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Open Access
Issue
A&A
Volume 675, July 2023
Article Number A85
Number of page(s) 17
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/202243528
Published online 10 July 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

Observations of the extragalactic background light (EBL) across the whole electromagnetic spectrum have shown that the integrated intensity in the far infrared (FIR) to millimeter regime is comparable to that of the optical portion (Puget et al. 1996; Fixsen et al. 1998; Dole et al. 2006; Cooray 2016). Attempting to resolve this cosmic infrared background into distinct sources led to the discovery of submillimeter galaxies (e.g., Smail et al. 1997; Hughes et al. 1998), which appear to be at high redshift and harbor dust-obscured star formation (Blain et al. 2002). Since these first detections, substantial progress has been made characterizing the also-called dusty star-forming galaxies (DSFGs; see Casey et al. 2014, for a review). The high spatial resolution and sensitivity of the Atacama Large Millimeter/submillimeter Array (ALMA) is making it possible to probe DSFGs in exquisite detail, giving more insights into the nature of star-forming galaxies at high redshift. Importantly, deep ALMA observations may help to resolve the faint end of the dusty galaxy population (i.e., sources having mm flux densities below 0.1 mJy), thus giving more clues to the origin of the EBL at millimeter wavelengths. While there are currently plenty of brighter DSFG detections (via single-dish and/or interferometric observations; see, e.g., Casey et al. 2014 and Hodge & da Cunha 2020), only a few studies have uncovered the faint-end population (Fujimoto et al. 2016; Aravena et al. 2016; González-López et al. 2017b, 2020).

One of these projects is the ALMA Frontier Fields Survey (PI: F. Bauer). Introduced in González-López et al. (2017b, hereafter Paper I), it looks for DSFGs using deep ALMA Band 6 observations toward Hubble Frontier Fields (FF) galaxy clusters (Lotz et al. 2017). This ALMA survey benefits from the unique ALMA capabilities and the strong gravitational lensing power of massive galaxy clusters, allowing us to detect (if magnified) sources beyond the confusion limit (e.g., rms depths on the order of tens of μJy beam−1 for current deep ALMA maps at 1.1 mm). It also exploits the public availability of deep multiwavelength data (including HST, Spitzer and VLA) and several detailed mass models for each galaxy cluster, both of which are at the core of the FF legacy project. The ALMA FF Survey has successfully found and characterized DSFGs both in the continuum (see Paper I) and emission lines (González-López et al. 2017a, hereafter Paper III). It has also included a multiwavelength photometric analysis of a sample of ALMA detections (Laporte et al. 2017, hereafter Paper II) and an ALMA stacking of Lyman-break galaxies in the FFs (Carvajal et al. 2020).

In Muñoz Arancibia et al. (2018, hereafter Paper IV), we reported 1.1 mm number counts for the first three FF galaxy clusters (see also Muñoz Arancibia et al. 2019, hereafter Paper IV Corrigendum) following a detailed treatment of the uncertainties that come from the lensing models. Based on 19 ALMA 1.1 mm continuum detections, we derived counts spanning around two orders of magnitude in demagnified flux density, down to ≈0.01 mJy. Although these counts were consistent at a 1σ level with most of deep ALMA observations reported by that time, we found that below ≈0.1 mJy our cumulative counts were lower by ≈0.5 dex, suggesting a flattening in the number counts. Encouragingly, this flattening was further confirmed independently by the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field Large Program (ASPECS-LP; González-López et al. 2020; Aravena et al. 2020). Based on the deepest 1.2 mm data to date in a contiguous area over the sky, González-López et al. (2020) derived number counts down to ≈0.03 mJy, noticing a flattening in the cumulative counts below 0.3 mJy. In this work, we expanded our census of the surface density of DSFGs detected at 1.1 mm in the FFs through galaxy number counts that include the remaining galaxy clusters that form our survey. We aim to derive number counts again down to flux densities fainter than 0.1 mJy, bolstering our previous results with better statistics.

This paper is organized as follows. Section 2 presents the ALMA observations, redshift estimates, and lensing models used. Section 3 describes the procedure used to derive lensing-corrected number counts. Section 4 reports our 1.1 mm counts and compares them with our previous work, recent literature data using ALMA, and galaxy evolution models. It also presents our estimates of cosmic variance and contribution to the EBL based on our number counts. Section 5 presents a summary of our findings. We adopt a flat ΛCDM cosmology with parameters H0 = 70 km s−1 Mpc−1, Ωm = 0.3, and ΩΛ = 0.7.

2. Data

2.1. Observations with ALMA

Five out of the six FF galaxy clusters were observed by ALMA in Band 6 as part of the ALMA Frontier Fields Survey (programs #2013.1.00999.S in cycle 2 and #2015.1.01425.S in cycle 3, PI: F. Bauer). These clusters are Abell 2744, MACSJ0416.1-2403, MACSJ1149.5+2223, Abell 370, and Abell S1063 (hereafter A2744, MACSJ0416, MACSJ1149, A370, and AS1063, respectively). Paper I introduces the 1.1 mm mosaic images, data reduction, and analysis of the cycle 2 observations, which cover the first three galaxy clusters. Paper III presents a summary of cycle 3 observations, which cover A370 and AS1063. For the five clusters, the inner ≈2′×2′ FF regions were covered with 126 pointings each, with a local oscillator frequency set to 263.14 GHz (≈1.1 mm). Datasets were reduced using the Common Astronomy Software Applications (CASA; McMullin et al. 2007) and imaged using the multifrequency synthesis algorithm in CLEAN, adopting a 0 . ″ 1 $ 0{{\overset{\prime\prime}{.}}}1 $ pixel size. Cleaned images were made using niter = 1000 and a threshold of four times the rms measured in the dirty images (which were generated using niter = 0), applying no masking during cleaning. Mosaics were created using natural weighting (see details in Paper I). Table 1 lists the cluster properties and natural-weighted mosaic details for the five clusters (images are shown in Fig. 1). Since the sixth FF cluster, MACSJ0717.5+3745, was only partially observed in cycle 3 with a much lower level of sensitivity, source extraction was not performed on it, and thus we do not include it in this study.

thumbnail Fig. 1.

ALMA continuum maps at 1.1 mm for five FF clusters, made with natural weighting. Images are displayed without PB sensitivity corrections. Color scale goes from blue to red, corresponding to −5σ to 5σ. A white contour in each map encloses the region with PB > 0.3, that is, where the PB sensitivity is at least 30% of the peak sensitivity. Source positions are displayed in blue as squares (high significance) or circles (low significance). A small box in the bottom left corner of each panel shows the synthesized beam (see parameters in Table 1). Green lines indicate the critical curves (i.e., infinite magnification) for a z = 2 source plane in the Diego v4.1 lens model (see references in Table 3).

Table 1.

ALMA mosaic properties of galaxy clusters.

We detect sources by searching for pixels with signal-to-noise ratio (S/N) larger than or equal to a given threshold, grouping pixels as individual sources with the DBSCAN python algorithm (Pedregosa et al. 2012). We choose a threshold of S/N = 4.5, as this is found to be an appropriate tradeoff between the purity and completeness correction factors that will be applied to our number counts (see Sect. 3). Detections in the first three FF clusters were already reported in Paper I and Paper IV, making a total of 19 sources. Here, we introduce our detections in A370 and AS1063. In each mosaic, we perform the source extraction within the region where the primary beam (PB) sensitivity is at least a given fraction of the peak sensitivity. In Paper IV, we considered PB > 0.5. However, in one of the last two FFs we find a high S/N source at PB = 0.34, A370-ID02. Therefore, in this study we used PB > 0.3 for all cluster fields. This does not add sources to the first three fields. Throughout this paper, all source flux densities and peak intensities correspond to PB-corrected values.

Extracting sources at S/N ≥ 4.5 in the last two FFs adds ten new detections to our catalog (see Table 2). In each of these cluster fields, we detect three high-significance (S/N ≥ 5) and two low-significance (4.5 ≤ S/N < 5) sources. Detection coordinates are shown in Fig. 1. Detections have peak intensities in the ∼0.28 − 6.30 mJy beam−1range. We measured integrated flux densities in the uv plane for our new high-significance detections as in Paper I using the UVMCMCFIT python algorithm (Bussmann et al. 2016). These Gaussian fits give integrated flux densities in the ∼1.00 − 6.81 mJy range, effective radii in the ≲ 0 . ″ 10 − 0 . ″ 45 $ {\lesssim}0{{\overset{\prime\prime}{.}}}10{-}0{{\overset{\prime\prime}{.}}}45 $ range (half-light radii1), and axial ratios in the ∼0.36 − 0.80 range.

