# A New Stellar Mass Proxy for Subhalo Abundance Matching

Chen-Yu Chuang Institute of Astronomy, National Tsing Hua University, Hsinchu
30013, Taiwan  Institute of Astronomy and Astrophysics, Academia Sinica,
Taipei 10617, Taiwan  Yen-Ting Lin Institute of Astronomy and Astrophysics,
Academia Sinica, Taipei 10617, Taiwan

###### Abstract

Subhalo abundance matching (SHAM) has played an important role in improving
our understanding of how galaxies populate their host dark matter halos. In
essence, the SHAM framework is to find a dark matter halo property that best
correlates with an attribute of galaxies, such as stellar mass. The peak value
of the maximum circular velocity ($V_{\rm max}$) a halo/subhalo has ever
attained throughout its lifetime, $V_{\rm peak}$, has been a popular choice
for SHAM. A recent study by Tonnesen & Ostriker (2021) suggested that quantity
$\phi$, which combines the present-day $V_{\rm max}$ and the peak value of
halo dark matter mass, performs better in predicting stellar mass than $V_{\rm
peak}$. Inspired by their approach, in this work, we find that further
improvement can be achieved by a quantity $\psi_{5}$ that combines the 90th
percentile of $V_{\rm max}$ a halo/subhalo has ever achieved with the 60th
percentile of the dark matter halo time variation rate. Tests based on the
simulation TNG300 of the IllustrisTNG project show that our new SHAM scheme,
with just three free parameters, can improve the stellar mass prediction and
mass-dependent clustering by 15% and 18% from $\phi$, respectively, over the
redshift range $z=0-2$.

Galaxy formation (595) — Galaxy physics (612) — Galaxy dark matter halos
(1880)

††journal: AAS Journals

##  1 Introduction

Among the methods employed to study the galaxy–halo connection (in simplest
terms, how galaxies populate dark matter halos; see Wechsler & Tinker 2018 for
a recent review), subhalo abundance matching (SHAM; see e.g., Kravtsov et al.
2004; Tasitsiomi et al. 2004; Vale & Ostriker 2004; Conroy et al. 2006;
Stiskalek et al. 2021; to name only a few) is popular for its simplicity –
typically only involving a couple of free parameters, the model galaxies
produced by the method by construction match the observed galaxy stellar mass
(or luminosity) function, as well as the stellar (or luminosity) dependence of
the two-point correlation function (2PCF).

In the SHAM framework, one essentially is seeking a dark matter halo property
$X$ that is most tightly correlated with a galaxy property (typically the
stellar mass, $M_{\star}$). It has been a popular choice to set $X=V_{\rm
peak}$, the peak value of the maximum circular velocity ($V_{\rm max}$) a halo
(or subhalo) has ever attained. With the inclusion of some scatter
$\sigma_{V}$ between $V_{\rm peak}$ and $M_{\star}$, the model is shown to be
able to best reproduce the observed 2PCFs of galaxies selected above different
stellar mass thresholds, compared to models that use other properties as $X$,
such as peak (sub)halo mass throughout its lifetime ($M_{\rm DM,peak}$ e.g.,
Reddick et al. 2013; Zentner et al. 2014; but see Masaki et al. 2022).

Lehmann et al. (2017) were among the first to consider $X$ to be a combination
of multiple halo properties. Recently, Tonnesen & Ostriker (2021, hereafter
TO21) proposed a new property, $X=\phi\equiv V_{\rm max,0}/v_{\rm
<EMAIL_ADDRESS>DM,peak}/(10^{12.7}M_{\odot}$) (please refer to equation 7
below for the definition of these terms), which is shown to be an excellent
stellar mass proxy. Inspired by their approach, here we explore variants to
$\phi$, and propose a new property $\psi_{5}$ that further improves the
accuracy in stellar mass prediction by 15% with respect to that of $\phi$.

This paper is structured as follows. In Section 2 we describe the simulation
and the loss metrics used for our abundance matching (AM) scheme. Then in
Section 3 we present our methodology for finding the best stellar mass
indicator, which is referred to as $\psi_{5}$. Our results are presented in
Section 4, where we show the performance of $\psi_{5}$, in terms of
reproducing the stellar mass and color dependencies of galaxy clustering for
mock galaxies from TNG300 of the IllustrisTNG project (Marinacci et al., 2018;
Naiman et al., 2018; Nelson et al., 2018; Pillepich et al., 2018; Springel et
al., 2018; Nelson et al., 2019, hereafter TNG300) from $z=0-2$. We discuss and
summarize our results in Section 5 and Section 6, respectively.

##  2 Simulation and Methods

We test alternatives to $\phi$ using the TNG300 simulation suites, and
quantify the performance by the prediction of $M_{\star}$ and stellar
mass/color-dependent 2PCF. In this section, we first describe how we select
galaxies from the simulation, then explain how the 2PCF is measured, as well
as how the performance of the variants of $\phi$ (which shall be referred to
as $\psi$) is evaluated.

###  2.1 TNG300

We extract the dark matter and baryonic features from TNG300, which has a box
of $300\,\rm{Mpc}$ on a side and contains $(2500)^{3}$ dark matter particles
and gas particles. To select the well-resolved galaxies, we choose model
galaxies with stellar mass111From the SubhaloMassType field in the TNG300
catalog. $M_{\star}\geq 10^{9}\,h^{-1}M_{\odot}$ and the dark matter subhalos
with dark matter mass $M_{DM}\geq 10^{11}\,h^{-1}M_{\odot}$ in the dark matter
only (DMO) counterpart of TNG300, where h=0.6774. Then, we separate the
galaxies into satellites and centrals using the subhalo catalog of TNG300.

Following TO21, we abundance match the stellar mass for the _all_ , _central_
, and _satellite_ groups of galaxies, while a combined sample in which
_central_ , _satellite_ are fit separately is labeled as the _mixed_ group. To
reproduce the color dependence of the 2PCFs (see Section 2.2, we also separate
the subhalos into blue and red samples by the color of the galaxies they host:

$(u-g)_{\rm cut}=-0.031M_{g}-0.065z+0.695$ (1)

where the $u-g$ is the Sloan Digital Sky Survey (SDSS) $u-g$ color, $M_{g}$ is
the $g$-band absolute magnitude, and $z$ is the redshift (following Skibba et
al., 2014). We obtain the magnitudes of subhalos from the catalog published by
Nelson et al. (2018).

###  2.2 2PCF

A 2PCF, $\xi(r)$, is a measure of excess probability of finding a pair of
galaxies separated by a distance $r$ over a random distribution.
Observationally, to avoid contamination from peculiar velocities along the
line of sight (LOS), one typically integrates along the redshift dimension and
obtains the projected 2PCF, defined as

$w_{p}(r_{p})=\int^{\infty}_{-\infty}\xi(r_{p},\pi)d\pi\approx
2\times\int^{\pi_{max}}_{0}\xi(r_{p},\pi)d\pi$ (2)

where $r_{p}$ is the separation on the plane of the sky and $\pi$ is the
distance along the LOS. The upper limit of LOS integration, $\pi_{max}$, is
chosen to be $40\,$Mpc, which is consistent with the choice of actual
measurements done with real data (e.g., Lin et al., 2016, 2022).

In Sections 4.2 & 4.3, we compare the 2PCFs from the true galaxy position and
that from the abundance-matched galaxy position in different $M_{\star}$ and
color bins. We make use of a high-performance package, Corrfunc (Sinha &
Garrison, 2020, 2019), for the $w_{p}$ calculations.

###  2.3 Loss Measures

The primary objective of our study is to predict the $M_{\star}$ of each
subhalo accurately while reproducing $w_{p}$ in the simulation. To measure the
difference between the prediction and the ground truth, we define three
metrics. We use the scatter to gauge the $M_{\star}$ predictions:

$\sigma=\sqrt{\frac{1}{N}\sum^{N}_{i}(\Delta y_{i}-\overline{\Delta y})^{2}}$
(3)

where $N$ is the total number of galaxies, $\Delta
y_{i}=\log(M_{\star,i})-\log\widehat{M_{\star,i}}$ is the residual of single
$M_{\star}$ prediction and $\overline{\Delta y}$ is the average of the
residual. To compare our results with TO21, we also calculate the error,
defined as

${\rm{Error}}=\frac{\sum^{N}_{i}|\log(M_{\star,i}/\widehat{M_{\star,i}})|}{N}$
(4)

For $w_{p}$ predictions, we obtain the average of the fractional error for the
$r_{p}$ and $M_{\star}$ bins.

$L=\frac{1}{N_{M_{\star}}\times
N_{r_{p}}}\sum_{N_{M_{\star}}}\sum_{N_{r_{p}}}|\frac{w_{p}(r_{p},M_{\star})-\widehat{w_{p}}(r_{p},M_{\star})}{w_{p}(r_{p},M_{\star})}|$
(5)

where $N_{M_{\star}}$ and $N_{r_{p}}$ are the number of $M_{\star}$ and
$r_{p}$ bins, respectively. The uncertainty is measured by jackknife
resampling. We calculate the $w_{p}$ with each jackknife sample and obtain the
uncertainty via

$\sigma_{w}(w_{p})=\sqrt{\frac{N-1}{N}\sum^{N}_{j}(w_{p}-\widehat{w_{p,j}})^{2}}$
(6)

where $N=27$ is the number of jackknife samples, and $\widehat{w_{p,j}}$ is
the projected 2PCF associated with the given jackknife sample. Here we do not
include scatter in our $w_{p}$ estimation.

