# The dependence of halo bias on the protohalo shape alignment with the
initial tidal field

Jounghun Lee 11footnotetext: Corresponding author. Jun-Sung Moon

###### Abstract

We present a numerical evidence supporting the primordial origin of secondary
halo bias even on the galactic mass scale. Analyzing the data from the
IllustrisTNG 300-1 simulations, we investigate the dependence of halo bias on
the degree of misalignment between the protohalo inertia and initial tidal
tensors, $\tau$, measured at redshift, $z_{i}=127$. From the TNG 300-1
galactic halos in logarithmic mass range of
$10.5<m\equiv\log[M/(h^{-1}M_{\odot})]\leq 13$ identified at $z=0,\ 0.5$ and
$1$, a clear signal of $\tau$ bias is detected. For the case that $\tau$ is
measured from the initial tidal field smoothed on the scale of
$R_{f}/(h^{-1}\,{\rm Mpc})\lesssim 1$, the halo $\tau$ bias is found to be
very similar in its tendency and amplitude to the spin bias at all of the
three redshifts, if the effects of backsplash halos are properly eliminated.
For the case of $R_{f}/(h^{-1}\,{\rm Mpc})=2$, the $\tau$ bias at $z=1$ turns
out to behave like the age bias, diminishing rapidly in the range of $m>12$.
At $z=0$ and $0.5$, however, the $\tau$ and age bias factors show large
differences in their overall strengths, which is attributed to the dominant
nonlinear effects that undermine the former but enhance the latter. Given
these numerical results along with the previous finding [1] that $\tau$ shares
a large amount of mutual information with the formation epochs and spin
parameters of galactic halos, it is concluded that the origins of halo age and
spin bias must be closely linked with the primordial factor, $\tau$, and that
the difference in the tendency between the two bias factors on the galactic
mass scale reflects the multi-scale influence of $\tau$ on the halo secondary
properties.

##  1 Introduction

The dark matter halos are biased tracers of the underlying dark matter (DM)
particles, as the former exhibits mass-dependent difference in their
clustering strengths from the latter [2, 3]. The halos with mass $M$ higher
than the characteristic mass scale $M_{\ast}$ tend to cluster more strongly
than the DM particles, while the opposite tendency is exhibited by the low-
mass counterparts with $M\lesssim M_{\ast}$. Here, $M_{\ast}$ satisfies the
condition of $\sigma(M_{\ast})=\delta_{c}$ with critical density contrast for
collapse, $\delta_{c}$, and rms density fluctuation, $\sigma$ [4]. This mass
dependence of halo bias is predicted by the standard excursion set theory [5,
6, 3], which effectively describes the growths of initial density peaks till
their gravitational collapse as random walking processes.

It was, however, discovered by multiple N-body simulations that the halo bias
depends not only on mass but also on secondary properties like formation
epoch, concentration, substructure abundance, velocity anisotropy, potential
depth, accretion rate, shape and spin parameter as well [7, 8, 9, 10, 11, 12,
13, 14, 16, 17, 15, 19, 18, 20, 21, 22, 23, 24, 25, 26, 27]. Among these
secondary properties, what turned out to exhibit the strongest bias signal is
the halo age for which the formation epoch and concentration parameter often
stand proxy [7, 9, 14, 19, 20, 21]. In the lowest-mass section ($M\ll
M_{\ast}$) is found a positive age bias trend that the older halos cluster
more strongly. While in the highest-mass section ($M\gg M_{\ast}$) is detected
the opposite tendency that the younger halos cluster more strongly, i.e.,
negative age bias. The inversion from negative to positive age bias trend was
witnessed around the characteristic mass scale $M_{\ast}$ [21]. Regarding the
other secondary properties, their bias trends were shown to be similar to that
of the halo ages, except for the spin bias, which inverts at a much lower mass
threshold $M\ll M_{\ast}$ [19, 21, 20, 28].

Ever since the first discovery of this phenomenon [7], much theoretical effort
was made to physically understand why the DM halos with different secondary
properties cluster differently. Although no standard theory has yet to be
fully established, it was conventionally accepted that the secondary bias of
low-mass halos differs in its origin from that of high-mass counterparts. In
the highest-mass section ($M\gg M_{\ast}$), the secondary halo bias was shown
by ref. [29] to have a primordial origin. Incorporating the peak curvature
dependence of initial density correlations into the excursion set theory, ref.
[29] analytically proved that the Lagrangian halo bias factor in the
asymptotic limit of $M\gg M_{\ast}$ is a function of the curvature of initial
density peaks, on which the halo formation epochs depend.

