# Galaxy and Mass Assembly (GAMA): Probing galaxy-group correlations in
redshift space with the halo streaming model

Qianjun Hang1,2, John A. Peacock1, Shadab Alam1, Yan-Chuan Cai1, Katarina
Kraljic1,3, Marcel van Daalen4, M. Bilicki5, B.W. Holwerda6, J.Loveday7
1Institute for Astronomy, University of Edinburgh, Royal Observatory
Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK
2Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, UK
3Aix Marseille Université, CNRS, CNES, UMR 7326, Laboratoire d’Astrophysique
de Marseille, Marseille, France
4Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The
Netherlands
5Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników
32/46, 02-668 Warsaw, Poland
6Department of Physics and Astronomy, University of Louisville, Natural
Science Building 102, 40292 KY Louisville, USA
7Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK E-mail:
<EMAIL_ADDRESS>
###### Abstract

We have studied the galaxy-group cross-correlations in redshift space for the
Galaxy And Mass Assembly (GAMA) Survey. We use a set of mock GAMA galaxy and
group catalogues to develop and test a novel ‘halo streaming’ model for
redshift-space distortions. This treats 2-halo correlations via the streaming
model, plus an empirical 1-halo term derived from the mocks, allowing accurate
modelling into the nonlinear regime. In order to probe the robustness of the
growth rate inferred from redshift-space distortions, we divide galaxies by
colour, and divide groups according to their total stellar mass, calibrated to
total mass via gravitational lensing. We fit our model to correlation data, to
obtain estimates of the perturbation growth rate, $f\sigma_{8}$, validating
parameter errors via the dispersion between different mock realizations. In
both mocks and real data, we demonstrate that the results are closely
consistent between different subsets of the group and galaxy populations,
considering the use of correlation data down to some minimum projected radius,
$r_{\rm min}$. For the mock data, we can use the halo streaming model to below
$r_{\rm min}=5{\,h^{-1}\,\rm Mpc}$, finding that all subsets yield growth
rates within about 3% of each other, and consistent with the true value. For
the actual GAMA data, the results are limited by cosmic variance:
$f\sigma_{8}=0.29\pm 0.10$ at an effective redshift of 0.20; but there is
every reason to expect that this method will yield precise constraints from
larger datasets of the same type, such as the DESI bright galaxy survey.

###### keywords:

gravitation; galaxies: groups: general; large-scale structure of Universe;

††pubyear: 2022††pagerange: Galaxy and Mass Assembly (GAMA): Probing galaxy-
group correlations in redshift space with the halo streaming model–E

##  1 Introduction

The large-scale structure in the galaxy distribution has a long history of
providing cosmological information. The first constituents of the
inhomogeneous galaxy density field to be identified were the rich clusters,
which today we see as marking the sites of exceptionally massive haloes of
dark matter. Proceeding down the halo mass spectrum, we find progressively
less rich groups of galaxies, leading to systems dominated by a single $L_{*}$
galaxy, such as the Local Group (e.g. Wechsler & Tinker 2018). All these
systems have been familiar constituents of the Universe since the first
telescopic explorations of the sky, but it took rather longer to appreciate
that they were connected as part of the cosmic web of voids & filaments (see
e.g. Peacock 2016 for some selective history). In part, the history here
showed a complex interaction of theory and observation, since redshift surveys
through the 1980s lacked the depth and sampling to reveal the cosmic web with
complete clarity. For a period, it was therefore a question of asking whether
the real Universe displayed the same structures that were predicted in
numerical simulations of structure formation in the Cold Dark Matter model
(Bond et al. 1996). But since those times, there has been an increasing
confidence that galaxy groups are indeed particularly extreme nonlinear points
in the general field of cosmic density fluctuations, and this makes them
interesting in two ways. First of all, groups are readily identified in galaxy
surveys, providing a relatively robust dataset (Eke et al., 2004; Robotham et
al., 2011). Secondly, their nonlinear nature makes them an informative probe
of theory. Modelling nonlinear behaviour is by its nature challenging compared
to linear theory, but by studying structure formation further into the
nonlinear regime, we have the chance to test the robustness of our
cosmological conclusions.

Our specific aim in this direction is to use galaxy groups as a probe of the
cosmological peculiar velocity field. Such deviations from uniform expansion
must exist through continuity, and density concentrations such as groups
should be associated with an average infall velocity in regions surrounding
the groups. The amplitude of these velocities depends in part on the strength
of gravity on cosmological scales, and the peculiar velocity field has thus
increasingly been seen as a means of probing the nature of gravity and testing
alternative theories (e.g. Jain & Khoury 2010). Although it is possible to
probe peculiar velocities directly using absolute distance indicators (Davis
et al., 2011), the most powerful tool has been Redshift Space Distortions
(RSD). These arise inevitably in the study of the 3D galaxy distribution
because the distances to galaxies observed on the sky are inferred from their
redshifts, $z$, via the standard relation

$d(z)=\int_{0}^{z}\frac{c\,dz^{\prime}}{H(z^{\prime})},$ (1)

where $c$ is the speed of light and $H(z)=\dot{a}/a$ is the Hubble parameter.
But this equation does not give the true distances, because Doppler shifts
from the peculiar velocities modify the observed redshift:
$1+z\rightarrow(1+z)(1+v_{r}/c)$, where $v_{r}$ is the radial component of the
peculiar velocity. If we then use the observed redshift as if it were a true
indicator of distance, we obtain a distribution of galaxies in ‘redshift
space’ – in which the apparent properties of galaxy clustering are distorted
in an anisotropic way.

These distortions have characteristics that depends on scale: outside large
density concentrations, galaxies fall coherently together under gravity; while
the orbital velocities inside dark-matter haloes are effectively randomised.
The latter effect convolves the redshift-space density field in the radial
direction, leading to the characteristic radial elongations of high-density
regions known as ‘Fingers of God’ (FoG). RSD due to coherent flows in the
linear regime were first studied by Kaiser (1987). The growth factor $f$ is
defined by

$f\equiv\frac{\partial\ln\delta}{\partial\ln a}\simeq\Omega_{m}(z)^{0.55},$
(2)

where $\delta$ is the matter overdensity, $a$ is the expansion factor, and
$\Omega_{m}$ is the matter fraction; The approximation for $f(\Omega_{m})$
only applies for flat $\Lambda$CDM models in standard gravity (Lahav et al.,
1991; Wang & Steinhardt, 1998; Linder, 2005). In Fourier space, and in the
small-angle limit of a distant observer, the matter power spectra in redshift
space and in real space are related by

$P^{s}_{m}(k,\mu)=P^{r}_{m}(k)\,(1+f\mu^{2})^{2},$ (3)

where $\mu$ is the cosine of the angle between the wave-vector $k$ and the
line of sight. This simple equation was highly influential from its first
appearance (Kaiser, 1987), as it offered the chance of measuring $\Omega_{m}$
from measuring the RSD anisotropy. But eventually goals shifted as
$\Omega_{m}$ became very well determined from other routes (especially the
Cosmic Microwave Background). Following Guzzo et al. (2008), the modern view
is therefore to emphasise that the growth rate for a given density is also
proportional to the strength of gravity, so that RSD can be used as a test of
theories of gravity.

The RSD signal has been measured by a number of surveys, including the 2dFGRS
at $z\simeq 0.2$ (Peacock et al., 2001; Hawkins et al., 2003); the 6dFGS at
$z\simeq 0.1$ (Beutler et al., 2012); the SDSS BOSS & eBOSS surveys at
$z\simeq 0.6$ (Reid et al., 2012; Alam et al., 2017, 2021a); and at $z\simeq
1$ by the 8m VVDS and VIPERS surveys (Guzzo et al., 2008; Pezzotta et al.,
2017). For the GAMA survey at $z\simeq 0.4$, aspects of RSD were studied by
Blake et al. (2013) and Loveday et al. (2018), who measured the pair-wise
velocity dispersion to small scales and as a function of luminosity. The above
studies all focused on galaxy auto-correlations.

The challenge in modelling RSD is that truly linear modes are rare. In
observation, large scales are affected by cosmic variance due to the finite
survey volume. McDonald & Seljak (2009) proposed the use of multiple tracers
in order to overcome cosmic variance, although in practice the improvement is
slight (Blake et al., 2013). To gain more information, one needs to probe
smaller scales, where the effect of non-linearity can systematically bias the
results (de la Torre & Guzzo, 2012).

One possible solution to this dilemma is to use galaxy groups to probe the
velocity field. Due to the small random virial velocity of the central galaxy
at the group centre, the coherent large-scale infall velocities of groups are
dominant down to intermediate and small scales. The group auto-correlation
would thus have reduced FoG, aiding the extraction of the linear growth rate
(Padilla et al., 2001; Mohammad et al., 2016). In practice, the group
catalogue in GAMA is sparse, with a number density of $4.3\times
10^{-3}h^{3}{\rm Mpc}^{-3}$ between $0.1<z<0.3$, and measurements of the auto-
correlation will have high statistical noise. The cross-correlation between
groups and galaxies is thus an intermediate route, which effectively improves
the statistical power while still reducing the non-linear pairwise velocities
at small scales. The clustering of GAMA groups has been recently studied in
Riggs et al. (2021), and the present work extends this study to further
subsets of the data, concentrating in more detail on their different RSD
signals.

Our aim here is thus to test the robustness of RSD methods down to small or
intermediate scales using multiple tracers involving galaxy groups. By cross-
correlating galaxies of different colours, and groups in different mass bins,
we examine the consistency of the inferred cosmological results between the
subsamples. In order to pursue this investigation, we develop a new model for
RSD in cross-correlation, involving a combination of the halo model and the
streaming model, which we implement by including some information taken from
mock data. Throughout the analysis, we adopt the WMAP7 Cosmology (Komatsu et
al., 2011) with $\sigma_{8}=0.81$, $\Omega_{m}=0.27$, $h=0.70$, and
$n_{s}=0.967$, consistent with the mock catalogue.

The GAMA data set and its mocks are detailed in Section 2 followed by Section
3.1 where we introduce the statistics for measuring the 2-point function in
the data. In Section 3.2 we present the resulting 2D correlation function
measurements for sub-samples, In Section 4 we discuss the theoretical
modelling of RSD in galaxy-group cross-correlations, and in section 5 we
confront this modelling with real and mock GAMA data. The models are validated
in Section 5.1 via detailed comparison with the GAMA mocks, where we establish
the scales to which the different theories can work without bias; we present
the fitting of the real GAMA data in Section 5.2. Finally, we summarize the
work in Section 6.

##  2 GAMA data and mocks

This analysis is based on the Galaxy And Mass Assembly (GAMA) spectroscopic
survey. This was conducted using the 2dF facility at the Anglo-Australian 4m
telescope over 210 nights between 2008 & 2014, accumulating spectra of 265 958
distinct galaxies. Together with existing data, this yielded a catalogue of
330 542 redshifts over five survey fields totalling 250 deg2, with a mean
redshift of $z\simeq 0.2$ (Driver et al., 2022). The three main fields near
the equator, G09, G12, and G15 are used here, each covering an area of
$12\times 5\,\deg^{2}$. The survey has an extinction-corrected $r$-band flux
limit of $r<19.8$, based on SDSS photometry.

The overall redshift completeness of the GAMA equatorial region is 98.5%: this
high completeness was achieved by a large number of repeated visits to 2dF
fields covering the survey area in different ways. This property is greatly
advantageous for small scale galaxy and group studies compared to much larger
surveys such as BOSS, where fibre collisions can lead to substantial
undercounting of close galaxy pairs and thus bias the measured galaxy 2-point
correlation function (Guo et al., 2012).

The present analysis uses the DR3 data release (Baldry et al., 2018), which
differs slightly from the final data release, DR4 (Driver et al., 2022). DR4
implements revised flux completeness limits through the use of new KiDS
photometry. The original SDSS limit of $r<19.8$ was trimmed in DR4 to
$r<19.58$ for 98% completeness in the equatorial fields. Galaxies for the
present study are selected from the SpecObjv27 DMU (Data Management Unit),
with CMB frame redshifts adopted from DistancesFramesv14. We apply the
following criteria: redshift quality nQ $\geq 3$, angular completeness mask
$>80\%$, and visual classification VIS_CLASS $=0,1,255$111VIS_CLASS $=0$: Not
visually inspected but suspicious based on SDSS flags; VIS_CLASS $=1$:
Visually inspected and a valid target; VIS_CLASS $=255$: Not visually
inspected but should be OK based on SDSS flags.. The spectroscopic redshifts
are computed by the code runz (Driver et al., 2011), which has a $1\sigma$
redshift error of 50 km s-1 in terms of peculiar velocity (Liske et al.,
2015). In order to compute correlation statistics, it is essential to
accompany the galaxy sample with a knowledge of the survey selection in angle
and redshift. As usual, this information is captured by a random catalogue
Randomsv02 of fictitious unclustered galaxies; this catalogue was generated by
Farrow et al. (2015) from the actual GAMA galaxy catalogue using a modified
method following Cole (2011). The idea of this method is to clone each galaxy
$n$ times and distribute them randomly within the maximum volume $V_{\rm max}$
that the galaxy can be observed given the survey magnitude limits,

$n=n_{\rm clones}\frac{V_{\rm max}}{V_{\rm max,dc}},$ (4)

where $n_{\rm clones}=400$ is the total number of randoms divided by data, and
$V_{\rm max,dc}$ is the maximum volume weighted by overdensity $\Delta(z)$.
This method is iterated until $\Delta(z)$ converges, and the redshift
distribution of the resultant random catalogue is smooth without large scale
features (see Fig. 4 in Farrow et al. 2015).

The official GAMA group catalogue (G3C) was constructed by Robotham et al.
(2011). Most of the groups are found within $z\lesssim 0.35$ (see Fig. 16 in
Robotham et al. 2011): thus we impose a redshift cut $0.1<z<0.3$ for the
groups. The group catalogue is derived using an anisotropic friends-of-friends
(FoF) algorithm calibrated against an $N$-body mock catalogue. However, in
order to have consistently defined groups in the GAMA mocks (see Section 2.3),
we do not use the official G3C catalogue. Instead, we apply a similar FoF
group finder algorithm due to Treyer et al. (2018) to both data and mocks (see
Kraljic et al. 2018 for an application to GAMA). The main difference between
the two algorithms is the parameterisation of the linking length, and a
detailed description of the algorithm and assessment of the group
reconstruction quality can be found in the Appendix of Treyer et al. (2018).

In addition to the above selections, we further split galaxies and groups into
subsamples based on galaxy colour and group mass. The number of selected
galaxies and groups in each GAMA field and for each subsample is summarised in
Table 1. We describe the selection in more detail below.