##  3 Rank Ordering Schemes

The simplest SHAM scheme is assuming a one-to-one correspondence between
galaxies and dark matter (sub)halos of the same number density, where the rank
order is through $M_{\star}$ for galaxies, and dark matter mass for
(sub)halos. However, the dark matter mass of subhalos usually suffers from
tidal stripping, and thus is not able to adequately reflect the rank order of
stellar mass. TO21 improved this by introducing two parameters, the $V_{\rm
max}$ at $z=0$ ($V_{\rm max,0}$) and $M_{\rm DM,peak}$. These properties are
shown to be less sensitive to tidal stripping than dark matter mass. By
combining these parameters into a single dimensionless parameter

$\phi=V_{\rm max,0}/V_{\rm<EMAIL_ADDRESS>DM,peak}/10^{12.95}\,\rm{M_{\odot}}$ (7)

(a _pivot mass_ of $M_{\rm DM,peak}=10^{12.95}\,M_{\odot}$ is introduced to
normalize $\phi$; $V_{\rm<EMAIL_ADDRESS>is $V_{\rm max,0}$ for a halo of the
pivot mass.222The mass is the sum of gravitationally bounded particles as
provided by the SUBFIND algorithm (Springel et al., 2001)), TO21 achieved an
improvement of 27% compared to using dark matter mass alone for the
$M_{\star}$ prediction.333As TO21 worked on TNG100 of the IllustrisTNG project
(with a box size of $100^{3}$ Mpc3), we have to refit the pivot mass. Using
their original pivot mass ($M_{\rm DM,peak}=10^{12.7}\,M_{\odot}$), the
improvement is 34% (TO21). In our work, we try to compare $\phi$ with the $X$
we developed with TNG300. Thus, we refit the pivot mass for $\phi$ in the
simulation and find $M_{\rm DM,peak}=10^{12.95}\,M_{\odot}$ for redshift 0.

Our goal is to seek other parameter combinations $\psi$ that perform better
than $\phi$, particularly for the clustering properties, an aspect not
addressed fully in TO21. To ensure a fair comparison between $\psi$ and
$\phi$, we do not include scatters between $M_{\star}$ and the dark matter
subhalo properties in our main results, as TO21 did not consider such a
scatter. However, we do present results with scatter in Appendix A, showing
that our best $\psi$ does not change when scatters are introduced. Although
the maxima of subhalo properties throughout their lifetimes such as $V_{\rm
peak}$ and $M_{\rm DM,peak}$ are shown to better trace $M_{\star}$, the
stochastic nature of those features as seen from the simulations can cause
additional scatter. Thus, instead of using the maximum (i.e., 100th
percentile) of those properties, we use a specific percentile for those
properties to avoid the scatter due to stochasticity. We tried the 10th, 20th,
… 100th ($=$ peak) percentiles for $V_{\rm max}$ and $M_{\rm DM}$ throughout
the lifetimes of subhalos, and conducted abundance matching using different
percentiles of each property. It is found that the 90th percentile can predict
$M_{\star}$ with the smallest scatter among all cases considered for $V_{\rm
max}$ and $M_{\rm DM}$ (hereafter $V_{\rm max,90\%}$ and $M_{\rm
DM,90\%}$).444With such a notation, $V_{\rm peak}\equiv V_{\rm max,100\%}$.
Replacing the properties in $\phi$ with the percentile properties, we found
that the scatter of abundance matching decreases by $\sim 12.5\%$.

Motivated by the finding that the peak value of the quantities often used in
SHAM does not always lead to the best results, we then conducted a thorough
examination of the performance of all the dark matter subhalo properties
provided by TNG300 of different percentiles. The best alternatives of $V_{\rm
max,90\%}$ is $V_{\rm disp,80\%}$ while those of $M_{\rm DM,90\%}$ are
$M_{R_{\rm max},\rm 80\%}$ and $|\dot{M}_{\rm DM}|_{\rm 60\%}$, where $V_{\rm
disp}$ is the velocity dispersion of a subhalo, $M_{R_{\rm max}}$ is the dark
matter mass within the radius that corresponds to $V_{\rm max}$, and
$|\dot{M}_{\rm DM}|$ is the absolute subhalo dark matter mass variation rate.
We will show in Section 4 that $X$ based on the combinations of these
quantities can improve the prediction of stellar mass and $w_{p}$, and define
six candidates for $X$ as

$\psi_{1}=\frac{V_{\rm max,90\%}}{V_{\rm<EMAIL_ADDRESS>DM,90\%}}{M_{\rm<EMAIL_ADDRESS>(8)

$\psi_{2}=\frac{V_{\rm disp,80\%}}{V_{\rm<EMAIL_ADDRESS>DM,90\%}}{M_{\rm<EMAIL_ADDRESS>(9)

$\psi_{3}=\frac{V_{\rm max,90\%}}{V_{\rm<EMAIL_ADDRESS>max},\rm 80\%}}{M_{R_{\rm max},\rm<EMAIL_ADDRESS>(10)

$\psi_{4}=\frac{V_{\rm disp,80\%}}{V_{\rm<EMAIL_ADDRESS>max},\rm 80\%}}{M_{R_{\rm max},\rm<EMAIL_ADDRESS>(11)

$\psi_{5}=\frac{V_{\rm max,90\%}}{V_{\rm<EMAIL_ADDRESS>DM}|_{\rm 60\%}}{|\dot{M}_{\rm DM}|_{\rm<EMAIL_ADDRESS>(12)

$\psi_{6}=\frac{V_{\rm disp,80\%}}{V_{\rm<EMAIL_ADDRESS>DM}|_{\rm 60\%}}{|\dot{M}_{\rm DM}|_{\rm<EMAIL_ADDRESS>(13)

where the parameters with a subscript $\rm@$ are the normalization factors at
a fitted pivot $M_{\rm DM,peak}$ right after. We tried a range of pivot masses
($M_{\rm DM,peak}=10^{10.0}-10^{14.0}\,M_{\odot}$) and select the one that
minimizes the scatter of $M_{\star}$ prediction for each $\psi$’s. The subhalo
properties along the main primary branch are used to calculate $\dot{M}_{\rm
DM}$ and all the parameters with the subscript of “peak” or a certain
percentile.

##  4 Results

###  4.1 Stellar Mass Prediction

Figure 1: Panels (a) - (e): the error of the stellar mass prediction
corresponding to stellar mass using the AM scheme of $\phi$ and $\psi_{1}$ for
panels (a) - (c), $M_{\rm DM,peak}$ and $M_{\rm DM,90\%}$ for panel (d),
$V_{\rm peak}$ and $V_{\rm max,90\%}$ for panel (e), and $\phi$ and $\psi_{5}$
for panel (f). The green, red, and blue contours represent the predictions of
“all”, “central” and “satellite” groups, respectively. There are two contour
lines for each AM scheme, which denote the data point density of 0.05 and 0.5
(normalized to maxima of 1). Panels (g) and (h): the overall scatter at
several redshifts with different AM schemes denoted on the top left of the
panel. The bars for each redshift bin are sorted so that the AM scheme with
the lowest scatter is on the left.

We show in Table 1 the performance of each AM scheme in stellar mass
prediction. In the upper-left part of each redshift of the table, the lifetime
peak properties are improved by selecting an optimized percentile for all four
groups of galaxies mentioned in Section 2.1. In the lower-left part of each
redshift, all $\psi$’s show a relatively better $M_{\star}$ prediction than
the original $\phi$ for the four groups. It is also shown that $\psi_{1}$
performs slightly worse than the other $\psi$’s, which means a better
prediction of $M_{\star}$ can be achieved by replacing $V_{\rm max,90\%}$ and
$M_{\rm DM,90\%}$ with different properties.

In Figure 1, we compare the prediction errors with respect to true $M_{\star}$
using different AM schemes. The contours show the density distribution of the
data points in the panels. A well-performed AM scheme should have contours
closer to the horizontal line at zero error. Panel (a) of the Figure shows
that the main improvement from $\phi$ to $\psi_{1}$ is for
$\log(M_{\star}/M_{\odot})<11.0$. Panels (b) and (c) show that the improvement
in central galaxies is limited to lower-mass galaxies, while that of the
satellites also occurs in higher-mass galaxies as well. Panels (d) and (e)
show the AM performances of individual parameters. Here, we can see that the
improvement from $M_{\rm peak}$ to $M_{\rm DM,90\%}$ is lower than that of
$V_{\rm peak}$ to $V_{\rm max,90\%}$. Panel (f) shows the error distribution
of $\psi_{5}$, which has the best performance among all AM schemes considered
here. Finally, panels (g) and (h) show the scatter of all AM schemes at
several different redshifts. We can see that the scatters of all the $\psi$’s
are smaller than that of $\phi$ at all redshifts.

###  4.2 Stellar Mass Dependence of Projected 2PCF

In Table 1, the upper-right part in each redshift of the table shows that the
peak properties have a larger error for projected 2PCF prediction than the
percentile properties, except for the $M_{\rm DM}$ properties and the
$|\dot{M}_{\rm DM}|$ properties at $z=0$. In the lower-right of each redshift,
we can also find that the $\psi$’s predictions of $w_{p}$ are better than that
of $\phi$. Figure 2 shows the ratio of the predicted $w_{p}$ over the true
$w_{p}$. The first row (panels (a) - (c)) and the second row (panels (d) -
(f)) compare the performances between the peak valued quantities to the
percentile quantities and that between $\phi$ to $\psi$’s, respectively. We
see that the differences in $w_{p}$ ratios are similar among all the
$M_{\star}$ bins. Panels (g) and (h) show that while $\psi_{5}$ and $\psi_{6}$
yield the best two $w_{p}$ predictions at $z=0$, the $|\dot{M}_{\rm DM}|_{\rm
60\%}$ outperforms $\psi_{5}$ and $\psi_{6}$ at all the other redshifts by a
small margin.

Figure 2: Panels (a) to (f): The data points denote the rational difference in
the projected 2PCF between the true and abundance-matched galaxy distribution
while the error bars are the jackknife uncertainties. The mass range of each
stellar mass bin is denoted at the top of each panel. (g) and (f) show the
fractional difference in the clustering averaged over all $r_{p}$ bins and
$M_{\star}$ bins in the “all” group. The bar at each redshift is sorted the
same way as Figure1. The data points with jackknife uncertainties larger than
0.2 are omitted in all panels.

