A dominant population of optically invisible massive galaxies in the early Universe

TL;DR

This study uncovers a large population of massive, dusty galaxies at high redshift that are invisible in ultraviolet light, significantly impacting our understanding of early galaxy formation and star-formation history.

Contribution

It reveals a previously overlooked abundant population of massive, dusty galaxies at z > 3 through submillimeter observations, challenging existing galaxy formation models.

Findings

Discovered 39 massive galaxies at z > 3 unseen in UV/near-IR

These galaxies have a space density 100 times higher than extreme starbursts

They account for ten times more star-formation-rate density than UV-bright galaxies

Abstract

Our current knowledge of cosmic star-formation history during the first two billion years (corresponding to redshift z >3) is mainly based on galaxies identified in rest-frame ultraviolet light. However, this population of galaxies is known to under-represent the most massive galaxies, which have rich dust content and/or old stellar populations. This raises the questions of the true abundance of massive galaxies and the star-formation-rate density in the early universe. Although several massive galaxies that are invisible in the ultraviolet have recently been confirmed at early epochs, most of them are extreme starbursts with star-formation rates exceeding 1000 solar masses per year, suggesting that they are unlikely to represent the bulk population of massive galaxies. Here we report submillimeter (wavelength 870um) detections of 39 massive star-forming galaxies at z > 3, which are unseen in the spectral region from the deepest ultraviolet to the near-infrared. With a space density of about $2 \times 10^{-5}$ per cubic megaparsec (two orders of magnitudes higher than extreme starbursts) and star-formation rates of 200 solar masses per year, these galaxies represent the bulk population of massive galaxies that have been missed from previous surveys. They contribute a total star-formation-rate density ten times larger than that of equivalently massive ultraviolet-bright galaxies at z >3. Residing in the most massive dark matter halos at their redshifts, they are probably the progenitors of the largest present-day galaxies in massive groups and clusters. Such a high abundance of massive and dusty galaxies in the early universe challenges our understanding of massive-galaxy formation.

Figures

Tables

Table 1. Extended Data Table. 1: Survey depths for each field

Field	Area	WFC3/F160W ($5\sigma$)	H-dropouts	ALMA-detected	ALMA-undetected
	${\mathrm{arcmin}}^2$		()	($S_{870\mu m}>0.6$ mJy)	( mJy)
CANDELS-GDS	184	–$29.7$	12	10	2
CANDELS-UDS	202	–$27.6$	33	14	19
CANDELS-COSMOS	208	–$27.8$	18	15	3

