Assembly History and Internal Structure of Cluster Cold Dark Matter Haloes

In the Phoenix project, a total of nine haloes were selected from the Millennium simulation (Springel et al., 2005). These haloes encompass a range of virial masses spanning from $5\times 10^{14}$ $h^{-1}{\rm M}_{\odot}$ to $2\times 10^{15}$ $h^{-1}{\rm M}_{\odot}$. These haloes were individually re-simulated using a zoom-in technique, and are denoted as Ph-X, where X varies from A to I. To investigate numerical convergence, the Ph-A halo was subjected to simulations at four different mass resolution levels, labeled as level-1 (highest resolution) through level-4 (lowest resolution). For the remaining haloes, Ph-B through Ph-I, two resolution levels were performed, namely level-2 and level-4. It is worth noting that in this study, we primarily focus on the results from level-2 and level-4 simulations.

The level-2 simulations feature a mass resolution ($m_{\rm p}$) of approximately $10^{6}$ $h^{-1}{\rm M}_{\odot}$ and employ a softening length of $\epsilon=0.32$ $h^{-1}{\rm kpc}$. In contrast, the level-4 simulations have a coarser mass resolution of around $10^{8}$ $h^{-1}{\rm M}_{\odot}$ and a larger softening length of $\epsilon=2.8$ $h^{-1}{\rm kpc}$. Within the virial radius of each level-2 and level-4 halo, there are approximately $10^{8}$ and $10^{6}$ particles, respectively.

Each Phoenix simulation consists of a total of 72 snapshots, labelled from 0 to 71, with snapshot 71 corresponding to the redshift of 0. For more information on the Phoenix project including the basic halo properties and convergence studies, readers are referred to Gao et al. (2012).

The Aquarius Project involved the re-simulation of six Milky Way-sized haloes, denoted Aq-A to Aq-F, in isolated environments within the lower resolution version of the Millennium II simulation (Boylan-Kolchin et al., 2009). These haloes exhibit virial masses ranging from $1\sim 2\times 10^{12}$ $h^{-1}{\rm M}_{\odot}$. The Aq-A halo, in particular, was simulated at five different levels of mass resolution, while the remaining five Aquarius haloes were subjected to simulations at two lower resolution levels. In our study, we primarily focus on the results from the level-2 Aquarius simulations.

In the level-2 Aquarius simulations, which are central to our investigation, the mass resolution ($m_{\rm p}$) is approximately $10^{4}$ ${\rm M}_{\odot}$, the softening length is set at $\epsilon=65.8$ ${\rm pc}$, and the number of particles contained within the virial radius is around $N_{200}\sim 10^{8}$. For more detailed information regarding the Aquarius project, readers are encouraged to refer to Springel et al. (2008).

We first look at the mass growth history, $M_{200}(z)$, of nine Phoenix haloes by tracing the main branches of their merger trees. In Figure 1, we show their halo masses at different redshift in coloured lines. The horizontal axis is redshift, and the vertical axis is the main branch progenitor halo mass scaled by the mass at $z=0$. For comparison, we have incorporated the overall mass evolution of the Aquarius haloes into our analysis, as shown as the black dotted line. To achieve this, we computed the average mass across all seven haloes at various redshifts. Readers seeking specific details regarding the individual halo’s mass evolution are encouraged to refer to Figure 1 in W11.

The growth history for haloes indicated as solid lines, are in general similar, with rapid mass growth after $z\sim 1$. This is especially true for Ph-G. It has only $<10\%$ of its final mass at $z=1$, and has not accreted half of its final mass until $z\sim 0.18$.