###  4.3 Stellar Mass and Color Dependence of Projected 2PCF

The upper-right part in each redshift of Table 1 shows the error of $w_{p}$
predictions for the red and blue galaxies. For $M_{\rm DM}$, replacing the
peak values with percentile quantities worsens the $w_{p}$ predictions for all
galaxies. However, for the other features, this replacement worsens the
$w_{p}$ predictions for the red galaxies at multiple redshifts, while
improving the predictions for the blue galaxies across all epochs. In the
bottom right in each redshift, the $\psi$’s outperforms $\phi$ in both types
of galaxies.

Figure 3 and 4 show the same quantities as Figure 2, but for red and blue
galaxies, respectively. The same as in Figure 2, the differences in $w_{p}$
ratios are similar among all the $M_{\star}$ bins. For the $w_{p}$ prediction
of red galaxies, $M_{\rm DM,peak}$ performs better than $M_{\rm DM,90\%}$ in
all $M_{\star}$ bins while $V_{\rm peak}$ performs roughly the same as $V_{\rm
max,90\%}$. In the middle $M_{\star}$ bin, $\psi_{1}$ performs roughly the
same as $\psi_{5}$, while $\psi_{5}$ performs better in other $M_{\star}$
bins. For most of the redshifts, $\psi_{5}$ and $\psi_{6}$ are the best in
predicting $w_{p}$ for red galaxies. For the $w_{p}$ prediction of blue
galaxies, $M_{\rm DM,90\%}$ and $V_{\rm max,90\%}$ perform better than $V_{\rm
peak}$ and $M_{\rm DM,peak}$ in the lower two $M_{\star}$ bins. The
performance of all properties in the highest $M_{\star}$ bins is not shown due
to the small number of galaxy pairs. For most of the redshifts, $|\dot{M}_{\rm
DM}|_{\rm 60\%}$ is the best for $w_{p}$ prediction of blue galaxies,
outperforming $\psi_{5}$ with a small margin. Overall, $\psi_{5}$ is still
reliable for reproducing color-dependent clustering.

Figure 3: Same as Figure 2, but for red galaxies.  Figure 4: Same as Figure 2,
but for blue galaxies. There are not enough galaxy pairs in the highest
$M_{\star}$ bin at $z=0$ for reliable error estimation, forcing us to skip the
bin.

##  5 Discussion

Using the halos and mock galaxies from TNG300, we have shown that our best
stellar mass proxy, $\psi_{5}$, performs better compared to both the popular
choice of $V_{\rm peak}$, as well as the parameter $\phi$ proposed by TO21, in
terms of predicting the stellar mass. Furthermore, when used in SHAM,
$\psi_{5}$ can reliably reproduce both the stellar mass and color dependencies
of the 2PCFs at $z=0-2$ better than $V_{\rm peak}$. Whereas the usual
implementation of $V_{\rm peak}$ model has one free parameter (the scatter
between $V_{\rm peak}$ and $M_{\star}$; see below), our $\psi_{5}$ model has
three (the percentiles for $V_{\rm max}$ and $|\dot{M}_{\rm DM}|$, and the
normalizing halo mass). Nonetheless, the better performances in $M_{\star}$
prediction (and hence the stellar mass dependence of $w_{p}$) should outweigh
the extra degrees of freedom.

In Section 3, we argued that the success of our method is attributed to the
use of percentile quantities, which effectively smooths the stochastic
behavior of dark matter subhalo properties. Here, we provide a more
quantitative analysis of the evolution of $M_{\rm DM},\ M_{\rm R_{max}},\
|\dot{M}_{\rm DM}|,\ V_{\rm max}$ and $V_{\rm disp}$ for the subhalos. For
each quantity mentioned above, we first obtain the moving average with a
window size of 10 snapshots along the time axis for each subhalo, and subtract
the original values with averaged values to obtain the residual of property
history. Next, we perform a fast Fourier transform to the residual to obtain
an amplitude and frequency relation. Lastly, we calculate the average
frequency for each subhalo weighted by the amplitude. We define this value as
the stochasticity of a subhalo for a certain dark matter property. Now, we can
compare the average stochasticity ($S$, Gyr-1) of the properties with the
relative improvement from the chosen percentile quantities instead of the peak
values for each property. In Table 1, the relative improvement in scatter in
the _All_ group for each property can be obtained directly
($I_{\sigma}=(\sigma_{\rm peak}-\sigma_{\rm perc})/\sigma_{\rm peak}$, where
$\sigma_{\rm peak}$ and $\sigma_{\rm perc}$ are the scatter with peak and
percentile quantities, respectively):

  1. 1.

$|\dot{M}_{DM}|$: $S=1.843,\ I_{\sigma}=0.542$

  2. 2.

$V_{\rm disp}$: $S=1.419,\ I_{\sigma}=0.236$

  3. 3.

$V_{\rm max}$: $S=1.509,\ I_{\sigma}=0.15$

  4. 4.

$M_{R_{max}}$: $S=1.487,\ I_{\sigma}=0.124$

  5. 5.

$M_{\rm DM}$: $S=1.284,\ I_{\sigma}=0.013$

The stochasticities are overall positively correlated with the improvements of
switching from peak to percentile quantities. We can see that the properties
with maximal ($|\dot{M}_{DM}|$) and minimal ($M_{\rm DM}$) stochasticity
correspond to the largest and least relative improvements, respectively. A
higher time resolution of the simulation might affect the best-fit percentiles
of various properties considered here, especially for those with high
stochasticities. However, due to the limitation of the TNG300 time resolution,
we are not able to further confirm this scenario.

One caveat for the success of the color dependence of clustering is that we
have separated red subhalos from blue ones a priori, based on the full
hydrodynamical run of TNG300, making our approach non-applicable to other DMO
simulations.555Please note that our $\psi_{5}$ can still accurately predict
the stellar mass dependence of 2PCFs at $z=0-2$. In C.-Y. Chuang et al.( 2022,
in preparation) from private communication, a graph neural network (GNN, e.g.
Jespersen et al., 2022) model is used to emulate the TNG300 results, which
enables one to _port_ the TNG hydrodynamical model to any _N_ -body
simulations of similar resolutions as TNG300. Of course, with the GNN model,
one then does not really require SHAM anymore, as the GNN model provides
accurate predictions for $M_{\star}$, $g-r$ color, star formation rate, and
several other properties, for galaxies with $M_{\star}\geq 10^{9}M_{\odot}$ at
$z=0-2$.

##  6 Conclusion

In recent years, the conditional abundance matching (Hearin et al. 2014; see
also Masaki et al. 2013) approach has been used to exploit secondary
connections between galaxy and halo properties, which allows one to assign
colors to model galaxies (e.g., Hearin et al. 2017). However, such models may
not be compatible with the observed color-dependence of the central galaxy
stellar mass–halo mass relation (SHMR; e.g., Mandelbaum et al., 2016). As our
model is calibrated against TNG300, it can only reproduce the observed galaxy
properties to the extent that the TNG300 model can. It is found that, while
the SHMR (irrespective of galaxy color) in TNG300 agrees with the measurements
of Leauthaud et al. (2012), when central galaxies are split by color, only the
red galaxy SHMR is reproduced. Finally, we note that there may well be an
intrinsic scatter for the $M_{\star}$ prediction through SHAM in the real
universe, while the scatter associated with the _best_ SHAM scheme could be
smaller than the intrinsic scatter. Thus, our $\psi_{5}$-based SHAM scheme may
inherit potential biases created by the numerical simulation we rely on.

Therefore, for a more realistic (and useful) application of the methodology
developed in this work, in a future paper, we plan to tune our $\psi$’s by
directly fitting to observations, such as the clustering measurements from
SDSS and its subsequent surveys, as well as deeper surveys like DEEP2, PRIMUS,
VIPERS, and UltraVISTA (e.g., Coil et al. 2008; Zehavi et al. 2011; Marulli et
al. 2013; Guo et al. 2014; Skibba et al. 2014; McCracken et al. 2015; Coil et
al. 2017).

We thank the anonymous referee for the comments that improved the manuscript.
We also appreciate the helpful comments from Shogo Masaki, Yao-Yuan Mao,
Christian Jespersen, and Hong Guo. We acknowledge support from the National
Science and Technology Council of Taiwan under grant Nos. MOST
110-2112-M-001-004 and MOST 109-2112-M-001-005. Y.T.L. thanks I.H., L.Y.L.,
and A.L.L. for constant encouragement and inspiration. The numerical work was
conducted on the high-performance computing facility at the Institute of
Astronomy and Astrophysics in Academia Sinica
(https://hpc.tiara.sinica.edu.tw). The IllustrisTNG simulations were
undertaken with compute time awarded by the Gauss Centre for Supercomputing
(GCS) under GCS Large-Scale Projects GCS-ILLU and GCS-DWAR on the GCS share of
the supercomputer Hazel Hen at the High Performance Computing Center Stuttgart
(HLRS), as well as on the machines of the Max Planck Computing and Data
Facility (MPCDF) in Garching, Germany. =7.5cm

