In the Λ cold dark matter cosmology, dark matter haloes grow primarily through the accretion of smaller haloes. Much of the mass in a halo of 1014 M⊙ comes in through accretion of ∼1013 M⊙ haloes. If each such halo hosted one luminous red galaxy (LRG) then the accretion of so many haloes is at odds with the observed number of LRGs in clusters unless these accreted LRGs merge or disrupt on relatively short time-scales (∼2 Gyr). These time-scales are consistent with classical dynamical friction arguments, and imply that two to three LRGs have merged or disrupted within each halo more massive than 1014 M⊙ by z= 0. The total amount of stellar mass brought into these massive haloes by z= 0 is consistent with observations once the intracluster light (ICL) is included. If disrupted LRGs build up the ICL, then the hierarchical growth of massive haloes implies that a substantial amount of ICL should also surround satellite LRGs, as suggested by recent observations of the Virgo cluster. Finally, we point out that these results are entirely consistent with a non-evolving clustering strength and halo occupation distribution, and note that observations of the latter in fact support the hypothesis that merging/disruption of massive galaxies does indeed take place at late times.
The formation and evolution of massive red galaxies provide a critical testing ground for modern theories of galaxy formation based on hierarchical merging of dark matter haloes. Ongoing growth of massive haloes via mergers is a generic feature of hierarchical models, such as cold dark matter (CDM). However, evidence for the ongoing assembly of massive galaxies is at best inconclusive. Evolution in the galaxy stellar mass and luminosity functions (LFs) at the massive/luminous end appears quite modest since z= 1 (e.g. Drory et al. 2004; Bundy, Ellis & Conselice 2005; Borch et al. 2006; Caputi et al. 2006; Faber et al. 2006; Fontana et al. 2006; Wake et al. 2006; Willmer et al. 2006; Brown et al. 2007) though estimates of the merger rate of massive galaxies present a less consistent picture (van Dokkum 2005; Bell et al. 2006; Masjedi et al. 2006; White et al. 2007). If, as theory predicts, massive haloes are constantly accreting haloes that are themselves hosts of massive galaxies, what is the fate of these accreted galaxies?
Two physical effects can cause satellite galaxies to ‘disappear’ from an observational sample. Tidal forces acting on a satellite as it orbits in the host halo potential can cause it to disrupt. At the same time, dynamical friction (DF) causes a satellite to lose energy to the background dark matter halo and eventually causes the satellite to sink towards the centre and merge with the central galaxy of the host halo. While such notions, and their relevance to the evolution of galaxies within clusters, have been known for decades (e.g. Chandrasekhar 1943; Ostriker & Tremaine 1975; Ostriker & Hausman 1977; Merritt 1984), accurate merger times of satellite galaxies have been historically hard to calculate, and are poorly constrained observationally. Unfortunately, the problem cannot at present by circumvented by brute-force simulations due both to severe resolution requirements and the uncertain effects of baryon condensation on the survival of satellite haloes (see e.g. Klypin et al. 1999; Moore et al. 1999; Diemand, Moore & Stadel 2004; Gao et al. 2004a, 2004b; Reed et al. 2005). In addition, while DF is usually considered in a collisionless medium (such as dark matter), DF acting in a collisional medium (such as intracluster gas) is stronger (weaker) than in the collisionless case for satellites travelling at supersonic (subsonic) speeds (Ostriker 1999). Observational constraints on the merging time-scale of satellites would hence provide valuable insight into this complex dynamical process.
This paper explores observational constraints on the average merging time-scale of luminous red galaxies (LRGs). We assign LRGs to dark matter haloes that have grown more massive than M∼ 1013 M⊙ and use an N-body simulation to follow their accretion on to larger dark matter haloes with z= 0 mass comparable to observed rich groups and clusters (M > 1014 M⊙). Comparison with the observed multiplicity function of LRGs at z∼ 0.3 implies that accreted LRGs must merge on time-scales comparable to those predicted by Chandrasekhar's formula (∼2 Gyr). While this may not be surprising, the relative flood of massive haloes on to more massive haloes implies that a substantial number of LRGs have disrupted over the history of the Universe. Though these numbers may at first glance appear large (on average two to three disrupted LRGs per z= 0 halo with M > 1014 M⊙) we show that the total stellar mass brought in by these accreted LRGs is consistent with the observed stellar mass in clusters so long as one counts both observed massive galaxies and the observed intracluster light (ICL).
