Understanding Large-scale Structure in the SSA22 Protocluster Region Using Cosmological Simulations^∗

Our sample consists of LBGs and LAEs with spectroscopic measurements within a $9^{\prime} \times 9^{\prime}$ region of the SSA22 field, centered on R.A. = 22:17:34, decl. = 00:15:04 $({\rm{J}}2000)$, as described by Steidel et al. (1998). The LBGs in our sample were selected as part of the survey of $z\sim 3$ star-forming galaxies presented in Steidel et al. (2003). The LAEs were first identified using broadband BV imaging from Subaru/Suprime-cam in addition to narrowband imaging from Keck/LRIS and Subaru/Suprime-cam using a filter tuned to the wavelength of Lyα at z = 3.09 (centered on 4985 Å with a bandwidth of 80 Å). The LAEs were selected based on BV−NB4985 colors indicating a narrowband excess, which ensures a sample of galaxies with large ($\gt 20$Å) Lyα EWs at redshifts coincident with the central density peak of SSA22 ($3.05\lesssim z\lesssim 3.12$). The spectroscopic measurements for galaxies in the SSA22 field were obtained using the LRIS spectrograph at the Keck observatory across multiple observing campaigns and instrumental configurations (Steidel et al. 2003; Nestor et al. 2011, 2013). A more detailed description of the redshift determinations can be found in Topping et al. (2016), and further details about the observations and data reduction can be found in Steidel et al. (2003) and Nestor et al. (2011, 2013).

We determined the systemic redshift of galaxies in the SSA22 field by measuring the redshift of Lyα emission, interstellar metal absorption lines, or both, and removing the effects of large-scale gas outflows. We applied the formulas presented in Trainor et al. (2015) for LAEs and Adelberger et al. (2003) for LBGs, to translate from the observed rest-UV emission and absorption redshifts to the true, systemic redshifts. We compiled the resulting systemic redshifts of galaxies within SSA22 into a redshift histogram (Figure 1). Galaxies in the SSA22 redshift histogram are clearly separated into peaks centered at z = 3.069 (blue) and z = 3.095 (red) with widths ${\sigma }_{z,b}=0.0047$ and ${\sigma }_{z,r}=0.0074$ respectively. Hereafter, we describe the total, blue, and red regions using the subscripts t, b, and r respectively.

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The significance of the SSA22 overdensity has been calculated in past work (Steidel et al. 1998, 2000). However, given our significantly larger sample of spectroscopic redshifts in SSA22 (Topping et al. 2016), and the updated LBG redshift selection function (Steidel et al. 2003), it is worth revisiting this calculation. To estimate the galaxy overdensity qualitatively, we compared the number of galaxies contained in the SSA22 redshift spike $({N}_{\mathrm{obs}})$ with the number of galaxies expected in the same redshift interval from the LBG average selection function $({N}_{\mathrm{expect}})$. For this calculation, we restricted ${N}_{\mathrm{obs}}$ to the LBGs in our observed sample and did not consider LAEs, since the LBGs have a well-defined redshift selection function. We define the galaxy overdensity, ${\delta }_{\mathrm{gal}}$, as

$$\begin{eqnarray}&&{\delta }_{\mathrm{gal}}=\displaystyle \frac{{N}_{\mathrm{obs}}}{{N}_{\mathrm{expect}}}-1.\end{eqnarray} \tag{ 1 }$$

The observed sample used for this calculation includes 82 LBGs in the redshift interval $2.6\leqslant z\leqslant 3.4$. The redshift histogram of these galaxies is shown in Figure 2, where the well-known overdensity at $z\sim 3.09$ is clearly visible.

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To construct the LBG selection function, we used the sample of LBGs from Steidel et al. (2003), with one key difference. The inclusion of SSA22 galaxies in the sample would increase the value of the selection function within the z = 3.09 spike interval, thus biasing the inferred overdensity toward lower values. Therefore, we excluded these galaxies, with 883 redshifts remaining. We fit a spline to the histogram of the remaining galaxies, which resulted in a smooth selection function. Finally, we normalized the selection to the SSA22 redshift histogram, which allowed us to directly compare the number of LBGs in a given redshift interval. Determining the correct normalization is a key step in calculating the galaxy overdensity. Specifically, we normalized the LBG selection function such that its integral over the redshift ranges $2.6\leqslant z\leqslant 3.03$ and $3.12\leqslant z\leqslant 3.4$, was equal to the number of observed galaxies in the SSA22 field in the same redshift intervals. These ranges were chosen to match the number of "field" galaxies in SSA22 and the overall LBG selection function. The resulting selection function is displayed in Figure 2 overlaid on the SSA22 LBG redshift histogram.

