ALMA Census of FIR Size and Luminosity

Demonstrating A New Census of INfrared Galaxies with ALMA (DANCING-ALMA). I. FIR Size and Luminosity Relation at z = 0-6 Revealed with 1034 ALMA Sources

Abstract

We present the large statistics of the galaxy effective radius in the rest-frame far-infrared (FIR) wavelength obtained from 1627 Atacama Large Millimeter/submillimeter Array (ALMA) 1-mm band maps that become public by 2017 July. Our ALMA sample consists of 1034 sources with the star-formation rate and the stellar mass at . We homogeneously derive and FIR luminosity of our ALMA sources via the -visibility method with the exponential disk model, carefully evaluating selection and measurement incompletenesses by realistic Monte-Carlo simulations. We find that there is a positive correlation between and at the significance level. The best-fit power-law function, , provides , and shows that at a fixed decreases toward high redshifts. The best-fit and the redshift evolution of are similar to those of in the rest-frame UV (optical) wavelength () revealed by Hubble  Space  Telescope (HST) studies. We identify that our ALMA sources have significant trends of and , which suggests that the dusty starbursts take place in compact regions. Moreover, of our ALMA sources is comparable to of quiescent galaxies at 13 as a function of stellar mass, supporting the evolutionary connection between these two galaxy populations. We also investigate rest-frame UV and optical morphologies of our ALMA sources with deep HST images, and find that 3040% of our ALMA sources are classified as major mergers. This indicates that dusty starbursts are triggered not only by the major mergers but also the other mechanism(s).

Subject headings:
galaxies: formation — galaxies: evolution — galaxies: high-redshift
6

1. Introduction

The study of galaxy sizes provides key insights into the galaxy formation and the evolution. The galaxy sizes are defined by the effective radius of . The value in the rest-frame ultra-violet (UV) bands, , traces the area of the young star formation with the small amount of dust. The value in the rest-frame optical bands, , shows the old star regions that allow us to understand the positional star-formation histories and the stellar migrations. The value in the rest-frame far-infrared (FIR) bands, , indicates the active star-forming regions that are obscured by the large amount of dust. The size-luminosity relation and the size evolution in all UV, optical, and FIR wavelengths are important to understand the cosmic stellar mass assembly in galaxies complementary.

The and have been measured for the galaxies selected at the rest-frame UV and optical bands with Advanced Camera for Surveys (ACS) and Wide Field Camera 3 (WFC3), respectively, on-board Hubble Space Telescope (HST) at (e.g., Shen et al. 2003; Ferguson et al. 2004; Hathi et al. 2008; Oesch et al. 2010; Ono et al. 2013; van der Wel et al. 2014; Shibuya et al. 2015). One of the most extensive size studies is conducted by Shibuya et al. (2015) with the sample of 190,000 galaxies at based on the deep HST images. The study shows that and rest-frame UV luminosity () have a positive power-law correlation of , and that there is a evolution of . The best-fit power-low slope and the redshift evolution are explained by the disk-formation model (e.g., Fall 1983, 2002; Mo et al. 1998) and the dark matter halo assembly (e.g., Ferguson et al. 2004; Hathi et al. 2008), respectively.

The Atacama Large Millimeter/submillimeter Array (ALMA) enables us to conduct the measurements with the capabilities of the high angular resolution and sensitivity. Bright submillimeter (submm) galaxies (SMGs; 1 mJy) have been observed with the high-resolution () ALMA 870 m, and spatially resolved (Ikarashi et al. 2015; Simpson et al. 2015; Hodge et al. 2016). Recent studies obtain the of kpc. Compared to the values of 4.4 kpc for the SMGs (Chen et al. 2015), the recent results indicate that the intense starbursts occur in the very compact regions (Simpson et al. 2015). Tadaki et al. (2016) and Barro et al. (2016) also report that the most of the rest-frame FIR sizes are smaller than the rest-frame optical sizes based on the similar high-resolution ALMA 870 m observations for massive star-forming galaxies at .

In the studies for FIR galaxies fainter than the SMGs, Rujopakarn et al. (2016) perform ALMA 1.3 mm blind deep imaging in Hubble Ultra Deep Field (HUDF; see also Dunlop et al. 2017). This HUDF study identifies 16 sources with the flux range of mJy, and estimates to be kpc. Another approach is carried out by González-López et al. (2017) who observe three massive galaxy clusters of Hubble Frontier Fields (HFF) with the ALMA 1.1 mm band for gravitationally lensed sources behind the clusters. Twelve sources are identified with the intrinsic flux of mJy, and the intrinsic values are estimated to be kpc under the assumption that these twelve sources reside at . Although these results in HUDF and HFF extend the studies to the faint end, the FIR luminosity () relation is still not well determined. This is because the source number is not enough to perform a reliable size statistic.

The evolution is also the remaining issue. Although the previous ALMA studies newly evaluate the values in the wide range, the sample with a wide redshift range is also necessary to investigate the evolution. Moreover, the technique of the size and flux measurements are different among the previous studies. A homogeneous measurement based on a large dataset is required to compare the different redshift samples and determine the evolution.

In this paper, we make the dataset from the ALMA archive by 2017 July. The 1627 deep ALMA maps in Band 6/7 provide us the largest dataset of multi-field ALMA ever made, which is composed of 1614 maps from the independent fields and 13 maps from the 3 lensing clusters. We analyze this large dataset to investigate the relation and the evolution by a systematic -visibility based technique. This is the first challenge for the statistical investigation of the properties in the systematic way. In Section 2, we show the observations and the data reduction. Section 3 describes the method of the source extraction, the flux and size measurements, our simulations, and our sample properties for deriving the relation and the evolution. We report the results of the redshift distribution, the relation, and the evolution, comparing with the previous studies in Section 4. In Section 5, we discuss the physical origins of the dusty starbursts. A summary of this study is presented in Section 6.

Throughout this paper, we assume a flat universe with , , , and km s Mpc. We use magnitudes in the AB system (Oke & Gunn 1983).

2. Data and Reduction

2.1. Data

To accomplish a complete study, we make full use of ALMA archival data in cycles that became public by 2017 July. We collect 1627 continuum maps in Band 6/7 from the ALMA archival data in the regions of HUDF, HFF, Cosmic Evolution Survey (COSMOS; Scoville et al. 2007), Subaru/ Deep Survey (SXDS; Furusawa et al. 2008), and The Great Observatories Origins Deep Survey South (GOODS-S; Vanzella et al. 2005), that have rich multi-wavelength data. Tables 1 and 2 present the summary and the details, respectively, for those of 1627 continuum maps.

We divide these 1627 continuum maps into 10 sub-datasets based on the angular resolutions (Table 1), because the angular resolutions would make different systematics in the estimates. Here, we regard the angular resolutions as the circularized beam size given by

(1)

where and represent the full width half maximums (FWHMs) of the major- and minor-axis of the synthesized beam, respectively. We list up the 1627 continuum maps with the observing details in Table 2.

Note that the 1627 continuum maps are taken in two types of the observation modes that are single pointing observations and mosaic observations with several pointing. We refer the single pointing and mosaic observations as ”single-field data” and ”mosaic data”, respectively, that are presented in Table 2.

Sub-Dataset Number of Maps
(1) (2) (3)
SB1 0 02 48
SB2 02 04 303
SB3 04 06 394
SB4 06 08 203
SB5 08 10 254
SB6 10 12 154
SB7 12 14 228
SB8 14 16 11
SB9 16 18 18
SB10 18 20 14

Notes: (1): ALMA maps with the circularized beam sizes from 0 to with a step of are referred to as SB1 to SB10, respectively. (2): Criterion of the range for each sub-dataset. (3): Number of the ALMA maps in each sub-dataset.