The following sections describe in more detail the salient accretion properties of massive dark matter haloes (Section 2), the inferred merging time-scale of LRGs, if LRGs correspond to massive haloes (Section 3), and the implied total stellar mass brought into massive z∼ 0 dark matter haloes by these accreted LRGs (Section 4). We conclude in Section 5. Throughout we assume a flat ΛCDM cosmology with, (Ωm, ΩΛ, h, σ8) = (0.25, 0.75, 0.72, 0.8) and use a virial mass definition, M200 corresponding to the mass contained within a region that has mean density equal to 200× the critical density (see e.g. Evrard et al. 2007).
2 THE ACCRETION HISTORY OF MASSIVE DARK MATTER HALOES
A robust expectation of a Universe dominated by cold dark matter is the hierarchical growth of structure, and in particular the growth of dark matter haloes via the accumulation of smaller haloes. An illustrative example of the accretion history of dark matter haloes is shown in Fig. 1. There we plot the multiplicity function for haloes more massive than, 1014 M⊙, that is, the distribution of the number of haloes with M > 1013 M⊙ that have accreted on to haloes with z= 0 mass greater than 1014 M⊙ (see Appendix A for details regarding the simulation used to compile this information). We refer to these more massive haloes as ‘hosts’ throughout. There are 2339 such hosts in our simulation at z= 0 corresponding to a number density of . Note that these distributions are not symmetric. On average, haloes more massive than 1014 M⊙ have been bombarded by 3.2 haloes with mass >1013 M⊙ over a Hubble time. Here we do not distinguish between haloes that were accreted directly on to the host halo and those that were accreted on to an intermediate halo that later accreted on to the host, although such a distinction will be utilized in the following sections. The accretion of such massive haloes is roughly equally important both at low and high redshift: on average two such haloes have been accreted at z < 0.8 (the Universe was about half its present age at z= 0.8 for the cosmology assumed herein).
Haloes more massive than ∼1013 M⊙ are expected to contain at least one massive galaxy at their centre (Zehavi et al. 2005b) even at moderate redshifts (e.g. Yan, Madgwick & White 2003; Yan, White & Coil 2004; Coil et al. 2006). From Fig. 1 we are led to the conclusion that in the absence of mergers, observed clusters with M > 1014 M⊙ should contain on average 3.2 massive galaxies (and certainly more if accreted haloes of lower masses also contain massive galaxies), with a significant tail towards much larger numbers. However, reproducing the observed clustering of massive galaxies at z∼ 0 (Zehavi et al. 2005a) would require closer to 1.2 galaxies in such haloes in our simulation, in agreement with other work (Masjedi et al. 2006; Kulkarni et al. 2007). While these statements are only qualitative, they will be confirmed in the more quantitative discussion that follows. In order to reconcile the accretion properties of haloes with observations, we are thus led to consider the fate of these massive haloes and the galaxies within them.
3 THE MERGING TIME-SCALE OF LRGs
LRGs are massive galaxies with very little ongoing star formation; they thus constitute the tip of the red sequence. They have uniform spectral energy distributions marked by numerous features and hence their redshifts are relatively straightforward to estimate photometrically (Padmanabhan et al. 2005, redshift uncertainties are δz∼ 0.03). Modelling of their spectral energy distributions has led to the conclusion that these galaxies formed the bulk of their stars at z > 2 (e.g. Trager et al. 2000; Thomas et al. 2005; Jimenez et al. 2006), and hence are expected to evolve largely dissipationlessly at z < 1. Their clustering strength is large, suggesting that they live in massive dark matter haloes M > 1013 M⊙ (Zehavi et al. 2005a).