Using the redshift histogram and the LBG selection function, we computed the galaxy overdensity of SSA22. In detail, we calculated the galaxy overdensity for three components of the SSA22 protocluster: the blue peak, the red peak, and the total volume. We carefully determined the boundaries of the redshift intervals in order to accurately calculate the overdensity. In contrast to previous work, here we found that the low and high-redshift boundaries of the total SSA22 interval were self-evident, as defined by a large gap on either side of the redshift distribution, with the boundaries occurring at the redshift of the last galaxy on each side of the overdensity. Therefore, we set the low and high-redshift boundaries to z = 3.0598, and z = 3.1048 respectively, and removed the galaxies that define these boundaries from our future calculations. To find the boundary that separates the red and blue peaks, we fit the sum of two Gaussians to the redshift histogram, and determined the redshift at the minimum of the trough between the two peaks. We measured this boundary to be at z = 3.0788.

Due to the effects of redshift-space distortions (Kaiser 1987), and the fact that the SSA22 protocluster is collapsing, the redshift intervals we defined are contracted compared to the ranges defined by the physical size of the protocluster in the Hubble flow. We used a correction factor C (Padmanabhan 1993) to quantify this effect, as defined by

$$\begin{eqnarray}&&C=1+f-f{(1-{\delta }_{m})}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&f=\displaystyle \frac{d\mathrm{ln}D}{d\mathrm{ln}a},\end{eqnarray} \tag{ 3 }$$

D is the linear growth factor, a is the cosmological scale factor, and ${\delta }_{m}$ is the matter overdensity, related to ${\delta }_{\mathrm{gal}}$ through:

$$\begin{eqnarray}&&1+b{\delta }_{m}=C(1+{\delta }_{\mathrm{gal}}).\end{eqnarray} \tag{ 4 }$$

We defined the LBG bias factor, b (Equation (5)), by comparing ${\sigma }_{8,\mathrm{gal}}$, the LBG number fluctuations, and ${\sigma }_{8,\mathrm{CDM}}\,=0.8228$, which corresponds to ${\sigma }_{8,\mathrm{CDM}}{| }_{z=3.09}=0.254$ at z = 3.09 (Planck Collaboration et al. 2014).

$$\begin{eqnarray}&&{b}^{2}={\left.\displaystyle \frac{{\sigma }_{8,\mathrm{gal}}^{2}}{{\sigma }_{8,\mathrm{CDM}}^{2}}\right|}_{z=3.09}.\end{eqnarray} \tag{ 5 }$$

We calculated the value of ${\sigma }_{8,\mathrm{gal}}$ using the correlation length, r₀, and the slope, γ, from the LBG autocorrelation function:

$$\begin{eqnarray}&&{\sigma }_{8,\mathrm{gal}}=\displaystyle \frac{72{\left(\tfrac{{r}_{0}}{8\mathrm{cMpc}}\right)}^{\gamma }}{{2}^{\gamma }(3-\gamma )(4-\gamma )(6-\gamma )}\end{eqnarray} \tag{ 6 }$$

(Peebles 1980). For these calculations, we found a value of f = 0.986, and adopted values of ${r}_{0}=6.0\pm 0.5{h}^{-1}\ \mathrm{Mpc}$ and $\gamma =1.5$ from Trainor & Steidel (2012), which result in a bias of $b=3.84\pm 0.25$. We estimated the errors of the bias from the uncertainties of the autocorrelation function parameters, r₀ and γ, and ${\sigma }_{8,\mathrm{CDM}}$. Table 1 shows the values of these parameters resulting from our calculation.

Table 1.  Properties of the SSA22 Redshift Peaks

	Blue	Red	Total
zmin	3.0598	3.0788	3.0598
zmax	3.0788	3.1048	3.1048
zpeak	3.069 ± 0.001	3.095 ± 0.001	−
Nexpect	1.71	2.66	4.40
Nobs	10	28	38
${\delta }_{\mathrm{gal}}$	4.83 ± 1.84	9.51 ± 1.99	7.64 ± 1.40
${\delta }_{m}$	0.9	1.509	1.285
C	0.765	0.647	0.688
Dimensions $[{h}^{-3}\ \mathrm{cMpc}\times \mathrm{cMpc}\times \mathrm{cMpc}]$	$\tfrac{3}{4}\times 12\times 14\times 18.38$	$12\times 14\times 24.95$	$12\times 14\times 43.338$
$V\ [{h}^{-3}\ {\mathrm{cMpc}}^{3}]$	2315.9	4191.6	7280.8
M	$(0.757\pm 0.171)\times {10}^{15}{h}^{-1}\ {M}_{\odot }$	$(2.146\pm 0.324)\times {10}^{15}{h}^{-1}\ {M}_{\odot }$	$(3.194\pm 0.401)\times {10}^{15}{h}^{-1}{M}_{\odot }$