Whereas, in the lowest-mass section ($M\ll M_{\ast}$), the secondary bias was
often attributed to the nonlinear physical processes during their assembly
histories. Ref. [23] scrupulously examined various theories suggested by the
previous works for the origin of low-mass age bias and found that the
following three mechanisms should make the most significant contributions:
first, the presence of backsplash halos in the vicinity of their former
higher-mass hosts [31, 14, 28]; second, the effects of large-scale web
environments which can differentiate the formation epochs of halos at fixed
mass by either facilitating or impeding mass accretion [15]; third, the shock
heat generated during the collapse of largest-scale structures, which can slow
down the mass accretion into the filament and sheet halos [30, 29].

In the analysis of ref. [23], however, it was also found that when the
backsplash halos were excluded, the fraction of low-mass halos yielding
positive age bias was quite low, which implied that not only the highest-mass
halos but also the lower mass halos may have had negative age bias of
primordial origin until they were vulnerably exposed to the aforementioned
physical mechanisms. This idea, the primordial origin of secondary halo bias
on the lower-mass scale of $M\lesssim M_{\ast}$, was also supported by the
result of ref. [24] that the secondary halo properties are more strongly
correlated with the linear density field rather than with the evolved density
field. According to ref. [24], the secondary halo bias should be a collateral
byproduct of the multi-scale competition between the external and internal
correlations of secondary halo properties with the linear density field, where
the external and internal correlations are measured on the scales larger and
smaller than the Lagrangian halo radius, respectively. In these previous
works, however, it was still inconclusive which primordial factor actually
drives the halo bias to acquire dependence on the secondary properties even on
the mass scale of $M\lesssim M_{\ast}$, and what causes the age and spin bias
to behave differently on this mass scale [21, 20, 28], if both of them share
the same origin [13, 20].

For the verification of the primordial origin of the secondary bias of lower
mass halos, what is required is to find out a primordial factor on which the
lower-mass halo bias exhibits the same strength of dependence as (or at least
similar to) on the other secondary properties, and to explain the difference
between the age and spin bias in terms of this singled out primordial factor.
In light of our prior work where the ages and spins of galactic halos were
found to share a large amount of mutual information with the degree of
misalignments, $\tau$, between the principal axes of protohalo inertia and
initial tidal tensors, we put forth a conjecture that $\tau$ may be the
primordial factor, and attempt to verify this conjecture by exploring the halo
$\tau$ bias. As in ref. [23], we will focus on the galactic mass scale,
$10.5<m\equiv\log\left[M/(h^{-1}\,M_{\odot})\right]\leq 13$, where the
existence of the secondary bias of primordial origin remains a crux, and the
trend of spin bias significantly deviates from that of age bias [19, 21, 20,
28].

The outlines of the upcoming sections are as follows. In section 2.1, the halo
age and spin bias factors are reviewed, and their mutual differences on the
galactic mass scales are reinspected. In section 2.2, the dependence of the
halo bias on the primordial factor $\tau$ and the variation of its strength
with mass and redshift are investigated, and compared with those of the halo
age and spin bias. In section 3, the key results are summarized and their
physical implications are discussed.

##  2 Detection of halo $\tau$ bias and its implication on the secondary bias

###  2.1 A brief review of halo age and spin bias trends

Figure 1:  Relative age bias, $\hat{b}(a_{f},m)$, from the $8$ $m$-selected
subsamples of the galactic halos belonging to the top (red filled circles) and
bottom (blue filled circles) quartiles of their ages versus the mean value of
$m$ averaged over each $m$-selected subsample. at three different redshifts.
In each panel, the horizontal the grey filled area represents the
$1\sigma_{b}$ Jackknife scatter around the relative halo bias $\hat{b}(m)$
from the whole population of the galactic halos in the logarithmic mass range
of $10.5\leq m\leq 13$. Figure 2:  Relative spin bias from the top (red filled
circles) and bottom (blue filled circles) quartiles of the halo spin
parameters. $\lambda$.  Figure 3:  Relative spin in the whole range of
$\lambda$ (red filled circles) from the galactic halos belonging to four
different $m$-bins at $z=0$, $0.5$ and $1$.

