Timing the last major merger of galaxy clusters with large halo sparsity

MNRAS 000, 1–18 (2021) Preprint 1 April 2022 Compiled using MNRAS LATEX style file v3.0

 Timing the last major merger of galaxy clusters with large halo sparsity
 T. R. G. Richardson,1★ P.-S. Corasaniti,1,2
 1 Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, Université de Paris, CNRS, F-92190 Meudon, France
 2 Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France

 Accepted XXX. Received YYY; in original form ZZZ
arXiv:2112.04926v2 [astro-ph.CO] 31 Mar 2022

 ABSTRACT
 Numerical simulations have shown that massive dark matter haloes, which today host galaxy clusters, assemble their mass over
 time alternating periods of quiescent accretion and phases of rapid growth associated with major merger episodes. Observations
 of such events in clusters can provide insights on the astrophysical processes that characterise the properties of the intra-cluster
 medium, as well as the gravitational processes that contribute to their assembly. It is therefore of prime interest to devise a fast
 and reliable way of detecting such perturbed systems. We present a novel approach to identifying and timing major mergers
 in clusters characterised by large values of halo sparsity. Using halo catalogues from the MultiDark-Planck2 simulation, we
 show that major merger events disrupt the radial mass distribution of haloes, thus leaving a distinct universal imprint on the
 evolution of halo sparsity over a period not exceeding two dynamical times. We exploit this feature using numerically calibrated
 distributions to test whether an observed galaxy cluster with given sparsity measurements has undergone a recent major merger
 and to eventually estimate when such an event occurred. We implement these statistical tools in a specifically developed public
 python library lammas, which we apply to the analysis of Abell 383 and Abell 2345 as test cases. Finding that, for example,
 Abell 2345 had a major merger about 2.1 ± 0.2 Gyr ago. This work opens the way to detecting and timing major mergers in
 galaxy clusters solely through measurements of their mass at different radii.
 Key words: galaxies: clusters: general – galaxies: fundamental parameters – galaxies: kinematics and dynamics – meth-
 ods:statistical

 1 INTRODUCTION have shown that the matter density profile at large radii is consistent
 with the universal Navarro-Frenk-White profile (NFW, Navarro et al.
 Galaxy clusters are the ultimate result of the hierarchical bottom-up
 1997), while deviations have been found in the inner regions (New-
 process of cosmic structure formation. Hosted in massive dark matter
 man et al. 2013; Annunziatella et al. 2017; Collett et al. 2017; Sartoris
 haloes that formed through subsequent phases of mass accretion
 et al. 2020). In relaxed systems, the gas falls in the dark matter domi-
 and mergers, galaxy clusters carry information on the underlying
 nated gravitational potential and thermalises through the propagation
 cosmological scenario as well as the astrophysical processes that
 of shock waves. This sets the gas in a hydrostatic equilibrium (HE)
 shape the properties of the intra-cluster medium (ICM) (for a review,
 that is entirely controlled by gravity. Henceforth, aside astrophysi-
 see e.g. Voit 2005; Allen et al. 2011; Kravtsov & Borgani 2012).
 cal processes affecting the baryon distribution in the cluster core,
 Being at the top of the pyramid of cosmic structures, galaxy clus-
 the thermodynamic properties of the outer ICM are expected to be
 ters are mostly found in the late-time universe. These can be observed
 self-similar (see e.g. Ettori et al. 2019; Ghirardini et al. 2019, 2021).
 using a variety a techniques that probe either the distribution of the
 This is not the case of clusters undergoing major mergers for which
 hot intra-cluster gas through its X-ray emission (see e.g. Vikhlinin
 the virial equilibrium is strongly altered (see e.g. Biffi et al. 2016).
 et al. 2005; Ebeling et al. 2010; Pierre et al. 2016; CHEX-MATE
 Such systems exhibit deviations from self-similarity such that scaling
 Collaboration et al. 2021), the scattering of the Cosmic Microwave
 relations between the ICM temperature, the cluster mass and X-ray
 Background radiation (CMB) due to the Sunyaev-Zeldovich effect
 luminosity differ from that of relaxed clusters (see e.g. Planelles &
 (see e.g. Staniszewski et al. 2009; Menanteau et al. 2013; Reichardt
 Quilis 2009; Rasia et al. 2011; Chen et al. 2019).
 et al. 2013; Planck Collaboration et al. 2014b; Bleem et al. 2015),
 through measurement of galaxy overdensities or the gravitational
 lensing effect caused by the cluster’s gravitational mass on back- A direct consequence of merger events is that the mass estimates
 ground sources (Rykoff et al. 2016; Maturi et al. 2019; Umetsu et al. inferred assuming the HE hypothesis or through scaling relations
 2011; Postman et al. 2012). may be biased. This may induce systematic errors on cosmological
 The mass distribution of galaxy clusters primarily depends on analyses that rely upon accurate cluster mass measurements. On the
 the dynamical state of the system. Observations of relaxed clusters other hand, merging clusters can provide a unique opportunity to
 investigate the physics of the ICM (Markevitch & Vikhlinin 2007;
 Zuhone & Roediger 2016) and test the dark matter paradigm (as in
 ★ E-mail: thomas.richardson@obspm.fr (TR) the case of the Bullet Cluster Clowe et al. 2004; Markevitch et al.