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Neglecting the effects of large-scale redshift-space distortions (i.e., infall) in estimating the number of LBGs expected from the LBG selection function causes us to underestimate the relevant redshift interval, and therefore the expected number of galaxies, N_expect, as well. We corrected for this effect by increasing the redshift interval by a factor of $1/C$ (see Table 1) when integrating the LBG selection function, and recalculating the number of galaxies expected within the interval, as well as the associated galaxy overdensity. One subtlety lies in the fact that our correction factor, C, was initially calculated based on an overdensity that was overestimated due to the underestimate of the selection function, resulting in a correction that is too large. We therefore recomputed the correction factor using the updated overdensity, and repeated the procedure of correcting the redshift interval of the selection function, and recalculating the overdensity. We iterated this process until the galaxy overdensity converged to its true value, which we adopted as our final value for the overdensity. We obtained overdensities of ${\delta }_{t,\mathrm{gal}}=7.6\pm 1.4$, ${\delta }_{r,\mathrm{gal}}=9.5\pm 2.0$, and ${\delta }_{b,\mathrm{gal}}=4.8\pm 1.8$, for the total, red, and blue regions respectively. Our updated total overdensity is larger than the value previously reported in Steidel et al. (1998; ${\delta }_{t,\mathrm{gal}}={3.6}_{-1.2}^{+1.4}$) but consistent with the value reported in Steidel et al. (2000; ${\delta }_{t,\mathrm{gal}}=6.0\pm 1.2$).

Using the updated estimates of the galaxy overdensity and appropriate volume for each section of the protocluster, corrected for the effects of redshift distortion, we computed the total, blue-peak, and red-peak protocluster masses using

$$\begin{eqnarray}&&M=\bar{\rho }{V}_{\mathrm{true}}(1+{\delta }_{m}),\end{eqnarray} \tag{ 7 }$$

where $\bar{\rho }$ is the mean density of the universe, and ${V}_{\mathrm{true}}\,={V}_{\mathrm{apparent}}/C$.

We calculated the mass overdensity, ${\delta }_{m}$, of each region using Equation (4), utilizing the values for the correction factors, C (see Table 1), that we obtained at the end of the iterative process described above. Using these correction factors, we calculated mass overdensities of ${\delta }_{t,m}=1.3\pm 0.4$, ${\delta }_{r,m}=1.5\;\pm 0.4$, and ${\delta }_{b,m}=0.9\pm 0.3$, for the total cluster, red peak, and blue peak respectively.

In order to estimate ${V}_{\mathrm{apparent}}$ (and the corresponding ${V}_{\mathrm{true}}$) for each region, we multiplied its line-of-sight extent and on-sky area. In the line-of-sight dimension, the spatial extent is represented by the difference in the radial comoving distance between the two redshift boundaries. We used the on-sky coverage of our observations, as described in Topping et al. (2016), as the area in the transverse dimensions, corresponding to a value of $12\times 14\ {h}^{-2}\ {\mathrm{cMpc}}^{2}$ for the area on the sky. For the blue peak, we reduced the area on the sky because the galaxies contained within this peak cover only $\sim 75 \%$ of the observing area (Topping et al. 2016). Our observations, and therefore the area used in our calculations, did not cover the full extent of the protocluster, as probed by, e.g., Matsuda et al. (2005) and Yamada et al. (2012). Therefore, increasing the volume to enclose the entire protocluster may result in an increased mass estimate. On the other hand, our observations were centered on the highest density region of the protocluster, so expanding the protocluster volume may dilute the overdensity, therefore negating the expected mass increase caused by using a larger volume. For example, using the positions presented in Hayashino et al. (2004) we determined that the average surface density of LAEs decreases by $\sim 20 \%$ if our observing window size is doubled. Analysis of protocluster membership in the Millennium Simulation shows that only $\sim 50 \%$ of the galaxies within this area will be gravitationally bound to the main cluster by z = 0 (Muldrew et al. 2015). The net result of these two effects is a predicted z = 0 mass higher than our estimate, but much more uncertain.

Based on the ${\delta }_{m}$ and ${V}_{\mathrm{true}}$ values described above, we calculated the mass of the total cluster to be $(3.19\,\pm 0.40)\times {10}^{15}\ {h}^{-1}\ {M}_{\odot }$, and calculated the mass of the red (blue) peak to be $(2.15\pm 0.32)\times {10}^{15}\ {h}^{-1}\ {M}_{\odot }$ ($(0.76\pm 0.17)\,\times {10}^{15}\ {h}^{-1}\ {M}_{\odot }$). We determined the errors on our mass calculation based on our uncertainties of the mass overdensity. The volumes encompassing the red and blue peaks do not fill the entire space of the total overdensity, so the sum of the red and blue peak masses is less than the mass of the entire structure.