Table 2.

Continuum detections at S/N ≥ 4.5.

As in Paper IV, we estimated the integrated flux densities of low-significance detections in A370 and AS1063 using source peak intensities; this choice is motivated by the large uncertainties in integrated source flux densities that are given at low S/N by two-dimensional Gaussian fits in the uv plane. We adopt as their observed effective radius and axial ratio the median values found for the high-significance sources. Including the five FFs, these are r eff , obs = 0 . ″ 23 $ r_{\mathrm{eff,obs}}=0{{\overset{\prime\prime}{.}}}23 $ and qobs = 0.58, respectively. We conducted source injection simulations (see Sect. 3), finding a typical ratio between the peak and integrated flux density for these size parameters of 0.80 and 0.71 in A370 and AS1063, respectively. These factors give image-plane integrated flux densities of the 4.5 ≤ S/N < 5 detections ranging from ∼0.35 to ∼0.55 mJy.

Each cluster field covers an observed area of ∼5.2 arcmin2 (PB > 0.3 region). The five clusters sum to a total image-plane area of ∼26 arcmin2. This corresponds to ∼6 times the area of the Hubble Ultra Deep Field (HUDF; Dunlop et al. 2017), ∼26 times the initial ALMA Spectroscopic Survey in the HUDF (ASPECS-Pilot; Walter et al. 2016; Aravena et al. 2016), and ∼6 times the ASPECS-LP (Aravena et al. 2020). We note, however, that this area is reduced when lens models are applied (see Sect. 2.3).

2.2. Source redshifts

For our ALMA detections in the first three FFs, we start by adopting the same redshifts as in Paper IV. These include spectroscopic redshifts from the GLASS survey using grism spectroscopy at the HST (Treu et al. 2015) and photometric redshifts (including full probability distributions) derived in Paper II for our high-significance detections via spectral energy distribution (SED) fitting. Since the publication of Paper IV, various new catalogs have been released, providing recent estimates for these galaxy clusters. These catalogs include: photometric redshifts reported by Ishigaki et al. (2018), the HFF-DeepSpace project (Shipley et al. 2018), Bhatawdekar et al. (2019), the KLASS survey (Mason et al. 2019), the SHARDS FF survey (Griffiths et al. 2021), Pagul et al. (2021), the ALCS survey (Kokorev et al. 2022), and the UNCOVER survey (Weaver et al. 2023), as well as spectroscopic redshifts by de La Vieuville et al. (2019), Vanzella et al. (2021), and Richard et al. (2021). We searched for counterparts in these catalogs to those of our sources that lack spectroscopic or photometric estimates. We checked that none of these sources have reliable counterparts within ≈1″ of their peak positions.

Bauer et al. (in prep.) performed an ALMA line scan targeting our high-significance ALMA detections in A2744 (program #2017.1.01219.S in cycle 5, PI: F. Bauer), obtaining several spectroscopic redshift estimates. We then replaced photometric redshifts for A2744 detections with secure spectroscopic redshifts when available. New included redshifts and secure line(s) identifications are as follows: A2744-ID02 has zspec = 2.585 from CO (3-2) and CO (5-4) lines (previously in Paper IV we adopted zspec = 2.482 from GLASS, which was based on a red continuum); A2744-ID03 has zspec = 3.058 from CO (4-3) and CO (5-4) lines; A2744-ID04 has zspec = 1.498 from CO (2-1), CO (4-3) and [C I] (1-0) lines; A2744-ID05 has zspec = 2.409 from CO (3-2), [C I] (1-0), CO (7-6) and [C I] (2-1) lines; and A2744-ID07 has zspec = 2.644 from CO (3-2), CO (5-4) and CO (8-7) lines.

We searched for counterparts to our ALMA detections in the last two FFs considering several public catalogs that report photometric and/or spectroscopic redshift estimates. These include photometric redshifts estimated by Monna et al. (2014), the CLASH team (Postman et al. 2012; Molino et al. 2017), Ishigaki et al. (2018), the HFF-DeepSpace project, the KLASS survey, the ASTRODEEP survey (Bradač et al. 2019), the SHARDS FF survey, Pagul et al. (2021), and the ALCS survey; catalogs of spectroscopic redshifts by Richard et al. (2010, 2014, 2021), Gómez et al. (2012), Boone et al. (2013), Johnson et al. (2014), Diego et al. (2016, 2018), Lagattuta et al. (2017, 2019, 2022), Vanzella et al. (2017), Strait et al. (2018), Vega-Ferrero et al. (2019), Walth et al. (2019), the GLASS survey, and the CLASH survey using VIMOS (Balestra et al. 2013) and MUSE (Karman et al. 2015, 2017; Caminha et al. 2016; Mercurio et al. 2021) at the VLT; and redshift estimates for Herschel detections (Rawle et al. 2016).

Two sources in A370 and two in AS1063 have spectroscopic redshifts; these are A370-ID01, A370-ID03, AS1063-ID02, and AS1063-ID04. A370-ID01 has a zspec = 1.06 (secure) counterpart within a radius of ≈ 0 . ″ 4 $ {\approx}0{{\overset{\prime\prime}{.}}}4 $ in the GLASS catalog, identifying multiple emission lines. A370-ID03 has a zspec = 1.258 (secure) counterpart within ≈ 0 . ″ 2 $ {\approx}0{{\overset{\prime\prime}{.}}}2 $ in Lagattuta et al. (2022) using MUSE at the VLT. They give a redshift confidence flag of 3 to this source, indicating high reliability; this means that the redshift is either based on multiple clear spectral features or on a single high-significance emission line with additional information. AS1063-ID02 has a zspec = 1.438 (possible or probable) counterpart within ≈ 0 . ″ 2 $ {\approx}0{{\overset{\prime\prime}{.}}}2 $ in GLASS. Treu et al. (2015) gave a redshift quality flag of 2.5 to this source, indicating that either a single strong emission line is robustly detected and the identification is supported by a photometric redshift or that more than one feature is marginally detected, or that a single line is detected with marginal quality. AS1063-ID04 has a zspec = 0.61 counterpart within ≈ 1 . ″ 1 $ {\approx}1{{\overset{\prime\prime}{.}}}1 $ in Gómez et al. (2012), obtained from Hβ and [O III] emission lines using GMOS at Gemini-South. It has a zspec = 0.609 (possible) counterpart within ≈ 0 . ″ 9 $ {\approx}0{{\overset{\prime\prime}{.}}}9 $ in GLASS, with a single line detection of marginal quality but strong [S III]λ9069 and [S III]λ9531 emission lines using the G141 grism. It also has a zspec = 0.611 (secure) counterpart within ≈ 0 . ″ 9 $ {\approx}0{{\overset{\prime\prime}{.}}}9 $ in CLASH using MUSE at the VLT. We adopted the latter value for this work. This counterpart is well resolved in HST images, displaying a clear spiral-like morphology. Using CLASH HST mosaics, Connor et al. (2017) derived a semi-major axis length of ≈ 3 . ″ 5 $ {\approx}3{{\overset{\prime\prime}{.}}}5 $ for this galaxy. AS1063-ID04 is found toward the disk of this galaxy, at ≈1″ from its center. In addition, this counterpart was listed in a follow-up study to 24 μm selected galaxies by Walth et al. (2019). They identified eight emission lines using LDSS-3 at Magellan-Clay in this galaxy and reported the discovery of a luminous kpc-sized HII region at one edge of its disk.