Table 1: Scatter and loss of stellar mass prediction via different abundance matching schemes. The best values are bold-faced and color shaded. Loss Metrics |  | Stellar Mass Prediction | Prediction of $w_{p}$ |  | Stellar Mass Prediction | Prediction of $w_{p}$
---|---|---|---|---|---|---
Galaxy Types |  | All | Central | Satellite | Mix | All | Blue | Red |  | All | Central | Satellite | Mix | All | Blue | Red
AM Scheme |  | Scatter | Loss | Scatter | Loss | Scatter | Loss | Scatter | Loss | Percentage Error |  | Scatter | Percentage Error
$M_{DM,peak}$ |  redshift 0 | 0.1952 | 0.1484 | 0.1755 | 0.1327 | 0.1942 | 0.1477 | 0.1800 | 0.1361 | 27.3$\pm$4.1 | 24.0$\pm$6.2 | 24.2$\pm$5.6 |  redshift 1.0 | 0.2053 | 0.1869 | 0.2082 | 0.1915 | 24.2$\pm$4.3 | 24.4$\pm$6.9 | 19.8$\pm$9.0
$M_{\rm DM,90\%}$ | 0.1926 | 0.1461 | 0.1691 | 0.1273 | 0.1917 | 0.1473 | 0.1745 | 0.1318 | 28.0$\pm$4.0 | 24.0$\pm$5.3 | 26.4$\pm$5.6 | 0.2003 | 0.1795 | 0.2027 | 0.1845 | 24.2$\pm$4.3 | 25.3$\pm$7.1 | 21.6$\pm$8.5
$M_{\rm R_{max},peak}$ | 0.2494 | 0.1908 | 0.2445 | 0.1871 | 0.2407 | 0.1838 | 0.2436 | 0.1864 | 21.2$\pm$3.8 | 24.2$\pm$7.0 | 16.5$\pm$5.4 | 0.2601 | 0.2521 | 0.2545 | 0.2526 | 22.4$\pm$4.3 | 25.2$\pm$6.6 | 14.0$\pm$8.7
$M_{\rm R_{max},80\%}$ | 0.2185 | 0.1660 | 0.2115 | 0.1604 | 0.2168 | 0.1653 | 0.2127 | 0.1615 | 19.3$\pm$3.9 | 18.8$\pm$6.2 | 15.8$\pm$5.6 | 0.2344 | 0.2267 | 0.2358 | 0.2286 | 17.3$\pm$4.3 | 18.5$\pm$7.0 | 14.9$\pm$8.1
$|\dot{M_{\rm DM}}|_{\rm peak}$ | 0.4318 | 0.3497 | 0.4385 | 0.3555 | 0.4060 | 0.3289 | 0.4313 | 0.3494 | 7.7$\pm$3.8 | 15.3$\pm$7.6 | 7.6$\pm$5.8 | 0.4296 | 0.4328 | 0.4123 | 0.4287 | 8.9$\pm$4.2 | 12.7$\pm$6.1 | 10.1$\pm$8.3
$|\dot{M_{\rm DM}}|_{\rm 60\%}$ | 0.1977 | 0.1516 | 0.1919 | 0.1464 | 0.1994 | 0.1532 | 0.1937 | 0.1480 | 9.3$\pm$4.3 | 7.1$\pm$5.9 | 13.5$\pm$5.4 | 0.1949 | 0.1926 | 0.1969 | 0.1935 | 5.8$\pm$4.4 | 5.7$\pm$6.0 | 9.9$\pm$8.4
$V_{\rm peak}$ | 0.1769 | 0.1363 | 0.1739 | 0.1338 | 0.1799 | 0.1394 | 0.1753 | 0.1351 | 8.8$\pm$3.9 | 13.9$\pm$6.0 | 5.9$\pm$5.1 | 0.1765 | 0.1725 | 0.1767 | 0.1734 | 12.1$\pm$4.1 | 15.5$\pm$6.3 | 9.5$\pm$8.7
$V_{\rm 90\%}$ | 0.1504 | 0.1149 | 0.1469 | 0.1120 | 0.1565 | 0.1200 | 0.1491 | 0.1138 | 6.1$\pm$3.8 | 11.0$\pm$5.6 | 4.7$\pm$5.1 | 0.1508 | 0.1471 | 0.1524 | 0.1482 | 9.9$\pm$4.5 | 12.2$\pm$7.0 | 10.6$\pm$8.5
$V_{\rm disp,peak}$ | 0.1914 | 0.1482 | 0.1883 | 0.1455 | 0.1957 | 0.1526 | 0.1900 | 0.1471 | 8.6$\pm$4.0 | 13.8$\pm$6.2 | 5.6$\pm$5.4 | 0.1953 | 0.1913 | 0.1974 | 0.1926 | 12.2$\pm$4.4 | 16.5$\pm$6.4 | 8.6$\pm$8.7
$V_{\rm disp,80\%}$ | 0.1463 | 0.1122 | 0.1444 | 0.1106 | 0.1499 | 0.1153 | 0.1457 | 0.1117 | 6.2$\pm$4.1 | 11.4$\pm$6.3 | 4.3$\pm$5.0 | 0.1551 | 0.1533 | 0.1523 | 0.1531 | 11.2$\pm$4.1 | 14.6$\pm$6.2 | 10.8$\pm$8.7
$\phi$ | 0.1667 | 0.1266 | 0.1448 | 0.1096 | 0.1791 | 0.1393 | 0.1533 | 0.1164 | 21.9$\pm$4.3 | 18.6$\pm$5.6 | 20.1$\pm$5.3 | 0.1748 | 0.1520 | 0.1916 | 0.1609 | 21.1$\pm$4.2 | 20.2$\pm$6.9 | 22.0$\pm$8.7
$\psi_{1}$ | 0.1458 | 0.1112 | 0.1402 | 0.1064 | 0.1499 | 0.1148 | 0.1425 | 0.1084 | 10.5$\pm$4.0 | 13.0$\pm$5.7 | 9.4$\pm$5.6 | 0.1506 | 0.1460 | 0.1514 | 0.1471 | 11.4$\pm$4.4 | 13.8$\pm$6.8 | 12.6$\pm$9.2
$\psi_{2}$ | 0.1436 | 0.1099 | 0.1406 | 0.1073 | 0.1461 | 0.1121 | 0.1418 | 0.1084 | 9.3$\pm$4.0 | 12.8$\pm$5.4 | 7.7$\pm$5.3 | 0.1548 | 0.1520 | 0.1517 | 0.1519 | 12.8$\pm$4.4 | 15.8$\pm$6.4 | 12.6$\pm$8.8
$\psi_{3}$ | 0.1457 | 0.1109 | 0.1419 | 0.1077 | 0.1501 | 0.1146 | 0.1438 | 0.1093 | 8.0$\pm$4.1 | 11.6$\pm$6.3 | 6.9$\pm$5.3 | 0.1515 | 0.1475 | 0.1523 | 0.1485 | 10.6$\pm$4.4 | 12.6$\pm$6.6 | 11.8$\pm$8.9
$\psi_{4}$ | 0.1444 | 0.1104 | 0.1425 | 0.1088 | 0.1471 | 0.1128 | 0.1436 | 0.1097 | 7.2$\pm$4.0 | 12.2$\pm$6.0 | 5.6$\pm$5.3 | 0.1568 | 0.1548 | 0.1532 | 0.1545 | 11.7$\pm$4.3 | 14.8$\pm$6.7 | 12.1$\pm$9.0
$\psi_{5}$ | 0.1412 | 0.1080 | 0.1391 | 0.1063 | 0.1464 | 0.1128 | 0.1408 | 0.1077 | 3.7$\pm$4.2 | 7.8$\pm$6.0 | 3.7$\pm$5.6 | 0.1450 | 0.1436 | 0.1458 | 0.1440 | 7.7$\pm$4.5 | 10.0$\pm$6.8 | 6.7$\pm$8.2