We use halo catalog and merger tree information drawn from the Small MultiDark Planck (SMDPL) simulation data set⁴ (Klypin et al. 2016) in order to compare the observed structure in SSA22 to what is found in cosmological N-body simulations (Behroozi et al. 2013a, 2013b; Klypin et al. 2016; Rodríguez-Puebla et al. 2016). We chose this simulation because its box size ($400\ {{\rm{h}}}^{-1}\ {Mpc}$) allows for a large enough sample (N = 19) of clusters that are within the estimated $3\sigma$ uncertainty of the mass of the red peak in SSA22 (i.e., ${10}^{15}\ {h}^{-1}\ {M}_{\odot }\leqslant M\,\leqslant 1.7\times {10}^{15}\ {h}^{-1}\ {M}_{\odot }$). Hereafter, we describe masses of halos using their virial mass, ${M}_{\mathrm{vir}}$, defined by Rodríguez-Puebla et al. (2016). The SMDPL simulation is also characterized by the following cosmological parameters: ${{\rm{\Omega }}}_{m}=0.307$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.693$, h = 0.678, n_s = 0.96, and ${\sigma }_{8}=0.829$. These parameters are consistent with current Planck results (Planck Collaboration et al. 2014), as opposed to those adopted for the Millennium simulation (Springel et al. 2005, ${{\rm{\Omega }}}_{m}=0.25$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.75$, h = 0.73, ${\sigma }_{8}=0.9$). In addition, with a particle mass of ${M}_{{part}}=9.63\times {10}^{7}\ {h}^{-1}$M_⊙, the mass resolution of the SMDPL simulation allows us to identify robust halos down to the mass that may host galaxies similar to those in our observations ($M\sim {10}^{10.6}\ {h}^{-1}$M_⊙). The halo catalogs are saved in a series of 117 snapshots, starting at Snapshot Number 0 (called snapnum in the catalogs) at z = 18.56, and ending with snapnum = 116 at z = 0. The snapshots are saved with a time resolution of ${\rm{\Delta }}z\approx 0.16$ at $z\sim 3$. This time resolution allows us to perform our analysis on halos at the epoch of the SSA22 protocluster observations. The difference between the cosmological parameters used in the SMDPL simulation and our analysis in Section 2 is not significant, and therefore our inferences based on the results are valid.

Based on the mass calculations presented in Section 2.3, we expect the SSA22 protocluster to evolve into a massive ($M\sim {10}^{15}\ {h}^{-1}{M}_{\odot }$) cluster at z = 0, so we start by selecting all z = 0 halos, determined using the ROCKSTAR spherical overdensity method (Behroozi et al. 2013a), in the simulation with masses $M\gt {10}^{15}\ {h}^{-1}$M_⊙ from the SMDPL halo catalog. We identify 19 systems that meet this criterion. After identifying these halos, we follow their histories through the merger trees constructed from the simulation (Behroozi et al. 2013b), in order to select the progenitor halos at a given epoch (z = 3.03, snapnum = 31). We chose this snapshot as it has the closest redshift to that of the SSA22 protocluster.

We present two methods to search for SSA22 analogs in the SMDPL simulation. First, we start by assuming that the observed structure in SSA22 will collapse into a massive cluster at z = 0. To mimic this regime in our analysis of the SMDPL simulation, we limit our sample to halos that collapse to a single massive structure at z = 0. We also employ an alternate, complementary approach in which we construct a sample of halos within a volume surrounding each of the $z\sim 3$ protoclusters with no requirement on their status as a member of the descendant cluster at z = 0. We then identify what kind of structures form from these halos by z = 0, and compare them to the current predictions for the fate of the SSA22 protocluster.

3.3.1. Progenitors Only

We begin by describing the method that selects our parent sample of halos based on their membership in a single massive structure at z = 0. In order to compare any structure present in the simulated protoclusters to the structure observed in SSA22, we constructed redshift histograms from the sample of cluster progenitor halos. We created redshift histograms by viewing each protocluster from multiple sight lines. By observing through many sight lines, we obtained a comprehensive view of each protocluster, and a better chance of detecting any structure that may be present. We expect adjacent sight lines to show similar evidence of structure, and since each sight line is a different random realization of the protocluster, sampling many sight lines allows us to differentiate between real structure and statistical flukes. For each protocluster, we observed 3600 sight lines, each of which is separated by 6^◦ in the azimuthal $\theta \in [0,2\pi )$ direction, and 3^◦ in the polar $\phi \in [0,\pi )$ direction.

For a given sight line, the simulated redshift histogram consists of calculated redshifts for 146 halos that are progenitors of a particular protocluster. We chose this number of halos to be the same as the number of galaxies (both LBGs and LAEs) that have spectroscopic redshifts in SSA22. To select these halos, we first narrowed down the sample based on their projected positions in the protocluster. We required that selected halos be within the observed area of SSA22, $\sim 12\times 14\ {h}^{-2}\ {\mathrm{cMpc}}^{2}$, centered on the highest density peak. To choose the 146 halos whose redshifts make up the redshift histogram for a given sight line, we first randomly selected 40 halos out of all cluster progenitor halos with masses above $M\gt {10}^{11.55}\ {h}^{-1}$M_⊙ (Trainor & Steidel 2012), corresponding to LBGs in our observed SSA22 sample. We then randomly selected 106 halos from among the remaining cluster progenitor halos with masses $M\gt {10}^{10.6}\ {h}^{-1}$M_⊙ (Gawiser et al. 2007), which represent the LAEs in our simulated redshift histogram. This selection process typically results in a sample that contains $\sim 10 \%$ of the total cluster progenitors. This analysis assumes that the LBGs and LAEs in our sample are the central galaxies of their host dark matter halos, as opposed to satellites. The similar number densities and clustering strengths of LBGs and their host halos (Conroy et al. 2008; Trainor & Steidel 2012), in addition to the low halo occupation fraction of LAEs (1%–10%; Gawiser et al. 2007), suggest that this assumption is valid.

