# On the Constraints of Galaxy Assembly Bias in Velocity Space

Kevin S. McCarthy1, Zheng Zheng1, Hong Guo2, Wentao Luo3, and Yen-Ting Lin4
1 Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA
2 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical
Observatory, Shanghai 200030, China
3 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa 277-8583, Japan
4 Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617,
Taiwan E-mail<EMAIL_ADDRESS>(KSM)E-mail<EMAIL_ADDRESS>(ZZ)

###### Abstract

If the formation of central galaxies in dark matter haloes traces the assembly
history of their host haloes, in haloes of fixed mass, central galaxy
clustering may show dependence on properties indicating their formation
history. Such a galaxy assembly bias effect has been investigated by Lin et
al. (2016), with samples of central galaxies constructed in haloes of similar
mass and with mean halo mass verified by galaxy lensing measurements, and no
significant evidence of assembly bias is found from the analysis of the
projected two-point correlation functions of early- and late-forming central
galaxies. In this work, we extend the the investigation of assembly bias
effect from real space to redshift (velocity) space, with an extended
construction of early- and late-forming galaxies. We carry out halo occupation
distribution modelling to constrain the galaxy-halo connection to see whether
there is any sign of the effect of assembly bias. We find largely consistent
host halo mass for early- and late-forming central galaxies, corroborated by
lensing measurements. The central velocity bias parameters, which are supposed
to characterise the mutual relaxation between central galaxies and their host
haloes, are inferred to overlap between early- and late-forming central
galaxies. However, we find a large amplitude of velocity bias for early-
forming central galaxies (e.g. with central galaxies moving at more than 50%
that of dark matter velocity dispersion inside host haloes), which may signal
an assembly bias effect. A large sample with two-point correlation functions
and other clustering measurements and improved modelling will help reach a
conclusive result.

###### keywords:

cosmology: observations – cosmology: large-scale structure of Universe –
galaxies: statistics

††pagerange: On the Constraints of Galaxy Assembly Bias in Velocity
Space–References

##  1 Introduction

Galaxy clustering data from large-volume galaxy redshift surveys has been
playing an important role in studying galaxy formation and evolution and in
learning about cosmology. As galaxies form and evolve inside dark matter
haloes and the halo population can be readily predicted by a given cosmology
model (e.g. through cosmological $N$-body simulations), models based on the
galaxy-halo connection have become the convenient and natural route to
interpret galaxy clustering data.

The commonly adopted descriptions of galaxy-halo relation (see Wechsler &
Tinker 2018 for a review), such as the framework of halo occupation
distribution (HOD; e.g. Berlind et al. 2003) and conditional luminosity
function (CLF; e.g. Yang et al. 2003), have been successfully applied to
galaxy surveys to infer the relation between galaxy properties and halo mass
(e.g. van den Bosch et al. 2003; Zheng 2004; Zehavi et al. 2005, 2011; Guo et
al. 2014; Xu et al. 2018). With such modelling, the high precision clustering
measurements at small to intermediate scales (i.e., sub-Mpc to tens of Mpc)
have the potential to tighten cosmological constraints (e.g. Abazajian et al.,
2005; Zheng & Weinberg, 2007; van den Bosch et al., 2013; Reid et al., 2014;
Zhai et al., 2019).

The implicit assumption in the above frameworks is that the statistical
properties of galaxies inside haloes only depend on halo mass, not on halo
environment or halo growth history, partly motivated by the excursion set
theory (Bond et al., 1991). However, it was later discovered (especially with
$N$-body simulations) that the clustering of haloes depends on not only halo
mass but also halo assembly history, which is termed halo assembly bias (e.g.
Sheth & Tormen, 2004; Gao et al., 2005) and has been under active
investigations (e.g. Paranjape et al., 2018; Xu & Zheng, 2018; Han et al.,
2019). If in haloes of fixed mass galaxy properties and halo assembly history
are closely connected, the inherited assembly bias from the host haloes would
be manifested as effects on galaxy clustering. While such galaxy assembly
bias, which can be conceptually defined as dependence of galaxy clustering on
galaxy properties at fixed halo mass, could enable us to learn more about
galaxy formation and evolution, it may introduce possible systematics in
cosmological constraints (e.g. Zentner et al., 2014) or limit the range of
clustering data to be used (e.g. McCarthy et al., 2019) if not properly
accounted for.

An empirical detection of galaxy assembly bias effect will advance our
understanding on galaxy formation and also guide the model development for
cosmological constraints with galaxy clustering. No definite conclusion has
been reached in the search of galaxy assembly bias effect in galaxy survey
data. Analyses of projected galaxy two-point correlation functions (2PCFs) in
the Sloan Digital Sky Survey (SDSS) show no significant evidence of galaxy
assembly bias (e.g. Lin et al., 2016; Vakili & Hahn, 2019; Zentner et al.,
2019; Salcedo et al., 2020). The reported strong assembly bias effect in the
projected 2PCFs of massive clusters (Miyatake et al., 2016) appears to be an
artefact caused by an insufficient control of projection effects (Zu et al.,
2017). Based on gravitational lensing measurements, Zu et al. (2020) find a
$\sim 10$ per cent difference in the large scale bias between low stellar
mass, low concentration and high stellar mass, high concentration galaxy
clusters of the same average mass, and it remains to be investigated how much
this difference in bias can be explained by the difference in the cluster mass
distribution. While Yuan et al. (2021) claim the detection of galaxy assembly
bias in redshift-space clustering of BOSS CMASS galaxies, it is in
contradiction to the result in Guo et al. (2015a).

One major difficulty in searching for assembly bias effect in observational
data lies in the halo mass determination, so that assembly bias effect can be
well separated from the effect of clustering from the halo-mass dependence. In
such regard, the assembly bias analysis in Lin et al. (2016; hereafter L16) is
distinct. They construct samples of central galaxies in SDSS in haloes of
similar mass, and the halo mass is verified by galaxy lensing measurements.
Their analysis with the projected 2PCFs of central galaxy samples with
different galaxy growth history and star formation properties shows no
significant evidence for assembly bias, which suggests that the correlation,
if any, between the chosen galaxy assembly indicator and halo formation
history may not be tight (Xu & Zheng, 2020).

The L16 analysis focuses on projected 2PCFs, effectively galaxy clustering in
real space. In this work, we extend the clustering analysis in L16 to redshift
space with an extended sample construction. Modelling of luminosity-threshold
samples of galaxies in BOSS and SDSS has revealed the existence of velocity
bias (Guo et al., 2015a, b), which is the difference between the motion of
galaxies and dark matter particles inside haloes. In particular, the central
galaxies do not appear to be at rest with respect to the host haloes (e.g., in
the halo center-of-mass frame), moving with a mean velocity about 20–30 per
cent of the velocity dispersion of dark matter inside haloes, which reflects
the mutual relaxation of galaxies and haloes (Ye et al., 2017). Motivated by
the expectation that central galaxies of different assembly/formation
histories may have different degrees of relaxation with respect to host
haloes, we perform investigations of the redshift-space clustering with the
extended L16 samples to see whether such a difference exists or whether there
is any hint for assembly bias in velocity space.

This paper is organised as follows. In Section 2, we present the details of
the construction of the various galaxy samples, extending those in L16. In
Section 3, we describe the clustering measurements and the modelling method.
The results are presented in Section 4, followed by a discussion in Section 5.
Finally in Section 6, we summarise the work and give the conclusion.

##  2 Data

For studying galaxy assembly bias, we would like to have samples of central
galaxies in haloes of fixed mass to investigate the dependence of clustering
on galaxy properties. L16 devise a procedure to construct early-forming and
late-forming galaxy samples to have similar mean host halo mass of central
galaxies, which is verified with galaxy lensing measurements. For simplicity,
in this work the early-forming (late-forming) galaxies are referred to as the
early (late) galaxies.

This work will entail an extended analysis of the early/late galaxy samples in
L16. The galaxies in these samples are supposed to be central galaxies,
selected base on the group catalogue of Yang et al. (2007; Y07) constructed
from SDSS (York et al. 2000) Data Release 7 (DR7; Abazajian et al. 2009),
specifically from DR7 modelC111https://gax.sjtu.edu.cn/data/Group.html. We use
‘LIN’ to label the original catalogue used in L16.

L16 applied a satellite decontamination algorithm to remove the incorrectly
labelled central galaxies from Y07. As will be shown in section 4.1, this
algorithm also removes a fraction of central galaxies, when they become close
in the direction transverse to the line of sight. It causes a paucity of
signal at small scales for the projected two-point correlation function. While
this has no effect on the L16 results based on large-scale projected
clustering, it would affect our analysis of redshift-space clustering.
Therefore we also have samples of galaxies selected with the same criteria as
in L16 but without the satellite decontamination algorithm. These samples are
labelled as ‘PCAT’, for parent catalogues. The satellite component will be
accounted for in our model.

Finally, to increase the signal-to-noise ratio of the clustering measurements
and hence improve the model constraints, we also construct galaxy catalogues
by loosening the selection criteria, which extends the halo mass range to
include more galaxies. We use ‘EXT’ to denote these extended samples.

In what follows, we describe the selection of the galaxy samples and the
division into the early and late galaxy samples according to star formation
history (SFH) and specific star formation rate (sSFR). The galaxy counts and
number density of each galaxy sample are summarised in Table 1.

###  2.1 SFH-based Construction

We first describe how to form early/late galaxy samples with the SFH
information of galaxies. The construction makes use of galaxy stellar mass,
galaxy colour, and galaxy SFH.