The two haloes, Ph-B and Ph-H, exhibit significant discontinuities in their mass growth histories, occurring notably at snapshot 69 ($z=0.04$) and snapshot 62 ($z=0.24$), respectively. Moreover, both of these haloes experience substantial mass losses as they evolve into redshifts below 1. They are marked with two dashed lines in Figure 1. We have conducted an investigation to understand the origins of these conspicuous mass jumps, which are associated with mergers between the primary halo and a smaller, yet denser halo. In such instances, the merger tree algorithm faces challenges in correctly identifying the main progenitor. This is primarily due to the fact that the most bound particle does not necessarily belong to the most massive halo.¹¹1This problem arises because the distinction between haloes and subhaloes can become difficult in major merger scenarios, making methods that use temporal information crucial in such cases (as pointed out by Behroozi et al. 2013, 2015 and references therein). Methods such as hbt (Han et al., 2012) can be employed to improve this, but it is beyond the scope of this work.

Interestingly, the Aquarius haloes do not encounter similar issues, possibly owing to their relatively limited involvement in major mergers that would significantly perturb their central regions in recent times. While it is conceivable to adopt an alternative definition and modify the merger tree algorithm to address this problem, such endeavors fall beyond the scope of this study.

Hence, for the entirety of this work, we have chosen to exclude these two haloes from our analysis. Nevertheless, it remains an intriguing avenue for future research to delve into the merging events that have impacted these particular haloes, shedding further light on their unique evolutionary trajectories.

It can be readily found from Figure 1 that the mass growth of clusters happens very late relative to galaxies. In general, more than 90% of the final halo mass is accreted after $z=2$. Compared to the Aquarius haloes (i.e. Figure 1 of W11, ), the slopes of $M_{200}(z)$ for the Phoenix haloes are in general steeper. This is expected in the hierarchical formation model, which predicts more recent growth for cluster haloes compared to galactic ones (see e.g. Navarro et al., 1996; Mo et al., 2010). Late assembly suggests that cluster haloes may not be fully virialized compared to galactic haloes. We investigate the mergers and accretions in more detail in the next sections.

In accordance with the methodology outlined by W11, we investigate merger event patterns for particles residing at varying radii within each Phoenix halo. At the redshift $z=0$, we meticulously investigate the history of each particle located in distinct radial shells. This investigation is conducted by tracking the particle’s history across different snapshots based on its unique particle ID. We discern the pivotal moment when each particle was first accreted into the main halo progenitor of the Phoenix halo, which serves as the accretion time for this particle.²²2Note that there are different definitions for particle accretion time (see W11 for details) and different halo definitions (e.g. FOF versus spherical overdensity), which can quantitatively affect the results presented in this work. FOF haloes may on occasion suffer from the ‘FOF bridge’ effect which misconnects two close haloes. To provide a straightforward comparison between Phoenix and Aquarius haloes, we adopt the same definitions of haloes and particle accretion events as W11. Furthermore, as we identify the point of accretion into the main halo progenitor, we also record the mass of the halo ($M_{\rm acc}$) responsible for bringing the particle into the main halo progenitor. This enables us to calculate the mass ratio between the main halo progenitor and the contributing halo, denoted as ${\rm ratio}=M_{200}(z_{\rm acc})/M_{\rm acc}$. Under this definition, a ratio closely approximating unity signifies that the particle was introduced into the halo progenitor through a major merger event, whereas a ratio significantly exceeding unity indicates that the particle entered through a minor merger event.

In Figure 2, we examine the radial distribution of merger mass ratios across seven Phoenix haloes. Our approach involves categorizing the mass ratios into distinct bins, each represented by a unique colour on the colourbar within each panel. Subsequently, we plot the fractions of particles at $z=0$ associated with various merger mass ratios within different radial bins. It is worth noting that if a particle lacks association with any resolved structure at the time of its accretion, we categorize this event as ‘diffuse’. A clump of mass must contain at least 20 particles to be resolved as a subhalo identified by subfind. Therefore, the category ‘diffuse’ in principle includes those unresolved mergers. However, we use the term ‘diffuse’ to refer to ‘unresolved mergers’ and ‘smoothed accretion’ without distinguishing between their physical contexts in this work. To maintain consistency with the analysis conducted by W11, we halt our trace-back analysis at $z=6$, designating particles already situated within the main halo progenitor at $z=6$ as ‘$z>6$’. As indicated in Figure 1, it is evident that only a small number of particles have already assumed their positions before the redshift of $z=6$.

