A Radio-to-millimeter Census of Star-forming Galaxies in Protocluster 4C 23.56 at z = 2.5: Global and Local Gas Kinematics

We used ALMA Band 4 receivers to obtain the redshifted CO (4–3) emission (ID: ADS/JAO.ALMA #2015.1.00152.S, PI: Minju Lee). We designed our observations to use two pointing positions with the primary-beam size of a ≈47'' diameter circle each, i.e., 50% of the full response of the antenna (Figure 1). A total of 16 out of 25 parent HAEs were targeted in this program. The field coverage was set differently from the previous CO (3–2) observations considering the observational efficiency. As a result, six galaxies (HAE 5, 14, 15, 18, 17, 22)  were not covered by the CO (4–3) observations. We highlight this point especially for HAE 5, which was detected in the CO (3–2) line.

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A total of either 38 or 39 antennas were used with the baseline length (L_baseline) between 15 m and 3.1 km (C40-6 configuration, which is equivalent to C36-(5)/6 in cycle 3). The total on-source time (T_integ.) was 2.7 hr with the two-pointing observations, thus ≃1.4 hr per pointing. We used four spectral windows (SPWs), with two placed in the upper and the lower sidebands each. One SPW was set in the Frequency-Division Mode (FDM) to detect the redshifted CO (4–3) line by having a channel width of 7.82 MHz (∼17.7 km s⁻¹) and a total bandwidth of 1.875 GHz. The remaining three SPWs were observed in the Time-Division Mode (TDM) with a 2.0 GHz bandwidth at 15.6 MHz resolution to cover the dust continuum at 2 mm. Two quasars, J2148+0657 and J2025+3343, were chosen for bandpass and flux calibration, respectively. The phase calibrator was J2114+2832.

We used CASA (McMullin et al. 2007) version 4.7.0 and 5.1.2 for the calibration of visibility data and imaging, respectively. We first processed with the pipeline script provided by the EA-ARC. Then, additional flagging of T_sys outliers was applied manually to improve the image qualities after the inspection of the obtained first images and the pipeline logs. We made final images using the recalibrated data sets after flagging.

Images were produced by CASA task tclean and deconvolved down to the 2σ noise level. Clean masks were created based on the positions of HAEs, using circular regions with a radius of 3× the beam size. The typical noise level in the final image is 0.14 mJy beam⁻¹ at 80 km s⁻¹. The synthesized beam is 052 × 032, which is a finer beam by a factor of 2, compared to the beam sizes of our previous ALMA observations of the CO (3–2) line and 1.1 mm continuum in Paper I (i.e., 091 × 066 and 078 × 068 for the CO (3–2) and the 1.1 mm observations, respectively). All images were analyzed after subtracting the continuum emission using uvcontsub. Images without continuum subtraction were also investigated to test whether the continuum has been misidentified as a (broad) line detection. This was intended for cross-checking the detection with large line widths and for justifying the velocity range used in the continuum subtraction.

We regard a galaxy as detected by imposing double-step criteria. First, we select candidates if (a) the signal-to-noise ratio (S/N) of the peak of the velocity-integrated intensity (I_CO43,peak) is equal to or greater than 4.5 (${\rm{S}}{/{\rm{N}}}_{{I}_{\mathrm{CO}43,\mathrm{peak}}}$ ≥ 4.5). The noise estimate for the peak line intensity is based on the calculation after taking into account the channel noise and integrating velocity range, following Hainline et al. (2004). We then investigate whether the peak position spectrum satisfies either of the following criteria: (b) a peak flux density (S_CO43,peak) ≥ 3.5σ, or (c) at least two continuous channels including a maximum peak-flux channel with fluxes >2.5σ, where σ is the average channel noise level, estimated from the line-free positions. These "two-step" criteria were successful in detecting CO (3–2) lines in Paper I. Provided the success of CO (3–2) line detection, we opt to apply a similar detection analysis for CO (4–3) as well. 

Column (9) in Table 1 shows the set of criteria satisfied for individual galaxies, which is used to determine the CO (4–3) line detection. One can also check whether or not criteria (a) and (b) are satisfied by referring to columns (4) and (5), respectively, in Table 1.

Table 1.  Identification of CO (4–3) Line for HAEs

ID	R.A.${}^{\diamond }$	Decl.${}^{\diamond }$	ICO43,peak	Speak,CO43	ICO43,Gaussian	zCO43	FWHM	Criteria†	CO (4–3)	CO (3–2)‡
	(J2000)	(J2000)	(Jy km s−1 beam−1)	(mJy beam−1)	(Jy km s−1)		(km s−1)	(CO43/CO32)	detection?	detection?
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
HAE 1	21:07:14.802	+23:31:44.593	0.32 ± 0.06	1.06 ± 0.25	0.45 ± 0.13	2.4945 ± 0.0009	414 ± 123	abc/none	y	n
HAE 2	21:07:21.739	+23:31:50.123	0.20 ± 0.04	0.79 ± 0.22	0.13 ± 0.05	2.4893 ± 0.0009	267 ± 114	ab/none	y	n
HAE 3	21:07:21.030	+23:31:13.922	0.36 ± 0.05	0.72 ± 0.20	0.48 ± 0.12	2.4889 ± 0.0009	456 ± 126	abc/ac	y	y
HAE 4	21:07:21.633	+23:31:41.273	0.12 ± 0.02	0.65 ± 0.17	0.15 ± 0.05	2.4916 ± 0.0009	199 ± 96	abc/abc	y	y
HAE 6	21:07:21.492	+23:31:19.524	⋯	<0.54	⋯	⋯	⋯	none/none	n	n
HAE 7	21:07:15.522	+23:31:37.543	0.47 ± 0.06	0.75 ± 0.18	0.48 ± 0.12	2.4825 ± 0.0010	966 ± 206	ab/none	y	n
HAE 8	21:07:15.943	+23:31:27.742	0.18 ± 0.02	1.18 ± 0.18	0.45 ± 0.07	2.4855 ± 0.0009	112 ± 91	abc/abc	y	y
HAE 9	21:07:22.586	+23:31:43.272	0.35 ± 0.05	0.94 ± 0.22	0.63 ± 0.16	2.4830 ± 0.0009	469 ± 126	abc/abc	y	y
HAE 10	21:07:15.725	+23:31:12.142	0.21 ± 0.04	0.58 ± 0.21	0.30 ± 0.11	2.4889 ± 0.0010	614 ± 201	ac/ac	y	y
HAE 12	21:07:14.904	+23:31:48.193	0.18 ± 0.04	1.05 ± 0.28	0.20 ± 0.07	2.4937 ± 0.0009	171 ± 99	abc/none	y	n
HAE 13	21:07:21.820	+23:31:41.747	⋯	<0.53	⋯	⋯	⋯	a/none	n	n
HAE 16	21:07:14.642	+23:31:15.593	0.31 ± 0.03	0.90 ± 0.18	0.53 ± 0.10	2.4848 ± 0.0009	348 ± 98	abc/abc	y	y
HAE 19	21:07:22.192	+23:31:45.995	⋯	<0.63	⋯	⋯	⋯	none/none	n	n
HAE 20	21:07:14.946	+23:31:20.572	⋯	<0.47	⋯	⋯	⋯	none/none	n	n
HAE 21	21:07:14.820	+23:31:42.856	⋯	<0.66	⋯	⋯	⋯	none/none	n	n
HAE 23	21:07:14.773	+23:31:18.443	0.23 ± 0.04	0.58 ± 0.16	0.23 ± 0.06	2.4774 ± 0.0010	631 ± 203	ab/none	y	n