Stellar mass is estimated based on absolute $r$-band magnitude ($M_{\rm r}$)
and colour ($g-r$) with the empirical formula of Bell et al. (2003),

$\log\left(\frac{M_{*}}{h^{-2}{\rm
M}_{\odot}}\right)=-0.406+1.097(g-r)-0.4(M_{\rm r}-5\log h-4.64),$ (1)

where $M_{\rm r}$ is the Petrosian magnitude $K$-corrected to $z=0$ from the
NYU value-added galaxy catalogue (Blanton et al., 2005). Galaxies are divided
into red and blue types following a dividing line defined as

$(g-r)_{\rm div}=0.67+0.03\log\left[M_{*}/(10^{10}h^{-2}{\rm
M}_{\odot})\right].$ (2)

Galaxy SFH comes from the versatile spectral analysis (VESPA;Tojeiro et al.
2007, 2009)222http://www-wfau.roe.ac.uk/vespa/, which adopts the stellar
population synthesis model of Bruzual & Charlot (2003), with the stellar
initial mass function of Chabrier (2003) and the Padova stellar evolution
tracks (Alongi et al., 1993; Bressan et al., 1993; Fagotto et al., 1994a, b;
Girardi et al., 1996). The data with the highest temporal resolution are used,
where the SFR is stored in 16 logarithmically-spaced bins with upper bin
boundary ranging from 0.02 to 14 Gyr in lookback time (fig.1 of Tojeiro et al.
2009). We divide the stellar mass formed in the earliest bin of lookback time
(from 9 to 14 Gyr, or $z\gtrsim 1.35$) by the total stellar mass. If this
fraction is $\geq 0.5$ ($<0.5$), the galaxy is classified as early-formed
(late-formed). VESPA employs two dust extinction models, and we only keep
galaxies with consistent classifications between the two models.

When constructing early and late galaxy samples with similar mean halo mass,
L16 find that there is a systematic effect in the halo mass estimates in the
Y07 group catalogue, as the colour dependent relation between halo mass and
stellar mass (More et al., 2011) is not taken into account. After accounting
for such a dependence, L16 use different stellar mass ranges for red and blue
galaxies to select halos of similar mass range. L16 further find that the SFH
of galaxies (early-formed/late-formed) also need to be considered. L16 choose
four ranges of stellar mass, here denoted as $\mathcal{M}_{i}$ ($i=$1, 2, 3,
and 4), corresponding to $\log[M_{*}/(h^{-2}{\rm M}_{\odot})]=[9.9,10.2]$,
[10.2, 10.45], [10.35, 10.6], and [10.52, 11.1], respectively. With the sets
of red/blue, $\mathcal{M}_{i}$, and early-formed/late-formed galaxies, the
construction of samples of early and late galaxies in L16 can be expressed as

${\rm Early}=\left[({\rm Red}\cap\mathcal{M}_{1})\cup({\rm
Blue}\cap\mathcal{M}_{2})\right]\cap\text{Early-formed}$ (3)

and

${\rm Late}=\left[({\rm Red}\cap\mathcal{M}_{3})\cup({\rm
Blue}\cap\mathcal{M}_{4})\right]\cap\text{\rm Late-formed},$ (4)

where ‘$\cup$’ and ‘$\cap$’ represent the union and intersection of sets,
respectively.

With the above construction procedure, the LIN catalogue (with satellite
decontamination using a friends-of-friends algorithm) has 18199/26070 galaxies
in the early/late sample. Consistent mean halo masses are found from galaxy
lensing measurements in L16, $M_{\rm 200c}=9.5^{+2.5}_{-2.0}\times
10^{11}h^{-1}{\rm M}_{\odot}$ and $8.4^{+2.2}_{-1.8}\times 10^{11}h^{-1}{\rm
M}_{\odot}$ for the early and late sample, respectively, where $M_{\rm 200c}$
is the halo mass within a radius within which the mean density is 200 times
the critical density of the universe.

Our PCAT catalogue, which has the same selection criteria as in L16 but
without the satellite decontamination, has 19711/30549 galaxies in the
early/late sample. In order to improve the signal-to-noise ratio of clustering
measurements, we increase the number of galaxies to form the EXT catalogue by
slightly extending the stellar mass ranges, with $\mathcal{M}_{i}$ ($i=$1, 2,
3, and 4) changing to $\log[M_{*}/(h^{-2}{\rm M}_{\odot})]=[9.9,10.4]$, [10.2,
10.75], [10.2, 10.6], and [10.5, 11.1], respectively. This results in
59871/37684 galaxies in the early/late sample.

###  2.2 sSFR-based Construction

In L16, the second way to constructing early and late galaxy samples is to use
the sSFR. Galaxies with higher sSFR are still actively forming stars and
therefore have more of their stellar mass formed recently, while galaxies with
lower sSFR have formed more of their stellar mass at earlier time.

The values of sSFR in Y07 are obtained from the The Max Planck Institute for
Astrophysics and Johns Hopkins University (MPA-JHU) value-added catalogues,
where a modified version of the Brinchmann et al. (2004) technique has been
employed. L16 find that different combinations of Y07 halo mass ($M_{\rm
200c}$) range and sSFR values can result in consistent mean halo mass, as
verified by galaxy lensing measurements. The sample of early galaxies are
selected to have low sSFR, $\log({\rm sSFR}/{\rm yr}^{-1})<-11.8$, in a Y07
halo mass range of $\log(M_{\rm 200c}/{\rm M}_{\odot})$=12.0–12.5. For the
sample of late galaxies, we have $\log({\rm sSFR}/{\rm yr}^{-1})>-11.8$ and
$\log(M_{\rm 200c}/{\rm M}_{\odot})$=12.75–13.1. With satellite
decontamination, the LIN catalogue ends up with 25837/29658 galaxies in the
early/late sample. The mean halo mass estimates from galaxy lensing are
$1.39^{+0.24}_{-0.21}\times 10^{12}h^{-1}{\rm M}_{\odot}$ and
$1.26^{+0.24}_{-0.20}\times 10^{12}h^{-1}{\rm M}_{\odot}$ for the two samples.

Our PCAT catalogue without the satellite decontamination has 29658/33420
galaxies in the early/late sample. Similar to the SFH-based construction, we
construct an EXT catalogue to increase the number of galaxies for improving
the signal-to-noise ratio of the clustering measurements. With more galaxies,
we are also able to shift the sSFR cut from $\log({\rm sSFR}/{\rm
yr}^{-1})=-11.8$ to $-11.0$ to better separate the quenched and active star-
forming galaxies. For the early EXT sample, we have $\log({\rm sSFR}/{\rm
yr}^{-1})<-11.0$ in a Y07 halo mass range of $\log(M_{\rm 200c}/{\rm
M}_{\odot})$=11.8–12.3, and for the late EXT sample, $\log({\rm sSFR}/{\rm
yr}^{-1})>-11.0$ with $\log(M_{\rm 200c}/{\rm M}_{\odot})$=12.6–13.2. This
results in 52014/37746 galaxies in the early/late sample in the EXT catalogue.

Table 1:  Early/late galaxy samples constructed based on SFH and sSFR. Listed are the total number ($N$) of galaxies and the comoving number density ($n_{\rm g}$; in units of $10^{-3}h^{3}{\rm Mpc^{-3}}$) for each sample.  | SFH-based | sSFR-based
---|---|---
Sample | $N$ | $n_{\rm g}$ | $N$ | $n_{\rm g}$
LIN–early | 18199 | 0.53 | 25837 | 0.60
LIN–late | 26070 | 0.24 | 29658 | 0.21
PCAT–early | 19711 | 0.64 | 29418 | 0.74
PCAT–late | 30549 | 0.30 | 33420 | 0.23
EXT–early | 59871 | 1.40 | 52014 | 1.56
EXT–late | 37684 | 0.50 | 37746 | 0.22

##  3 Methodology

###  3.1 Galaxy Clustering Measurements

The primary measurements considered in this work are the projected two-point
correlation functions (2PCFs) $w_{\rm p}$ and the multipoles $\xi_{l}$ of the
redshift-space 2PCFs. To obtain $w_{\rm p}$, we measure the three-dimensional
(3D) 2PCF $\xi(r_{\rm p},r_{\pi})$ in terms of the line-of-sight and
transverse separations, $r_{\rm p}$ and $r_{\pi}$, of galaxy pairs. To compute
$\xi_{l}$, we measure the 3D 2PCF $\xi(s,\mu)$ in terms of the pair separation
$s$ and the cosine of the angle between the line-of-sight direction and the
pair separation vector. In detail, for two galaxies located at
$\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in redshift space, we define the line-
of-sight vector $\mathbf{l}\equiv(\mathbf{v}_{1}+\mathbf{v}_{2})/2$ and
separation vector $\mathbf{s}=\mathbf{v}_{2}-\mathbf{v}_{1}$ (e.g. Zehavi et
al., 2011). We then have

$s=|\mathbf{s}|,\,\,\,\,\,\mu\equiv\cos\theta=\mathbf{s}\cdot\mathbf{l}/|\mathbf{s}||\mathbf{l}|.$
(5)

and

$r_{\pi}\equiv|\mathbf{s}\cdot\mathbf{l}|/|\mathbf{l}|,\,\,\,\,\,r_{\rm
p}\equiv\sqrt{s^{2}-r_{\pi}^{2}}.$ (6)

We use the Landy-Szalay estimator (Landy & Szalay, 1993) to measure
$\xi(r_{\rm p},r_{\pi})$ and $\xi(s,\mu)$,

$\xi=\frac{\rm DD-2DR+RR}{\rm RR},$ (7)

where DD, DR, and RR are the normalised numbers of data–data, data–random, and
random–random galaxy pairs in the corresponding ($r_{\rm p}$, $r_{\pi}$) or
($s$, $\mu$) bins. The random catalogue for each sample has the same footprint
and redshift distribution as the sample, with about 5 million points. We adopt
uniform $r_{\rm p}$ bins in logarithmic space with a bin width $\Delta\log
r_{\rm p}=0.2$, with $\log[r_{\rm p}/(\,h^{-1}{\rm{Mpc}})]$ ranging from $-1$
to $1.4$. The $r_{\pi}$ bins are uniform in linear space from 0 to
40$\,h^{-1}{\rm{Mpc}}$ with a bin width $\Delta r_{\pi}=2\,h^{-1}{\rm{Mpc}}$.
The setup of $s$ bins are the same as $r_{\rm p}$, and 20 linear bins are used
for $\mu$ from 0 to 1 (i.e. with a bin width $\Delta\mu=0.05$).