To calculate the observed redshift of a halo, we first required its 3D position and velocity, given in the SMDPL halo catalog. We defined the center of the protocluster as the center of mass of all cluster progenitor halos, and set the center of each protocluster to be at z = 3.09. We calculated the redshift of each halo by determining its line-of-sight distance away from the protocluster center, and the corresponding velocity using the Hubble flow. In addition, we adjusted the estimated redshift to take into account the line-of-sight peculiar velocity, ${\rm{\Delta }}v$, of each halo, using ${\rm{\Delta }}z={\rm{\Delta }}v/c\times (1+{z}_{{\rm{H}}})$, where z_H is the redshift of the halo after taking into account the Hubble flow. We then collected the redshifts of all 146 halos into a redshift histogram.

We began by using the two-sample Kolmogorov–Smirnov (KS) test as a metric for comparison between each simulated protocluster redshift distribution and the observed SSA22 distribution. For each KS test, we determined the probability that the two distributions were drawn from the same parent distribution, a p-value. We introduced a p-value cutoff of $p\geqslant 0.4$, which distinguished redshift histograms that were well represented by two peaks, and those that presented only a single peak. We determined the value for this cutoff by trial-end-error. We adjusted the cutoff and visually inspected each qualifying histogram and its best-fit models to determine at what p-value the histograms are typically double peaked. This cutoff allowed us to exclude those redshift histograms from further analysis that did not show similar structure to that in the SSA22 protocluster.

After we determined the existence of structure in a given sight line, we compared the simulated redshift histogram to the one observed in SSA22. We first fit the sum of two Gaussians to the simulated redshift histogram. We then required the associated best-fit parameters to be comparable to parameters found for the SSA22 redshift histogram. The requirements for the parameters of the larger (l) and smaller (s) peaks are as follows:

$$\begin{eqnarray}0.341 & \leqslant & \displaystyle \frac{{N}_{s}}{{N}_{l}}\leqslant 0.493\\ 0.004 & \leqslant & {\sigma }_{l}\leqslant 0.01\\ 0.004 & \leqslant & {\sigma }_{s}\leqslant 0.01\\ 0.02 & \leqslant & {\rm{\Delta }}z\leqslant 0.032.\end{eqnarray} \tag{ 8 }$$

In these expressions, N_s and N_l are the number of galaxies in the smaller and larger peaks, respectively. We determined a boundary at the trough between the two peaks, and counted the number of galaxies on either side. We define ${\sigma }_{l}$ and ${\sigma }_{s}$ as the best-fit standard deviations, in redshift units, of the large and small peaks, respectively, and ${\rm{\Delta }}z$ as the redshift difference between the centers of the two peaks. The existence of redshift histograms that fit these criteria would suggest that the observed structure in SSA22 may collapse into a single massive cluster at z = 0.

3.3.2. Halos in Surrounding Volume

We first tested the assumption that the double-peaked redshift histogram is representative of the progenitors of a single massive ($M\geqslant {10}^{15}\ {h}^{-1}$M_⊙) protocluster at z = 0. Under this assumption, we expect that the majority of the galaxies in SSA22 will collapse into a massive cluster at z = 0. By investigating the z = 3 cluster progenitors of massive clusters at z = 0, we are able to identify which, if any, parts of the structure will be a component of the cluster once it has collapsed.

Using the methods described in Section 3.3.1, we determined whether there is any structure comparable to that of the SSA22 protocluster, in any of the 19 massive protoclusters in the SMDPL simulation. We found that none of the protoclusters had any sight lines that show evidence for a double-peaked morphology with best-fit parameters similar to those in SSA22, as defined in Equation (8). Figure 3 displays the redshift histogram that, out of all sight lines of the 19 protoclusters, shows the greatest similarity with SSA22 as defined by the KS-test p-value ($p=1.8\times {10}^{-4}$). Even this distribution does not show a double-peaked morphology. By observing the spatial distribution of progenitor halos, we can understand why there is very little extended structure present. The range of redshifts present in the SSA22 protocluster corresponds to a spatial separation of ${\rm{\Delta }}z=0.045$ ($\sim 40\ {h}^{-1}\ \mathrm{cMpc}$), while the $z\sim 3$ halo progenitors of a single massive z = 0 cluster in the SMDPL simulation typically extend over ${\rm{\Delta }}z=0.015$ ($\sim 13\ {h}^{-1}\ \mathrm{cMpc}$). Sufficiently high peculiar velocities could perturb the redshifts outside the primary structure; however, the collapsing nature of these protoclusters tends to compress the redshift distribution on such scales, not expand it. In summary, comparison with the SMDPL simulation demonstrates that the double-peaked morphology observed in the SSA22 redshift histogram does not comprise the coalescing progenitors of a single z = 0 structure.