To numerically explore if the secondary halo bias has a primordial origin even
on the galactic mass scale of $10.5<m\leq 13$, we utilize the data from the
300 Mpc volume run (TNG300-1) of the IllustrisTNG suite of cosmological gravo-
magneto-hydrodynamical simulations [32, 33, 34, 35, 36, 37]. The IllustrisTNG
300-1 run was conducted on a box of volume $V_{\rm
tot}\equiv(205)^{3}\,h^{-3}{\rm Mpc}^{3}$ with the Planck initial condition
[41] to track the dynamical evolution of $2\times 2500^{3}$ particles. One
half of the particles in the TNG 300-1 are baryonic cells with individual mass
of $1.1\times 10^{7}\,M_{\odot}$, while the other half are collisionless DM
particles each of which is as massive as $5.9\times 10^{7}\,M_{\odot}$. The
standard friends-of-friends (FoF) with the linkage length parameter of $0.2$
and SUBFIND algorithm [42] were employed to resolve the bound halos and their
substructures, respectively, in the TNG 300-1 snapshots [34]. Selecting the
most massive central subhalos of the FoF groups, we extract information on
their comoving positions (${\bf x}$), total masses ($M$), formation epochs
($a_{f}$) as defined in ref. [38, 39] and the spin parameters ($\lambda$) as
defined in ref. [40] from the publicly released dataset of the TNG 300-1
simulations. Throughout this paper, we will refer to the selected central
subhalos as galactic halos, and consider only two secondary properties,
$a_{f}$ and $\lambda$, of them.

In the standard excursion set theory [3], the Eulerian halo bias factor,
$b(m)$, can be expressed at a given redshift as

$b(m)\equiv\frac{n(m|\delta>0)}{n(m)\Delta V}\,.$ (2.1)

Here $n(m)$ is the mean number density of DM halos in a differential interval
of $[m,m+dm]$, observed at $z$, while $n(m|\delta>0)$ is the number density of
those halos in the same differential interval of $m$ but embedded in a large
over-dense region of Eulerian volume $\Delta V$ with matter density contrast
$\delta$. Note that eq. (2.1) is equivalent to defining the bias factor as
$b(m)\equiv\delta_{h}(m)/\delta$, where $\delta_{h}(m)$ is the number density
contrast of the halos in the mass range of $[m,m+dm]$ within the volume
$\Delta V$ [3].

The following flow chart describes how we determine the relative age and spin
bias factors from the TNG 300-1 data in accordance with eq.(2.1).

  1. 1.

Divide the simulation volume, $V_{\rm tot}$, into $64^{3}$ grids of equal
volume $\Delta V\equiv V_{\rm tot}/(64^{3})$. Construct the matter density
contrast field, $\delta$, on the grids by applying the cloud-in-cell algorithm
to the particle snapshot at each of the three redshifts.

  2. 2.

Select the galactic halos in the logarithmic mass range of
$10.5<m\equiv\ln[M/(h^{-1}\,M_{\odot})]\leq 13$ to make a main sample.

  3. 3.

Split the main sample into eight subsamples of equal size according to the
values of $m$. Treat the galactic halos belonging to each $m$-selected
subsample as particles, construct the halo number density contrast field,
$\delta_{h}(m)$, with the help of the cloud-in-cell method.

  4. 4.

Examine the $\delta$ value at each of $64^{3}$ grids and select those grids at
which $\delta\geq\sigma$ to minimize the shot noise. Take the ratio,
$\delta_{h}(m)/\delta$, at each of the selected grids, and determine $b(m)$ as
the ensemble average of this ratio over the selected grids for each of the $8$
$m$-selected subsamples.

  5. 5.

Create eight jackknife resamples by dividing the selected grids points
according to the grid locations, and compute the Jackknife errors from the
eight resamples (one standard deviation scatter of the bias factor obtained
from each resample from the original $b(m)$ obtained over $V_{\rm tot}$.

  6. 6.