We restrict our photometric redshift search to those data providing full posterior probability distributions, that is, the HFF-DeepSpace project. Shipley et al. (2018) derived photometric redshifts using up to 22 filters in the wavelength range 0.2 − 8 μm and the EAZY fitting code (Brammer et al. 2008). Among the sources that lack spectroscopic redshifts, one in A370 and two in AS1063 have counterparts with reliable photometric redshifts within 0 . ″ 5 $ 0{{\overset{\prime\prime}{.}}}5 $. We used the redshift quality parameter pΔz provided by the catalog as a measure of the zphot reliability, with pΔz ≈ 1, indicating that most of the total integrated probability lies within Δz = 0.2 of the zphot estimate. Listing the best (i.e., zm2) redshift values and 1σ errors of the closest counterpart as a reference, these sources are A370-ID02 ( z phot = 2 . 581 − 0.180 + 0.193 $ z_{\mathrm{phot}}=2.581_{-0.180}^{+0.193} $), AS1063-ID01 ( z phot = 2 . 511 − 0.113 + 0.111 $ z_{\mathrm{phot}}=2.511_{-0.113}^{+0.111} $), and AS1063-ID03 ( z phot = 1 . 468 − 0.063 + 0.065 $ z_{\mathrm{phot}}=1.468_{-0.063}^{+0.065} $). All these estimates have a redshift quality parameter of pΔz ≈ 1. AS1063-ID05 has a counterpart within ≈ 0 . ″ 3 $ {\approx}0{{\overset{\prime\prime}{.}}}3 $ with z phot = 2 . 256 − 1.302 + 1.766 $ z_{\mathrm{phot}}=2.256_{-1.302}^{+1.766} $. However, it has pΔz ≈ 0.2, indicating a broad and/or multi-modal probability distribution. Therefore, we do not include this redshift estimate in this work.

We note that one detection in A370 (A370-ID03) and three in AS1063 (AS1063-ID01, AS1063-ID02, and AS1063-ID04) have counterparts in the Herschel catalog within ≈1″ (source coordinates given by either VLA or HST position references). A370-ID03 has a FIR-estimated redshift of ∼0.9 from SED fitting using five Herschel bands (see Rawle et al. 2016); we did not use this estimate as we prioritized the spectroscopic redshift found for A370-ID03. We also searched for counterparts to our sources in available catalogs of FF cluster members, as well as in catalogs used as input by the lens models included in this work. We find that none of our ALMA detections have counterparts in them.

We list the source redshifts chosen for this work in Table 2. When available, we use spectroscopic redshifts and (if not) photometric redshift probability distributions. Nine detections have no reliable counterparts in the aforementioned catalogs. As in Paper IV, for these sources we assume a Gaussian redshift distribution centered at z = 2 with σ = 0.5, which is consistent within 1σ with average redshift values found for faint dusty galaxies from the literature (e.g., Aravena et al. 2016, 2020; Dunlop et al. 2017).

2.3. Lensing models

We correct our number counts by gravitational lensing effects making use of the models publicly available on the FF website2, which are produced by independent teams. These models provide maps of the normalized mass surface density and shear of the galaxy cluster for a redshift z = ∞ background. We used the full set of individual mass reconstructions released by the teams (which sample the range of uncertainties), reprojected to the size and resolution of the ALMA maps. From these maps and adopting a given source redshift, we can compute magnification maps, deflection fields around each galaxy cluster, and thus effective source-plane areas for each detection (i.e., the angular area where a map pixel with a given peak intensity can be detected over a S/N threshold). We refer the reader to Paper IV for more details regarding how these quantities were obtained from the aforementioned maps.

For the first three FFs, we used the same lens models as in Paper IV, as no newer versions have been publicly released3. Table 3 lists these models, together with the models considered for the two FF clusters introduced in this work; we adopt for use eight, nine, seven, nine, and nine lens models for A2744, MACSJ0416, MACSJ1149, A370, and AS1063, respectively. We consider only v4 or newer models, as these use the best data to date as constraints4. We also restrict our set only to models that cover the whole region where our ALMA sources lie. A370-ID03 is not covered by GLAFIC v4 model, and therefore we do not include it for A370. A fraction of the region where the ALMA maps have PB > 0.3 is not fully covered by the models GLAFIC v4 (∼2% and 3% for A2744 and AS1063, respectively), Sharon v4 (∼0.7% for A370), Williams v4 (∼3%, 19%, 10% and 14% for A2744, MACSJ0416, MACSJ1149, and A370, respectively), and Williams v4.1 (∼14% for A370). We consider no magnification effects (μ = 1) for the missing pixels in magnification maps.

Table 3.

Lensing models considered in this work.

3. Methodology

We computed demagnified number counts at 1.1 mm considering ALMA detections down to S/N = 4.5. We followed the procedure that is described in detail in Paper IV. There, we used a Monte Carlo approach to take into account the uncertainties in observed flux densities, adopted redshifts, and magnifications given by the lens models. After running simulation realizations for all lens models, we computed median values as the best estimates of the true values for our number counts.

We conducted this procedure for the five galaxy clusters with no modifications other than the PB limit, restricting us to the PB > 0.3 region instead of using PB > 0.5 (see Sect. 2.1). For the first three FFs, this implied recomputing the quantities where the PB limit is involved. These are i) completeness, that is, the proportion of sources that were not detected because their noise level shifted their peak S/N below our chosen threshold; ii) flux deboosting (Hogg & Turner 1998; Weiß et al. 2009); iii) fraction of spurious sources, that is, generated by noise; and iv) the effective area as a function of demagnified peak intensity. The first three of these corrections were done in the image plane.

We used source injection simulations, in the image-plane, to estimate the completeness and deboosting corrections (see Paper IV Sect. 3.1 for details). After source extraction, we computed the completeness as a function of image-plane integrated flux density Sobs and separated it in bins of image-plane scale radius. Completeness corrections for the five cluster fields are shown in Fig. 2. For point sources, a value of 50% is reached at image-plane flux densities of 0.32 mJy for A370 and 0.35 mJy for AS1063. However, the completeness level drops to 19% and 7%, respectively, at the same flux densities for image-plane source sizes in the 0 . ″ 20 − 0 . ″ 25 $ 0{{\overset{\prime\prime}{.}}}20-0{{\overset{\prime\prime}{.}}}25 $ range (i.e., for the image-plane size assumed for our low-significance detections). Figure 3 shows the ratio between the extracted and injected flux densities for our simulations. We find that at S/N = 4.5 the noise boosts the flux densities by 6% for both A370 and AS1063. We correct for this effect both the observed peak intensities and integrated flux densities of all our detections.

thumbnail Fig. 2.

Completeness correction C as a function of image-plane integrated flux density and separated in bins of image-plane scale radius (half-light radius). Error bars indicate binomial confidence intervals.

We obtain a rough estimate of the Eddington bias (i.e., an overestimate of the derived faint-end number counts due to noise fluctuations and a steep underlying flux density distribution; Eddington 1913), as done in Paper IV, making no assumptions regarding the true underlying distribution of flux densities, since the number density of ALMA sources in the FFs is low even after adding two clusters. We created sets of 104 simulated point sources that follow the SIDES galaxy formation model (Béthermin et al. 2017) distributions of both redshift and 1.1 mm flux density, lensed these sources using the “best” CATS v4 model for each cluster, injected them in and extracted them from our ALMA mosaics down to S/N = 4.5, and obtained the ratio between output and input demagnified flux density as a function of S/N (see Paper IV Sect. 3.1 for details). At S/N = 4.5, we find flux enhancements by 21%, 29%, 22%, 19%, and 33% for A2744, MACSJ0416, MACSJ1149, A370, and AS1063, respectively. These ratios are consistent with the deboosting corrections obtained in Fig. 3. We thus considered it safe to skip Eddington bias corrections in this work.

thumbnail Fig. 3.

Deboosting correction as a function of S/N. We display the ratio between the extracted and injected flux densities for our simulated sources as light magenta dots. Thick dark magenta lines correspond to median values, while thin dark magenta lines indicate the 16th and 84th percentiles.

We computed the fraction of spurious sources as a function of S/N for each galaxy cluster, pfalse, as the average ratio between the number of detections over a peak S/N in a set of 300 simulated noise maps and in the actual ALMA mosaic (see Paper IV and Paper IV Corrigendum). Figure 4 shows this fraction for each of the five FFs. At S/N ≥ 4.5, pfalse is ≈18% for A370 and ≈28% for AS1063. At S/N ≥ 5, it drops to ≈4% and ≈5%, respectively. Based on the source extraction on the simulated noise maps, the average number of spurious sources at S/N ≥ 4.5 is 0.93 ± 1.33 for A370 and 1.45 ± 1.79 for AS1063. This is consistent within 1σ with both the amount of spurious sources from the negative mosaics (zero and one, respectively) and the number of sources without reliable counterparts at other wavelengths (two and one, respectively).

thumbnail Fig. 4.

Fraction of spurious sources at a given S/N. We display curves for A2744, MACSJ0416, MACSJ1149, A370, and AS1063 in dark magenta, dark teal, magenta, teal, and purple, respectively. A vertical dotted line indicates our S/N threshold of 4.5.