The projected 2PCF $w_{\rm p}$ is calculated as

$w_{\rm p}(r_{\rm p})=2\sum_{i}\xi(r_{\rm p},r_{\pi,i})\Delta r_{\pi}.$ (8)

The redshift-space 2PCF multipoles $\xi_{l}$ is calculated as

$\xi_{l}(s)=(2l+1)\sum_{i}\xi(s,\mu_{i})\mathcal{P}_{l}(\mu_{i})\Delta\mu,$
(9)

where $\mathcal{P}_{l}$ is the $l$-th Legendre polynomial. We limit our
analysis and modelling to the monopole $\xi_{0}$ and the quadrupole $\xi_{2}$,
as the measurements of the hexadecapole $\xi_{4}$ are usually noisy and
contains little information about velocity bias we are after.

In comparison to L16, we extend the sample construction and make 2PCF
measurements in redshift space. In addition, we apply a correction to the
fibre collision effect. In SDSS, because of the finite size of the fibre
plugs, two fibres could not be placed at a separation smaller than
$55\arcsec$, corresponding to a projected separation of $\sim 0.1h^{-1}{\rm
Mpc}$ at the median redshift $z\sim 0.12$. The effect extends to larger scales
in redshift space with the Finger-of-God (FoG) effect. While fibre collision
has little influence on the large-scale $w_{\rm p}$ analysis in L16, it can
become important for small-scale and redshift-space clustering analyses
presented in this work. We employ the technique developed by Guo et al. (2012)
to correct the fibre collision effect. In short, it makes use of galaxy pairs
in tile overlap regions to recover the counts of collided pairs. It has been
successfully applied in previous work (e.g. Guo et al., 2015a, b).

Finally, we also perform galaxy-galaxy lensing measurements for the galaxy
samples. Following Luo et al. (2017), from the galaxy shear measurements we
obtain the excess surface density (ESD) profile for each galaxy sample,

$\Delta\Sigma(R)=\bar{\Sigma}(<R)-\Sigma(R),$ (10)

where $\bar{\Sigma}(<R)$ is the mean mass surface density within a projected
radius $R$ around galaxies in the sample and $\Sigma(R)$ is the mass surface
density at radius $R$. We note that these are all new ESD measurements, and
even for the LIN samples, they are independent of those in L16 (with slightly
different cosmological models adopted). The ESD measurements are not included
in our modelling. Instead, they are used to provide a cross-check of the
modelling results.

###  3.2 Modelling the Clustering Measurements

The projected and redshift-space galaxy 2PCFs are modelled within the HOD
framework. Here we introduce the HOD parameterisation and the 2PCF
calculations.

While the construction of galaxy samples focuses on selecting central
galaxies, they include a small fraction of satellites, in particular for the
samples without satellite decontamination. So we parameterise the mean
occupation function of galaxies as the sum of central and satellite
components.

For each of the late/early samples, the central galaxies are supposed to
reside in haloes within a small range of mass. Different from the step-like
function for a luminosity-bin sample (e.g. Zheng et al., 2005), the mean
occupation function of central galaxies is similar to that for a luminosity-
bin sample or follows closely to the form of the conditional luminosity
function of central galaxies (e.g. Yang et al., 2008). We model the central
component of the mean occupation function for each sample as a Gaussian bump
in terms of the logarithmic halo mass,

$\langle N_{\rm cen}(M_{\rm h})\rangle=A_{\rm c}\exp\left[-\frac{(\log M_{\rm
h}-\log M_{\rm c})^{2}}{2\sigma_{\rm c}^{2}}\right],$ (11)

where the three parameters $A_{\rm c}$, $\log M_{\rm c}$, and $\sigma_{\rm c}$
represent the amplitude, centre, and width of the Gaussian profile.

For satellites, with the SFH-based construction they have a stellar mass range
similar to that of central galaxies. It implies that they live in haloes more
massive than the hosts of central galaxies. With the sSFR-based construction,
a halo mass cut is imposed based on the Y07 halo mass estimate, so the
satellites may reside in haloes of the same mass range as central galaxies. It
is also possible that they are in more massive haloes, given the inaccuracy in
the Y07 halo mass estimate. As our purpose of the study is on central
galaxies, the satellite parameters are treated as nuisance parameters. We
perform tests to model the satellite mean occupation function with either a
power-low form or a Gaussian form, and we find consistent constraints on the
HOD of central galaxies. We present the results by adopting a Gaussian form of
the satellite mean occupation function, similar to that of the central
galaxies, with the three parameters being $A_{\rm s}$, $\log M_{\rm s}$, and
$\sigma_{\rm s}$,

$\langle N_{\rm sat}(M_{\rm h})\rangle=A_{\rm s}\exp\left[-\frac{(\log M_{\rm
h}-\log M_{\rm s})^{2}}{2\sigma_{\rm s}^{2}}\right].$ (12)

The mean occupation function for all the galaxies is

$\langle N(M_{\rm h})\rangle=\langle N_{\rm cen}(M_{\rm h})\rangle+\langle
N_{\rm sat}(M_{\rm h})\rangle.$ (13)

We jointly model the 2PCFs of early and late galaxies. As we parameterise the
mean occupation functions of early and late central galaxies separately, there
is a possibility that the sum of the mean occupation numbers of early and late
central galaxies exceeds unity, which becomes nonphysical. In such cases, we
renormalise the amplitudes of the two samples, $A_{\rm c,early}\leftarrow
A_{\rm c,early}/(A_{\rm c,early}+A_{\rm c,late})$ and $A_{\rm
c,late}\leftarrow A_{\rm c,late}/(A_{\rm c,early}+A_{\rm c,late})$.

To model the redshift-space clustering, we also have two velocity bias
parameters ($\alpha_{\rm c}$ and $\alpha_{\rm s}$) to describe the differences
between the motions of galaxies and dark matter (e.g. Guo et al., 2015a). In
the rest frame (centre-of-mass frame) of a halo, the central galaxy is not
necessarily at rest, and we assume its line-of-sight velocity to follow a
Laplacian distribution (Guo et al., 2015c) with velocity dispersion
$\sigma_{\rm v,c}=\alpha_{\rm c}\sigma_{\rm v,h}$. Here $\sigma_{\rm v,h}$ is
the one-dimensional (1D) velocity dispersion of dark matter particles inside
the halo, and the central velocity bias parameter $\alpha_{\rm c}$
characterises the status of mutual relaxation between central galaxies and
their host haloes (e.g. Ye et al., 2017). For satellites, we parameterise
their velocity dispersion to be $\sigma_{\rm v,s}=\alpha_{\rm s}\sigma_{\rm
v,h}$, with $\alpha_{\rm s}$ the satellite velocity bias parameter. In the
case that galaxies follow the motion of dark matter particles, we have
$\alpha_{\rm c}=0$ and $\alpha_{\rm s}=1$. Our main purpose of this study is
to see whether there is difference between the motions of early and late
central galaxies. That is, we focus on the constraints on $\alpha_{\rm c}$,
and the satellite velocity bias $\alpha_{\rm s}$ is treated as a nuisance
parameter.

With the HOD parameters, we compute the model 2PCFs following the simulation-
based method developed in Zheng & Guo (2016). All the necessary information in
fine halo mass bins relevant to 2PCF calculations is tabulated based on haloes
identified in an N-body simulation. With the tables, the galaxy 2PCFs are
obtained through summation over halo mass bins with galaxy occupation
distribution accounted for. The method allows for efficient and accurate
clustering predictions without the need to create mock galaxy catalogues. In
this work, the tables are computed with rockstar haloes (Behroozi et al.,
2013) from the MultiDark MDPL2 simulation (Klypin et al., 2016). The
simulation assumes a spatially flat cosmology with $\Omega_{\rm m}=0.307$,
$\Omega_{\rm b}=0.048$, $h=0.678$, $n_{\rm s}=0.96$, and $\sigma_{8}=0.823$.
The simulation has a box size of $1h^{-1}$Gpc (comoving) on a side with
$3840^{3}$ particles, and the corresponding mass resolution is $1.51\times
10^{9}h^{-1}{\rm M}_{\odot}$.

Given a set of HOD parameters, the predicted 2PCFs are equivalent to the
average measurements from different realizations of mock catalogues. In each
mock catalogue, central galaxies are placed at the centres of host haloes, and
random dark matter particles inside haloes are selected to represent
satellites. The possibility of a halo of mass $M_{\rm h}$ to host a central
galaxy follows $\langle N_{\rm cen}(M_{\rm h})\rangle$ (equation 11), and the
number of satellites follows a Poisson distribution with mean $\langle N_{\rm
sat}(M_{\rm h})\rangle$ (equation 12). The velocities of central and
satellites are modified according to the velocity bias parameters $\alpha_{\rm
c}$ and $\alpha_{\rm s}$.

We employ the Markov Chain Monte Carlo (MCMC) to explore the HOD parameter
space from modelling the 2PCFs and galaxy number densities. The likelihood
function is based on the value of $\chi^{2}$. For each sample (early or late
galaxies), the $\chi^{2}$ is constructed as

$\chi^{2}=\mn@boldsymbol{(\xi-\xi^{*})}^{\rm
T}\mn@boldsymbol{\mathsf{C^{-1}}}\mn@boldsymbol{(\xi-\xi^{*})}+\frac{(n_{\rm
g}-n_{\rm g}^{*})^{2}}{\sigma_{n_{\rm g}}^{2}},$ (14)

where $\mn@boldsymbol{\xi}$ is the 2PCF data vector with covariance matrix
$\mn@boldsymbol{\mathsf{C}}$, $n_{\rm g}$ is the galaxy number density with an
uncertainty $\sigma_{n_{\rm g}}$ (assumed to be $10\%$ of the measured value
shown in Table 1). Quantities with (without) a superscript ‘$*$’ represents
the measurements (model predictions). We perform joint modelling, with the
total $\chi^{2}$ the sum of those pertaining to the early and the late galaxy
sample.