Table 1: Number of selected galaxies and groups from GAMA fields with redshifts $0.1<z<0.3$ and flux limit $r<19.8$. Galaxies are split into two colour classes, red and blue, by Eq. 5. Groups are split into three stellar mass bins: 40% (LM), 50% (MM), and 10% (HM) by mass ranking from low to high, covering the mass range $\log_{10}(M_{*}/h^{-2}M_{\odot})=9.5-12.5$. Number of |  | G09 | G12 | G15
---|---|---|---|---
Galaxies | Blue | 17 335 | 18 719 | 19 053
| Red | 20 584 | 22 155 | 21 141
| Total | 37 919 | 40 874 | 40 194
Groups | LM | 1877 | 2084 | 2054
| MM | 2347 | 2606 | 2569
| HM | 470 | 522 | 514
| Total | 4694 | 5212 | 5137
G3C | Total | 4937 | 5367 | 5358

###  2.1 GAMA galaxy colour selection

Galaxies are divided into two populations that are known to have distinct
clustering properties: the ‘red’ galaxies, which tend to be older, with little
or no active star formation, and the ‘blue’ cloud, where galaxies are younger
with active star formation. To obtain the galaxy colours, we use the
extinction corrected SDSS magnitudes from the TilingCatv46 DMU. It is however
non-trivial to separate the galaxy population into these subsets, because the
colour distribution is continuous without gaps: elaborate approaches have been
discussed in e.g. Taylor et al. (2015). For the purpose of this study, we
adopt a simple quadratic cut in the apparent $g-i$ colour versus redshift
plane:

$g-i=6.220z^{2}+1.383z+0.831.$ (5)

A cut of this form is motivated empirically by the apparent bimodality in the
colour-redshift plane, as shown in Fig. 1. The precise location of the cut was
adjusted in order to match the red and blue fraction at each redshift in the
GAMA data with the corresponding result in the mocks (which are discussed
below in Section 2.3). The overall fraction of red or blue galaxies is very
close to 0.5, and it changes only slightly with redshift: at the low redshift
end, the red and blue fractions are similar, while towards higher redshifts,
the fraction of red galaxies increases mildly until $z\sim 0.2$, and the
difference in the red and blue fraction then becomes small at $z\sim 0.3$. We
create random catalogues for the red and blue galaxy subsamples where required
by applying this smoothly-varying colour balance to the redshift distribution
of the main random catalogue.

Figure 1: Distribution of the $g-i$ colour of galaxies in the redshift range
$0.1<z<0.3$, for the real GAMA data (upper panel) and the average of 25 mocks
combined (lower panel). Red and blue populations are separated by the dashed
cyan lines. The cut in GAMA is chosen such that both GAMA and mocks have
similar red and blue fractions at any given redshift. The dotted lines show
the cut for an alternative ‘contaminated’ red sample (see text).

###  2.2 GAMA group mass selection

Groups are accepted with $\geq 2$ group members, and the centre of the group
is determined by the most (more) massive member in terms of stellar mass. The
2-member systems make up $66\%$ of the total groups in the GAMA data, but are
likely to have poor fidelity. Thus, we emphasise that having the same group
finder algorithm for the data and mocks is vital in order for these low-
fidelity groups to be comparable. There are several approaches for determining
the group centre. The simplest choice is to select the most massive member to
be the central galaxy, and assume that it overlaps with the halo centre. Other
approaches include determining a weighted centre by averaging over the
positions of the group members, or iteratively excluding members that are most
distantly separated (see e.g. Robotham et al. 2011). The iterative centres are
used in the G3C catalogue, and it is shown in Robotham et al. (2011) that the
agreement with using the brightest group galaxy (BCG) as group centre is
$95\%$ for groups with $N\geq 5$, and that both BCG and iterative centres give
highly consistent results for $2\leq N\leq 4$ compared with the mock, and that
the BCG centres are only degraded by about $3\%$ compared to the iterative
centres. The effects of different group centre choices on the group-galaxy
cross-correlation concern mainly the 1-halo regime at $r\leq 1{\,h^{-1}\,\rm
Mpc}$, and the correlation functions converge on larger scales (Yang et al.,
2005).

The halo mass of GAMA groups was found to be tightly correlated with the group
total luminosity (Han et al., 2015; Viola et al., 2015; Rana et al., 2022),
based on using stacked weak lensing measurements to determine the mass
distribution of the GAMA groups. The halo mass of groups is related to the
$r$-band luminosity $L_{\rm grp}$ via

$M_{h}=M_{p}\left(\frac{L_{\rm grp}}{L_{0}}\right)^{\alpha},$ (6)

where $L_{0}=2\times 10^{11}h^{-2}L_{\odot}$,
$\log_{10}(M_{p}/h^{-1}M_{\odot})=13.48-0.08\pm 0.12$, and
$\alpha=1.08+0.01\pm 0.22$ (Han et al., 2015). The group halo mass can be used
to compute the expected mean group bias, as shown in Section 5.3, where we
also consider alternative mass calibrations. It should be noted that in these
works, only groups with three members or above are used. Robotham et al.
(2011) showed that the mass function is noisier, but not biased when including
two-member groups. Although these systems are individually of low reliability,
this aspect should be allowed for by the mock catalogue, allowing us to gain
the statistical advantage of using a larger sample. The luminosity is computed
from the apparent $r$-band magnitude:

$-2.5\log(L/L_{\odot})=m-K(z)-5\log(d_{L})-25-M_{\odot},$ (7)

where $K(z)$ the $k$-correction up to $z=0$ (kcorr_z00), $d_{L}$ is the
luminosity distance, and $M_{\odot}=4.67$ is the $r$-band absolute magnitude
of the sun. The luminosity distance is expressed with unit ${\,h^{-1}\,\rm
Mpc}$ so that the luminosity has units of $h^{-2}L_{\odot}$.

The total luminosity of the group is computed in Robotham et al. (2011) via

$L_{\rm FoF}=BL_{\rm ob}\frac{\int_{-30}^{-14}10^{-0.4M_{r}}\phi_{\rm
GAMA}(M_{r})\,dM_{r}}{\int_{-30}^{M_{r-{\rm lim}}}10^{-0.4M_{r}}\phi_{\rm
GAMA}(M_{r})\,dM_{r}},$ (8)

where $L_{\rm ob}$ is the total observed luminosity in the $r_{\rm AB}$ band,
$B=1.04$ is the correction for median unbiased estimate for $N\geq 5$ groups,
and $M_{r-{\rm lim}}$ is the absolute magnitude limit of the group depending
on the redshift $z$. $\phi_{\rm GAMA}$ is the luminosity function defined in
Robotham et al. (2011). The luminosity function at the faint end for GAMA
galaxies is well approximated by $\phi\propto L^{-1}\exp(-L/L_{*})$ (Loveday
et al., 2012). Thus in practice we take the simpler approach of estimating a
total group luminosity by scaling the observed luminosity by a redshift-
dependent correction factor $\exp(z^{2}/z_{*}^{2})$ with $z_{*}=0.33$, where
$z$ is the mean redshift of the group members. This correction factor has been
checked using the G3C groups to produce a total luminosity consistent with the
official TotFluxProxy.

The total stellar mass is another proxy for the total group mass. We take the
StellarMassesv19 DMU from Taylor et al. (2011), where stellar population
synthesis is used to model the optical photometry of the GAMA galaxies.
Because the modelling uses rest frame luminosities, which depends on distance,
the stellar mass is expressed in units of $h^{-2}M_{\odot}$222Notice that this
is only approximately true, because the stellar mass to light ratio, $M/L$,
which is used to obtain the stellar mass, depends on age and is therefore
specific to the choice of $h$. The stellar mass used here assumes $h=0.72$.
.Furthermore, for each group, we correct the total stellar mass by the same
redshift dependent factor as the total luminosity. Notice that we do not apply
the fluxscale correction here, which accounts for the missing flux from
matched aperture photometry, because our results do not rely on the absolute
stellar mass of the groups. This correction therefore does not affect our
primary aim of splitting the groups into a few bins based on their ranking in
mass.

The calibration of the total stellar mass and the halo mass from weak lensing
of the GAMA groups is shown in Fig. 2 for the official G3C groups from the
G3CFoFGroupv09 DMU (dashed line) and the group catalogue used in this work
(solid line). The contours show 95%, 50%, and 20% of the total sample, and are
highly consistent between the two group catalogues. We choose to divide groups
into three stellar mass bins based on percentiles: the Low Mass (LM) bin
consists of the least massive 40% groups, the Medium Mass (MM) bin corresponds
to the middle 50%, and the High Mass (HM) bin contains the most massive 10%.
The signal-to-noise of high mass haloes is expected to be high, despite the
low number in the HM bin.

Figure 2: Upper panel: Correlation between the stellar mass, corrected to
total by a factor $\exp(z^{2}/z_{*}^{2})$ and the halo mass estimate derived
from the total group luminosity mass proxy, together with a lensing-based
absolute calibration from Han et al. (2015), for the GAMA groups with two or
more members between redshifts $0.1<z<0.3$. The contours denote 95%, 50%, and
20% of the total sample. The solid lines show the groups used in this work
using the group finder algorithm in Treyer et al. (2018), and the dashed lines
show the the official G3C groups (Robotham et al., 2011). Lower panel: The
same relation for the mock catalogue. In this case, $M_{h}$ is not estimated
from the luminosity, but directly taken as the arithmetic mean host halo mass
of the group member. The difference in the distributions indicates that the
stellar populations in the mock data are not entirely realistic, but it also
warns us that the exact values of halo mass corresponding to the different
GAMA group subsets must be treated empirically, and should not be treated as
being known precisely.

###  2.3 Mocks

We include mock catalogues for two reasons: (1) to validate the RSD models and
assess the bias on the recovered growth rate, and (2) to quantify the impact
of cosmic variance via the construction of covariance matrices. We used 25
realisations of a lightcone mock catalogue based on the GALFORM semi-
analytical galaxy formation (Gonzalez-Perez et al., 2014). The catalogue
exploits the Millennium Simulation (Boylan-Kolchin et al., 2009) with the
WMAP7 cosmology. These mocks can be obtained from the Durham hosted Virgo-
Millennium
Database1333http://virgodb.dur.ac.uk:8080/MyMillennium/Help?page=databases/gama_v1/lc_multi_gonzalez2014a
(Lemson & Virgo Consortium, 2006). For more details regarding the mock
catalogue, see Farrow et al. (2015). By Eq. 2, the fiducial value of growth
rate at the mean redshift of the mocks, $z=0.195$, is $f_{\rm fid}=0.593$. The
lightcone is constructed using the methods in Merson et al. (2013), where,
given an observer, the galaxy is placed at the epoch where it first enters the
past lightcone of the observer. The galaxy trajectories are interpolated
between snapshots. Each mock covers the five GAMA fields with the SDSS
$r$-band apparent magnitude ${\texttt{SDSS\\_r\\_obs\\_app}<21}$, and $z<0.9$.

We use galaxies in the G09, G12, and G15 fields and apply the same selection
in redshifts $0.1<z<0.3$ and the apparent $r$-band magnitude cut
${\texttt{SDSS\\_r\\_obs\\_app}}<19.8$. We also apply the same survey mask
generated using the random catalogue. The masked areas are obtained by binning
random galaxies in each field with an average of $\sim 2000$ counts in each
bin. Pixels with counts smaller than five times the Poisson noise are masked.
The total masked area in the three fields is about $0.14\,{\rm deg}^{2}$.
Because the mock redshift distribution is not matched exactly with GAMA data
and random (see Fig. 3), we create a random catalogue for these mocks by down-
sampling the random catalogue for the GAMA data, such that the $n(z)$ matches
the mean of 25 mocks.

Figure 3: The mean redshift distribution of the 25 GAMA mocks (square) is
offset from that of the random sample (dotted line). A random catalogue is
created for the mocks to have matched redshift distribution as the mock mean.
The redshift distribution of the GAMA galaxy sample is also shown (histogram)
for comparison.

The red and blue subsamples for the mean of the mocks are separated by the
empirical line given by

$g-i=0.46+3.2z,$ (9)

as shown in the lower panel of Fig. 1. The line is chosen to go through the
green valley of the mock galaxy $g-i$ colour. The GAMA galaxies have a more
concentrated red sequence overlapping with an extended blue population,
without a distinct green valley in between. On the contrary, the mocks have a
broader red population which is well separated from the blue population by a
green valley. Since the mock catalogues have more distinctive separation for
the two populations, we find the corresponding colour cut in the GAMA data by
matching red and blue fractions in the two catalogues for 20 redshift bins in
$0.1<z<0.3$. The cut is smoothed by fitting a second order polynomial, as
shown in the upper panel of Fig. 1.

The contamination of the red and blue sub-samples in the GAMA data resulting
from the colour cut is quantified in the following way: for each redshift bin,
the red and blue sub-samples are fitted by a double Gaussian. It is a
reasonable fit except for the green valley in the mocks, as shown in Fig. 4.
Given a colour cut, the contamination of the red sub-sample is defined as the
area under the blue Gaussian over the area under the red Gaussian, and
similarly for the contamination of the blue sub-sample. Clearly, GAMA data
contain a contaminated red sample and a pure blue sample. Therefore, we create
a contaminated red sub-sample using the mock catalogues by placing the mock
colour cut such that extra blue galaxies are included with the same level of
contamination as GAMA data. The contaminated red cut in the mocks (see Fig. 1)
is smoothed by fitting a quadratic polynomial of the form

$g-i=2.43z^{2}+1.55z+0.388.$ (10)

Figure 4: The $g-i$ colour distribution of GAMA and 25 mocks combined at 5 of
the 20 redshift bins in $0.1<z<0.3$. The black dashed lines are double
Gaussian fits to the distributions, characterizing the blue and red
populations. The yellow cuts show a linear cut in the green valley in the
mocks, and the corresponding cuts in GAMA which give the same red and blue
fraction. The green dotted cuts in mock is the cut for the impure red sample.

For mock groups, the stellar mass is computed by the sum of diskstellarmass
and bulgemass of all group members, and corrected by the same redshift-
dependent factor as the data. We do not estimate the group halo mass from the
same mass-luminosity relation in Eq. 6. Instead, we use the host halo mass of
the mock galaxy directly. Because some haloes contain more than one galaxy,
for each group, we test the largest, the arithmetic mean, and the median halo
mass of the group member, and find that they give similar results. We also
test using the sum of unique host haloes in the group. This increases the
total group halo mass in the lower mass end, but does not affect the higher
mass end. The stellar-halo mass relation of the groups using the total stellar
mass and the arithmetic mean host halo mass of the group members is shown in
the lower panel of Fig. 2. It is clear that the mocks show a much larger
scatter in the $M_{h}-M_{*}$ plane and the slope is smaller compared to data,
i.e., at fixed stellar mass, the halo mass is larger. The total stellar mass
of the mock groups is also smaller by about 0.5 dex compared to data. The
clear difference between data and the mocks shows that estimating the halo
mass from luminosity using Eq. 6 is not very reliable. The luminosity is
itself strongly correlated with stellar mass via the luminosity-mass relation,
thus the upper panel of Fig. 2 does not show the true scatter of $M_{h}$ at
fixed $M_{*}$ faithfully (or vice versa). A comparison between the group and
the halo catalogue in the GAMA mocks reveals that the 2-member groups have low
fidelity, also discussed in Robotham et al. (2011). This again emphasises the
importance of using a consistent group finder algorithm between the GAMA data
and the mock catalogues. The mock group catalogues are separated into three
stellar mass bins based on the 40%, 50%, and 10% percentiles as measured in
the data.