Repeating the above steps (2)-(5), but with only those youngest (oldest)
galactic halos whose formation epochs belong to the top (bottom) quartile
values of $a_{f}$, determine the halo age bias, $b(a_{f},m)$, and take its
ratio to $b(m)$ to obtain the relative age bias, $\hat{b}(a_{f},m)\equiv
b(a_{f},m)/b(m)$.

  7. 7.

Repeating the above steps (2)-(5), but with only those rapidly (slowly)
spinning galactic halos whose spin parameters belong to the top (bottom)
quartile of $\lambda$, determine the spin bias, $b(\lambda,m)$ and then take
its ratio to $b(m)$ to obtain the relative spin bias,
$\hat{b}(\lambda,m)\equiv b(\lambda,m)/b(m)$.

A total of $618777$, $637406$ and $644753$ central subhalos are selected at
$z=0$, $0.5$ and $1$, respectively. Figure 1 plots $\hat{b}(a_{f},m)$ versus
the mean values of $m$ of eight $m$-selected subsamples of the galactic halos
with $a_{f}$ belonging to its top and bottom quartiles (blue and red filled
circles, respectively). As can be seen, our results are quite consistent with
what was found in the previous works [7, 8, 9, 11, 16, 19, 20, 21], capturing
two characteristic tendencies of the age bias: the older halos ($25\%$ lowest
$a_{f}$) cluster more strongly than the younger ones ($25\%$ highest $a_{f}$)
[21] and this age dependence of the halo bias becomes weaker at higher $z$
[18]. Figure 2 plots the same as figure 2 but from the top and bottom
quartiles of $\lambda$, confirming the existence of spin bias. As can be seen,
the inversion from the negative to positive spin bias is witnessed around
$m\simeq 11.5$ only at $z=0$. Meanwhile, at higher $z$ the spin bias appears
to become more significant as $m$ increases in the mass range considered,
showing no inversion. The overall trends of $\hat{b}(a_{f},m)$ and
$\hat{b}(\lambda,m)$ are all quite consistent with the previous findings [7,
8, 9, 10, 11, 14, 16, 19, 20, 21].

As can be seen, the spin bias is less significant than the age counterpart on
the galactic mass scale. To see more closely how the halo bias changes with
$\lambda$ at each redshift, we split the main sample into four subsamples
corresponding to the $m$-ranges of $[10.5,11)$, $[11,11.5)$, $[11.5,12)$ and
$[12,13]$. Dividing the range of $\lambda$ of the galactic halos belonging to
each of the four $m$-selected subsample into eight bins, we compute
$\hat{b}(\lambda)$ separately at each $\lambda$ interval via the above flow
chart. Figure 3 shows $\hat{b}(\lambda)$ versus $\lambda$ from the four
$m$-selected subsamples at the three redshifts, revealing that the inversion
of $\hat{b}(\lambda)$ occurs not only in a $m$-dependent way but also in a
$z$-dependent way. The galactic halos of mass $10.5\leq m\leq 11$ decreases
(increases) with $\lambda$ at $z=0$ ($z=1$), while at $z=0.5$, they show
little variation with $\lambda$.

It is worth mentioning here that we estimate the secondary halo bias by taking
the excursion set approach based on eq. (2.1) rather than by employing the
conventional methods based on the density correlations or power spectra [7, 9,
12, 19, 20, 21, 23]. True as it is that the conventional methods are quite
efficient for the estimation of secondary halo bias in a wide mass range, it
suffers from an ambiguity in the choice of the range of separation distance
$r$, which may make it difficult to properly investigate the dependence of
halo bias on a scale-dependent secondary property like $\tau$, the primary
focus of the current analysis. Nevertheless, our method turns out to reproduce
quite well the overall trends of age and spin bias estimated by the
conventional correlation function method in the galactic mass range.

###  2.2 Scale and redshift dependence of halo $\tau$-bias

As mentioned in section 1, it was found by our prior work [1] that the basic
traits of galactic halos share a large mount of mutual information with a
primordial factor, $\tau$, defined as the degree of misalignment222Since the
principal axes of a protohalo inertia tensor are aligned with it shape axes,
it is equivalent to saying that $\tau$ describes how weakly the protohalo
shape axes are anti-aligned with the principal axes of the initial tidal
field. between the principal axes of protohalo inertia and initial tidal
tensors. For the case of massive galactic halos with $m\geq 11.5$, the
primordial factor $\tau$ was found to contain larger amounts of mutual
information about the formation epochs and spin parameters than the halo mass,
local density and environmental shears (see figure 3 in ref. [1]).
Furthermore, the probability density functions of both of $\lambda$ and $\tau$
were also shown to be best approximated by the same Gamma distribution [1].
These prior results naturally lead us to expect the existence of $\tau$ bias
and to speculate its possible links with the age and spin bias.