The covariance matrix of each sample is estimated with the jackknife sampling
technique, with $101$ and $400$ jackknife samples for the LIN/PCAT and EXT
sample, respectively. To account for the mean bias in the precision matrix
(inverse of the covariance matrix), we employ a correction to the precision
matrix following Hartlap et al. (2007). That is, the precision matrix is
multiplied by $(n-p-2)/(n-1)$, where $n$ is the number of jackknife samples
and $p$ the number of 2PCF data points. The correlations between 2PCFs of
early and late galaxy samples are neglected, as they cannot be reliably
estimated with the limited sample size.

Finally, as we measure galaxy-galaxy lensing signals for each galaxy sample,
we also compare the lensing measurements to the predictions from the best-fit
model as a consistency check. For such a comparison, we construct mock galaxy
catalogues according to the best-fit model. With dark matter particles in the
MDPL2 simulation box, we directly measure the ESD profile (equation 10) around
each mock galaxy and take the average. To speed up the calculation without
losing precision, we down-sample the dark matter particles by randomly
selecting 1% of them. For future investigations it would be beneficial to
include the lensing measurements in addition to the galaxy clustering
measurements to constrain model parameters.

Figure 1:  Measurements and modelling results of the projected 2PCFs
$\xi=[w_{\rm p}]$ for the LIN, PCAT, and EXT samples (from left to right),
with an SFH-based division into early and late galaxies. Measurements are
shown as points in upper panels with jackknife error bars, and the predictions
from the best-fitting HOD model (fitting to observations above $2h^{-1}{\rm
Mpc}$) are shown as solid curves with the shaded region indicating the
$1\sigma$ range. Ratios of early-to-late galaxy $w_{\rm p}$ measurements and
best model fits are in the lower panels, where the $\chi^{2}$ values of the
best fits (with 2 degrees of freedom) and the resultant satellite fractions
$f_{s}$ are also displayed.  Figure 2: Constraints on mean mass of central
galaxy hosting haloes from modelling only the projected 2PCFs $\xi=[w_{\rm
p}]$ for the LIN, PCAT, and EXT samples (from left to right), with an SFH-
based division into early and late galaxies. For each sample, the curve show
the PDF of the constraints, with the shaded region marking the central 68.3%
distribution. In the legend are the mean and $1\sigma$ uncertainties of the
mean halo mass of central galaxies for each sample, together with the
significance of the split between the distributions indicated by
$\sigma_{split}$. Figure 3:  Measurements and modelling results of the
projected 2PCFs and redshift-space multipoles, $\xi=[w_{\rm
p},\xi_{0},\xi_{2}]$ (from left to right), for the LIN, PCAT, and EXT samples,
with an SFH-based division into early and late galaxies. The values of
$\chi^{2}$ are from jointly modelling $[w_{\rm p},\xi_{0},\xi_{2}]$. For
clarity, vertical offsets are applied to data points and model curves, as
indicated in the upper panels. Ratios of early-to-late galaxy measurements and
best model fits are in the lower panels. The format is similar to that in Fig.
1. Figure 4:  Same as Fig. 2, but from modelling the projected 2PCFs and
redshift-space multipoles, $\xi=[w_{\rm p},\xi_{0},\xi_{2}]$ for the LIN,
PCAT, and EXT samples (from left to right), with an SFH-based division into
early and late galaxies. Figure 5:  Similar to Fig. 4, but for constraints on
central galaxy velocity bias $\alpha_{\rm c}$ for the SFH-based LIN, PCAT, and
EXT samples (from left to right). For each sample, the curve show the PDF of
the constraints, with the shaded region marking the central 68.3%
distribution. In the legend are the mean and $1\sigma$ uncertainties of the
central galaxy velocity bias for each sample, together with the significance
of the split between the distributions indicated by $\sigma_{split}$.  Figure
6:  ESD from lensing measurements (data points) and from best-fitting model
predictions (curves) for the SFH-based LIN, PCAT, and EXT samples (from left
to right). Predictions from fits to $\xi=[w_{\rm p}]$ and $\xi=[w_{\rm
p},\xi_{0},\xi_{2}]$ are shown as solid and dashed curves, respectively.

##  4 Results

In this section, we present the modelling results for the SFH-based and sSFR-
based samples, respectively. For each of the LIN, PCAT, and EXT catalogue
constructions (section 2 and Table 1), the early and late samples are jointly
modelled. To reduce any residual effect caused by fibre collision correction,
we limit the modelling to scales above $2h^{-1}{\rm Mpc}$.

We first consider the case of modelling only the projected 2PCFs $\xi=[w_{\rm
p}]$ (and number densities). In this case, the free parameters do not include
the velocity bias parameters (with $\alpha_{\rm c}$ and $\alpha_{\rm s}$ fixed
at 0 and 1, respectively). We focus on the constraints on the mean mass of the
haloes hosting central galaxies to study the assembly bias signature.

We then model both the projected 2PCFs and redshift-space 2PCF multipoles,
$\xi=[w_{\rm p},\xi_{0},\xi_{2}]$, together with the number densities. We
focus on presenting the results on both the mean mass of central host haloes
and the central velocity bias parameter $\alpha_{\rm c}$.

We also show the consistency check with the weak lensing measurements by
comparing the ESD measured from the data and predicted from the best-fitting
models.

###  4.1 Results for SFH-based Samples

We start with the early/late samples split based on SFH. Fig. 1 shows the
projected 2PCF $w_{\rm p}$ measurements (points) and the results from
modelling $w_{\rm p}$ only (curves for bestfit and shaded regions for the
1–$\sigma$ ranges), with the LIN, PCAT, and EXT samples (left to right
panels). We use the notation that the results for the early (late) sample are
in red (blue).

In all cases, by eye, the modelling curves are able to reproduces the
measurements. However, the values of $\chi^{2}$ appear to be large, given the
degrees of freedom (dof). For each set of early/late samples, there are 14
data points (12 for $w_{\rm p}$ and 2 for $n_{\rm g}$) and the model has 12
free parameters (3 for the mean occupation function of central galaxies and 3
for that of satellites for the early sample, and the same numbers for the late
sample; see equations 11 and 12). We then have dof=2. The large values of
$\chi^{2}$ could result from the noise in the estimated covariance matrix
and/or the neglect of the correlation between the measurements of early and
late samples.

In each case, the amplitude of $w_{\rm p}$ of the late sample appears to be
slightly higher than that of the early sample. Their ratio (bottom panels of
Fig. 1) are consistent with unity, given the uncertainty. The left panels
simply reproduce the L16 results that no clear evidence is found for the
assembly bias with the early and late galaxies (with the measurements improved
with fibre collision correction). As discussed in the sample construction, the
LIN sample removed some central galaxies (mislabelled as satellites),
especially for those in pairs having similar lines of sight. This results in
the substantial drop of projected 2PCFs towards smaller scales (e.g. below
1$h^{-1}{\rm Mpc}$). The PCAT and EXT samples have recovered those mislabelled
central galaxies, and the corresponding projected 2PCFs do not show the drop
towards small scales. With a fraction of central-central pairs along similar
lines of sight missing, the redshift-space clustering for LIN samples would
become hard to interpret and model. Hereafter, we show the modelling results
with LIN samples for complete purposes and will not attempt to draw
conclusions from them. We will focus on presenting the results from the PCAT
and EXT samples.

The PCAT and EXT samples specifically include satellites by construction, and
the modelling results show a low satellite fraction. To see whether the small
difference in the $w_{\rm p}$ amplitude signals assembly bias effect, we turn
to the constraints on the mean mass of haloes hosting central galaxies.

The probability distribution functions (PDF) of the mean halo mass for central
galaxies are shown in Fig. 2. For the LIN samples, the 1$\sigma$ ranges of the
mean host halo masses for central early/late galaxies overlap, and the values
are consistent with those inferred from lensing measurements in L16
($\log\langle M_{\rm h}/(h^{-1}{\rm M}_{\odot})\rangle=11.98^{+0.14}_{-0.10}$
and $11.92^{+0.10}_{-0.10}$, respectively). For the PCAT samples, the
difference between the mean halo masses of central early/late galaxies becomes
larger, although still being consistent. For the EXT samples, the mean halo
mass PDFs overlap and the constraints become tighter, resulting from the
larger samples.

Similar to the projected 2PCFs, for redshift-space clustering, the
measurements of the 2PCF monopoles (middle panel of Fig. 3) for early galaxies
appear to be lower than those for late galaxies, while being consistent within
the uncertainties. The quadrupoles of the two samples are similar (right panel
of Fig. 3). As shown in Fig. 4, the trend in the constraints on the mean halo
mass of central galaxies is similar to that with the projected 2PCFs, with
halo mass for late galaxies being generally more massive. The addition of
redshift-space 2PCF multipoles have produced tighter constraints, and the
split between mass scales of early and late central galaxies becomes more
significant (with $\sim 2.5\sigma$ and $\sim 1.8\sigma$ for the PCAT and EXT
samples).

In Fig. 5 the constraints on the central-galaxy velocity bias parameter
($\alpha_{\rm c}$) are compared. As mentioned before, because the LIN samples
miss a fraction of central-central galaxy pairs along similar lines of sight,
we do not consider redshift-space constraints from the LIN samples to be
informative and simply show them for consistency in the presentation of the
results. The $\alpha_{\rm c}$ constraints for early and late central galaxies
are consistent for both the PCAT and EXT samples, with those for PCAT sample
having larger uncertainties. While there is a trend that early central
galaxies have higher velocity bias (with lower mean halo mass), the large
uncertainties prevent a definitive conclusion.