Table 2. Extended Data Table. 2: Physical properties of H-dropouts

ID	R.A.	Decl.	[4.5]	$S_{870\mu m}$	$z$	Log $M_{\ast }$
	(J2000)	(J2000)		(mJy)		$M_{\odot }$
GDS-25526	03:32:47.97	-27:54:16.4	22.05	8.34$\pm$0.18	4.74${}_{-0.30}{}^{}{}_{}{}^{+0.28}$	10.84${}_{-0.17}{}^{}{}_{}{}^{+0.05}$
GDS-27571	03:32:30.62	-27:42:24.3	22.44	0.82$\pm$0.16	4.64${}_{-1.64}{}^{}{}_{}{}^{+0.19}$	10.89${}_{-0.48}{}^{}{}_{}{}^{+0.04}$
GDS-40613	03:32:11.44	-27:52:07.1	23.17	1.83$\pm$0.18	3.04${}_{-0.35}{}^{}{}_{}{}^{+0.22}$	10.64${}_{-0.22}{}^{}{}_{}{}^{+0.02}$
GDS-43215	03:32:20.34	-27:42:28.8	23.02	1.52$\pm$0.16	2.91${}_{-0.22}{}^{}{}_{}{}^{+0.19}$	10.25${}_{-0.53}{}^{}{}_{}{}^{+0.03}$
GDS-44539	03:32:28.59	-27:48:50.2	23.41	0.69$\pm$0.10	4.22${}_{-0.67}{}^{}{}_{}{}^{+0.77}$	10.94${}_{-0.31}{}^{}{}_{}{}^{+0.03}$
GDS-47375	03:32:14.62	-27:43:06.0	23.48	1.89$\pm$0.12	3.60${}_{-0.66}{}^{}{}_{}{}^{+0.66}$	10.27${}_{-0.12}{}^{}{}_{}{}^{+0.13}$
GDS-48764	03:32:32.31	-27:54:26.9	23.31	2.54$\pm$0.42	5.16${}_{-1.73}{}^{}{}_{}{}^{+3.08}$	10.38${}_{-0.03}{}^{}{}_{}{}^{+0.51}$
GDS-48885	03:32:47.17	-27:45:25.1	23.57	0.87$\pm$0.11	4.62${}_{-0.17}{}^{}{}_{}{}^{+0.19}$	10.56${}_{-0.08}{}^{}{}_{}{}^{+0.02}$
GDS-49094	03:32:31.85	-27:43:12.7	23.59	0.89$\pm$0.13	3.69${}_{-0.28}{}^{}{}_{}{}^{+0.29}$	10.29${}_{-0.05}{}^{}{}_{}{}^{+0.10}$
GDS-52734	03:32:10.10	-27:50:33.1	24.06	1.41$\pm$0.15	5.13${}_{-1.18}{}^{}{}_{}{}^{+1.72}$	10.71${}_{-0.21}{}^{}{}_{}{}^{+0.17}$
GDS-54513	03:32:04.99	-27:41:56.5	23.71	0.6	4.33${}_{-0.37}{}^{}{}_{}{}^{+0.32}$	10.30${}_{-0.37}{}^{}{}_{}{}^{+0.00}$
GDS-58560	03:32:40.11	-27:42:55.3	23.85	0.6	5.35${}_{-2.18}{}^{}{}_{}{}^{+2.33}$	10.58${}_{-0.43}{}^{}{}_{}{}^{+0.39}$
COS-16199	10:00:25.41	+02:25:43.9	21.96	3.91$\pm$0.09	6.54${}_{-1.54}{}^{}{}_{}{}^{+1.43}$	10.90${}_{-0.27}{}^{}{}_{}{}^{+0.14}$
COS-19762	10:00:15.89	+02:24:45.9	22.94	4.35$\pm$0.10	3.52${}_{-0.19}{}^{}{}_{}{}^{+5.36}$	10.79${}_{-0.08}{}^{}{}_{}{}^{+0.88}$
COS-23718	10:00:28.95	+02:25:05.3	22.82	2.25$\pm$0.10	5.77${}_{-0.88}{}^{}{}_{}{}^{+0.80}$	11.02${}_{-0.24}{}^{}{}_{}{}^{+0.13}$
COS-23913	10:00:23.03	+02:21:55.0	22.87	1.63$\pm$0.09	3.65${}_{-0.29}{}^{}{}_{}{}^{+0.35}$	10.65${}_{-0.11}{}^{}{}_{}{}^{+0.10}$
COS-24466	10:00:38.07	+02:28:06.2	23.22	1.38$\pm$0.09	3.35${}_{-0.38}{}^{}{}_{}{}^{+0.39}$	10.37${}_{-0.26}{}^{}{}_{}{}^{+0.34}$
COS-25270	10:00:23.62	+02:13:57.4	23.47	0.6	3.78${}_{-0.56}{}^{}{}_{}{}^{+0.55}$	10.48${}_{-0.20}{}^{}{}_{}{}^{+0.04}$
COS-25363	10:00:26.68	+02:31:26.2	23.15	3.0 $\pm$0.2	5.113${}_{-0.005}{}^{}{}_{}{}^{+0.001}$	10.52${}_{-0.19}{}^{}{}_{}{}^{+0.09}$
COS-25881	10:00:27.03	+02:24:24.0	22.96	1.30$\pm$0.10	6.58${}_{-1.38}{}^{}{}_{}{}^{+1.43}$	11.12${}_{-0.30}{}^{}{}_{}{}^{+0.17}$
COS-27285	10:00:27.79	+02:25:52.2	23.74	1.58$\pm$0.09	4.32${}_{-0.22}{}^{}{}_{}{}^{+0.23}$	10.31${}_{-0.06}{}^{}{}_{}{}^{+0.14}$
COS-27392	10:00:27.98	+02:25:29.7	23.42	0.6	3.61${}_{-0.49}{}^{}{}_{}{}^{+0.51}$	10.38${}_{-0.08}{}^{}{}_{}{}^{+0.11}$
COS-30182	10:00:14.70	+02:28:01.7	23.08	1.59$\pm$0.09	6.37${}_{-2.15}{}^{}{}_{}{}^{+1.16}$	10.94${}_{-0.28}{}^{}{}_{}{}^{+0.22}$
COS-30614	10:00:14.69	+02:30:04.6	23.39	0.85$\pm$0.10	3.97${}_{-0.29}{}^{}{}_{}{}^{+0.19}$	10.22${}_{-0.14}{}^{}{}_{}{}^{+0.04}$
COS-31278	10:00:26.09	+02:12:31.6	23.36	0.93$\pm$0.09	3.37${}_{-0.35}{}^{}{}_{}{}^{+0.43}$	10.14${}_{-0.15}{}^{}{}_{}{}^{+0.13}$
COS-31483	10:00:46.50	+02:23:09.1	23.89	0.70$\pm$0.11	2.97${}_{-0.41}{}^{}{}_{}{}^{+0.40}$	9.91${}_{-0.13}{}^{}{}_{}{}^{+0.13}$
COS-31661	10:00:41.83	+02:25:47.0	23.28	2.88$\pm$0.12	3.72${}_{-0.19}{}^{}{}_{}{}^{+0.17}$	10.36${}_{-0.20}{}^{}{}_{}{}^{+0.05}$
COS-32409	10:00:15.84	+02:23:04.0	23.66	0.70$\pm$0.11	3.91${}_{-1.16}{}^{}{}_{}{}^{+1.66}$	9.87${}_{-0.06}{}^{}{}_{}{}^{+0.48}$
COS-32932	10:00:22.44	+02:23:41.1	23.22	0.6	3.0971${}_{-0.001}{}^{}{}_{}{}^{+0.0002}$	9.96${}_{-0.11}{}^{}{}_{}{}^{+0.15}$
COS-34487	10:00:35.34	+02:28:26.7	23.36	4.3 $\pm$0.15	3.15${}_{-0.60}{}^{}{}_{}{}^{+0.52}$	10.18${}_{-0.20}{}^{}{}_{}{}^{+0.26}$