For a given galaxy cluster, source redshift and individual mass reconstruction of a lens model, the effective source-plane area over which a source can be detected was obtained as follows. We used the surface mass density and shear maps to obtain a magnification map. Both this and the PB-corrected rms map for the cluster were deflected to the source plane. At each demagnified peak intensity, Sdemag, peak, the effective area Aeff was obtained from the source-plane pixels that satisfy Sdemag, peak/σdemag ≥ 4.5, with σdemag being the PB-corrected rms corrected for magnification. Since we needed to create several source-plane maps for each lens model (sampling the source redshift distributions) but also wanted our Monte Carlo simulations to run fast, we precomputed source-plane maps for all mass reconstructions and lens models over the redshift range of zmin = 0.4 to zmax = 4 using steps of Δz = 0.2. We used these maps to obtain curves of Aeff as a function of Sdemag, peak for the five FFs.

We ran 1000 realizations of the number counts per lens model per cluster field. Each was computed from a simulated source catalog using the same centroid coordinates and observed source sizes as the 29 ALMA detections. Simulated catalogs had, for each source i, integrated flux densities Sobs, i and peak intensities Sobs, peak, i drawn from Gaussian distributions centered in the measured values (see Table 2), and redshifts zi drawn from the values and distributions selected in Sect. 2.2. From the simulated Sobs, i and Sobs, peak, i values, we obtained completeness Ci (for each source size) and deboosting corrections, (S/N)i values (later considering only (S/N)i ≥ 4.5 sources) and fractions of spurious sources pfalse, i. This was done by interpolating the curves shown in Figs. 2–4.

From the source centroid pixel coordinates and zi, and randomly selecting one of the mass reconstructions provided by the lens model, we obtained each source magnification, μi. We neglected the effects of differential magnification, as all our detections lied far enough away (i.e., more than four synthesized beams away) from critical lines in most of the adopted redshifts and lens models5. With all these quantities, we obtained demagnified integrated flux densities Sdemag, i = Sobs, i/μi, demagnified peak intensities Sdemag, peak, i = Sobs, peak, i/μi, and source effective areas Aeff, i. The latter was computed by interpolating the Aeff versus Sdemag, peak curves that corresponded to the same mass reconstruction used to obtain μi.

We then computed differential (cumulative) number counts at each demagnified flux density bin (limit) as

d N d log ( S ) = 1 Δ log ( S ) ∑ i X i $$ \begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}\log (S)}=\frac{1}{\Delta \log (S)}\sum \limits _{i}X_i \end{aligned} $$(1)

and

N ( > S ) = ∑ i X i , $$ \begin{aligned} N(>S)=\sum \limits _{i}X_i, \end{aligned} $$(2)

where we summed the contribution Xi to the counts by the sources with Sdemag, i within (larger than) that flux density bin (limit), and

X i = 1 − p false , i C i A eff , i . $$ \begin{aligned} X_i=\frac{1-p_{\mathrm{false},i}}{C_i\,A_{\mathrm{eff},i}}. \end{aligned} $$(3)

We finally obtained combined differential (cumulative) counts, both in each separate cluster and including the five FFs, as the median values per flux density bin (limit). We computed the associated uncertainties using the 16th and 84th percentiles, added in quadrature with scaled Poisson confidence levels for 1σ lower and upper limits (Gehrels 1986).

We also used our Monte Carlo simulation realizations to obtain median estimates and uncertainties (from the 16th and 84th percentiles) in several source properties, as done in Paper IV. Source magnifications, integrated flux densities (both observed and demagnified), and effective areas for the last two FFs are shown in Figs. 5–7. Median (combined) magnification values for these clusters range from 1.3 to 11. Within the uncertainties, combined magnifications range from one to 29. For most sources, the discrepancies in the median values predicted by different models exceed their individual uncertainties, as Fig. 5 shows. Median (combined) lensing-corrected flux densities range from ∼0.04 to 4.13 mJy. Within the uncertainties, combined demagnified flux densities cover around 2.6 orders of magnitude. In most of our cluster fields, our sources observed at low S/Ns have the largest magnifications, and thus the faintest observed sources are also the faintest intrinsically (see Fig. 6). Median (combined) effective areas range from ∼0.03 to 2.19 arcmin2. Within the uncertainties, combined effective areas cover around 2.3 orders of magnitude. Our faintest sources have the smallest effective areas (see Fig. 7).

thumbnail Fig. 5.

Median magnification per source for lens models (colored symbols), and also combining all models for each cluster field (large black circles). Error bars indicate the 16th and 84th percentiles. Values for each model have been offset around the source ID for clarity.

thumbnail Fig. 6.

Median demagnified integrated flux density as a function of observed integrated flux density. Median values are obtained combining all models for each cluster field. Error bars in demagnified fluxes correspond to the 16th and 84th percentiles, while for observed fluxes these are 1σ statistical uncertainties. As a reference, black lines indicate magnification values of one (solid), five (dotted), ten (dashed), and 50 (dot-dashed).

thumbnail Fig. 7.

Median effective area as a function of demagnified integrated flux density. Median values are obtained combining all models for each cluster field. Error bars correspond to the 16th and 84th percentiles. For comparing uncertainty values, both axes cover the same interval in order of magnitude. Within the errors, both demagnified flux densities and effective areas span around 2.6 and 2.3 orders of magnitude, respectively.

4. Results and discussion

4.1. Number counts

We show our differential and cumulative number counts at 1.1 mm in Fig. 8, combining all models and cluster fields. Our counts combining models for each cluster field separately and altogether are presented in Table A.1 and Fig. A.1. We present counts down to the flux density bin centered on 0.024 mJy, as this is the faintest flux density bin where at least one cluster field has nonzero combined differential counts at the 84th percentile. We find variations across lens models in the median differential counts per flux bin up to ≈0.6 dex (although consistent within the error bars). Uncertainties coming from our Monte Carlo simulations differ by a factor of ∼0.04 − 1.65 (∼0.04 − 3.58) from that predicted from Poisson statistics in the differential (cumulative) counts.

thumbnail Fig. 8.

Differential (top) and cumulative (bottom) counts at 1.1 mm compared to ALMA results and galaxy evolution model predictions from the literature. Our counts (large, black, filled circles) correspond to median values combining all models for all cluster fields. Error bars indicate the 16th and 84th percentiles, adding the scaled Poisson confidence levels for 1σ lower and upper limits, respectively, in quadrature. Arrows indicate 3σ upper limits for flux densities with zero median counts and nonzero values at the 84th percentile. We show our best double power-law fit as a black solid line (see Sect. 4.5). We display the counts we derived for the first three FF clusters using PB > 0.5 (Muñoz Arancibia et al. 2019, Paper IV Corrigendum) as large, gray, open circles. We show previous results reported by Ono et al. (2014) as dark magenta squares, Carniani et al. (2015) as dark teal diamonds, Fujimoto et al. (2016) as magenta triangles, Hatsukade et al. (2016) as blue crosses, Aravena et al. (2016) as purple squares, Umehata et al. (2018) as dark magenta diamonds, Franco et al. (2018) as dark teal triangles, Hatsukade et al. (2018) as magenta crosses, González-López et al. (2020) as blue squares (with their best fit from a P(D) analysis shown as a black dashed line), Béthermin et al. (2020) as purple diamonds, Gómez-Guijarro et al. (2022) as dark magenta triangles, Chen et al. (2023) as dark teal crosses, and Dunlop et al. (2017) as a black, dot-dashed curve. We show number counts predicted by the galaxy evolution models from Cowley et al. (2015) (yellow line), Béthermin et al. (2017) (orange line), Schreiber et al. (2017) (red line), Popping et al. (2020) (dark red line), Lagos et al. (2020) (light magenta line), and Hayward et al. (2021) (light blue line). We scale the counts derived at other wavelengths as S1.1 mm = 1.59 × S1.3 mm, S1.1 mm = 1.27 × S1.2 mm, and S1.1 mm = 0.51 × S850 μm, assuming a modified black body (see Sect. 4.3).

Combining all cluster fields, our differential counts span two orders of magnitude in demagnified flux density. However, at their limiting flux density bins (centered at 0.075 mJy and 2.371 mJy) we report only 3σ upper limits, as their median counts are zero but they have nonzero values at the 84th percentile. We note that at 0.024 mJy, only ∼8% of all realizations have nonzero counts, and thus combined differential counts are zero both in the median and the 84th percentile (thus explaining the absence of a data point at 0.024 mJy in Fig. 8, top panel). These realizations have a slight influence on the combined cumulative counts at 0.013 mJy, which are ≈0.1 dex higher than those at 0.042 mJy.