Table 1Summary of Our ALMA Maps
Map ID Project ID Target (Band) Ant. Dist. Beam Size
(mm) (GHz) (m) (mJy beam) ()
(1) (2) (3) (4) (5) (6) (7) (8)
Single-Field Data
1 2011.1.00097.S COSMOSLowz640 0.88 342 (7) 21384 0.45 0.58 0.42 0.49 (SB3)
2 2011.1.00097.S COSMOSLowz641 0.88 342 (7) 21384 0.44 0.59 0.42 0.49 (SB3)
3 2011.1.00097.S COSMOSLowz6410 0.88 342 (7) 21384 0.47 0.61 0.41 0.5 (SB3)
4 2011.1.00097.S COSMOSLowz6411 0.88 342 (7) 21384 0.46 0.61 0.42 0.5 (SB3)
5 2011.1.00097.S COSMOSLowz6412 0.88 342 (7) 21384 0.5 0.62 0.41 0.5 (SB3)
6 2011.1.00097.S COSMOSLowz6413 0.88 342 (7) 21384 0.47 0.62 0.41 0.5 (SB3)
7 2011.1.00097.S COSMOSLowz6414 0.88 342 (7) 21384 0.48 0.62 0.42 0.51 (SB3)
8 2011.1.00097.S COSMOSLowz6415 0.88 342 (7) 21384 0.46 0.63 0.41 0.5 (SB3)
9 2011.1.00097.S COSMOSLowz6416 0.88 342 (7) 21384 0.47 0.63 0.41 0.5 (SB3)
10 2011.1.00097.S COSMOSLowz6417 0.88 342 (7) 21384 0.53 0.64 0.42 0.51 (SB3)
11 2011.1.00097.S COSMOSLowz6418 0.88 342 (7) 21384 0.51 0.7 0.4 0.52 (SB3)
12 2011.1.00097.S COSMOSLowz6419 0.88 342 (7) 21384 0.49 0.71 0.4 0.53 (SB3)
13 2011.1.00097.S COSMOSLowz642 0.88 342 (7) 21384 0.47 0.59 0.42 0.49 (SB3)
14 2011.1.00097.S COSMOSLowz6420 0.88 342 (7) 21384 0.51 0.71 0.4 0.53 (SB3)
15 2011.1.00097.S COSMOSLowz6421 0.88 342 (7) 21384 0.51 0.71 0.4 0.53 (SB3)
16 2011.1.00097.S COSMOSLowz6422 0.88 342 (7) 21384 0.5 0.71 0.4 0.53 (SB3)
17 2011.1.00097.S COSMOSLowz6423 0.88 342 (7) 21384 0.53 0.71 0.4 0.53 (SB3)
18 2011.1.00097.S COSMOSLowz643 0.88 342 (7) 21384 0.47 0.59 0.42 0.49 (SB3)
19 2011.1.00097.S COSMOSLowz644 0.88 342 (7) 21384 0.45 0.59 0.42 0.49 (SB3)
20 2011.1.00097.S COSMOSLowz645 0.88 342 (7) 21384 0.44 0.59 0.42 0.49 (SB3)
21 2011.1.00097.S COSMOSLowz646 0.88 342 (7) 21384 0.47 0.61 0.42 0.5 (SB3)
22 2011.1.00097.S COSMOSLowz647 0.88 342 (7) 21384 0.46 0.61 0.42 0.5 (SB3)
23 2011.1.00097.S COSMOSLowz648 0.88 342 (7) 21384 0.47 0.61 0.42 0.5 (SB3)
24 2011.1.00097.S COSMOSLowz649 0.88 342 (7) 21384 0.5 0.61 0.41 0.5 (SB3)
25 2011.1.00097.S COSMOSlowz640 0.88 342 (7) 21382 0.28 0.58 0.47 0.52 (SB3)
26 2011.1.00097.S COSMOSlowz641 0.88 342 (7) 21382 0.28 0.58 0.47 0.52 (SB3)
27 2011.1.00097.S COSMOSlowz6410 0.88 342 (7) 21382 0.31 0.57 0.47 0.51 (SB3)
28 2011.1.00097.S COSMOSlowz6411 0.88 342 (7) 21382 0.31 0.57 0.47 0.51 (SB3)
29 2011.1.00097.S COSMOSlowz6412 0.88 342 (7) 21382 0.31 0.57 0.48 0.52 (SB3)
30 2011.1.00097.S COSMOSlowz6413 0.88 342 (7) 21382 0.31 0.57 0.48 0.52 (SB3)
31 2011.1.00097.S COSMOSlowz642 0.88 342 (7) 21382 0.31 0.58 0.47 0.52 (SB3)
32 2011.1.00097.S COSMOSlowz643 0.88 342 (7) 21382 0.31 0.58 0.47 0.52 (SB3)
33 2011.1.00097.S COSMOSlowz644 0.88 342 (7) 21382 0.29 0.58 0.47 0.52 (SB3)
34 2011.1.00097.S COSMOSlowz645 0.88 342 (7) 21382 0.3 0.58 0.47 0.52 (SB3)
35 2011.1.00097.S COSMOSlowz646 0.88 342 (7) 21382 0.31 0.58 0.47 0.52 (SB3)
36 2011.1.00097.S COSMOSlowz647 0.88 342 (7) 21382 0.31 0.57 0.47 0.51 (SB3)
37 2011.1.00097.S COSMOSlowz648 0.88 342 (7) 21382 0.29 0.57 0.47 0.51 (SB3)
38 2011.1.00097.S COSMOSlowz649 0.88 342 (7) 21382 0.32 0.57 0.47 0.51 (SB3)
39 2013.1.00151.S COSMOSmedz1070 0.88 342 (7) 21382 0.17 0.57 0.48 0.52 (SB3)
40 2013.1.00151.S COSMOSmedz1071 0.88 342 (7) 21382 0.26 0.57 0.48 0.52 (SB3)
41 2013.1.00151.S COSMOSmedz1072 0.88 342 (7) 21382 0.25 0.56 0.48 0.51 (SB3)
42 2013.1.00151.S COSMOSmedz1073 0.88 342 (7) 21382 0.24 0.56 0.48 0.51 (SB3)
43 2013.1.00151.S COSMOSmedz1074 0.88 342 (7) 21382 0.25 0.56 0.48 0.51 (SB3)
44 2011.1.00097.S COSMOShighz1130 0.88 342 (7) 21375 0.21 0.68 0.49 0.57 (SB3)
45 2011.1.00097.S COSMOShighz1131 0.88 342 (7) 21375 0.21 0.68 0.49 0.57 (SB3)
46 2011.1.00097.S COSMOShighz1132 0.88 342 (7) 21375 0.22 0.68 0.5 0.58 (SB3)
47 2011.1.00097.S COSMOShighz1133 0.88 342 (7) 21375 0.22 0.68 0.5 0.58 (SB3)
48 2011.1.00097.S COSMOShighz1134 0.88 342 (7) 21375 0.23 0.68 0.51 0.58 (SB3)
49 2011.1.00097.S COSMOShighz1135 0.88 342 (7) 21375 0.25 0.68 0.51 0.58 (SB3)
50 2011.1.00097.S COSMOShighz1136 0.88 342 (7) 21375 0.24 0.68 0.51 0.58 (SB3)
51 2011.1.00097.S COSMOShighz1380 0.88 342 (7) 21384 0.2 0.64 0.51 0.57 (SB3)
52 2011.1.00097.S COSMOShighz1381 0.88 342 (7) 21384 0.2 0.63 0.51 0.56 (SB3)
53 2011.1.00097.S COSMOShighz1382 0.88 342 (7) 21384 0.19 0.62 0.51 0.56 (SB3)
54 2011.1.00097.S COSMOShighz1383 0.88 342 (7) 21384 0.2 0.62 0.51 0.56 (SB3)
55 2011.1.00097.S COSMOShighz1384 0.88 342 (7) 21384 0.19 0.61 0.51 0.55 (SB3)
56 2011.1.00097.S COSMOShighz1385 0.88 342 (7) 21384 0.19 0.61 0.51 0.55 (SB3)
57 2011.1.00097.S COSMOShighz1386 0.88 342 (7) 21384 0.18 0.61 0.51 0.55 (SB3)
58 2011.1.00097.S COSMOShighz1410 0.88 342 (7) 21384 0.2 0.63 0.48 0.54 (SB3)
59 2011.1.00097.S COSMOShighz1411 0.88 342 (7) 21384 0.2 0.63 0.49 0.55 (SB3)