##  3 Cross-correlation measurements

###  3.1 Correlation statistics

We estimate the 2-point correlation function by counting pairs of galaxies and
randoms using the Davis–Peebles estimator (Davis & Peebles, 1983):

$\hat{\xi}(r_{p},\pi)=\frac{D_{1}D_{2}}{D_{1}R_{2}}-1,$ (11)

where the subscript $i=1,2$ denotes the two samples to be correlated (in case
of auto-correlation, the same sample), $D_{i}$ denotes data, and $R_{i}$
denotes the corresponding random points. Each term in the equation (e.g.,
$D_{1}D_{2}$) is the normalised pair count between data and (or) random
points, measured in bins of the pair separation $(r_{p},\pi)$ defined below.
For two objects located at $\mathbf{s}_{1}$ and $\mathbf{s}_{2}$, their
separation is given by $\mathbf{s}=\mathbf{s}_{1}-\mathbf{s}_{2}$. The line of
sight is defined along the mean position of the pair,
$\mathbf{r}=(\mathbf{s}_{1}+\mathbf{s}_{2})/2$. One can then decompose the
separation into components parallel and perpendicular to the line of sight:

$\pi=\frac{\mathbf{s}\cdot\mathbf{r}}{|\mathbf{r}|};\quad
r_{p}=\sqrt{\mathbf{s}^{2}-\pi^{2}}.$ (12)

The random catalogue $R$ captures various properties of the actual data, such
as the survey mask and the sample redshift distribution $n(z)$, but has no
spatial correlation. Thus, these estimators essentially measure the excess
clustering of the data points compared to a random distribution. For the red
and blue galaxy samples, the random $n(z)$ is modulated by the redshift-
dependent red and blue fraction respectively (see Section 2.1).

For the case of auto-correlations, the Landy–Szalay estimator (Landy & Szalay,
1993) is known to be superior to the Davis–Peebles approach, and there is a
natural generalisation to cross-correlation:

$\hat{\xi}(r_{p},\pi)=\frac{D_{1}D_{2}-D_{1}R_{2}-D_{2}R_{1}+R_{1}R_{2}}{R_{1}R_{2}}.$
(13)

However, implementing this estimator would require a random catalogue for
galaxy groups in different mass ranges, and we prefer to avoid this
complication. In contrast, the Davis–Peebles estimator requires a random
catalogue for only one of the populations being correlated. Both Mohammad et
al. (2016) and Riggs et al. (2021) estimated cross-correlations using a form
of Landy–Szalay where $R_{2}$ was replaced by $R_{1}$, but this has no
justification, and will yield incorrect results when the selection functions
of the two tracers are very different. We measured the galaxy auto-
correlations using both estimators, and found negligible difference for our
sample. Throughout the analysis, the size of random catalogue used is 20 times
that of data.

The 2D correlation functions are measured out to a maximum scale of
$40{\,h^{-1}\,\rm Mpc}$ for both the $r_{p}$ and $\pi$ directions in bins of
$1{\,h^{-1}\,\rm Mpc}$. This maximum scale is chosen due to the limited volume
of the GAMA survey. It may appear to be a concern that at this scale
perturbation theory starts to break down (thus the scale is often chosen as
the cut-off scales for larger data sets). However, we shall show later that
empirical models based on linear theory can still give relatively unbiased
results below this scale. For the halo streaming model, which we will
elaborate in Section 4.2, we use a 1-halo template to absorb the deviation of
non-linear clustering from the perturbative 2-halo term. Although in
principle, $r_{p}$ and $\pi$ should be measured in the range
$[-40,40]{\,h^{-1}\,\rm Mpc}$, in practice, pair counts in the positive and
negative bins are combined and our correlation functions have two mirror
planes of symmetry. This is the standard practice for $r_{p}$, because the
correlation function is symmetric around the transverse direction. Along the
line of sight, positive and negative $\pi$ measurements can in fact be
distinctive in cross-correlations due to secondary gravitational effects. For
example, gravitational redshift can give rise to a non-vanishing dipole
between two samples that differ significantly in mass (Bonvin et al., 2014;
Wojtak et al., 2011; Cai et al., 2017; Beutler & Dio, 2020). However, given
the size of the sample, we shall not investigate this issue in the present
study. The combination of the $\pi$ bins also improves the signal-to-noise for
our measurements.

Figure 5: False colour images of auto- and cross-correlation functions in
redshift space for the actual GAMA data and the corresponding average over the
set of 25 GAMA mocks. $r_{p}$ denotes transverse separation; $\pi$ is radial
separation. LM, MM, and HM denote the three group mass bins. A number of
trends are apparent: both the bias (amplitude of clustering) and the small-
scale Finger of God (FoG) dispersion increase with group mass, and are larger
for red galaxies than for blue. The mock contaminated red sample is shown as
dotted contours on the second column, with
$\log_{10}|\xi|=\\{-1.0,-0.5,0,0.7\\}$.

###  3.2 Results

In the following analysis, we will refer to the group subsamples as LM (low
mass bin), MM (medium mass bin), and HM (high mass bin). We thus have six
configurations for the group-galaxy cross-correlations: LMred, MMred, HMred,
LMblue, MMblue, and HMblue. In addition, we also measure the red and blue
galaxy auto-correlations. The inclusion of galaxy auto-correlations in the
analysis helps in breaking the near degeneracy between $b_{\rm gal}$ and
$b_{\rm grp}$. Ideally, one would also include the auto-correlations for the
group catalogue. These are excluded here: as mentioned above, we do not
construct random versions of the group catalogue.

Fig. 5 shows the red and blue galaxy auto-correlations (top row) and the
cross-correlation functions for the six configurations (lower three rows)
measured from the GAMA data (first and third column), and the corresponding
mocks average (second and forth column). The first noticeable feature is that
the red configurations (left two columns) have larger clustering signals
compared to the blue ones (right two columns). This is most obvious in the two
galaxy auto-correlations. On relatively large scales, the squashing is also
stronger in the blue configurations. This effect is controlled by the Kaiser
distortion parameter $\beta=f/b$ (see Section 4.1 below), and is expected to
be stronger for samples with a smaller bias $b$ (vice versa), given that the
growth rate $f$ is fixed. The fact that red galaxies have a larger galaxy bias
compared to blue galaxies implies that red galaxies are preferentially
associated with more massive dark matter haloes, in agreement with other
studies (e.g. Guo et al., 2014; Mandelbaum et al., 2016; Bilicki et al.,
2021). On smaller scales, the red configurations also show a much more
prominent FoG signal compared to the blue ones. This is intuitively sensible
because more massive haloes are associated with a larger velocity dispersion.
Comparing the correlation functions across different groups for a specific
galaxy selection, we see a similar trend on both small and large scales: with
the increase in group mass, a larger clustering amplitude, group bias, and FoG
effect are observed. Notice the high signal-to-noise at small scales in the HM
groups, despite that the sample size in this mass range is only about $1/4$ of
the other mass ranges. These observations confirm that the identification of
the galaxy groups, as well as the separation of group masses based on the
effective halo mass (total luminosity) are successful for the purpose of this
study.

The agreement between the mock average and the GAMA data is good in general –
the same trends in galaxy colour and group mass are captured. In regions where
$r_{p}$ is close to zero, the mock average seem to produce weaker clustering
compared to the actual data in the blue configurations. The match in the red
configuration, on the other hand, is excellent. The mock contaminated red
sample is also shown as dotted contours (second column). The inclusion of
extra blue galaxies has the effect of slightly reducing the overall amplitude
in this contaminated sample compared to the pure red sample. On larger scales,
the signal in data is noise and cosmic variance dominated. This is most
noticeable in the LM subsamples, where the signal greatly exceeds the mock
average on $r\geq 20{\,h^{-1}\,\rm Mpc}$. Inspecting the measurement in each
mock sample, this level of fluctuation in the data is expected. It should be
noted that cosmology adopted in the mock catalogue is $\Omega_{m}=0.27$, which
is lower than the current constraint from Planck, $\Omega_{m}=0.315\pm 0.007$
(Planck Collaboration et al., 2020). Thus, one may expect some difference in
clustering between the mock average and the data. However, given the noise in
the GAMA data, a percent-level shift in the growth rate
$f\propto\Omega_{m}^{0.55}$ is hard to discern. It is also found in Farrow et
al. (2015) (e.g. their Figure 8) that the mock can capture similar clustering
trends as the GAMA data, when split into bins of redshift and stellar mass.
Notice that there are significant deviations at small scales
($r_{p}<1{\,h^{-1}\,\rm Mpc}$) in the shape of the projected correlation
functions, but these scales are not explored in this analysis.

##  4 RSD models

###  4.1 Quasilinear dispersion model

To describe the RSD in galaxy density field, one can extend Eq. 3 by including
the galaxy bias $b$:

$P^{s}_{g}(k,\mu)=b^{2}P^{r}(k)\,(1+\beta\mu^{2})^{2},$ (14)

where $\beta=f/b$ is often referred to as the distortion parameter. The above
formalism is valid for galaxy auto-correlation, but it is straightforward to
generalise to cross-correlation:

$P_{c}^{s}(k,\mu)=\frac{b_{\rm gal}^{2}}{b_{12}}(1+\beta_{\rm
gal}\mu^{2})(1+b_{12}\beta_{\rm gal}\mu^{2})P^{r}(k),$ (15)

where $b_{\rm gal}$ is the galaxy bias, and $b_{12}$ is the ratio between
galaxy and group bias:

$b_{12}\equiv b_{\rm gal}/b_{\rm grp}.$ (16)

The 2-point correlation function is the Fourier transform of the power
spectrum. It is convenient to express the correlation functions in terms of
Legendre polynomials $\mathcal{P}_{\ell}(\mu)$ with $\ell=0,2,4$ (Hamilton,
1992):

$\xi^{s}_{g}=\xi_{0}(r)\mathcal{P}_{0}(\mu)+\xi_{2}(r)\mathcal{P}_{2}(\mu)+\xi_{4}(r)\mathcal{P}_{4}(\mu).$
(17)

In this expression, the coefficients of the Legendre polynomials,
$\xi_{0}(r)$, $\xi_{2}(r)$, and $\xi_{4}(r)$ are referred to as the monopole,
quadrupole, and hexadecapole. In linear theory, only even modes are present up
to the forth order because of the RSD effect modifies the power spectrum by
the factor $(1+\beta\mu^{2})^{2}$. The specific form of these multipoles are
computed by Hamilton (1992) for auto-correlation, and Mohammad et al. (2016)
for cross-correlation. We summarise these formulae in Appendix A.

The FoG effect is accounted for by a convolution of the correlation function
with some distribution of the non-linear random peculiar velocity along the
line of sight (Peacock & Dodds, 1994). $N$-body simulations show that the
actual distribution is non-Gaussian (e.g. Sheth, 1996; Scoccimarro, 2004;
Cuesta-Lazaro et al., 2020). Thus, we adopt:

$D(\pi)=\frac{1}{\sqrt{2}\sigma_{12}}\exp\left(-\sqrt{2}H_{0}\pi/\sigma_{12}\right),$
(18)

where $\sigma_{12}$ is the pairwise velocity dispersion. In Fourier space,
this function takes the form of a Lorentzian function,
$\tilde{D}(k\mu)=[1+(k\mu\sigma_{12})^{2}/2]^{-1}$, which damps the high-$k$
modes of the anisotropic power spectrum.

At quasilinear scales, non-linearity may introduce systematic biases in the
inferred cosmological parameters (de la Torre & Guzzo, 2012). There are
multiple challenges in extending the model beyond linear regime. There, the
peculiar velocities can be large, and the formalism described breaks down at
linear order. Nonlinearities alter the small scale shape of the matter power
spectrum and correlate the density and velocity fluctuations. Accounting for
these effects requires higher order expansion in the Perturbation Theory and
the inclusion of the velocity spectrum, $P_{\theta\theta}(k)$, and the
density-velocity cross spectrum, $P_{\delta\theta}(k)$, e.g. the TNS model by
Taruya et al. (2010). Galaxy bias can also be nonlinear and stochastic on
small scales (Dekel & Lahav, 1999). Furthermore, the approximate velocity
dispersion in equation 18 fails to fit auto-correlation data on the smallest
scales. More elaborate velocity distributions are proposed by e.g., Reid &
White (2011); Zu & Weinberg (2013); Bianchi et al. (2015) based on
simulations.

One simple approach is the replacement of the linear power spectrum in the
linear Kaiser model (Eq. 14) by the non-linear power spectrum. This is
reasonable because the redshift space power spectrum should match that in real
space at $\mu=0$. Blake et al. (2011) showed that this combination is actually
among the best-performing RSD models when fitting down to $k_{\rm
max}=0.2\,h\,{\rm Mpc}^{-1}$ with fixed cosmology444Although it should be
noted that if the model could introduce bias to $\Omega_{m}$ if the cosmology
is not fixed, as shown in Parkinson et al. (2012).. For this model, we adopt
the non-linear power spectrum from Halofit (Smith et al., 2003; Takahashi et
al., 2012). In the nonlinear regime we should in principle allow for a scale-
dependent bias. But in practice it is a good approximation to assume that the
nonlinear galaxy and matter power spectra are in a constant ratio (see Camacho
et al. 2019). In the following analysis, we refer to this model as the
‘quasilinear dispersion’ (QD) model.

###  4.2 RSD in the halo model

The main deficiency of the QD model is that it does not address post-linear
couplings between density and velocity, which will modify the simple Kaiser
angular anisotropy. There is an extensive literature of attempts to improve
such modelling, based on various forms of perturbation theory. The model of
Taruya et al. (2010) is widely used, although more recent efforts have
concentrated on the Effective Field Theory approach. This adds additional
terms dictated by symmetry in a way that can also capture bias effects,
including non-linearity and non-locality (e.g. Carrasco et al. 2012; Senatore
2015; d’Amico et al. 2020). These results are impressive, but have the
limitation that they are presented in Fourier space and are not reliable
beyond $k\simeq 0.3\,h\,{\rm Mpc}^{-1}$. For a robust prediction of
correlation functions, we need a formalism that still behaves correctly in the
large-$k$ limit.

For this reason, we have developed a model that seeks to access the highly
nonlinear regime by using the halo model. In real space, this involves
correlations that count pairs of galaxies in the same halo or in different
haloes:

$\xi(r)=\xi_{1h}(r)+\xi_{2h}(r).$ (19)

The 1-halo term is determined by the form of the halo density profile, and the
2-halo term is close to a linearly biased version of the matter two-point
function. The bias in turn is determined by the halo occupation number,
$N(M)$, of galaxies in haloes as a function of their mass. This halo model has
proved a highly effective way to understand the relation between the
clustering of galaxies and of mass (Seljak, 2000; Peacock & Smith, 2000;
Cooray & Sheth, 2002), and for the case of dark matter alone has led to the
highly precise HALOFIT framework (Smith et al., 2003; Takahashi et al., 2012).

The halo-model separation into two independent pair contributions must also
apply for the redshift-space correlations, namely

$\xi(r_{p},\pi)=\xi_{1h}(r_{p},\pi)+\xi_{2h}(r_{p},\pi),$ (20)

but it should be clear from the outset that the 1-halo and 2-halo
contributions would be expected to have rather different anisotropy signals.
The characteristic quadrupole plus hexadecapole Kaiser distortion arises from
the coherent component of the velocity field, and this will apply to the
2-halo term only, since pairs from within the same halo are unaffected by bulk
motion of the halo. This redshift-space decomposition using the halo model was
advocated by Hand et al. (2017), who invested much effort in trying to predict
the two distinct components using perturbation theory. Our work bears some
resemblance to their approach, with two distinct differences: we work directly
in configuration space, and we base the 1-halo term on empirical simulation
results, rather than attempting to calculate it a priori.