 © 2021 The Authors

2 Richardson & Corasaniti
2004). This underlies the importance of identifying merging events be predicted from prior knowledge of the halo mass functions at
in large cluster survey catalogues. the overdensities of interests, which allows to infer cosmological
 The identification of unrelaxed clusters relies upon a variety of parameter constraints from cluster sparsity measurements (see e.g.
proxies specifically defined for each type of observations (for a re- Corasaniti et al. 2018, 2021).
view see e.g. Molnar 2016). As an example, the detection of radio As haloes grow from inside out such that newly accreted mass
haloes and relics in clusters is usually associated with the presence is redistributed in concentric shells within a few dynamical times
of mergers. Similarly, the offset between the position of the brightest (see e.g. Wang et al. 2011; Taylor 2011, for a review), it is natural to
central galaxy and the peak of the X-ray surface brightness, or the expect that major mergers can significantly disrupt the onion structure
centroid of the SZ signal are used as proxy of merger events. This is of haloes and result in values of the sparsity that significantly differ
because the merging process alters differently the distribution of the from those of the population of haloes that have had sufficient time
various matter constituents of the cluster. to rearrange their mass distribution and reach the virial equilibrium.
 The growth of dark matter haloes through cosmic time has been Here, we perform a thorough analysis of the relation between halo
investigated extensively in a vast literature using results from N-body sparsity and the halo mass accretion history using numerical halo
simulations. Zhao et al. (2003) found that haloes build up their mass catalogues from large volume high-resolution N-body simulations.
through an initial phase of fast accretion followed by a slow one. We show that haloes which undergo a major merger in their recent
Li et al. (2007) have shown that the during the fast-accretion phase, history form a distinct population of haloes characterised by large
the mass assembly occurs primarily through major mergers, that is sparsity values. Quite importantly, we are able to fully characterise
mergers in which the mass of the less massive progenitor is at least the statistical distributions of such populations in terms of the halo
one third of the more massive one. Moreover, they found that the sparsity and the time of their last major merger. Thus, building upon
greater the mass of the halo the later the time when the major merger these results, we have developed a statistical tool which uses cluster
occurred. In contrast, slow accretion is a quiescent phase dominated sparsity measurements to test whether a galaxy cluster has undergone
by minor mergers. Subsequent studies have mostly focused on the re- a recent major merger and if so when such event took place.
lation between the halo mass accretion history and the concentration The paper is organised as follows. In Section 2 we describe the
parameter of the NFW profile (see e.g. Neto et al. 2007; Zhao et al. numerical halo catalogues used in the analysis, while in Section 3 we
2009; Ludlow et al. 2012, 2016; Lee et al. 2017; Rey et al. 2019). present the results of the study of the relation between halo sparsity
Recently, Wang et al. (2020) have shown that major mergers have and major mergers. In Section 4 we present the statistical tests devised
a universal impact on the evolution of the median concentration. In to identify the imprint of mergers in galaxy clusters and in discuss
particular, after a large initial response, in which the concentration the statistical estimation of the major merger epoch from sparsity
undergoes a large excursion, the halo recovers a more quiescent dy- measurements. In Section 5 we discuss the implications of these
namical state within a few dynamical times. Surprisingly, the authors results regarding cosmological parameter estimation studies using
have also found that even minor mergers can have a non-negligible halo sparsity. In Section 6 we validate our approach using similar
impact on the mass distribution of haloes, contributing to the scatter data, assess its robustness to observational biasses and describe the
of the concentration parameter. application of our methodology to the analysis of known galaxy
 The use of concentration as a proxy of galaxy mergers is never- clusters. Finally, in Section 7 we discuss the conclusions.
theless challenging for multiple reasons. Firstly, the concentration
exhibits a large scatter across the merger phase and the value inferred
from the analysis of galaxy cluster observations may be sensitive to 2 NUMERICAL SIMULATION DATASET
the quality of the NFW-fit. Secondly, astrophysical processes may al-
ter the mass distribution in the inner region of the halo, thus resulting 2.1 N-body Halo catalogues
in values of the concentration that differ from those estimated from
 We use N-body halo catalogues from the MultiDark-Planck2
N-body simulations (see e.g. Mead et al. 2010; King & Mead 2011),
 (MDPL2) simulation (Klypin et al. 2016) which consists of 38403
which could be especially the case for merging clusters.
 particles in (1 ℎ−1 Gpc) 3 comoving volume (corresponding to a par-
 Alternatively, a non-parametric approach to characterise the mass
 ticle mass resolution of = 1.51 · 109 ℎ−1 M ) of a flat ΛCDM
distribution in haloes has been proposed by Balmès et al. (2014) in
 cosmology run with the Gadget-21 code (Springel 2005). The cos-
terms of simple mass ratios, dubbed halo sparsity:
 mological parameters have been set to the values of the Planck cos-
 Δ1 mological analysis of the Cosmic Microwave Background (CMB)
 Δ1 ,Δ2 = , (1)
 Δ2 anisotropy power spectra (Planck Collaboration et al. 2014a): Ω =
 0.3071, Ω = 0.0482, ℎ = 0.6776, = 0.96 and 8 = 0.8228.
where Δ1 and Δ2 are the masses within spheres enclosing re- Halo catalogues and merger trees at each redshift snapshot were gen-
spectively the overdensity Δ1 and Δ2 (with Δ1 < Δ2 ) in units of erated using the friend-of-friend (FoF) halo finder code rockstar2
the critical density (or equivalently the background density). This (Behroozi et al. 2013a,b). We consider the default set up with the
statistics presents a number of interesting properties that overcome detected haloes consisting of gravitationally bound particles only.
many of the limitations that concern the concentration parameter. We specifically focus on haloes in the mass range of galaxy groups
First of all, the sparsity can be estimated directly from cluster mass and clusters corresponding to 200c > 1013 ℎ−1 M .
estimates without having to rely on the assumption of a specific For each halo in the MDPL2 catalogues we build a dataset con-
parametric profile, such as the NFW profile. Secondly, for any given taining the following set of variables: the halo masses 200c , 500c
choice of Δ1 and Δ2 , the sparsity is found to be weakly dependent and 2500c estimated from the number of N-body particles within
on the overall halo mass with a much reduced scatter than the con- spheres enclosing overdensities Δ = 200, 500 and 2500 (in units of
centration (Balmès et al. 2014; Corasaniti et al. 2018; Corasaniti &
Rasera 2019). Thirdly, these mass ratios retain cosmological infor-
mation encoded in the mass profile, thus providing an independent 1 https://wwwmpa.mpa-garching.mpg.de/gadget/
cosmological proxy. Finally, the halo ensemble average sparsity can 2 https://code.google.com/archive/p/rockstar/

MNRAS 000, 1–18 (2021)

