L_{1.4 GHz}/W Hz^{-1})]$. The dashed line and shaded region represent the median and $\pm 1\sigma$ values for SPIRE sources from Ivison et al. (2010). Squares and triangles represent sources with reliable redshift information in GOODS-N and Lockman Hole North, respectively. Good agreement is seen with the Ivison et al. (2010) values. No significant evolution with redshift is observed, suggesting our single $q_{IR}$ model should introduce no redshift-dependent bias to the photometric redshift estimates.</figcaption></figure>
Importantly, no evolution with redshift is observed, similar to Ivison et al. (2010), and hence our single *q* model should introduce no redshift-dependent bias. As none of the sources in Fig. 3 lies significantly off the radio-to-far-IR correlation, we are confident that potential contributions to the radio flux density from active galactic nucleus (AGN) activity may be ignored.

To get the best estimate of the redshift, we simultaneously fit the $L_{IR}-T_D$ and $L_{IR}-L_{1.4\text{GHz}}$ relations, assuming the far-IR emission is described by a modified blackbody. After including the radio-to-far-IR correlation, the photometric redshift accuracy improves to $|\Delta z|/(1+z) = 0.15$ (or $\langle|\Delta z|\rangle = 0.45$). If we consider only those sources with spectroscopic redshifts, the accuracy improves again to $|\Delta z|/(1+z) = 0.12$ (or $\langle|\Delta z|\rangle = 0.35$). Interestingly, this same improvement is not seen for the $L_{IR}-T_D$ only method when the sample is restricted to only those with spectroscopic redshifts ($|\Delta z|/(1+z) = 0.16$ or ($\langle|\Delta z|\rangle = 0.55$)).

While finding the redshift that gives the maximum likelihood redshift (i.e. the best fit) is the primary aim, it is also of interest to examine the probability distribution to see how wide a range in redshift also gives a reasonable fit. For each source, we calculate the redshift range in which 68 per cent of the probability is enclosed. These points are shown as error bars in Fig. 4. Using this approach, the typical estimated redshift error for photometric redshifts using only the $L_{IR}-T_D$ is $\sigma_z = 0.6$, while including the radio-to-far-IR correlation reduces this to $\sigma_z = 0.4$. Encouragingly these numbers are quite similar to the true error measured by comparing to known redshifts.

Many previous works have attempted to use a combination of radio and ground-based submm/mm data to estimate photometric

redshifts. Aretxaga et al. (2007) estimated submm-radio photometric redshifts for 120 850-µm sources detected in the SHADES survey (Mortier et al. 2005), 58 of which had previously known spectroscopic or robust optical/near-IR photometric redshifts. They make use of two distinct photo-z techniques: a simple 850 µm to 1.4 GHz spectral index estimator, similar to that proposed by Carilli & Yun (1999), and full fitting of the far-IR to radio SED to a set of well-defined templates. Comparing the results of these two techniques to the known redshifts gives a mean photometric redshift error of $|\Delta z| = 0.9$ for the spectral index method and $|\Delta z| = 0.6$ for the template fitting method.

More recently, Biggs et al. (2011) used a similar spectral index approach to estimate photometric redshifts for 870-µm sources detected in the LESS survey of ECDFS. Comparing the 48 sources which have spectroscopic redshifts, or reliable 17-band photometric redshifts (Wardlow et al. 2011), the photometric redshift accuracy is found to be $|\Delta z|/(1+z) = 0.25$ or a mean deviation of $\langle|\Delta z|\rangle = 0.68$.