At 0.13 mJy and below, Monte Carlo-simulation-based uncertainties dominate the combined cumulative counts over Poisson errors. The adoption of different values for the image-plane correction factors (i.e., completeness, flux deboosting, and fraction of spurious sources) can systematically change the shape of the number counts as a whole; nevertheless, the main drivers of uncertainty in the faint-end number counts, for an individual lens model in the Monte Carlo simulations, are the uncertainties in the source-plane effective areas. For each ALMA detection, these are given by the uncertainties in their demagnified intensities (which in turn are given by those in both source magnification and observed intensity), but also by the steepness of the effective area versus demagnified peak intensity curve. The slope of this curve may vary with lens model, source redshift, and demagnified peak intensity regime (see Paper IV). The steeper the curve, the larger the variation of Aeff with Sdemag, peak. The precisions in both intensity measurements and source magnifications given by the lens models play an important role when applying our method for computing number counts.

4.2. Cosmic variance

Number counts obtained in deep pencil beam surveys as the FF are limited by cosmic variance, that is, the uncertainty in observational estimates of the number density of extragalactic objects due to the underlying large-scale structure, which could become the dominant source of systematic error (e.g., Somerville et al. 2004; Trenti & Stiavelli 2008; Moster et al. 2011). We obtained 1σ cosmic variance estimates using three different approaches. First, we derived the field-to-field variance directly from our cumulative counts, computing the variance from the median counts per cluster field down to 0.013 mJy. Dividing its square root by the number of clusters and the corresponding median combined counts, we obtained a relative uncertainty of 17%. Secondly, we obtained an approximate cosmic variance using the expression determined by Driver & Robotham (2010) from galaxies in the Sloan Digital Sky Survey Data Release 7 and generalized over all redshifts. We adopted 2.28′×2.28′ as the (observed) survey area for each of our cluster fields, obtained comoving transverse lengths for these values at z = 2, and used them in their Eq. (4) for one line of sight. We obtained a cosmic variance of 31% (20%) for the comoving volume contained in the range z = 1 − 3 (z = 0 − 4). Adopting a number of independent volumes equal to our number of cluster fields gave a cosmic variance of 14% and 9% for said redshift ranges. We note that Driver & Robotham (2010) assumed the same cosmological parameters adopted in this work.

Finally, we obtained another estimate using the Cosmic Variance Calculator v1.036 by Trenti & Stiavelli (2008), where cosmic variance is computed via the two-point correlation function in extended Press-Schechter theory. Adopting a survey area of 2.28′×2.28′, mean z = 2, an intrinsic number of objects of four (as this is the median number of sources in our combined cumulative counts down to 0.013 mJy), a halo filling factor of 1.0, completeness of 1.0, rms matter fluctuation today in linear theory σ8 = 0.9, and a Sheth-Tormen bias, we obtained a relative cosmic variance of 18% (14%) for the redshift interval z = 1 − 3 (z = 0 − 4). Adopting a survey area 11.4′×2.28′ (i.e., as if our five cluster fields were displayed contiguously) and an intrinsic number of objects of 29 (as this is the total number of sources) gave a cosmic variance of 13% and 10% for said redshift ranges. However, we note that some of the cosmological parameters used by this calculator are slightly different (Ωm = 0.26, ΩΛ = 0.74) to those we adopted.

At our faintest flux limit, uncertainties in the median combined cumulative counts coming from our Monte Carlo simulations are in the 75–281% range, while those from Poisson statistics are in the 48–79% range. From these we note that cosmic variance estimates from the three approaches mentioned before are all exceeded by these uncertainties.

Popping et al. (2020) successfully reproduced the ASPECS-LP number counts (González-López et al. 2020) using a semi-empirical model. They estimated the cosmic variance for the ASPECS-LP survey using 100 random sub-areas of their simulated sky, each covering an area the size of the ASPECS-LP region. They found that below 1 mJy, the typical 2σ scatter due to field-to-field variance is a factor of 1.5 at 1.1 mm, while sources brighter than 1 mJy are typically missed by surveys of this angular size. Moreover, their model predicts that sources fainter than 1 mJy mostly have stellar masses below 1011 M⊙, while sources brighter than 1 mJy mostly have stellar masses above 5 × 1010 M⊙. Our cumulative counts have a notable agreement with both the observed and modeled counts in ASPECS-LP (see Sects. 4.3 and 4.4). Therefore, from their estimates we expect the bright end of our number counts (> 1 mJy) to be more affected by cosmic variance, missing a fraction of the objects with stellar mass in the 5 × 1010 − 1011 M⊙ range and most of the objects at > 1011 M⊙ that a wider area survey may detect.

4.3. Comparison to previous works

We compare our results with the counts we derived using the first three FFs and adopting PB > 0.5 (see Paper IV Corrigendum). Combining five FFs using the PB > 0.3 region and incorporating new spectroscopic redshifts for five sources in A2744, current counts are consistent with our previous values to 1σ, showing a similar flattening toward fainter flux densities. Our median differential (cumulative) counts change by a factor 0.6–1.8 (0.6–2.6). Uncertainties in the differential (cumulative) counts obtained from the Monte Carlo simulations change by a factor 0.6–4.0 (0.3–6.5), while scaled Poisson uncertainties change by a factor 0.8–1.4 (0.7–1.7). We note that in Paper IV, combined counts below 0.133 mJy were contributed by two sources from A2744 and one from MACSJ1149. Now two sources from A370 and two from AS1063 are added to this flux density regime (see Fig. 7), and the flattening in the counts derived from them reinforces our previous results (see also Fig. 9).

thumbnail Fig. 9.

Differential (left) and cumulative (right) counts at 1.1 mm for different assumptions regarding the image-plane source scale radii (half-light radii) for low-significance sources, namely, a) adopting r eff , obs = 0 . ″ 23 $ r_{\mathrm{eff,obs}}=0{{\overset{\prime\prime}{.}}}23 $ (black filled circles, fiducial), b) assuming they are point sources (dark magenta filled diamonds), and c) adopting r eff , obs = 0 . ″ 5 $ r_{\mathrm{eff,obs}}=0{{\overset{\prime\prime}{.}}}5 $ (dark teal filled squares). Our counts correspond to median values. Error bars indicate the 16th and 84th percentiles, adding the scaled Poisson confidence levels for 1σ lower and upper limits, respectively, in quadrature. Arrows indicate 3σ upper limits for flux densities with zero median counts and nonzero values at the 84th percentile. We show previous results reported by Fujimoto et al. (2016) as magenta triangles, Aravena et al. (2016) as purple squares, and González-López et al. (2020) as blue squares. We show number counts predicted by the galaxy evolution models from Cowley et al. (2015) (yellow line), Béthermin et al. (2017) (orange line), Schreiber et al. (2017) (red line), Popping et al. (2020) (dark red line), Lagos et al. (2020) (light magenta line), and Hayward et al. (2021) (light blue line). We scale the counts derived at other wavelengths as in Fig. 8.

As in Paper IV, we explore the effect of adopting different source redshifts in the predicted counts, considering the following cases for all detections: a) assuming a Gaussian redshift distribution centered at z = 2 ± 0.5, b) adopting exactly z = 2, c) assuming a uniform redshift distribution between the cluster redshift and z = 4, and d) assuming a Gaussian redshift distribution centered at z = 3 ± 0.5. Combining all clusters, our fiducial counts are consistent with those obtained in all these cases, with variations in the median values up to ≈0.1 dex. Case d), however, adds a 3σ upper limit at 0.024 mJy.