The apparent high values of central velocity bias in Fig. 5 appear surprising.
For the PCAT and EXT samples, the PDFs of $\alpha_{\rm c}$ peak around 0.5 or
above. This is different from the central velocity bias constraints with
luminosity-threshold samples Guo et al. (2015c), which is usually lower than
$\alpha_{\rm c}=0.5$ with median values of 0.2–0.3. In the galaxy formation
model investigated by Ye et al. (2017), high $\alpha_{\rm c}$ is possible but
only for central galaxies with low stellar mass to halo mass ratio. However,
if we account for the full distribution of $\alpha_{\rm c}$, $\alpha_{\rm
c}=0.2$–0.3 falls into the 3$\sigma$ range of the constraints. One possible
cause of the tendency of high $\alpha_{\rm c}$ is that some satellite galaxies
(with higher velocity dispersion) are misidentified as central galaxies.
However, we do include satellites in our model, which is supposed to capture
the satellite contribution. In the parameterisation, the mean occupation
function of satellites assumes a log-normal form (equation 12). We perform
tests by changing it to be a power-law form, and the resultant $\alpha_{\rm
c}$ constraints remain similar. The other possibility is that the tendency of
high $\alpha_{\rm c}$ may be some manifestation of the assembly bias effect
(see discussion in section 5). This possibility needs to be addressed with
more sophisticated models. But given the uncertainties seen in the constraints
here, the difference in the assembly bias effects of the early and late
central galaxies are unlikely to be established with high statistical
significance.

In Fig. 6, we show the ESD measurements from gravitation lensing on scales
below 2$h^{-1}{\rm Mpc}$. In each case, at $r_{\rm p}\sim 1.5h^{-1}{\rm Mpc}$,
the ESD around late galaxies appears to be higher than that around early
galaxies. It is not clear whether this is a statistical fluctuation or not.
Except for this point, the ESD measurements around early and late galaxies
agree with each other, which indicates similar mean host halo mass for
populations dominated by central galaxies. The curves show the predictions
from mocks generated by populating haloes in the MultiDark MDPL2 $N$-body
simulation from the HOD that best-fits $w_{\rm p}$ (solid) and [$w_{\rm p}$,
$\xi_{0}$, $\xi_{2}$] (dashed).

For the PCAT case, the model reproduces the ESD for late galaxies, but
underpredicts that for early galaxies. The underprediction is in line with the
lower mean halo mass of early galaxies, as shown in the middle panels of Fig.
2 and Fig. 4. The results of low halo mass, high $\alpha_{\rm c}$, and low ESD
for the PCAT early galaxies, if taken together, may hint assembly bias effect
– for example, these early central galaxies may reside in haloes of higher
mass but late-formed (thus lower clustering amplitude); therefore, the model
(without assembly bias) puts them in lower mass haloes (thus lower ESD); the
resultant lower velocity dispersion inside haloes requires a higher
$\alpha_{\rm c}$ to fit the redshift-space clustering.

For the EXT case, the model ESD curves agree with the ESD measurements for
both late and early galaxies. The predicted ESD for early galaxies is slightly
lower. The overall trend in the results of mean halo mass, $\alpha_{\rm c}$,
and ESD for early galaxies is similar to the PCAT case but much weaker.

As a whole, with the SFH-based early and late samples with different
constructions (PCAT and EXT), we find that the HOD modelling results support
that the central galaxies reside in haloes of similar mass. The differences
seen in mean halo mass and central velocity bias parameter may be attributed
to assembly bias effect, but the differences in the constraints are not large
enough to make definitive conclusions. With a model incorporating assembly
bias descriptions, we would be able to more quantitatively assess the assembly
bias effect by jointly fitting the projected and redshift-space 2PCFs and the
ESD.

Figure 7:  Same as Fig. 1, but with an sSFR-based division into early and late
galaxies.  Figure 8:  Same as Fig. 2, but with an sSFR-based division into
early and late galaxies.

###  4.2 Results for sSFR-based Samples

Figure 9:  Same as Fig. 3, but with an sSFR-based division into early and late
galaxies.  Figure 10:  Same as Fig. 4, but with an sSFR-based division into
early and late galaxies. Figure 11:  Same as Fig. 5, but with an sSFR-based
division into early and late galaxies.  Figure 12:  Same as Fig. 6, but with
an sSFR-based division into early and late galaxies.

The $w_{\rm p}$ measurements and the corresponding model fits for early/late
samples constructed according to sSFR are shown in Fig. 7. Opposite to the
SFH-based samples, the projected 2PCFs for early galaxies in the LIN and PCAT
cases have higher clustering amplitudes than late galaxies, while the
amplitudes are similar in the EXT case (on scales above $2h^{-1}{\rm Mpc}$).
However, the amplitude ratio in each case is consistent with unity
(particularly so in the EXT case), given the uncertainties. The HOD model
provides good fits to the measurements in all the cases.

Compared to the SFH-based samples, the mean halo mass for central galaxies is
better constrained (Fig. 8). While the halo mass scale for early central
galaxies appears to be slightly higher than that for late central galaxies,
the central $1\sigma$ constraints for the mass scales of early and late
central galaxies overlap with each other. In particular, the mass scale
constraints for early/late galaxies in the EXT case fall on top of each other,
a manifestation of the similar amplitudes in $w_{\rm p}$ (Fig. 7). In the LIN
case, the mass scales are consistent with those from lensing measurements in
L16 ($\log\langle M_{\rm h}/(h^{-1}{\rm M}_{\odot})\rangle=12.14\pm 0.07$ and
$12.10\pm 0.08$ for early and late central galaxies).

Fig. 9 shows the measurements of both the projected and redshift-space 2PCFs,
$\xi$=[$w_{\rm p}$, $\xi_{0}$, $\xi_{2}$], and the fitting results. For the
same reason as with the SFH-based samples, we focus on the PCAT and EXT cases.
We note that for both the PCAT and EXT late galaxy samples, the quadrupoles
below $2h^{-1}{\rm Mpc}$ become negative, opposite to the expected Finger-of-
God effect. We perform tests and discuss the possible origin in section 5. On
scales above $2h^{-1}{\rm Mpc}$, the amplitudes of the projected and redshift-
space 2PCFs for early and late galaxies agree with each other. The HOD model
results in reasonable values of $\chi^{2}$ when fitting jointly to
$\xi$=[$w_{\rm p}$, $\xi_{0}$, $\xi_{2}$].

The halo mass constraints (Fig. 10) from jointly modelling $\xi$=[$w_{\rm p}$,
$\xi_{0}$, $\xi_{2}$] are similar to those from modelling $w_{\rm p}$ only
(Fig. 8), but becoming much tighter with the benefit of the redshift-space
clustering.

For the constraints on the central velocity bias (Fig. 11), similar to the
trend seen in the SFH-based samples, both the PCAT and EXT cases have a higher
$\alpha_{\rm c}$ for early galaxies. The 1$\sigma$ range of $\alpha_{\rm c}$
for the two cases are (0.39, 0.69) and (0.70, 0.86), respectively. We note
that the velocity bias for the EXT early galaxy case is substantially higher
than that for luminosity-threshold samples (Guo et al., 2015c).

For the sSFR-based samples, the ESD measurements and the predictions from the
best-fitting HOD models are shown in Fig. 12. The broad agreement between the
measurements and predictions lends further support that the samples
constructed have similar host halo masses for central galaxies. For the EXT
case, the measured ESD around early galaxies appears to be higher than the
predictions. In combination with the high $\alpha_{\rm c}$ for the EXT early
galaxies, it may point toward a possibility that a fraction of satellites in
high mass haloes are misidentified as central galaxies in haloes of $\sim
10^{12}h^{-1}{\rm M}_{\odot}$. However, our model includes contributions from
such satellites, and like the SFH-based case our tests with an alternative
parameterisation for satellite occupation function does not change the
results. Assembly bias remains as a possible explanation, which will be
discussed in section 5.

As a whole, with the sSFR-based samples, we obtain results similar to those
with the SFH-based samples. Although there are overall agreement between the
modelling results of early and late central galaxies, early central galaxies
tend to have higher values of velocity bias.

##  5 Discussion

The main model constraints we are interested in this work are the mass scales
of central galaxy host haloes and the velocity bias of central galaxies.
Unlike the samples used in Lin et al. (2016), called LIN samples in this
paper, we do not perform the satellite decontamination algorithm in defining
the extended PCAT and EXT samples in order to study the redshift-space
clustering. While we have parameters for satellite components in our HOD
model, for the purpose of studying central galaxies, they are treated as
nuisance parameters to absorb the effect of satellites. We focus our
discussion on the results with the PCAT and EXT samples.

For the PCAT samples, the mass scales of central hosting haloes of early and
late galaxies show a split. While in the SFH-based samples early galaxies tend
to reside in haloes of lower mass than late galaxies, the trend is opposite in
the sSFR-based samples. However, the split in either SFH-based or sSFR-based
samples is not statistically significant from the marginalised distributions
of the mass scale. Compared with the constraints from $w_{\rm p}$ measurements
only, including redshift-space multipoles reduces the uncertainty in the mass
scale by a factor of about two, and the split remains insignificant.

When constructing the EXT samples, we not only exclude the satellite
decontamination procedure but also adjust the range of galaxy properties to
substantially extend the samples. As a consequence, the EXT samples lead to
tight constraints on the mass scale of central hosting haloes, and adding the
redshift-space clustering further tightens the constraints. In our full
analyses with projected and redshift-space clustering, the uncertainties in
the mean mass scale are about 0.05 dex for either the SFH-based or the sSFR-
based samples. Even with such tight constraints, no significant difference
between the mass scales of early and late central galaxies are found.

As a consistency check, we compare the HOD modelling results to the lensing
measurements. In general, the predicted ESD values measured from mock galaxy
catalogues based on the best-fitting HOD show a broad agreement with those
measured from the data. The main exception is with the SFH-based early galaxy
sample. The predicted ESD curves from the $w_{\rm p}$-only fit are about 1 dex
lower than the measurements. This is in line with the difference in the halo
mass constraints between early and late galaxies. Accounting for the large
uncertainty ($\sim$0.33 dex) in the mass scale constraints reduces the
discrepancy. With the inclusion of redshift-space clustering, the predicted
ESD curve for early galaxies is shifted upward by $\sim$0.5 dex. It is likely
that the apparent mismatch between the predicted and measured ESD for the SFH-
based early galaxy sample reflects the loose constraint on the halo mass scale
and that the current best-fitting model for the SFH-based early galaxies is
somewhat biased. Including ESD in the data vector in future investigations
will tighten model constraints and assist in selecting the appropriate models.