Thus, it is clear that photometric redshifts estimated using the $L_{IR}-T_D$ relation alone offer a slight improvement over radio-submm redshifts, while using a combination of the $L_{IR}-T_D$ relation and radio-far-IR correlation, the accuracy of submm/mm-radio photometric redshifts can be increased by almost a factor of 2. By contrast, a similar study by Dannerbauer et al. (2010) using PACS 100 and 160 µm, in combination with mid-IR, radio and submm/mm (but not SPIRE), data find that the accuracy of photometric redshifts using IR templates is significantly worse if PACS data are included. This highlights both the need for SPIRE data, which is at or close to the peak wavelength of the far-IR emission at these

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Spectroscopic versus photometric redshift estimates for submm/mm photometry alone, assuming a universal $L_{IR}-T_D$ relation (left), and combined with radio constraints, assuming a universal radio-to-far-IR correlation (right). Filled symbols designate sources with spectroscopic redshifts, while open symbols designate those sources with optical/near-IR photometric redshifts only. For submm/mm photometric redshifts, we achieve a mean photo-z accuracy of $|\Delta z|/(1+z) = 0.16$ ($\langle|\Delta z|\rangle = 0.51$), while if radio constraints are included this is reduced to $|\Delta z|/(1+z) = 0.15$ ($\langle|\Delta z|\rangle = 0.45$). Error bars are calculated by finding the redshift range which contains 68 per cent of the probability.</figcaption></figure>
redshifts, as well as the need for improved SED models in the mid-IR (5–30 µm).

## 5.2 Redshift distribution of submm/mm sources

The redshift distribution of sources selected at different submm and mm wavelengths has been a subject of much debate. Many previous works have claimed that sources selected at mm wavelengths are preferentially found at higher redshifts than those at ~850 µm (Dannerbauer et al. 2004; Younger et al. 2007; Greve et al. 2008; Coppin et al. 2009). In contrast, others have claimed that the ~850 µm and mm-selected populations are at similar redshifts, with the apparent differences originating from selection effects (Wardlow et al. 2011).

Thus, it is of interest to re-examine this using our sample of SPIRE–mm sources. While only 22 of the 46 SPIRE-detected mm sources have reliable spectroscopic or photometric redshifts, as we have seen in Section 5.1, it is possible to estimate the redshifts of SPIRE–mm sources with reasonable accuracy using a combination of submm/mm and radio data alone.

As the submm photometric redshift errors are large, and our sample of SPIRE–mm sources small, we prefer to use the full probability distribution function (PDF) for each source rather than simply taking the maximum likelihood estimate of the photometric redshift fit in constructing the redshift distribution. This has the advantage of giving a more robust reconstruction of the redshift distribution (Sheth & Rossi 2010), although it should be remembered that the distribution produced in this way is essentially a convolution of the true redshift distribution and the typical photo-z error.

In addition to the SPIRE–mm sources, we can also estimate the redshifts of the 16 SPIRE-undetected mm sources, using the well-

established radio–mm spectral index (Carilli & Yun 1999). For the sources in Lockman Hole North, Lindner et al. (2011) quote estimates of the redshift for all 39 1.2-mm sources with radio identifications, using, where possible, a combination of 1.2-mm, 1.4 GHz and lower frequency GMRT 325 and 610 MHz observations. We use their redshift estimates for the nine sources without SPIRE counterparts in the Lockman Hole North field. In GOODS-N, we again use the spectral index approach as described by Carilli & Yun (1999) to estimate redshifts for the seven SPIRE-undetected mm sources, assuming the same values for $\alpha_{radio}$ and $\alpha_{submm}$ as Lindner et al. (2011), $\alpha_{radio} = -0.58$ and $\alpha_{submm} = 3.2$, to ensure consistency between the two samples. One exception to this is G14, otherwise known as HDF850.1, for which we used the radio–far-IR photometric redshift of $z = 4.1 \pm 0.5$ determined by Dunlop et al. (2004). To assess how reliable the mm–radio spectral index technique is for this sample, we compare redshift estimates produced via this method to the known redshifts for the 22 sources in our sample with good redshift information. We find a photo-z accuracy of $|\Delta z|/(1+z) = 0.24$, or a mean difference $\langle|\Delta z|\rangle = 0.6$, for photometric redshifts estimated from the radio–mm spectral index alone.