We compare our combined counts to results from recent ALMA observations that probe down to the sub-milliJansky level, as well as to predictions by state-of-the-art galaxy evolution models. To the literature data already compared in Paper IV at 1.1–1.3 mm (for details, see Paper IV and references therein), we add number counts derived from seven new dedicated surveys and three new models: a 20 arcmin2 survey at 1.1 mm that contains a z = 3.09 protocluster (ALMA deep field in SSA22; Umehata et al. 2018); a 69 arcmin2 survey at 1.1 mm in the deepest HST-WFC3 H-band part of the GOODS-South field (GOODS-ALMA; Franco et al. 2018); a 26 arcmin2 survey at 1.2 mm in the GOODS-South field (ASAGAO; Hatsukade et al. 2018); a 4.2 arcmin2 survey at 1.2 mm in the HUDF (ASPECS-LP; González-López et al. 2020); a 25 arcmin2 serendipitous survey at 850 μm around a sample of spectroscopically confirmed star-forming galaxies in the COSMOS and CDF-South fields (ALPINE; Béthermin et al. 2020); a 72 arcmin2 survey at 1.1 mm in the GOODS-South field, including both small and large spatial scales (GOODS-ALMA 2.0; Gómez-Guijarro et al. 2022); a 0.3 deg2 serendipitous survey at 1.2 mm toward 1001 ALMA calibrators (ALMACAL; Chen et al. 2023); a semi-empirical model for the dust continuum number counts of galaxies (Popping et al. 2020); a semi-analytic model of galaxy formation that includes Band 6 number count predictions (Lagos et al. 2020); and a cosmological hydrodynamical simulation that predicts 850 μm number counts (Hayward et al. 2021). Since we include counts derived at wavelengths other than 1.1 mm in our comparison, we scale their estimates as S1.1 mm = 1.59 × S1.3 mm, S1.1 mm = 1.27 × S1.2 mm, and S1.1 mm = 0.51 × S850 μm. These factors are obtained assuming a modified black-body model with a dust emissivity index of β = 1.5, dust temperature of Tdust = 35 K, and z = 2.5, which are values typically found for dusty galaxies (e.g., Swinbank et al. 2014).

Our combined counts are consistent within 1σ with most of the previous ALMA studies and galaxy evolution model predictions, as shown in Fig. 8. Moreover, our differential counts are consistent within ≈1.6σ with those of Hatsukade et al. (2016) at 0.25 mJy, and our cumulative counts are consistent within ≈1.1σ with those of Fujimoto et al. (2016) at 0.13 mJy. At 0.04 mJy, however, our median cumulative counts are lower than the estimates of both Fujimoto et al. (2016) and Aravena et al. (2016) by ≈0.4 dex.

Extrapolating the cumulative counts of Fujimoto et al. (2016) to 0.01 mJy, our combined counts disagree beyond 1σ. This difference may arise from a combination of several factors, including the assumed redshift distribution for sources contributing to counts at these flux densities (a Gaussian centered at z = 2 ± 0.5 by us versus adopting exactly z = 2.5 by Fujimoto et al. 2016), depth in cluster ALMA data (55 − 71 μJy beam−1 versus 38 − 41 μJy beam−1), and lens modeling uncertainties. We note that our cumulative counts in A2744 agree within 1σ when extrapolating their data to 0.01 mJy (see Fig. 9). Together with the different number of cluster fields (five versus one), this suggests that field-to-field variance may play a role in the difference with respect to our combined counts (see also Sect. 4.2).

Since we have assumed a given image-plane source size for low-significance detections (11/29 of our sample), we explore the effect of varying this source size by testing two extreme cases. As in Paper IV, these are a) assuming these are point sources; and b) adopting an observed effective radius of reff, obs = 0.5″, which gives integrated flux densities of 4.5 ≤ S/N < 5 sources scaling the peak intensities by the new ratios 0.46 and 0.35 in A370 and AS1063, respectively. These estimates for both differential and cumulative number counts are presented in Fig. 9, for each cluster field separately and altogether, compared to our fiducial case.

When testing these two cases, median counts combining all clusters change up to ≈0.3 dex compared to the fiducial case and are found to be consistent within 1σ with galaxy evolution models. Under the assumption that low-significance detections are point sources, median cumulative counts are consistent with those of Fujimoto et al. (2016) within 1σ at 0.42 mJy and brighter, lower by ≈0.6 dex at 0.13 mJy (consistent within ≈1.9σ), and lower by ≈0.6 dex at 0.04 mJy (consistent within ≈1.3σ). They are consistent with the results of Aravena et al. (2016) within 1σ down to 0.13 mJy and within ≈1.1σ at 0.04 mJy. They agree within 1σ with the results of González-López et al. (2020). Assuming reff, obs = 0.5″ for 4.5 ≤ S/N < 5 sources, median cumulative counts agree within 1σ with the results of Fujimoto et al. (2016), Aravena et al. (2016), and González-López et al. (2020).

4.4. Number count flattening

In line with results from Paper IV, our findings continue to suggest a flattening in the number counts below ≈0.1 mJy. This flux density is consistent within ≈2.6σ with the parameterization we provide in Sect. 4.5 ( S 0 = 3 . 6 − 2.3 + 2.5 × 10 − 2 mJy $ S_0=3.6_{-2.3}^{+2.5}\times10^{-2}\,\mathrm{mJy} $). Popping et al. (2020) predicted the dust continuum flux density of model galaxies by coupling an observationally motivated model for their stellar-mass and star-formation-rate (SFR) distributions with empirical scaling relations. Their model reproduces the flattening in the counts below 0.3 mJy found by González-López et al. (2020). We compare our results to the cumulative number counts of Popping et al. (2020) at 1.1 mm for their entire simulated light cone down to ∼10−3 mJy. We highlight the remarkable agreement between our median cumulative counts and the González-López et al. (2020) observational estimates (both their direct number counts and best fit from a P(D) analysis, and even extrapolating their cumulative counts to 0.01 mJy), as well as with the Popping et al. (2020) predictions.

Popping et al. (2020) also presented model number counts decomposed in bins of redshift, stellar mass, SFR, and dust mass. In this scenario, main contributors to 1.1 mm cumulative number counts in the 0.01 − 0.1 mJy range are galaxies that lie at z = 1 − 2 and have a stellar mass in the range of 5 × 109 − 1010 (1010 − 5 × 1010) M⊙ below (above) 0.03 mJy, SFR in the range of 10 − 50 M⊙ yr−1, and a dust mass in the range of 107 − 108 (108 − 109) M⊙ below (above) 0.08 mJy. Their predictions suggest that the number count flattening is driven by the shape (specifically, the “knee” at S1.1 mm ≈ 0.2 − 0.5 mJy, and shallow faint-end slope) of the (sub)millimeter luminosity function at z = 1 − 2. We note, however, that their model does not account for gravitational lensing effects. Moreover, only one of the sources that contribute to our faint-end number counts has a reliable counterpart (AS1063-ID04, see Table 2 and Fig. 6), preventing a detailed comparison with the results of Popping et al. (2020).

Further characterization of the physical properties of faint-end sources in the five FFs (e.g., using SED fitting and/or models to estimate their photometric redshift, stellar mass, and SFR, from ALMA data plus non-detections at other wavelengths) may help to constrain the validity of these predictions. This may also be aided via assembling a source catalog from a larger sample of cluster fields at millimeter wavelengths. One such work in progress is the ALMA Lensing Cluster Survey (ALCS; Kohno et al., in prep.), as among its goals is deriving number counts toward 33 ALMA-observed galaxy clusters (Fujimoto et al., in prep.).

4.5. Extragalactic background light

Using the Monte Carlo realizations of our differential number counts, we computed the contribution to the extragalactic background light (EBL) provided by each of them by adding up the contribution contained in each flux bin. We found a median contribution of 14 . 3 − 7.4 + 12.2 $ 14.3_{-7.4}^{+12.2} $ ( 7 . 5 − 3.9 + 6.4 $ 7.5_{-3.9}^{+6.4} $) Jy deg−2 resolved in our demagnified sources at 1.1 mm down to 0.013 (0.133) mJy; here, uncertainties were computed from the 16th and 84th percentiles.

We compare our estimates to that by the Planck Collaboration (using their best-fit extended halo model; see Planck Collaboration XXX 2014) and a recent determination based on COBE/FIRAS and Planck observations (Odegard et al. 2019). In both cases, we first found their EBL values at 263.14 GHz (the set local oscillator frequency for our observations) through interpolation of their data. At this frequency, we obtained EBL estimates of 19 . 1 − 0.7 + 0.8 Jy deg − 2 $ 19.1_{-0.7}^{+0.8}\,\mathrm{Jy}\,\mathrm{deg}^{-2} $ and 16 . 8 − 6.1 + 9.6 Jy deg − 2 $ 16.8_{-6.1}^{+9.6}\,\mathrm{Jy}\,\mathrm{deg}^{-2} $ for the Planck Collaboration XXX (2014) and Odegard et al. (2019), respectively. The contribution provided by our demagnified sources at 1.1 mm down to 0.013 (0.133) mJy represents 74 . 5 − 38.4 + 64.0 % $ 74.5_{-38.4}^{+64.0}\% $ ( 38 . 9 − 20.1 + 33.9 % $ 38.9_{-20.1}^{+33.9}\% $) of the EBL by Planck Collaboration XXX (2014) and 86 . 4 − 50.0 + 120.4 % $ 86.4_{-50.0}^{+120.4}\% $ ( 45 . 6 − 26.2 + 57.3 % $ 45.6_{-26.2}^{+57.3}\% $) of that by Odegard et al. (2019). We note that both EBL estimates chosen for comparison involved several assumptions, for instance, when calibrating the brightness scale and subtracting the contribution of both cosmic microwave background and Galactic foreground emission, which may increase the uncertainties in the quoted EBL beyond reported values.