A particular point to clarify in this decomposition is the treatment of
Fingers of God. Random motions within a halo are treated in the dispersion
model by a radial convolution – but in fact the appropriate convolution will
be different for the 1-halo and 2-halo terms. The main reason for this is that
the 1-halo and 2-halo terms weight contributions as a function of halo mass
differently, with a higher weight given to high-mass haloes in the 1-halo term
(see e.g. equations 8 & 10 of Seljak 2000). Since the pairwise dispersion
$\sigma_{12}$ increases with halo mass, we expect larger FoG effects to apply
to the 1-halo term. This is further complicated by the existence of central
and satellite galaxies, since the weighting of these is different in the
1-halo and 2-halo terms. For example, suppose each halo contains either just a
single central or one central and one satellite, where the velocity dispersion
of satellites is $\sigma$. The 1-halo contribution must pair a central with a
satellite, so the pairwise dispersion is $\sigma$. But the 2-halo term can
also pair centrals with centrals (assumed to have negligible pairwise
dispersion – although Reid et al. 2014 showed that the actual pairwise
velocity could be up to $30\%$ of $\sigma$) and satellites with satellites
(pairwise dispersion $\smash{\sqrt{2}\sigma)}$, so the average rms pairwise
dispersion depends on the fraction of haloes that contain a satellite. If most
haloes are central-only (as in BOSS CMASS, for example), the pairwise
dispersion for the 2-halo term will be $\ll\sigma$. In the opposite direction,
one can argue that the velocity field of haloes will contain some stochastic
component in addition to the coherent velocities that generate the Kaiser
distortion.

With this perspective, an improved simple model for the cross-power between
tracers $a$ and $b$ would be as follows:

$\displaystyle{P_{ab}(k,\mu)=}$ | $\displaystyle{{}P_{1h}(k)\,D_{1}(k\mu)}$
---|---
| $\displaystyle{{}+b_{a}b_{b}P_{\rm
lin}(k)\,(1+\beta_{a}\mu^{2})(1+\beta_{b}\mu^{2})\,D_{2}(k\mu).}$
(21)

Leaving aside the 1-halo term for the moment, one way in which we can seek to
improve this expression further is in terms of quasilinear effects on the
2-halo term. A first requirement is that the real-space spectrum (at $\mu=0$)
should have the full nonlinear form. When discussing the dispersion model, we
achieved this by replacing $P_{\rm lin}$ by the nonlinear spectrum. In the
halo model, we should not do this, since the 2-halo term in real space is
close to linear theory, and the 1-halo term supplies most of the nonlinear
corrections (Smith et al., 2003). We do however adopt the HALOFIT 2-halo term,
with scale-independent bias, as the best model for the real-space 2-halo term.

The next step is to seek improvement in the density-velocity coupling that
leads to the Kaiser distortion factors. An attractive approach here is the
streaming model (e.g. Fisher 1995; Vlah et al. 2016), in which we consider the
quasilinear relative velocity distribution as a function of pair separation,
and use this to transform to redshift space while exactly conserving pair
counts. The details of the construction of this model are given in Appendix B.
As with the linear model for the 2-halo term, there are three main free
parameters, the tracer biases and the growth rate: $(b_{a},b_{b},f)$. This
assumes that the mass power spectrum is known exactly, whereas it depends on
all fundamental $\Lambda$CDM parameters. The main variation of the power is
$\propto\sigma_{8}^{2}$, so it is common to factor out this degree of freedom
and take the main RSD parameters to be
$(b_{a}\sigma_{8},b_{b}\sigma_{8},f\sigma_{8})$. However, there is a weaker
further dependence on $\sigma_{8}$ when we adopt the HALOFIT prediction of the
2-halo matter power spectrum, rather than taking this to be pure linear theory
(although the difference is not important in practice).

In addition to these three main RSD parameters, we have parameters connected
to the FoG damping. As described earlier, it is conventional to model FoG
effects by radial convolution, taking the velocity PDF to be a Lorentzian and
using a velocity dispersion as the single free parameter. But in the present
context, it is important to be clear that the empirical evidence for the
Lorentzian form comes mainly from the 1-halo term. This is $D_{1}(k_{z})$ in
Eq. 21; but we want the effect on the 2-halo term, $D_{2}(k_{z})$. We have
argued that this will be characterised by a different dispersion, but in
addition there is no strong reason to assume it will have a Lorentzian form.
As a more general alternative, we considered a modified Lorentzian:

$D_{2}(k\mu)=\left(1+(k\mu\sigma_{12})^{2}/2\gamma\right)^{-\gamma}$ (22)

and experimented with different values of $\gamma$. But in practice the
results were rather insensitive to the choice of this parameter, so we
retained the Lorentzian $\gamma=1$. It is shown in e.g. Scoccimarro (2004);
Bianchi et al. (2015); Cuesta-Lazaro et al. (2020) that the shape of the PDF
is only relevant where the correlation function changes significantly over a
scale comparable to the width of the smoothing function. But the issue of the
exact form of the PDF for FoG corrections to the 2-halo term is a problem that
merits further study.

In summary, we therefore have two models with similar real-space correlations,
but different degrees of RSD: (1) QD: quasilinear dispersion model and (2) HS:
halo+streaming model. Both of these converge to the linear Kaiser model on
large scales, so what is of interest is the smallest scale to which their
predictions are reliable. We will assess these by comparison with mock data.

Figure 6: Illustrating the decomposition of the measured mock correlation data
(left panel) into a 2-halo fit (middle panel) and an empirical 1-halo term in
the form of the residual of the fit (right panel), for the particular case of
red-MM cross-correlation. The 2-halo term is computed using the streaming
model, and is matched to the data at radii $r>10{\,h^{-1}\,\rm Mpc}$, with the
additional criterion that $r_{p}>3{\,h^{-1}\,\rm Mpc}$.

####  4.2.1 The 1-halo term

The real-space 1-halo term can in principle be computed in the usual halo
model framework, given the occupation numbers for the tracers and the halo
radial profile. But there is also a case for taking an empirical approach,
given that the real-space correlations are in principle observable directly,
in a manner free of RSD effects, via the projected correlation function
$w_{p}(r_{p})$. One might for example model the real-space 1-halo term by a
power-law of free amplitude and slope, or via an NFW profile.

But whatever approach is taken in real space, there is then the question of
how the 1-halo term appears in redshift space. As described above, the
simplest approach is to assume that the transition to redshift space consists
of a radial convolution with a single FoG function. However, it is not hard to
see that this must be an oversimplification. The 1-halo term arises from
random orbital velocities within the halo, but the velocity dispersion is
unlikely to be constant. If for example we consider the case of isotropic
orbits, then the dispersion would need to fall to zero at the virial radius of
the halo, beyond which the density is assumed to vanish.

Here we address this concern directly by using the mocks. Given a hypothesis
for the 2-halo term, we can subtract the 2-halo prediction from the mock data
to obtain an empirical $\xi_{1h}(r_{p},\pi)$ that sums with the 2-halo term to
give exactly the mock data (specifically, we apply this approach to the
average of all the mocks). The 2-halo term can be deduced by fitting to the
mock data in a regime where we assume the 1-halo contribution to be
negligible. The exact cuts adopted in the process are not critical; in
practice, we chose to match to the data at radii $r>10{\,h^{-1}\,\rm Mpc}$,
with the additional criterion that $r_{p}>3{\,h^{-1}\,\rm Mpc}$. The operation
of this procedure is illustrated in Figure 6. The resulting residual 1-halo
term is clearly well localised near the origin, and indeed it can be seen that
the RSD effects in the 1-halo term are complicated, with the FoG effect being
largest at $r_{p}=0$, whereas the function appears more isotropic close to its
outer limit at $r_{p}\simeq 5{\,h^{-1}\,\rm Mpc}$. This interesting behaviour
is clearly worthy of being modelled in detail, but we shall not do that here.

We now have a decomposition of the redshift-space correlations that by
construction exactly matches the average of the mocks. However, each mock
realization will be different, as will be the real data, so can these
different datasets be fitted in this framework? The 2-halo term is already
parameterised, and these parameters can be varied for any given dataset. But
the 1-halo term must also have some variation. Our approach is to assume that
the mocks are sufficiently realistic that the effective 1-halo term in any
given case will be close to the mock average, and that the difference can be
captured by two nuisance parameters:

$\xi_{1h}(r_{p},\pi)\to\alpha\xi_{1h}(r_{p},\eta\,\pi).$ (23)

In other words, assume that we have roughly the right functional form, but
that the amplitude may be off (scale by $\alpha$), and that the FoG strength
may be off (stretch in the radial direction by $1/\eta$). Physically, the
amplitude parameter $\alpha$ can be related to the way in which galaxies
populate haloes: there may be different numbers of satellite galaxies in a
given halo compared to the mock, leading to a different small-scale clustering
signal. We have previously seen that an empirical rescaling of the 1-halo
amplitude can yield an accurate fit to correlation data in real space (Hang et
al. 2021). The $\eta$ parameter attempts to capture the velocity dispersion of
the galaxies in the halo: a smaller $\eta$ produces a larger FoG effect.
Again, this can be understood in terms of an uncertain halo occupation, which
can alter the mean mass of the haloes that contribute the 1-halo term.

As we show below, this approach is able to succeed in matching the individual
mock realizations, and so we see no reason not to apply the same model to the
real data. We emphasise that we do not need to assume that the mocks are
completely realistic, as long as they are qualitatively similar to reality.
The reliability of this approach can be judged by whether or not the fitted
values of $\alpha$ and $\eta$ are close to unity (as indeed turns out to be
the case). We only consider this minimal set of two empirical nuisance
parameters in the current analysis; for forthcoming large data sets of higher
statistical precision, more parameters may be required in order to make the
model acceptably accurate. Eventually, we will need to validate the model by
deriving 1-halo templates from a given set of mocks and showing that they can
fit data derived from mocks produced according to different assumptions. We
intend to pursue this high-precision robustness test in a future study.

###  4.3 Fitting methodology

####  4.3.1 Covariance matrix and likelihood inference

In the above discussion, we have not been explicit about exactly what it means
to fit the averaged mock data. In principle, one would like to have an
understanding of the errors on the data, so that the likelihood can be
computed as a figure of merit that is used to optimise the fit. For an
individual dataset, this can be done in the standard way by using an ensemble
of mocks to estimate the covariance matrix of the data, and then appealing the
to central limit theorem to compute the likelihood in the Gaussian
approximation. For fitting the stacked mocks, the appropriate covariance
matrix is less obvious, but in any case it is less important to have a
likelihood in that case, where the aim is simply to estimate a 1-halo
contribution as a basis for further modelling. We are not interested in
placing errors on the best-fitting parameters of the 2-halo term, for which a
likelihood would be required. In practice, therefore, we took the simple
approach of seeking a least-squares fit in $\ln(1+\xi)$ to the mock average.
The exact figure of merit chosen is unimportant as regards the 1-halo
residual.

The covariance matrix for a single data realization is most often estimated in
one of two ways: either directly via the scatter over a number of mock
realizations, or via Jackknife resampling of a single realization. Both of
these approaches have their limitations, but the best strategy is when they
are combined: an expanded set of mock realizations is created by Jackknife
resampling of each one, yielding an improved estimate of the covariance matrix
(Alam et al., 2021b). For a data vector with component $x_{i}$ and a model
vector $y_{i}$, where $i=1,...,m$, the $\chi^{2}$ is defined as

$\chi^{2}=\sum_{i,j}^{m}\,[x_{i}-y_{i}]C_{ij}^{-1}[x_{j}-y_{j}].$ (24)

In the above equation, $C_{ij}$ is the covariance matrix, estimated from $N$
independent realizations of mock data:

$\hat{C}_{ij}=\frac{1}{N-1}\sum_{k=1}^{N}\,[x^{k}_{i}-\langle
x^{k}_{i}\rangle][x^{k}_{j}-\langle x^{k}_{j}\rangle].$ (25)

Given a model with $p$ parameters, there are $m-p$ degrees of freedom in the
$\chi^{2}$-fitting. Due to the small number of mocks, we apply Jackknife re-
sampling on the mocks by dividing each survey field into 18 sub-regions,
giving a total of $N_{J}=54$ samples for each mock. The covariance matrix for
an individual mock sample is estimated using equation 25, with an extra factor
$(N_{J}-1)$ to account for correlations between the Jackknife samples. We
average over the covariance matrices of the 25 mocks to obtain the final
covariance matrix. It is pointed out in Escoffier et al. (2016) that this
method can reduce the noise on the covariance estimation, and fast approach
the truth. However, we caution that these mocks are not completely
independent, because they are constructed from the same N-body simulation
(Gonzalez-Perez et al., 2014).

The posterior of the model parameters $\theta$ given data $D$ is estimated in
a Bayesian way:

$P(\theta|D)=\frac{P(D|\theta)P(\theta)}{P(D)},$ (26)

where $P(\theta)$ is the prior and $P(D)$ is treated as a normalization. The
term $P(D|\theta)$ is proportional the likelihood $\mathcal{L}$, which we
assume to be Gaussian:

$\mathcal{L}\propto\exp(-\chi^{2}/2).$

We use Monte-Carlo Markov Chain (MCMC) sampling of the parameter space,
implementing the python package emcee555http://dfm.io/emcee/.

####  4.3.2 Data compression

Instead of fitting the whole 2D correlation function, which requires an
$N(r_{p})\times N(\pi)$ – dimensional covariance matrix, we compress the 2D
information into the multipoles defined as

$\xi_{\ell}(r)=\frac{2\ell+1}{2}\int_{-1}^{1}\xi_{c}(r,\mu)\,\mathcal{P}_{\ell}(\mu)\,d\mu;\;\;\;\ell=0,2,4.$
(27)

We ignore higher order multipoles because they are typically noisy and more
sensitive to non-linearity. Multipoles are computed by interpolating the 2D
correlation function, and this is done consistently for both the measurements
and the models. In the QD model, we exclude $\xi_{4}$ from the fitting,
because the non-zero signal at scales $r\geq 10{\,h^{-1}\,\rm Mpc}$ cannot be
well reproduced by this model.

We also considered adding the projected correlation function $w_{p}$:

$w_{p}(r_{p})=\int_{-\pi_{\rm max}}^{\pi_{\rm max}}\xi(r_{p},\pi)\,d\pi.$ (28)

This has the merit that it is in principle independent of RSD for large enough
$\pi_{\rm max}$, and a knowledge of the true real-space clustering should be
advantageous if we are focusing on redshift-space effects that cause
deviations from this. However, we found in practice that it was not possible
to choose a large enough $\pi_{\rm max}$ to achieve results that converged to
the true real-space clustering without the results being too noisy to be
useful. The $w_{p}$ statistic may be useful at small separations,
$r<10{\,h^{-1}\,\rm Mpc}$, as a means of probing the real-space 1-halo term,
but as discussed above we do not need to do this in the present work. We
included $w_{p}$ in the fitting for the QD model; however, due to the limited
information it could provide in addition to the multipoles, the $w_{p}$ was
excluded in the HS model fitting.