$\psi_{6}$ | 0.1423 | 0.1092 | 0.1407 | 0.1079 | 0.1463 | 0.1129 | 0.1420 | 0.1090 | 4.3$\pm$4.1 | 8.8$\pm$5.9 | 3.3$\pm$4.9 | 0.1510 | 0.1505 | 0.1495 | 0.1503 | 8.6$\pm$4.4 | 11.6$\pm$5.9 | 6.5$\pm$8.0
$M_{DM,peak}$ |  redshift 0.5 | 0.2096 | 0.1604 | 0.1945 | 0.1492 | 0.2144 | 0.1660 | 0.1981 | 0.1521 | 20.0$\pm$5.1 | 20.0$\pm$6.9 | 18.5$\pm$11.1 |  redshift 2.0 | 0.2196 | 0.2093 | 0.2268 | 0.2116 | 8.0$\pm$5.6 | 8.8$\pm$7.2 | 7.2$\pm$13.1
$M_{\rm DM,90\%}$ | 0.2032 | 0.1527 | 0.1866 | 0.1396 | 0.2089 | 0.1608 | 0.1907 | 0.1433 | 20.7$\pm$5.1 | 22.5$\pm$7.1 | 20.1$\pm$11.5 | 0.2066 | 0.1968 | 0.2171 | 0.1995 | 8.5$\pm$5.8 | 9.0$\pm$6.4 | 7.4$\pm$12.8
$M_{\rm R_{max},peak}$ | 0.2607 | 0.2021 | 0.2528 | 0.1956 | 0.2566 | 0.1998 | 0.2535 | 0.1963 | 20.9$\pm$5.3 | 22.6$\pm$7.3 | 13.2$\pm$11.9 | 0.2631 | 0.2570 | 0.2612 | 0.2575 | 8.2$\pm$6.6 | 9.6$\pm$7.2 | 7.4$\pm$14.4
$M_{\rm R_{max},80\%}$ | 0.2428 | 0.1847 | 0.2356 | 0.1784 | 0.2436 | 0.1878 | 0.2370 | 0.1801 | 17.3$\pm$5.1 | 19.5$\pm$6.9 | 13.7$\pm$10.7 | 0.2515 | 0.2485 | 0.2500 | 0.2487 | 7.8$\pm$6.3 | 8.2$\pm$7.5 | 6.8$\pm$11.3
$|\dot{M_{\rm DM}}|_{\rm peak}$ | 0.4189 | 0.3362 | 0.4206 | 0.3369 | 0.4056 | 0.3297 | 0.4180 | 0.3356 | 8.6$\pm$5.3 | 12.2$\pm$7.2 | 15.7$\pm$12.1 | 0.4002 | 0.4010 | 0.3903 | 0.3997 | 4.2$\pm$5.8 | 6.1$\pm$7.0 | 22.6$\pm$17.7
$|\dot{M_{\rm DM}}|_{\rm 60\%}$ | 0.1988 | 0.1493 | 0.1975 | 0.1478 | 0.2016 | 0.1532 | 0.1982 | 0.1487 | 3.7$\pm$5.1 | 4.3$\pm$6.4 | 11.9$\pm$12.2 | 0.2164 | 0.2169 | 0.2130 | 0.2164 | 2.3$\pm$5.4 | 2.8$\pm$6.4 | 4.3$\pm$13.8
$V_{\rm peak}$ | 0.1783 | 0.1371 | 0.1744 | 0.1341 | 0.1765 | 0.1363 | 0.1748 | 0.1345 | 13.6$\pm$5.2 | 15.4$\pm$6.9 | 13.7$\pm$11.2 | 0.1948 | 0.1924 | 0.1874 | 0.1918 | 6.9$\pm$6.1 | 8.0$\pm$7.4 | 5.0$\pm$12.5
$V_{\rm 90\%}$ | 0.1558 | 0.1153 | 0.1525 | 0.1121 | 0.1536 | 0.1159 | 0.1527 | 0.1127 | 11.6$\pm$5.1 | 12.5$\pm$7.2 | 18.2$\pm$12.6 | 0.1757 | 0.1742 | 0.1657 | 0.1732 | 6.7$\pm$6.1 | 7.6$\pm$7.2 | 6.6$\pm$14.3
$V_{\rm disp,peak}$ | 0.2007 | 0.1548 | 0.1972 | 0.1519 | 0.2001 | 0.1549 | 0.1977 | 0.1524 | 13.8$\pm$5.0 | 16.2$\pm$7.1 | 11.2$\pm$11.5 | 0.2205 | 0.2184 | 0.2156 | 0.2180 | 7.4$\pm$6.2 | 8.5$\pm$7.1 | 6.0$\pm$13.3
$V_{\rm disp,80\%}$ | 0.1663 | 0.1234 | 0.1648 | 0.1215 | 0.1586 | 0.1194 | 0.1638 | 0.1211 | 12.0$\pm$5.3 | 14.3$\pm$7.5 | 16.8$\pm$11.1 | 0.1911 | 0.1911 | 0.1766 | 0.1893 | 6.9$\pm$6.0 | 7.5$\pm$6.5 | 8.2$\pm$14.2
$\phi$ | 0.1800 | 0.1333 | 0.1613 | 0.1212 | 0.1957 | 0.1493 | 0.1678 | 0.1261 | 19.4$\pm$5.0 | 20.2$\pm$7.3 | 20.6$\pm$11.7 | 0.2013 | 0.1917 | 0.1955 | 0.1922 | 10.2$\pm$5.9 | 10.9$\pm$6.9 | 8.5$\pm$12.8
$\psi_{1}$ | 0.1569 | 0.1162 | 0.1527 | 0.1122 | 0.1542 | 0.1165 | 0.1530 | 0.1130 | 12.7$\pm$5.4 | 14.4$\pm$6.7 | 20.1$\pm$12.3 | 0.1762 | 0.1740 | 0.1672 | 0.1731 | 6.7$\pm$6.2 | 7.2$\pm$6.4 | 7.4$\pm$13.4
$\psi_{2}$ | 0.1652 | 0.1230 | 0.1626 | 0.1202 | 0.1588 | 0.1197 | 0.1619 | 0.1201 | 12.9$\pm$5.3 | 14.9$\pm$6.9 | 17.0$\pm$11.8 | 0.1816 | 0.1786 | 0.1757 | 0.1783 | 7.6$\pm$6.4 | 8.5$\pm$7.4 | 8.1$\pm$13.3
$\psi_{3}$ | 0.1583 | 0.1175 | 0.1547 | 0.1140 | 0.1554 | 0.1174 | 0.1548 | 0.1146 | 12.1$\pm$5.5 | 13.7$\pm$6.7 | 18.6$\pm$12.9 | 0.1780 | 0.1762 | 0.1691 | 0.1753 | 6.5$\pm$5.8 | 7.1$\pm$6.8 | 6.9$\pm$13.9
$\psi_{4}$ | 0.1691 | 0.1256 | 0.1673 | 0.1234 | 0.1609 | 0.1213 | 0.1662 | 0.1230 | 12.6$\pm$5.5 | 15.4$\pm$6.9 | 11.3$\pm$10.8 | 0.1940 | 0.1938 | 0.1800 | 0.1921 | 6.8$\pm$6.3 | 7.4$\pm$7.1 | 7.0$\pm$13.3
$\psi_{5}$ | 0.1511 | 0.1121 | 0.1501 | 0.1107 | 0.1484 | 0.1123 | 0.1498 | 0.1110 | 9.3$\pm$5.1 | 10.4$\pm$7.0 | 10.7$\pm$12.1 | 0.1697 | 0.1695 | 0.1610 | 0.1684 | 4.9$\pm$5.9 | 5.5$\pm$6.9 | 4.9$\pm$13.4
$\psi_{6}$ | 0.1603 | 0.1190 | 0.1602 | 0.1184 | 0.1554 | 0.1175 | 0.1594 | 0.1183 | 8.8$\pm$5.2 | 9.9$\pm$7.1 | 14.7$\pm$13.0 | 0.1800 | 0.1804 | 0.1713 | 0.1793 | 4.8$\pm$6.1 | 5.1$\pm$6.6 | 5.0$\pm$12.1