UDS-24945	02:16:59.77	-05:11:52.8	22.16	0.6	3.50${}_{-0.61}{}^{}{}_{}{}^{+0.76}$	10.76${}_{-0.11}{}^{}{}_{}{}^{+0.22}$
UDS-29006	02:17:05.52	-05:08:45.8	22.71	0.6	3.79${}_{-0.53}{}^{}{}_{}{}^{+0.54}$	10.46${}_{-0.21}{}^{}{}_{}{}^{+0.12}$
UDS-31037	02:18:07.67	-05:13:26.8	22.58	1.83$\pm$0.13	3.62${}_{-0.65}{}^{}{}_{}{}^{+0.70}$	10.49${}_{-0.07}{}^{}{}_{}{}^{+0.26}$
UDS-31072	02:17:43.32	-05:11:57.4	22.29	3.63$\pm$0.25	4.20${}_{-0.44}{}^{}{}_{}{}^{+4.17}$	11.31${}_{-0.09}{}^{}{}_{}{}^{+0.33}$
UDS-31959	02:18:11.36	-05:16:23.7	22.98	0.6	3.88${}_{-0.73}{}^{}{}_{}{}^{+0.68}$	10.22${}_{-0.14}{}^{}{}_{}{}^{+0.19}$
UDS-34637	02:18:05.80	-05:11:23.1	22.89	1.07$\pm$0.37	2.84${}_{-0.44}{}^{}{}_{}{}^{+0.54}$	10.33${}_{-0.17}{}^{}{}_{}{}^{+0.16}$
UDS-37344	02:18:02.86	-05:15:05.4	23.36	0.6	2.89${}_{-1.48}{}^{}{}_{}{}^{+1.43}$	10.18${}_{-0.63}{}^{}{}_{}{}^{+0.08}$
UDS-37423	02:18:10.02	-05:11:31.5	23.07	1.12$\pm$0.15	7.47${}_{-0.64}{}^{}{}_{}{}^{+0.63}$	11.17${}_{-0.08}{}^{}{}_{}{}^{+0.20}$
UDS-37560	02:17:03.44	-05:15:51.3	22.72	4.40$\pm$0.14	3.95${}_{-0.34}{}^{}{}_{}{}^{+0.35}$	10.35${}_{-0.17}{}^{}{}_{}{}^{+0.22}$
UDS-37649	02:17:36.95	-05:16:07.3	23.14	1.25$\pm$0.31	2.82${}_{-0.48}{}^{}{}_{}{}^{+0.44}$	10.31${}_{-0.07}{}^{}{}_{}{}^{+0.11}$
UDS-40772	02:17:36.56	-05:12:52.0	23.19	1.96$\pm$0.31	4.00${}_{-0.98}{}^{}{}_{}{}^{+0.98}$	10.87${}_{-0.40}{}^{}{}_{}{}^{+0.00}$
UDS-41502	02:17:18.03	-05:11:03.9	23.23	1.59$\pm$0.20	3.73${}_{-0.43}{}^{}{}_{}{}^{+0.44}$	10.59${}_{-0.33}{}^{}{}_{}{}^{+0.09}$
UDS-41525	02:16:59.59	-05:14:15.4	23.34	0.6	6.13${}_{-1.01}{}^{}{}_{}{}^{+1.08}$	11.17${}_{-0.16}{}^{}{}_{}{}^{+0.15}$
UDS-41773	02:18:07.02	-05:09:18.1	23.11	1.54$\pm$0.15	3.52${}_{-0.84}{}^{}{}_{}{}^{+1.35}$	10.20${}_{-0.24}{}^{}{}_{}{}^{+0.56}$
UDS-42280	02:18:11.16	-05:10:27.1	23.88	1.16$\pm$0.37	4.21${}_{-0.17}{}^{}{}_{}{}^{+0.18}$	10.06${}_{-0.16}{}^{}{}_{}{}^{+0.09}$
UDS-42875	02:18:21.15	-05:09:42.5	23.04	1.74$\pm$0.15	7.20${}_{-1.83}{}^{}{}_{}{}^{+1.20}$	11.75${}_{-0.34}{}^{}{}_{}{}^{+0.05}$
UDS-43941	02:17:43.65	-05:14:23.9	23.46	1.91$\pm$0.11	3.39${}_{-0.23}{}^{}{}_{}{}^{+0.23}$	10.17${}_{-0.10}{}^{}{}_{}{}^{+0.02}$
UDS-44515	02:18:20.89	-05:11:11.1	23.48	0.6	4.20${}_{-1.35}{}^{}{}_{}{}^{+1.35}$	10.59${}_{-0.27}{}^{}{}_{}{}^{+0.07}$
UDS-44594	02:17:20.20	-05:11:55.4	23.79	0.66$\pm$0.14	4.44${}_{-0.39}{}^{}{}_{}{}^{+0.52}$	10.60${}_{-0.17}{}^{}{}_{}{}^{+0.04}$
UDS-45868	02:18:15.00	-05:10:02.7	23.61	0.6	3.66${}_{-0.88}{}^{}{}_{}{}^{+0.88}$	10.45${}_{-0.13}{}^{}{}_{}{}^{+0.13}$
UDS-46241	02:17:58.31	-05:15:00.3	23.44	0.6	2.18${}_{-0.53}{}^{}{}_{}{}^{+0.78}$	9.72${}_{-0.38}{}^{}{}_{}{}^{+0.22}$
UDS-46513	02:18:17.87	-05:11:53.9	23.84	0.58$\pm$0.11	3.54${}_{-0.93}{}^{}{}_{}{}^{+0.95}$	10.31${}_{-0.19}{}^{}{}_{}{}^{+0.11}$
UDS-46648	02:17:08.17	-05:15:37.8	23.52	2.27$\pm$0.20	6.88${}_{-1.74}{}^{}{}_{}{}^{+1.67}$	11.37${}_{-0.43}{}^{}{}_{}{}^{+0.30}$
UDS-46693	02:17:59.07	-05:09:37.5	23.47	0.6	3.56${}_{-0.26}{}^{}{}_{}{}^{+0.37}$	10.24${}_{-0.29}{}^{}{}_{}{}^{+0.05}$
UDS-48514	02:17:29.83	-05:14:23.5	23.65	0.6	2.59${}_{-0.83}{}^{}{}_{}{}^{+0.93}$	10.04${}_{-0.42}{}^{}{}_{}{}^{+0.03}$
UDS-49119	02:17:07.14	-05:12:54.0	23.77	1.34$\pm$0.30	4.60${}_{-1.83}{}^{}{}_{}{}^{+1.94}$	10.65${}_{-0.53}{}^{}{}_{}{}^{+0.01}$
UDS-49199	02:18:21.40	-05:11:46.3	23.6	0.6	3.96${}_{-0.97}{}^{}{}_{}{}^{+0.84}$	10.66${}_{-0.24}{}^{}{}_{}{}^{+0.04}$
UDS-49594	02:18:01.13	-05:13:45.7	23.65	0.6	3.77${}_{-0.55}{}^{}{}_{}{}^{+0.48}$	9.79${}_{-0.26}{}^{}{}_{}{}^{+0.03}$
UDS-49784	02:17:37.48	-05:09:47.7	23.6	0.6	3.95${}_{-2.21}{}^{}{}_{}{}^{+1.63}$	9.91${}_{-0.77}{}^{}{}_{}{}^{+0.41}$
UDS-51119	02:17:58.29	-05:11:44.7	23.75	0.6	3.45${}_{-1.16}{}^{}{}_{}{}^{+1.16}$	10.25${}_{-0.13}{}^{}{}_{}{}^{+0.12}$
UDS-52324	02:17:06.27	-05:09:48.3	23.49	2.69$\pm$0.16	4.95${}_{-1.61}{}^{}{}_{}{}^{+1.74}$	10.62${}_{-0.26}{}^{}{}_{}{}^{+0.12}$
UDS-54074	02:17:09.70	-05:15:11.3	23.75	0.81$\pm$0.10	5.89${}_{-2.37}{}^{}{}_{}{}^{+1.90}$	10.65${}_{-0.35}{}^{}{}_{}{}^{+0.17}$
UDS-63094	02:17:53.06	-05:11:25.5	23.98	0.6	5.08${}_{-0.26}{}^{}{}_{}{}^{+0.24}$	10.00${}_{-0.27}{}^{}{}_{}{}^{+0.08}$