60 2011.1.00097.S COSMOShighz1412 0.88 342 (7) 21384 0.21 0.63 0.49 0.55 (SB3)
. . . . . . . . .
. . . . . . . . .
Table 2Examples of Our ALMA Maps
Map ID Project ID Target (Band) Ant. Dist. Beam Size
(mm) (GHz) (m) (mJy beam) ()
. . . . . . . . .
. . . . . . . . .
1569 2015.1.00039.S ALESS103.3 0.99 304 (7) 15310 0.1 1.25 0.85 1.03 (SB6)
1570 2015.1.00039.S ALESS116.2 0.99 304 (7) 15310 0.1 1.26 0.85 1.03 (SB6)
1571 2015.1.00039.S ALESS124.1124.4 0.99 304 (7) 15310 0.1 1.26 0.84 1.02 (SB6)
1572 2015.1.00039.S ALESS14.1 0.99 304 (7) 15310 0.11 1.24 0.84 1.02 (SB6)
1573 2015.1.00039.S ALESS2.2 0.99 304 (7) 15310 0.11 1.24 0.84 1.02 (SB6)
1574 2015.1.00039.S ALESS23.1 0.99 304 (7) 15310 0.11 1.24 0.84 1.02 (SB6)
1575 2015.1.00039.S ALESS3.1 0.99 304 (7) 15310 0.11 1.24 0.84 1.02 (SB6)
1576 2015.1.00039.S ALESS37.2 0.99 304 (7) 15310 0.1 1.24 0.84 1.02 (SB6)
1577 2015.1.00039.S ALESS55.2 0.99 304 (7) 15310 0.1 1.24 0.84 1.02 (SB6)
1578 2015.1.00039.S ALESS69.269.3 0.99 304 (7) 15310 0.11 1.25 0.84 1.02 (SB6)
1579 2015.1.00039.S ALESS72.1 0.99 304 (7) 15310 0.1 1.25 0.84 1.02 (SB6)
1580 2015.1.00039.S ALESS76.1 0.99 304 (7) 15310 0.1 1.24 0.84 1.02 (SB6)
1581 2015.1.00039.S ALESS79.4 0.99 304 (7) 15310 0.11 1.25 0.84 1.02 (SB6)
1582 2015.1.00039.S ALESS87.3 0.99 304 (7) 15310 0.1 1.25 0.84 1.02 (SB6)
1583 2015.1.00039.S ALESS88.2 0.99 304 (7) 15310 0.1 1.25 0.84 1.02 (SB6)
1584 2015.1.00039.S ALESS9.1 0.99 304 (7) 15310 0.11 1.24 0.84 1.02 (SB6)
1585 2015.1.00039.S ALESS99.1 0.99 304 (7) 15310 0.1 1.25 0.85 1.03 (SB6)
1586 2015.1.00540.S UVISTA-169850 1.29 233 (6) 15331 0.03 1.43 1.08 1.24 (SB7)
1587 2015.1.00540.S UVISTA-238225 1.29 233 (6) 15331 0.03 1.44 1.08 1.24 (SB7)
1588 2015.1.00540.S UVISTA-279127 1.29 233 (6) 15331 0.03 1.43 1.09 1.24 (SB7)
1589 2015.1.00540.S UVISTA-304384 1.29 233 (6) 15331 0.03 1.43 1.09 1.24 (SB7)
1590 2015.1.00540.S UVISTA-304416 1.29 233 (6) 15331 0.03 1.44 1.08 1.24 (SB7)
1591 2015.1.00540.S UVISTA-65666 1.29 233 (6) 15331 0.03 1.44 1.08 1.24 (SB7)
1592 2015.1.00664.S KMOS3DCOS4-10347 1.13 265 (6) 15460 0.07 0.79 0.69 0.73 (SB4)
1593 2015.1.00664.S KMOS3DCOS4-13174 1.13 265 (6) 15460 0.07 0.79 0.69 0.73 (SB4)
1594 2015.1.00664.S KMOS3DCOS4-13701 1.13 265 (6) 15460 0.07 0.79 0.69 0.73 (SB4)
1595 2015.1.00664.S KMOS3DCOS4-15813 1.13 265 (6) 15460 0.07 0.79 0.69 0.73 (SB4)
1596 2015.1.00664.S KMOS3DCOS4-15820 1.13 265 (6) 15460 0.07 0.78 0.69 0.73 (SB4)
1597 2015.1.00664.S KMOS3DCOS4-19680 1.13 265 (6) 15460 0.07 0.78 0.69 0.73 (SB4)
1598 2015.1.00664.S KMOS3DCOS4-24763 1.13 265 (6) 15460 0.07 0.78 0.69 0.73 (SB4)
Mosaic Data7
1599 2012.1.00756.S Field-10 1.13 265 (6) 20650 0.05 0.51 0.41 0.45 (SB3)
1600 2012.1.00756.S Field-11 1.13 265 (6) 20650 0.05 0.5 0.41 0.45 (SB3)
1601 2012.1.00756.S Field-12 1.13 265 (6) 20650 0.05 0.5 0.41 0.45 (SB3)
1602 2012.1.00756.S Field-13 1.13 265 (6) 20650 0.05 0.5 0.42 0.45 (SB3)
1603 2012.1.00756.S Field-14 1.13 265 (6) 20650 0.05 0.5 0.42 0.45 (SB3)
1604 2012.1.00173.S HUDF0 1.36 221 (6) 151268 0.03 0.47 0.38 0.42 (SB3)
1605 2012.1.00173.S HUDF1 1.36 221 (6) 151268 0.03 0.47 0.38 0.42 (SB3)
1606 2012.1.00173.S HUDF10 1.25 241 (6) 151268 0.01 0.93 0.69 0.8 (SB5)
1607 2012.1.00173.S HUDF2 1.36 221 (6) 151268 0.03 0.48 0.38 0.42 (SB3)
1608 2012.1.00173.S HUDF3 1.36 221 (6) 151268 0.03 0.47 0.38 0.42 (SB3)
1609 2012.1.00173.S HUDF4 1.25 241 (6) 151268 0.01 0.65 0.51 0.57 (SB3)
1610 2012.1.00173.S HUDF5 1.36 221 (6) 151268 0.03 0.47 0.38 0.42 (SB3)
1611 2012.1.00173.S HUDF6 1.36 221 (6) 151268 0.03 0.47 0.38 0.42 (SB3)
1612 2012.1.00173.S HUDF7 1.25 241 (6) 151268 0.01 0.87 0.66 0.75 (SB4)
1613 2012.1.00173.S HUDF8 1.25 241 (6) 151268 0.01 0.71 0.56 0.63 (SB4)
1614 2012.1.00173.S HUDF9 1.25 241 (6) 151268 0.01 1.08 0.75 0.9 (SB5)
1615 2013.1.00999.S Abell27440 1.14 263 (6) 15820 0.06 0.65 0.5 0.57 (SB3)
1616 2013.1.00999.S Abell27441 1.14 263 (6) 15820 0.06 0.59 0.47 0.52 (SB3)
1617 2013.1.00999.S Abell27442 1.14 263 (6) 15820 0.06 0.66 0.5 0.57 (SB3)
1618 2013.1.00999.S Abell27443 1.14 263 (6) 15820 0.06 0.63 0.5 0.56 (SB3)
1619 2013.1.00999.S Abell27444 1.14 263 (6) 15820 0.06 0.64 0.5 0.56 (SB3)
1620 2013.1.00999.S Abell27445 1.14 263 (6) 15820 0.06 0.65 0.5 0.57 (SB3)
1621 2013.1.00999.S Abell27446 1.14 263 (6) 15820 0.06 0.63 0.5 0.56 (SB3)
1622 2013.1.00999.S Abell27447 1.14 263 (6) 15820 0.06 0.63 0.5 0.56 (SB3)
1623 2013.1.00999.S MACSJ0416.1-24030 1.14 263 (6) 15349 0.07 1.53 0.85 1.14 (SB6)
1624 2013.1.00999.S MACSJ0416.1-24031 1.14 263 (6) 15349 0.06 1.54 0.86 1.15 (SB6)
1625 2013.1.00999.S MACSJ0416.1-24032 1.14 263 (6) 15349 0.06 1.54 0.85 1.14 (SB6)
1626 2013.1.00999.S MACSJ0416.1-24033 1.14 263 (6) 15349 0.06 1.5 0.86 1.13 (SB6)
1627 2013.1.00999.S MACSJ1149.5+22230 1.14 263 (6) 15349 0.07 1.2 1.13 1.16 (SB6)