The Phoenix haloes exhibit similar patterns to the Aquarius haloes, as depicted in Figure 3 of W11. Specifically, we observe that particles accreted prior to approximately $z\sim 6$ and those introduced through relatively major mergers, characterized by mass ratios falling in the range of $1\sim 10$, constitute the primary contributors to the central regions of the haloes. In contrast, particles from diffuse accretion are infrequently found in the inner regions but assume a more substantial presence in the outskirts. Minor mergers, characterized by mass ratios exceeding 10, also play a significant role in shaping the outer regions of the haloes. In general, mass accreted before the redshift of $z\sim 6$ does not make a substantial contribution.

A notable overarching trend in the merger ratio distribution is the correlation between larger merger ratios and greater radial distances from the halo centre. While there is considerable variability among haloes regarding particles already in place before $z\sim 6$, differences between the haloes are relatively minor. Notably, if we exclude particles that were already present before redshift 6, the patterns of merger ratios across all radii exhibit a high degree of uniformity across all seven Phoenix haloes. This echoes the universal unevolved subhalo mass function and merger ratio distributions found over a wide range of halo masses (see e.g. van den Bosch et al., 2005; Han et al., 2016; Han et al., 2018; Dong et al., 2022).

We subsequently normalize the shell radii by the virial radius, denoted as $r_{200}$, for each individual halo. This normalization facilitates a meaningful comparison of the averaged radial fractions of merger ratio profiles between the Phoenix haloes and the Aquarius haloes, as showcased in Figure 3. To ensure a fair assessment, the haloes within each project are averaged within each radius bin. Given that galactic haloes tend to form at earlier cosmic epochs, it is not surprising that the particles originating at redshifts exceeding 6 constitute a more significant fraction in comparison to cluster haloes. However, when we focus our analysis on the contributions from particles at lower redshifts and disregard the high-redshift particles, an intriguing revelation emerges: these two types of haloes actually exhibit strikingly similar merger ratio profiles. Specifically, the central regions predominantly consist of major mergers with ratios ranging from 1 to 10, while progressively larger merger ratios become more prevalent in the outskirts. Furthermore, diffuse particles primarily participate in mergers in the outskirts, where they exert a considerable influence.

To gain a deeper understanding of how different merger patterns vary with radius, we classify particles into three distinct accretion modes based on their merger mass ratios. Specifically, we define major mergers as events with mass ratios falling within the range of 1 to 10, minor mergers as events with mass ratios exceeding 10, and diffuse accretions as particles that were not associated with any resolved structures at the time of their accretion ($z_{\rm acc}$). In Figure 4, we present the mean fractions of these three accretion modes as a function of re-scaled radius ($r/r_{200}$) for both Aquarius and Phoenix haloes. It is important to note that in this analysis, we do not impose a redshift cutoff when tracing the particle histories. Consequently, particles accreted at redshifts exceeding 6 are also considered, and their merger ratios are calculated.

The comparison reveals similar merger patterns between the Aquarius and Phoenix haloes. Major mergers predominantly contribute to the centralmost regions, whereas minor mergers and diffuse accretions are more prevalent in the outskirts. Notably, in the case of Phoenix haloes, this trend is even more pronounced, with major mergers playing a larger role in the central regions compared to Aquarius haloes. Additionally, in the outskirts of Phoenix haloes, minor mergers contribute more than their Aquarius counterparts. Within the virial radius, diffuse accretion dominates minor mergers in Aquarius haloes, whereas for Phoenix haloes, the mass contribution from minor mergers exceeds that of diffuse accretion at radial distances beyond approximately $0.02r_{200}$.

The findings presented above highlight that, on the whole, Phoenix haloes exhibit qualitatively similar patterns of structure formation as Aquarius haloes. However, there are notable quantitative distinctions in their merger pathways. In particular, it appears that merger events exert more pronounced effects on Phoenix haloes compared to their Aquarius counterparts. This effect is especially pronounced for particles located in the central regions, where the Phoenix haloes tend to have undergone more recent activity and experienced a higher frequency of major mergers.