In the following, we concisely describe the computational steps via which the
relative $\tau$ bias factor, $\hat{b}(\tau,m)$, is measured:

  1. 1.

Trace back the constituent particles of each galactic halo back to the
earliest epoch of redshift $z_{i}=127$ to find their Lagrangian positions,
${\bf q}\equiv(q_{i})$, and their center of mass ${\bf q}_{c}\equiv(q_{c,i})$,
as well. Then, compute the inertia tensor of a located protohalo, $I_{jk}({\bf
q}_{c})$, as [1]

$\hat{I}_{jk}\equiv\sum_{\mu=1}^{n_{p}}m_{\mu}(q_{\mu,j}-q_{c,j})(q_{\mu,k}-q_{c,k})\,,$
(2.2)

where $m_{\nu}$ is the mass of $\nu$th particle located at the initial
position ${\bf q}_{\mu}\equiv(q_{\mu,i})$, and $n_{p}$ is the number of the
particles that constitute the protohalo.

  2. 2.

Apply the cloud-in-cell method to the particle snapshot at redshift
$z_{i}=127$ to determine the initial matter density contrast field,
$\delta({\bf q})$, on $512^{3}$ grid points as in [1]. Find the Fourier space
initial density contrast field, $\delta({\bf k})$, via a Fast Fourier
transformation (FFT) of $\delta({\bf q})$ with the Fourier space wave vector
of ${\bf k}$.

  3. 3.

Compute the Fourier space initial tidal field convolved by a Gaussian kernel
with scale radius $R_{f}$, as $\tilde{T}_{ij}({\bf k})\equiv
k_{i}k_{j}\tilde{\delta}({\bf k})\exp(-|{\bf k}|^{2}R^{2}_{f}/2)/|{\bf
k}|^{2}$. Then, perform an inverse FFT of it to obtain the real-space initial
tidal field $[{T}_{ij}({\bf q})]$. At the grid point in which ${\bf q}_{c}$
falls, diagonalize $T_{ij}({\bf q}_{c})$ to find its three orthonormal
eigenvectors, $\\{{\bf e}_{1},\ {\bf e}_{2},\ {\bf e}_{3}\\}$.

  4. 4.

Construct a rotation matrix, ${\bf R}$, whose three columns correspond to
$\\{{\bf e}_{1},\ {\bf e}_{2},\ {\bf e}_{3}\\}$, and perform a similarity
transformation of the protohalo inertia tensor as
$I^{\prime}_{jk}=R^{-1}_{jl}I_{ls}R_{sk}$.

  5. 5.

For each galactic halo, compute the degree of misalignment, $\tau(r_{f})$,
between the principal axes of the protohalo inertia and initial tidal tensors
smoothed on the scale of $r_{f}\equiv R_{f}/(h^{-1}{\rm Mpc})$ as [1]

$\tau(r_{f})\equiv\left[\frac{(I^{\prime}_{12})^{2}+(I^{\prime}_{23})^{2}+(I^{\prime}_{31})^{2}}{(I^{\prime}_{11})^{2}+(I^{\prime}_{22})^{2}+(I^{\prime}_{33})^{2}}\right]^{1/2}\,.$
(2.3)

  6. 6.

Follow the flow chart described in section 2.1 to obtain the relative $\tau$
bias factor for three different cases of $r_{f}=0.5$, $1$ and $2$, denoted by
$\hat{b}(\tau_{0.5},m)$, $\hat{b}(\tau_{1},m)$ and $\hat{b}(\tau_{2},m)$, at
each redshift.