The most important results of this work are the constraints on the central
galaxy velocity bias ($\alpha_{\rm c}$), shown in Fig. 5 and Fig. 11 for the
SFH-based and sSFR-based samples, respectively. Again our focus here are on
the PCAT and EXT sample constructions (middle and right panels). There appears
to be a split between the values of central velocity bias of early and late
samples, in the sense of higher velocity bias for early galaxies. For the SFH-
based samples, one may associate this with the apparent split in halo mass
scale (fig. 4) and argue for a halo mass dependent velocity bias. However, the
trend of higher $\alpha_{\rm c}$ in lower mass haloes appears to be opposite
to that inferred from luminosity-threshold samples (Guo et al., 2015c). For
the sSFR-based samples, while there is a split in $\alpha_{\rm c}$, there is
no clear split in the halo mass scale. Being unlikely attributed to the halo-
mass dependence, the split in $\alpha_{\rm c}$ of early and late galaxies may
signal a galaxy assembly bias effect. In the model, the central velocity bias
characterises the mutual motion (relaxation) between central galaxies and
their host haloes. If early galaxies reside in early-formed haloes (Xu &
Zheng, 2020), the split would indicate that galaxies in early-formed haloes
are less relaxed, contrary to the naive expectation. As investigated in Ye et
al. (2017) with the Illustris simulation, the magnitude of velocity bias is
related to the history of major and minor mergers, and the above split would
provide information on the merging history of haloes and galaxies. However,
the constraints on central velocity bias are loose and the split is not
statistically significant, with the biggest split being about 1.7$\sigma$ in
the sSFR-based EXT samples. Taking the face values, we find no evidence of
significant difference in the central velocity bias of early and late
galaxies. Then we would reach the conclusion that no assembly bias effect is
detected in the motion of central galaxies relative to their dark matter
haloes.

However, the surprise lies in the value of central velocity bias. If we use
the central velocity bias constraints from luminosity-threshold samples (Guo
et al., 2015c) as a comparison point, the constraints on the central velocity
bias for late galaxies in this work are in the ballpark of those for
luminosity-threshold samples. However, those for early galaxies are much
higher, roughly lying in the range of $\alpha_{\rm c}\sim$ 0.5–1.0. That is,
in the halo rest frame early central galaxies would on average move at a
velocity on the order of 35–70% that of the halo virial velocity. The stellar
mass to halo mass ratio of these central galaxies is around 1 per cent
(section 2), and in the theoretical investigation of Ye et al. (2017), the
expected velocity bias is about 0.32 (see their fig.2). Could this high
velocity bias result from mistaking satellites as central galaxies? First, we
account for satellite galaxies in the model, and we find similar central
velocity bias constraints with different parameterisations of the mean
satellite occupation function. Second, given the selection criteria, galaxies
in the samples are dominated by central galaxies in haloes around
$10^{12}h^{-1}{\rm M}_{\odot}$ and their stellar mass is a few times
$10^{10}h^{-2}{\rm M}_{\odot}$ (section 2). It would be rare for such haloes
to host such massive satellites. Such satellites can reside in more massive
halos, but given the low satellite fraction they would not play a role in
significantly interfering with the central velocity bias constraints.
Therefore, the apparent high velocity bias inferred for early central galaxies
is unlikely caused by mistaking satellites as central galaxies.

The high value of velocity bias for early central galaxies may be a signal of
assembly bias effect, but here the inferred velocity bias does not necessarily
reflect the mutual motion between central galaxies and host haloes. With early
and late central galaxies in the PCAT or EXT samples residing in haloes of
similar mass, we make the assumption that early central galaxies on average
are found in early formed haloes (e.g. Hearin & Watson, 2013; Xu & Zheng,
2020). Earlier formed haloes have a higher pairwise velocity dispersion – in
haloes of $\sim 10^{12}{\rm M}_{\odot}$, haloes in the quartile of earliest
formation can have a pairwise velocity dispersion 1.5 $\times$ higher than the
average at this mass scale at a halo pair separation of $\sim 1h^{-1}{\rm
Mpc}$ (see middle panels in fig.11 of Xu & Zheng 2018). In modelling the
redshift-space clustering using the HOD model that neglects assembly bias
effect, such a high halo pairwise velocity dispersion would be captured. But
it would be interpreted as pairwise velocity between central galaxies, causing
the inferred central galaxy velocity bias to be high. While the above scenario
may lead to difference in the clustering amplitude, the difference depends on
how strong the correlation between galaxy properties and halo assembly history
and it may be buried in the measurement uncertainties if the correlation is
not strong.

So the assembly bias effect in halo pairwise velocity could offer an
explanation on the velocity bias constraints. In haloes of fixed mass, the
spatial clustering and the pairwise velocity are correlated (see Xu & Zheng
2018; e.g. their fig.12). If the results in early galaxy velocity bias suggest
assembly bias effect in halo pairwise velocity, other constraints from the HOD
modelling would be re-interpreted. For example, the inferred mass scales for
central galaxies would be modulated by the assembly bias effect and the
constraints would not reflect the true mass scale.

There may be other signs of assembly bias in the clustering measurements. As
mentioned in section 4.2, the quadrupole of redshift-space 2PCFs of sSFR-based
late galaxies (in both the PCAT and EXT samples) appears to be negative on
scales below $2h^{-1}{\rm Mpc}$ (Fig. 10), opposite to the expected Finger-of-
God effect. We find that this feature is not caused by fibre-collision
correction. To further test the fibre collision, we divide the samples into to
redshift bins and perform clustering measurements. As fibre collision has a
fixed angular scale, the fibre-collision effect would show up at different
comoving scales for the two redshift bins. However, we obtain consistent
quadrupole measurements. The negative quadrupole could result from halo
exclusion effect for samples dominated by central galaxies, but we do not see
such a feature in the early samples. To see whether a simple prescription of
assembly bias effect can lead to such a feature, we measure the quadrupole for
the blue central galaxies in the mock galaxy catalogue constructed by Hearin &
Watson (2013), where galaxy colour is tied to halo formation time. No negative
quadrupole is found on scales below $2h^{-1}{\rm Mpc}$. It remains possible
that this feature is related to assembly bias, but the link between galaxy
properties and halo assembly variable is not as simple as in the above model.

To summarise, based on the results of analysing the redshift-space clustering
of the PCAT and EXT samples of early and late galaxies (divided either by SFH
or sSFR), we suspect that there are hints of galaxy assembly bias effect.
Clearly a more sophisticated model incorporating assembly bias with an
effective description of galaxy properties and halo assembly history is needed
to further investigate the possibility and put concrete constraints on the
assembly bias effect. Given the level of uncertainties seen in our constraints
(especially on $\alpha_{\rm c}$), the assembly bias may remain subtle to be
detected. Therefore, a variety of observables may be needed to reveal the
assembly bias signal, including the projected and redshift-space 2PCFs, the
lensing measurements, and the count-in-cell statistics (e.g. Wang et al.,
2019).

##  6 Summary and Conclusion

In this work, we extend the investigations of galaxy assembly bias effect in
Lin et al. (2016) from real space (with projected 2PCFs) to redshift
(velocity) space (with reshift-space 2PCF multipoles). To achieve such a goal,
we make an extension to the galaxy sample construction and divide the sample
into early- and late-forming galaxies (named as early and late galaxies in
this work) according to either their SFH or their sSFR. With the projected and
redshift-space clustering measurements, we carry out HOD modelling to
constrain the galaxy-halo relation to see whether there is any sign of
assembly bias effect. Our particular interest lies in the velocity bias
parameters of central galaxies, which is supposed to characterise the mutual
relaxation between central galaxies and their host haloes. Differences in the
velocity bias of early and late central galaxies may signal assembly bias
effect.

In Lin et al. (2016), the early and late galaxies (called LIN samples here)
are constructed to consist of central galaxies and weak lensing measurements
show that they have consistent mean host halo mass. In this paper, based on
the HOD modelling of the $w_{\rm p}$ measurements, we find that both the SFH-
based and sSFR-based LIN samples have low satellite fraction (at per cent
level). Within the uncertainties, the mean masses of central hosting haloes
for early and late galaxies constrained in the HOD model broadly agree with
each other and are consistent with the lensing measurements in Lin et al.
(2016). Therefore, our results based on $w_{\rm p}$ lend support to the
findings in Lin et al. (2016).

The sharp drop in the $w_{\rm p}$ measurements toward the smallest scales in
the LIN samples indicates that a fraction of central-central galaxy pairs
along similar lines of sight ($r_{\rm p}\lesssim$ 0.5–0.8$h^{-1}{\rm Mpc}$)
are removed with the satellite decontamination algorithm when constructing the
LIN samples. While this has little effect on analysing $w_{\rm p}$
measurements in Lin et al. (2016) and in this paper, it would affect our
analyses of the redshift-space clustering. Therefore we focus our study on the
redshift-space clustering with the extended PCAT and EXT samples, constructed
without the satellite decontamination procedure. The PCAT samples are simply
the LIN samples before performing the satellite decontamination. The EXT
samples extend the range of allowed stellar mass of galaxies, leading to a
factor of 1.94 and 1.43 times more galaxies in the catalogue. In addition, the
sSFR-based EXT samples shift the early/late galaxy dividing cut from
$\log({\rm sSFR}/{\rm yr}^{-1})\sim-11.8$ to $\log({\rm sSFR}/{\rm
yr}^{-1})\sim-11.0$ to better separate the quenched and active star-forming
galaxies.