The full set of SPIRE–mm–radio and mm–radio photometric redshifts is presented in Tables 3 and 4 for the GOODS-N and Lockman Hole North fields, respectively.

The redshift distribution for our SPIRE–mm sources and SPIRE-undetected mm sources is shown in Fig. 5. SPIRE–mm sources show a strong peak at $z \sim 2–3$, similar to other known submm/mm samples. Interestingly, the redshift distribution for SPIRE-undetected sources shows that they are predominately found at $z > 3$. Only four of the SPIRE-detected sample are estimated to be at $z > 3.5$, while 13 of the 16 SPIRE-undetected mm sources are found above this redshift. This is not surprising as we have already found in

<figure><img src="image_6.png" /></figure>

**Figure 5.** Left: the redshift distribution of SPIRE–mm sources as determined by use of submm/mm–radio photometric redshifts. SPIRE-detected mm sources are shown as a grey shaded histogram. The distribution of SPIRE–mm sources with spectroscopic redshifts or reliable optical/near-IR photometric redshifts from Tables 1 and 2 is shown by the dashed line. Redshift estimates from the radio–mm spectral index for sources from our parent mm sample without SPIRE detections are shown as a solid line. Right: the redshift distribution of SPIRE–mm sources (grey shaded) and all mm sources (grey hatched) compared to 850 µm selected sources (solid blue line) from a compilation by Chapman et al. (in preparation), 1.1-mm selected sources (dotted red line) from C09 and 870 µm sources (dashed line) from Wardlow et al. (2011). In each case, the histogram is scaled to the total number of sources in our sample. The mm-selected sources are seen to peak at higher redshifts than those selected at ~850 µm, with a significant excess at $z > 3$ made up of primarily mm-only (i.e. SPIRE undetected) sources.
Section 4.2 that the sources with the largest mm-to-SPIRE colours are found to be at the highest redshifts.

However, this large population of SPIRE-undetected mm sources at high redshift is somewhat at odds with the redshift distributions of previous submm surveys, which are not limited by SPIRE. Shown for comparison is the redshift distribution of 850 µm selected sources, from a compilation by Chapman et al. (in preparation), 1.1-mm sources in GOODS-N from C09 and 870 µm sources in the LESS (Weiß et al. 2009) from Wardlow et al. (2011, hereafter W11). Comparing the redshift distribution of SPIRE-mm sources and the submm (850/870 µm) selected samples of Chapman et al. (2005) and W11, we find a clear deficit of $z < 2.5$ SPIRE-mm sources. In addition, once the mm-only sources are included, we find many more $z > 3.5$ sources than either Chapman et al. or W11. Much better agreement is seen with the C09 sample, although this work shows somewhat fewer sources at low $z$ (i.e. $z < 1$), potentially due to the smaller area probed in GOODS-N compared to our larger sample. A Kolmogorov-Smirnov test suggests that the C09 redshifts and our mm-selected sample have a >99 per cent chance of being drawn from the same underlying redshift distribution.

One potential reason for the disagreement between the redshift distribution of the mm and 850/870 µm sources is the depth of radio data available; in this work, we have used the two deepest blank-field images of the sky at 1.4 GHz available, with a typical rms noise of ~3–4 µJy. This is critical as of the 13 $z > 3.5$ SPIRE-undetected mm sources, nine are fainter than 30 µJy at 1.4 GHz. C09 also make use of the Morrison et al. (2010) radio data in GOODS-N. In ECDFS, the 1.4 GHz imaging has an rms of 6.5 µJy at its deepest, and thus W11 will miss the counterparts for many of the highest redshift sources. Similarly, the Chapman et al. list is based on lower quality radio data over several fields, and thus is also unlikely to identify the highest redshift submm sources.

W11 attempt to account for submm sources which do not have robust optical/near-IR or radio counterparts by investigating the redshift distribution of optical-near-IR sources neighbouring their submm-detected sample, and this correction is included in the redshift distribution shown in Fig. 5. However, while this means that the W11 redshift distribution is not limited by the depth of the radio (or mid-IR) imaging used for identification, this correction is limited by the depth of the available optical/near-IR imaging. W11 estimate that even after this correction, 20 per cent of LESS sources are still unaccounted for.