Figure 4:  Relative $\tau_{1}$ bias from the galactic halos whose primordial
$\tau_{1}$ values belong to the top (red filled circles) and bottom (blue
filled circles) quartiles, where $\tau_{1}$ quantifies how weakly the triaxial
proto-halos shapes are aligned with the principal axes of the initial tidal
field smoothed on the scale of $R_{f}/(h^{-1}{\rm Mpc})=1$ (see eq.(2.2)).
Note a robust similarity between $\hat{b}(\tau_{1},m)$ and
$\hat{b}(\lambda,m)$ (see figure 2) at $z=1$. Figure 5:  Relative $\tau_{1}$
in the whole range of $\tau_{1}$ (red filled circles) from the galactic halos
belonging to four different $m$-bins at $z=0$, $0.5$ and $1$. Figure 6:  Same
as figure 4 but for the case of $\tau_{2}$ that quantifies how weakly the
triaxial shapes of protohalos are aligned with the principal axes of the
initial tidal field smoothed on the larger scale, $R_{f}/(h^{-1}{\rm Mpc})=2$.
Note a similarity between the relative $\tau_{2}$ and $a_{f}$ bias (see figure
1) at $z=1$. Figure 7:  Same as figure 4 but for the case of $\tau_{0.5}$ that
quantifies how weakly the triaxial shapes of protohalos are aligned with the
principal axes of the initial tidal field smoothed on the smaller scale,
$R_{f}/(h^{-1}{\rm Mpc})=0.5$.

Figure 4 shows the relative $\tau_{1}$ bias, revealing how
$\hat{b}(\tau_{1},m)$ behaves as $m$ changes at different redshifts. The
factor by which $\hat{b}(\tau_{1},m\gtrsim 12)$ enhances as $\tau_{1}$ changes
from the bottom to top quartile reaches up to $\sim 1.4$ at $z=0$ but slightly
diminishes down to $\sim 1.3$ at $z=1$. In the lower mass range of $m\leq
11.5$, the relative $\tau_{1}$ bias has insignificantly low values at $z=0$
but tends to become significant as $z$ increases. The overall trend is that
the $\tau_{1}$ bias mildly increases as $m$ increases at all of the three
redshifts. Note that the behavior and amplitude of $\hat{b}(\tau_{1},m)$ at
$z=1$ is quite similar to that of $\hat{b}(\lambda,m)$ shown in figure 2. But,
at $z=0.5$ and $0$, a notable difference exists between $\hat{b}(\tau_{1},m)$
and $\hat{b}(\lambda,m)$ especially in the low mass range $m\leq 11.5$. Unlike
$\hat{b}(\lambda,m)$, no inversion to the negative bias at $m\simeq 11.5$ is
exhibited by $\hat{b}(\tau,m)$ at $z=0$.

As mentioned in section 1, however, it was already proved by ref. [28] that
the inversion of $\hat{b}(\lambda,m\simeq 11.5)$ at $z\simeq 0$ was entirely
due to the locations of abundant backsplash halos in the vicinity of massive
former hosts. When the backsplash halos were properly excluded from the
calculation of $\hat{b}(\lambda,m)$ at $z=0$, the inversion signal disappeared
even at $z=0$. What ref. [28] also found that at $z\geq 1$ the effect of
backsplash halos on $\hat{b}(\lambda,m)$ was negligible, since the abundance
of backsplash halos decreases sharply with the increment of $z$. Comparing
$\hat{b}(\tau_{1},m)$ with the results of ref. [28], i.e.,
$\hat{b}(\lambda,m)$ measured after the elimination of backsplash halos (see
figure 3 in ref. [28]), we note that a robust similarity between the spin and
$\tau_{1}$ bias factors exists even at $z=0$.

To gain a better insight on the dependence of halo bias on $\tau_{1}$ at each
redshift, we also compute $\hat{b}(\tau_{1})$ separately from the four
$m$-selected subsamples by following the same procedure used to measure
$\hat{b}(\lambda)$ in section 2.1. Figure 5 shows the same as figure 3 but
with $\tau_{1}$ substituting for $\lambda$. The comparison with figure 3
reveals that at $z=0.5$ and $1$, the overall trends of $\hat{b}(\tau_{1})$ and
$\hat{b}(\lambda)$ are indeed quite similar to each other not only in the top
and bottom quartiles but also in the whole ranges of $\tau_{1}$ and $\lambda$.
Note that even at $z=0$ where the effects of backsplash halos are dominant the
two bias factors show similar behaviors in the range of $12\leq m\leq 13$.