Notes: (1) ALMA project code. (2) Target name in the initial ALMA observation. (3) Wavelength in the observed frame. (4) Frequency in the observed frame. (5) Range of the antenna distances. (6) One sigma noise measured after the CLEAN process. (7) Synthesized beam size (weighting=”natural”). (8) Circularized beam size. Classification of the sub-dataset is presented in the parentheses. We cut out the mosaic maps based on the source positions identified in the dirty maps to perform the data analyses (Section 2.2). (The complete table is available in a machine-readable form in the online journal.)

Figure 1.— Histograms of the circularized beam size (left), the sensitivity (middle), and the frequency in the observed frame (right) for our ALMA maps. The blue histograms denote our 1627 ALMA maps. The red histograms represent our ALMA maps with the ALL5S source(s) whose peak pixel value(s) exceeds the confidence level.

2.2. Data Reduction

Basically, the data is reduced in the same manner as Fujimoto et al. (2016) In this process, we use the CASA versions from 4.2.0 to 4.7.2. Our CASA reduction has three major steps: bad data flagging, bandpass calibration, and gain calibration with flux scaling. These major steps are performed with the scripts provided by the ALMA observatory. We apply additional flaggings if we find problems in the final images that the noise level is significantly higher than the calibrated products provided by the ALMA observatory, or that there remain striped patterns. For the cycle 0 data, we update the flux model to ’Bulter-JPL-Horizons 2012’, if the data is calibrated with the old version of the flux model of ’Bulter-JPL-Horizons 2010’.

Because the HUDF region is observed in two different ALMA projects (#2013.1.00718.S and #2013.1.00173.S), we combine these two data sets with the concat task. Here, we recalculate the data weights with the statwt task based on its visibility scatters which include the effects of integration time, channel widths, and systematic temperature.

We perform Fourier transformations for the reduced -data to create ”dirty” continuum maps. With the dirty maps, we estimate the standard deviation, , of the pixel values.

For the single-field data, we process the maps with the CLEAN algorithm in the following three steps: 1) identifying peak pixels with the 50 level, 2) making CLEAN boxes with the boxit task for the peak pixels given in the step 1), and 3) performing the CLEAN algorithm down to the depth of the 3 level with the natural weighting in the CLEAN boxes. We repeat 1)-3) for four times, changing from the 50 level to the 20, 10, and 5 levels in the step 1). We then obtain the final maps. The process from the dirty maps to the final maps is referred to as ”CLEAN process”. We define the 1 noise level of each ALMA map, , with the standard deviation of the pixel values in the residual map.

For the mosaic data, we find that the CLEAN process cannot be performed due to the large data size. We thus reduce the data size of the mosaic data, having the following three steps. I) We extract the sources whose peak flux values exceed the levels in the dirty maps. Here, we identify the source positions. II) We select pointings whose central positions are located within from the source positions. III) We create a new dirty map only with the -visibility from these pointings. Completing above three steps, we carry out the CLEAN process to make final maps in the same manner as the single-field data.

Since our scope is the dust continuum, it is desired to remove the contamination of atomic and molecular transition lines in the FIR wavelength. There are two cases that the FIR lines are included in the channels of ALMA Band 6/7. First case is that the initial ALMA data is taken for the FIR lines. In our ALMA data, the ALMA projects of #2012.1.00076.S, #2013.1.00668.S, and #2012.1.00523.S are taken for the CO or [C ii] 158 m lines from high- star-forming galaxies. For those of three ALMA data, we remove the line channels if robust line profiles are identified in the data cubes. Second case is that the FIR lines are contaminated by chance. Among the FIR lines, the [C ii] 158 m line is one of the brightest FIR lines in the star-forming galaxies (Stacey et al. 1991), which is potentially contaminated in ALMA Band 6/7 with the source redshifts at 47. In fact, Swinbank et al. (2012) have identified two [C ii] 158 m lines from two SMGs at and in an ALMA survey of submillimeter galaxies in the Extended Chandra Deep Field-South (ALESS) with 126 ALMA data cubes. This indicates that the [C ii] 158-m line contamination is not significant with the chance of 12%. Moreover, the flux densities of the [C ii] 158-m lines identified in Swinbank et al. (2012) are calculated to contribute only % of those of the dust continuum. Therefore, we assume that the second case is negligible in our statistical results, and do not take it into account through this paper.

The 1627 final maps of the pointing and mosaic data achieve angular resolutions of and the sensitivities of 0.011.4 mJy beam before the primary beam corrections. We summarize the properties of the final maps in Table 2. In Figure 1, we also show the histograms of the circularized beam size, the sensitivity, and the observed frequency for the final maps.

3. Data Analysis

3.1. Source Detection

We conduct source extractions for our ALMA maps before primary beam corrections with sextractor version 2.5.0 (Bertin & Arnouts 1996). The source extraction is carried out in the high sensitivity regions. For the single-field data, we use the regions with the primary beam sensitivity greater than 50%. For the mosaic data, we perform the source extractions where the relative sensitivity to the deepest part of the mosaic map is greater than 50%.

We identify sources with a positive peak count () above the 3 level. We then select only the sources with to make a 5-peak catalog. We obtain 1034 sources in the 5-peak catalog. The details of these 1034 sources are presented in Table 4. To perform reliable size measurements, we select 780 sources with total flux densities () above the 10 level from the 5-peak catalog, making a 10-total catalog. The total flux densities are measured with the imfit task in CASA that carries out the fitting routine with the 2D Gaussian model on the image plane. We refer the -peak and -total catalogs as ”ALL5S” and ”ALL10S”, respectively. The summary of the source catalogs is presented in Table 3.

Catalog Name Selection Criteria Source Number
(1) (2) (3)
ALL5S 1034
ALL10S 780
OC5S ALL5S optical-NIR counterpart 577
OC10S ALL10S optical-NIR counterpart 444
OC5S-mmT OC5S FIR-mm target obs. 131
OC5S-optT OC5S optical target obs. 446

Notes: (1) Name of our ALMA source catalog. (2) Selection criteria of the ALMA sources for the catalog (see text). (3) Source number in the catalog.

Table 3Our Catalogs
{turnpage}
Source ID R.A. Dec. SN flag offset -mag
(J2000) (J2000) (mm) (mJy/beam) (mJy) (mJy) (mJy) () () () (mag)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
1 150.203888 2.50107 0.87 5.6 1.13 0.2 2.22 0.39
2 150.192612 2.219835 0.87 17.5 3.61 0.21 5.51 0.21 5.23 0.66 4.32 0.55 0.23 0.05 0.21 0.04 2
3 149.882294 2.507151 0.87 9.3 1.92 0.21 3.85 0.35 3.66 0.84 3.16 0.73 0.13 0.07 0.14 0.08 2 0.03 2.38 23.1
4 150.028015 2.43566 0.87 5.2 0.68 0.13 0.8 0.15
5 150.064514 2.32913 0.87 9.6 1.24 0.13 1.33 0.13 1.33 0.28 1.15 0.24 0.09 0.07 0.10 0.08 1
6 150.092575 2.398174 0.87 9.1 1.22 0.13 1.68 0.15 1.61 0.37 1.4 0.32 0.11 0.07 0.12 0.08 0
7 150.104691 2.243663 0.87 6.5 0.6 0.09 0.58 0.09
8 150.018112 2.34992 0.87 25.6 2.38 0.09 3.94 0.14 4.02 0.36 3.74 0.33 0.14 0.03 0.12 0.02 0 0.09 2.27 28.3
9 150.373688 2.112025 0.87 5.5 0.5 0.09 0.46 0.09 0.13 -98.99 20.7
10 149.957428 2.028263 0.87 9.1 0.87 0.1 1.37 0.13 1.39 0.34 1.2 0.29 0.16 0.08 0.17 0.08 0