For each of the six cross-correlation configurations, we fit the measurement
simultaneously with its corresponding galaxy auto-correlation. This allows us
to break the degeneracy between the galaxy and group bias.

####  4.3.3 Scale cuts

Below quasilinear scales $r\sim 10{\,h^{-1}\,\rm Mpc}$, both models may fail
to capture the full non-linear features. Fitting data points at these scales
may introduce significant bias into the measured growth rate. Therefore, we
test the models on a set of minimum fitting scales, $r_{\rm
min}=2,5,10,15,20{\,h^{-1}\,\rm Mpc}$ using the mock catalogues, and adopt the
most appropriate cut for each subsample. For the HS model, because the model
is designed to be able to fit smaller scales, we only test the model at
$r_{\rm min}=2,5,10{\,h^{-1}\,\rm Mpc}$.

####  4.3.4 Integral constraints

To account for the missing power for modes larger than the GAMA survey scale,
we include the integral constraint $I$, which is a small constant added to the
2D correlation function. The expected integral constraint is given by:

$I\equiv\frac{b_{1}b_{2}}{3}\sigma_{\rm
eff}^{2}=\frac{b_{1}b_{2}}{3}\int\Delta^{2}(k)\,W^{2}(k,r=r_{\rm eff})\,d\ln
k,$ (29)

where $\Delta^{2}(k)$ is the dimensionless linear matter power spectrum,
$b_{1}$ and $b_{2}$ are the tracer biases,
$W(k,r)=3[\sin(kr)/(kr)^{3}-\cos(kr)/(kr)^{2}]$, and $r_{\rm eff}$ is the
effective radius of the survey volume in one of the GAMA fields, $V=4\pi
r_{\rm eff}^{3}/3$. The factor $1/3$ accounts for the fact that we combine
measurements over three GAMA fields. Eq. 29 gives $I=0.0017b_{1}b_{2}$. This
can also be measured in the mock data directly, by comparing the projected
correlation function $w_{p}$ at large scales between the average of 25 samples
and the combination of all mock samples. The measured values are consistent
with the expectation given the statistical errors. In the QD model, we allowed
$I$ to be a free parameter, and found that it has little impact on other
parameters, with a posterior consistent with zero (e.g. see Table 4 and 5).
The integral constraints are then fixed to the measured values from mock for
the streaming model, as shown in Table 6 and 7.

####  4.3.5 Priors

The parameters used in the two models and their uniform prior ranges can be
found in Table 2. For each of the group-galaxy subsample, the galaxy auto-
correlation is fitted simultaneously with the cross-correlation. Parameters
with subscript ‘$a$’ are used for auto-correlations and ‘$c$’ for cross-
correlations. There is another cosmological parameter that should be
considered: the normalisation of the (linear) matter power spectrum
$\sigma_{8}$. From Eq. 14, it is clear that on linear scales, $\sigma_{8}$ and
$b$ are completely degenerate, hence RSD measurements are usually quoted in
the combination $f\sigma_{8}$. At large $k$, the shape of the non-linear power
spectrum is actually sensitive to $\sigma_{8}$. However, such dependence is
weak for the scales probed here, and we fix $\sigma_{8}=0.81$ throughout the
analysis.

Table 2: Range of the uniform priors of the RSD fitting parameters. For growth rate, the usual constraint from RSD is $f\sigma_{8}$, but we fix $\sigma_{8}=0.81$ in this analysis. The $\alpha$, and $\eta$ parameters are the 1-halo parameters applied in the GSM model only. Parameter | Prior (QD) | Prior (HS)
---|---|---
$b_{\rm gal}$ | $[0.1,2.5]$ | $[0.5,2]$
$b_{12}$ | $[0.1,2.5]$ | $[0.25,3]$
$f$ | $[0,2]$ | $[0,2]$
$\sigma_{a}$ [km s-1] | $[143,1140]$ | $[30,800]$
$\sigma_{c}$ [km s-1] | $[143,1140]$ | $[30,800]$
$I_{a}$ | $[0,0.1]$ | fixed
$I_{c}$ | $[0,0.1]$ | fixed
$\alpha_{a}$ | - | $[0.1,2]$
$\alpha_{c}$ | - | $[0.1,2]$
$\eta_{a}$ | - | $[0.5,2.5]$
$\eta_{c}$ | - | $[0.5,2.5]$

##  5 Results

###  5.1 Mocks

Figure 7: The means and scatter of the best-fit growth rate $f$ from 25 mocks
as a function of the minimum fitting scale, $r_{\rm min}$, for the quasilinear
dispersion (QD: top) and the halo streaming (HS: bottom) models. Data points
at each $r_{\rm min}$ are displaced by $0.1{\,h^{-1}\,\rm Mpc}$ for clarity.
The grey band marks the $10\%$ regions around the mock fiducial value
$f=0.593$ at $z=0.195$. Note that the error bars are for a single survey, so
that the errors on the mean of the mocks are 5 times smaller than shown.

We fit both models to each of the 25 mock samples, and compute the mean and
scatter of the best-fit parameters. The aim is to assess the scale at which an
unbiased growth rate can be recovered. The result is shown in Fig. 7 for the
set of $r_{\rm min}$ as mentioned in previous sections, and for each of the
six configurations. The fiducial value of $f$ with $\pm 10\%$ range is marked
by the grey band in each panel. The error bar is comparable to, but should not
be taken directly as the expected error size on the GAMA sample. The specific
values of all model parameters are summarised in Tab. 4 and Tab. 6.

Notice that in the case of halo streaming model, there is a caveat that the
1-halo templates are obtained from the average of the same set of mocks as
they are tested on. Ideally, we would like to have access to multiple sets of
simulations covering different cosmology and HOD prescription, with matched
survey configurations as GAMA. Then, we would test the halo streaming model on
by extracting the 1-halo templates from one set of simulations and apply it to
the measurements from the others. In this way, we can assess whether the model
is robust against bias due to a different cosmology or change of the
simulation settings. Such test will be particularly relevant for the forth-
coming large data sets, where the demand of the precision of the model is
high. However, this is beyond the scope of this paper given the noise level of
the GAMA data. We would like to defer such detailed comparison to a future
study.

The top panel of Fig. 7 shows the recovered growth rate $f$ using the QD model
(Section 4.1). As expected, when the small scales are included ($r_{\rm
min}\leq 5{\,h^{-1}\,\rm Mpc}$), the fitted growth rates are significantly
biased in all configurations, while at larger scales ($r_{\rm min}\geq
15{\,h^{-1}\,\rm Mpc}$), they converge to the fiducial value. The overall
growth rate seems to be under-estimated by about $5-10\%$ for most scale cuts,
but this is much smaller compared to the statistical error of the GAMA sample.
It is noticeable that the blue configurations are less biased down to smaller
scales, with $f$ recovered to within 10% at $r_{\rm min}=5-10{\,h^{-1}\,\rm
Mpc}$, compared to the red configurations which are only unbiased at $r_{\rm
min}=15-20{\,h^{-1}\,\rm Mpc}$. This may be due to the smaller FoG effect in
the blue configurations compared to the red. From this test, we choose to
adopt $r_{\rm min}=\\{10,10,10,15,20,20\\}{\,h^{-1}\,\rm Mpc}$ for the LMblue,
MMblue, HMblue, LMred, MMred, and HMred subsamples respectively for the QD
model in the application to the GAMA data. The bottom panel of Fig. 7 shows
the the recovered growth rate $f$ using the HS model, where the results are
impressively consistent. The growth rates for the different subsets are
consistent to within an rms of 3% in the mock average results, and the global
average of these different subsets is within 2% of the fiducial value. This
successful performance holds down to even $r_{\rm min}=2{\,h^{-1}\,\rm Mpc}$,
although the estimated errors at that point are little different to those at
$r_{\rm min}=5{\,h^{-1}\,\rm Mpc}$, so we conservatively adopt the larger
figure in our HS analysis.

Fig. 8 shows the linear and streaming model with the mean best-fit parameters
from the mock subsamples, at the respective $r_{\rm min}$ as mentioned above.
The mock average measurement as well as the $1\sigma$ error on the mean is
also shown. In addition, we also show the corresponding 2-halo term of the
streaming model in dotted black lines. On large scales ($r\geq
15{\,h^{-1}\,\rm Mpc}$), all of the model curves converge, and match well with
the mock average. It is noticeable that the full streaming model (with the
addition of the 1-halo template) and its 2-halo term do not coincide exactly
on these scales: the extracted 1-halo template still has some residuals in the
monopole and quadrupole. The largest difference is seen in the hexadecapole.
The slightly positive values seem to be produced only by the 1-halo FoG, which
both the linear and the 2-halo terms of the streaming model fail to capture.
Looking at smaller scales ($r\leq 10{\,h^{-1}\,\rm Mpc}$), it seems that the
QD model under-predicts the power in the red configurations, and over-predicts
that in the blue configurations.

Figure 8: The multipoles of the group-galaxy cross-correlation functions in
the mock average (open diamond). The coloured bands show the scatter on the
mean. The best-fit QD models are shown in dashed black line, with $r_{\rm
min}=\\{10,10,10,15,20,20\\}{\,h^{-1}\,\rm Mpc}$ for the LMblue, MMblue,
HMblue, LMred, MMred, and HMred subsamples respectively. The best-fit Gaussian
streaming models with a 1-halo template are shown in solid black lines, with a
fixed $r_{\rm min}=5{\,h^{-1}\,\rm Mpc}$ for all sub-samples, with the
corresponding 2-halo term shown in dotted black lines. For the presentation
purpose, the multipoles have been multiplied by $r^{1.5}$.

###  5.2 GAMA

Figure 9: Same as 8 but for the actual GAMA data, with the same $r_{\rm min}$
adopted. The coloured bands show Jack-knife errors. The $\chi^{2}$ for each of
the models can be found in Table 5 and 7. Figure 10: The MCMC parameter
constraints for the actual GAMA data using the QD model (open symbols) and the
streaming model with a 1-halo template (filled symbols). Each model is fitted
with the respective $r_{\rm min}$ as in Fig. 8. The filled purple points show
the constraints obtained using the 1-halo template from the ‘contaminated’ red
sample in the mock data. The 1-halo parameters are marginalised over in the HS
model. On the top panel, we have converted the constraints back to
$f\sigma_{8}$ by multiplying back the fiducial $\sigma_{8}(z=0.195)$. The
black line and the grey band on the top panel mark $f\sigma_{8}=0.47\pm 0.01$,
the fiducial growth rate at $z=0.195$ using $\Omega_{m}$ and $\sigma_{8}$
constraints from Planck Collaboration et al. (2020). The specific values for
all model parameters are shown in Table 5 and 7.

Fig. 9 shows the measured GAMA multipoles (filled circles), the best-fit QD
models (black dashed lines), and the HS models (black solid lines). In
addition, the corresponding HS model 2-halo term is shown in the dotted black
lines. The same scale cuts, $r_{\rm min}$, are adopted as in the mock case for
each of the models. The $\chi^{2}$ and parameter values are shown in Table 5
and 7. The full HS model 2D models are contrasted with the GAMA data in
Appendix C.

We see that the QD model provides a reasonable fit to the monopole and
quadrupole at given $r_{\rm min}$ in most configurations. The only exception
is the LMred and LMblue subsamples, where the monopole power is boosted at
large scales and the quadrupole power is consistent with zero. Despite the
visual discrepancy, the $\chi^{2}$ of these models are consistent with the
degrees of freedom of the data. The HS model is able to capture the shape of
the multipoles down to smaller scales, especially the hexadecapole at scales
$r>5{\,h^{-1}\,\rm Mpc}$. At smaller scales ($r<5{\,h^{-1}\,\rm Mpc}$),
although excluded from fitting, the mock 1-halo template continues to provide
a reasonable fit to the red configurations. But this is not the case for the
blue configurations, where the non-linear velocity dispersion seems to be
stronger in the actual data compared to the mock catalogues. One possible
explanation could be the impact of redshift measuring errors. These are not
included in the mocks, and so any measured velocity dispersion in the real
data only will include the redshift error in quadrature. The typical GAMA
error is 50 km -1, but in detail Liske et al. (2015) showed that redshift
errors can depend on spectral and target properties. The redshift error for
galaxies classified as the ‘absorption’ type (i.e. the spectrum is dominated
by absorption features) is $101$ km s-1, compared to the ‘emission’ type,
which is $33$ km s-1. But red galaxies have a larger measured velocity
dispersion, so the impact of redshift errors on the total measured dispersion
will be less in that case.

Fig. 10 shows the the mean and $1\sigma$ error on the model parameters from
the MCMC posterior for the GAMA data, fitted at respective $r_{\rm min}$. The
open and filled symbols denote parameter constraints from the QD model and the
HS model respectively. In the latter case, we also show the constraints
measured using the 1-halo template from the ‘contaminated’ red galaxy sample
(purple filled symbols). All sets of constraints show good consistency. Notice
that the size of the error bar in the blue configurations is similar in both
models, although the QD model has a scale cut at $10{\,h^{-1}\,\rm Mpc}$ while
the HS model at $5{\,h^{-1}\,\rm Mpc}$. This is the consequence of the
additional nuisance parameters added in the latter model. The specific
parameter values, including the 1-halo parameters in the HS model, can be
found in Table 5 and 7. In Fig. 15-18, we further show the full posteriors
from MCMC for all parameters in both models, grouped by the red and blue
configurations. In the HS model, the 1-halo parameters $\alpha$ and $\eta$
have no primary degeneracy with the growth rate, although the growth rate can
be shifted slightly through their small degeneracy with the velocity
dispersion parameters. In practice, one would always marginalise over the
1-halo parameters.

The middle two panels show the measured group and galaxy biases in both
models. The LM, MM, and HM group biases measured consistently between the red
and blue configurations in both models. For our selection of galaxies, we find
that $b_{\rm gal}\approx 1$ for the blue galaxies, and $b_{\rm gal}\approx
1.3$ for the red galaxies; for the groups, we find that $b_{\rm grp}\approx
1.2,1.4,1.8$ for the low, medium and high mass ranges. The $b_{\rm grp}$
measurements are in qualitative agreement with that in Riggs et al. (2021) on
large scales (e.g. see their Fig. 8), although direct comparison is non-
trivial due to difference in the group selection. The consistency between the
two models is good in general, although for the blue configuration, the HS
model gives systematically lower biases compared to the QD model by
$0.5\sigma-1.5\sigma$. The lower two panels show the measured auto- and cross-
correlation velocity dispersion, $\sigma_{a}$ and $\sigma_{c}$. The two models
measure consistent velocity dispersion, despite slightly different form for
the FoG term. Notice that for the red configuration in the QD model, because
of the large scale cut, the velocity dispersion posterior is prior driven.
There is a tentative trend (at $\sim 2\sigma$) that the red configurations
have larger velocity dispersion compared to the blue configurations, with
$\sigma_{a,c}\sim 400-500\,{\rm kms}^{-1}$ for the red configurations, and
$\sigma_{a,c}\sim 300\,{\rm kms}^{-1}$ for the blue configurations in both
auto- and cross-correlations. There is, however, no clear dependence on the
group mass. These measured galaxy biases and velocity dispersion are in good
agreement with other measurements from GAMA (e.g. Blake et al., 2013; Loveday
et al., 2018). The marginalised posterior for $f$, $b_{\rm gal}$, and $b_{12}$
for the six configurations is shown in Fig. 11 and 12 for the QD and HS models
respectively.