Note. This table lists the CO (4–3) line information analyzed in the map and spectrum with the original beam size of 052 × 032 and at 80 km s⁻¹ resolution, respectively. Columns: (1) HAE ID; (2) right ascension (R.A.) for the peak position of either CO (4–3) (if detected) or Hα emissions (if not); (3) declination (decl.) for the peak position of either CO (4–3) (if detected) or Hα emissions (if not); (4) peak intensity of CO (4–3); (5) peak flux of CO (4–3); (6) CO (4–3) line intensity based on a Gaussian fit with an aperture of 07 using CASA imfit, and the errors include both the fitting errors and image noise; (7) the redshift estimated from the 1D Gaussian fit of the CO (4–3) spectrum, and the errors are fitting errors including the assumed spectral resolution, i.e., 80 km s⁻¹; (8) the estimated line width with fitting errors; (9) detection criteria that are satisfied, with the same (loosened) criteria used for the CO (4–3) and CO (3–2) detections; (10) information on whether the CO (4–3) line is detected (y = yes) or not (n = no); (11) information on whether the CO (3–2) line is detected (y = yes) or not (n = no). $\diamond$: we use the position of CO (4–3) if detected in the CO (4–3) line. Otherwise, we adopt the position of Hα from Subaru/MOIRCS observations (Tanaka et al. 2011; I. Tanaka 2019, in preparation), †: Detection criteria (a) peak intensity (column 4) S/N > 4.5; (b) peak flux (column 5) >3.5σ; (c) at least two continuous channels including a maximum peak flux with S/N > 2.5σ. ‡: CO (3–2) detection using the same criteria applied within the paper, reconfirming the previous detection of HAE 3, 4, 8, 9, 10, and 16 presented in Paper I, at matched 80 km s⁻¹ resolution with CO (4–3).

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We note that the S/N requirements used here are slightly different from those used in Paper I. Compared to Paper I, the requirements are loosened by 0.5 (for conditions "a" and "b") and 1 (for condition "c"), considering the smaller beam size and only a little improvement on the point-source sensitivity. It is intended to enhance and not to miss the detection for resolved galaxies with lower fluxes. The loosened criteria do not change the total number of CO (3–2) detections at the same time with matched spectral resolution (Section 2.3).

We tested the data cubes with a spectral solution of 80 km s⁻¹, considering typical line widths due to the galaxy rotation. Hereafter, we use the velocity resolution of 80 km s⁻¹ as a default resolution, or otherwise specified. We searched for a peak position within a circle of radius 10, centered on the positions of HAEs. The radius is a conservative choice considering the S/N in the intensity map and the beam size.

For the first criterion (a), we determined the integral velocity range by exploring both natural-weighted and lower resolution images. The lower resolution images were created by giving less weights to longer-baseline visibilities (uv tapering) in order to have a synthesized beam size that is comparable to previous CO (3–2) observations. We made the tapered images by modifying the uvtaper argument in the CASA task tclean with a parameter of 05 to produce a synthesized beam of 074 × 058. Hereafter, we call this lower-resolution image the "tapered" image and the natural-weighted image the "original" image.

The velocity-integration ranges are determined based on the S/N. We investigated a wide velocity-integration range with a redshift prior set as the central velocity (v₀ = 0) if the galaxy has, for instance, redshift information determined by the CO (3–2) line detection. If not, z = 2.485 is set to the central velocity as an initial value. We investigated a velocity-integration range as large as 2000 km s⁻¹ with a shift in the central velocity of up to ±1800 km s⁻¹ (v₀ ± 1800 km s⁻¹), which is large enough to cover the expected protocluster members. The final S/Ns for both integrated flux and peak flux were confirmed after visual inspection of the data cubes and velocity-integrated maps.

We reanalyzed our CO (3–2) images to have a common spectral resolution with the CO (4–3) images, because we used the spectral resolution of 100 km s⁻¹ in Paper I. The details of the calibration are given in Paper I. New images were created using CASA version 5.1.2.

The same detection criteria used for CO (4–3) line detection (Section 2.2) are applied for the CO (3–2) images. The total detection number of the CO (3–2) line has not changed (n = 7) with the new spectral resolution and the loosened criteria, giving support to the justification (or the robustness) of the loosened criteria.

A total of 11 HAEs are detected in the CO (4–3) line, out of the 16 targeted. The number of CO-detected galaxies has grown by a factor of ∼2 with the CO (4–3) observations. Figure 2 shows the CO (4–3) intensity maps and the CO (4–3) spectra  obtained from the original images at a resolution of 052 × 032 overlying the CO (3–2) spectra in dashed lines. We list the line intensities, the peak flux values, the redshifts, and the line widths for the CO (4–3) detection, as well as the 3σ upper limits of the peak fluxes for nondetections, in Table 1. We used the primary-beam-corrected images for all these measurements. The line intensities are measured by performing a two-dimensional Gaussian fit on their respective maps with a circular aperture of 07 using CASA imfit. The aperture size is determined by investigating the flux growth curves at various aperture sizes. With this aperture, the flux starts to flatten and/or the S/N is the highest value. The uncertainties of the integrated line flux include both the fitting error and the image noise. The listed line widths and the redshifts are estimated from a one-component Gaussian-model fitting with errors from the fitting and the assumed spectral resolution.

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All six CO (3–2)-detected HAEs (Paper I) that are located within the observed fields are detected in the CO (4–3) line (see Section 3.2). This confirms the detections of two CO lines associated with the HAEs and the protocluster. Tables 1 and 2 present two kinds of redshifts obtained from the detected CO (4–3) line, where the former lists the redshifts estimated from the original data cubes, while the latter shows the ones estimated from the tapered images. The comparison between these two measurements are used for later discussions on the merger nature of HAE 9 (see Sections 3.2 and 5.1).

In Appendix A, we summarize the five newly detected galaxies (HAE 1, 2, 7, 12, and 23) in the CO (4–3) line without a previous CO (3–2) line detection. These galaxies need additional observations in different CO transitional lines or deeper CO (3–2) integrations to confirm the CO (4–3) line detection. We note that three of these HAEs (HAE 1, 2, 7) have Hα redshifts from Subaru grism spectroscopy (Tanaka et al. 2011; I. Tanaka et al. 2019, in preparation), and except for HAE 1, the redshifts from CO and Hα are consistent within errors (see Appendix A for a potential reason for the difference).