In order to estimate the EBL derived from our counts at 1.1 mm for any flux density limit, we fitted a double power-law function (e.g., Scott et al. 2002) to our differential number counts. This is given by

ϕ ( S ) d S = N 0 S 0 [ ( S S 0 ) α + ( S S 0 ) β ] − 1 d S , $$ \begin{aligned} \phi (S)\mathrm{d}S=\frac{N_0}{S_0}\left[\left(\frac{S}{S_0}\right)^{\alpha }+\left(\frac{S}{S_0}\right)^{\beta }\right]^{-1}\mathrm{d}S, \end{aligned} $$(4)

where N0, S0, α, and β are free parameters, and the integrated flux density down to a flux limit Slim can be computed as

S integ = ∫ S lim ∞ S ϕ ( S ) d S . $$ \begin{aligned} S_{\mathrm{integ}}=\int _{S_{\mathrm{lim}}}^{\infty }S\phi (S)\mathrm{d}S. \end{aligned} $$(5)

The logarithmic form of Eq. (4) is given by

n ( S ) d log ( S ) = ln ( 10 ) N 0 S S 0 [ ( S S 0 ) α + ( S S 0 ) β ] − 1 d log ( S ) , $$ \begin{aligned} n(S)\mathrm{d}\log (S)=\ln (10)\frac{N_0S}{S_0}\left[\left(\frac{S}{S_0}\right)^{\alpha }+\left(\frac{S}{S_0}\right)^{\beta }\right]^{-1}\mathrm{d}\log (S), \end{aligned} $$(6)

with n(S)dlog(S) = ϕ(S)dS. We considered the ranges of N0 = 103 − 106 mJy deg−2, S0 = 0.01 − 1 mJy, α = 0 − 5, and β = −1 to 1. We fitted Eq. (6) to each realization of our differential number counts, including only flux density bins that have nonzero values (thus allowing us to extrapolate our counts to flux densities outside the regime covered by our ALMA sources).

From these fits, we found median parameters and associated uncertainties (from the 16th and 84th percentiles) of N 0 = 2 . 7 − 1.7 + 5.0 × 10 5 mJy deg − 2 $ N_0=2.7_{-1.7}^{+5.0}\times10^5\,\mathrm{mJy}\,\mathrm{deg}^{-2} $, S 0 = 3 . 6 − 2.3 + 2.5 × 10 − 2 mJy $ S_0=3.6_{-2.3}^{+2.5}\times10^{-2}\,\mathrm{mJy} $, α = 2 . 8 − 0.8 + 1.0 $ \alpha=2.8_{-0.8}^{+1.0} $, and β = 0 . 6 − 1.6 + 0.4 $ \beta=0.6_{-1.6}^{+0.4} $. Using Eq. (5) with the parameters found for each realization and Slim = 0 mJy, we estimated a median total EBL associated with 1.1 mm galaxies in the FFs of 21 . 2 − 9.6 + 40.4 Jy deg − 2 $ 21.2_{-9.6}^{+40.4}\,\mathrm{Jy}\,\mathrm{deg}^{-2} $. For Slim = 0.013 (0.133) mJy, we obtained a median contribution of 18 . 8 − 8.8 + 40.4 $ 18.8_{-8.8}^{+40.4} $ ( 5 . 0 − 3.1 + 42.1 $ 5.0_{-3.1}^{+42.1} $) Jy deg−2, corresponding to 94 . 2 − 14.8 + 5.3 % $ 94.2_{-14.8}^{+5.3}\% $ ( 27 . 3 − 17.0 + 63.0 % $ 27.3_{-17.0}^{+63.0}\% $) of our estimate of the total EBL. This suggests that we may be resolving most of the EBL at 1.1 mm down to 0.013 mJy.

5. Concluding remarks

We estimated lensing-corrected number counts at 1.1 mm. By adding S/N ≥ 4.5 ALMA detections in two FF galaxy clusters, A370 and AS1063, we extended the study conducted in Paper IV with A2744, MACSJ0416, and MACSJ1149. The combination of these five FF clusters covered a total observed area of ∼26 arcmin2 (PB > 0.3 region), which was reduced once lens models were applied (e.g., by ∼2.7 times after applying the CATS v4 model for a source-plane z = 2).

Based on 29 ALMA detections over a PB > 0.3 region, our counts combining all cluster fields span two orders of magnitude in demagnified flux density. They are consistent with the counts we obtained for the first three FFs using the PB > 0.5 region to 1σ (see Paper IV Corrigendum). They also agree within 1σ with most of the previous estimates using both ALMA observations and galaxy evolution model predictions. Below ≈0.1 mJy, however, our cumulative number counts are lower by ≈0.4 dex compared to deep ALMA studies except for ASPECS-LP, supporting the flattening in the counts we already reported in Paper IV. Also, at this flux density and fainter, uncertainties coming from our Monte Carlo simulations dominate the cumulative counts over Poisson uncertainties.

We find systematic variations in the median differential counts across lens models up to ≈0.6 dex, although these are consistent within the error bars. For each individual lens model in our Monte Carlo simulations, the main drivers of uncertainty in the faint-end number counts are the uncertainties in the source-plane effective areas, which are dependent on source demagnified intensities. This highlights the importance, when applying our method, of having both precise intensity measurements and precise source magnifications given by the lens models.

Cosmic variance estimates are all exceeded by uncertainties in our median combined cumulative counts coming from both our Monte Carlo simulations and Poisson statistics. Using the Monte Carlo realizations of our differential number counts directly, we derive a median contribution to the EBL of 14 . 3 − 7.4 + 12.2 $ 14.3_{-7.4}^{+12.2} $ ( 7 . 5 − 3.9 + 6.4 $ 7.5_{-3.9}^{+6.4} $) Jy deg−2 resolved in our demagnified sources at 1.1 mm down to 0.013 (0.133) mJy. Using double power-law function fits to these realizations instead, we obtain a median contribution of 18 . 8 − 8.8 + 40.4 $ 18.8_{-8.8}^{+40.4} $ ( 5 . 0 − 3.1 + 42.1 $ 5.0_{-3.1}^{+42.1} $) Jy deg−2; comparing this contribution to our estimate of the total EBL associated with 1.1 mm galaxies in the FFs suggests that we may be resolving most of the EBL at this wavelength down to 0.013 mJy.

Among current number count estimates using ALMA data, our work is the one based on the largest set of ALMA-observed galaxy clusters reported to date. This number will increase once number counts from surveys such as ALCS are released.

1

The half-light size of a source is given by 2.35 times its effective radius.

3

We note that the FF website released corrected Sharon v4 models after acceptance of Paper IV. In this work (as well as in the Paper IV Corrigendum), we used updated models.

4

On top of those listed, Zitrin & Merten (e.g., Zitrin et al. 2015) also supplied lens models for the FF clusters, but their latest update was v3, so these were not used here.

5

However, see Paper IV for a discussion about A2744-ID09 and A2744-ID11.