Equations

$$\omega (\theta )=\frac{HG(\theta )-HR(\theta )-GR(\theta )+RR(\theta )}{RR(\theta )}$$

$$\Delta \omega (\theta )=\frac{1+\omega (\theta )}{\sqrt{HG(\theta )}}.$$

$$\omega (\theta )=A_{\omega }{\theta }^{-\beta }-IC,$$

$$IC=\frac{\sum RR(\theta )A_{\omega }{\theta }^{-\beta }}{\sum RR(\theta )}.$$

$${\sigma }_{8,\mathrm{gal}}^2=\frac{72}{(3-\gamma )(4-\gamma )(6-\gamma )2^{\gamma }}{\left(\frac{r_0}{8h^{-1}\mathrm{Mpc}}\right)}^{\gamma }$$

$$b=\frac{{\sigma }_{8,\mathrm{gal}}}{{\sigma }_8(z)},$$

Full text

{affiliations}

Institute of Astronomy, Graduate School of Science, The University of Tokyo, Tokyo, Japan

AIM, CEA, CNRS, Université Paris Diderot, Saclay, Sorbonne Paris Cité, Gif-sur-Yvette, France

National Astronomical Observatory of Japan, Mitaka, Tokyo, Japan

Leiden Observatory, Leiden University, Leiden, The Netherlands

Department of Physics, University of Oxford, Oxford, UK

Research Center for the Early Universe, Graduate School of Science, Tokyo, Japan

Department of Physics, Anhui Normal University, Wuhu, China

Faculty of Physics, Ludwig-Maximilians-Universität, Munich, Germany

National Astronomical Observatories of China, Chinese Academy of Sciences, Beijing, China

Academia Sinica Institute of Astronomy and Astrophysics, Taipei,, Taiwan

Graduate Institute of Astrophysics, National Taiwan University, Taipei, Taiwan

Observations of galaxies across cosmic time have revealed that more massive galaxies have assembled their stellar masses at earlier epochs, with a significant population of massive ellipticals already in place at redshifts $z\sim 3-4$ Glazebrook:2017, Schreiber:2018c, Spitler:2014. The early assembly of these massive galaxies has posed serious challenges to current galaxy formation theories. Understanding their formation processes requires studies of their progenitors formed at even higher redshifts. However, most currently known high-redshift galaxies, including mainly Lyman-break galaxies (LBGs) and few extreme starbursts, are found inadequate to account for the large population of these early formed ellipticals, due to either low stellar masses and star formation rates, SFRs (for LBGsWilliams:2014) or low space densities (for the extreme starbursts). This suggests that the main progenitors of massive galaxies at $z>3$ remain to be found. Identification of these currently missing massive galaxies is key to our understanding of both massive-galaxy formation and the cosmic SFR density in the early universe.

The main targets of this study are a population of galaxies that are Spitzer/Infrared Array Camera (IRAC)-bright yet undetected in even the deepest near-infrared (NIR: H-band) imaging with Hubble Space Telescope (HST), that is, H-dropouts. (Throughout this Letter we use the short form “Telescope/Instrument” to represent usage of a particular instrument on a particular telescope.) In total, we have identified 63 $H-$dropouts with IRAC 4.5-$\mu$m magnitude, [4.5], less than 24 mag, within a total survey area of $\sim 600$ arcmin2 in deep CANDELS fields with typical depth of H $>27$ mag (5$\sigma$)(Fig. 1, Extended Data Table 1, Methods). Although previous studies have shown that these bright and red IRAC sources are promising candidates for massive galaxies atHuangJ:2011, WangT:2016a $z>3$, confirming their nature has been difficult so far owing to the limited sample size, the poor resolution of Spitzer and the lack of multiwavelength information. Here we explore their nature with high-resolution, 870 $\mu$m continuum imaging with the Atacama Large Millimeter/submillimeter Array (ALMA). With only 1.8 min of integration per object, 39 of them (detection rates of 62%) are detected down to an integrated flux of 0.6 mJy (4$\sigma$, Extended Data Fig. 1, Extended Data Table 2). Their 870-$\mu$m fluxes range from 0.6 mJy to 8 mJy, with a median of $S_{870\leavevmode\nobreak\ \mu m}$ =1.6 mJy (Extended Data Fig. 2). Hence most of them are fainter than the 2-mJy confusion limit of the single dish instruments that discovered submillimeter galaxies (SMGs), and much fainter than most SMGs studied until now with typicalSwinbank:2014 $S_{870\leavevmode\nobreak\ \mu m}\gtrsim 4$ mJy. The sky density of these ALMA-detected H-dropouts is approximately $5.3\times 10^{2}$ deg*-2* after correction for incompleteness (Methods), two orders of magnitude higher than Herschel/SPIRE-selected extreme starbursts (with SFR $\gtrsim 1000\leavevmode\nobreak\ M_{\odot}$ yr*-1*) Riechers:2013, Dowell:2014.

The ALMA detections confirm unambiguously that most of the H-dropouts are dusty star-forming galaxies at high redshifts, consistent with their admittedly uncertain photometric redshifts–from optical spectral energy distribution (SED) fitting–with median redshift $z_{\rm median}=4$ (Extended Data Fig. 3). Further insights into their properties are obtained from the stacked infrared (IR) SED of the 39 ALMA-detected H-dropouts from MIPS 24 $\mu$m up to ALMA 870 $\mu$m. The stacked SED peaks between the observed 350 and 500 $\mu$m (Extended Data Figure 3), consistent with being at $z\sim 4$. With a median stellar mass of $M_{*}\sim 10^{10.6}M_{\odot}$ and a characteristic IR luminosity (over $8-1000$ $\mu$m) of $L_{\it IR}=2.2\pm 0.3\times 10^{12}L_{\odot}$ ($L_{\odot}$, solar luminosity) derived from the stacked SED, these ALMA-detected H-dropouts are fully consistent with being normal massive star-forming galaxies atSchreiber:2018a $z=4$ (Fig. 2). Moreover, the ALMA detections also provide crucial constrains on the redshift of individual galaxies. Combined with SCUBA-2 450 $\mu$m and VLA 3 GHz data, the majority of the ALMA-detected H-dropouts exhibit red $S_{870\mu m}/S_{450\mu m}$ and $S_{1.4GHz}/S_{870\mu m}$ colors that are suggestive of redshifts of $z>3$ (Extended Data Fig. 4). Similarly, the non-detections at 24 $\mu$m (5$\sigma$ detection limit of 20 $\mu$Jy) for most of the sources implies red $S_{870\leavevmode\nobreak\ \mu m}/S_{24\mu m}$ colors that are also consistent with $z>3$ assuming typical SED templatesDaddi:2009a. We hence conclude that whereas the estimated redshifts for individual galaxies exhibit a large uncertainty, all the available data points to the ALMA-detected H-dropouts being massive, dusty star-forming galaxies at $z>3$.