All CO (3–2)-detected HAEs other than HAE 5, which was outside the fields of view (FoVs; Figure 1), are detected in the CO (4–3) line. We matched the angular resolution and the spectral resolution, as explained in Sections 2.2 and 2.3 to investigate the consistency. Table 2 lists the line information after the matching processes.¹⁶  

The spectroscopic redshifts derived independently from the CO (4–3) (tapered) and CO (3–2) data cubes are consistent within $| {\rm{\Delta }}{z}_{43-32}| \,=\,| {z}_{\mathrm{CO}43,\mathrm{tapered}}\,-\,{z}_{\mathrm{CO}32,\mathrm{original}}| \,\leqslant \,0.0017$, corresponding to the velocity difference of ±150 km s⁻¹. The line FWHM values are also consistent within the  errors. Therefore, we conclude that both CO lines are simultaneously detected and consistent with respect to the line redshifts and the line widths. 

Unlike other galaxies, for HAE 9, we find that the redshift defined by CO (3–2) is not consistent with the value obtained from the original map of CO (4–3), which is indicated in Figure 2 visually. The spectrum of the CO (3–2) line is shown as a teal dashed line, which is inconsistent with that of the CO (4–3) line, shown as a black solid line.  The redshift is different by $| {z}_{\mathrm{CO}43,\mathrm{natural}}-{z}_{\mathrm{CO}32,\mathrm{natural}}| =0.0028$.  We explain the details of this discrepancy in Section 3.3 and discuss  the nature of HAE 9 in Section 5.1. 

The positional differences (Δ) between the peaks of different CO lines are within 03 for most cases. We conclude that the CO (4–3) and CO (3–2) line emissions are coming from the same region, at least globally. The positional differences are all consistent within the positional errors considering the S/Ns and the synthesized beam of the tapered map.

One exception is HAE 4 (Δ ≈ 06). In HAE 4, the peak position of the CO (4–3) emission is closer to the peak of the Hα (NB) emission (Figure 2) than that of the CO (3–2) emission. As discussed in Paper I, the Hα emission shows an extended structure and we attributed the peak offset between the CO (3–2) and the Hα emissions toward either the internal structure of the galaxy or dust extinction. The difference in the peak position between CO (4–3) and CO (3–2) might have originated from the different excitation conditions within the galaxy structure. The origin of the positional difference between the line emissions needs to be confirmed by deeper, higher angular resolution observations to understand the nature of this massive galaxy.

Finally,  the observed line intensity ratios between CO (4–3) and CO (3–2), I_co43/I_co32 are spread around unity, i.e., I_co43/I_co32 = 0.63–1.26 with a median of 0.86. The median is similar to the typical line ratio observed for high-z normal disk-like galaxies (e.g., Daddi et al. 2015) or local LIRGs, which is slightly lower than that observed in local ULIRGs (e.g., Papadopoulos et al. 2012). In this paper, we defer the discussions on CO spectral line energy distributions (SLEDs) and gas excitation for the protocluster galaxies because these cannot be done with only two CO detections in close transition (J = 4–3 and J = 3–2).

We highlight two gas-rich (f_gas > 0.7) galaxies detected at a relatively high significance,  HAE 9 and HAE 16. We find evidence of gas-rich mergers by exploring the data cubes with different integration ranges around the positions of the HAEs. 

For HAE 9, we confirmed two distinct components which are spatially and spectroscopically resolved. The CO (3–2) observations, which have coarser spatial resolution, could not resolve this galaxy. Given the discrepancies between the CO (3–2) and the CO (4–3) lines in the original data cubes, we investigated the CO (4–3) cubes again. An additional component just next to the "main" component that still satisfies our detection criteria is identified. Related to this, we emphasize that our detection procedure is optimized to search for a peak and a channel range that delivers the largest S/N. As such, additional "sub"components at lower significance, if any, would be missed. The blueshifted component of CO (3–2) has been identified as a "main component" of CO (4–3) in the first run of the detection analysis. By comparing the CO (4–3) spectrum with that of CO (3–2) and searching for the redshifted component in the image, we could confirm another distinct, spatially offset component in the western direction, which still satisfies our detection criteria with the peak intensity ${\rm{S}}{/{\rm{N}}}_{{I}_{\mathrm{CO}43,\mathrm{peak}}}$ of 6.4 but was missed in our first detection run. We  name this a subcomponent of HAE 9 hereafter. We show the main and subcomponents and the spectra from tapered/original CO (4–3) cubes as well as from CO (3–2) in Figure 3.

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Another galaxy, HAE 16, has a potential minor merging subcomponent, which appears to be gas rich given the detection of CO (4–3). The spectrum of the companion is shown in Figure 4. This potential minor merging companion is found southwest of HAE 16, which is detected with a S/N_I(CO43,peak) of ≈6 and with a spatial separation of 13, corresponding to ∼10 kpc. The redshift difference between HAE 16 and the subcomponent is Δz = 0.0013 (i.e., velocity offset ≃100 km s⁻¹).

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We investigated the chances a of false-positive detection using astrodendro.¹⁷ We ran this with positive and inverse (negative) S/N maps obtained by dividing the original data cubes and the inverse cubes (original data cube ×(−1)) with individual channel noises, respectively. We tested both cubes with searching parameters of min_value = 2.5 (i.e., minimum peak S/N), min_delta = 1.0 (i.e., the significance of the identified "clump" above min_value), and min_npix = 10 (i.e., the minimum number of pixels = 10) with a pixel size of 005 × 005, which are set similarly to our detection criteria as much as possible. We found that the number of line candidates with ${\rm{S}}{/{\rm{N}}}_{{I}_{\mathrm{CO}43,\mathrm{peak}}}$ ≥ 6.0 with a line width ≥160 km s⁻¹, is seven for the positive cube, while the total number with the same conditions in the negative cube is one; thus, the chance of a false positive is ≈14% (=1/7) for this field with the given S/N threshold. Although this kind of blind line-detecting analysis is not the same as our two-step detection criteria and needs future confirmation, we conclude it is less likely that the minor merging companion for HAE 16 is a false-positive detection.

A halo-mass estimate for protoclusters is valuable to test whether or not the protocluster can evolve into a present-day massive cluster. Under the assumption of virialization, velocity dispersion offers a rough measure of the halo mass of the protocluster. We estimate the velocity dispersion based on the velocity differences with respect to the protocluster redshift.

To define the protocluster redshift, we combined the redshift information from our CO (4–3) and CO (3–2) observations, where a total of 12 CO redshifts are available. We used the mean value of the redshifts listed in Table 2 for galaxies with simultaneous CO detection. It is validated by small differences between the redshifts estimated individually from two CO lines (see Section 3.2). HAE 9 is a special case in that we used the main (the brightest or secure) component listed in Table 1 instead of taking the mean (see Sections 3.3 and 5.1). The discussion of the halo mass does not change even if we add the subcomponent of HAE 9 for the order estimate.

We list the redshift information in Table 3 that we used for the estimate of the halo mass. In the table, we also included the Hα redshifts if available from the Subaru grism observations (Tanaka et al. 2011; I. Tanaka et al. 2019, in preparation). The redshift uncertainty for Hα is several factors larger than the CO measurements, corresponding to several hundred km s⁻¹.