Recently, Ho et al. (2007) have measured the multiplicity function of LRGs extracted from the Sloan Digital Sky Survey (SDSS; Adelman-McCarthy et al. 2006) for 43 clusters over the redshift range 0.2 < z < 0.5.1 Cluster virial masses were derived from ROSAT X-ray data and range 1014.1 < M200/M⊙ < 1014.9. The average stellar mass of the LRGs in this sample is, Mstar= 1011.6 M⊙ as determined from a colour-based stellar mass estimator [for a Chabrier initial mass function (IMF); Bell et al. (2003)]. These clusters contain on average 2.5 LRGs. The reader is referred to Ho et al. (2007) for further details regarding these observations.
In this sample there are approximately five clusters that contain no LRGs.2 When plotting the observed multiplicity function below we include both the one reported in Ho et al. and one where clusters with N < 1 are artificially assigned N= 1. This is done to afford a more robust comparison to our simple model (see below) where we assume that each cluster halo contains at least one LRG at the centre. As discussed below, our conclusions are insensitive to this distinction.
The dark matter halo accretion history of massive haloes (e.g. Fig. 1) is closely related to the LRG multiplicity function. The former can be converted into the latter if one knows both the minimum halo mass (measured at the epoch of accretion) associated with accreted LRGs, Mmin and the average time it takes for LRGs to merge and/or disrupt3 once accreted. Below we argue for reasonable values of Mmin and then attempt to directly constrain the average LRG merging time-scale. We parametrize the probability that an LRG will have merged by a time tacc since accretion on to the host halo via
The minimum LRG stellar mass in the observed sample is 1011.3 M⊙. A minimum halo mass associated with LRGs can be estimated by assuming the universal baryon fraction fb= 0.17 and an efficiency factor η of converting baryons into stars. While this factor is only poorly constrained even at low redshift, values of the order of ∼0.1 are likely reasonable for these massive galaxies (Hoekstra et al. 2005; Mandelbaum et al. 2006) and imply a minimum LRG hosting halo mass of Mmin∼ 1013 M⊙. Note that the minimum halo mass for hosting LRGs is here used to identify those accreted haloes that are likely to host an LRG, with a significant fraction of the accretion occurring at z∼ 1 or higher. Thus this minimum mass will likely not directly correspond to the minimum halo mass hosting LRGs at z∼ 0 since the LRGs that have survived to the present epoch will have accreted much more dark matter, resulting in a larger value for Mmin at the present epoch. In our simulations haloes today are on average five times more massive than they were when they first crossed, Mmin making our estimate consistent with that inferred from z∼ 0 clustering (Zehavi et al. 2005a; Kulkarni et al. 2007). As we discuss in Section 4, Mmin much larger than 1013 M⊙ would require unreasonably long DF times and Mmin either much larger or much smaller would be in conflict with stellar mass estimates in clusters.
Fig. 2 plots the resulting LRG multiplicity function for both LRGs in accreted haloes more massive than 1013 M⊙ (solid lines) and observations (dashed lines). We also include predictions for LRGs associated with haloes twice as massive as our fiducial minimum mass (dotted lines) in order to illustrate the sensitivity to our assumed LRG halo mass threshold. Each panel in Fig. 2 is the multiplicity function for a different merger time-scale. The top four panels assume that the merger time-scale, τ is constant. It is apparent from these panels that if Mmin∼ 1013 M⊙ then LRGs must merge on a characteristic time-scale of ∼2 Gyr. This time-scale implies an average number of LRGs per cluster of 2.5, satisfyingly close to the observed value of 2.6.
Note that in order to compare to the observations we have weighted host haloes at z= 0.3 (of which there are 460 in our simulation within the observed mass range) in such a way as to reproduce the mass distribution of the observed clusters. It is the combination of these two effects (higher redshift and different mass distribution) that does not allow a direct comparison between Figs 1 and 2.