Timing the last major mergers of galaxy clusters 3

Table 1. Characteristics of the selected halo samples at = 0, 0.2, 0.4 and (Jiang & van den Bosch 2016; Wang et al. 2020),
0.6 (columns from left to right). Quoted in the rows are the number of haloes √ ∫ √︁
 2 Δvir ( )
in the samples and the redshift of the last major merger LMM used to select ( | LMM ) = , (2)
the haloes for each sample. LMM + 1
 where Δvir ( ) is the virial overdensity, which we estimate using the
 Merging Halo Sample ( > −1/2) spherical collapse model approximated formula Δvir ( ) = 18 2 +
 82[Ω ( ) − 1] − 39[Ω ( ) − 1] 2 (Bryan & Norman 1998). Hence,
 = 0.0 = 0.2 = 0.4 = 0.6
 one has = 0 for haloes which undergo a major merger at the time
 #-haloes 23164 28506 31903 32769 they are investigated (i.e. LMM = ), and < 0 for haloes that had
 LMM < 0.113 < 0.326 < 0.540 < 0.754 their last major merger at earlier times (i.e. LMM > ). Notice that the
 definition used here differs by a minus sign from that of Wang et al.
 Quiescent Halo Sample ( < −4)
 (2020), where the authors have found that merging haloes recover a
 = 0.0 = 0.2 = 0.4 = 0.6 quiescent state within | | ∼ 2 dynamical times.
 In Section 3.1, we investigate the differences in halo mass pro-
 #-haloes 199853 169490 140464 113829 file between merging haloes and quiescent ones, to maximise the
 LMM > 1.15 > 1.50 > 1.86 > 2.22 differences we select haloes samples as following:
 • Merging haloes: a sample of haloes that are at less than one half
 the dynamical time since their last major merger ( > −1/2), and
the critical density) respectively; the scale radius, , of the best- therefore still in the process of rearranging their mass distribution;
fitting NFW profile; the virial radius, vir ; the ratio of the kinetic to • Quiescent haloes: a sample of haloes for which their last major
the potential energy, / ; the offset of the density peak from the merger occurred far in the past ( ≤ −4), thus they had sufficient
average particle position, off ; and the scale factor (redshift) of the time to rearrange their mass distribution to an equilibrium state;
last major merger, LMM ( LMM ). From these variables we addi- In the case of the = 0 catalogue, the sample of merging haloes
tionally compute the following set of quantities: the halo sparsities with > −1/2 consists of all haloes for which their last major merger
 200,500 , 200,2500 and 500,2500 ; the offset in units of the virial ra- as tagged by the rockstar algorithm occurred at LMM > 0.897
dius, Δ = off / vir , and the concentration parameter of the best-fit ( LMM < 0.115), while the samples of quiescent haloes with ≤ −4
NFW profile, 200c = 200c / , with 200c being the radius enclos- in the same catalogue are characterised by a last major merger at
ing an overdensity Δ = 200 (in units of the critical density). In our LMM < 0.464 ( LMM > 1.155). In order to study the redshift
analysis we also use the mass accretion history of MDPL2 haloes. dependence, we perform a similar selection for the catalogues at =
 In addition to the MDPL2 catalogues, we also use data from the 0.2, 0.4 and 0.6 respectively. In Table 1 we quote the characteristics
Uchuu simulations (Ishiyama et al. 2021), which cover a larger cos- of the different samples selected in the various catalogues.
mic volume with higher mass resolution. We use these catalogues
to calibrate the sparsity statistics that provides the base for practical
applications of halo sparsity measurements as cosmic chronometers
of galaxy cluster mergers. The Uchuu simulation suite consists of 3 HALO SPARSITY & MAJOR MERGERS
N-body simulations of a flat ΛCDM model realised with GreeM 3.1 Halo Sparsity Profile
code (Ishiyama et al. 2009, 2012) with cosmological parameters set
to the values of a later Planck-CMB cosmological analysis (Planck Here, we seek to investigate the halo mass profile of haloes undergo-
Collaboration et al. 2016): Ω = 0.3089, Ω = 0.0486, ℎ = 0.6774, ing a major merger as traced by halo sparsity and evaluate to which
 = 0.9667 and 8 = 0.8159. In particular, we use the halo cat- extent the NFW profile can account for the estimated sparsities at
alogues from the (2 Gpc ℎ−1 ) 3 comoving volume simulation with different overdensities. To this purpose, we compute for each halo in
128003 particles (corresponding to a particle mass resolution of selected samples the halo sparsities 200,500 and 200,2500 from their
 = 3.27 · 108 ℎ−1 M ) that, as for MDPL2, were also generated SOD estimated masses, as well as the values obtained assuming the
using the rockstar halo finder. NFW profile using the best-fit concentration parameter 200c , which
 NFW
 we denote as 200,500 NFW
 and 200,2500 respectively. These can be in-
 It is important to stress that the major merger epoch to which we
refer in this work is that defined by the rockstar halo finder, that ferred from the sparsity-concentration relation (Balmès et al. 2014):
is the time when the particles of the merging halo and those of the
 200c Δ
parent one are within the same iso-density contour in phase-space. ln (1 + 200c Δ ) − 1+ 
 3 Δ 
 200c Δ
Hence, this should not be confused with the first core-passage time Δ = 200c , (3)
 200 ln (1 + 200c ) − 1+ 
usually estimated in Bullet-like clusters. 200c

 where Δ = Δ / 200c with Δ being the radius enclosing Δ times
 the critical density. Hence, for any value of Δ and given the value of
 200c for which the NFW-profile best-fit that of the halo of interest,
2.2 Halo Sample Selection we can solve Eq. (3) numerically to obtain Δ and then derive the
 value of the NFW halo sparsity given by:
We aim to study the impact of merger events on the halo mass profile.
To this purpose we focus on haloes which undergo their last major NFW 200 −3
 200,Δ = . (4)
merger at different epochs. In such a case, it is convenient to introduce Δ Δ
a time variable that characterises the backward time interval between It is worth emphasising that such relation holds true only for haloes
the redshift (scale factor ) at which a halo is investigated and that whose density profile is well described by the NFW formula. In such a
of its last major merger LMM ( LMM ) in units of the dynamical time case the higher the concentration the smaller the value of sparsity, and

 MNRAS 000, 1–18 (2021)

4 Richardson & Corasaniti

 Merging Merging
 14 Quiescent Quiescent
 = 500 z = 0.0 = 500 z = 0.2
 = 2500 = 2500
 12

 10
 NFW /s200, )

 8
 (1 s200,

 6

 4

 2

 0
 Merging Merging
 14 Quiescent Quiescent
 = 500 z = 0.4 = 500 z = 0.6
 = 2500 = 2500
 12

 10
 NFW /s200, )

 8
 (1 s200,

 6

 4

 2

 0 0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4
 1 s200,
 NFW /s200, 1 s200,
 NFW /s200,

 NFW / 
Figure 1. Distribution of the relative deviations of individual halo sparsities with respect to the expected NFW value for 200,500 = 1 − 200,500 200,500 (dashed
 NFW
lines) and 200,2500 = 1 − 200,2500 / 200,2500 (solid lines) in the case of the merging (blue lines) and quiescent (orange lines) haloes at = 0.0 (top left panel),
0.2 (top right panel), 0.4 (bottom left panel) and 0.6 (bottom right panel) respectively

inversely the lower the concentration the higher the sparsity. Because = 0. This corresponds to an average bias of the NFW-estimated
of this, the mass ratio defined by Eq. (1) provides information on NFW
 sparsity 200,500 of order ∼ 4% at = 0. A similar trend occurs
the level of sparseness of the mass distribution within haloes, that for the distribution of 200,2500 , though with a larger scatter and a
justifies being dubbed as halo sparsity. Notice that from Eq. (3) we can larger shift in the location of the peak of the distribution at = 0,
compute 200,Δ for any Δ > 200, and this is sufficient to estimate the NFW
 which corresponds to an average bias of 200,2500 of order ∼ 14% at
sparsity at any other pair of overdensities Δ1 ≠ Δ2 > 200 as given by = 0. Such systematic differences are indicative of the limits of the
 Δ1 ,Δ2 = 200,Δ1 / 200,Δ2 . Haloes whose mass profile deviates from NFW-profile in reproducing the halo mass profile of haloes both in
the NFW prediction will have sparsity values that differ from that the outskirt regions and the inner ones. Moreover, the redshift trend
given by Eq. (3). is consistent with the results of the analysis of the mass profile of
 This is emphasised in Fig. 1, where we plot the distribution of the stacked haloes presented in (Child et al. 2018), which shows that the
relative deviations of individual halo sparsities with respect to the NFW-profile better reproduce the halo mass distribution at = 3 than
 NFW / 
expected NFW value for 200,500 = 1 − 200,500 at = 0 (see top panels of their Fig. 8). Very different is the case of the
 200,500 (dashed
 NFW
 merging halo sample, for which we find the distribution of 200,500
lines) and 200,2500 = 1 − 200,2500 / 200,2500 (solid lines) in the and 200,2500 to be highly non-Gaussian and irregular. In particular,
case of the merging (blue lines) and quiescent (orange lines) haloes in the case of 200,500 we find the distribution to be characterised
at = 0.0, 0.2, 0.4 and 0.6 respectively. We can see that for quiescent by a main peak located near the origin with a very heavy tail up to
haloes the distributions are nearly Gaussian. More specifically, in the relative differences of order 20%. The effect is even more dramatic
case 200,500 we can see that the distribution has a narrow scatter with for 200,2500 , in which case the distribution looses the main peak and
a peak that is centred at the origin at = 0.6, and slightly shifts toward become nearly bimodal, while being shifted over a positive range of
positive values at smaller redshifts with a maximal displacement at