Acknowledgments

We gratefully acknowledge support from ANID through FONDECYT Postdoctoral Fellowship 3160776 (A.M.M.A.); FONDECYT Regular 1171710 (A.M.M.A., E.I.), 1221846 (E.I.), 1190818 (F.E.B.), 1200495 (F.E.B.), 1190335 (T.A.); Millennium Science Initiative Program – ICN12_009 (F.E.B., T.A.), CATA-Basal – AFB-170002 (F.E.B., R.D.), ACE210002 (F.E.B., R.D.) and FB210003 (F.E.B., T.A., R.D.); Grant No. 2020750 from the United States-Israel Binational Science Foundation and Grant No. 2109066 from the United States National Science Foundation, and by the Ministry of Science & Technology, Israel (A.Z.). A.M.M.A. acknowledges the AstroHackWeek 2018 event for helpful discussions that contributed to improve the code used in our analysis. The ALMA observations were carried out under programs ADS/JAO.ALMA#2013.1.00999.S, ADS/JAO.ALMA#2015.1.01425.S, and ADS/JAO.ALMA#2017.1.01219.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work is based on data and catalog products from HFF-DeepSpace, funded by the National Science Foundation and Space Telescope Science Institute (operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555). This work utilizes gravitational lensing models produced by PIs Bradač, Natarajan & Kneib (CATS), Merten & Zitrin, Sharon, Williams, Keeton, Bernstein and Diego, and the GLAFIC group. This lens modeling was partially funded by the HST Frontier Fields program conducted by STScI. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. The lens models were obtained from the Mikulski Archive for Space Telescopes (MAST).

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Appendix A: Number counts for each Frontier Fields galaxy cluster

Here, we present our differential and cumulative number counts at 1.1 mm, combining models for each cluster field separately and together in Table A.1. These data are shown in Fig. A.1, together with number counts obtained for each lens model.

thumbnail Fig. A.1.

Differential (left) and cumulative (right) counts at 1.1 mm (large black circles) for each cluster (see legends in the top left) and combining all cluster fields (bottom panel). In the first five panels, we also show counts for each lens model (colored symbols) offset in flux around the combined counts for clarity. All values correspond to median counts. Error bars indicate the 16th and 84th percentiles, adding the scaled Poisson confidence levels for 1σ lower and upper limits, respectively, in quadrature. Arrows indicate 3σ upper limits for flux densities having zero median counts and nonzero values at the 84th percentile.

Table A.1.

Demagnified 1.1 mm number counts.

All Tables

Table 1.

ALMA mosaic properties of galaxy clusters.

In the text
Table 2.

Continuum detections at S/N ≥ 4.5.

In the text
Table 3.

Lensing models considered in this work.

In the text
Table A.1.

Demagnified 1.1 mm number counts.

In the text

All Figures

thumbnail Fig. 1.

ALMA continuum maps at 1.1 mm for five FF clusters, made with natural weighting. Images are displayed without PB sensitivity corrections. Color scale goes from blue to red, corresponding to −5σ to 5σ. A white contour in each map encloses the region with PB > 0.3, that is, where the PB sensitivity is at least 30% of the peak sensitivity. Source positions are displayed in blue as squares (high significance) or circles (low significance). A small box in the bottom left corner of each panel shows the synthesized beam (see parameters in Table 1). Green lines indicate the critical curves (i.e., infinite magnification) for a z = 2 source plane in the Diego v4.1 lens model (see references in Table 3).
In the text
thumbnail Fig. 2.

Completeness correction C as a function of image-plane integrated flux density and separated in bins of image-plane scale radius (half-light radius). Error bars indicate binomial confidence intervals.
In the text
thumbnail Fig. 3.

Deboosting correction as a function of S/N. We display the ratio between the extracted and injected flux densities for our simulated sources as light magenta dots. Thick dark magenta lines correspond to median values, while thin dark magenta lines indicate the 16th and 84th percentiles.
In the text
thumbnail Fig. 4.

Fraction of spurious sources at a given S/N. We display curves for A2744, MACSJ0416, MACSJ1149, A370, and AS1063 in dark magenta, dark teal, magenta, teal, and purple, respectively. A vertical dotted line indicates our S/N threshold of 4.5.
In the text
thumbnail Fig. 5.

Median magnification per source for lens models (colored symbols), and also combining all models for each cluster field (large black circles). Error bars indicate the 16th and 84th percentiles. Values for each model have been offset around the source ID for clarity.
In the text
thumbnail Fig. 6.

Median demagnified integrated flux density as a function of observed integrated flux density. Median values are obtained combining all models for each cluster field. Error bars in demagnified fluxes correspond to the 16th and 84th percentiles, while for observed fluxes these are 1σ statistical uncertainties. As a reference, black lines indicate magnification values of one (solid), five (dotted), ten (dashed), and 50 (dot-dashed).
In the text
thumbnail Fig. 7.

Median effective area as a function of demagnified integrated flux density. Median values are obtained combining all models for each cluster field. Error bars correspond to the 16th and 84th percentiles. For comparing uncertainty values, both axes cover the same interval in order of magnitude. Within the errors, both demagnified flux densities and effective areas span around 2.6 and 2.3 orders of magnitude, respectively.
In the text
thumbnail Fig. 8.

Differential (top) and cumulative (bottom) counts at 1.1 mm compared to ALMA results and galaxy evolution model predictions from the literature. Our counts (large, black, filled circles) correspond to median values combining all models for all cluster fields. Error bars indicate the 16th and 84th percentiles, adding the scaled Poisson confidence levels for 1σ lower and upper limits, respectively, in quadrature. Arrows indicate 3σ upper limits for flux densities with zero median counts and nonzero values at the 84th percentile. We show our best double power-law fit as a black solid line (see Sect. 4.5). We display the counts we derived for the first three FF clusters using PB > 0.5 (Muñoz Arancibia et al. 2019, Paper IV Corrigendum) as large, gray, open circles. We show previous results reported by Ono et al. (2014) as dark magenta squares, Carniani et al. (2015) as dark teal diamonds, Fujimoto et al. (2016) as magenta triangles, Hatsukade et al. (2016) as blue crosses, Aravena et al. (2016) as purple squares, Umehata et al. (2018) as dark magenta diamonds, Franco et al. (2018) as dark teal triangles, Hatsukade et al. (2018) as magenta crosses, González-López et al. (2020) as blue squares (with their best fit from a P(D) analysis shown as a black dashed line), Béthermin et al. (2020) as purple diamonds, Gómez-Guijarro et al. (2022) as dark magenta triangles, Chen et al. (2023) as dark teal crosses, and Dunlop et al. (2017) as a black, dot-dashed curve. We show number counts predicted by the galaxy evolution models from Cowley et al. (2015) (yellow line), Béthermin et al. (2017) (orange line), Schreiber et al. (2017) (red line), Popping et al. (2020) (dark red line), Lagos et al. (2020) (light magenta line), and Hayward et al. (2021) (light blue line). We scale the counts derived at other wavelengths as S1.1 mm = 1.59 × S1.3 mm, S1.1 mm = 1.27 × S1.2 mm, and S1.1 mm = 0.51 × S850 μm, assuming a modified black body (see Sect. 4.3).
In the text
thumbnail Fig. 9.

Differential (left) and cumulative (right) counts at 1.1 mm for different assumptions regarding the image-plane source scale radii (half-light radii) for low-significance sources, namely, a) adopting r eff , obs = 0 . ″ 23 $ r_{\mathrm{eff,obs}}=0{{\overset{\prime\prime}{.}}}23 $ (black filled circles, fiducial), b) assuming they are point sources (dark magenta filled diamonds), and c) adopting r eff , obs = 0 . ″ 5 $ r_{\mathrm{eff,obs}}=0{{\overset{\prime\prime}{.}}}5 $ (dark teal filled squares). Our counts correspond to median values. Error bars indicate the 16th and 84th percentiles, adding the scaled Poisson confidence levels for 1σ lower and upper limits, respectively, in quadrature. Arrows indicate 3σ upper limits for flux densities with zero median counts and nonzero values at the 84th percentile. We show previous results reported by Fujimoto et al. (2016) as magenta triangles, Aravena et al. (2016) as purple squares, and González-López et al. (2020) as blue squares. We show number counts predicted by the galaxy evolution models from Cowley et al. (2015) (yellow line), Béthermin et al. (2017) (orange line), Schreiber et al. (2017) (red line), Popping et al. (2020) (dark red line), Lagos et al. (2020) (light magenta line), and Hayward et al. (2021) (light blue line). We scale the counts derived at other wavelengths as in Fig. 8.
In the text
thumbnail Fig. A.1.

Differential (left) and cumulative (right) counts at 1.1 mm (large black circles) for each cluster (see legends in the top left) and combining all cluster fields (bottom panel). In the first five panels, we also show counts for each lens model (colored symbols) offset in flux around the combined counts for clarity. All values correspond to median counts. Error bars indicate the 16th and 84th percentiles, adding the scaled Poisson confidence levels for 1σ lower and upper limits, respectively, in quadrature. Arrows indicate 3σ upper limits for flux densities having zero median counts and nonzero values at the 84th percentile.
In the text

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