First, we define the protocluster redshift, and this will be used as v = 0 or the systemic velocity of the protocluster. We use the (one-step) biweight average (Beers et al. 1990) of the member galaxy redshifts (${\bar{z}}_{\mathrm{BI}}$). The biweight mean is more resistant (i.e., hardly changeable) in the presence of outliers compared to the simple mean, assuming that the observed radial velocities are drawn from the Gaussian parent population. With CO detections, the protocluster redshift is defined as ${\bar{z}}_{\mathrm{BI},\mathrm{CO}}=2.4874$. For comparison, if the Hα redshifts in Table 3 are included, the protocluster redshift is ${\bar{z}}_{\mathrm{BI},\mathrm{all}}=2.4880$. Hereafter, we use the protocluster redshift obtained from the CO measurements, i.e., ${z}_{\mathrm{proto}}={\bar{z}}_{\mathrm{BI},\mathrm{CO}}$ for the estimate of the halo mass.

Then, a line-of-sight (LOS) proper (or peculiar) velocity v_i (hereafter referred to as the LOS velocity) is calculated using the definition v_i = c(z_i−z_proto)/(1 + z_proto) (Harrison 1974; Danese et al. 1980), which is shown in Figure 5. Though the number is limited, the cumulative distribution of the LOS velocity is similar to a cumulative Gaussian distribution (see the top-right panel in Figure 5).

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Based on the derived LOS velocities, we derive the velocity dispersion, which is estimated by using two statistically different methods: (1) the biweight scale¹⁸ and (2) the gapper estimator. The reason for testing two estimators is due to the limited number of available redshifts (which is n = 12 for CO detection), in light of the discussions addressed in Beers et al. (1990) and Girardi et al. (1993) that the gapper estimation is preferable for systems with fewer (n ≃ 10) redshift information.

The velocity dispersion is σ_los,CO = 421 ± 148 km s⁻¹ (biweight) and 365 ± 128 (gapper) km s⁻¹. The errors are derived by taking the root sum of the squares of the statistical error calculated following the estimate in Ruel et al. (2014) and the error of 80 km s⁻¹ coming from the assumed spectral resolution. With 12 CO detections, individual estimates agree well with each other within the errors. Therefore, we adopt the biweight scale for the order estimate of the protocluster mass in the following. We note that the derived velocity dispersion of the protocluster including all redshift information is σ_los = 451 km s⁻¹ (biweight) and σ_los = 498 km s⁻¹ (gapper); this is consistent with CO-only base measurements though the errors can be much larger (up to ≈600 km s⁻¹) than the CO-based estimates, caused by the coarser spectral resolution of the grism spectroscopy.

We finally derive the halo mass of the protocluster from the derived velocity dispersion, using two different calculations used in Finn et al. (2005) and Evrard et al. (2008). Finn et al. (2005) derived the halo mass based on the modified formulation of the virial mass at a given virial radius, which is expressed as

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{cl}} & = & 1.2\times {10}^{15}{\left(\displaystyle \frac{{\sigma }_{\mathrm{los}}}{1000\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{3}\\ & & \times \ \displaystyle \frac{1}{\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{0}{\left(1+z\right)}^{3}}}\,{h}_{100}^{-1}\,{M}_{\odot },\end{array}\end{eqnarray} \tag{ 1 }$$

where h₁₀₀ = H₀/100 km s⁻¹ Mpc⁻¹. Evrard et al. (2008) derived this based on the model fit between the (DM) velocity dispersion and M_cl in the DM-only simulation, using the following formula

$$\begin{eqnarray}&&{M}_{\mathrm{cl}}={10}^{15}{\left[\displaystyle \frac{{\sigma }_{\mathrm{DM}}(M,z)}{{\sigma }_{\mathrm{DM},15}}\right]}^{1/\alpha }h{\left(z\right)}^{-1}\,{M}_{\odot },\end{eqnarray} \tag{ 2 }$$

where σ_DM,15 is the normalization at mass 10¹⁵ h⁻¹ M_⊙, α is the logarithmic slope, and h(z) = H(z) /100 km s⁻¹ Mpc⁻¹. The best-fit values are σ_DM,15 = 1082.9 ± 4.0 km s⁻¹ and α = 0.3361 ± 0.0026 from the simulation, which we adopt for our calculation. We use the (biweight) velocity dispersion to be equivalent to σ_DM for this calculation. We note that Equation (2) is not changed even when baryonic physics are included (Munari et al. 2013).

The range of halo masses from these two methods is log (M_cl/M_⊙) = 13.4–13.6 ∼ ${ \mathcal O }({10}^{13})$, though the value should be taken as an upper limit for an unvirialized system. The progenitors of Virgo-like clusters (M_halo ∼ (3–10) × 10¹⁴ M_⊙) or the most massive one like Coma clusters (M_halo > 10¹⁵ M_⊙) at z ∼ 2.5 are expected to have a halo mass of >10¹³ M_⊙ in simulations (Chiang et al. 2013). Therefore, if the protocluster follows the typical evolutionary track as the simulation expects, our estimate suggests that the protocluster is likely to become a Virgo-like, intermediate-mass cluster at z = 0 or even a more massive cluster.

When discussing the halo masses of protoclusters, one should bear in mind that there is a large uncertainty regarding the assumption of virialization as a whole. This is because protoclusters are by definition, unvirialized, premature clusters at high redshift, though many studies have assumed they were (locally) virialized to estimate their halo mass (e.g., Shimakawa et al. 2014; Kubo et al. 2016; Miller et al. 2018). Measurements based on X-ray detection or the Sunyaev–Zel'dovich (SZ) effect can be alternative methods to estimate the halo mass, but we do not have deep-enough  exposures to test whether the hot intracluster medium due to virial shocks is already in place. While deeper X-ray or SZ follow-up observations are required, we take a different approach in the next section to alleviate the uncertainties related to the estimated halo mass by evaluating the mass of the substructures of the protocluster that might be locally virialized.

Protocluster 4C 23.56 is known to have two density peaks in the projected map (Tanaka et al. 2011; Paper I), and hence, the protocluster center is difficult to define. Based on this distribution, we define two clumps around these two peaks,  namely, the East Clump (EC) and the West Clump (WC). The sizes of these clumps are defined in the projected map using the expected typical protocluster core radius ($\langle {R}_{200}\rangle$) at z ∼ 2, which is expected to evolve into a core of M_cl > 10¹⁴ M_⊙ at z = 0 from simulations (Chiang et al. 2017). The radius is 0.8 Mpc in comoving scale or ∼0.2 Mpc in physical scale.

The EC includes seven HAEs (HAE 2, 4, 6, 9, 13, 18 and 19) with three (five) CO (CO+Hα) spectroscopic redshifts, while the WC includes nine HAEs (HAE 1, 7, 8, 10, 12, 16, 20, 21, and 23) with seven (eight)  redshifts. With the increased redshift information, we now confirm that two substructures are kinematically connected with only a small systemic velocity offset of ≈220 (40) km s⁻¹ if limited to galaxies with CO (CO+Hα) redshifts. The small difference implies two galaxy groups in a merging phase within the protocluster.

Using the same method as addressed in the previous section, the estimated halo masses for both clumps are log (M₂₀₀/M_⊙) = 13.0 and 13.8 for the EC and the WC from σ_EC,CO-only = 324 ± 274 km s⁻¹ and σ_WC,CO-only = 567 ± 171 km s⁻¹, respectively. Therefore, it is $\sim { \mathcal O }({10}^{13})$, which is consistent with the estimate in the previous section. The uncertainty is larger due to the smaller number of CO detections per clump. 