To see how the relative $\tau$ bias changes with the smoothing scale $r_{f}$,
we repeat the whole analysis but for the case of $r_{f}=2$ and $0.5$. Figure 6
plots the same as figure 4 but for the case of $r_{f}=2$, revealing a striking
difference in the behavior of $\hat{b}(\tau_{2},m)$ from that of
$\hat{b}(\tau_{1},m)$. In contrast to $\hat{b}(\tau_{1},m)$, a mild decrease
of $\hat{b}(\tau_{2},m)$ with the increment of $m$ is found in the whole mass
range considered. Note that at $z=1$ the overall trend of the relative
$\tau_{2}$ bias is quite similar to the relative age bias shown in figure 1.
Recalling the results of [23] that the three nonlinear mechanisms had an
effect of deviating the halo age bias from the primordial trend by augmenting
it especially in the low-mass range ($m\lesssim 12.5$), we interpret the
similarity between $\hat{b}(\tau_{2},m)$ and $\hat{b}(a_{f},m)$ at $z=1$ as an
evidence supporting the primordial origin of the halo age bias [23, 24]. We
also take the low $\tau_{2}$ bias signal at $z=0$ as a result indicating that
the three nonlinear processes undermines the $\tau_{2}$ bias at low redshift,
deviating further $\hat{b}(\tau_{2},m)$ from $\hat{b}(a_{f},m)$. In other
words, while the halos undergo the three nonlinear processes, they tend to
lose the primordially generated connections between $\tau_{2}$ and $a_{f}$.

Figure 7 plots the same as figure 4 but for the case of $r_{f}=0.5$. Although
the $\tau_{0.5}$ bias is similar in its overall trend to the $\tau_{1}$
counterpart, the strength of the former seems to be significant even in the
low-mass range of $m\lesssim 11.5$ at $z=0$, in contrast to the latter. The
comparison of the results shown in figures 1-7 indicates that the behavior and
strength of $\tau(r_{f})$ depends quite sensitively on $r_{f}$ and that there
is two characteristic scales, say $r_{a}\simeq 2$ and $r_{s}\simeq 1$, on
which the behaviors of $\tau(r_{f}=r_{a})$ and $\tau(r_{f}=r_{s})$ bias
factors become quite similar to those of the spin and age bias, respectively,
at $z\simeq 1$. Any departure of $r_{f}$ from these characteristic scales in
either direction cause $\hat{b}(\tau)$ to behave differently from
$\hat{b}(a_{f})$ and $\hat{b}(\lambda)$, respectively. We speculate that the
primordial effect of $\tau(r_{f})$ on $a_{f}$ and $\lambda$ should be the
strongest on these particular scales $r_{a}$ and $r_{s}$, respectively.
Moreover, given that $r_{a}$ is larger than $r_{s}$, it can be inferred that
$a_{f}$ and $\lambda$ are affected by the larger and smaller scale effects of
$\tau$ in the primordial epoch, respectively. In other words, the differences
between the age and spin bias may be primordially caused by the multi-scale
effects of $\tau(r_{f})$. At lower redshifts $z\lesssim 0.5$ when the
secondary bias factors are severely modified by the nonlinear physical
processes [23], however, the primordial $\tau(r_{f})$ effects seem to dwindle
away especially in the low-mass range, no matter what $r_{f}$ is.

##  3 Summary and conclusion

From the TNG 300-1 galactic halos in the mass range of
$10.5<m\equiv\log[M/(h^{-1}M_{\odot})]\leq 13$ at $z=0,\ 0.5$ and $1$, we have
detected the significant dependence of halo bias on the primordial factors,
$\tau_{0.5}$, $\tau_{1}$ and $\tau_{2}$, which denote the degree of
misalignments of the triaxial protohalo shapes with the initial tidal tensors
smoothed on the scales of $r\equiv R_{f}/(h^{-1}{\rm Mpc})=0.5,\ 1$ and $2$,
respectively (see figures 4-7). Comparing the relative bias of the three
primordial factors, $\hat{b}(\tau_{0,5},m)$, $\hat{b}(\tau_{1},m)$, and
$\hat{b}(\tau_{2},m)$, with the relative age and spin bias, $\hat{b}(a_{f},m)$
and $\hat{b}(\lambda,m)$ (see figures 1-2), we have found that at $z=1$, the
overall trend of $\hat{b}(\tau_{1},m)$, is very similar to that of
$\hat{b}(\lambda,m)$ in the entire mass range considered. Although an apparent
difference has been found between $\hat{b}(\tau_{1},m)$ and
$\hat{b}(\lambda,m)$ in the low-mass range ($m\leq 11.5$) at $z=0$ and $0.5$,
it has been realized that the exclusion of backsplash halos fully recovers the
robust similarity between them [28] even in the low-mass range of $m\leq 11.5$
at all of the three redshifts.