The top panel shows the measured growth rate, consistent across the six
subsamples for both models. Here, we have presented the results in the more
general form of $f\sigma_{8}(z)$. The rationale for this is that our modelling
assumes that the background cosmology (WMAP7 parameters) is known exactly.
This is not precisely true, and the observed distortion parameter,
$\beta=f/b$, is actually $\propto f\sigma_{8}$ (since $b\sigma_{8}$ is
observable). We therefore multiply our fitted $f$ by the fiducial
$\sigma_{8}(z)$ in order to obtain a combination that should be insensitive to
the exact fiducial model.

We also note that RSD analyses commonly also allow for the Alcock-Paczynski
effect (Alcock & Paczynski, 1979; Ballinger et al., 1996), which introduces
additional distortions of the 2D correlation function from distance
measurements using a ‘wrong’ cosmology. This degree of freedom can boost the
errors on $f\sigma_{8}$ substantially if the cosmological model is left free.
But the AP effect is unimportant if the model is constrained by precise
external CMB data as here. A further reason that this is reasonable is that
the interest in RSD comes from the desire to test gravity: the CMB data give a
precise prediction of $f\sigma_{8}$ and we want to know if this is what we
measure.

In detail, then, we take $\sigma_{8}(z)=g(z)\sigma_{8}(0)=0.73$, where
$z=0.20$, $\sigma_{8}(0)=0.81$, and $g(z)$ is the time-dependence of the
(linear) density fluctuation in linear theory, normalised to $g(0)=1$. The
measurements give a mean of $f\sigma_{8}=0.27$ with uncertainties ranging from
$0.07-0.20$ for the QD model, and $f\sigma_{8}=0.29$ with uncertainties
ranging from $0.07-0.14$ for the HS model. We combine our measurements from
the six cross-correlation configurations for the HS model, accounting for
their correlations using the mock catalogues. We compute the scatter on the
average as well as the covariance of the best-fit $f$ for the six
configurations in the 25 mock samples. Ideally, one would like to use the the
full posterior. However, this would require the time consuming step of running
MCMC for each of the mock sample, thus the simple average of the maximum
likelihood values is adopted. Our combined constraint from the HS model thus
gives

$f\sigma_{8}(z=0.20)=0.29\pm 0.10.$ (30)

The corresponding figure for the QD model is $0.27\pm 0.14$, showing the extra
information gained through the smaller scales that the HS model is able to
probe. We note that the limited number of mock samples means that our
covariance matrices will be imprecise, so that the errors on the growth rate
for an individual sample may be underestimated (Hartlap et al., 2007;
Sellentin & Heavens, 2016). However, the empirical dispersion in the mean of
the maximum-likelihood values should be robust.

The striking thing about this GAMA-based figure is that it is rather low
compared to the fiducial Planck figure of $f\sigma_{8}(z=0.20)=0.47\pm 0.01$,
derived from the Planck TT, TE, EE+lowE+lensing cosmological parameters
(Planck Collaboration et al., 2020): our figure is $1.8\sigma$ below this
Planck value. This discrepancy is certainly unexpected given how well our
modelling was able to account for the RSD signal in the different mock
realizations, and how the recovered growth rates were consistent between
different methods of model fitting. Furthermore, the figures recovered from
the different GAMA subsamples show the same level of consistency with each
other as is seen in subsamples within the mocks.

There are a number of things that can be said about the low observed figure.
The first is that there is some evidence that the fiducial Planck figure may
be too high, with local gravitational lensing data consistently arguing for a
reduction of about 10% (see e.g. Hang et al. 2021). Our measurement would then
be in $1.3\sigma$ disagreement with a revised fiducial value of 0.42, implying
that GAMA is an unusual dataset, but not unreasonably so. And we do have
evidence that this is the case: inspection of Fig. 3 shows that $N(z)$ has a
substantial dip at $z\simeq 0.24$, which is seen consistently in all three
fields. One might suspect a problem with the redshift pipeline, but this
feature is absent in a subsequent fourth GAMA field, not used here; the three
main GAMA fields are simply rather unusual regions of space. Finally, note
that a multitracer analysis of RSD in GAMA by Blake et al. (2013) gave
$f\sigma_{8}(z=0.18)=0.36\pm 0.09$, which is also slightly lower than Planck,
albeit not inconsistently so.

Figure 11: Marginalised MCMC posteriors for the QD model. Figure 12:
Marginalised MCMC posteriors for the HS model. The best-fit bias parameters
are very different among different sub-samples, but the recovered $f$ values
are consistent.

###  5.3 Group bias

Table 3: Bias for the groups in the LM, MM, and HM stellar mass bins for the mock average and GAMA. The first column shows the mean bias value computed from the fitting formula Tinker et al. (2010), and the second column shows the corresponding bias computed at the mean halo mass in each case. The next two columns marked with a ‘*’ show the bias computed in the same way, but with the GAMA group halo mass computed from a more up-to-date mass-luminosity relation (Rana et al., 2022). The rest of the columns show the fitted biases from the six cross-correlation configurations in the mocks and the GAMA data. Group bias |  | T10 | T10 $b(\bar{M}_{h})$ | T10∗ | T10∗ $b(\bar{M}_{h})$ | QD-red | QD-blue | HS-red | HS-blue
---|---|---|---|---|---|---|---|---|---
Mocks | LM | $1.02$ | $0.92$ | - | - | $1.20\pm 0.04$ | $1.20\pm 0.02$ | $1.18\pm 0.03$ | $1.20\pm 0.02$
| MM | $1.26$ | $1.13$ | - | - | $1.48\pm 0.05$ | $1.46\pm 0.02$ | $1.42\pm 0.02$ | $1.38\pm 0.02$
| HM | $1.83$ | $1.65$ | - | - | $1.90\pm 0.06$ | $2.09\pm 0.03$ | $1.96\pm 0.03$ | $1.92\pm 0.03$
GAMA | LM | $1.00$ | $0.96$ | 1.12 | 1.11 | $1.07\pm 0.24$ | $1.27\pm 0.10$ | $1.14\pm 0.11$ | $1.24\pm 0.08$
| MM | $1.20$ | $1.17$ | 1.41 | 1.39 | $1.58\pm 0.30$ | $1.52\pm 0.10$ | $1.41\pm 0.10$ | $1.34\pm 0.07$
| HM | $1.52$ | $1.49$ | 1.85 | 1.81 | $1.71\pm 0.46$ | $1.98\pm 0.19$ | $1.81\pm 0.20$ | $1.79\pm 0.14$

Finally, it is interesting to ask if the group biases that we measure are in
accord with what is expected for haloes of these masses. We compute the
expected group bias in GAMA from the calibrated halo mass for the groups based
on Eq. 6. We adopt the Tinker et al. (2010) fitting formula for the linear
halo bias. The halo bias is expressed in terms of the peak height parameter
$\nu\equiv\delta_{c}/\sigma_{R}$, where $\delta_{c}\approx 1.686$, and
$\sigma_{R}$ is the rms of the linear power spectrum filtered with a spherical
top hat function with radius $R$ (cf. Eq. 29). This is related to the halo
mass via $M_{h}=200\bar{\rho}_{m}4\pi R^{3}/3$, where $\bar{\rho}_{m}$ is the
mean background density. The mean group bias in a stellar mass range is
estimated by

$\hat{b}_{\rm grp}=\frac{\sum_{i}N(\log M^{i}_{h})\,b(\log
M^{i}_{h})}{\sum_{i}N(\log M^{i}_{h})},$ (31)

where $N(\log M^{i}_{h})$ is the number of groups in the logarithmic halo mass
bin $i$. The halo mass adopted in case of mock and GAMA are as shown in Fig.
2. For GAMA, we also include the uncertainty in the calibrated halo mass due
to the uncertainties in $\log M_{p}$ and $\alpha$ in Eq. 6. We combine the
error via:

$\Delta_{\log M_{h}}=\Delta_{\log M_{p}}+|\Delta_{\alpha}\log(L_{\rm
grp}/L_{0})|.$ (32)

For the mass range concerned here, $\Delta_{\log M_{h}}=0.8-0.2$ dex from low
mass to high mass. We account for this scatter by convolving the number of
objects with a Gaussian distribution with width $\Delta_{\log M_{h}}$ in each
$\log M_{h}$ bin. The predicted and measured group biases using mocks and GAMA
data are summarised in Table 3. We also show biases computed at the
logarithmic mean halo mass $\bar{M}_{h}$. These values are close to the
average bias computed from Eq. 31 in the case of GAMA, but they deviate from
the mock average significantly. This indicates that estimating the bias from
the mean halo mass depends heavily on the distribution of the halo mass of the
sample considered. In addition, we include the case where the more up-to-date
mass-luminosity relation from Rana et al. (2022) is used to compute the group
halo mass. The equivalent parameters to Eq. 6 are $M_{p}=(8.1\pm 0.4)\times
10^{13}h^{-1}M_{\odot}$, $L_{0}=10^{11.5}h^{-2}L_{\odot}$, and $\alpha=1.01\pm
0.07$. The halo mass computed with this calibration is larger than using the
fiducial Han et al. (2015), resulting in good consistency with the fitted
group bias from both the QD and HS models as shown in Table 3.

In the mock catalogues, the predicted group bias for the LM, MM, and HM
subsamples are all lower than the fitted values. These differences are
significant given the error bar on the average of the mock measurements. In
all cases, the group bias has apparently been under-estimated by about $20\%$.
This deviation in the group bias may arise because the arithmetic mean host
halo mass of the group members is used as a proxy for the group halo mass.
However, if one uses the total mass of unique host haloes in the group as
$M_{h}$, then the bias in each mass range only increases by $\sim 10\%$. Since
the mock group masses are calibrated using ‘real’ simulation halo masses, the
difference illustrates clearly that our galaxy groups are not in 1-to-1
correspondence with single haloes, emphasising once again the importance of
analysing real and mock data with the same group finder.

##  6 Summary and Conclusions

In this work, we have investigated the Redshift-Space Distortions (RSD) of
group-galaxy cross-correlations, with the aim of understanding the robustness
with which measurements of the density fluctuation growth rate can be
extracted from such measurements. We have focused on the differences in the
measured RSD using different types of galaxy and group, and developed new
methods for fitting such data down to the small-scale nonlinear regime.

We have used data from the GAMA survey in the redshift range $0.1<z<0.3$ to
measure the 2D cross-correlation function $\xi(r_{p},\pi)$ between groups and
galaxies. The groups were found using a FoF algorithm from Treyer et al.
(2018), and were subdivided into three stellar mass bins (LM: 40%, MM: 50%,
and HM: 10%). The corresponding halo mass for the groups was calibrated using
the relation in Han et al. (2015), and the groups are expected to have typical
masses of $(10^{12.2},10^{12.7},10^{13.2})\,M_{\odot}$. For Rana et al.
(2022), the mean halo masses are $(10^{12.6},10^{13.1},10^{13.5})\,M_{\odot}$.
The galaxies were split into red and blue subsets using a cut in the $g-i$ vs
$z$ plane, yielding in total six cross-correlation configurations: LMred,
MMred, HMred, LMblue, MMblue, and HMblue.

We have used 25 GAMA lightcone mocks from Farrow et al. (2015) to test RSD
models and to construct Jackknife covariance matrices for likelihood fitting.
Mock group catalogues were generated using the identical algorithm that was
applied to the real GAMA data. The mock catalogues are distinct from
observation in several aspects: the mean redshift distribution, the bimodal
$g-i$ colour distribution, and the total stellar mass of the groups. We
discuss the appropriate empirical selection that yields the best match between
the mocks and the data subsamples. The measured 2D correlation functions show
good consistency between the data and the mocks down to small scales, and the
same variation of the signals with galaxy colours and group masses are
observed. The different cross-correlation results yield group biases that
increase with mass, as expected. For GAMA, the predicted group bias from
Tinker et al. (2010) is lower but consistent with the fitted values using the
halo mass calibration in Han et al. (2015), whereas that from Rana et al.
(2022) agrees well with the fitted values. For mocks, however, these values
tend to be higher than predicted. This difference illustrates that the groups
found in redshift space do not constitute a pure halo sample.

We have compared these measurements with two RSD models: (1) a Quasilinear
Dispersion (QD) model; (2) a novel Halo Streaming (HS) model. The QD model is
a generalization of the linear dispersion model of Mohammad et al. (2016) to
use the non-linear real-space power spectrum. We found from testing on the
mocks that this model provides unbiased measurements of the growth rate at
$r_{\rm min}=10-20{\,h^{-1}\,\rm Mpc}$ depending on the subsample. The HD
model uses a halo model decomposition of the correlations, where a streaming
model 2-halo term is combined with an empirical 1-halo template adopted from
the mock average. This promising model, with the addition of two nuisance
parameters, allows unbiased results on the growth rate down to $r_{\rm
min}=5{\,h^{-1}\,\rm Mpc}$ when fitting individual mock realizations, and for
all group-galaxy combinations. For the GAMA measurements, an MCMC analysis was
used to obtain the posterior of our model parameters. We found that given the
scale cuts, all of the subsamples recover consistent growth rates in both
models. The average growth rate from the six subsamples using the HS model is
$f\sigma_{8}=0.29\pm 0.10$ at $z=0.20$, where the error should be robust as it
is taken directly from the dispersion in maximum-likelihood values for the
mock data. This figure is $1.8\sigma$ lower than the fiducial Planck value of
$f\sigma_{8}=0.47\pm 0.01$, and we have considered the implications of this
result. At face value, the low GAMA result is consistent with the suggestions
from gravitational lensing that the true value of $f\sigma_{8}$ may be about
10% lower than the Planck central figure (e.g. Hang et al. 2021. But there are
objective reasons to believe that the GAMA dataset may be a statistical
outlier, based on known anomalies in the redshift distribution in the GAMA
fields.

Therefore, the real test of the RSD modelling presented here will be when it
can be applied to much larger and more precise datasets, such as the Bright
Galaxy Sample from the Dark Energy Spectroscopic Instrument (DESI) survey
(Martini et al., 2018) and the the Wide Area VISTA Extra-Galactic Survey
(WAVES) (Driver et al., 2016). We are greatly encouraged by the success of our
halo streaming model in reproducing mock cross-correlations down to the
smallest scales, and in yielding consistent values of $f\sigma_{8}$ from
different tracers, to a tolerance of better than 3%. This hybrid approach,
taking advantage of ever more realistic mock data, therefore seems an
attractive way of obtaining robust constraints on the growth of cosmological
density fluctuations, and we look forward to seeing it applied to next-
generation surveys.

## Acknowledgements

QH was supported by Edinburgh Global Research Scholarship and the Higgs
Scholarship from Edinburgh University. JAP and SA were supported by the
European Research Council under the COSFORM grant no. 670193. YC acknowledges
the support of the Royal Society through a University Research Fellowship and
an Enhancement Award. YC thanks the hospitality of the Astrophysics and
Theoretical Physics groups of the Department of Physics at the Norwegian
University of Science and Technology during his visit. MB is supported by the
Polish National Science Center through grants no. 2020/38/E/ST9/00395,
2018/30/E/ST9/00698, 2018/31/G/ST9/03388 and 2020/39/B/ST9/03494, and by the
Polish Ministry of Science and Higher Education through grant DIR/WK/2018/12.