MNRAS 000, 1–18 (2021)

Timing the last major mergers of galaxy clusters 5

 43 2 1
 z
 03 2 1
 z
 0 2.0
 3.0 zLMM= 0.2
 zLMM= 0.4
 2.5 1.8 zLMM= 0.6
 zLMM= 0.8
 zLMM= 1.0
s200, 500

 2.0 1.6

 s200, 500
 1.5 1.4
 6
 1.2
 5

 4 4
 s500, 2500

 3

 2 3

 s500, 2500
 10
 9
 8
 7 2
 6
 s200, 2500

 5
 4
 3 7
 2
 10.2 0.4 0.6 0.8 1.0
 1 0.8 0.6 0.4 6
 a a

 5
 s200, 2500

 Figure 2. Evolution with scale factor (redshift ) of the median sparsity
 200,500 (top panels), 500,2500 (middle panels) and 200,2500 (bottom panels) 4
 for a sample of 104 randomly selected haloes from the MDPL2 halo catalogue
 at = 0 and the sample of all haloes with a last major merger event at
 LMM = 0.67 (right panels). The solid lines corresponds to the median 3
 sparsity computed from the mass accretion histories of the individual haloes,
 while the shaded area corresponds to the 68% region around the median. 2

 values that extend up to relative variations up ∼ 40%. Overall this 14 2 0 2 4
 suggests that sparsity provides a more reliable proxy of the halo mass T(a|aLMM)
 profile than that inferred from the NFW concentration.
 Figure 3. Median sparsity histories as function of the backward time interval
 since the major merger events (in units of dynamical time) for halo samples
 3.2 Halo Sparsity Evolution
 from the MDPL2 catalogue at = 0 with different last major merger redshifts
 Differently from the previous analysis, we now investigate the evo- LMM = 0.2, 0.4, 0.6, 0.8, 0.8 and 1 (curves from bottom to top). Notice that
 lution of the halo mass profile as traced by halo sparsity, which the backward time interval used here differ by a minus sign from that given
 we reconstruct from the mass accretion histories of the haloes in the by Eq. (2) to be consistent with the definition by Wang et al. (2020).
 MDPL2 catalogue at = 0. In Fig. 2, we plot the median sparsity evo-
 lution of 200,500 (top panel), 500,2500 (middle panel) and 200,2500
 (bottom panel) as function of the scale factor. In the left panels we of the fraction of mass in a shell of radii Δ1 and Δ2 relative to the
 shown the case of a sample of 104 randomly selected haloes, thus mass enclosed in the inner radius Δ2 , i.e. Eq. (1) can be rewritten
 behaving as quiescent haloes in the redshift range considered, while Δ1 ,Δ2 = Δ / Δ2 + 1. As such 200,500 is a more sensitive probe
 in the right panels we plot the evolution of the sparsity of all haloes of the mass distribution in the external region of the halo, while
 in the = 0 catalogue undergoing a major merger at LMM = 0.67. 500,2500 and 200,2500 are more sensitive to the inner part of the
 The shaded area corresponds to the 68% sparsity excursion around halo.
 the median, while the vertical dashed line marks the value of the As we can see from Fig. 2, the evolution of the sparsity of merg-
 scale factor of the last major merger. ing haloes matches that of the quiescent sample before the major
 It is worth remarking taht the sparsity provides us with an estimate merger event. In particular, during the quiescent phase of evolution,

 MNRAS 000, 1–18 (2021)