We can derive the expected virial radius based on the velocity dispersion (Finn et al. 2005), which is 0.2–0.4 Mpc in our case. Therefore, we may reasonably conclude that each clump defined by a projected radius of ∼0.2 physical Mpc is virialized locally. The estimates of halo masses for individual clumps support the idea that the 4C 23.56 protocluster is composed of two virialized groups, which are likely to merge and may evolve into a present-day Virgo-like cluster. Such merging of smaller scale groups could be more common for protoclusters, considering the merger trees of present-day clusters.

As a final remark for this section, we note two additional uncertainties that were not fully taken into account in the estimate of the halos mass and the potential descendant. First, the order estimate of the protocluster is not based on the "complete samples" in terms of galaxy populations. Here, we rely on the Hα-selected galaxies, which are identified in the flux-limited observations, and the follow-up observations of the CO lines are also flux limited. We do miss less massive galaxies and quenched populations, if any, without significant star-forming activity. They might be distributed differently over the protocluster region, which may change our interpretation of mass of the protocluster and the descendant, as well as their evolutionary picture (Muldrew et al. 2015). In this regard, we are also limited by the survey area, wherein the entire structure of the protocluster can be extended up to R_e ∼ 5–10 Mpc comoving from simulations (e.g., Chiang et al. 2013). Second, we rely on simulated results in which scatter is large, as shown in Chiang et al. (2013; see Figure 2 therein), though this is our best conjecture on the descendant of the protocluster in the local universe. We may be able to solve the problem observationally be collecting a larger number of (proto)clusters and assessing them statistically in terms of their number density as a function of redshift.

In Section 3.3, we found that HAE 9 and HAE 16 have subcomponents that are likely associated with them. Here, we present further discussions on whether HAE 9 and HAE 16 are experiencing gas-rich mergers, based on ancillary data sets.

For HAE 9, we utilize the misalignment diagnostics (Franx et al. 1991). We measure the difference between the geometrical position angle (PA_geo) and the kinematical position angle (PA_kin). The kinematical position angle is defined by the line that connects the peak positions of the blue (main) and red (sub) components in HAE 9. We used the 1.1 mm continuum as a probe of PA_geo, where the emission is extended and marginally resolved at 078 × 068 resolution. We used CASA task uvmodelfit for measuring PA_geo, giving PA_geo = −60° ± 19° (shown as a white straight line in Figure 3). The misalignment between the morphological and the kinematical PAs for HAE 9 is ≈30°. As a reference, the misalignment angle between the morphological and kinematic PAs for disk-like star-forming galaxies at high redshift is typically less than 30° with a median of 12° (e.g., Wisnioski et al. 2015). The misalignment could have originated from external forces either from a merger, gas stripping, or galaxy harassment in the protocluster.

There is supporting evidence that the misalignment originated from a gas-rich merger. The line width of the CO (3–2) emission, observed at 091 × 066, exceeds 700 km s⁻¹, which is barely seen in a typical isolated galaxy. Further, the 1.1 mm flux was one of the highest among the detections (Paper I), close to the flux of dusty star-forming galaxies (DSFGs) or submillimeter-bright galaxies (SMGs), some of which are found to be mergers. The peak position for the subcomponent is offset by 034 from the peak of the main component, which is larger than the minor axis of the beam, implying that these are perhaps distinct galaxies. The K_p-band AO image for this galaxy, using the IRCS/Subaru, shows two separable components in the smoothed map (Paper I), further supporting a merger origin for the CO emissions in HAE 9.

What about HAE 16? We searched for multiwavelength counterparts of the companion, including the CO (3–2) and the near-infrared data sets in hand. Unfortunately, we could not confirm the existence of a counterpart at other wavelengths. Nevertheless, this does not reject the minor gas-rich merger scenario because observations are not deep enough to detect galaxy mergers with a stellar mass ratio less than ∼1:10. The 3σ limiting magnitudes of the Ks and J bands with Subaru/MOIRCS are 24.47 (AB magnitude) and 24.79 (AB magnitude), respectively, and the depth may have detected galaxies with stellar mass as low as log (M_⋆/M_⊙) ∼ 9.5 in principle, though the estimate can vary with assumptions on the IMF, metallicity, and star formation history. If there is a galaxy lower than this stellar limit, it would have been missed with the current Subaru observations. Also, previous CO (3–2) observations lacked the point-source sensitivity of the CO (4–3) observations. We need deeper observations to confirm the existence of a gas-rich minor merging counterpart.  However, the gas-rich nature (f_gas ≈ 0.8, Paper I) of HAE 16 may be reasonably explained by gas-rich accreting materials considering the existence of a metal-enriched, gas-rich component accreting from (or merging with) the outer filaments of the large-scale structure (Emonts et al. 2016; Ginolfi et al. 2017).

For the discussion of the remaining CO-detected galaxies, we compare the CO line widths (FWHM) and CO luminosities (${L}_{\mathrm{CO}}^{{\prime} }$) with other CO studies both in (proto)clusters and general fields. The line width is sensitive to both dynamical mass and any inclination effects, while ${L}_{\mathrm{CO}}^{{\prime} }$ is sensitive to the total mass of molecular gas. We discuss the characteristic properties of 4C 23.56 protocluster galaxies based on this.

For protocluster galaxies, we obtained the measurements from Aravena et al. (2012), Casasola et al. (2013), Ivison et al. (2013), Tadaki et al. (2014), Chapman et al. (2015), Rudnick et al. (2017), Dannerbauer et al. (2017), Hayashi et al. (2017), Noble et al. (2017, 2019), Wang et al. (2018), Coogan et al. (2018), Tadaki et al. (2019), and Gómez-Guijarro et al. (2019). All of these studies targeted galaxies associated with protoclusters at 1.5 ≲ z ≲ 2.5 and detected low-/mid-J CO lines (up to J = 4). If the SFR and stellar mass measurements are available, we also checked these values. Galaxies from the (proto)cluster studies are scattered around the main sequence, while some galaxies (9/84, 84 corresponds to the number of galaxies with M_⋆ and SFR in the literature) have exceptionally high SFR ≳ 500 M_⊙ yr⁻¹ (or the corresponding L_IR) that largely deviate from the main sequence.

There are several measurements for the same galaxies in different angular resolutions and/or different CO lines in the protocluster studies. We used the CO (1–0) measurements in Tadaki et al. (2014) and Dannerbauer et al. (2017; n = 2 and 1, respectively) instead of taking the values from the CO (3–2) measurements in Tadaki et al. (2019). For Noble et al. (2017, 2019; n = 4), we used the updated values in Noble et al. (2019) for the CO (2–1) line detection. For overlapping sources between Wang et al. (2018) and Gómez-Guijarro et al. (2019; n = 10), we used the CO (1–0) measurements from Wang et al. (2018). We note that the line widths in different transitions are comparable with each other for these works. For Coogan et al. (2018), we used the CO (4–3) measurements instead of those for CO (1–0), due to the low S/N in CO (1–0). Adding our CO measurements of CO (4–3) and CO (3–2) (i.e., 11 from CO (4–3) and one from CO (3–2)), a total of 103 galaxies are considered (proto)cluster galaxies.