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##  Appendix A Scatter between Stellar Mass and Subhalo Properties

In Section 3, we assigned model galaxies to subhalos strictly following the
rank order (in $\psi$) to facilitate a direct comparison between our results
and those of TO21. However, in the usual implementation of SHAM, a stellar
mass function is used to assign $M_{\star}$ to subhalos of the same number
density. In so doing, a scatter between dark matter subhalo properties ($X$)
and $M_{\star}$ is found to be necessary. Effectively, our approach assumes
zero scatter. Here, we examine the effect of introducing a scatter on our
results. Specifically, we artificially introduce an uncertainty in stellar
mass through the use of log-normal probability distributions with standard
deviations of $\sigma_{\star}=0.1$ and $0.4$ dex for $M_{\star}$,
respectively. Subsequently, we evaluate the loss metrics for all adopted AM
schemes utilizing this uncertainty. To ensure the robustness of our results,
this calculation was repeated 1000 times for all AM schemes and the averages
of the loss metrics were subsequently reported.

Figure 5\- 8 present the same information as that of panels (g) and (h) of
Figure 1\- 3, and panels (e) and (f) of Figure 4, respectively, but with
different $\sigma_{\star}$’s. Specifically, the figures compare the results
obtained with $\sigma_{\star}=0.1$ and $0.4$ dex. It is noteworthy that the
relative performances among the various AM schemes evaluated, as depicted in
these figures, are overall consistent with the relative performances obtained
in the absence of uncertainties ($\sigma_{\star}=0$).

Figure 5: The panels located on the left and right sides of this figure depict
the same data as that presented in Figure 1(h) and Figure 1(g), respectively.
However, these representations differ in the log-normal scatter values that
are indicated in the rightmost of each row of panels. Figure 6: The panels
located on the left and right sides of this figure depict the same data as
that presented in Figure 2(h) and Figure 2(g), respectively. However, these
representations differ in the log-normal scatter values that are indicated in
the rightmost of each row of panels. Figure 7: Same as Figure 6, but with red
galaxies. Figure 8: Same as Figure 6, but with blue galaxies.