While it appears that our work here confirms that radio-detected, mm-selected samples appear at higher redshift than those selected at ~850 µm, it is difficult to determine whether this result is not simply a product of our use of deeper 1.4 GHz radio data for identifications. To investigate the possibility that the K-correction between submm (i.e. 850 µm) and mm wavelengths (i.e. 1.2 mm) is the cause for the difference in the redshift distribution, in Fig. 6 we plot the limiting IR luminosity assuming a single modified blackbody with dust temperature 30 and 40 K as a function of redshift. The flux limits at 350, 850 and 1.2 mm are taken to be 10, 5 and 2 mJy, respectively. The 350-µm and 1.2-mm limits were chosen to be representative of the sample under study here, while the 850-µm limit was chosen to be representative of current generation surveys at, or near, this wavelength (Mortier et al. 2005; Weiß et al. 2009).

From Fig. 6 it is clear that, irrespective of dust temperature, 850 µm is marginally more sensitive in the redshift range $1 < z < 3$, while 1.2 mm is more sensitive at $z > 3$. We can also see that the 350-µm limiting luminosity increases rapidly as a function of redshift, crossing the 1.2-mm limit at $z \sim 3$. This explains the lack of SPIRE-mm sources seen above this redshift.

While the difference in luminosity is small, the shape of the luminosity function at these extreme luminosities is very steep (Chapman et al. 2005; W11). At $1 < z < 2$, the 850-µm and 1.2-mm sensitivities differ by ~0.1 decades in $L_{IR}$. Using the Chapman et al. (2005) determination of the IR luminosity function, extrapolating to the limiting luminosity shown in the left-hand panel of Fig. 6 for a $T_D = 30$ K SED ($L_{IR} = 12.4 L_\odot$; $\log_{10}\phi \sim -5.3$ Mpc$^{-3}$ per decade) and assuming the full volume of our parent surveys between $z = 1$ and $3$ (0.24 deg$^2$; $5.4 \times 10^6$ Mpc$^3$), we could miss approximately

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Limiting IR luminosity as a function of redshift for flux-limited selection at three submm/mm wavelengths: $S_{350} > 10$ mJy (solid line), $S_{850} > 5$ mJy (dotted line) and $S_{1.2\,\text{mm}} > 2$ mJy (dashed line). In each case, we have assumed a modified blackbody SED with dust temperature $T_D = 30$ (left) and 40 K (right). In both panels, 850 µm is more sensitive in the redshift range $1 < z < 3$, while 1.2 mm is more sensitive at $z > 3$.</figcaption></figure>
five 850 µm detected sources via a 1.2-mm selection. Similarly, in the redshift range $3 < z < 5$, assuming the sensitivities differ by ~0.2 decades in $L_{IR}$ and again using the Chapman et al. (2005) luminosity function estimate at $L_{IR} = 12.4 L_{\odot}$, we could detect an extra ~10 sources via a 1.2-mm selection at the quoted flux limits. These numbers are crudely in line with the excess, and decrement, of sources we observe in redshift distribution of our mm-selected sources at $1 < z < 3$ and $z > 3$, respectively. Performing a similar exercise assuming $T_D = 40$ K would give a decrement of about one and excess of about two sources at $1 < z < 3$ and $3 < z < 5$, respectively. Thus, if the evolution of the 850 µm–1.2 mm K-correction is the cause of the observed differences in the redshift distribution, it must be relatively cold ($T_D \sim 30$ K) sources which make up the discrepant population.

Of course, these calculations assume a particular optical depth $\tau = (\nu/nu_0)^\beta$, although it is worth noting that the differences in sensitivity between 850 µm and 1.2 mm are enhanced for lower values of $\beta$, and are relatively insensitive to $v_0$.