We compiled the field data points from Tacconi et al. (2013), Daddi et al. (2015), Bourne et al. (2019), and Aravena et al. (2019). For field galaxies, we filtered out one galaxy at z > 3, yielding the field control sample of n = 80 at 1 ≲ z ≲ 3. We note that these field galaxies are also located around the main sequence, except for one with an SFR of 630 M_⊙ yr⁻¹.

If we only focus on the CO line widths, the median value for the 4C 23.56 protocluster galaxies is 433 km s⁻¹, compared to 422 km s⁻¹ for other protocluster galaxies, which is indicated in the left panel of Figure 6. Considering the typical errors for fitting and assumed spectral resolution (∼100 km s⁻¹), we conclude that CO line widths are consistent with each other. The result does not change even if we take all CO measurements for overlapping sources.

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For a fair comparison with field galaxies, we converted all CO line widths (=FWHM) to the rotation velocity by taking the isotropic virial estimate of the circular velocity (FWHM × $\sqrt{3/(8\,\mathrm{ln}2})$ to match with galaxies in the PHIBSS survey (Tacconi et al. 2013), which provides the largest number of field galaxies near the main sequence (n = 52). The authors reported the characteristic rotation velocity (v_rot) either from the line FWHM for unresolved sources or the inclination-angle-corrected velocity gradient for resolved sources. Because the line FWHMs, the velocity gradients, and the inclination angles are not listed in their tables, we apply the prescription to unresolved cases for 103 protocluster galaxies and for the remaining field galaxies (n = 28). All data points are shown in the right panel of Figure 6.

There is a hint of higher rotation velocity (from the CO line widths) for protocluster galaxies. We find a higher median value for the protocluster (v_rot,pc = 304 km s⁻¹) compared to the field galaxies (v_rot,fd = 192 km s⁻¹), which is shown in Figure 6. There is a clear tail toward higher rotation velocity in the histogram for protocluster galaxies. Such a difference becomes even larger if we exclude galaxies from Tacconi et al. (2013). Considering random inclination angles that affect the observed line widths for both field galaxies and protocluster galaxies, the difference between protocluster and field galaxies is likely the intrinsic feature.

We also converted the CO line flux into the CO (1–0) luminosity as a probe of molecular gas mass in protoclusters. We assumed the line luminosity ratio of $R1J(={L}_{\mathrm{CO}1-0}^{{\prime} }/{L}_{\mathrm{CO}J-(J-1)}^{{\prime} })=1.2,1.8,$ and 2.4 for J = 2, 3, and 4, respectively, which are the values often adopted for typical star-forming galaxies (e.g., Dannerbauer et al. 2009; Aravena et al. 2010; Bolatto et al. 2015; Daddi et al. 2015). While the CO emissions may be highly excited compared to field galaxies for some of the galaxies (e.g., Coogan et al. 2018), we take the nominal ratios as we do not have secure constraints for many of the protocluster galaxies. This is why we tried to use the CO line width and the flux from CO (1–0) measurements, when available. We also note that the depth of field and protocluster studies varies, which makes a fair comparison difficult. However, most observations except for those targeting extreme starbursts have similar depth in terms of the ${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ limit, i.e., one to a few ×10⁹ (K km s⁻¹ pc⁻²). Therefore, the general discussions below would not be largely affected by the different sensitivities of the different studies that we consider.

In terms of the CO (1–0) luminosity, we find that there is a 0.11 dex difference in terms of the median value between the 4C 23.56 protocluster ($\mathrm{log}({L}_{\mathrm{CO}10}^{{\prime} }$/(K km s⁻¹ pc⁻²)) = 10.04) and other protocluster galaxies ($\mathrm{log}({L}_{\mathrm{CO}10}^{{\prime} }/({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{-2}))\,=10.15$), which is negligible considering the typical errors. The consistency does not change even if we take all available CO measurements. Therefore, we conclude that all (proto)cluster galaxies considered have consistent line widths and CO (1–0) luminosities on average.

Similarly, the difference between field galaxies and all protocluster galaxies in terms of ${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ is also negligible ($\mathrm{log}({L}_{\mathrm{CO}10}^{{\prime} }/$(K km s⁻¹ pc⁻²)) = 10.13 versus $\mathrm{log}({L}_{\mathrm{CO}10}^{{\prime} }/$ $({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{-2}))=10.01$). This suggests that we do not see a higher gas fraction per galaxy on average for protocluster galaxies that are mostly on the main sequence, confirming our previous argument in Paper I.

There are two possible ways to explain the difference in the line width. First, protocluster galaxies are kinematically different, perhaps owing to higher merger rates, which are unresolved with the large beam size, broadening the line width. Related to this aspect, it is worth noting that most of these measurements are based on observations at spatial resolutions of ≈1''–6''. Therefore, most sources are unresolved. Second, protocluster galaxies have intrinsically large line widths, suggestive of larger dynamical masses or different mass distributions. The current sensitivity is not sufficient to conclude whether one dominates over the other, or both contribute equally.

Regarding the possibility of unresolved mergers, several protocluster studies have reported enhanced merger rates based on rest-frame optical images compared to fields (e.g., Hine et al. 2016; Coogan et al. 2018; Watson et al. 2019). Here, the kinematical property of a larger line width may consistently indicate that many protocluster galaxies are undergoing (gas-rich) mergers that are not spatially resolved.

For the 4C 23.56 protocluster, indeed, HAE 9 has a broader line width in the CO (3–2) emission, which was not resolved with the large beam (∼08). The CO (4–3) observations at higher angular resolution suggest (Figure 3) that this galaxy may be a gas-rich merger (Sections 3.3 and 5.1). Other potential merger candidates with large line widths (FWHM > 400 km s⁻¹; Table 1) from the CO (4–3) detections are HAE 1, HAE 3, HAE 7, HAE 10, and HAE 23. We discussed in Paper I that HAE 3 may be a merger from its resolved stellar map. For HAE 1 and HAE 7, there are hints of elongated emissions in CO (4–3) as indicated in Figure 2, which needs future deeper observations to confirm whether they are "very" close pairs or outflowing components. The current sensitivity limits such confirmation.

For the second possibility of an intrinsically large line width, it is important to constrain the size of the galaxies first. We fitted the CO intensity with an elliptical Gaussian using the CASA task ${\mathtt{imfit}}$ for the 4C 23.56 protocluster galaxies. We could only constrain the sizes of the galaxies for HAE 8, 9 (main galaxy), and 16 (main galaxy) from the fitting, yielding semimajor beam-deconvolved sizes of 021–031 (with errors 008–012), meaning that the galaxies are marginally resolved with the beam (052 × 032). This corresponds to ≈1.7–2.6 kpc. The S/Ns for other galaxies are insufficient to make useful measurements.

Noble et al. (2019) estimated the CO (2–1) sizes with a beam size of ∼04, yielding R_1/2,CO ∼ 2–7 kpc. They argued for smaller sizes in CO emission for cluster galaxies at z = 1.6 compared to field galaxies, though the statistical difference was small (∼0.9σ). For other cluster galaxies at slightly higher redshift z = 1.99, Coogan et al. (2018) also measured the size, giving the CO (4–3) size constraints for two galaxies with FWHM/2 ≈ 02–03, comparable to our measurements, and the upper limit for three galaxies (<016).