Next, we turn our attention to examining the radial profiles of particle accretion times within Phoenix haloes. Analogous to the particle mass ratio profile discussed in the previous subsection, we segment the particle accretion redshift, denoted as $z_{\rm acc}$, into distinct bins. For each radius within each Phoenix halo, we calculate the proportion of particles falling into various $z_{\rm acc}$ bins. The resultant profiles for all seven Phoenix haloes are visually depicted in Figure 5.

In a broad sense, all haloes demonstrate an inside-out formation pattern, characterized by early accretion of mass in the inner regions, followed by later accretion in the outskirts. Nevertheless, it is worth noting that there are substantial variations in this pattern among the Phoenix haloes. To shed light on this variability, we reference the formation redshift, denoted as $z_{\rm form}$, for these seven haloes, as obtained from Gao et al. (2012). The formation redshift values are as follows: $z_{\rm form}=1.17$ for Ph-A, $0.76$ for Ph-C, $0.46$ for Ph-D, $0.91$ for Ph-E, $1.1$ for Ph-F, $0.18$ for Ph-G, and $0.56$ for Ph-I.

For haloes Ph-D and Ph-G, the majority of particles are accreted after the redshift of $z=0.65$. In contrast, for the remaining haloes, particles in this redshift range also contribute significantly to the final halo mass, although their contributions fall short of the 50% mark. Notably, haloes Ph-A, Ph-D, Ph-F, and Ph-G exhibit a particularly pronounced inside-out formation pattern, especially after the redshift of $z=3$, with later accreted particles predominantly located in the halo outskirts.

For haloes Ph-C, Ph-E, and Ph-I, this inside-out pattern commences at a later stage. In these cases, particles accreted at $z>3$ do not occupy the centralmost region at the present day. In the case of halo Ph-E, particles accreted at $z>3$ are offset from the halo centre. In the case of halo Ph-C, particles accreted during the redshift ranges $1.07<z<3$ and $0.65<z<1.07$ collectively contribute to nearly 50% of the mass in the central region, while particles accreted at $z>3$ play a marginal role in shaping the halo’s mass and structure.

Note that the radial profile of particles with $z_{\rm acc}>3$ might sensitively depend on the definition of the halo centre. To check this, we have re-done Figure 5 by setting the $z=0$ halo centre as the centre of mass position of the particles within the half-mass radius. We found that the particles with $z_{\rm acc}>3$ in the Ph-E halo occupy the centralmost region. However, the Ph-C and Ph-I haloes still exhibit similar profiles as in Figure 5. Comparing with the Aquarius haloes shown in W11, which always have their $z_{\rm acc}>6$ particles in the centralmost region, this reflects that cluster haloes are less relaxed, and therefore it is more difficult to identify a robust centre. It might also be worth investigating the effect of using alternative halo finder algorithms, but this is beyond the scope of this study.

In Figure 6, we present stacked radial particle accretion time profiles for both Aquarius and Phoenix haloes, with the radius scaled by the virial radius of each respective halo. In a general sense, both Aquarius and Phoenix haloes exhibit qualitatively similar inside-out accretion patterns. For Aquarius haloes, a stable central region has already formed before the redshift of $z=6$ and has persisted to the present day. Conversely, for Phoenix haloes, a significant portion of the mass in the central regions is accreted after the redshift of $z=3$. This difference highlights the varying temporal evolution of the central regions in these two sets of haloes.