Through HOD modelling of the projected 2PCFs and redshift-space 2PCF
multipoles measured with the PCAT and EXT early/late galaxies, we find that
the halo mass scales of central early and late galaxies, within the
uncertainties, agree with each other and with those from lensing measurements
inferred in Lin et al. (2016). If there is any split in the mass scale, it is
only seen in the PCAT samples (middle panels of Fig. 4 and Fig. 10). The trend
of the split depends on the construction of the early and late galaxy samples
– for SFH-based PCAT samples, early galaxies tend to reside in lower mass
haloes, with the significance of the split being $\sim 2.5\sigma$; for sSFR-
based PCAT samples, early galaxies tend to be in higher mass haloes, with the
significance of the split being $\sim 1.2\sigma$. For each galaxy sample, we
also perform lensing measurements for consistency check, and the ESD
measurements largely agree with the predictions from the best-fitting HOD
model.

The central galaxy velocity bias constraints from modelling both the projected
and the redshift-space clustering measurements appear to be puzzling. In
either the PCAT or the EXT samples, early central galaxies tend to have a
higher velocity bias $\alpha_{\rm c}$ than late galaxies. A naive explanation
would be that in haloes of similar mass, the mutual relaxation between central
galaxies and their host haloes is weaker for early galaxies or in older
haloes, a trend opposite to expectation. The differences in $\alpha_{\rm c}$
of early and late galaxies, however, are below 1.7$\sigma$ in all cases.
Therefore, if the trend hints assembly bias effect, we do not have a
statistically significant detection. The puzzle lies in the magnitude of the
$\alpha_{\rm c}$ constraints, which appear to be high compared to previous
inference based on luminosity-threshold samples (Guo et al., 2015c). While
$\alpha_{\rm c}$ constraints for late galaxies largely fall into the range
found in previous work, those for early galaxies are high, roughly in the
range of $\alpha_{\rm c}\sim$ 0.5–1.0. We suspect that this reflects assembly
bias effect. For example, instead of the central galaxy motion in the frame of
host haloes, the velocity bias constrained here captures the high pairwise
velocity dispersion among early-formed haloes (Xu & Zheng, 2018). In addition,
the negative quadrupole on small scales measured in the EXT late galaxy
samples may also signal assembly bias effect that cannot be described by a
simple model.

To further investigate the possible assembly bias signatures hinted in our
results, a halo model with assembly bias description (e.g. Hadzhiyska et al.,
2021; Yuan et al., 2021) is necessary, and to reach a conclusive result a
variety of clustering measurements may need to be modelled, which can include
the redshift-space clustering, the lensing measurements, the environment-
dependent clustering, the count-in-cell statistics, and void-related
statistics (e.g. Wang et al., 2019; Hadzhiyska et al., 2021). In addition,
such investigations will benefit from substantially larger galaxy samples than
studied here, such as the Bright Galaxy Sample in the Dark Energy
Spectroscopic Instrument survey (DESI Collaboration, 2016).

## Acknowledgements

KSM acknowledges the support by a fellowship from the Willard L. and Ruth P.
Eccles Foundation. The work is supported by a seed grant at the University of
Utah. The support and resources from the Center for High Performance Computing
at the University of Utah are gratefully acknowledged. ZZ and HG acknowledge
the support of the National Science Foundation of China (no. 11828302).

The CosmoSim database used in this paper is a service by the Leibniz-Institute
for Astrophysics Potsdam (AIP). The MultiDark database was developed in
cooperation with the Spanish MultiDark Consolider Project CSD2009-00064. The
authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.
(www.gauss-centre.eu) and the Partnership for Advanced Supercomputing in
Europe (PRACE, www.prace-ri.eu) for funding the MultiDark simulation project
by providing computing time on the GCS Supercomputer SuperMUC at Leibniz
Supercomputing Centre (LRZ, www.lrz.de).

## Data Availability

No new data were generated or analysed in support of this research.

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11institutetext: Saint Petersburg State University, Saint Petersburg, Russia

# Strongly intensive fluctuations and correlations in ultrarelativistic
nuclear collisions in the model with string fusion

Vladimir Kovalenko 11<EMAIL_ADDRESS>
###### Abstract

The several types of strongly intensive correlation variables are studied in
nuclear collisions at LHC energy. These quantities are expected not to depend
on centrality class width. They have been calculated in the dipole-based
parton-string Monte Carlo model with string fusion. The centrality dependence
of the mean transverse momentum correlation coefficient and strongly intensive
quantity $\Sigma$ between multiplicity and $P_{T}$ have been obtained.
Dynamical charge fluctuation $\nu_{dyn}$ has been also calculated and compared
with experimental data. It is shown that string fusion improves agreement with
the experiment.

##  1 Introduction

The exploration of the QCD phase diagram and a search for new phenomena is one
of the main subjects of heavy ion physics. The experimental investigation of
the phase diagram of a strongly interacting matter is related to the study of
the fluctuations and correlations of observables in the nuclear collisions at
high energyJiang:2015hri . Around the critical point, anomalies, such as
enhanced fluctuations are expected to appear Luo:2017faz . Physical program of
NA61/SHINE at SPS Gazdzicki:2008pu ; Gazdzicki:2011fx  and MPD at NICA
Toneev:2007yu  is focused on these investigations. Correlations and
fluctuations are also studied at LHC energy Abelev:2012pv ; ATLAS:2012as ;
Abelev:2009ag  because they are sensitive to the very first stages of the
collision Tribedy:2017yxb . The so-called long-range correlations
Altsybeev:2016uci  between observables in separated rapidity intervals play a
special role. Causality implies that they are developed in the initial state.
In addition, a large enough rapidity gap can suppress the background
contribution of resonance decays.

In the present work, the string-parton Monte Carlo model Kovalenko:2012ye ;
Kovalenko:2012nt ; Kovalenko:2014tca  is applied to the relativistic nuclear
collisions at high energy to calculate long-range correlations and
fluctuations. The model includes the string fusion Braun:1992ss ; Braun:1991dg
; Amelin:1993cs ; Braun:1997ch  effects and the finite rapidity length of
strings, and also charge conservation in string fragmentation. We have
calculated several types of strongly intensive event-by-event variables and
long-range correlations for the experiment.

##  2 Model description

The present model Kovalenko:2012ye ; Kovalenko:2012nt  describes nucleon-
nucleon collisions at the partonic level. For the nuclear density of $Pb$ and
$Xe$ the Woods-Saxon distribution of nucleons is taken:

$\rho(r)=\dfrac{\rho_{0}}{1+\exp[(r-R)/d]},$ (1)

$\text{\newline with }R=6.62\text{ fm, }d=0.546\text{ fm for }Pb;R=5.42\text{
fm, }d=0.57\text{ fm }\text{for }Xe.$

Each nucleon contains a valence quark-diquark pair and a certain number of sea
quark-antiquark pairs, distributed around the center of nucleon according to
two-dimensional Gauss distributions. The number of sea pairs is distributed
according to Poisson law. The combined exclusive distribution of parton
coordinates in the transverse plane ($r_{j}$) preserves the conservation law
for the center of energy (Boyer_2005, ):

$\sum\limits_{j=1}^{N}\vec{r}_{j}\cdot x_{j}=0,$ (2)

where $x_{j}$ is a longitudinal momentum fraction of parton $j$. These
variables follow the exclusive distribution, which for an arbitrary number of
quark-antiquark pairs has the following form:

$\rho(x_{1},...x_{N})=c\cdot\prod\limits_{j=1}^{N-1}x_{j}^{-\frac{1}{2}}\cdot
x_{N}^{\alpha_{N}}\delta(\sum\limits_{i=1}^{N}x_{i}-1).$ (3)

The valence quark is labeled by N-1, the diquark – by N, and the other indexes
refer to sea quarks and antiquarks. ${\alpha_{N}}$ = 3/2 (ud-diquark),
${\alpha_{N}}$ = 5/2 (uu-diquark). This distribution corresponds to the
inclusive distributions, used in the model with quark-gluon strings
Kaidalov:1985jg ; Arakelian:2002iw . Corresponding algorithms for sampling
with the exclusive distributions mentioned above are discussed in
Kovalenko:2012nt .

An elementary interaction is realized in the model of colour dipoles
Flensburg:2008ag ; Gustafson:2009qz . Taking also into account also the
confinement scale ($r_{max}$), the probability amplitude for a scattering the
dipoles with transverse coordinates $(\vec{r}_{1},\vec{r}_{2})$ and
$(\vec{r}_{3},\vec{r}_{4})$ is given by:

$f=\frac{\alpha_{s}^{2}}{2}\left[K_{0}\left(\frac{|\vec{r}_{1}-\vec{r}_{3}|}{r_{max}}\right)+K_{0}\left(\frac{|\vec{r}_{2}-\vec{r}_{4}|}{r_{max}}\right)-K_{0}\left(\frac{|\vec{r}_{1}-\vec{r}_{4}|}{r_{max}}\right)-K_{0}\left(\frac{|\vec{r}_{2}-\vec{r}_{3}|}{r_{max}}\right)\right]^{2}$
(4)

Here $\alpha_{S}$ is assumed to be an effective coupling constant and treated
as a model parameter. The total probability of inelastic interaction of two
protons in the eikonal form appears as ${p=1-e^{-\sum\limits_{i,j}f_{ij}}}$,
where the summation is done over all the dipoles.

Then it is assumed, that if there is a collision between two dipoles, two
quark-gluon strings are stretched between the ends of the dipoles, and the
process of string fragmentation gives observable particles. The string in the
rapidity space is formed with rapidity ends $y_{\text{min}}$ and
$y_{\text{max}}$ Kovalenko:2012nt ; Kovalenko:2014tca , which emits the
produced particles uniformly in rapidity between the string ends with some
mean number of charged particles per rapidity $\mu_{0}$. The emission happens
independently in each rapidity interval, with Poisson distribution. Each
parton can interact with another one only once, forming a pair of quark-gluon
strings, hence, producing particles. In the model, the energy-momentum of the
nucleons is conserved on the partonic level if we take into account both
interacting and non-interacting partons. It is also fully valid at the level
of strings, however during the production of particles it is approximate.