For the remaining approximately $40\%$ of H-dropouts that are not detected with ALMA, photometric redshift estimates based on their optical SEDs suggest a similar redshift distribution to that of ALMA-detected ones, with $z_{\rm median}=3.8$(Extended Data Figure 2). Their stacked ALMA 870 $\mu$m image yields a $6\sigma$ detection with $S_{870\mu m}=0.24\pm 0.04$ mJy, approximately 8 times lower than that of ALMA-detected ones, suggesting lower specific SFRs compared to ALMA-detected ones, which is also confirmed by a full fitting of the stacked optical-to-IR SEDs (Extended Data Fig. 5).

Spectroscopic confirmation of H-dropouts has been so far limited to a few sources, which are all found at $z>3$. Most of these confirmed cases are extreme SMGs with $S_{870\mu m}\gtrsim 10$ mJy, for example$\cite[cite]{\@@bibref{Number}{Walter:2012}{}{}}.,HDF-850at$z = 5.18. An H-dropout galaxy with submillimeter flux similar to that of our sample ($S_{744\mu m}=2.3\pm 0.1$ mJy) has been recently confirmedSchreiber:2018b to be at $z=3.709$: it was discovered serendipitously near a quiescent galaxy at the same redshift Glazebrook:2017. By targeting 3 H-dropouts in our sample that show significant excess ($>4\sigma$, Methods) in Subaru medium bands in the optical ($\sim$ 3500-6000 Å) with VLT/X-shooter, we have successfully detected Lyman-$\alpha$ for two of them and confirmed their redshifts to be $z>3$ ($z=3.097$ and $z=5.113$, Extended Data Fig. 6). These spectroscopic redshifts ($z_{\rm spec}$ are in good agreement with their photometric redshift ($z_{\rm phot}$) based on UV-to-NIR SED fitting, with $\sigma_{\Delta z/(1+z_{spec})}\sim 0.1$.

Having established that most of the H-dropouts are massive galaxies at $z>3$, we now derive their contribution to the cosmic SFR density and stellar mass function. Whereas populations of similarly red galaxy populations are known to exist at lower redshiftsRiguccini:2015, these largely overlap with the stellar-mass-limited sample used to estimate the SFR density at $z<3$. Assuming that the intrinsic infrared SED of the ALMA-detected H-dropouts is the same as the SED derived from stacking, the SFR density of ALMA-detected H-dropouts (in 10${}^{-3}M_{\odot}$ yr*-1* Mpc*-3*) reaches about 2.9, 2.1, and 0.9 at $z=$ 4, 5, 6, respectively, or approximately 1.6$\times 10^{-3}M_{\odot}$ yr*-1* Mpc*-3* when averaged over the three bins (Fig. 3). This corresponds to about 10% of the SFR density from LBGs at similar redshiftsBouwens:2012. However, if we focus only on LBGs with masses similar to those of $H-$dropouts with $M_{*}>10^{10.3}M_{\odot}$, the SFR densities of $H-$dropouts are one to two orders of magnitude higher, demonstrating that H-dropouts dominate the SFR density in massive galaxies. This dominance is further reflected in the stellar mass functions, as shown in Fig. 3. The fraction of $H$-dropout becomes progressively higher at higher masses. At $M_{*}\gtrsim 10^{10.5}M_{\odot}$, the number density of $H$-dropout surpasses that of LBGs. Moreover, if we also include galaxies detected in $H-$band but which show similar red colors ($H-[4.5]>2.5$, Extended Data Fig. 7)Spitler:2014, WangT:2016a, they make up more than than 80% of the most massive galaxies at $z>4$. Taken together, these results suggest that the majority of the most massive galaxies at $z>3$ have indeed been missed from the LBG selection, and are optically dark.

To put the H-dropouts in the context of the cosmic evolution of massive galaxies, we probe their clustering properties through their cross-correlation with $H$-detected galaxies at $3.5<z<5.5$ from the CANDELS survey in the same three fields (Extended Data Fig. 8, Methods). The derived galaxy bias, that is, the relationship between the spatial distribution of galaxies and the underlying dark matter density field, for the H-dropouts is $b=8.4\pm 1.5$, corresponding to a dark matter halo mass of $M_{\rm h}\sim 10^{13\pm 0.3}h^{-1}M_{\odot}$ at $z=4$ (Fig. 4, Methods). This halo mass of H-dropouts is consistent with them being progenitors of the most massive quiescent galaxies at $z=2-3$, as well as progenitors of today’s ellipticals that reside in the central region of massive groups and clusters.

The discovery and confirmation of these H-dropouts as massive galaxies at $z\approx 3-6$ alleviates greatly the tension between the small number of massive LBGs at $z>3$ and the rapid emergence of massive (and quiescent) galaxies at $z\approx 2-3$. Assuming an average redshift of $z\approx 4$ and SFR $\approx 220\leavevmode\nobreak\ M_{\odot}$ yr*-1*, these H-dropouts will grow in stellar mass by $1.3\times 10^{11}M_{\odot}$ before $z\sim 3$. Their number density, $n\sim 2\times 10^{-5}$ Mpc*-3*, is also comparable to that of the most massive, quiescent galaxies at $z\sim 3$ withStraatman:2014 $M_{*}>10^{11}M_{\odot}$. The early formation of such a large number of massive, dusty galaxies is unexpected with current semi-analytical modelsHenriques:2015, which underestimates their density by one to two orders of magnitude (Fig. 3). Similarly, a deficit of such galaxies is also present in hydrodynamic simulations, which contain no such galaxies at $z>3$ in mock deep fields ($\sim 23.5$ arcmin2) from the Illustris Project Snyder:2017. Moreover, even considering LBGs alone, the number of massive galaxies already appears too large when compared to the number of massive halos at $z>4$ predictedSteinhardt:2016 by our current understanding of galaxy evolution in the Lambda Cold Dark Matter (LCDM) framework. Together, this unexpected large abundance of massive galaxies in the early Universe suggests that our understanding of massive-galaxy formation may require substantial revision. Spectroscopic follow-up of the whole population of H-dropouts would be key to providing further insights into this question, which calls for mid-infrared spectroscopy with James Webb Space Telescope in the near future.