6 Richardson & Corasaniti
we notice that 200,500 remains nearly constant, while 500,2500 and catalogues at different redshift. Here, we revert to the definition of
 200,2500 are decreasing functions of the scale factor. This is consis- given by Eq. (2), where the time interval is measured relative
tent with the picture that haloes grow from inside out, with the mass to the time the haloes are investigated, that is the redshift of the
in the inner region (in our case 2500c ) increasing relative to that in halo catalog. Hence, = 0 for haloes undergoing a major merger at
the external shell (Δ = Δ1 − 2500c , with Δ1 = 200 and 500 in LMM = and < 0 for those with LMM > .
units of critical density), thus effectively reducing the value of the For conciseness, here we only describe the features of the joint dis-
sparsity. This effect is compensated on 200,500 , thus resulting in a tribution ( 200,500 , ) shown in Fig. 4 in the form of iso-probability
constant evolution. We can see that the onset of the major merger contours in the 200,500 − plane at = 0 (top left panel), 0.2 (top
event induce a pulse-like response in the evolution of halo sparsities right panel), 0.4 (bottom left panel) and 0.6 (bottom right panel). We
at the different overdensities with respect to the quiescent evolution. find a similar structure of the distributions at other redshift snapshots
These trends are consistent with the evolution of the median concen- and for the halo sparsities 200,2500 and 500,2500 . In each panel the
tration during major mergers found in Wang et al. (2020), in which horizontal solid line marks the characteristic time interval | | = 2. As
the concentration rapidly drops to a minimum before bouncing again. shown by the analysis of the evolution of the halo sparsity, haloes with
Here, the evolution of the sparsity allows to follow how the merger | | > 2 have recovered a quiescent state, while those with | | < 2 are
alters the mass profile of the halo throughout the merging process. still undergoing the merging process. The marginal conditional prob-
In fact, we may notice that the sparsities rapidly increase to a maxi- ability distributions ( 200,500 | < −2) and ( 200,500 | > −2) are
mum, suggesting the arrival of the merger in the external region of shown in the inset plot.
the parent halo, which increases the mass Δ in the outer shell rel- Firstly, we may notice that the joint probability distribution has a
ative to the inner mass. Then, the sparsities decrease to a minimum, universal structure, that is the same at different redshift snapshots.
indicating that the merged mass has reached the inner region, after Moreover, it is characterised by two distinct regions. The region with
which the sparsities increases to a second maximum that indicates ≤ −2, that corresponds to haloes which are several dynamical
that the merged mass has been redistributed outside the 2500c ra- times away since their last major merger event (| | ≥ 2), as such
dius. However, notice that in the case of 200,2500 and 500,2500 the they are in a quiescent state of evolution of the sparsity; and a re-
second peak is more pronounced than the first one, while the opposite gion with −2 < < 0, corresponding to haloes that are still in the
occurs for 200,500 , which suggests that the accreted mass remains merging processes (| | < 2). In the former case, the pdf has a rather
confined within 500c . Afterwards, a quiescent state of evolution is regular structure that is independent of , while in the latter case the
recovered. pdf has an altered structure with a pulse-like feature shifted toward
 In Fig. 3 we plot the median sparsities of haloes in the MDPL2 higher sparsity values. The presence of such a feature is consistent
catalogue at = 0 that are characterised by different major merger with the evolution of the median sparsity inferred from the halo mass
redshifts LMM as function of the backward interval of time (in accretion histories previously discussed. This is because among the
units of the dyanmical time) since the last major merger. Notice that haloes observed at a given redshift snapshot, those which are within
 used in this plot differs by a minus sign from that given by Eq. (2) two dynamical times from the major merger event are perturbed,
to conform to the definition by Wang et al. (2020). We can see that thus exhibiting sparsity values that are distributed around the median
after the onset of the major merger (at ≥ 0), the different curves shown in Fig. 3. In contrast, those which are more than two dynamical
superimpose on one another, indicating that the imprint of the major times since their last major merger had time to redistributed the ac-
merger on the profile of haloes is universal, producing the same creted mass and are in a quiescent state, causing a regular structure of
pulse-like feature on the evolution of the halo sparsity. Furthermore, the pdf. From the inset plots, we can see that these two regions iden-
all haloes recover a quiescent evolution within two dynamical times, tify two distinct population of haloes, quiescent haloes with ≤ −2
i.e. for ≥ 2. Conversely, on smaller time scale < 2, haloes are still and merging (or perturbed) ones with −2 < < 0 characterised by
perturbed by the major merger event. These result are consistent with a stiff tail toward large sparsity values and that largely contributes to
the findings of Wang et al. (2020), who have shown that the impact the overall scatter of the halo sparsity of the entire halo ensemble. It
of mergers on the median concentration of haloes leads to a time is worth stressing that the choice of = 2 as threshold to differentiate
pattern that is universal and also dissipates within two dynamical between the quiescent haloes and the perturbed ones at a given red-
times. Notice, that this distinct pattern due to the major merger is shift snapshot is not arbitrary, since it is the most conservative value
the result of gravitational interactions only. Hence, it is possible that of the dynamical time above which haloes that have undergone a
such a feature may be sensitive to the underlying theory of gravity or major merger recover a quiescent evolution of their halo mass profile
the physics of dark matter particles. as shown in Fig. 3.
 As we will see next, the universality of the pulse-like imprint of Now, the fact that two populations of haloes have different prob-
the merger event on the evolution of the halo sparsity, as well as its ability distribution functions suggests that measurements of cluster
limited duration in time, have quite important consequences, since sparsity can be used to identify perturbed systems that have under-
these leave a distinct feature on the statistical distribution of sparsity gone a major merger.
values, which can be exploited to use sparsity measurements as a
time proxy of major mergers in clusters.
 4 IDENTIFYING GALAXY CLUSTER MAJOR MERGERS
3.3 Halo Sparsity Distribution
 Given the universal structure of the probability distributions charac-
We have seen that the sparsity of different haloes evolves following terising merging haloes and quiescent ones, we can use the numerical
the same pattern after the onset of the major merger, such that the halo catalogues to calibrate their statistics at different redshifts and
universal imprint of the merger event is best highlighted in terms of test whether a cluster with a single or multiple sparsity measurements
the backward interval time . Hence, we aim to investigate the joint has had a major merger in its recent mass assembly history.
statistical distribution of halo sparsity values for haloes characterised In the light of these observations, we first design a method to assess
by different time since their last major merger in the MDPL2 whether a cluster has or hasn’t been recently perturbed by a major

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Timing the last major mergers of galaxy clusters 7

Figure 4. Iso-probability contours of the joint probability distribution in the 200,500 − plane for the haloes from the MDPL2 catalogues at = 0.0, 0.2, 0.4
and 0.6 respectively. The solid horizontal line marks the value = −2. The inset plots show that marginal probability distribution for haloes with > −2 (blue
histograms) and < −2 (beige histograms) respectively.

merger. To do this we design a binary test, as defined in detection statistic and associated thresholds. This may appear cumbersome,
theory (see e.g. Kay 1998), to differentiate between the different however it is necessary to unambiguously define thresholds according
cases. Formally, this translates into defining two hypotheses denoted to probabilistic criteria rather than arbitrary ones, while the variety
as H0 , the null hypothesis and, H1 , the alternate hypothesis. In our of approaches we adopt allow us to check their robustness.
case these are, H0 : The halo has not been recently perturbed and
H1 : The halo has undergone a recent major merger. Formally the
distinction between the two is given in terms of the backward time 4.1 Frequentist Approach
interval , We start with the simplest possible choice that is using 200,500 as
(
 H0 : ( LMM | ( )) < −2 our test statistic. Separating our data set into the realisations of the
 (5) two hypotheses, we estimate their respective likelihood functions,
 H1 : ( LMM | ( )) ≥ −2
 which we model using a generalised 0 probability density function
if we consider the halo to no longer be perturbed after 2 dyn . In Fig. 4 (pdf),
we have delimited these two regions using black horizontal lines.   −1     − − 
 In the context of detection theory (see e.g. Kay 1998), one defines 1 + 
some test statistic, ( , , , , ) = , (7)
 ( , )
 H1
Γ ≷ Γth , (6) where ( , ) is the Beta function and where = 200,500 − 1.
 H0 From our two samples we then fit this model using a standard least
from the observed data such that when compared to a threshold, Γth , squares method to obtain the set of best fitting parameters under
allows us to distinguish between the two hypotheses. both hypotheses, these are reproduced in Tab. 2 and particular fits
 In the following we will explore multiple ways of defining the test are shown in Fig. 5 for the halo catalogues at = 0. In both cases

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8 Richardson & Corasaniti

Table 2. Best fitting parameters for the distribution of sparsities at = 0 2.6
 p = 0.05
 p = 0.01
under both hypotheses. Here, we quote each parameter with its 95 percent
 2.4
 p = 0.005
confidence interval estimated over 1000 bootstrap iterations.

 Parameter H0 H1 2.2

 500(z)
 1.4+0.1 1.5+0.2
 −0.1 −0.2 2.0

 s200,
 0.61+0.03 0.71+0.10

 th
 −0.03 −0.08
 7.7+0.3 4.1+0.4
 
 −0.3
 0.304+0.002
 −0.3
 0.370+0.008 1.8
 −0.003 −0.008

 1.6
 101 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
 0 z
 100 1
 th
 Figure 6. Sparsity thresholds 200,500 as function of redshift for p−values=
 (s 1, , , p, q)

 10 1
 0.05 (purple solid line), 0.01 (orange solid line) and 0.005 (green solid line)
 10 2 computed using the Frequentist Likelihood-Ratio approach.