Note. – (1) Wavelength in the observed frame. (2) Peak SNR in the ALMA maps before primary beam correction. (3) Peak value in the ALMA maps before primary beam correction. (4) Integrated flux density measured with the IMFIT task after primary beam correction. (5) Integrated flux density measured with the UVMULTIFIT task after primary beam correction. (6) Integrated flux density measured with the UVMULTIFIT task after the corrections of the primary beam and the Monte-Carlo simulation. (7) FWHM of the source size obtained with the UVMULTIFIT task. (8) FWHM of the source size derived with the UVMULTIFIT task after the correction of the Monte-Carlo simulation. (9) Flag for the reliability of the UVMULTIFIT fitting. (10) Offset between the centers of the ALMA source and the optical-NIR counterpart in unit of arcsecond. (11) Photometric redshift of the optical-NIR counterpart. (12) -band AB magnitude of the optical-NIR counterpart. (The complete table is available in a machine-readable form in the online journal.)

Table 4Examples of the 5-peak catalog (ALL5S)

3.2. Flux and Size Measurement

We estimate the flux densities and the sizes for our 780 objects in ALL10S. We perform uvmultifit (Martí-Vidal et al. 2014) that is a simultaneous fitting tool for multiple objects on -visibility. For the visibility fitting, we use the flux density and the size measurements obtained by the imfit task as the initial values, while the source positions are fixed. Here we adopt a symmetric exponential disk model. Hodge et al. (2016) estimate the median Srsic index for 15 SMGs as , which is close to our assumption of the exponential disk with . The fitting result of the FWHM values are converted to the values via the relation of in the case of (MacArthur et al. 2003). Hereafter, the values are referred to as our size measurements. Figure 2 shows the several examples of our best-fit results with the uvmultifit task. The flux densities for the rest of the 254 (1034780) sources in ALL5S are obtained by the imfit task, and we do not estimate the sizes for those of the 254 sources.

To remove the bad visibility data, we flag the ALL10S sources by visual inspection on the visibility amplitude plots such shown in Figure 2. We define flag = 0, 1, and 2 as a source whose fitting result is reliable, tentative, and bad, respectively. We use only the results with flag = 0 in the following analyses in this paper. In Table 4, we summarize the flux and size measurements obtained by the uvmultifit task together with the flag values. For the sources with flag = 1 and 2, we use the flux densities that are obtained by the imfit task, and do not include the size measurement in this paper.

Figure 2.— Examples of the best-fit result of the uvmultifit task for ALL10S sources in the sub-datasets of SB1SB9 on the -visibility. The black points denote the visibility amplitudes obtained from our ALMA data. The red dots indicate our best-fit results with the exponential disk model. Because there are no ALL10S sources with flag = 0 (reliable) in SB10, we do not show the example of the best-fit result of SB10.

3.3. Simulation and Correction

We investigate the systematics in the flux density and size measurements with the uvmultifit task for the ALL10S sources, performing Monte-Carlo simulations for the flux density and the size measurements. Because the angular resolution of the ALMA maps may produce different systematics in these measurements, we conduct the Monte-Carlo simulations based on each sub-dataset (Table 1).

First, we select one ALMA map from each sub-dataset, making 10 representative ALMA maps. The representative ALMA map is randomly selected from the ALMA maps in which no sources are detected. Second, we create artificial sources with the uniform distribution of the total fluxes in the levels and the circularized source sizes of . We then inject the artificial sources individually into the -visibilities of each representative ALMA map at random positions within a 10 radius from the centers. Finally, we obtain the flux densities and sizes of the artificial sources in the same manner as the flux density and the size measurements for the real sources. Figure 3 shows two example results of the input and output values of the flux density and the size for the sub-datasets of SB1 and SB7.

As shown in Figure 3, we find that the output values of both flux density and size well recover the input values within the uncertainty. However, we also find that output values have a slight offset from the input values in some sub-datasets, which suggests that we need the corrections for the output values.

To carry out the corrections for the flux density and size measurements, we model the input values of the flux density and the size in Figure 3 as a function of output value,

(2)
(3)

where () and () are the output (input) values of the flux density and the size, respectively. and are the free parameters. In the model fitting, we divide our simulation results into three bins with the output peak signal to noise ratio (SNR) of 510, 1020, and 2030. In Figure 3, we plot the best-fit models for these three peak SNR bins. Based on the peak SNR of each source and those of the best-fit models, we correct the values of the flux density and the size for the ALL10S sources. For the bright ALL10S sources with the peak SNR 30, we use the best-fit model result of the peak SNR 20 30. Table 4 lists the values of the flux density and the size after the corrections.

Figure 3.— Relationship between input and output values of the flux density (left) and size (right) in two sub-datasets of SB1 and SB7, as for example. The black, blue, and red circles (curves) indicate the values (the best-fit functions) of the samples with the output peak SNR ranges of 510, 1020, and 2030, respectively. In the left panel, the flux density of the input and output values are normalized by . In the right panel, the open squares indicate the output size measurements that are below the reliable size measurement limits (Section 3.4). We do not include the open squares for the fitting.

3.4. Comparison with Previous Measurements

To investigate the potential systematics in our method, we compare our flux and size measurements with the previous ALMA results in the literature (Ikarashi et al. 2015; Simpson et al. 2015; Tadaki et al. 2016; Barro et al. 2016; Hodge et al. 2016; Rujopakarn et al. 2016; González-López et al. 2017).

Figure 4 presents our flux and size measurements and the previous ALMA results. For a fair comparison, the size values obtained in the previous ALMA results are converted into the values. In Figure 4, our flux measurements show a good agreement with the previous ALMA results in the wide flux range. Our size measurements are also consistent with the majority of the previous ALMA results within the 12 errors, and no systematic offsets are identified in the size comparison. Although the scatter in the size comparison seems to be larger than that in the flux comparison, the large scatter indicates that the size measurement is sensitive to the way of measurements, the assumptions of the model profile, the parameter range, and the initial values in the fitting process. Therefore, we conclude that both of our flux and size measurements are consistent with the previous ALMA results, and not biased by any systematics.

Figure 4.— Flux (top) and size (bottom) comparison between our measurements and the previous ALMA results in the literature. The abscissa and ordinate axes provide our measurements (, ) and the previous ALMA results (, ), respectively, with black squares (H16; Hodge et al. 2016), triangles (I15; Ikarashi et al. 2015), sideways triangles (S15; Simpson et al. 2015), diamonds, (T16; Tadaki et al. 2016), stars (B16; Barro et al. 2016), crosses (G17; González-López et al. 2017), and inverse triangles (R16; Rujopakarn et al. 2016). For the gravitationally lensed sources in HFF, we use the flux and size measurements before the lensing correction in the comparison.

3.5. Selection and Measurement Completeness

We examine the selection and measurement completenesses for the ALL10S sources. Figure 5 shows the ALL10S sources in the 10 sub-datasets of SB1SB10 on the flux density and size plane.

Figure 5.— Selection and measurement completeness for the ALL10S sources in the 10 sub-datasets of SB1SB10. The background gray scale denotes the selection completeness, which is defined with the color bar shown on the right-hand side. The dashed lines represent the limits of the reliable size measurements . For the estimates from Equation 4, we adopt and for a cutoff, and assume that corresponds to the ratio of the total flux density to the pixel noise. The hatched areas show the regions where the measurements are not reliable. The red circles show the ALL10S sources with the flux density and the size measurements after the corrections. The red shades indicate the areas below the completeness thresholds (see text).