GAMA is a joint European-Australasian project based around a spectroscopic
campaign using the Anglo-Australian Telescope. The GAMA input catalogue is
based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared
Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained
by a number of independent survey programmes including GALEX MIS, VST KiDS,
VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio
coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and
the participating institutions. The GAMA website is http://www.gama-
survey.org/.

## Data Availability

All of the GAMA data are publicly available on http://www.gama-
survey.org/dr3/schema/. The mock catalogues and the GAMA group catalogue are
available upon reasonable request.

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##  Appendix A The linear model for redshift-space cross-correlations

The redshift-space 2-point correlation function can be expanded in terms of
Legendre polynomials $\mathcal{P}_{\ell}(\mu)$ with $\ell=0,2,4$ (Hamilton,
1992):

$\xi^{s}_{g}=\xi_{0}(r)\mathcal{P}_{0}(\mu)+\xi_{2}(r)\mathcal{P}_{2}(\mu)+\xi_{4}(r)\mathcal{P}_{4}(\mu).$
(33)

The multipoles are given by

$\displaystyle\xi_{0}(r)$
$\displaystyle=\Big{(}1+\frac{2}{3}\beta+\frac{1}{5}\beta^{2}\Big{)}\,\xi_{g}(r),$
(34) $\displaystyle\xi_{2}(r)$
$\displaystyle=\Big{(}\frac{4}{3}\beta+\frac{4}{7}\beta^{2}\Big{)}\,[\xi_{g}(r)-\bar{\xi}_{g}(r)],$
(35) $\displaystyle\xi_{4}(r)$
$\displaystyle=\frac{8}{35}\,\beta^{2}\,\Big{[}\xi_{g}(r)+\frac{5}{2}\,\bar{\xi}_{g}(r)-\frac{7}{2}\,\bar{\bar{\xi}}_{g}(r)\Big{]},$
(36)

where,

$\displaystyle\xi(r)$ $\displaystyle=\int_{0}^{\infty}d\ln
k\,\Delta^{2}(k)\,\frac{\sin(kr)}{kr},$ (37) $\displaystyle\overline{\xi}(r)$
$\displaystyle=(3/r^{3})\int_{0}^{r}\xi(x)\,x^{2}\,dx$ (38)
$\displaystyle=\int{3\Delta^{2}(k)\over(kr)^{3}}\,d\ln k\,(\sin kr-kr\,\cos
kr)\;;$ (39) $\displaystyle\overline{\overline{\xi}}(r)$
$\displaystyle=(5/r^{5})\int_{0}^{r}\xi(x)\,x^{4}\,dx$ (40)
$\displaystyle=\int{5\Delta^{2}(k)\over(kr)^{5}}\,d\ln
k\,(kr(6-k^{2}r^{2})\cos kr-(6-3k^{2}r^{2})\sin kr),$ (41)

and the subscript ‘$g$’ indicates an extra factor of $b^{2}$. For practical
purposes it is useful to have an explicit direct expression:

$\displaystyle{\xi_{g}^{s}(r,\mu)}$ | $\displaystyle{{}=\xi_{g}+\left({2\beta\over 3}+{\beta^{2}\over 2}\right)\,\overline{\xi}_{g}-{3\beta^{2}\over 10}\,\overline{\overline{\xi}}_{g}}$
---|---
|
$\displaystyle{{}+\left(2\beta(\xi_{g}-\overline{\xi}_{g})+3\beta^{2}(\overline{\overline{\xi}}_{g}-\overline{\xi}_{g})\right)\,\mu^{2}}$
|
$\displaystyle{{}+\beta^{2}\left(\xi_{g}+5\overline{\xi}_{g}/2-7\overline{\overline{\xi}}_{g}/2\right)\,\mu^{4}.}$
(43)

Since this expression is linear in the different contributions to $P$, we
immediately see that Hamilton’s expression can be generalised by the
replacements $2\beta\to(\beta_{1}+\beta_{2})$ and
$\beta^{2}\to\beta_{1}\beta_{2}$. In the case of two tracers with biases
$b_{\rm gal}$ and $b_{\rm grp}=b_{\rm gal}/b_{12}$, equations 34 – 36 are
therefore modified to (Mohammad et al., 2016)

$\displaystyle\xi_{0,c}(r)$ $\displaystyle=\Big{(}1+\frac{1}{3}\beta_{\rm
gal}(1+b_{12})+\frac{1}{5}\beta^{2}_{\rm gal}b_{12}\Big{)}\xi_{c}(r),$ (44)
$\displaystyle\xi_{2,c}(r)$ $\displaystyle=\Big{(}\frac{2}{3}\beta_{\rm
gal}(1+b_{12})+\frac{4}{7}\beta^{2}_{\rm
gal}b_{12}\Big{)}[\xi_{c}(r)-\bar{\xi}_{c}(r)],$ (45)
$\displaystyle\xi_{4,c}(r)$ $\displaystyle=\frac{8}{35}\beta^{2}_{\rm
gal}b_{12}\Big{[}\xi_{c}(r)+\frac{5}{2}\bar{\xi}_{c}(r)-\frac{7}{2}\bar{\bar{\xi}}_{c}(r)\Big{]},$
(46)

where $\beta_{\rm gal}=f/b_{\rm gal}$, and $\beta_{\rm grp}=fb_{12}/b_{\rm
gal}$. The various moments with a subscript ‘$c$’ are defined as in equations
37 – 41, but with a factor $b^{2}_{\rm gal}/b_{12}$.

##  Appendix B The streaming model for redshift-space cross-correlations

Redshift-space correlations are frequently discussed using the streaming model
(e.g. Fisher 1995), in which the total observed pair counts are related to the
true counts in real space by a scale-dependent shift and smoothing:

$1+\xi_{s}(\sigma,x)=\int\left(1+\xi_{r}(\sigma,y)\right)\,{\cal P}(x-y\mid
y)\,dy,$ (47)

where $\sigma$ is the transverse separation and $x$ & $y$ are the radial
separations in respectively redshift space and real space. For the correct
true choice of the distribution function $\cal P$, this equation should be
exact by definition as it just expresses conservation of pairs. The same
equation should apply to both the auto- and cross-correlation case.

Commonly, the distribution function is assumed to be a Gaussian:

${\cal P}(x-y\mid
y)={1\over\sqrt{2\pi}\,\sigma(y)}\,\exp\left(-{(x-y-u(y))^{2}\over
2\sigma^{2}(y)}\right),$ (48)

where $u(y)$ is the line of sight comoving offset caused by the mean pairwise
streaming velocity: $u(y)=\mu v_{12}(r)/aH$, where $\mu=y/r$ and
$r^{2}=\sigma^{2}+y^{2}$. Although we do not need to assume that the
correlations are small, it is convenient to adopt the standard linear-theory
prescription for the mean pairwise velocity that results from a given
correlation, in which the mean proper relative infall velocity of a pair of
biased tracers at comoving separation $r$ is

${v_{12}(r)\over aHr}=-{1\over
3}\,f(b_{1}+b_{2})\,\overline{\xi}_{m}(r)=-{1\over
3}\,(\beta_{1}+\beta_{2})\,\overline{\xi}_{x}(r),$ (49)

where the cross-correlation for the tracers is related to the autocorrelation
of the mass by $\overline{\xi}_{\times}=b_{1}b_{2}\overline{\xi}_{m}$. The
appearance of only $\beta$ factors without any explicit bias parameter then
makes it plausible that direct consideration of pairwise peculiar velocities
will yield the correct Kaiser–Hamilton form in the linear $\xi\ll 1$ limit,
although proving this is not straightforward (see Fisher 1995).

To achieve the correct linear limit, it is necessary to include not only the
mean streaming, but also the scale-dependent dispersion in pairwise velocity
that must arise in a Gaussian velocity field. Note that the velocity field
smooths $1+\xi$, and any effects on $\xi$ itself from the smoothing will
introduce corrections beyond linear order in $\xi$. Therefore, almost
paradoxically, the linear corrections to $\xi$ come from smoothing the unity
term in $1+\xi$: this gives a non-unity result because the smoothing is not
exactly a convolution. Therefore we have

$1+\xi_{s}(\sigma,x)=\xi_{r}(\sigma,x)+\int{\cal P}(x-y\mid y)\,dy.$ (50)

Incidentally, this shows that there is an error in Fisher’s final equation
(26), which omits the critical unity term.

To evaluate the integral for the linear correction term, one can Taylor expand
the integrand. But this needs to be done with care, since the argument of the
exponential will not always be small. We have to consider the effects of the
scale-dependent streaming, $u(y)$, and the scale-dependent squared dispersion,
$V(y)\equiv\sigma^{2}(y)$. But both of these quantities scale in proportion to
$\xi$, so to linear order we will never need to consider terms in which they
are multiplied. Therefore, the two effects can be considered in turn.
Variations in $u$ so the leading effect is to scale $z$ by a factor
$1+u^{\prime}$ (where $u^{\prime}\equiv du/dy$), lead to a correction to $\xi$
of

$\Delta\xi=-u^{\prime},$ (51)

where $u^{\prime}\equiv du/dy$. This is just a Jacobian correction.

Allowing for variation in the dispersion is more complicated. We need to
expand ${\cal P}$ up to order $z^{4}$, where $z\equiv y-x+u(x)$, and use the
Gaussian result that $\langle z^{4}\rangle=3\langle z^{2}\rangle^{2}$. This
then yields

$\Delta\xi=V^{\prime\prime}/2,$ (52)

in agreement with Fisher’s equation (24). This term is critical, as it
generates the hexadecapole distortion.

The pairwise dispersion

Implementing the streaming model thus requires two steps: (1) allowing for the
mean infall as a function of scale; (2) smoothing by a scale-dependent
pairwise dispersion, in order to achieve the linear Kaiser limit correctly.
The second step sounds like the FoG convolution, but that would be a separate
correction that allows for a random ‘dither’ in the large-scale velocity
field. The smoothing here is present purely through the statistics of the
linear velocity field. For the streaming model, we require the line of sight
pairwise velocity dispersion:

$\sigma^{2}=\langle(v_{z}^{b}-v_{z}^{a})^{2}\rangle-\langle(v_{z}^{b}-v_{z}^{a})\rangle^{2}.$
(53)

Recall that the averages here are density weighted, otherwise the mean
relative velocity would vanish. But the result of this is that
$\langle(v_{z}^{b}-v_{z}^{a})\rangle$ is of order the linear correlation
function, so $\langle(v_{z}^{b}-v_{z}^{a})\rangle^{2}$ is of second order in
$\xi_{\rm lin}$. If we keep only the lowest order terms, then

$\sigma^{2}=2\langle v_{z}^{2}\rangle-2\langle v_{z}^{a}v_{z}^{b}\rangle$ (54)

(and the effects of density weighting are neglected as also being of higher
order). These terms can be evaluated using the theory of correlated velocity
fields as presented in Gorski (1988):

$\sigma^{2}=2\Psi_{\perp}(0)-2\Psi_{\perp}(r)-2[\Psi_{\parallel}(r)-\Psi_{\perp}(r)]\,\mu^{2},$
(55)

where $\mu=r_{\parallel}/r$ and

$\Psi_{\parallel,\perp}(r)=a^{2}H(z)^{2}f^{2}(z)\int\Delta^{2}(k,z)\,k^{-2}\,d\ln
k\,K(kr),$ (56)

with the kernels being respectively

$\displaystyle{K_{\perp}(x)}$ | $\displaystyle{{}=[\sin(x)-x\cos(x)]/x^{3};}$
---|---
$\displaystyle{K_{\parallel}(x)}$ | $\displaystyle{{}=\sin(x)/x-2[\sin(x)-x\cos(x)]/x^{3},}$
(57)

both of which tend to 1/3 as $x\to 0$.

A hybrid streaming + halo model

In real space, it is common to decompose the matter power spectrum as

$P(k)=P_{2h}(k)+P_{1h}(k),$ (58)

Where the 2-halo term is close to linear theory and the 1-halo term is a
discreteness term that reflects the finite number density of haloes – of the
form of shot noise but filtered on small scales to reflect the extended nature
of haloes. Hang et al. (2021) proposed an effective model for decomposing the
auto-power of biased tracers:

$P(k)=b_{2h}^{2}P_{2h}(k)+b_{1h}^{2}P_{1h}(k),$ (59)

where $P_{2h}$ and $P_{1h}$ are provided by HALOFIT. In the same spirit, then,
one might propose a model for cross-correlations:

$\xi_{12}=b_{1}b_{2}\xi_{2h}+b_{1h}^{2}\xi_{1h}.$ (60)

It is not clear in advance that the 1-halo term is appropriate here, since
$\xi_{1h}$ is defined for autocorrelation and averaging over all halo masses,
so the appropriate shape of the 1-halo correction for cross-correlation and
with halo mass bins may not be universal. But this simple 1-parameter form for
the 1-halo term seems worth investigating as a first approximation. More
generally, we can add an empirical 1-halo term, $\xi_{12,1h}$, to the large-
scale linear power, which could be described via a number of nuisance
parameters: the 1-parameter $b_{1h}$ prescription is the simplest, but one
might have two parameters describing the slope and amplitude of a power law
etc.

To convert to redshift space, we first have to allow for large-scale peculiar
velocities affecting halo positions, so that $b_{1}b_{2}\xi_{2h}$ is replaced
by the quasilinear $\xi_{12,s}$ discussed above:

$\xi_{12,s}=\xi_{12,{\rm stream}}(r_{\perp},r_{\parallel})+\xi_{12,1h}(r)$
(61)

The advantage of using the streaming model over linear theory is that the
function includes non-trivial quasilinear corrections even when the linear
coupling between density and velocity is assumed. We can go further than this
and add an assumed quasilinear correction to the pairwise infall itself, by
making the reasonable assumption that the peculiar velocities will
approximately cancel the Hubble flow if the clustering is sufficiently
nonlinear. This motivates the form

${v_{12}(r)\over
aHr}=-{(\beta_{1}+\beta_{2})\,\overline{\xi}_{\times}(r)/3\over
1+(\beta_{1}+\beta_{2})\,\overline{\xi}_{\times}(r)/3}.$ (62)

Finally, we need to allow for nonlinear peculiar velocities that convolve the
galaxy distribution in the radial direction, so that we convolve the
correlation in $r_{\parallel}$ by some FoG damping factor, $D(\Delta
r_{\parallel})$:

$\xi_{12,s}\to\xi_{12,s}*D(\Delta r_{\parallel}).$ (63)

The FoG term is generally taken to be an exponential,

$D(\Delta r_{\parallel})=(\sqrt{2}\sigma_{\rm FoG})^{-1}\exp(-\sqrt{2}|\Delta
r_{\parallel}|/\sigma_{\rm FoG}),$ (64)

corresponding to a Lorentzian filtering of the power spectrum:

${\tilde{D}}(k_{\parallel})=\left(1+(k_{\parallel}\sigma_{\rm
FoG})^{2}/2\right)^{-1}.$ (65)

As we argue in the body of the paper, it is necessary to use a different
convolution for the 1-halo and 2-halo terms. But in any case we have not
derived a full model for the 1-halo term in redshift space, and in the present
paper we take this empirically from simulations, supplementing the above
2-halo term described via the streaming model.