 10 3
 relation. In Fig. 6 we have estimated the threshold corresponding
 10 4 to three key p-values at increasingly higher redshifts. Here, each
 point is estimated using the sparsity distributions from the numerical
 10 10
 5
 2 10 1 100 101 halo catalogues. This figure allows to quickly estimate the values of
 s200, 500 1 sparsity above which a halo at some redshift should be considered
 as recently perturbed.
Figure 5. Estimated probability distribution functions for H0 (purple solid It is worth noticing that these thresholds are derived from spar-
line) and H1 (orange solid line) hypotheses at = 0 along with best fitting sity estimates from N-body halo masses. In contrast, sparsities of
generalised beta prime distribution functions (dotted black lines). The shaded observed galaxy clusters are obtained from mass measurements that
area corresponds to the 95 percent confidence interval estimated over 1000 may be affected by systematic uncertainties that may differ depend-
bootstrap iterations. ing on the type of observations. The impact of mass biases is reduced
 in the mass ratio, but it could still be present. As an example, using
 results from hydro/N-body simulations for an extreme model of AGN
we additionally report the 95 percent confidence intervals estimated
 feedback model, Corasaniti et al. (2018) have shown that baryonic
using 1000 bootstrap iterations.
 processes on average can bias the estimates of the sparsity 200,500
 The quality of the fits degrades towards the tails of the distribu-
 up to . 4% and 200,2500 up to . 15% at the low mass end. This
tions, most notably under H1 due to the fact we do not account for
 being said, as long as the mass estimator is unbiased we expect our
the pulse feature. Nonetheless, they still allow us to obtain an esti-
 analysis to hold, albeit with a modification to the fitting parameters.
mate, Σ̃( ), of the corresponding likelihood ratio (LR) test statistic
 In Section 6 we present a preliminary analysis of the impact of mass
Σ( ) = ( |H1 )/ ( |H0 ). Under the Neyman-Pearson lemma (see
 biasses on our approach, however we will leave more in depth in-
e.g. Kay 1998) the true LR test statistic constitutes the most powerful
 vestigations into this topic, as well as modifications that could arise
estimator for a given binary test. We can express this statistic in terms
 from non-gravitational physics, to upcoming work.
of the fitted distribution, for = 0 this reads as:
 (1 + ( / 1 ) 1 ) − 1 − 1
Σ̃( ) ∝ 1 1 − 0 0 (8) 4.2 Bayesian approach
 (1 + ( / 0 ) 0 ) − 0 − 0
 (1 + ( /0.370) 4.1 ) −2.2 An alternate way of tackling this problem is through the Bayesian
 = −4.6 (9)
 (1 + ( /0.304) 7.7 ) −2.0 flavour of detection theory. In this case, instead of looking directly
 at how likely the data is described by a model characterised by the
from which we can obtain an approximate expression, Σ̃( ) ∝ 1.8 , model parameters in terms of the likelihood function ( | ), one
for large values,  0.3. What one can observe is that for large is interested in how likely is the model given the observed data, that
values of sparsity the LR test statistic is a monotonously increasing is the posterior function ( | ).
function of = 200,500 −1, indicating that in this regime the sparsity Bayes theorem allows us to relate these two quantities:
itself will have comparable differentiating power to the LR test. A
 ( | ) ( )
similar dependence holds at > 0. What this indicates is that we ( | ) = , (11)
can use Γ = 200,500 to efficiently differentiate between haloes that ( )
have undergone a recent major merger from an at rest population. In where ( ) is the prior distribution for the parameter vector and
addition to this result, one can estimate a simple p−value, ∫
 ∫ 200,500 −1 ( ) = ( | ) ( ) , (12)
p = Pr (Γ > 200,500 |H0 ) = 1 − ( |H0 ) (10)
 0 is a normalisation factor, known as evidence.
i.e. the probability of finding a higher value of 200,500 in a halo While this opens up the possibility of estimating the parameter
at equilibrium. And conversely, one can also estimate the value of vector, which we will discuss in sub-section 4.4, this approach also
the threshold corresponding to a given −value, by inverting this allows one to systematically define a test statistic known as the Bayes

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Timing the last major mergers of galaxy clusters 9

 1.0
Factor,
 ∫
 1
 ( | ) ( ) 
 f = ∫ , (13)
 0
 ( | ) ( ) 
 0.8
associated to the binary test. Here, we have denoted 1 and 0 the

 th| 1)
volumes of the parameter space respectively attributed to hypothesis
H1 and H0 . 0.6
 In practice, to evaluate this statistic we first need to model the

 Pr ( >
likelihood. Again we use the numerical halo catalogues as calibrators.
We find that the distribution of 200,500 for a given value of the scale 0.4
factor at the epoch of the last major merger, LMM , is well described S 1D
by a generalised 0 pdf. In particular, we fit the set of parameters
 0.2 BF 1D
 = [ , , , ] > that depend solely on LMM by sampling the SVM 1D
posterior distribution using Monte-Carlo Markov Chains (MCMC) BF 3D
with a uniform prior for LMM ∼ U (0; ( )) 3 . This is done using the
 0.0 SVM 3D
emcee4 library (Foreman-Mackey et al. 2013). The resulting values
of f can then be treated in exactly the same fashion as the Frequentist 0.0 0.2 0.4 0.6 0.8 1.0
statistic. It is however important to note that the Bayes factor is often Pr ( > th| 0)
associated with a standard “rule of thumb” interpretation (see e.g.
Trotta 2007) making these statistic particularly interesting to handle. Figure 7. ROC curves associated with the binary tests studied in this work:
 One way of comparing the efficiency of different tests is to draw the Frequentist sparsity test (S 1D, solid orange line), the Bayes Factor based
their respective Receiver Operating Characteristic (ROC) curves on a single sparsity value (BF 1D, dashed green line) and using three values
(Fawcett 2006), which show the probability of having a true de- (BF 3D, dash-dotted magenta line), the Support Vector Machines with one
 sparsity value (SVM 1D, dotted purple line) and three sparsities (SVM 3D,
tection, Pr (Γ > Γth |H1 ), plotted against the probability of a false
 dotted yellow line). What can be observed is that all 1D tests are equivalent
one, Pr (Γ > Γth |H0 ) for the same threshold. In other words, this
 at small false alarm rates and the only way to significantly increase the power
means we are simply plotting the probability of finding a value of Γ of the test is to increase the amount of input data, i.e. adding a third mass
that is larger than the threshold under the alternate hypothesis against measurement as in the BF 3D and SVM 3D cases.
that of finding a value of Γ larger than the same threshold under the
null hypothesis. The simplest graphical interpretation of this type
of figure is, the closer a curve gets to the top left corner the more quickly observes that a switching spherical-like coordinate system,
powerful the test is at differentiating between both cases. r = [ , , ] > , allows for a much simpler description. The resulting
 In Fig. 7 we plot the ROC curves corresponding to all the tests likelihood model,
we have studied in the context of this work. These curves have been  
evaluated using a sub sample of 104 randomly selected haloes from ( ; ) 1
 (r; , , C) = √︁ exp − ( − ) > C−1 ( − ) , (15)
the MDPL2 catalogues at = 0 with masses 200c > 1013 ℎ−1 M . 2 | C | 2
Let us focus on the comparison between the Frequentist direct spar-
 treats as independent from the two angular coordinates that are
sity approach (S 1D) against the Bayes Factor obtained using a single
 placed within the 2-vector = [ , ] > . Making the radial coor-
sparsity measurement (BF 1D). We can see that both tests have very
 dinate independent allows us to constrain ( , ) simply from the
similar ROC curves for low false alarm rates. This indicates that
 marginalised distribution. Doing so we found that the latter is best
we do not gain any substantial power from the additional computa-
 described by a Burr type XII (Burr 1942) distribution,
tional work done to estimate the Bayes factor using a single value of
sparsity. "  # − −1
 − −1 − 2
   