## 5.3 Contribution of mm-selected sources to the star formation rate density

Now that we have some estimate of the redshift for all of our parent of radio-detected mm-selected sample, we can estimate their contribution to the IR luminosity density ($\rho_{IR}$) and hence the star formation rate density (SFRd) of the Universe. Given the wide redshift distribution of mm sources, we estimate $\rho_{IR}$ in four redshift bins: $0.5 < z < 1.5$, $1.5 < z < 2.5$, $2.5 < z < 3.5$ and $3.5 < z < 5$. The redshift bins were chosen so as to match previous measurements (i.e. C05; W11) while still maintaining a reasonable number of sources per bin. The contribution of each source is estimated by taking the measured IR luminosity ($L_{IR}$) and dividing by the maximum volume a source could be detected in ($V_{max}$). Incompleteness in the mm sample is taken into account by dividing each object by the completeness at its mm-wavelength flux density. Completeness estimates for the 1.2-mm source list in Lockman North are given by Lindner et al. (2011). Unfortunately, Penner et al. (2011) do not provide completeness estimates for their source list in GOODS-N, so we calculate the completeness as a function of 1.16-mm flux density by injecting sources at a range of flux densities into their combined MAMBO–AzTEC image. Sources are considered recovered if they are found within 3 arcsec of the injected position with a peak S/N of $\sigma > 3.8$.

For each source, we wish to use the most reliable redshift available. For the 22 sources with spectroscopic or good optical/near-IR photometric redshifts, we use these measurements, while for the 25 SPIRE–mm without prior redshift estimates we use our submm/mm–radio photo-z values from Section 5.1. For the remaining 16 mm-only sources, we use estimates from the radio–mm spectral index, as described in Section 5.2.

However, given the large photometric redshift errors of the submm/mm–radio and radio–mm spectral index estimates which form the bulk of our sample, we estimate $\rho_{IR}$ via a Monte Carlo approach in which realizations are produced by placing each source at a redshift randomly drawn from the PDF of the submm/mm–radio fitting. In each of the realizations, sources with good spectroscopic redshifts are held fixed. Sources with optical/near-IR photometric redshifts are assumed to have Gaussian PDFs with the width determined by the quoted errors from Table 2. Similarly, mm-only sources with radio–mm spectral index redshifts are assumed to have Gaussian PDFs with width given by the typical $\Delta z$ found in Sec-

<table><caption>**Table 5.** Contribution of mm-selected sources to the IR luminosity density.</caption><thead><tr><th>Redshift range</th><th>ρ<sub>IR</sub><br>(log<sub>10</sub> L<sub>☉</sub> Mpc<sup>-3</sup>)</th><th>ρ<sub>SFR</sub><br>(log<sub>10</sub> M<sub>☉</sub> yr<sup>-1</sup> Mpc<sup>-3</sup>)</th></tr></thead><tbody><tr><td>0.5 &lt; z &lt; 1.5</td><td>7.1 ± 0.1</td><td>-2.7 ± 0.1</td></tr><tr><td>1.5 &lt; z &lt; 2.5</td><td>7.72 ± 0.07</td><td>-2.05 ± 0.07</td></tr><tr><td>2.5 &lt; z &lt; 3.5</td><td>7.86 ± 0.08</td><td>-1.91 ± 0.06</td></tr><tr><td>3.5 &lt; z &lt; 5</td><td>7.7 ± 0.1</td><td>-2.1 ± 0.1</td></tr></tbody></table>

tion 5.2 ($\Delta z = 0.6$). In addition, mm-only sources are assumed to be at a dust temperature of 35 K.