=7.5cm

Table 2: Same as Table 1, but with $\sigma_{\star}=0.1$ dex.  Loss Metrics |  | Stellar Mass Prediction | Prediction of $w_{p}$ |  | Stellar Mass Prediction | Prediction of $w_{p}$
---|---|---|---|---|---|---
Galaxy Types |  | All | Central | Satellite | Mix | All | Blue | Red |  | All | Central | Satellite | Mix | All | Blue | Red
AM Scheme |  | Scatter | Loss | Scatter | Loss | Scatter | Loss | Scatter | Loss | Percentage Error |  | Scatter | Percentage Error
$M_{DM,peak}$ |  redshift 0 | 0.2225 | 0.1729 | 0.2055 | 0.1595 | 0.2211 | 0.1715 | 0.2092 | 0.1622 | 27.4$\pm$4.1 | 24.1$\pm$6.2 | 24.2$\pm$5.6 |  redshift 1.0 | 0.2316 | 0.2155 | 0.2338 | 0.2194 | 24.3$\pm$4.3 | 24.3$\pm$6.8 | 19.9$\pm$8.9
$M_{\rm DM,90\%}$ | 0.2205 | 0.1710 | 0.2004 | 0.1549 | 0.2190 | 0.1709 | 0.2048 | 0.1586 | 28.3$\pm$4.0 | 24.4$\pm$5.5 | 26.5$\pm$5.7 | 0.2279 | 0.2100 | 0.2292 | 0.2141 | 24.3$\pm$4.3 | 25.5$\pm$7.0 | 21.5$\pm$8.6
$M_{\rm R_{max},peak}$ | 0.2717 | 0.2116 | 0.2673 | 0.2084 | 0.2635 | 0.2045 | 0.2664 | 0.2075 | 21.5$\pm$3.8 | 24.3$\pm$7.1 | 16.5$\pm$5.4 | 0.2819 | 0.2746 | 0.2764 | 0.2750 | 22.5$\pm$4.3 | 25.0$\pm$6.6 | 14.0$\pm$8.5
$M_{\rm R_{max},80\%}$ | 0.2436 | 0.1890 | 0.2375 | 0.1842 | 0.2416 | 0.1875 | 0.2384 | 0.1850 | 19.5$\pm$3.9 | 19.2$\pm$6.4 | 15.7$\pm$5.6 | 0.2588 | 0.2521 | 0.2595 | 0.2537 | 17.6$\pm$4.3 | 18.8$\pm$6.8 | 15.1$\pm$8.1
$|\dot{M_{\rm DM}}|_{\rm peak}$ | 0.4458 | 0.3622 | 0.4522 | 0.3679 | 0.4213 | 0.3417 | 0.4453 | 0.3619 | 7.8$\pm$3.8 | 15.4$\pm$7.6 | 7.5$\pm$5.8 | 0.4437 | 0.4467 | 0.4275 | 0.4429 | 9.1$\pm$4.1 | 12.9$\pm$6.1 | 10.3$\pm$8.3
$|\dot{M_{\rm DM}}|_{\rm 60\%}$ | 0.2249 | 0.1754 | 0.2200 | 0.1712 | 0.2259 | 0.1761 | 0.2214 | 0.1723 | 8.8$\pm$4.3 | 7.1$\pm$5.9 | 13.1$\pm$5.4 | 0.2230 | 0.2212 | 0.2242 | 0.2218 | 5.7$\pm$4.4 | 5.8$\pm$5.9 | 10.1$\pm$8.4
$V_{\rm peak}$ | 0.2064 | 0.1618 | 0.2040 | 0.1599 | 0.2085 | 0.1637 | 0.2050 | 0.1608 | 9.0$\pm$4.0 | 13.7$\pm$6.0 | 5.9$\pm$5.1 | 0.2061 | 0.2029 | 0.2059 | 0.2035 | 12.3$\pm$4.1 | 15.8$\pm$6.3 | 9.4$\pm$8.7
$V_{\rm 90\%}$ | 0.1842 | 0.1437 | 0.1816 | 0.1416 | 0.1885 | 0.1473 | 0.1832 | 0.1429 | 6.3$\pm$3.9 | 10.8$\pm$5.6 | 4.9$\pm$5.2 | 0.1852 | 0.1825 | 0.1855 | 0.1831 | 10.1$\pm$4.5 | 12.5$\pm$6.8 | 10.8$\pm$8.5
$V_{\rm disp,peak}$ | 0.2190 | 0.1721 | 0.2165 | 0.1701 | 0.2224 | 0.1750 | 0.2179 | 0.1712 | 8.7$\pm$4.0 | 13.4$\pm$6.2 | 5.4$\pm$5.4 | 0.2226 | 0.2192 | 0.2240 | 0.2202 | 12.5$\pm$4.4 | 16.8$\pm$6.3 | 8.8$\pm$8.7
$V_{\rm disp,80\%}$ | 0.1807 | 0.1413 | 0.1794 | 0.1403 | 0.1829 | 0.1432 | 0.1802 | 0.1409 | 6.2$\pm$4.1 | 11.4$\pm$6.2 | 4.3$\pm$4.9 | 0.1888 | 0.1876 | 0.1854 | 0.1872 | 11.4$\pm$4.1 | 14.4$\pm$6.1 | 11.1$\pm$8.8
$\phi$ | 0.1977 | 0.1538 | 0.1797 | 0.1399 | 0.2078 | 0.1633 | 0.1865 | 0.1452 | 22.0$\pm$4.3 | 18.5$\pm$5.7 | 20.0$\pm$5.3 | 0.2047 | 0.1856 | 0.2191 | 0.1929 | 21.2$\pm$4.2 | 20.4$\pm$7.1 | 22.0$\pm$8.8
$\psi_{1}$ | 0.1804 | 0.1407 | 0.1762 | 0.1371 | 0.1829 | 0.1429 | 0.1777 | 0.1385 | 10.6$\pm$4.0 | 12.7$\pm$5.7 | 9.5$\pm$5.6 | 0.1850 | 0.1815 | 0.1847 | 0.1822 | 11.6$\pm$4.4 | 13.6$\pm$6.7 | 12.8$\pm$9.2
$\psi_{2}$ | 0.1785 | 0.1395 | 0.1763 | 0.1377 | 0.1797 | 0.1406 | 0.1771 | 0.1383 | 9.5$\pm$3.9 | 12.7$\pm$5.5 | 8.1$\pm$5.3 | 0.1884 | 0.1865 | 0.1849 | 0.1862 | 13.0$\pm$4.4 | 15.9$\pm$6.3 | 12.8$\pm$8.7
$\psi_{3}$ | 0.1803 | 0.1405 | 0.1775 | 0.1382 | 0.1831 | 0.1428 | 0.1788 | 0.1392 | 8.4$\pm$4.1 | 11.6$\pm$6.3 | 7.3$\pm$5.3 | 0.1857 | 0.1828 | 0.1855 | 0.1834 | 10.8$\pm$4.3 | 12.8$\pm$6.5 | 11.9$\pm$8.9
$\psi_{4}$ | 0.1792 | 0.1399 | 0.1779 | 0.1389 | 0.1806 | 0.1411 | 0.1785 | 0.1394 | 7.4$\pm$4.0 | 12.0$\pm$6.0 | 5.8$\pm$5.3 | 0.1902 | 0.1888 | 0.1862 | 0.1883 | 11.9$\pm$4.3 | 14.8$\pm$6.7 | 12.0$\pm$9.0
$\psi_{5}$ | 0.1765 | 0.1380 | 0.1751 | 0.1368 | 0.1799 | 0.1410 | 0.1762 | 0.1378 | 3.5$\pm$4.2 | 7.7$\pm$6.0 | 3.9$\pm$5.6 | 0.1804 | 0.1795 | 0.1800 | 0.1796 | 7.8$\pm$4.5 | 10.0$\pm$6.8 | 6.5$\pm$8.2
$\psi_{6}$ | 0.1774 | 0.1388 | 0.1764 | 0.1380 | 0.1798 | 0.1410 | 0.1772 | 0.1387 | 4.3$\pm$4.1 | 8.8$\pm$6.0 | 3.3$\pm$5.1 | 0.1852 | 0.1852 | 0.1830 | 0.1848 | 8.8$\pm$4.4 | 11.1$\pm$6.9 | 6.6$\pm$8.1
$M_{DM,peak}$ |  redshift 0.5 | 0.2356 | 0.1839 | 0.2222 | 0.1738 | 0.2396 | 0.1876 | 0.2253 | 0.1762 | 20.1$\pm$5.1 | 20.2$\pm$6.9 | 19.5$\pm$11.1 |  redshift 2.0 | 0.2450 | 0.2357 | 0.2515 | 0.2377 | 10.4$\pm$5.5 | 11.0$\pm$6.1 | 7.3$\pm$13.4
$M_{\rm DM,90\%}$ | 0.2312 | 0.1773 | 0.2169 | 0.1657 | 0.2353 | 0.1831 | 0.2202 | 0.1687 | 21.0$\pm$5.1 | 22.2$\pm$7.3 | 20.3$\pm$11.2 | 0.2356 | 0.2273 | 0.2438 | 0.2294 | 10.5$\pm$5.8 | 12.0$\pm$6.4 | 8.7$\pm$13.5
$M_{\rm R_{max},peak}$ | 0.2828 | 0.2220 | 0.2755 | 0.2162 | 0.2789 | 0.2189 | 0.2761 | 0.2167 | 21.2$\pm$5.3 | 22.9$\pm$7.2 | 13.5$\pm$11.6 | 0.2856 | 0.2799 | 0.2838 | 0.2804 | 12.8$\pm$6.1 | 14.4$\pm$6.5 | 4.8$\pm$13.6
$M_{\rm R_{max},80\%}$ | 0.2672 | 0.2062 | 0.2609 | 0.2008 | 0.2673 | 0.2080 | 0.2620 | 0.2020 | 17.5$\pm$5.1 | 19.6$\pm$6.9 | 14.9$\pm$10.7 | 0.2767 | 0.2741 | 0.2742 | 0.2741 | 11.0$\pm$6.1 | 12.1$\pm$6.5 | 7.9$\pm$12.9
$|\dot{M_{\rm DM}}|_{\rm peak}$ | 0.4336 | 0.3495 | 0.4352 | 0.3504 | 0.4212 | 0.3425 | 0.4328 | 0.3490 | 8.6$\pm$5.2 | 12.1$\pm$7.1 | 15.9$\pm$12.2 | 0.4162 | 0.4171 | 0.4070 | 0.4158 | 5.4$\pm$5.5 | 8.0$\pm$5.9 | 30.5$\pm$14.4
$|\dot{M_{\rm DM}}|_{\rm 60\%}$ | 0.2269 | 0.1739 | 0.2260 | 0.1728 | 0.2288 | 0.1766 | 0.2265 | 0.1735 | 3.5$\pm$5.0 | 4.2$\pm$6.7 | 12.4$\pm$12.1 | 0.2443 | 0.2449 | 0.2400 | 0.2443 | 3.7$\pm$5.8 | 3.9$\pm$6.1 | 3.7$\pm$12.4
$V_{\rm peak}$ | 0.2078 | 0.1629 | 0.2046 | 0.1605 | 0.2058 | 0.1613 | 0.2048 | 0.1606 | 13.8$\pm$5.2 | 16.1$\pm$7.1 | 14.4$\pm$11.3 | 0.2224 | 0.2204 | 0.2155 | 0.2198 | 8.8$\pm$5.9 | 10.2$\pm$6.3 | 5.8$\pm$13.5
$V_{\rm 90\%}$ | 0.1901 | 0.1448 | 0.1877 | 0.1424 | 0.1870 | 0.1441 | 0.1876 | 0.1427 | 11.8$\pm$5.0 | 13.5$\pm$7.3 | 18.9$\pm$12.3 | 0.2083 | 0.2072 | 0.1982 | 0.2061 | 9.0$\pm$6.3 | 10.3$\pm$6.8 | 6.1$\pm$14.0
$V_{\rm disp,peak}$ | 0.2275 | 0.1785 | 0.2245 | 0.1761 | 0.2265 | 0.1776 | 0.2248 | 0.1764 | 14.0$\pm$5.0 | 16.3$\pm$7.0 | 11.9$\pm$11.5 | 0.2455 | 0.2437 | 0.2408 | 0.2433 | 9.1$\pm$6.0 | 10.6$\pm$6.9 | 4.7$\pm$13.6
$V_{\rm disp,80\%}$ | 0.1991 | 0.1515 | 0.1981 | 0.1502 | 0.1914 | 0.1471 | 0.1969 | 0.1497 | 12.2$\pm$5.2 | 14.6$\pm$7.4 | 17.7$\pm$11.1 | 0.2221 | 0.2224 | 0.2080 | 0.2206 | 9.6$\pm$6.1 | 10.9$\pm$6.7 | 6.2$\pm$14.3
$\phi$ | 0.2093 | 0.1606 | 0.1934 | 0.1498 | 0.2228 | 0.1731 | 0.1988 | 0.1539 | 19.5$\pm$5.0 | 20.4$\pm$7.1 | 20.5$\pm$11.2 | 0.2282 | 0.2197 | 0.2229 | 0.2201 | 12.6$\pm$6.5 | 13.8$\pm$6.9 | 9.3$\pm$13.7
$\psi_{1}$ | 0.1910 | 0.1455 | 0.1879 | 0.1425 | 0.1875 | 0.1446 | 0.1878 | 0.1428 | 12.9$\pm$5.4 | 15.1$\pm$7.1 | 19.2$\pm$12.5 | 0.2087 | 0.2070 | 0.1995 | 0.2061 | 8.9$\pm$6.1 | 10.0$\pm$7.1 | 6.3$\pm$13.8
$\psi_{2}$ | 0.1981 | 0.1511 | 0.1961 | 0.1490 | 0.1915 | 0.1473 | 0.1953 | 0.1487 | 13.2$\pm$5.3 | 15.1$\pm$6.9 | 17.8$\pm$11.4 | 0.2135 | 0.2112 | 0.2070 | 0.2107 | 10.1$\pm$6.3 | 11.6$\pm$6.8 | 8.0$\pm$12.2
$\psi_{3}$ | 0.1922 | 0.1465 | 0.1895 | 0.1439 | 0.1885 | 0.1454 | 0.1893 | 0.1441 | 12.4$\pm$5.5 | 14.1$\pm$6.6 | 16.1$\pm$12.2 | 0.2103 | 0.2090 | 0.2011 | 0.2080 | 8.9$\pm$6.4 | 10.0$\pm$6.6 | 6.1$\pm$13.4
$\psi_{4}$ | 0.2014 | 0.1534 | 0.2001 | 0.1518 | 0.1934 | 0.1487 | 0.1990 | 0.1513 | 13.1$\pm$5.5 | 15.7$\pm$6.9 | 11.6$\pm$10.6 | 0.2247 | 0.2248 | 0.2109 | 0.2231 | 9.7$\pm$6.1 | 10.7$\pm$6.6 | 7.2$\pm$12.9
$\psi_{5}$ | 0.1861 | 0.1419 | 0.1855 | 0.1411 | 0.1826 | 0.1410 | 0.1850 | 0.1411 | 9.6$\pm$5.1 | 10.9$\pm$7.0 | 12.4$\pm$12.2 | 0.2030 | 0.2030 | 0.1941 | 0.2020 | 6.5$\pm$6.7 | 7.3$\pm$7.1 | 5.1$\pm$13.2
$\psi_{6}$ | 0.1938 | 0.1477 | 0.1940 | 0.1475 | 0.1887 | 0.1454 | 0.1931 | 0.1471 | 9.0$\pm$5.2 | 10.1$\pm$7.2 | 14.0$\pm$12.5 | 0.2123 | 0.2129 | 0.2032 | 0.2117 | 6.6$\pm$6.1 | 7.2$\pm$6.3 | 5.8$\pm$12.7