We evaluate the selection completeness, performing the selection procedures same as those of ALL10S based on the Monte-Carlo simulation results (Section 3.3). The background gray scale in Figure 5 denotes the selection completenesses of the 10 representative ALMA maps. In Figure 5, there are two kinds of the selection incompletenesses. The first incompleteness is caused by one of ALL10S criteria, (Table 3). Figure 5 shows the first incompleteness in the vertical region at the faint end. The second incompleteness is caused by another criterion of ALL10S, (Table 3). If a source is extended, the peak flux of the source does not reach the criterion especially in the high-resolution ALMA maps. Thus, the second incompleteness affects the identification of the extended sources in the high-resolution observations. In Figure 5, the second incompleteness is shown in the diagonal region in each panel. Even with these two types of incompletenesses, our ALMA sources are placed in the parameter space with the selection completeness of 60%.

We next investigate the measurement completeness. A small size of the synthesized beam, , enables us to measure a compact source size. Moreover, if a source is bright, we can measure the small size accurately. In fact, the reliable size measurement limit with an interferometer, , is calculated as

(4)

where is the SNR of the averaged visibilities, is a coefficient that typically takes the value in the range of weakly depending on the spatial distribution of the telescopes, and is related to the probability cutoff for a false size-detection (Martí-Vidal et al. 2012). We cannot obtain the reliable size measurements for the compact and/or faint sources in the low-resolution observations. In Figure 5, the hatched areas show the regions in which the size values are below the reliable size measurement limit. The SB2SB8 panels show that the source distributions overlap the hatched area, where the measurement completeness is not satisfied.

To select the sources that are not biased by the selection nor measurement completenesses, we define the completeness threshold in each sub-dataset. In Figure 5, the red shades represent the areas below the completeness thresholds. We use the completeness thresholds to verify the FIR sizeluminosity relation in Section 4.2.

3.6. Optical and NIR Counterpart

We search the optical-NIR counterparts for the ALL5S and the ALL10S sources. Table 5 summarizes the -band limiting magnitudes of the optical-NIR source catalogs of HUDF, HFF, COSMOS, SXDS, and GOODS-S that are used in this study. Recent studies with ALMA and HST have reported two types of the offsets between the ALMA and optical-NIR source centers. One is the intrinsic offset of between the rest-frame FIR and UV-optical emission (Chen et al. 2015). The other is the astrometric uncertainty of HST in GOODS-S (Dunlop et al. 2017; Rujopakarn et al. 2016, see also Section 5.3), which we apply the corrections of = mas and = mas (Rujopakarn et al. 2016) for the optical-NIR source centers. Since no one knows whether other fields also have the astrometric uncertainty same as GOODS-S, we do not apply this astrometry correction to the optical-NIR sources in other fields. Taking these potential offsets into account, we define the optical-NIR counterparts as the optical-NIR sources that locate within a search radius of 0 from our ALMA source centers. If there exist several optical-NIR objects within the search radius of , we regard the nearest one as the optical-NIR counterpart. We identify a total of 577 and 444 optical-NIR counterparts in ALL5S and the ALL10S that are referred to as ”OC5S” and ”OC10S”, respectively. The OC5S and the OC10S catalogs are presented in Table 3. We take the values of and -band magnitude from the optical source catalogs in Table 4, and estimate the spatial offset between the source centers in our ALMA maps and the optical catalogs. The values is calculated with a modified black body following the analytical expression,

(5)

where and are Planck and Boltzmann constant, respectively, is the light velocity, and are the rest- and observed-frame frequency, respectively, is the observed-flux density, is the luminosity distance, , , are the black body, the Gamma, and the Riemann zeta function, respectively, is the spectral index, and is the dust temperature. Here, we assume (e.g., Chapin et al. 2009, Planck Collaboration 2011) and K (e.g., Kovács et al. 2006; Coppin et al. 2008), and take the cosmic microwave background (CMB) effect (da Cunha et al. 2013) into account. Table 4 summarize , spatial offset, -band, and .

Note that our ALMA sources identified behind the HFF clusters are gravitationally lensed. For these lensed ALMA sources, we use the intrinsic , , and -band magnitudes estimated by dividing the observed values with the magnification factors that are obtained from Castellano et al. (2016). For the other ALMA sources identified in the fields, we assume that the contamination of the lensed sources is negligible, because Simpson et al. (2016) show that the contamination of the potential lensed sources is 8%. We examine the effect of the potential lensed sources to our results in Section 4.2.

Field Catalog Reference Detection Limit (ap.)
(1) (2) (3)
SXDS Santini et al. (2015)
GOODS-S Wide Santini et al. (2015)
GOODS-S Deep Santini et al. (2015)
HUDF Santini et al. (2015)
COSMOS Wide 8 Ilbert et al. (2013) 9
COSMOS Deep 10 Momcheva et al. (2016) 11
HFF Castellano et al. (2016) 28.529.0

Notes: (1) Field name. (2) Optical-NIR catalog that is used in our optical-NIR counterpart identification. (3) Optical-NIR source detection limit. The aperture size is presented in the parentheses. The catalog is constructed with a detection image from the sum of the UltraVISTA DR1 images, where the 5 detection limit of the -band image is in a aperture. The catalog is constructed with the HST -band that has the 5 detection limit of in a aperture.

Table 5Optical Catalog Summary

3.7. Our Sample on the SFR plane

We examine the star-formation properties of our ALMA sources. Figure 6 presents the stellar mass () star formation rate (SFR) relation for the OC5S sources. For comparison, Figure 6 also displays the 3D-HST sources whose SFR and values are estimated in Momcheva et al. (2016).

For our ALMA sources, the values are obtained from the optical source catalogs in Table 5. In this paper, we assume the Chabrier (2003) initial mass function (IMF). To obtain the Chabrier IMF values of , we divide the Salpeter (1955) IMF values that are used in Castellano et al. (2016) by a factor of 1.8. The SFR values are estimated from the sum of the dust obscured () and un-obscured () star-formation rates, and given by

(6)

The SFR and SFR values are calculated with the fomulae of Murphy et al. (2011),

(7)
(8)

Note that the different detection limits in our ALMA maps () produces the different limits of the SFR values. We calculate the SFR limit of each ALMA map by Equation 8, where the value is estimated in the same manner as Section 3.6 from the ALMA detection limit. In Figure 6, the background gray-scale corresponds to the fraction of our ALMA maps whose SFR limits are below the SFR values on the ordinate axis, referred to as ”ALMA data completeness”.

In Figure 6, most of our ALMA sources fall in the ranges of and SFR yr. Due to the ALMA data completeness, our ALMA sources are limited by the SFR values, which indicates that our ALMA sources are not the complete sample at a fixed .

To investigate the star-formation mode of our ALMA sources, we compare the specific SFR ( SFR/; sSFR) of our ALMA sources with the main sequence in four redshift bins of 01, 12, 24, and 46. We define the starbursts as the sources with sSFR over a factor of 4 larger than that of the main sequence (Rodighiero et al. 2011). The sSFR value of the main sequence is calculated by the median sSFR value of the 3D-HST sources in each redshift bin. We find that 100%, 58%, 49%, and 26% of our ALMA sources are classified as the starbursts at 01, 12, 24, and 46, respectively. With the redshift bins all together, the starburst fraction is 52%. The decreasing trend of the starburst fraction toward high redshifts indicates that the massive sources on the main-sequence are also detected above the SFR limits at the high redshifts. This is because the SFR values of the main sequence are increased at high redshifts. The starburst fraction and the trend are generally consistent with the ALESS result that shows the starburst fraction of 49% and 27% at 1.52.5 and 2.54.5, respectively (da Cunha et al. 2015). Therefore, we conclude that our ALMA sources are the general population of the dusty starbursts similar to the ALESS sample, having the half of them are the starbursts, while the rest of half are the high-mass end of the main sequence.

Figure 6.— SFR relation at 06. The red and blue circles denote the OC5S and the 3D-HST sources, respectively. The background gray-scale represents the completeness of the ALMA data depending on the flux density limits that correspond to SFRs by Equation 8.