In each Monte Carlo realization, we assess the contribution of each source to each redshift bin by taking the sum $\sum_i^n L_{IR}^i / V_{max}^i$, where $n$ runs over the sources which happen to fall in that redshift bin in a given realization, and $L_{IR}$ and $V_{max}$ for source $i$ are calculated using the best-fitting modified blackbody. To calculate $V_{max}$, we find the minimum and maximum redshift at which an observed source would still be found in our sample, i.e. $> 3.8\sigma$ at 1.16 or 1.2 mm, depending on the field, and $> 5\sigma$ at 1.4 GHz, taking into account variations in both the luminosity distance and K-correction. The posterior probability of $\rho_{IR}$ and the mean redshift in each bin are then inferred from a histogram of the Monte Carlo realizations, fitting a Gaussian to return the peak likelihood and variance.

Based on this analysis, our best estimate of the contribution of mm-selected sources to the IR luminosity density is given in Table 5.

Fig. 7 compares these values to previous measures of the contribution to the IR luminosity density by 850 µm and LESS 870 µm sources (Chapman et al. 2005; W11), and a prediction of the contribution from $L_{IR} > 10^{12.5} L_{\odot}$ from the model of Béthermin et al. (2010). At $z \sim 2.5$, our estimates appear lower than both the contribution from ~850 µm sources and the predictions for $L_{IR} > 10^{12.5} L_{\odot}$ sources. However, there are a number of ways to account for this. First, as we have seen in Section 5.2, mm-selected surveys are less sensitive than their 850 µm counterparts at $1 < z < 3$. However, mm surveys must be less sensitive in general as the areal density of mm-detected sources is much lower than equivalent 850 µm samples. Including sources without radio identifications, our parent sample contains 82 mm-detected sources in $0.24 \text{ deg}^2$, while the LESS survey detect 126 sources at 870 µm in a $0.25 \text{ deg}^2$ and SHADES detected 120 sources at 850 µm across two fields totalling $0.22 \text{ deg}^2$.

Given this, it is puzzling that our estimate at $z \sim 4$ is somewhat larger than that found for 870 µm sources by W11; however, as we have seen in Section 5.2, there are good reasons to expect a significantly larger contribution at $z > 3$, as mm surveys are more sensitive at these redshifts than their 850 µm counterparts, and we have the advantage of significantly deeper radio data with which to identify our mm sources. The W11 analysis is missing the 45 per cent of LESS SMGs that do not have identifications; similarly, Chapman et al. (2005) only have radio identifications for ~70 per cent. Here we have radio identifications for 78 per cent of our parent mm-detected sample.

It is of interest that our measurement of the SFRd at $z \sim 4$, while higher than the W11 estimate for 870 µm sources, is still somewhat short of that predicted for IR luminous galaxies ($\log_{10} L_{IR} > 12.5 L_{\odot}$) by Béthermin et al. (2010), and ~100 lower than the total SFRd estimated from a combination of UV, radio and IR measurements (Hopkins & Beacom 2006). Thus, while selecting sources at mm wavelengths offers an effective way to isolate
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Contribution of mm-selected sources to the IR luminosity density evolution. For comparison, we show the estimates of Chapman et al. (2005) and W11 at $z = 0-4$. As ECDFS is underdense at submm wavelengths, W11 present two estimates: one based on the observed number of sources in ECDFS and another which has been scaled to the SHADES 850 µm number density (Coppin et al. 2006). Also shown is the predicted contribution by $>10^{12.5}$ L⊙ ultraluminous IR galaxies to the IR luminosity density from the empirical model of Béthermin et al. (2010). At $z < 3$, our estimate is significantly lower than those found in both previous studies and models, while higher than previous studies at $z \sim 4$. These disagreements can be explained when considering the flux limits and biases of the comparison samples.</figcaption></figure>

luminous starbursts at high redshift ($z > 3$), it is clear that mm-selected sources, at least at the imaging depth possible with current facilities, do not make a significant contribution to the SFRd at any redshift.