1 Observations

1.1 Selection of $H-$dropouts and incompleteness correction

We have crossmatched the F160W-selected catalog from the three CANDELS fields (Table 1) with an IRAC 3.6 and 4.5 $\mu$m selected catalogAshby:2013 from the SEDS survey. The SEDS survey covers the three fields of H-dropouts to a depth of 26 AB mag (3$\sigma$) at both 3.6 and 4.5 $\mu$m and is 80% complete down to [4.5] $\sim$ 24 mag. We first matched sources with $[4.5]<24$ mag in the SEDS catalog to the F160W-selected catalog and identified those without H-band counterparts within a 2” radius (corresponding roughly to the PSF size of IRAC 3.6 and 4.5 $\mu$m). This 4.5 $\mu$m magnitude cut was applied to enable sufficient color range to identify extremely red objects while keeping a complete 4.5 $\mu$m selected sample. We then visually inspected the IRAC images and excluded sources whose flux is contaminated by bright neighbors as well as those falling on the edge of the F160W image. With knowledge of their positions, some of these H-dropouts are marginally detected in the F160W band but exhibit extended profiles and are unidentifiable as real sources without that prior knowledge. This left us 63 sources with 2 of them serendipitously detected in previous band-7 continuum observations with ALMA.

The criterion of no HST counterparts within 2” radius ensures a clean selection of H-dropouts with reliable constrains of IRAC fluxes. However, given the high density of HST sources in these deep fields, the chance probability of an IRAC-HST coincidence (with distance $<$ 2”) is non-negligible . This means that we may have missed some $H-$dropouts simply due to the presence of a random HST source falling within the 2” search radius of the IRAC source. To correct for this effect, we calculate the completeness of this selection approach, which is defined as at a given position the probability of finding zero galaxies in the 2” radius, $p(n=0)={\rm exp}(-N*\pi*$ radius2), with $N$ representing the surface density of HST sources. Averaging over the three CANDELS fields yields $N=0.05$ arcsec*-2*, implying $p(n=0)=0.53$. This suggests that while our approach yields a clean selection of $H-$dropouts, roughly half of the true $H-$dropouts have been missed simply due to chance superposition of sources, which needs to be corrected. In fact, this completeness correction is consistent with recent findings from a blind ALMA survey, which reveals four H-dropouts (with $[4.5]<24$) that were not picked up by our approach within an area of 1/3 of the GOODS-South filed Franco:2018, Yamaguchi:2019, in comparison to 12 sources selected by our approach in the whole GOODS-South field. Among these four sources, 3 of them have at least one HST counterpart within 2” (with the remaining one absent from our IRAC catalog, which is shallower than the one used in Franco:2018), which is inconsistent with being the right counterpart of the ALMA emission based on the redshift and other physical properties. Albeit with small number statistics, this implies a completeness of our searching approach of $\sim 57\%$, consistent with our estimated value. In addition to this correction, we need to also correct for the incompleteness of the IRAC imaging from the SEDS survey, which ranges from 93% at [4.5] =22 to 75% -80% at $[4.5]=24$ in the three fields. Combining the two corrections, a factor of 2 to 2.4 has been applied to the number density (including also star formation rate and stellar mass density) of H-dropouts depending on their IRAC fluxes.

1.2 Multiwavelength photometry

In each field, we gathered mosaics in a large number of bands, including all the images used to build the 3DHST Skelton:2014 and ZFOURGE Straatman:2016 catalogs. All our galaxies therefore had rich and deep photometry from the UV to the NIR, reaching typical $5\sigma$ depths (AB) of $27$ in u to i, $26$ in z to H, and $25$ in Ks. We provide the full detail of the used mosaics below.

For GOODS-South, we used VLT/VIMOS images in the U and R bands Nonino:2009, ESO/WFI images in the U, U38, B, V, R, I bands from GaBoDS Hildebrandt:2006, CTIO/MOSAIC image in the z band from MUSYC Gawiser:2006, Subaru images in 15 medium bands from MUSYC Cardamone:2010, Hubble images in the F395W, F606W, F775W, F8514W, F850LP, F105W, F125W, F160W bands from GOODS and CANDELS programs Giavalisco:2004, Grogin:2011, Koekemoer:2011, VLT/ISAAC images in the J, H, Ks bands Retzlaff:2010, CFHT/WIRCam images in the J and Ks bands from TENIS Hsieh:2012, Magellan/FOURSTAR images in the J1, J2, J3, Hs, Hl, Ks bands from ZFOURGE Straatman:2016, a VLT/HawK-I image in the Ks band from HUGS Fontana:2014, and Spitzer IRAC images from SEDS Ashby:2013.

For UDS, we used a CFHT/Megacam image in the u band produced by the 3DHST team Skelton:2014, Subaru images in the B, V, R, i, z bands Furusawa:2008, Hubble images in the F606W, F814W, F125W, F140W, F160W bands from the CANDELS and 3DHST programs Koekemoer:2011, Momcheva:2016, UKIRT/WFCAM images in the J, H, K bands from UKIDSS Lawrence:2007, Magellan/FOURSTAR images in the J1, J2, J3, Hs, Hl, Ks bands from ZFOURGE Straatman:2016, VLT/HawK-I images in the Y and Ks bands from HUGS Fontana:2014, and Spitzer IRAC images from SEDS Ashby:2013 and SpUDS (PI: J. Dunlop).

For COSMOS, we used CFHT/Megacam images in the u and i bands from CFHTLS Cuillandre:2012, Subaru images in the B, g, V, r, i, z bands as well as 10 medium bands Taniguchi:2007, Hubble images in the F606W, F814W, F125W, F140W, F160W bands from the CANDELS and 3DHST programs Koekemoer:2011, Momcheva:2016, CFHT/WIRCam images in the H and Ks bands mccracken:2010, Magellan/FOURSTAR images in the J1, J2, J3, Hs, Hl, Ks bands from ZFOURGE Straatman:2016, VISTA/VIRCAM images in the Y, J, H, Ks from UltraVISTA DR3 Mccracken:2012, and Spitzer IRAC images from SEDS Ashby:2013 and S-COSMOS Sanders:2007.