##  Appendix C Fits to individual datasets

In this Appendix, we present the full 2D correlation function model fits to
the GAMA data, as obtained using the HS (halo streaming) model. Fig. 13 shows
the results for red galaxies and Fig. 14 shows the results for blue galaxies.

Figure 13: False colour images of auto- and cross-correlation functions in
redshift space for the actual red-galaxy GAMA data and the corresponding model
derived according to the H (halo streaming) approach. $r_{p}$ denotes
transverse separation; $\pi$ is radial separation. LM, MM, and HM denote the
three group mass bins. Figure 14: False colour images of auto- and cross-
correlation functions in redshift space for the actual blue-galaxy GAMA data
and the corresponding model derived according to the H (halo streaming)
approach. $r_{p}$ denotes transverse separation; $\pi$ is radial separation.
LM, MM, and HM denote the three group mass bins.

##  Appendix D Summary of model parameters

We summarise the mean and standard deviation of the model parameters for the
quasilinear dispersion (QD) model (Tab. 4 and 5) and the halo streaming (HS)
model (Tab. 6 and 7) for the auto-correlation only and the six cross-
correlation configurations. Tab. 4 and 6 show the mock results, where the mean
and scatter from the best-fit of each mock sample is presented. Tab. 5 and 7
show the GAMA data results, where the mean and scatter from MCMC is presented.

Table 4: Mock measurements using the quasilinear dispersion (QD) model. The values are at smallest $r_{\rm min}$ which gives $f$ below $10\%$ bias compared to the fiducial value. $\sigma_{8}$ has been fixed to $0.81$. | All | LMred | LMblue | MMred | MMblue | HMred | HMblue
---|---|---|---|---|---|---|---
$r_{\rm min}$ | 10 | 15 | 10 | 20 | 10 | 20 | 10
$\bar{\chi}^{2}/{\rm dof}$ | 61.9/101 | 44.7/75 | 62.7/101 | 29.8/49 | 62.4/101 | 30.1/49 | 62.8/101
$f$ | 0.55 $\pm$ 0.19 | 0.56 $\pm$ 0.26 | 0.57 $\pm$ 0.16 | 0.58 $\pm$ 0.36 | 0.55 $\pm$ 0.19 | 0.63 $\pm$ 0.37 | 0.54 $\pm$ 0.18
$b_{\rm gal}$ | $1.22\pm 0.08$ | $1.47\pm 0.14$ | $0.89\pm 0.06$ | $1.44\pm 0.20$ | $0.90\pm 0.07$ | $1.41\pm 0.21$ | $0.90\pm 0.06$
$b_{12}$ | $0.86\pm 0.02$ | $1.26\pm 0.17$ | $0.74\pm 0.05$ | $0.98\pm 0.10$ | $0.62\pm 0.03$ | $0.74\pm 0.09$ | $0.43\pm 0.03$
$\sigma_{a}$ | $487\pm 106$ | $496\pm 237$ | $392\pm 107$ | $550\pm 340$ | $379\pm 130$ | $601\pm 417$ | $378\pm 117$
$\sigma_{c}$ | $473\pm 109$ | $489\pm 179$ | $437\pm 116$ | $533\pm 356$ | $410\pm 120$ | $556\pm 357$ | $350\pm 129$
$I_{a}$ | $0.019\pm 0.012$ | $0.028\pm 0.020$ | $0.008\pm 0.007$ | $0.029\pm 0.023$ | $0.008\pm 0.006$ | $0.023\pm 0.022$ | $0.009\pm 0.006$
$I_{c}$ | $0.021\pm 0.014$ | $0.026\pm 0.031$ | $0.014\pm 0.020$ | $0.029\pm 0.026$ | $0.015\pm 0.013$ | $0.028\pm 0.029$ | $0.023\pm 0.016$
$b_{\rm grp}$ | $1.42\pm 0.09$ | $1.20\pm 0.22$ | $1.20\pm 0.11$ | $1.48\pm 0.23$ | $1.46\pm 0.09$ | $1.90\pm 0.30$ | $2.09\pm 0.17$
Table 5: Same as Table 4, but for the GAMA measurements. The error bars come from MCMC posterior. | All | LMred | LMblue | MMred | MMblue | HMred | HMblue
---|---|---|---|---|---|---|---
$r_{\rm min}$ | 10 | 15 | 10 | 20 | 10 | 20 | 10
$\bar{\chi}^{2}/{\rm dof}$ | 76/101 | 39/75 | 78/101 | 21/49 | 79/101 | 36/49 | 76/101
$f$ | 0.33 $\pm$ 0.15 | 0.37 $\pm$ 0.26 | 0.35 $\pm$ 0.10 | 0.37 $\pm$ 0.25 | 0.33 $\pm$ 0.10 | 0.43 $\pm$ 0.28 | 0.35 $\pm$ 0.11
$b_{\rm gal}$ | $1.32\pm 0.08$ | $1.19\pm 0.16$ | $1.03\pm 0.05$ | $1.31\pm 0.19$ | $1.06\pm 0.05$ | $1.32\pm 0.19$ | $1.04\pm 0.06$
$b_{12}$ | $0.87\pm 0.02$ | $1.11\pm 0.20$ | $0.81\pm 0.05$ | $0.83\pm 0.11$ | $0.70\pm 0.03$ | $0.77\pm 0.18$ | $0.53\pm 0.04$
$\sigma_{a}$ | $378\pm 95$ | $670\pm 250$ | $268\pm 85$ | $631\pm 283$ | $258\pm 78$ | $595\pm 276$ | $273\pm 83$
$\sigma_{c}$ | $363\pm 99$ | $684\pm 276$ | $298\pm 90$ | $622\pm 252$ | $297\pm 72$ | $636\pm 263$ | $278\pm 81$
$I_{a}$ | $0.015\pm 0.005$ | $0.014\pm 0.011$ | $0.005\pm 0.004$ | $0.011\pm 0.009$ | $0.003\pm 0.003$ | $0.011+0.009$ | $0.003\pm 0.003$
$I_{c}$ | $0.006\pm 0.005$ | $0.010\pm 0.009$ | $0.007\pm 0.006$ | $0.036\pm 0.018$ | $0.026\pm 0.025$ | $0.038\pm 0.023$ | $0.037\pm 0.016$
$b_{\rm grp}$ | $1.52\pm 0.10$ | $1.07\pm 0.24$ | $1.27\pm 0.10$ | $1.58\pm 0.30$ | $1.52\pm 0.10$ | $1.71\pm 0.46$ | $1.98\pm 0.19$
Table 6: Mock measurements using the halo streaming (HS) model. $\sigma_{8}$ has been fixed to $0.81$. The error bars come from the standard deviation of best-fit values over the 25 mock samples. | LMred | LMblue | MMred | MMblue | HMred | HMblue
---|---|---|---|---|---|---
$r_{\rm min}$ | 5 | 5 | 5 | 5 | 5 | 5
$\bar{\chi}^{2}/{\rm dof}$ |  $77\pm 15$/118 |  $77\pm 11$/118 |  $81\pm 12$/118 |  $78\pm 11$/118 |  $81\pm 15$/118 |  $76\pm 11$/118
$f$ | $0.61\pm 0.17$ | $0.60\pm 0.13$ | $0.57\pm 0.19$ | $0.62\pm 0.16$ | $0.59\pm 0.16$ | $0.62\pm 0.15$
$b_{\rm gal}$ | $1.38\pm 0.08$ | $0.87\pm 0.04$ | $1.38\pm 0.07$ | $0.87\pm 0.05$ | $1.37\pm 0.07$ | $0.87\pm 0.05$
$b_{12}$ | $1.17\pm 0.09$ | $0.73\pm 0.06$ | $0.97\pm 0.04$ | $0.63\pm 0.04$ | $0.70\pm 0.06$ | $0.46\pm 0.04$
$\sigma_{a}$ | $482\pm 76$ | $358\pm 67$ | $459\pm 95$ | $369\pm 84$ | $467\pm 71$ | $371\pm 66$
$\sigma_{c}$ | $524\pm 87$ | $458\pm 77$ | $484\pm 99$ | $417\pm 94$ | $469\pm 81$ | $383\pm 85$
$\alpha_{a}$ | $0.99\pm 0.08$ | $1.00\pm 0.18$ | $0.99\pm 0.11$ | $1.04\pm 0.22$ | $0.98\pm 0.11$ | $1.02\pm 0.23$
$\alpha_{c}$ | $0.96\pm 0.09$ | $0.95\pm 0.19$ | $0.96\pm 0.10$ | $1.00\pm 0.13$ | $1.01\pm 0.08$ | $1.02\pm 0.12$
$\eta_{a}$ | $1.02\pm 0.05$ | $1.01\pm 0.04$ | $1.00\pm 0.05$ | $1.00\pm 0.03$ | $1.01\pm 0.06$ | $0.99\pm 0.05$
$\eta_{c}$ | $1.04\pm 0.04$ | $1.02\pm 0.04$ | $1.02\pm 0.04$ | $1.00\pm 0.04$ | $0.99\pm 0.04$ | $0.98\pm 0.05$
$I_{a}$ | $0.0035$ | $0.0015$ | $0.0035$ | $0.0015$ | $0.0035$ | $0.0015$
$I_{c}$ | $0.0020$ | $0.0018$ | $0.0032$ | $0.0018$ | $0.0036$ | $0.0019$
$b_{\rm grp}$ | $1.18\pm 0.13$ | $1.20\pm 0.12$ | $1.42\pm 0.09$ | $1.38\pm 0.09$ | $1.96\pm 0.17$ | $1.92\pm 0.16$
Table 7: Same as Tab. 6 but for the GAMA data. The error bars come from MCMC posterior. | red | blue | LMred | LMblue | MMred | MMblue | HMred | HMblue
---|---|---|---|---|---|---|---|---
$r_{\rm min}$ | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5
$\bar{\chi}^{2}/{\rm dof}$ |  $40$/59 |  $41$/59 |  $68$/118 |  $88$/118 |  $80$/118 |  $89$/118 |  $85$/118 |  $84$/118
$f$ | $0.49\pm 0.24$ | $0.39\pm 0.11$ | $0.40\pm 0.20$ | $0.33\pm 0.08$ | $0.41\pm 0.20$ | $0.39\pm 0.11$ | $0.45\pm 0.22$ | $0.37\pm 0.11$
$b_{\rm gal}$ | $1.34\pm 0.09$ | $1.00\pm 0.04$ | $1.26\pm 0.08$ | $1.01\pm 0.04$ | $1.40\pm 0.08$ | $1.02\pm 0.04$ | $1.36\pm 0.09$ | $1.02\pm 0.04$
$b_{12}$ | - | - | $1.10\pm 0.08$ | $0.81\pm 0.04$ | $0.99\pm 0.04$ | $0.76\pm 0.03$ | $0.75\pm 0.07$ | $0.57\pm 0.04$
$\sigma_{a}$ | $454\pm 130$ | $242\pm 73$ | $350\pm 109$ | $197\pm 60$ | $400\pm 101$ | $270\pm 77$ | $459\pm 121$ | $247\pm 80$
$\sigma_{c}$ | - | - | $391\pm 122$ | $325\pm 67$ | $440\pm 99$ | $281\pm 69$ | $378\pm 145$ | $263\pm 83$
$\alpha_{a}$ | $1.07\pm 0.14$ | $0.82\pm 0.26$ | $0.95\pm 0.11$ | $0.70\pm 0.18$ | $0.89\pm 0.09$ | $0.83\pm 0.23$ | $1.05\pm 0.14$ | $0.78\pm 0.28$
$\alpha_{c}$ | - | - | $0.88\pm 0.20$ | $0.80\pm 0.29$ | $0.93\pm 0.14$ | $0.69\pm 0.15$ | $1.04\pm 0.22$ | $0.94\pm 0.18$
$\eta_{a}$ | $0.76\pm 0.14$ | $1.54\pm 0.21$ | $0.77\pm 0.12$ | $1.67\pm 0.21$ | $1.03\pm 0.14$ | $1.51\pm 0.20$ | $0.79\pm 0.14$ | $1.56\pm 0.24$
$\eta_{c}$ | - | - | $0.87\pm 0.16$ | $1.21\pm 0.21$ | $1.11\pm 0.13$ | $1.62\pm 0.19$ | $0.99\pm 0.13$ | $1.32\pm 0.17$
$I_{a}$ | $0.0035$ | $0.0015$ | $0.0035$ | $0.0015$ | $0.0035$ | $0.0015$ | $0.0035$ | $0.0015$
$I_{c}$ | - | - | $0.0020$ | $0.0018$ | $0.0032$ | $0.0018$ | $0.0036$ | $0.0019$
$b_{\rm grp}$ | - | - | $1.14\pm 0.11$ | $1.24\pm 0.08$ | $1.41\pm 0.10$ | $1.34\pm 0.07$ | $1.81\pm 0.20$ | $1.79\pm 0.14$
Table 8: Same as Tab. 7 but for the 1-halo templated adopted from the mock contaminated red samples. | LMred∗ | MMred∗ | HMred∗
---|---|---|---
$r_{\rm min}$ | 5 | 5 | 5
$\bar{\chi}^{2}/{\rm dof}$ |  $119$/118 |  $137$/118 |  $144$/118
$f$ | $0.52\pm 0.17$ | $0.48\pm 0.17$ | $0.42\pm 0.16$
$b_{\rm gal}$ | $1.24\pm 0.07$ | $1.35\pm 0.06$ | $1.33\pm 0.06$
$b_{12}$ | $1.17\pm 0.10$ | $0.95\pm 0.03$ | $0.74\pm 0.05$
$\sigma_{a}$ | $424\pm 85$ | $427\pm 76$ | $427\pm 93$
$\sigma_{c}$ | $454\pm 114$ | $492\pm 75$ | $362\pm 116$
$\alpha_{a}$ | $1.51\pm 0.16$ | $1.37\pm 0.10$ | $1.51\pm 0.16$
$\alpha_{c}$ | $1.16\pm 0.21$ | $1.14\pm 0.11$ | $1.22\pm 0.18$
$\eta_{a}$ | $0.81\pm 0.10$ | $1.03\pm 0.10$ | $0.90\pm 0.12$
$\eta_{c}$ | $0.86\pm 0.15$ | $1.12\pm 0.07$ | $1.04\pm 0.10$
$I_{a}$ | $0.0031$ | $0.0031$ | $0.0031$
$I_{c}$ | $0.0011$ | $0.0046$ | $0.0049$
$b_{\rm grp}$ | $1.05\pm 0.11$ | $1.42\pm 0.08$ | $1.78\pm 0.15$

##  Appendix E MCMC posteriors

We present the full parameter posterior for the QD and HS model applied to the
GAMA measurements for the six configurations in this section. Fig. 15 shows
the three red configurations in the QD model, and Fig. 16 shows the blue
configurations. Fig. 17 shows the three red configurations in the HS model,
and Fig. 18.

Figure 15: The posteriors for fitting the QD model to the red galaxy cross-
correlations. Figure 16: The posteriors for fitting the QD model to the blue
galaxy cross-correlations. Figure 17: The posteriors for fitting the HS model
to the red galaxy cross-correlations. Figure 18: The posteriors for fitting
the HS model to the blue galaxy cross-correlations.