The transverse position of a string is assigned to the arithmetic mean of the
transverse coordinates of the partons at the ends of the string. Due to the
finite transverse size of the strings they overlap, that in the framework of
the string fusion model Braun:1992ss ; Braun:1991dg ; Amelin:1993cs ;
Braun:1997ch  gives a source with higher tension. The mean multiplicity of
charged particles and mean $p_{t}$ originated from the cell where $k$ strings
are overlapping look as follows:

$\left\langle\mu\right\rangle_{k}=\mu_{0}\sqrt{k},\hskip
28.45274pt\left\langle p_{t}\right\rangle_{k}=p_{0}\sqrt[4]{k}.$

Here $\mu_{0}$ and $p_{0}$ are the mean charged multiplicity from one single
string per rapidity unit and mean transverse momentum from one single string.
For the overlapping of the strings of a finite rapidity length, the rapidity
space is divided into the regions where the number of overlapped strings is an
integer, and they are processed separately. All parameters of the model are
constrained from the data on inelastic cross section and multiplicity in pp,
p-Pb and Pb-Pb collisions Kovalenko:2014tca .

The charge differentiation of the particles in this model is performed in the
following way. If we neglect the production of the double charge at string
fragmentation, each emitted particle can change the charge going along
rapidity only by 0 or 1 (emitting of neutral or charged pion). After removing
the neutral particles, we see that the charge of particles emitted from one
string (or a cluster of fused strings) must alternate between 1 to -1 along
the rapidity axis. This procedure is fulfilled for each string cluster.

##  3 Definition of observables

Firstly, we define the correlation coefficient of long range correlations
between observables in two separated rapidity windows as follows:

$b=\dfrac{\langle BF\rangle-\langle B\rangle\langle F\rangle}{\langle
F^{2}\rangle-\langle F\rangle^{2}},$ (5)

where each of ($B$, $F$) could be either $N_{\mathrm{ch}}$ – the number of
charged particles in the rapidity window or $p_{t}$ – the mean event
transverse momentum of charged particles in the given window:

$p_{t}=\dfrac{1}{N_{\mathrm{ch}}}\sum\limits_{i=1}^{N_{\mathrm{ch}}}{{p_{t}}_{i}}.$
(6)

Correspondingly, three types of correlation coefficients can be studied: the
n-n, pt-n, pt-pt correlations.

Unfortunately, the n-n and pt-n correlation coefficients are shown to be very
dependent of the centrality class width and details of the centrality
determination method Kovalenko:2013jya  (note pt-pt has not this property). To
overcome this dependence, the so-called strongly intensive variables can be
used Gorenstein:2011vq ; Gazdzicki:2013ana ; Andronov:2015bqn ;
Andronov:2018xom . In this paper, we apply only $\Sigma[F,B]$ defined as:

$\Sigma(F,B)\equiv\frac{\langle F\rangle\,\omega_{B}+\langle
B\rangle\,\omega_{F}-2\,\left(\langle FB\rangle-\langle F\rangle\langle
B\rangle\right)}{\langle F\rangle+\langle B\rangle}\ ,$ (7)

where $\omega_{F}\equiv\dfrac{\langle F^{2}\rangle-\langle
F^{2}\rangle}{\langle F\rangle}\ ,\omega_{B}=\dfrac{\langle
B^{2}\rangle-\langle B^{2}\rangle}{\langle B\rangle}\ $ – scaled (or relative)
variance.

Variables $F$ and $B$ here must be extensive. For them, we have used the total
multiplicity in a rapidity window ($N$), and total transverse momentum
($P_{T}$).

To study the charge fluctuations, we have used dynamical fluctuation measure,
$\nu_{dyn}$, defined as follows:

$\nu_{dyn}=\left\langle\left(\dfrac{N_{+}}{\langle
N_{+}\rangle}-\dfrac{N_{-}}{\langle
N_{-}\rangle}\right)^{2}\right\rangle-\left(\dfrac{1}{\langle{N_{+}}\rangle}+\dfrac{1}{\langle
N_{-}\rangle}\right).$ (8)

It can be shown that the variable
$N_{ch}\nu_{dyn}=\langle{N_{+}}+{N_{-}}\rangle\nu_{dyn}$ is also strongly
intensive.

The increase and non-monotonic behavior of the mentioned strongly intensive
variables can be an indication of the critical point of the QCD phase
transition Stephanov:2004wx , so they are used in the experimental
investigations Aduszkiewicz:2017mei ; Andronov:2018ukf ; Andronov:2018ccl .

##  4 Results

Figure 1 shows the centrality dependence of the mean transverse momentum
correlation coefficient. We have calculated it for two colliding systems – Pb-
Pb and Xe-Xe at the LHC energy to demonstrate also the system size dependence
of the pt-pt correlation coefficient. In case of Pb-Pb collisions, the model
predicts Kovalenko:2013jya  non-monotonic behaviour with centrality, which was
also confirmed by experimental data Altsybeev:2017eoj , but it is hard to
reproduce in the models Kovalenko:2016bcx . The position of the peak can be
attributed to the point of the saturation of the color field.

In the case of a smaller system, the saturation region seems not to be reached
because even in most central Xe+Xe collisions the required density is not
achieved. Similar behavior has been also obtained in p-Pb collisions
Kovalenko:2014ika .

Figure 1: Mean transverse momentum correlation coefficient as a function of
centrality at LHC energy, calculated in the Monte Carlo model with string
fusion for Pb-Pb and Xe-Xe nuclei.

In figure 2 the forward-backward strongly intensive measure $\Sigma$ is shown
for sufficiently large separated rapidity intervals (-3.8, -3), (3, 3.8). We
see that above the very peripheral collisions the value of $\Sigma$ decreases
with centrality. This behavior has been recently observed in the experimental
data; however other models cannot reproduce this effect IwonaHotQuarks ;
Sputowska:2016vmz .

Figure 2: Strongly intensive quantity $\Sigma_{NN}$ as a function of
centrality, calculated in the Monte Carlo model with string fusion for Pb-Pb
collisions at 2.76 TeV. Two centrality class binning (2% and 5%) are shown by
different markers.

Figure 3: Strongly intensive quantities $\Sigma_{P_{T}N}$ (left) and
$\Sigma_{P_{T}P_{T}}$ (right) as a function of centrality, calculated in the
Monte Carlo model with string fusion for Pb-Pb collisions at 2.76 TeV. As in
fig. 2, two centrality class binning (2% and 5%) are shown by different
markers.

Figure 3 shows the centrality dependence of the two other types of the
strongly intensive quantities between the observables in the separated
rapidity intervals, namely, $\Sigma[P_{T},N]$ and $\Sigma[P_{T},P_{T}]$.

$\Sigma[P_{T},N]$ is rather flat, what reminds one the dependence of the
average transverse momentum with multiplicity Abelev:2013bla . In pp and p-Pb
collisions the latter is less trivial so it is worth checking the multiplicity
dependence of a strongly intensive pt-n measure there.

The strongly intensive transverse momentum variable $\Sigma[P_{T},P_{T}]$
(right plot of figure 3) shows, in particular, the non-monotonic behavior.
This reminds us the behavior of $b_{p_{t}-p_{t}}$. However, the position of
the minimum here is at a different point than the maximum of $b_{p_{t}-p_{t}}$
(here it lies in the range of a less central collision).

The maxima appearing in figures 2 and 3 in the region of very peripheral
collisions could be related with the transition into a region of diffusive
edges of nuclei. But, this centrality range is hardly accessible by the
experiment Abelev:2013qoq .

To summarize, all the three variants of the forward-backward strongly
intensive measure indeed does not depend on the centrality class width; they
provide additional non-trivial information which can not be extracted only
from the correlation coefficient and thus should be widely experimentally
studied, at LHC energy and also at lower such as MPD at NICA.

The figure 4 shows the charge fluctuations in terms of $\nu_{dyn}$ (left plot)
and $N_{ch}\nu_{dyn}$ (right plot) with the comparison to the experimental
data Abelev:2012pv . The results have shown that taking into account the
string fusion improves the agreement with the experimental data. Without
fusion, the model behaves as hadron resonance gas, wheres an adding of a
string collectivity modifies $N_{ch}\nu_{dyn}$ and makes it centrality-
dependent.

Figure 4: Dynamical charge fluctuation $\nu_{dyn}$ as a function of centrality
(number of participant nucleons) for Pb-Pb collisions at 2.76 TeV, calculated
in the Monte Carlo model with string fusion and compared with experimental
data Abelev:2012pv . The solid line corresponds to the model with string
fusion, dashed – without fusion. The right panel shows $\nu_{dyn}$ multiplied
by multiplicity. Areas that correspond to hadron resonance gas (HRG) and
quark-gluon plasma (QGP) are hatched.

##  5 Conclusions

Various strongly intense quantities are studied in AA collisions at the LHC
energy, depending on the centrality compared (where possible) with
experimental data of ALICE. All analyzed strongly intensive variables and
correlations between intensive observables have shown robust behavior against
the volume fluctuations and the details of centrality determination.

It is shown that the string fusion model correctly qualitatively describes the
$\Sigma_{NN}$ dependence on the centrality. Charge fluctuations in Pb-Pb
collisions at the energy of 2.76 TeV have been investigated. The correct
description of the experimental ALICE data for $\nu_{dyn}$ has been obtained
depending on the centrality of the collisions (string fusion effects improve
the agreement with the experiment). The string fusion model explains the non-
monotonic behavior of pt-pt forward-backward correlations in Pb-Pb collisions
and predicts that at a smaller system this should not be observed.

The studied observables provide a reference to the properties of the initial
state of AA collisions and are useful to study both at the LHC and NICA
energies.

##  6 Acknowledgments

The research was supported by the grant of the Russian Foundation for Basic
Research (project 18-32-01055 mol_a).

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