 While this may seem as the end of the line for the method based ( , , , , ) = 1+ , (16)
on the Bayes factor, the latter does present the significant advantage 
of being easily expanded to include additional data. In our case this
 with additional displacement, , and scale, , parameters. In total
comes in the form of adding additional sparsity measurements at
 the likelihood function is described by 9 parameters, 3 of which are
different overdensities. Simply including a third mass measurement,
 constrained by fitting the marginalised distribution of realisations
here we use 2500c , gives us access to two additional sparsities from
 assuming = 0 and 5, 2 in and 3 in C, are measured through
the three possible pairs, 200,500 , 200,2500 and 500,2500 . This leads
 unbiased sample means and covariances.
us to defining each halo as a point in a 3-dimensional space with
 In a similar fashion as in the single sparsity case, we evaluate these
coordinates
 parameters as functions of LMM and thus recover a posterior like-


 = 200,500 − 1 lihood for the epoch of the last major merger using MCMC, again


 = 200,2500 − 1 (14) applying a flat prior on LMM . This posterior in turn allow us to mea-

 = 500,2500 − 1
 sure the the corresponding Bayes Factor. We calculate these Bayes
 factors for the same test sample used previously and evaluate the cor-
After estimating the likelihood in this coordinate system, one responding ROC curve (BF 3D in Fig. 7). As intended, the additional
 mass measurement has for effect of increasing the detection power
 of the test, thus raising the ROC curve with respect to the 1D tests.
3 The upper bound is the scale factor at the epoch at which the halo is Increasing the true detection rate from 40 to 50 percent for a false
observed. positive rate of 10 percent. We have tested that the same trends hold
4 https://emcee.readthedocs.io/en/stable/ valid at > 0.

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10 Richardson & Corasaniti
4.3 Support Vector Machines
 10
An alternative to the Frequentist – Bayesian duo is to use machine s200, 500 = 1.2
learning techniques designed for classification. While Convolutional
 8 s200, 500 = 1.7
Neural Networks (see eg. Lecun et al. 2015, for a review) are very s200, 500 = 2
 s200, 500 = 3

 (aLMM|s200, 500)
efficient and have been profusely used to classify large datasets, both
in terms of dimensionality and size, recent examples in extra-galactic 6
astronomy include galaxy morphology classification (e.g. Hocking
et al. 2018; Martin et al. 2020; Abul Hayat et al. 2020; Cheng et al. 4
2021; Spindler et al. 2021) detection of strong gravitational lenses
(e.g. Jacobs et al. 2017, 2019; Lanusse et al. 2018; Cañameras et al. 2
2020; Huang et al. 2020, 2021; He et al. 2020; Gentile et al. 2021;
Stein et al. 2021) galaxy merger detection (Ćiprijanović et al. 2021) 0 0.2 0.4 0.6 0.8 1.0
and galaxy cluster merger time estimation (Koppula et al. 2021). aLMM
They may not be the tool of choice when dealing with datasets of
small dimensionality, like the case at a hand. A simpler option for Figure 8. Posterior distributions for different values of the sparsity 200,500 =
this problem is to use Support Vector Machines (SVM) (see e.g. 1.2 (dash-dotted green line), 1.7 (dashed orange line), 2 (dotted purple line)
Cristianini & Shawe-Taylor 2000) as classifiers for the hypotheses and 3 (solid magenta line). We can see that for large sparsity values, the
defined in Eq. (5), using as training data the sparsities measured distributions are bimodal at recent epoch, while low values produce both a
from the halo catalogues. continuous distribution at low scale factor values as well as a single peak at
 A SVM works on the simple principle of finding the boundary recent epochs corresponding to a confusion region. This induce a degeneracy
 that needs to be broken if we are to accurately estimate LMM .
that best separates the two hypotheses. In opposition to Random
Forests (see e.g. Breiman 2001) which can only define a set of hor-
izontal and vertical boundaries, albeit to arbitrary complexity, the
SVM maps the data points to a new euclidean space and solves for
the plane best separating the two sub-classes. This definition of a see, in the case of large sparsity values ( 200,500 ≥ 1.7), we find a bi-
new euclidean space allows for a non-linear boundary between the modal distribution in the posterior, caused by the pulse-like feature in
classes. For large datasets however the optimisation of the non-linear the structure of the joint distribution shown in Fig. 4 and which is con-
transformation can be slow to converge, and thus we restrict ourselves sequence of universal imprint of the major merger on the halo sparsity
to a linear transformations. To do so we make use the scikit-learn5 evolution shown in Fig. 3. In particular, we notice that the higher the
(Pedregosa et al. 2011) python package. The “user friendly” design measured sparsity and the lower the likelihood of having the last ma-
of this package allows for fast implementations with little required jor merger to occur in the distant past. A consequence of this pulse-
knowledge of python and little input from the user, giving this method like feature is that a considerable population of haloes with a recent
an advantage over its Frequentist and Bayesian counterparts. major mergers, characterised by −1/2 < ( LMM ; ( )) < −1/4,
 In order to compare the effectiveness of the SVM tests, with 1 and have sparsities in the same range as those in the quiescent regime.
3 sparsities, against those previously presented we again plot the cor- This confusion region results in a peak of the posterior distribution
responding ROC curves6 in Fig. 7. What can be seen is that the SVM for the 200,500 = 1.2 case, that is located on the minimum of the bi-
tests reach comparable differentiating power to both the Bayesian nomial distributions associated to the values of 200,500 ≥ 1.7. This
and Frequentist test for 1 sparisty and is only slightly out performed suggests the presence of a degeneracy, such that quiescent haloes
by the Bayesian test using 3 sparsities. This shows that designing a may be erroneously identified as haloes having undergone a recent
statistical test based on the sparsity can be done in a simple fashion merger, or on the contrary haloes having undergone a recent merger
without significant loss of differentiation power. Making sparsity an are misidentified as quiescent haloes.
all the more viable proxy to identify recent major mergers. The presence of this peak in the ( LMM | 200,500 ) posterior for
 low sparsity values biases the Bayesian estimation towards more re-
 cent major mergers when using a single sparsity measurement. As
4.4 Estimating cluster major merger epoch a result the previously mentioned Bayes factors, which depend on
 such a posterior, will also be biased towards recent mergers resulting
In the previous section we have investigated the possibility of using in higher measured values. Moreover, this impacts our choice when
halo sparsity as a statistic to identify clusters that have had a recent it comes to the estimation we use, indeed a maximum likelihood
major merger. We will now expand the Bayesian formulation of the estimation will be very sensitive to this peak. Therefore, we prefer to
binary test to estimate when this last major merger took place. This use a median likelihood estimation that is significantly more robust.
can be achieved by using the posterior distributions which we have The credible interval is then estimated iteratively around the median
previously computed to calculate the Bayes Factor statistics. These as to encompass 68 percent of the total probability. The end result of
distributions allow us to define the most likely epoch for the last this procedure is shown in Fig. 9 where we plot inferred posteriors
major merger as well as the credible interval around this epoch. along with the corresponding credible intervals (shaded areas) and
 Beginning with the one sparsity estimate, in Fig. 8 we plot the re- median likelihood measurements (vertical lines) obtained assuming
sulting posterior distributions ( LMM | 200,500 ) obtained assuming one (orange curves) and three sparsity (purple curves) values from
four different values of 200,500 = 1.2, 1.7, 2 and 3 at = 0. As we can three haloes selected from the numerical halo catalogue at = 0.
 The black vertical dashed lines indicate the true LMM value of the
 haloes. We can clearly see that the inclusion of an additional mass
5 https://scikit-learn.org/stable/ measurement (or equivalently two additional sparsity estimates) al-
6 Note that the test data used for the ROC curves was excluded from the lows to break the 200,500 degeneracy between quiescent and merging
training set. haloes with low sparsity values. In such a case the true LMM value