Given that cluster haloes tend to have formation redshifts later than galactic haloes, we introduce the concept of the normalized accretion time, denoted as $t_{\rm acc}-t_{\rm form}$. Here, $t_{\rm acc}$ represents the cosmic time, or equivalently the age of the universe, corresponding to $z_{\rm acc}$, while $t_{\rm form}$ corresponds to $z_{\rm form}$. In Figure 7, we present the radial profiles of mean normalized particle accretion times for all seven Phoenix haloes. These profiles can be effectively categorized into two distinct groups. Haloes Ph-D, Ph-G, and Ph-I exhibit behaviours distinct from the other four haloes, which exhibit more similar patterns. This observation aligns with expectations, as the formation times of the former three haloes are later than those of the latter four. Furthermore, when comparing the normalized accretion time profiles of Phoenix haloes to those of Aquarius haloes (as shown in Figure 5 of W11), it becomes evident that Phoenix haloes tend to exhibit larger halo-to-halo variations in their accretion time profiles.

In Figure 8, we present a comparison of stacked $t_{\rm acc}-t_{\rm form}$ profiles averaged across six Aquarius haloes and seven Phoenix haloes. This comparison reveals that, in contrast to galactic haloes, cluster haloes tend to accumulate their innermost particles at an earlier stage relative to the halo formation time $t_{\rm form}$. Conversely, they tend to accrete their outermost particles within a shorter time interval in relation to $t_{\rm form}$.

We closely examine the evolution of density profiles within Phoenix haloes spanning from redshift 2 to 0. In Figure 9, we present the density profiles within a radius of $1\ h^{-1}{\rm Mpc}$ for all seven Phoenix haloes at redshifts 2, 1, 0.5, 0.2, and 0. The upper panels feature the physical halo radius on the horizontal axis and the spherically-averaged density in units of $\rm M_{\odot}\,kpc^{-3}$ on the vertical axis. The lower panels depict the ratio between the density profiles at different redshifts and the density profile at $z=0$.

Notably, the density profiles exhibit remarkably strong evolution, and substantial halo-to-halo variations. As previously mentioned, Ph-A and Ph-F have formation times ($z_{\rm form}>1$) earlier than the other Phoenix haloes. Correspondingly, the density profiles for Ph-A and Ph-F display greater stability than the other five haloes. In the case of Ph-A, the density profile undergoes minimal evolution after reaching a redshift of $z=0.5$. The profile ratios at $z=0.5$ and $0.2$ hover around 1, signifying a stable structure. Similarly, the density profile of Ph-F undergoes very little evolution after reaching a redshift of $z=1$, particularly at radii $r<500\ h^{-1}{\rm kpc}$. In contrast, all other haloes demonstrate dramatic evolution in their density profiles. In fact, some of them (Ph-D, Ph-G, and Ph-I) do not exhibit stable density profiles even at a redshift as low as $0.2$. The pronounced halo-to-halo variations are consistent with the significant scatter in their formation redshifts.

We now turn our attention to investigating the inner particle distribution and its temporal evolution. To accomplish this, we focus on two central spheres with radii of $R_{0}=50\ {\rm kpc}$ and $R_{0}=150\ {\rm kpc}$, respectively. We trace the enclosed mass within these spheres from $z=0$ until the virial radius of the Phoenix halo progenitor becomes smaller than $2R_{0}$, i.e., when $r_{200}(z)<2R_{0}$. The results of this analysis are displayed in Figure 10, where the horizontal axis represents the age of the Universe, and the vertical axis depicts the ratio between the enclosed mass at various redshifts and that at $z=0$. Different coloured lines in the legend represent the evolution of the enclosed mass within different spherical regions.

For reference, we also overlay the results for $R_{0}=500\ {\rm kpc}$. Notably, this analysis reveals intriguingly distinct behaviours when compared to the Aquarius haloes studied in W11. Aquarius haloes generally exhibit stable inner structures after approximately 7 Gyr of the Universe’s age³³3A similar investigation of the evolution of enclosed mass at different physical radii was conducted by Diemand et al. (2007) using a Milky Way-sized galaxy from the Via Lactea simulation, and the authors reached a similar conclusion., with the exception of Aq-F, which experiences a major merger very recently.

In contrast, all Phoenix haloes display substantial evolution in their inner mass distribution after 7 Gyr, and some haloes, such as Ph-D and Ph-G, even undergo fairly active changes in the recent past. Once again, the significant halo-to-halo variation is evident.