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The approach described in the previous section was based on a starting assumption that the entire double-peaked structure in SSA22 corresponds to the progenitor of a single $M\geqslant {10}^{15}\ {h}^{-1}$M_⊙ cluster at z = 0. Therefore, we restricted our analysis to include only the $z\sim 3$ progenitor halos of such z = 0 clusters. Using the alternative approach described in Section 3.3.2, we attempt to find structures within the volumes surrounding protoclusters in the simulation at $z\sim 3$ that, when "observed" (as we observe the SSA22 field), produce redshift histograms at $z\sim 3$ that are similar to that in SSA22. We then used the SMDPL simulation to characterize the evolution of such structures to z = 0.

When examining the distributions of halos in the more extended volumes surrounding massive cluster progenitors, we do find sight lines yielding redshift histograms similar to that of the SSA22 protocluster based on the SMDPL merger trees (Behroozi et al. 2013b; Klypin et al. 2016). Figure 4 shows an example of a redshift histogram (p = 0.85) computed for a single sight line of one simulated protocluster that fits our criteria for similarity to the SSA22 redshift histogram. In addition to finding sight lines that satisfy our similarity criteria stated in Equation (8), we find that many of these "good" sight lines occur from similar viewing angles, suggesting that they are due to real structure, and not statistical flukes.

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We separate the 19 protoclusters into three categories based on the number of distinct sight line groups present in each protocluster. The three categories are "no sight lines," "single sight line group," and "multiple sight line groups." To assign a protocluster to one of the categories, we first looked at the p-values distributed throughout the sight lines. Figures 5(a)–7(a) show a projection of the p-value distribution as a function of sight-line. We then looked in detail at the redshift histogram produced when observing along a sight line with a high p-value that passes through a possible cluster progenitor to confirm that it satisfied the similarity criteria of Equation (8). Figure 7(c) shows an example where the double-peaked structure of the redshift distribution can clearly be seen. If several of these sight lines are clustered around a specific viewing angle, we consider the viewing angles to be a sight-line group. Finally, we categorize each protocluster volume based on the number of sight-line groups. Below we describe the three categories to which we assign each protocluster, with an example from each category detailing the important features in each case.

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4.2.1. No Sight Lines

4.2.2. Single Sight-line Group

Another subset of protocluster volumes each yield a single group of closely packed sight lines that produce double-peaked histograms. Figure 6 shows an example of a protocluster in this category. In this example, many sight lines near the southern pole have high p-values suggesting that there is some structure arising in the redshift histograms. In addition, many of these sight lines also fit our criteria for similarity to the SSA22 redshift histogram, given in Equation (8). The viewing angle of these sight lines is coincident with the progenitor of a second, massive ($M={10}^{14.4}\ {h}^{-1}$M_⊙) protocluster. Figure 6(b) shows this protocluster toward the bottom of the panel. We display the positions of halos from one sight line that shows a similar redshift histogram to that of SSA22 in Figure 6(c). At $z\approx 3$, the main and adjacent structures appear as two separate groups of halos, separated by a lower-density gap. In many cases, the smaller group of halos is the progenitor of a cluster with mass comparable to the expected mass of the blue (smaller) redshift peak in SSA22 at z = 0 ($M\sim 0.7\times {10}^{15}\ {h}^{-1}$M_⊙). Most of the halos that make up the larger and smaller redshift peaks are progenitors of either the main or neighboring cluster. At z = 0, the two structures have collapsed into two distinct clusters. We find that $9/19=47 \%$ protoclusters in our sample fall into this category.

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4.2.3. Multiple Sight-line Groups

The last category consists of protocluster volumes that each contain more than one distinct group of adjacent sight lines. Each of these groups is composed of many closely packed sight lines that produce a double-peaked histogram. Figure 7 shows an example of a protocluster in this category. The KS p-value distribution (Figure 7(a)) shows similar properties to the distribution presented in the "single sight-line" case. Protoclusters in this category, however, show multiple separate viewing angles comprised of many sight lines with elevated p-values, as seen by the different groups of green points. Each one of these separate viewing angles corresponds to the presence of another nearby massive protocluster. Similar to the adjacent structures in the "single sight-line" group, many of the neighboring structures in the "multiple sight-line" category have masses comparable to the predicted z = 0 mass of the blue redshift peak in SSA22. The centers of these neighboring protoclusters typically lie 10–20 ${h}^{-1}\ \mathrm{cMpc}$ away from the main protocluster. All neighboring protoclusters are separate from each other at $z\sim 3$, and the majority of halos that make up double-peaked histograms are members of the main protocluster, and a single neighboring protocluster, as no sight lines intersect multiple neighboring protoclusters. The neighbors, in addition to the main protocluster, all remain distinct as they collapse to separate structures at z = 0. We find that $8/19=42 \%$ of the protoclusters in our sample fall into this category.