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Timing the last major mergers of galaxy clusters 11

 14
 s200, 500 = 1.3 s200, 500 = 1.4 s200, 500 = 1.5
 12 s200, 2500 = 3.8 s200, 2500 = 3.0 s200, 2500 = 2.9
 s500, 2500 = 3.0 s500, 2500 = 2.2 s500, 2500 = 1.9
 10
 8
 p( |x)

 6
 4
 2
 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
 aLMM aLMM aLMM
Figure 9. Posterior distributions of the last major merger epoch for three selected haloes with different sparsity values from the = 0 halo catalogue. The shaded
areas corresponds to the 68% credible interval around the median (coloured vertical line) assuming a single (orange) and three sparsity (purple) measurements.
The black vertical dashed lines mark the true fiducial value of LMM for each of the selected haloes.

is found to be within the 1 credible regions. Hence, this enable us to at sub-percent level? To assess this question, we can make use of
also identify merging haloes that are located in the confusion region. the sparsity thresholds defined in Section 4.1 based on the p-value
 statistics. As an example in the right panel of Fig. 10, we plot the
 mean sparsities h 200,500 i, h 500,2500 i and h 200,2500 i as function
 of redshift computed using the full halo sample (blue curves), and a
5 COSMOLOGICAL IMPLICATIONS
 selected halo sample from which we have removed haloes with spar-
Before discussing practical applications on the use of large halo spar- sities above the sparsity thresholds, such as those shown in Fig. 6,
sities as tracers of major merger events in clusters, it is worth high- associated to p-values of ≤ 0.01 (green curves) and ≤ 0.005
lighting the impact that such systems can have on average sparsity (orange curves) respectively. We can see that removing outliers alter
measurements that are used for cosmological parameter inference. the estimated mean sparsity h 200,500 i at sub-percent level over the
 Halo sparsity depends on the underlying cosmological model range 0 < < 2, and in the case of h 500,2500 i and h 200,2500 i up to
(Balmès et al. 2014; Ragagnin et al. 2021), and it has been shown a few per-cent level only in the high-redshift range 1 . < 2.
(Corasaniti et al. 2018, 2021) that the determination of the average
sparsity of an ensemble of galaxy clusters estimated at different red-
shifts can provide cosmological constraints complementary to those
 6 TOWARDS PRACTICAL APPLICATIONS
from standard probes. This is possible thanks to an integral relation
between the average halo sparsity at a given redshift and the halo mass We will now work towards applying the statistical analysis pre-
function at the overdensities of interest, which allows to predict the sented in Section 4 to observational data. To this purpose
average sparsity for a given cosmological model (Balmès et al. 2014). we have specifically developed the numerical code lammas7 .
Hence, the average is computed over the entire ensemble of haloes Given the mass measurements 200c , 500c and 2500c of a
as accounted for by the mass functions. In principle, this implies galaxy cluster, the code computes the sparsity data vector =
that at a given redshift the mean sparsity should be computed over { 200,500 , 200,2500 , 500,2500 } (the last two values only if the es-
the available cluster sample without regard to their state, since any timate of 2500c is available) and performs a computation of the
selection might bias the evaluation of the mean. This can be seen in frequentist statistics discussed in Section 4.1 and the Bayesian com-
the left panel of Fig. 10, where we plot the average sparsity h 200,500 i putation presented in Section 4.2. The code computes the frequentist
(top panel), h 500,2500 i (central panel) and h 200,2500 i as function of p-value only for 200,500 and it’s associated uncertainty. Bayesian
redshift in the case of haloes which are within two dynamical times statistics are measure for both 1 and 3 sparsities, these include the
since the last major merger (blue curves), for those which are more posterior distributions ( LMM | ) and their associated marginal
than two dynamical times since the last major merger (orange curves) statistics, along with the Bayes factor, f , using the available data.
and for the full sample (green curves). As we can see removing the We implement the statistical distributions of merging and quiescent
merging haloes induces a ∼ 10% bias on h 200,500 i at = 0, which halo populations calibrated on the halo catalogues from the Uchuu
decreases to ∼ 4% at = 1, while in the same redshift range the bias simulations (Ishiyama et al. 2021) rather than MDPL2, thus bene-
is at ∼ 20% level for h 500,2500 i and ∼ 30% for h 200,2500 i. fiting from the higher mass resolution and redshift coverage of the
 However, the dynamical time is not observable and in a realistic Uchuu halo catalogues. A description of the code lammas is given
situation, one might have to face the reverse problem, which is that in Appendix A. In the following we will first validate this code by
of having a number of outliers characterised by large sparsity values presenting it with haloes from N-body catalogues that were not used
in a small cluster sample, potentially biasing the estimation of the for calibration. We will then quantify the robustness of our analysis
mean compared to that of a representative cluster ensemble. Which to observational mass biases using empirical models. In particular,
clusters should be considered as outliers, and which should be re-
moved from the cluster sample such that the estimation of the mean
sparsity will remain representative of the halo ensemble average, say 7 https://gitlab.obspm.fr/trichardson/lammas

 MNRAS 000, 1–18 (2021)