It has also been found that $\hat{b}(\tau_{2},m)$ and $\hat{b}(a_{f},m)$
exhibit similar behaviors at $z=1$, but evolve in opposite directions at lower
redshifts: as $z$ decreases, the former becoming weaker while the latter
stronger. We have ascribed the deviation of $\hat{b}(a_{f},m)$ from
$\hat{b}(\tau_{2},m)$ at $z\leq 0.5$ to the three physical processes that ref.
[23] already showed were mainly responsible for the age bias of low-mass halos
at $z=0$: the large-scale web environments, shock heats generated during the
formation of large scale structures, and clustering of the backsplash halos.
These nonlinear processes seem to have two prolonged effects: diminishing the
dependence of halo bias on the primordial factor, $\tau_{2}$, and enhancing
the age bias of low-mass halos at lower redshifts.

We interpret $\tau(r_{f})$ as the protohalo response to the multi-scale
influences of the initial tidal field, ${\bf T}(r_{f},z_{i})$. A higher
(lower) value of $\tau(r_{f})$ corresponds to a slower (faster) response of a
protohalo to ${\bf T}(r_{f},z_{i})$. If protohalos of same mass $M$ respond
more slowly (rapidly) to ${\bf T}(r_{f},z_{i})$, then they would be less
(more) susceptible to its effect, which will eventually produce a difference
between them in their secondary properties linked with ${\bf T}(r_{f},z_{i})$.
For the case of the spin parameter, it is produced under the primary influence
of ${\bf T}(r_{f}\simeq r_{\rm L},z_{i})$ [43] where
$r_{L}\equiv\left[4\,M/(3\pi\,\bar{\rho})\right]^{1/3}$ is the Lagrangian
virial radius of the protohalo [24]. Any difference in the response to ${\bf
T}(r_{f}\simeq r_{\rm L},z_{i})$ would cause a difference in $\lambda$ among
the protohalos of same mass, originating the halo spin bias and its similarity
to the $\tau(r_{f}\simeq r_{L})$ bias until the effects of backsplash halos
modify the spin bias in the lower-mass range of $m\leq 11.5$ at low redshifts.
Whereas, for the case of the formation epochs, it is determined largely by
${\bf T}(r_{f}>r_{\rm L},z)$ (i.e., the initial tidal effects on the scales
larger than $r_{L}$). If protohalos of same mass respond more slowly (rapidly)
to ${\bf T}(r_{f}>r_{\rm L},z)$ due to weak (strong) alignments of their
shapes with ${\bf T}(r_{f}>r_{\rm L},z)$, then their formations will be
delayed (expedited), which in turn will create the age bias similar to the
$\tau(r_{f}>r_{L})$ bias. The memory for the primordial $\tau(r_{f}>r_{L})$
factor will be retained in the age bias until it is severely undermined by the
nonlinear effects found by ref. [23].

Backing up the conjecture previously suggested by refs. [23, 24] that the
secondary halo bias must have a primordial origin regardless of halo mass, the
current work has made two new contributions. Firstly, we single out the
primordial factor, $\tau$, the bias of which behaves like the age and spin
bias on the galactic mass scale at $z\simeq 1$. Secondly, we have provided an
explanation for the difference in behavior and tendency between the spin and
age bias of galactic halos in terms of the primordial $\tau$ factors defined
on different scales. Our future work is in the direction of deriving an
analytic model for the halo $\tau$ bias in terms of linear quantities from the
first principle, and exploring the $\tau$ bias in a much wider range of mass.

## Acknowledgments

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. JL
acknowledges the support by Basic Science Research Program through the NRF of
Korea funded by the Ministry of Education (No.2019R1A2C1083855). JSM
acknowledges the support by the National Research Foundation (NRF) of Korea
grant funded by the Korean government (MEST) (No. 2019R1A6A1A10073437).

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