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We found that the double-peaked redshift histogram of the SSA22 protocluster is the reflection of the presence of less-massive ($\gt {10}^{14}\ {h}^{-1}$M_⊙) protoclusters in its vicinity. We can use a simple analytic approach to explain quantitatively the prevalence of neighboring, less-massive clusters around $\gt {10}^{15}\ {h}^{-1}$M_⊙. Due to halo biasing, we expect the most massive clusters at z = 0, which lie on an enhanced density peak, to be surrounded by smaller, but still massive, nearby clusters (Kaiser 1984; Barkana & Loeb 2004). Using the halo–halo correlation function and the halo mass function, we calculated the number of clusters at a given distance away from some of the most massive clusters. In this section, in order to more accurately compare to simulations, we adopt a cosmology consistent with the SMDPL simulation: ${{\rm{\Omega }}}_{m}=0.308$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}\,=0.692$, h = 0.677, ${\sigma }_{8}=0.8228$, (Planck Collaboration et al. 2014).

We define the halo–halo correlation as the excess probability of finding a neighbor at a distance r and in the volume $\delta V$ as

$$\begin{eqnarray}&&\delta P=n\delta V(1+\xi (r)),\end{eqnarray} \tag{ 9 }$$

where n is the average number density of halos (Peebles 1980). We use linear bias to relate the linear matter correlation function, ${\xi }_{\mathrm{lin}}(r)$, to the two-point correlation function of halos with masses M₁ and M₂, ${\xi }_{{hh}}({M}_{1},{M}_{2},r)$, by

$$\begin{eqnarray}&&{\xi }_{{hh}}({M}_{1},{M}_{2},r)=b({M}_{1})b({M}_{2}){\xi }_{\mathrm{lin}}(r).\end{eqnarray} \tag{ 10 }$$

To calculate the linear bias factor, b(M), we adopt the definition given by Quadri et al. (2007):

$$\begin{eqnarray}&&{b}_{h}(M)\\ &&=\,1+\displaystyle \frac{1}{{\delta }_{c}}\left[{\nu }^{{\prime} 2}+b{\nu }^{{\prime} 2(1-c)}-\displaystyle \frac{{\nu }^{{\prime} 2c}/\sqrt{a}}{{\nu }^{{\prime} 2c}+b(1-c)(1-c/2)}\right],\end{eqnarray} \tag{ 11 }$$

where ${\nu }^{{\prime} }=\sqrt{a}{\delta }_{c}/\sigma (M,z)$, $\sigma (M,z)$ is the mass variance on scales of $R={\left(\tfrac{3M}{4\pi \bar{\rho }}\right)}^{1/3}\ {h}^{-1}\ \mathrm{Mpc}$, and $\bar{\rho }$ is the mean matter density of the universe. As in Quadri et al. (2007), we use values of ${\delta }_{c}=1.686$, a = 0.707, b = 0.5, and c = 0.6.

We calculate the linear mass correlation function from the power spectrum of fluctuations, P(k), using

$$\begin{eqnarray}&&{\xi }_{\mathrm{lin}}(R)=\displaystyle \frac{1}{2{\pi }^{2}}{\int }_{0}^{\infty }P(k)\displaystyle \frac{\sin {kR}}{{kR}}{k}^{2}{dk}\end{eqnarray} \tag{ 12 }$$

and derive a power spectrum based on the methods described in Naoz & Barkana (2005).

Using the halo–halo correlation function, we predicted the mean number of halos, with masses M≥M₂, within a surrounding volume centered on a halo with mass M₁ using

$$\begin{eqnarray}&&\langle N(R)\rangle =n{\int }_{0}^{R}4\pi {r}^{2}[1+{\xi }_{{hh}}(r)]{dr}.\end{eqnarray} \tag{ 13 }$$

Where n is the average number density of halos, calculated using the halo mass function of Sheth & Tormen (1999).⁵

Using the method described here, we obtain an analytic prediction for the prevalence of $\gt {10}^{14}\ {h}^{-1}$M_⊙ clusters as a function of distance from a $\gt {10}^{15}\ {h}^{-1}$M_⊙ cluster at z = 0. We then compare our analytic prediction with the results from the SMDPL simulations, and finally with our observations of the SSA22 protocluster. We calculated the number of halos of a given mass within a given distance, R from the center of a cluster with a mass corresponding to the mass of one of the 19 clusters present in the SMDPL simulation. This process was repeated for each of the 19 $M\gt {10}^{15}\ {h}^{-1}$M_⊙ clusters in the simulation, and then we averaged the resulting total number. Figure 8 shows this analytic result, calculated using the method described above (dashed lines). For comparison, Figure 8 displays the number of halos with a given mass and within a given radius, R computed directly from the SMDPL simulation by counting the average number of halos in a sphere with radius R centered on each of the 19 $M\gt {10}^{15}\ {h}^{-1}\ {M}_{\odot }$ clusters at z = 0 (solid lines).