The number of galaxies with CO size constraints is also limited for field galaxies at z ≳ 2. Even from the PHIBSS, we could only obtain one value from the list, which is 3.8 kpc, estimated from the CO (3–2) line. In Aravena et al. (2019), they measured the CO (3–2) emitting sizes from two galaxies at z ∼ 2.5 using (see their Appendix C), which is ≈4 kpc from the Gaussian fit. The CO sizes of HAE 8, 9, and 16, are smaller than those of the field galaxies at similar redshift.

If we take the size and line width differences as a generic feature of protocluster galaxies with the caveats of inhomogeneous selection and small sample sizes, this suggests that the dynamical mass ($\propto {{rv}}_{\mathrm{rot}}^{2}$) traced by CO lines is comparable to or slightly higher (by ∼20%) than the field galaxies, and the mass is more centrally concentrated for protocluster galaxies. We note that the above field and cluster galaxies have stellar masses comparable to HAE 8, 9, and 16, and thus the size difference may not be due to the difference in the stellar masses. Different CO transitions can have different emitting sizes, but we compare close CO transitions at similar redshifts, and the low-J and mid-J CO emissions are known to have similar distributions, at least for typical main-sequence galaxies (e.g., Bolatto et al. 2015).

It may be worth noting that two main galaxies of HAE 9 and HAE 16 are likely to be undergoing gas-rich mergers and have smaller CO sizes (∼2×) compared to the field. Considering this, two scenarios that explain the large line width for protocluster galaxies may not be independent but intertwined mechanisms. In other words, mergers may result in the intrinsically large line width with smaller sizes by redistributing the gas.

In Section 4, we found that the 4C 23.56 protocluster is kinematically composed of two merging galaxy groups that are loosely connected. In such environment where hierarchical structure formation is underway, galaxy mergers should also be prevalent, where the velocity dispersion between galaxies is smaller compared to local virialized clusters. HAE 16 and HAE 9 are indeed located in two dense substructures of the protocluster (see Figure 5 and Section 4) with moderate galaxy–galaxy dispersions (i.e., 324 and 567 km s⁻¹), which increase the probability of mergers.

If mergers are frequent, then the next question would be the role of these mergers in the formation of ETGs. In Figure 7, we plot the local empirical relations between the specific angular momentum and the stellar mass for the three morphological types of spiral, S0, and elliptical galaxies from Romanowsky & Fall (2012). Fall & Efstathiou (1980) found that elliptical and spiral galaxies form parallel sequences, with the former having a factor of ≈6 smaller specific stellar AM (j_⋆) than the latter. Romanowsky & Fall (2012) updated the work from Fall & Efstathiou (1980) and showed that galaxy mergers can naturally explain the positions of elliptical galaxies in the j_stars–M_stars plane and that disks and bulges follow fundamentally different j_stars–M_stars relations.

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In light of this, we derive a crude estimate of the specific angular momentum (j_disk = 2 × v_circ × R_disk = 1.19 × v_circ × r_e, where v_circ is circular velocity; e.g., Romanowsky & Fall 2012; Obreschkow & Glazebrook 2014; Burkert et al. 2016) based on the size and line widths for field and protocluster galaxies. It gives for field star-forming galaxies a value comparable to or slightly smaller (by ≈30%) than that for protocluster galaxies in general.

For HAE 16, we could measure the specific angular momentum of the disk from kinematical modeling with GALPAK^3D (Bouché et al. 2015; see Appendix B for the detailed kinematic modeling of HAE 16 and discussions on the derived values). While the galaxy may have a gas-rich companion, it shows a smooth velocity gradient that allows us to model the galaxy. The estimated specific angular momentum after correcting for the redshift dependence of the Hubble constant in log scale are in a range between 2.8 and 3.0 (Figure 7). For the circular velocity, we took into account the pressure component from turbulent motion, which makes it larger than the rotation velocity v_rot alone, ${v}_{\mathrm{circ}}^{2}$ = ${v}_{\mathrm{rot}}^{2}$ + $2{\sigma }_{0}^{2}{r}_{e}/{R}_{\mathrm{disk}}$ (e.g., Übler et al. 2017). For our case, this increases the specific angular momentum value by 6%–16%, depending on the assumed velocity profiles for fitting. We consider that the stellar specific angular momentum of the local universe is comparable to the (cold) gas specific angular momentum (e.g., Tadaki et al. 2017), because high-z galaxies are gas rich (f_gas ∼ 0.5).

Our findings for HAE 16 suggest that this galaxy was, at the very least, born in a similar way to the local spirals. Even if the galaxy might have experienced mergers, the specific angular momentum has not decreased. The galaxy must lose a significant amount of angular momentum via "dissipational" processes to become local elliptical galaxies that are abundant in cluster regions.

Lagos et al. (2018) recently presented a quantified assessment for the impact of mergers in galaxy evolution, including dry, wet, minor, and major mergers using the EAGLE simulation. They found that regardless of whether they are minor or major mergers, "wet" mergers tend to increase the specific angular momentum, thus spinning up the galaxy rotation, while "dry" mergers tend to decrease the specific angular momentum. Even if additional dissipative processes from the environment is acting, angular momentum might not change but might remain similar values to field galaxies, perhaps with compact rotating star-forming cores at high redshift (e.g., Barro et al. 2017; Tadaki et al. 2017; Talia et al. 2018). This picture is also consistent with the Illustris simulation in Genel et al. (2015; see also Teklu et al. 2015) that ETGs—including slow rotators, which are round and have lower angular momentum at z = 0—reside at high redshift on the late-type (spiral) relation and need active galactic nucleus (AGN) feedback or gas-poor mergers to lose the angular momentum.

The next step to confirm mergers and to quantify the exact role of gas-rich mergers is to have higher angular resolution observations with good sensitivity. As the number of CO-detected galaxies in fields and protoclusters has now reached ∼100 at high z (z > 1), we need to verify the above scenario with a larger number of samples.

The derived range of maximum rotational velocity, V_max = 301 and 433 km s⁻¹ (for the mass and tanh profiles, respectively), suggests that HAE 16 has a slightly higher rotational velocity than what has been observed in field galaxies with a mean value of ∼200 km s⁻¹ at z ∼ 2 for those with comparable mass in Hα emissions (e.g., Förster Schreiber et al. 2009, 2018; Wisnioski et al. 2015) and in CO emissions (e.g., Tacconi et al. 2013; Daddi et al. 2015; Aravena et al. 2019; Bourne et al. 2019; see also Figure 6). We note that the FWHMs of the CO lines from 1D Gaussian fitting are also high, i.e., 452 ± 101 km s⁻¹ and 547 ± 138 km s⁻¹ for the CO (4–3) tapered map and the CO (3–2) map, respectively, supporting the large rotational velocity.