The photometry was obtained with a procedure very similar to that previously used in deep surveys Straatman:2016, Momcheva:2016, which we summarize here. Fluxes in UV-to-NIR were extracted on re-gridded and PSF-matched images in fixed apertures of 2*′′* diameter. Because of the broader PSF in Spitzer images, fluxes in the IRAC bands were extracted separately, with a 3” aperture and without PSF matching. The asymmetric IRAC PSF was rotated to match the telescope roll angle for each field. Prior to extracting the fluxes, all the neighboring sources within a 10” radius were subtracted from the images. This was done by identifying the sources from a stacked detection image, and using the HST F160W profile of each source as a model. These models were convolved by the PSF of each image, where they were fit simultaneously using a linear solver. Most often the dropouts were not found in the stacked detection image, and were therefore modeled as point-sources at the coordinates of their IRAC centroid during the de-blending stage. Once the flux was extracted, additional ”sky” apertures were placed randomly around each dropout. The median flux in these sky apertures was subtracted from the dropout’s flux, to eliminate any remaining background signal, while the standard deviation of these fluxes was used as flux uncertainty. Lastly, fluxes and uncertainties were aperture-corrected using the matched PSF’s light curve, assuming point-like morphology.

1.3 ALMA observation and data reduction

Our ALMA band-7 continuum observations of H-dropouts are performed during January and July 2016. The observations were centered on the IRAC positions with a spectral setup placed around a central frequency of 343.5 GHz. While we asked 0.7*′′-resolution observation for all the three fields, only the CANDLES-COSMOS field was observed as requested, yielding a synthesis beam of 0.6\times 1\hbox{{}^{\prime\prime}}. The other two fields were observed at 0.2-0.3′′* resolution. The integration time is roughly 1.8 mins per object with a total observing time of $\sim$2 h. We reduced the data using the CASA pipeline (version 4.3.1). To reach an homogeneous angular resolution, we tapered the baselines for these two fields to an angular resolution of 0.6*′′*. This resolution corresponds to $\sim$ 4 kpc at $z=4$, compared to typical sizes of $\sim$2 kpc for SMGs Hodge:2016.

We measured the total flux of all our targets directly in the $(u,v)$ plane using the uvmodelfit procedure from the CASA pipeline. The sources were modeled with a circular Gaussian profile of variable total flux, centroid, width, axis ratio and position angle. 39 H-dropouts are detected at S/N$>4$ with $S_{870\mu m}>0.6$ mJy, including two galaxies that were serendipitously detected in a previous ALMA programSchreiber:2016 targeting $H$-detected $z\sim 4$ galaxies, which has reached similar depth as this observation. The positions of the 870 $\mu$m emission as measured from ALMA are in good agreement with IRAC, with $\Delta$RA=0.081\pm 0.128\hbox{{}^{\prime\prime}} and $\Delta$DEC= -0.13\pm 0.16\hbox{{}^{\prime\prime}}.

1.4 SCUBA-2 450 $\mu$m and VLA observations

One of the three $H$-dropout fields, CANDELS-COSMOS, is covered by deep SCUBA-2 450 $\mu$m and 870 $\mu$m observations from the STUDIES surveyWangW:2017. Previous observations with JCMT/SCUBA-2 at the same regionCasey:2013, Geach:2013, Geach:2017 have also been combined to produce an extremely deep 450 $\mu$m image and a confusion-limited 850 $\mu$m image. The instrumental noises at 450 $\mu$m and 850 $\mu$m at the deepest regions reach $\sim$0.65 mJy and $\sim 0.1$ mJy, respectively.

The SCUBA2-450 $\mu$m and -850 $\mu$m fluxes for H-dropouts are measured at the position of the IRAC 3.6 and 4.5 $\mu$m emission with the prior-based PSF-fitting code FASTPHOTBethermin:2010b. We further restrict all the extracted fluxes to be positive with bounded value least-square minimization. During the fit we have included all the MIPS 24 $\mu$m- and VLA detections as priors to perform source extraction. The VLA 3 GHz observation in COSMOSSmolcic:2017 reaches a rms of 2.3 $\mu$Jy/beam at an angular resolution of 0.75*′′, which is deep enough to put useful constraints on their redshifts. The flux measurement for H-dropouts in the far-IR suffers minimum source confusion due to our selection criterion (no close neighbours within a 2′′* radius). A comparison of 870 $\mu$m fluxes measured by ALMA and SCUBA-2 yields excellent agreement with a median value of $S_{\rm ALMA}/S_{\rm SCUBA-2}=1.05$.

1.5 X-SHOOTER spectra

In the COSMOS field, deep medium band images in the optical were obtained with the Subaru telescopeTaniguchi:2007. We visually inspected these images at the location of each dropout in our sample and found three galaxies with flux excesses in one of these images, with a significance above 4 sigma. Examples are shown on Extended Data Figure 6. Such flux excess can be interpreted as coming from a bright emission lineSobral:2018. For these three dropouts, the line could be identified as Ly$\alpha$ at $z=5.0$, $3.2$ and $4.1$, respectively. Even though H-dropouts are typically very obscured, Ly$\alpha$ may still be detected through un-obscured sight lines, or by scatteringFinkelstein:2008. Judging from the spatial offsets of about 1\hbox{{}^{\prime\prime}} we observed between this optical flux excess and the Spitzer–IRAC or ALMA emission, scattering appears to be the most plausible explanation.

We thus followed up these objects with VLT/X-SHOOTER to confirm the presence of an emission line. Each dropout was observed in May 2018 in the UVB and VIS arms for 50 minutes in stare mode (no nodding), split in three exposures. The 2D spectra were reduced using the standard pipeline, and 1D spectra were produced by fitting a Gaussian profile to each spectral slice. Uncertainties were controlled by computing the standard deviation of spectral elements in regions without sky lines; we found that the 1D uncertainty spectrum had to be rescaled upwards by a factor 1.27 to match the observed noise.

We then searched for emission lines in the spectra, considering only the wavelength range covered by the Subaru medium band in which the flux excess was previously identified. The result of this search is displayed on Extended Data Figure 6. We found a 10$\sigma$ detection at 0.498$\mu$m for the dropout 32932, corresponding to $z_{\rm spec}=3.0971\pm 0.0002$, and a more marginal but still significant 4.3$\sigma$ detection at 0.739$\mu$m for the dropout 25363, corresponding to $z_{\rm spec}=5.113^{+0.001}_{-0.005}$. Because our search space is tightly limited by the Subaru passband, the latter only has a 0.4% chance of being spurious, and we therefore consider it a reliable detection. The third dropout showed no significant line emission above $2\sigma$.