If LRGs never merged, there would be on average 5.8 LRGs per z= 0.3 host halo (averaged over the observed distribution of halo masses). Comparing this number to the observed 2.5 LRGs per cluster highlights the importance and prevalence of LRG mergers.times the critical density. It is important to note that tDF gets shorter both at higher redshift and for merger mass ratios closer to unity. In this lower panel we have additionally included results for haloes of mass M > 5 × 1012 M⊙ (dot–dashed line) for comparison.
This comparison with simple DF estimates provides a satisfying cross-check to the results in the upper panels. In particular, for our fiducial minimum LRG halo mass of 1013 M⊙ the DF time-scale averaged over all the accreted haloes is 2.4 Gyr (median time is 1.8 Gyr – the distribution is highly asymmetric), which is quite similar to the constant merger time-scale that best matches the observed multiplicity function (∼2 Gyr). From Fig. 2 it is apparent that the simple DF time-scale would not have reproduced the observed LRG multiplicity function if the minimum haloes mass capable of hosting LRGs were substantially more or less massive than 1013 M⊙. The implication here is clear: if Mmin is in fact considerably larger or smaller than 1013 M⊙ then simple DF arguments do not apply to the LRG population.
In fact, it is not at all clear that equation (3) should apply here or in general to the dynamical evolution of satellite galaxies, as it is strictly valid for a point mass moving in an infinite, uniform background density field. Indeed, much work has gone into both testing the validity of equation (3) with simulations (e.g. White 1983; van den Bosch et al. 1999; Velazquez & White 1999; Read et al. 2006) and developing extensions to it (e.g. Tremaine & Weinberg 1984; Colpi, Mayer & Governato 1999), including numerically following the evolution of the satellite orbit and the mass-loss due to tidal forces (Benson et al. 2002; Taffoni et al. 2003; Taylor & Babul 2004; Zentner et al. 2005). This body of work has shown that equation (3) is at best a crude approximation to the realistic, time-dependent problem. For these and other reasons it is quite surprising, if not entirely coincidental, that the classical Chandrasekhar DF time-scale adequately captures the merging time-scale of LRGs.
As can be seen from the upper four panels in Fig. 2, there is a degeneracy between the merger time-scale and the minimum halo mass associated with LRGs in the sense that a larger Mmin coupled to a larger time-scale can produce roughly the same multiplicity function. Thus, if one thought that LRGs lived in more massive haloes than what we have assumed here, then one would infer a longer merging time-scale for LRGs. However, this is exactly opposite to what one would infer from DF arguments since tDF∝M−1s. Furthermore, increasing Mmin to 5 × 1013 M⊙ would result in far too few LRGs in massive haloes compared to observations, even if τ=∞; in this case the average number of LRGs per halo would be 1.6. As we describe in Section 4, Mmin is further constrained by observations of the stellar light in massive haloes.
These merger time-scales can easily be cast into a discussion of LRG merger rates. For this discussion we consider the full host halo population at z= 0 rather than the population at z= 0.3 meant to coincide with the data from Ho et al., in order to draw more general conclusions about LRG mergers. Fig. 3 plots the cumulative distribution of merged LRGs as a function of redshift, per host halo. The figure includes constant merger time-scales of 0 and 2 Gyr and a time-scale set by DF. The 0-Gyr case can equivalently be thought of as the distribution of accreted LRGs, since in this case the accretion and merging epochs are coincident. In this figure we only count LRGs that merge within the main progenitor of the z= 0 host halo. This figure is thus not directly comparable to Fig. 1. In other words, if a halo merges within a halo that itself later mergers with the host halo then it is not counted here. This plot is thus meant to capture the number of mergers actually occurring within the main progenitor of the host halo.
The DF time-scale is shorter than 2 Gyr at high redshift and longer than 2 Gyr at low redshift; this results in a more gradual increase in the merger rate per unit redshift compared to a constant merger time of 2 Gyr. Since the constant and DF time-scales are different at redshifts both greater and less than, 0.3 comparisons to the multiplicity function at different epochs can in principle rule out either (or both) of these time-scales. Both the constant 2-Gyr time-scale and that determined by DF imply that 2–2.5 LRGs have merged with the host halo by z= 0. Moreover, the figure indicates that a substantial number of LRGs are merging/disrupting at z < 1. In the next section we set this in the context of recent observational results of the stellar mass budget in groups and clusters.