There are a few limitations suggested by the inspection of the best-fit parameters. While the maximum rotational velocity may be larger than that of field galaxies, the maximum rotational velocity largely depends on the assumption of different velocity profiles. The lack of sensitivity in the outer disk region is one of the main causes that introduce such a difference. In addition, there is a covariance between the maximum rotation velocity and inclination, which shows the tightest negative correlation among the joint distributions of the parameters. The maximum rotational velocity V_max is higher for the tahn profile, where the inclination is lower than that of the mass profile. We need both deeper and higher angular resolution imaging observations for HAE 16 to constrain the velocity profile and the maximum velocity without any degeneracies. The estimated size of the galaxy is consistent with each other, which is 2.8 ± 0.1 kpc, which is slightly larger than the image-based deconvolved FWHM (1.8 ± 0.7 kpc). We note that the FWHM from the Gaussian fitting corresponds to the n = 0.5 case, which is different from what we assumed for the kinematical modeling (n = 1). The errors are not measured in the same way, and the systematic errors may also contribute to the different estimates. But, all of these values are still marginally smaller than the typical CO sizes for the field galaxies we discussed in Section 5.2. Bearing these degeneracies and uncertainties in mind, we discuss the nature of HAE 16 with the best-fit model parameters and taking the difference as systematic errors.

How different is HAE 16 compared to other galaxies in fields and clusters in kinematical properties? First, we evaluate how this disk galaxy is dominated by the ordered rotation. The ratio between the rotation velocity and the intrinsic dispersion, V_max/σ₀, provides a measure of the strength of the ordered rotation relative to random motions. In general, a galaxy is rotation dominated when the ratio is larger than unity. Within the uncertainties of the model constraints, the high values of V_max/σ₀ = 5.7 (tanh) and 3.6 (mass) suggest that the galaxy is rotation dominated, and the values are plotted in Figure 10.

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For comparison, the results of field surveys are plotted in Figure 10, which used either Hα, [O iii], or [O ii] lines using the IFU, or CO lines using millimeter interferometry. For this plot, we used the results from MUSE+KMOS (Swinbank et al. 2017), KROSS (Harrison et al. 2017), KMOS^3D (Wisnioski et al. 2015), and KDS (Turner et al. 2017), which were compiled in the KDS paper. The remaining surveys of GHASP (Epinat et al. 2008a, 2008b), DYNAMO (Green et al. 2014), MASSIV (Epinat et al. 2012), PHIBSS-I (Tacconi et al. 2013), SINS/zC-SINF (Förster Schreiber et al. 2009, 2018), and AMAZE-LSD (Gnerucci et al. 2011b) are retrieved from the literature.

We compare with the SINS/zC-SINF survey, which probed field galaxies at a similar redshift and mass range. It revealed that the average ratio of V_max/σ₀ is 4.1 with the median of 3.2 (Förster Schreiber et al. 2018). Based on these values, we can conclude that HAE 16 shows a degree of rotation similar to or slightly higher than that of average field population, though the measurements are based on Hα, instead of CO. Though we need to verify whether the ionized gas and the molecular gas trace the same gas dynamics, they may not be significantly different globally for field galaxies at least (Übler et al. 2018).

For another comparison, we take the values of V_max/σ₀ for cluster galaxies from the results of ZFIRE (Alcorn et al. 2018), which is a slit spectroscopic survey using Keck with a subset of cluster galaxies at z ∼ 2. We note that the ZFIRE cluster galaxies are even less massive than comparable field populations and HAE 16. This is the only available literature for (proto)cluster galaxies at z ≳ 2 that allow for a comparison, providing the largest sample available. Interestingly, the average V_rot/σ₀ for these less massive galaxies in a z ∼ 2 cluster is also slightly higher than that for z ∼ 2 field galaxies, but with a large scatter.

We note that the ratio tends to be  lower for less massive galaxies and for those at higher redshifts, which is known as kinematical downsizing (e.g., Simons et al. 2016). Based on the high value (especially for the tanh profile) found in HAE 16 and the ZFIRE clusters, it is a tantalizing trend that suggests earlier and perhaps rapid formation of "disk" galaxies in high-z clusters, such that kinematical downsizing in the overdense regions of high-z (proto)clusters precedes that of general fields, along with hierarchical structural formation as expected. However, the uncertainty in our estimate (e.g., the lower value from the mass profile) makes this less convincing. We need a larger number of samples and deeper observations to confirm this.

Given the constraints of V_max, σ₀, and the gas fraction f_gas, we can calculate the Toomre Q parameter (Q_Toomre; Toomre 1964) to constrain the stability of the (rotation-dominated) disk. The Toomre Q parameter is expressed as

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{gas}} & = & \displaystyle \frac{{\sigma }_{0}\kappa }{\pi G{{\rm{\Sigma }}}_{\mathrm{gas}}}=\left(\displaystyle \frac{{\sigma }_{0}}{{V}_{\max }}\right)\left(\displaystyle \frac{a({V}_{\max }^{2}{R}_{\mathrm{disk}}/G)}{\pi {R}_{\mathrm{disk}}^{2}{{\rm{\Sigma }}}_{\mathrm{gas}}}\right)\\ & = & \ \left(\displaystyle \frac{{\sigma }_{0}}{{V}_{\max }}\right)\left(\displaystyle \frac{{{aM}}_{\mathrm{tot}}}{{M}_{\mathrm{gas}}}\right)=\left(\displaystyle \frac{{\sigma }_{0}}{{V}_{\max }}\right)\left(\displaystyle \frac{a}{{f}_{\mathrm{gas}}}\right),\end{array}\end{eqnarray} \tag{ 5 }$$

where κ is the epicyclic frequency, Σ_gas is the gas surface density, G is the gravitational constant, R_disk is the disk radius, and M_tot is the total mass. We use the baryonic mass (M_tot = M_gas + M_star) for now. We use $a=\sqrt{2}$ for a disk with constant rotational velocity. A simple calculation based on the best-fit V_max/σ₀ and gas fraction gives Q_gas ≪ 1, ranging between 0.3 and 0.5, and thus an even lower value than the critical Q_crit applied for a thick disk (Goldreich & Lynden-Bell 1965).

This implies that the galaxy is gravitationally unstable, even though the galaxy shows a disk-like feature with a smooth velocity gradient and high V_max/σ₀ value. The gravitationally unstable disk may be a natural consequence of the gas-rich disk.

In Figure 10, we show a toy model in which the redshift evolution of V_max/σ₀ is explained by the gas fraction evolution assuming a marginally stable disk, i.e., Toomre Q is close to 1 (Wisnioski et al. 2015). HAE 16 slightly deviates from this model, such that the linear Toomre-stability analysis may no be longer valid. This recalls a disk with violent gravitational instability (e.g., van den Bergh et al. 1996; Elmegreen & Elmegreen 2005; Bournaud et al. 2008; Genzel et al. 2008; Genel et al. 2012; Agertz et al. 2009; Ceverino et al. 2010). In this scenario, the disk can no longer support the rotation by itself and fragments into clumps. This is consistent with the Kp-band image (Paper I), which shows clumpy structures. Such clumps may lead the gas (clumps) to fall into the central region, creating a dense stellar component in the near future to stabilize the structure. The smaller size of the CO (4–3)-emitting region may also support this picture, and a large amount of gas might already have transferred into the central region.