1.6 Lyman-break galaxy selection

In order to compare the properties of $H-$dropouts and LBGsSteidel:1996, we have selected LBGs using the ZFOURGE catalogs in the same three CANDELS fieldsStraatman:2016. The advantage of the ZFOURGE catalog is that it is essentially a $K{{}_{s}}-$band selected catalog, for which the deep $K_{s}$ band data provides critical constraints on the redshift and stellar masses estimates at $z>4$. We select our $z=4-6$ LBG galaxy sample using the selection criterion in Bouwens:2015. Due to the lack of B-band data from HST, the $z\sim 4$ LBG sample is only limited to GOODS-South field while the $z\sim 5$ and $z\sim 6$ LBG sample include galaxies from all the three fields. To enable a clean selection of galaxies with reliable flux density measurements, we have further limited the selection to galaxies with $use=1$ as recommendedStraatman:2016. This reduces the effective area to 132.2, 139.2, and 135.6 arcmin2 for GOODS-South, COSMOS, and UDS, respectively. To identify total SFRD from massive LBGs with $M_{*}>10^{10.3}M_{\odot}$, we utilized the latest determination of the UV luminosity function at $z\sim 4-6$Ono:2018. Taking into account variations in the $M_{*}-M_{UV}$ relation, this mass cut corresponds to $M_{UV}<[-21.55,-22.04,-22.27]$ at $z=[4,5,6]$, respectively. We then derive the dust-corrected SFR for these brightest UV-selected galaxies following the approach in ref.Bouwens:2012.

2 Determination of Physical Properties

2.1 Stacked UV-to-NIR SEDs

To produce the stacked UV-to-NIR SEDs, we took the fluxes of each galaxy in our photometric catalog and normalized them by their respective IRAC $4.5\,\mu$m flux. We then computed the mean flux in each band, using inverse variance weighting, and finally multiplied the resulting stacked fluxes by the average $4.5\,\mu$m flux of the stacked sample. In the stack, we combined bands that have similar effective wavelengths, even though the true passbands could be slightly different; for example we stacked together all the $K$s bands from UKIDSS, UltraVISTA, FOURSTAR, WIRCam, and ISAAC into a single $K$s band. The uncertainties on the stacked fluxes were derived by formally combining the uncertainties of each stacked galaxy. We note that, since we obtained our photometry using fixed-size apertures, this method is strictly equivalent to stacking the images.

2.2 Photometric redshift and stellar mass determination

Using the aforementioned multiwavelength photometry, including bands with formal non-detections, photometric redshifts were computed with EAzY Brammer:2008 using the full set of template SEDs, namely, including the old-and-dusty template and the extreme emission line template. The prior on the observed magnitudes was not used. Using these redshifts, we then ran FAST Kriek:2009a to estimate the stellar masses. We assumed a delayed-exponentially-declining star-formation history, with a range of age and exponential timescale. Dust attenuation was modeled with the Calzetti:2001 prescription, allowing $A_{V}$ up to 6 magnitudes. Metallicity was fixed to solar during the fitting. We also used the infrared luminosities inferred from the ALMA fluxes to further constrain the fits. This was implemented as follows. From the stacked Herschel SED (see Figure 3), we measured the mean dust temperature of our sample: $T_{\rm dust}=36.7\pm 2.1\,{\rm K}$. Based on Herschel and ALMA observations of $z>2$ galaxies Schreiber:2018a, we expect a typical scatter of $5\,{\rm K}$ around the average temperature at any given redshift. Assuming this distribution of temperatures holds for the dropouts, we generated probability distributions for $L_{\rm IR}$ using a Monte Carlo procedure: the measured ALMA flux was randomly perturbed with Gaussian noise of amplitude set by the flux uncertainty, and the dust temperature was drawn from a Gaussian distribution centered on $36.7\,{\rm K}$ and with a width of $5\,{\rm K}$; the resulting dust SED was then used to extrapolate $L_{\rm IR}$ from the ALMA measurement. For galaxies whose ALMA flux has $S/N<2$, the resulting probability distribution of $L_{\rm IR}$ was close to Gaussian, while for the detections the probability distribution was close to log-normal. We modeled these two regimes accordingly in the fit, by assuming either Gaussian noise on $L_{\rm IR}$ or $\log_{10}(L_{\rm IR})$, respectively. The observed infrared luminosity was then compared to the modeled value, which we computed as the difference of bolometric luminosity before after applying dust attenuation. This resulted in an additional contribution to the $\chi^{2}$, which was then used for standard model selection.

Uncertainties on the photometric redshifts were derived from the 16th and 84th percentiles of the probability distribution produced by EAzY. This accounts for uncertainty in the photometry as well as on the model galaxy templates. Uncertainties on the derived physical parameters, including the stellar mass, were derived using Monte Carlo simulations, where the observed photometry was randomly perturbed with Gaussian noise of amplitude determined by the estimated photometric uncertainties. This was repeated 200 times. The error bars on physical parameters were then derived from the 16th and 84th percentiles of the distribution of the values obtained in the Monte Carlo simulations. For each fit, the redshift was left free to vary within the 68% confidence interval reported by the photometric redshift code. Therefore the resulting error bars account for uncertainties on the photometry and on the redshift.

2.3 Clustering measurements

Since the number of H-dropouts is small, we calculate two-point angular cross-correlation function (CCF) with a much larger population of galaxies sharing the same cosmic volume (redshifts) in order to enhance the statistics. Specifically we select all the galaxies with $3.5<z<5.5$ from the H-selected catalog in the same three CANDELS fields (“the galaxy sample”, hereafter), and then calculate CCF using the estimator as follows Landy:1993:

[TABLE]

where $HG$, $HR$, $GR$, $RR$ are Hdropout-galaxy, Hdropout-random, galaxy-random and random-random pair counts respectively. The random galaxy sample are created within the same CANDELS footprint as the H-dropouts (we exclude HUDF in the GOODS-S field because of its much deeper integration than other regions). The uncertainties of CCF are estimated as:

[TABLE]

We then fit the derived CCF with a power-law model:

[TABLE]

where $A_{\omega}$ is the correlation amplitude and $\beta$ is the power-law index fixed to 0.8 and $IC$ is the integral constraint. Integral constraints is an offset due to the clustering measurement over the limited area and is calculated by

[TABLE]

The derived correlation amplitude can be converted to three-dimensional correlation length $r_{0}$ by Limber equationLimber:1953 modified by Croom:1999 for the cross-correlation.

The correlation length is related to galaxy bias $b$, such that

[TABLE]

and

[TABLE]

where $\sigma_{8,\mathrm{gal}}$ is a galaxy fluctuation, $\gamma=1+\beta$, and $\sigma_{8}(z)$ is a matter fluctuationPeebles:1993. The halo mass is then derived from the estimated galaxy biasMoH:2002.