=7.5cm

Table 3: Same as Table 1, but with $\sigma_{\star}=0.4$ dex.  Loss Metrics |  | Stellar Mass Prediction | Prediction of $w_{p}$ |  | Stellar Mass Prediction | Prediction of $w_{p}$
---|---|---|---|---|---|---
Galaxy Types |  | All | Central | Satellite | Mix | All | Blue | Red |  | All | Central | Satellite | Mix | All | Blue | Red
AM Scheme |  | Scatter | Loss | Scatter | Loss | Scatter | Loss | Scatter | Loss | Percentage Error |  | Scatter | Percentage Error
$M_{DM,peak}$ |  redshift 0 | 0.4840 | 0.3841 | 0.4759 | 0.3773 | 0.4818 | 0.3825 | 0.4772 | 0.3785 | 25.7$\pm$4.0 | 25.5$\pm$7.8 | 20.1$\pm$5.4 |  redshift 1.0 | 0.4904 | 0.4826 | 0.4902 | 0.4842 | 23.4$\pm$4.2 | 22.9$\pm$6.0 | 19.8$\pm$9.4
$M_{\rm DM,90\%}$ | 0.4835 | 0.3832 | 0.4738 | 0.3751 | 0.4811 | 0.3819 | 0.4755 | 0.3766 | 26.4$\pm$3.8 | 26.6$\pm$7.4 | 22.0$\pm$5.3 | 0.4896 | 0.4811 | 0.4881 | 0.4825 | 23.2$\pm$4.1 | 23.9$\pm$6.6 | 20.8$\pm$8.9
$M_{\rm R_{max},peak}$ | 0.5130 | 0.4080 | 0.5112 | 0.4065 | 0.5076 | 0.4033 | 0.5103 | 0.4058 | 23.6$\pm$3.8 | 26.0$\pm$7.6 | 17.4$\pm$5.4 | 0.5216 | 0.5181 | 0.5173 | 0.5180 | 23.9$\pm$4.2 | 25.2$\pm$7.0 | 17.4$\pm$8.9
$M_{\rm R_{max},80\%}$ | 0.4963 | 0.3940 | 0.4938 | 0.3918 | 0.4943 | 0.3925 | 0.4939 | 0.3920 | 21.6$\pm$3.8 | 22.1$\pm$7.3 | 16.1$\pm$5.2 | 0.5083 | 0.5055 | 0.5070 | 0.5058 | 20.0$\pm$4.2 | 20.5$\pm$6.6 | 19.5$\pm$8.6
$|\dot{M_{\rm DM}}|_{\rm peak}$ | 0.6355 | 0.5133 | 0.6398 | 0.5171 | 0.6213 | 0.5011 | 0.6356 | 0.5134 | 10.3$\pm$3.8 | 15.3$\pm$7.9 | 5.8$\pm$5.7 | 0.6350 | 0.6368 | 0.6270 | 0.6348 | 12.1$\pm$4.0 | 15.0$\pm$6.4 | 6.7$\pm$7.8
$|\dot{M_{\rm DM}}|_{\rm 60\%}$ | 0.4853 | 0.3850 | 0.4840 | 0.3837 | 0.4850 | 0.3850 | 0.4842 | 0.3840 | 6.6$\pm$4.1 | 6.7$\pm$6.8 | 11.3$\pm$5.6 | 0.4866 | 0.4868 | 0.4856 | 0.4866 | 5.8$\pm$4.0 | 6.1$\pm$6.1 | 9.1$\pm$8.5
$V_{\rm peak}$ | 0.4750 | 0.3772 | 0.4749 | 0.3770 | 0.4749 | 0.3775 | 0.4749 | 0.3771 | 11.5$\pm$3.9 | 17.7$\pm$7.5 | 8.2$\pm$5.3 | 0.4763 | 0.4757 | 0.4747 | 0.4755 | 14.1$\pm$4.0 | 17.2$\pm$6.2 | 11.6$\pm$8.4
$V_{\rm 90\%}$ | 0.4643 | 0.3679 | 0.4642 | 0.3676 | 0.4648 | 0.3688 | 0.4644 | 0.3679 | 9.2$\pm$3.8 | 9.7$\pm$5.8 | 7.6$\pm$5.3 | 0.4668 | 0.4667 | 0.4648 | 0.4663 | 12.0$\pm$4.3 | 15.0$\pm$7.1 | 12.7$\pm$9.1
$V_{\rm disp,peak}$ | 0.4817 | 0.3828 | 0.4814 | 0.3825 | 0.4825 | 0.3837 | 0.4817 | 0.3828 | 11.2$\pm$4.0 | 18.8$\pm$7.2 | 7.6$\pm$5.7 | 0.4852 | 0.4844 | 0.4846 | 0.4844 | 14.2$\pm$4.2 | 17.0$\pm$6.2 | 11.7$\pm$9.2
$V_{\rm disp,80\%}$ | 0.4623 | 0.3663 | 0.4628 | 0.3666 | 0.4618 | 0.3665 | 0.4626 | 0.3666 | 8.6$\pm$4.0 | 8.7$\pm$5.6 | 7.4$\pm$5.2 | 0.4685 | 0.4691 | 0.4647 | 0.4682 | 12.9$\pm$3.9 | 15.0$\pm$6.3 | 13.3$\pm$8.8
$\phi$ | 0.4705 | 0.3732 | 0.4627 | 0.3667 | 0.4746 | 0.3769 | 0.4655 | 0.3690 | 21.9$\pm$4.1 | 20.5$\pm$7.8 | 19.5$\pm$5.4 | 0.4754 | 0.4669 | 0.4818 | 0.4700 | 21.4$\pm$4.2 | 22.2$\pm$7.4 | 24.0$\pm$8.9
$\psi_{1}$ | 0.4623 | 0.3663 | 0.4615 | 0.3654 | 0.4620 | 0.3665 | 0.4616 | 0.3656 | 12.4$\pm$3.8 | 18.0$\pm$7.5 | 12.2$\pm$5.8 | 0.4667 | 0.4662 | 0.4643 | 0.4658 | 12.8$\pm$4.3 | 15.9$\pm$7.1 | 14.3$\pm$9.4
$\psi_{2}$ | 0.4612 | 0.3655 | 0.4612 | 0.3654 | 0.4602 | 0.3652 | 0.4610 | 0.3653 | 11.3$\pm$3.9 | 11.7$\pm$5.9 | 11.3$\pm$5.5 | 0.4683 | 0.4685 | 0.4644 | 0.4677 | 13.6$\pm$4.2 | 16.0$\pm$6.3 | 15.4$\pm$9.0
$\psi_{3}$ | 0.4623 | 0.3662 | 0.4621 | 0.3659 | 0.4621 | 0.3666 | 0.4621 | 0.3661 | 10.5$\pm$4.0 | 11.8$\pm$6.9 | 9.9$\pm$5.5 | 0.4670 | 0.4668 | 0.4647 | 0.4664 | 12.6$\pm$4.3 | 15.0$\pm$6.7 | 14.0$\pm$9.5
$\psi_{4}$ | 0.4615 | 0.3657 | 0.4620 | 0.3660 | 0.4607 | 0.3656 | 0.4617 | 0.3659 | 9.7$\pm$4.0 | 9.8$\pm$6.1 | 8.9$\pm$5.6 | 0.4691 | 0.4696 | 0.4650 | 0.4687 | 12.8$\pm$4.1 | 14.8$\pm$6.8 | 14.2$\pm$9.5
$\psi_{5}$ | 0.4602 | 0.3646 | 0.4607 | 0.3649 | 0.4603 | 0.3653 | 0.4606 | 0.3650 | 5.4$\pm$4.0 | 7.1$\pm$6.3 | 5.7$\pm$5.4 | 0.4643 | 0.4650 | 0.4619 | 0.4644 | 10.1$\pm$4.4 | 11.6$\pm$6.6 | 9.3$\pm$8.5
$\psi_{6}$ | 0.4605 | 0.3649 | 0.4611 | 0.3653 | 0.4602 | 0.3652 | 0.4609 | 0.3653 | 6.3$\pm$4.0 | 8.4$\pm$6.5 | 5.1$\pm$5.0 | 0.4666 | 0.4678 | 0.4633 | 0.4668 | 10.7$\pm$4.1 | 12.5$\pm$6.3 | 9.6$\pm$8.1
$M_{DM,peak}$ |  redshift 0.5 | 0.4946 | 0.3931 | 0.4878 | 0.3875 | 0.4954 | 0.3935 | 0.4892 | 0.3886 | 21.6$\pm$4.9 | 21.5$\pm$6.1 | 27.3$\pm$10.3 |  redshift 2.0 | 0.5044 | 0.4992 | 0.5070 | 0.5002 | 14.6$\pm$5.3 | 14.6$\pm$5.7 | 17.8$\pm$13.3
$M_{\rm DM,90\%}$ | 0.4944 | 0.3901 | 0.4874 | 0.3841 | 0.4941 | 0.3912 | 0.4886 | 0.3853 | 21.5$\pm$4.7 | 21.9$\pm$6.3 | 25.3$\pm$8.7 | 0.5028 | 0.4987 | 0.5043 | 0.4994 | 14.5$\pm$5.1 | 14.6$\pm$5.8 | 17.9$\pm$13.2
$M_{\rm R_{max},peak}$ | 0.5248 | 0.4177 | 0.5210 | 0.4146 | 0.5214 | 0.4143 | 0.5211 | 0.4145 | 24.4$\pm$5.1 | 25.5$\pm$6.7 | 23.4$\pm$11.5 | 0.5316 | 0.5284 | 0.5293 | 0.5285 | 16.2$\pm$6.0 | 16.9$\pm$6.1 | 19.7$\pm$14.0
$M_{\rm R_{max},80\%}$ | 0.5166 | 0.4083 | 0.5138 | 0.4057 | 0.5146 | 0.4076 | 0.5139 | 0.4060 | 20.0$\pm$4.9 | 20.5$\pm$6.5 | 26.9$\pm$11.1 | 0.5292 | 0.5284 | 0.5242 | 0.5279 | 15.8$\pm$5.8 | 16.3$\pm$6.5 | 21.0$\pm$13.3
$|\dot{M_{\rm DM}}|_{\rm peak}$ | 0.6300 | 0.5068 | 0.6309 | 0.5075 | 0.6241 | 0.5031 | 0.6297 | 0.5067 | 13.0$\pm$5.0 | 15.7$\pm$6.8 | 14.1$\pm$12.2 | 0.6232 | 0.6237 | 0.6183 | 0.6230 | 10.6$\pm$5.8 | 12.3$\pm$5.9 | 17.7$\pm$16.5
$|\dot{M_{\rm DM}}|_{\rm 60\%}$ | 0.4910 | 0.3876 | 0.4914 | 0.3878 | 0.4902 | 0.3877 | 0.4912 | 0.3878 | 6.5$\pm$5.0 | 5.6$\pm$6.5 | 14.5$\pm$11.4 | 0.5069 | 0.5080 | 0.5012 | 0.5072 | 8.7$\pm$5.8 | 8.3$\pm$6.4 | 10.8$\pm$13.3
$V_{\rm peak}$ | 0.4790 | 0.3805 | 0.4782 | 0.3798 | 0.4760 | 0.3781 | 0.4778 | 0.3795 | 16.3$\pm$4.9 | 17.7$\pm$6.6 | 12.6$\pm$10.1 | 0.4903 | 0.4897 | 0.4846 | 0.4891 | 12.7$\pm$5.5 | 13.2$\pm$6.0 | 12.0$\pm$13.6
$V_{\rm 90\%}$ | 0.4718 | 0.3717 | 0.4717 | 0.3713 | 0.4673 | 0.3694 | 0.4709 | 0.3710 | 14.1$\pm$5.0 | 15.1$\pm$7.1 | 18.4$\pm$10.0 | 0.4862 | 0.4864 | 0.4773 | 0.4853 | 12.3$\pm$5.7 | 12.7$\pm$6.0 | 11.5$\pm$13.9
$V_{\rm disp,peak}$ | 0.4897 | 0.3894 | 0.4889 | 0.3887 | 0.4874 | 0.3875 | 0.4887 | 0.3885 | 17.1$\pm$4.8 | 18.5$\pm$6.6 | 18.9$\pm$11.4 | 0.5039 | 0.5033 | 0.4995 | 0.5028 | 12.5$\pm$5.4 | 12.5$\pm$6.2 | 13.4$\pm$13.7
$V_{\rm disp,80\%}$ | 0.4766 | 0.3754 | 0.4771 | 0.3755 | 0.4698 | 0.3710 | 0.4759 | 0.3747 | 14.4$\pm$4.8 | 15.7$\pm$6.6 | 16.1$\pm$8.5 | 0.4948 | 0.4958 | 0.4832 | 0.4942 | 13.0$\pm$5.7 | 13.8$\pm$6.4 | 14.7$\pm$14.2
$\phi$ | 0.4795 | 0.3803 | 0.4721 | 0.3744 | 0.4854 | 0.3854 | 0.4744 | 0.3763 | 20.6$\pm$5.1 | 20.7$\pm$6.4 | 24.7$\pm$8.2 | 0.4931 | 0.4888 | 0.4887 | 0.4888 | 15.5$\pm$6.0 | 15.8$\pm$6.3 | 17.0$\pm$13.5
$\psi_{1}$ | 0.4722 | 0.3721 | 0.4717 | 0.3713 | 0.4675 | 0.3696 | 0.4710 | 0.3710 | 15.0$\pm$5.1 | 16.3$\pm$6.3 | 15.7$\pm$10.1 | 0.4865 | 0.4863 | 0.4779 | 0.4853 | 12.6$\pm$5.8 | 12.9$\pm$6.2 | 13.1$\pm$13.6
$\psi_{2}$ | 0.4759 | 0.3750 | 0.4760 | 0.3748 | 0.4698 | 0.3711 | 0.4749 | 0.3741 | 15.3$\pm$5.1 | 16.8$\pm$7.1 | 15.5$\pm$9.6 | 0.4894 | 0.4889 | 0.4823 | 0.4881 | 14.0$\pm$5.6 | 14.4$\pm$6.4 | 15.2$\pm$13.1
$\psi_{3}$ | 0.4728 | 0.3725 | 0.4725 | 0.3720 | 0.4680 | 0.3700 | 0.4717 | 0.3716 | 14.4$\pm$5.1 | 15.6$\pm$6.7 | 14.6$\pm$10.5 | 0.4874 | 0.4874 | 0.4788 | 0.4864 | 12.5$\pm$5.8 | 13.1$\pm$6.0 | 13.5$\pm$13.7
$\psi_{4}$ | 0.4777 | 0.3763 | 0.4781 | 0.3764 | 0.4708 | 0.3718 | 0.4768 | 0.3756 | 14.9$\pm$5.1 | 16.5$\pm$7.2 | 15.7$\pm$10.1 | 0.4963 | 0.4971 | 0.4848 | 0.4956 | 13.3$\pm$5.4 | 13.9$\pm$6.1 | 15.6$\pm$13.5
$\psi_{5}$ | 0.4695 | 0.3700 | 0.4703 | 0.3703 | 0.4649 | 0.3676 | 0.4694 | 0.3699 | 12.2$\pm$4.9 | 12.5$\pm$6.8 | 12.4$\pm$10.6 | 0.4831 | 0.4839 | 0.4748 | 0.4827 | 11.2$\pm$6.0 | 11.1$\pm$6.5 | 13.2$\pm$13.3
$\psi_{6}$ | 0.4735 | 0.3731 | 0.4747 | 0.3737 | 0.4682 | 0.3699 | 0.4736 | 0.3731 | 12.0$\pm$5.1 | 12.3$\pm$6.8 | 17.2$\pm$10.9 | 0.4889 | 0.4900 | 0.4803 | 0.4888 | 11.5$\pm$5.5 | 11.2$\pm$6.2 | 13.4$\pm$12.6