4. Results

Figure 7.— Normalized redshift distribution of the OC5S sources. The source number in each sub-sample is presented in the parentheses. Left: The red, green, and blue histograms denote the samples with the ranges of , , and , respectively. The color arrows indicate the median redshifts of the three samples. Middle: The red and blue histograms present the samples of ALMA Band 6 and 7, respectively. The color arrows show the median redshifts of the two samples. Right: Redshift distribution of OC5S-mmT (solid histogram) and OC5S-optT (dashed histogram). The filled (open) arrow denotes the median redshift of OC5S-mmT (OC5S-optT).

4.1. Redshift Distribution

We estimate the redshift distribution for our ALMA sources, and verify the potential bias of the target selection in the initial ALMA observations by comparing the redshift distributions. In this analysis, we divide the OC5S sources into two types of the sources. The sources whose central targets in the initial ALMA observations are selected in the FIR-mm and optical-NIR bands are referred to as ”OC5S-mmT” and ”OC5S-optT”, respectively. We also regard the sources as OC5S-mmT, if the initial ALMA observation is the blind survey. On the other hand, if the central target selection include the redshift properties even based on the FIR-mm bands, we classify the sources as OC5S-optT. We assume that OC5S-mmT is not biased by the target selection in the initial ALMA observations. We summarize the source number in the new catalogs of OC5S-mmT and OC5S-optT in Table 3.

First, we derive the redshift distribution with the OC5S-mmT sources. In the left and middle panels of Figure 7, the redshift distributions are plotted in three bins of and two bins of ALMA Band 6/7, respectively. We perform the Kolmogorov-Smirnov test (KS-test) to investigate whether the redshift distributions of our ALMA sources are changed by or ALMA Band 6/7. The result of the KS-test suggests that we cannot rule out the possibility that the redshift distributions are produced from the same parent sample in the both cases. We thus conclude that neither nor ALMA Band 6/7 significantly affect the redshift distribution for the ALMA sources. The median redshift is estimated to be , which is consistent with the previous results of the blind submm/mm surveys (Chapman et al. 2005; Yun et al. 2012; Simpson et al. 2014; Dunlop et al. 2017).

Second, we also derive the redshift distribution of the OC5S-optT sources to test whether the OC5S-optT sources are biased by the target selection in the initial ALMA observations. The right panel of Figure 7 shows the redshift distributions of the OC5S-mmT and OC5S-optT sources. We perform the KS-test for these two redshift distributions. The result of the KS-test suggests that we cannot reject the possibility that these two redshift distributions are made from the same parent sample. The median redshift of the OC5S-optT sources is estimated to be that is almost same as obtained from the OC5S-mmT sources. The little difference of the redshift properties between the OC5S-mmT and OC5S-optT sources implies that the OC5S-optT sources are not biased by target selection in the initial ALMA observations. Therefore, we use both OC5S-mmT and OC5S-optT in the following analyses with except for Section 5.5.

4.2. FIR Size and Luminosity Relation

Before we investigate the and relation, we perform two tests to verify whether there exist potential biases in the and measurements our ALMA sources.

Figure 8.— Comparison of the FIR sizeluminosity relations. The filled and open circles indicate our ALMA sources above and below the completeness thresholds (Section 3.5), respectively. Top: The red and black circles represent the OC10S and ALL10S sources, respectively. For the and estimates, we assume that all of the OC10S and ALL10S sources reside at . Bottom: The red and black circles denote the OC10S sources whose values are derived with the assumptions of 35 K and the relation (see text), respectively.

First, we examine whether there is a difference between OC10S and ALL10S on the plane. In the top panel of Figure 8, we show the estimates as a function of for the ALL10S and OC10S sources with the black and red circles, respectively. To calculate the and values for the ALL10S and OC10S sources, we assume that all of the sources reside at (Section 4.1). To evaluate the distributions of the ALL10S and OC10S sources, we fit a power-law function to the and values. The power-law function is defined as

(9)

where and are free parameters. The value presents the effective radius at a luminosity of . We select the value to the best-fit Schechter parameter at . For the fitting, we adopt the completeness thresholds (Section 3.5) to select the ALMA sources that are not biased neither by the selection nor measurement completeness. In Figure 8, open and filled red circles represent our ALMA sources below and above the completeness thresholds, respectively. We use our ALMA sources above the completeness thresholds alone in the following analyses in this paper. We estimate the 1-error range of the power-law function by the perturbation method (Curran 2014). We generate 1000 data sets of the and for the ALL10S and OC10S sources based on the random perturbations following the Gaussian distribution whose sigma is defined by the observational errors. We obtain 1000 best-fit power-law functions for the 1000 data sets. We define the error of the power-law function as the range of 68% distribution for the values in the 1000 best-fit power-law functions. We refer this process to evaluate the 1-error range as the perturbation method. In the top panel of Figure 8, the black and red shades denote the estimated 1-error ranges of the power-law functions for the ALL10S and OC10S sources, respectively. We find that the red shade shows a agreement with the black shade, suggesting that there is no significant difference in the distributions between the ALL10S and OC10S sources. We thus conclude that the OC10S sources generally represent the ALL10S sources. Because the ALL10S sources with no optical-NIR counterparts do not allow us to examine the physical properties, we investigate the OC10S sources alone with the individual photometric redshifts in the following analyses in this subsection.

Second, we test whether our assumption of the single dust temperature, K, makes a bias on the plane. Symeonidis et al. (2013) report that has a positive correlation with for the -selected galaxies at . We estimate the values with the relation for the OC10S sources, and compare the source distribution on the plane to that obtained with K. For evaluating the relation, we model as a function of with a linear function,

(10)

where and are the free parameters. Note that it is not clear that - and ALMA-selected galaxies have the same relation, because the wavelength coverage of ALMA Band 6/7 is sensitive to the galaxies with the dust temperatures colder than that of does at a fixed (e.g., Simpson et al. 2016). We thus evaluate the relation with following 3 steps: a) fitting the linear function to the -selected galaxies in Symeonidis et al. (2013), b) fixing the value of the best-fit linear function obtained in a), and c) fitting the linear function with the fixed to the ALMA-selected galaxies whose and values are well determined with ALMA bands in Swinbank et al. (2014) and Simpson et al. (2016). We then obtain the and values from our best estimate of the relation and Equation 5. In the bottom panel of Figure 8, we show the OC10S sources whose values are estimated from the relation and K with the black and red circles, respectively. The color shades denote the 1-error ranges of the power-law functions for the and values obtained by the perturbation method, which shows that the power-law functions are consistent within the 1 errors. We thus conclude that our assumption of K does not produce the significant systematics on the plane. Because the evaluation and the application processes of the relation may contain the systematics in the estimates rather than the assumption of the single dust temperature, we adopt the values estimated by K in the following analyses.

Figure 9.— FIR sizeluminosity relation of our best estimates with OC10S representing ALL10S (see text). The red filled and open circles are the OC10S sources above and below the completeness thresholds (Section 3.5), respectively. The shade region indicates 1 uncertainty range of the best-fit power-law function that is calculated by the perturbation method. In the power-law fitting, we do not include the OC10S sources shown with the open circles. The black dashed lines denote the constant surface SFR density =10, 100, and 1000 estimated from Equation 8 and the assumption of the uniform surface density with . Other symbols denote the previous ALMA results in the same assignment as Figure 4, except for the blue pentagons (L16; Lindroos et al. 2016). The symbol colors indicate that the types in the initial ALMA observations of the submm/mm bands selected target (black), the optical/NIR bands selected target (blue), and the blind survey (green).

Figure 9 shows our best estimates of the relation with the red circles. In Figure 9, our ALMA sources fall in the range of kpc. From Equation 8 and the assumption of the uniform surface density with , the SFR surface densities of our ALMA sources are estimated to be 8800 .