4 THE TOTAL CLUSTER STELLAR MASS
The merger rate of LRGs found in the previous section indicates that a significant number of LRGs must have disrupted over the history of massive haloes. In particular, a merger time-scale of τ= 2 Gyr implies that on average two to three LRGs have disrupted within each host halo more massive than 1014 M⊙ by z= 0 (and approximately 2.1 since z= 1). In this section we discuss how reasonable such a disruption rate is in light of the total stellar mass observed in clusters at z∼ 0.
The amount of stellar mass brought into massive haloes by LRGs can be estimated in the following way. We consider haloes that have a mass at z= 0 greater than 1014 M⊙ and assign a stellar mass of 1011.6 M⊙, the mean stellar mass of the observed LRG sample, to the centre of each. Then, each halo that was accreted on to these z= 0 haloes is assigned the same amount of stellar mass if the halo mass at accretion is >1013 M⊙. This exercise is thus meant to count the total amount of stellar mass that was at some point associated with LRGs. We make no distinction between disrupted and non-disrupted LRGs except in one case: if, according to our best guess LRG merger time-scale (2 Gyr), an LRG halo would have disrupted not in the main progenitor of the z= 0 halo but rather in some smaller halo that would itself later accrete on to the main progenitor but does not merge, then we do not count this LRG in the final stellar mass budget. In this case the disrupted LRG contributes to the satellite's ICL (i.e. ICL that surrounds the satellite and is distinct from the central ICL). As it turns out, only 20 per cent of all accreted haloes more massive than 1013 M⊙ fall into this category and including these haloes in the stellar mass budget does not appreciably change our conclusions.
We compare to data presented in Lin, Mohr & Stanford (2004) who have compiled information on 93 clusters at z < 0.1 including X-ray observations used to derive cluster virial masses and luminosities of cluster members derived from Two Micron All-Sky Survey (2MASS) photometry. From these data Lin et al. have estimated the LF of each cluster, assuming that the faint-end slope is fixed at α=−1.1. Using their LFs we are able to estimate the total luminosity in galaxies brighter than LK= 2.8 × 1011 L⊙ which corresponds to the minimum luminosity of the Ho et al. LRG sample. This total luminosity is converted into stellar mass by assuming a mass-to-light ratio of Mstar/LK= 0.72 which is appropriate for red galaxies with a Chabrier IMF (Bell et al. 2003). The LFs reported in Lin et al. (2004) do not include the brightest cluster galaxy (BCG); we thus add these in separately. Finally, we have assumed that each cluster contains ICL with a stellar mass equal to the mass of the BCG identified by Lin et al. (i.e. LICL=LBCG). This amount of light associated with the ICL is consistent with recent observations (Gonzalez, Zabludoff & Zaritsky 2005; Zibetti et al. 2005). In this ‘total’ cluster luminosity we do not include any possible ICL associated with satellites; it is for this reason that we did not include LRGs that disrupted in haloes which themselves later merged with the host.
Fig. 4 presents a comparison between the data from Lin et al. (2004) and the stellar mass associated with accreted haloes. The agreement is encouraging. Note that varying any one of our assumptions can change the results from both the data and our model; the important point to take away from this comparison is that the influx of massive galaxies embedded within accreted haloes appears to roughly agree with the total stellar mass within observed clusters at z∼ 0. This provides further support to our identification of haloes with mass >1013 M⊙ as being host to LRGs and suggests that disrupted LRGs deposit their stars into a combination of the central galaxy and ICL. Increasing or decreasing Mmin by a factor of 2 would result in substantial disagreement with the observations shown in Fig. 4. This is due to the fact that the number of accreted haloes does not scale linearly with the accreted halo mass, and provides further support for our choice of Mmin= 1013 M⊙.