Due to the strong dust temperature bias, mm-selected sources never contain the bulk of IR luminous sources at any redshift. Taking the Béthermin et al. (2010) prediction as a guide, our mm-selected sample never makes up more than ~50 per cent of the expected SFRd of IR luminous sources ($\log_{10} L_{IR} > 12.5$). We can see the reason for this clearly in Fig. 6: at $T_D = 30$ K we could expect our mm sample to reach this level, while at $T_D = 40$ K we are only sensitive to IR sources above a luminosity of $\log_{10} L_{IR} \gtrsim 12.7$. Thus, for the Béthermin et al. (2010) prediction to hold, this shortfall must be made up of IR sources with $\log_{10} L_{IR} \lesssim 12.7$ and $T_D \gtrsim 30$ K.

# 6 CONCLUSIONS

We have presented an analysis of 63 radio-detected, mm-selected sources. Our parent sample of mm sources was drawn from the recent MAMBO 1.2-mm imaging of Lockman Hole North by Lindner et al. (2011) and the combined MAMBO-AzTEC 1.16-mm image of GOODS-N presented by Penner et al. (2011). Radio identifications are found for the Penner et al. (2011) sample, with counterparts for 24 of the 41 1.16-mm sources presented. These new identifications were combined with the 39 (out of 41) radio identifications presented by Lindner et al. (2011) for their MAMBO

sample to give a sample of 63 radio-detected mm sources. We have cross-matched this mm-selected sample with SPIRE data from the HerMES project, making use of a map-based technique which reduces the problem of confusion in the SPIRE data by using radio and 24 µm positions as a prior. The main results of this work are as follows.

(i) After comparing with the SPIRE data, we found 47 mm sources with a detection in one or more SPIRE band, giving a detection rate of 74 per cent.

(ii) For SPIRE-detected mm sources with spectroscopic or good quality optical/near-IR photometric redshifts, we found a tight correlation between dust temperature ($T_D$) and IR luminosity ($L_{IR}$). This correlation is found to be offset to colder dust temperatures from the local relationship (e.g. Chapman et al. 2003).

(iii) Using the observed tight relationship between dust temperature ($T_D$) and IR luminosity ($L_{IR}$) to break the degeneracy between redshift and dust temperature, we have demonstrated that photometric redshifts from submm/mm data alone offer an accuracy of $|\Delta z|/(1+z) = 0.16$ ($|\Delta z| = 0.51$). If constraints from the radio-far-IR correlation are included, this is improved to $|\Delta z|/(1+z) = 0.15$ ($|\Delta z| = 0.45$).

(iv) Via a combination of spectroscopic, optical/near-IR photometric and submm/mm-radio photometric redshifts, the redshift distribution of mm-selected sources was presented. We found that the redshift distribution of mm-selected sources peaks at higher redshifts than those selected at ~850 µm, with a long tail to high $z$ ($z > 3.5$) made up of primarily mm-only (i.e. SPIRE undetected) sources. This difference may be explained by a combination of the depth of the radio data used to identify the sources, as well as the evolving K-correction between submm and mm wavelengths. Good agreement was found between this study and previous mm-selected samples.

(v) We measured the contribution of mm-selected sources to the SFRd of the Universe. We found that mm-selected sources do not contribute significantly to the global SFRd, nor do they make up the bulk of IR luminous sources, at any redshift.

# ACKNOWLEDGMENTS

We thank the anonymous referee for suggestions which greatly enhanced this work.

LW and SJO were supported by the Science and Technology Facilities Council (ST/F002858/1).

JSD acknowledges the support of the Royal Society via a Wolfson Research Merit award, and the support of the European Research Council via the award of an Advanced Grant.

AF, GM, LM and MV were supported by the Italian Space Agency (ASI Herschel Science Contract I/005/07/0).

SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including University of Lethbridge (Canada), NAOC (China), CEA, LAM (France), IFSI, University of Padua (Italy), IAC (Spain), Stockholm Observatory (Sweden), Imperial College London, RAL, UCL-MSSL, UK ATC, University of Sussex (UK), Caltech, JPL, NHSC, University of Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK) and NASA (USA).

The data presented in this paper will be released through the Herschel Database in Marseille HeDaM (hedam.oamp.fr/HerMES)
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