We compare our estimates to previous ALMA results. In Figure 9, we show the previous ALMA results of the FIR size studies with various symbols. The black symbols denote the high-resolution (01603) observations for the SMGs at (Ikarashi et al. 2015; Simpson et al. 2015; Hodge et al. 2016). The FIR size for the SMGs are estimated to be kpc. The green symbols represent the results of the deep observations for one blank field of HUDF and three gravitational lensing clusters of HFF (Rujopakarn et al. 2016; González-López et al. 2017). These deep observations explore the FIR size measurements for the fainter sources than SMGs. The typical FIR sizes of these faint sources are estimated to be kpc. The optically-selected galaxies with blue symbols are also examined for the FIR size properties. Lindroos et al. (2016) estimate the FIR sizes by the -stacking method for the star-forming galaxies at . The stacking result shows the typical FIR sizes of kpc. Tadaki et al. (2016) and Barro et al. (2016) conduct the ALMA observations for massive () star-forming galaxies at , and report the rest-frame FIR size range of kpc.

These results of the previous studies indicate that the relation has a large scatter. In Figure 9, we find that our and estimates are generally consistent with the previous studies within the scatter.

We next evaluate the relation. We divide our sample into four redshift samples of 01, 12, 24, and 46. Spearman’s rank test is used to examine the correlation between and . We find that there exist positive correlations between and with 9098% significance levels in the four redshift samples. With the redshift samples all together, the significance level becomes 99%. To estimate the slope of the positive correlation, we fit the power-law function of Equation 9 to the relation. We estimate the value from the redshift sample of 24 that has the largest source number among the redshift samples. We obtain . With the redshift samples all together, the value is estimated to be 0.28 0.07, suggesting that is in any cases. Hereafter, we adopt a fiducial value of . Note that we obtain the value that is consistent with the fiducial value within the error, if we perform the power-law fitting for the redshift samples all together with no completeness thresholds. We also confirm that the value is unchanged from the fiducial value within the error, if we adopt another fitting model with the Srsic index different from in the size measurements in Section 3.2.

To test the influence by the AGNs, the gravitationally lensed sources, and the potentially low quality data in early ALMA cycles, we also derive the relation without these sources. For the AGN identifications, we cross-match the OC10S sources with the X-ray AGNs in the literature (Ueda et al. 2008; Xue et al. 2011; Civano et al. 2012). If the OC10S sources correspond to the central target AGNs in the initial ALMA observations, we also classify these OC10S sources as AGNs. For the lensed sources, we regard the OC10S sources as the potential lensed sources, if the optical-NIR counterparts of the OC10S sources have the photometric redshifts at . This is because the lensed sources typically have the lensing pairs at 1 (e.g., Simpson et al. 2016; Sonnenfeld et al. 2017). For the data in early ALMA cycles, we assume the all of the ALMA data in cycle 0/1 as the potentially low-quality data. Figure 10 presents the relations for the OC10S sources without the AGNs, the potential lensed sources, the potentially low-quality data of ALMA cycle 0/1 with the blue, green, and black shades, respectively, that are obtained from the perturbation method. For comparison, we also show our best estimate of the relation with the red hatched region. In Figure 10, we find that our best estimate of the relation is consistent with any of these three cases within the errors. We thus conclude that the relation is unchanged by the contaminations of the AGNs, the lensed sources, and the early ALMA cycles. Interestingly, in Figure 10, we also find that the X-ray AGNs are likely to be placed at the region where the values are higher than the general values of the other ALMA sources. Although there remain the uncertainties such as the measurements and the completeness of the X-ray source catalogs, it may indicate that the high values are related to the fueling mechanism of the AGNs.

Figure 10.— Comparison of the FIR sizeluminosity relations. The red circles present the OC10S sources in the same assignment as Figure 9. The blue, green, and black shade regions denote the 1 uncertainty range of the best-fit relation for the OC10S sources without the AGNs (blue squares), potential lensed sources (green diamonds), and the data in ALMA cycles 0/1 (black inner circles), respectively, that are estimated by the perturbation method. The red hatched region represents the 1 uncertainty range of the best-fit relation that is shown in Figure 9. The dashed lines are the constant surface SFR density in the same assignment as Figure 9.

We compare the value with that of UV wavelength, . One of the most extensive UV size study is performed by Shibuya et al. (2015) with star-forming galaxies at 08 from deep HST images. This comprehensive study obtain the value to be 0.27 0.01. Our estimate of shows a good agreement with the measurement. We discuss the physical origins of this value in Section 5.1.

Figure 11.— Redshift evolution of the rest-frame FIR sizeluminosity relation for the OC10S sources at . The colors indicate the redshift ranges described at the bottom right of each panel. The solid lines and the shade regions present the best-fit power law functions and the associated 1 errors, respectively. In the power law function fitting, the vlaue of each redshift is fixed to that is the best-fit value form the redshift samples all together.

We also investigate the redshift evolution of the relation. Because the source number is not enough in the redshift sample of 01, here we use the redshift samples of 12, 24, and 46. To overcome the small statistics in the individual redshift samples, we fix the value to estimate the values for all of the redshift samples. Figure 11 shows the best-fit functions and the associated errors with solid lines and the shade regions, respectively. For comparison, the 1-error region of 12 sample is presented in all of the redshift panels. We find that the best-fit values are tend to decrease toward high redshift. This trend is also consistent with that of the rest-frame UV and optical studies (e.g., Shibuya et al. 2015).

Note that the CMB effect may produce a bias in the measurements towards high redshifts (Zhang et al. 2016).  To test whether the decreasing trend is contributed by the CMB effect, we compare the values between the sources identified in ALMA Band 6 and Band 7 in the redshift sample of 46. This is because ALMA Band 6 is sensitive to the CMB effect more than Band 7, especially at . We find that there is no significant difference in the values between ALMA Band 6 and Band 7, indicating that the CMB effect is negligible in our results.

5. Discussion

In this section, we discuss the physical origins of the dusty starbursts based on the properties in the rest-frame FIR, optical, and the UV wavelengths.

5.1. Slope of FIR Size and Luminosity Relation

Here we discuss the physical origins of the slope of that is obtained in our rest-frame FIR study of the relation. In the rest-frame UV and optical studies, the sizemass relation shows the different power-law slopes in the star-forming and the compact quiescent galaxies. (e.g., Shen et al. 2003; Huang et al. 2013; van der Wel et al. 2014; van Dokkum et al. 2015). For the star-forming galaxies, the power-law slope is estimated to be that is consistent with the predictions from the disk formation models (Shen et al. 2003; Huang et al. 2013; van der Wel et al. 2014). For the compact quiescent galaxies, the relatively steep power-law slope of is obtained, which can be explained by the repeated mergers (Shen et al. 2003; van Dokkum et al. 2015). Our best estimate of shows the good agreement with the rest-frame UV sizeluminosity relation for the star-forming galaxies with (e.g., Huang et al. 2013; Shibuya et al. 2015), suggesting that the origin of the slope is also explained by the disk formation. In fact, Hodge et al. (2016) and Barro et al. (2016) report that the dusty star-forming galaxies has the disk-like morphologies in the rest-frame FIR wavelength. The CO observations support the existence of the gas disks with the rotating kinematics in the dusty star-forming galaxies (e.g., Hodge et al. 2012). Therefore, our result may indicate that our ALMA sources occur in the disk formation process that makes the slope of .

It is worth mentioning that we find no ALMA sources with 800 /yr/kpc. Based on the balance between the radiation pressure from the star-formation and the self-gravitation, Simpson et al. (2015) estimate the Eddington limit of the SMGs to be 1000 /yr/kpc, which is generally consistent with the maximum values among our ALMA sources. Although not all of our ALMA sources are likely to reach the Eddington limit, the slope of the relation may also be contributed by the Eddington limit.

Note that Lutz et al. (2016) show the negative slope in the relation for local (U)LIRGs. In the local (U)LIRGs, the strong gas compression such as major mergers is needed to occur the dusty starbursts due to the low gas fractions (e.g., Casey et al. 2014). The different mechanisms of the dusty starbursts between local and high redshifts probably provides the different slopes.

5.2. Sizes in UV, Optical, and FIR

Figure 12.— Left: Redshift evolution of the average values of (red cycles), (blue shade) and