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In the case of the SSA22 protocluster, the adjacent structure lies at a distance of $D\approx 20\ {h}^{-1}\ \mathrm{cMpc}$ calculated from the difference in peak redshifts neglecting the effects of infall. Within this distance, our analysis predicts $\sim 1-2$ clusters with a mass comparable to the mass of the blue peak of SSA22. This number increases by $\sim 20 \%$ when the mass of the central cluster is doubled. This analytic prediction is consistent with our results, which place more protoclusters in the "single sight-line" category, compared to the other categories. We also predict ∼10 clusters with masses $\sim {10}^{14}\ {h}^{-1}$M_⊙ within this distance, which is again consistent with the simulations. However, the neighboring clusters that give rise to double-peaked redshift histograms typically have masses of $\gtrsim 3\times {10}^{14}\ {h}^{-1}$M_⊙.

We have determined which, if any, sight lines in a given simulated protocluster produce redshift histograms that present a double-peaked morphology, and whether they are similar to the observed redshift histogram in SSA22. In this section, we discuss the probability of observing a double-peaked redshift histogram, based on our analysis of protoclusters in the SMDPL simulation. For this analysis, we calculated the density of protoclusters that, when observed, would result in a redshift histogram that contains two peaks similar to that of SSA22, or any structure beyond a single redshift peak. By searching through sight lines across all protoclusters in the simulation, we determined the frequency at which observations of massive protoclusters would yield double-peaked redshift histograms.

We started by calculating the covering fraction of sight lines that produced double-peaked redshift histograms. For an individual protocluster, we calculated the total covering fraction by summing up the contribution from each sight line that we have determined to be double peaked. The area on the sky covered by a single sight line is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Omega }}={\int }_{\theta }^{\theta +{\rm{\Delta }}\theta }{\int }_{\phi }^{\phi +{\rm{\Delta }}\phi }d\theta d\phi ,\end{eqnarray} \tag{ 14 }$$

where ${\rm{\Delta }}\theta =6\deg$, ${\rm{\Delta }}\phi =3\deg$, and $(\theta ,\phi )$ is the angle of the sight line. For a given protocluster, the covering fraction of sight lines with redshift histograms similar to that of SSA22 is

$$\begin{eqnarray}&&F=\displaystyle \frac{{\rm{\Omega }}}{4\pi },\end{eqnarray} \tag{ 15 }$$

where ${\rm{\Omega }}=\sum {\rm{\Delta }}{\rm{\Omega }}$ is the total solid angle covered by the relevant sight lines.

On average, the covering fraction of sight lines for a given protocluster is $F=0.025\pm 0.017$, with values for individual protoclusters ranging from F = 0 for protoclusters in the "no sight lines" category, to F = 0.065 for a protocluster in the "multiple sight lines" category. We also consider counting sight lines that are better fit by two peaks, but whose fitting parameters may not fit the criteria presented in Equation (8). Such sight lines contain evidence of structure beyond the main protocluster, but, when observed, do not produce redshift histograms similar to that of SSA22. The average covering fraction of such additional sight lines is F = 0.13. To determine the occurrence rate of structures similar to those observed in SSA22, we multiply the covering fraction of sight lines producing double-peaked histograms by the number density of massive protoclusters in the SMDPL simulation, $19/{400}^{3}\ {h}^{3}\ {\mathrm{cMpc}}^{-3}=296\ {h}^{3}\ {\mathrm{Gpc}}^{-3}$. We therefore calculated the cosmic abundance of observing structure similar to that of the SSA22 protocluster to be $n=7.4\ {h}^{3}\ {\mathrm{Gpc}}^{-3}$. This density suggests that the observed structure in the SSA22 protocluster is rare, and its discovery unexpected within the $1.07\times {10}^{-3}\ {h}^{3}\ {\mathrm{Gpc}}^{-3}$ volume of the survey that discovered it (Steidel et al. 2003). Even placing a less stringent similarity requirement for the simulated redshift histograms (i.e., some evidence for structure ($p\gt 0.4$), as defined in Section 3.3.1 without strictly satisfying Equation (8)), we find a cosmic abundance of only $n=38\ {h}^{3}\ {\mathrm{Gpc}}^{-3}$, which still makes the discovery of SSA22 extremely fortuitous within the LBG survey volume. Hints of bimodality have been seen in other protoclusters (e.g.; Venemans et al. 2007; Kuiper et al. 2011). However, better spectroscopic sampling as well as evidence of a spatial offset between redshift peaks are required to determine the similarity of these structures to the observed large-scale structure in SSA22.