The simple model presented here also provides a straightforward means for understanding the observed trend of decreased scatter in Mstar/Mhost with increasing Mhost. This arises because the number of accreted haloes with mass >1013 M⊙ is a weak function of Mhost. This is in contrast to the observed number of satellites, which appears to be closer to linear in Mhost (Lin et al. 2004; Popesso et al. 2007). The difference implies that fractionally more haloes/LRGs are merging in lower mass haloes compared to higher mass haloes.
Interestingly, we find that it is not uncommon for z= 0 host haloes to contain disrupted LRGs that did not disrupt within the host halo (and hence were not counted in the above figure) but rather disrupted in a smaller halo that later accreted on to the host (and remained as a satellite to z= 0). If these disrupted LRGs are depositing some fraction of their stars into ICL, then this suggests that there could be a significant amount of ICL that is not centred on the central galaxy but is instead centred on cluster satellites. Such a scenario is corroborated by recent observations of the Virgo cluster that show significant amounts of ICL surrounding several of the most massive satellites (Mihos et al. 2005).
The results of the previous sections suggest the following picture. If LRGs are associated with haloes more massive than 1013 M⊙ at the time when they are accreted on to more massive host haloes, then the observed multiplicity function of LRGs at z∼ 0.3 implies that LRGs must merge and/or disrupt on time-scales of ∼2 Gyr. Such a merger rate implies that two to three such LRGs have disrupted in haloes more massive than 1014 M⊙ by z= 0. This merger time-scale is consistent with classical DF arguments and suggests that a rather simplistic dynamical prescription for the evolution of LRGs is applicable when considering ensemble averages.
Moreover, the amount of total stellar mass in clusters that was at one point associated with these infalling LRGs (ignoring for the moment whether or not this stellar mass is locked up in satellite galaxies) is consistent with observations when the observed amount of stars in the ICL is accounted for. This in turn suggests that the disrupting LRGs are depositing their stars into a combination of the ICL and central galaxy, which is consistent with previous modelling (Monaco, Fontanot & Taffoni 2007; Conroy, Wechsler & Kravtsov 2007; Murante et al. 2007; Purcell, Bullock & Zentner 2007). Finally, there appears to be a significant number of LRGs that have disrupted within haloes that only later accreted on to (but did not merge with) what would become the z= 0 host halo. This suggests that there could be a significant amount of ICL surrounding cluster satellites, in addition to what is known to be associated with the central galaxy.
It has been historically challenging to constrain the merger rate of galaxies. Previous studies have relied on either morphological disturbances (e.g. Conselice et al. 2003; van Dokkum 2005; Bell et al. 2006) or close pair counts (e.g. Masjedi et al. 2006) as probes of the merger rate of massive galaxies. Unfortunately, both methods are rather indirect since the connection between either morphological disturbances or close pair counts and merger rates is uncertain. The most recent inferred LRG–LRG merger rate is from Masjedi et al. (2006) who find a rate of 0.6 × 104 Gyr−1 Gpc−3. Averaging over all haloes between z= 0.5 and z= 0.2, the model presented herein implies an LRG–LRG merger rate of Gyr−1 Gpc,−3 depending on whether the constant 2 Gyr or DF time-scale is used. The agreement with Masjedi et al. (2006) is encouraging, especially given the (different) uncertainties in both approaches. These rates are also consistent with current predictions from cosmological hydrodynamic simulations (Maller et al. 2006).
Many studies have attempted to constrain the stellar mass growth of massive galaxies from their inferred merger rates. However, as argued in Conroy et al. (2007) and herein, the merging of massive galaxies will often not correspond to significant growth of the resulting galaxy because a substantial amount of stars can be transferred to the ICL. We hence caution against using merger rates to constrain the stellar mass growth of galaxies.4 In fact, significant growth of the ICL via merging at late times provides a means for reconciling two apparently contradictory facts: on one hand, observations at z < 1 indicate that central massive red galaxies grow little in mass (e.g. Bundy et al. 2006; Fontana et al. 2006; Wake et al. 2006; Brown et al. 2007), while on the other hand, merging/disruption of galaxies within groups and clusters at late times appears relatively common (e.g. White et al. 2007).
White et al. (2007) outlined an approach for measuring the merging rate of massive galaxies similar to the one presented herein. Using the observed evolution in the clustering of massive galaxies, these authors concluded that ∼1/3 of massive satellites merge/disrupt between z≃ 0.9 and 0.5. In the present work we find roughly 50 per cent of massive satellites have disrupted over similar epochs. Our fraction is slightly higher because we have focused on more massive galaxies than in White et al. (2007). The more general conclusion from these two studies is, however, robust – the population of massive galaxies experiences significant amounts of merging/disruption, even at z < 1.
There is an important implication of considering the evolution of galaxies within the context of the hierarchical growth of haloes. At first glance, the lack of evolution in the observed correlation function of massive galaxies and their halo occupation distribution at z < 1 suggests that massive galaxies do not disrupt or merge over this epoch. However, these galaxies are embedded within dark matter haloes that are continually merging and accreting new galaxies, which instead suggests that massive galaxies must merge in order that these observed quantities not evolve appreciably at late times. This statement is further corroborated by dissipationless simulations which show explicitly that the average number of subhaloes within host haloes does not evolve appreciably at z < 1 because the accretion and disruption rate of subhaloes are approximately equal (Reed et al. 2005; Conroy, Wechsler & Kravtsov 2006). If satellite galaxies reside within these subhaloes then the observed non-evolution of the clustering and halo occupation of massive galaxies at z < 1 is in fact consistent with significant amounts of merging at late times.
Our results highlight the power of using purely dissipationless simulations coupled to simple relations between galaxies and dark matter to infer the evolution of galaxies and their relation to the underlying dark matter with time. The approach outlined herein can easily be extended to other data sets to provide additional constraints on the merger rate of galaxies.
We thank David Hogg, Mike Boylan-Kolchin, Morad Masjedi, Eliot Quataert and Jerry Ostriker for comments on an earlier draft. SH thanks David Spergel, Jim Gunn, Jerry Ostriker, Chris Hirata and Nikhil Padmanabhan for insightful discussions and comments. CC thanks nature for being complicated, but not too complicated. MW is supported by NASA. The simulations were performed on the supercomputers at the National Energy Research Scientific Computing Centre.
APPENDIX A: THE SIMULATION, HALO CATALOGUE AND MERGER TREES
We use a high-resolution simulation of a ΛCDM cosmology (ΩM= 0.25 = 1 −ΩΛ, ΩB= 0.043, h= 0.72 ns= 0.97 and σ8= 0.8). The linear theory power spectrum is computed by evolution of the coupled Einstein, fluid and Boltzmann equations using the code described in White & Scott (1996). This code agrees well with cmbfast (Seljak & Zaldarriaga 1996), see, for example, Seljak et al. (2003). The simulation employs 10243 particles of mass 8 × 109 h−1 M⊙ in a periodic cube of side 500 h−1 Mpc using a treepm code (White 2002). The Plummer equivalent softening is (comoving).
The phase-space data for the particles exists at 50 outputs, spaced equally in conformal time between z≃ 3 and 0. For each output we generate a catalogue of haloes using the Friends-of-Friends (FoF) algorithm (Davis et al. 1985) with a linking length of 0.168× the mean interparticle spacing. This procedure partitions the particles into equivalence classes, by linking together all particle pairs separated by less than a distance b. The haloes correspond roughly to particles with ρ > 3/(2πb3) ≃ 100 times the background density. For each halo we compute a number of properties, including the mass M200 interior to r200 within which the mean density is 200× the critical density. M200 is computed from a fit of an NFW profile (Navarro, Frenk & White 1997) to the particles in the FoF group.
Merger trees are computed from the set of halo catalogues by identifying for each halo a ‘child’ at a later time. The child is defined as that halo which contains, at the later time-step, more than half of the particles in the parent halo at the earlier time-step (weighting each particle equally). For the purposes of tracking haloes this simple linkage between outputs suffices (note that we do not attempt to track subhaloes within larger haloes, which generally requires greater sophistication). From the merger trees it is straightforward to compute the time when a halo ‘falls in’ to a larger halo, the number and masses of the progenitors, etc.