# The impact of constrained interacting dark energy on the bound-zone velocity
profile

Jounghun Lee 11footnotetext: Corresponding author. Marco Baldi

###### Abstract

We numerically study the effects of constrained interacting dark energy
(CIDER) on the bound-zone velocity profiles around massive dark matter halos.
Analyzing the CIDER simulations performed by ref. [1] for three different
cases of dark sector coupling ($\beta=0.03$, $0.05$ and $0.08$) as well as for
the standard $\Lambda$CDM cosmology ($\beta=0$), we determine the mean
peculiar velocity profiles in the bound zones around the friends-of-friends
halos with masses larger than $M_{\rm cut}=3\times 10^{13}\,h^{-1}M_{\odot}$
at three redshifts, $z=0$, $0.5$ and $1$. It is found that the universal
power-law formula proposed by ref. [2] originally for the $\Lambda$CDM case
still describes well the bound-zone velocity profiles, $V(r)$, even in the
CIDER models. The slope of $V(r)$, turns out to be significantly affected by
the CIDER, progressively decreasing as $\beta$ increases. Meanwhile, the
amplitude of $V(r)$ exhibits little dependence on $\beta$, which is ascribed
to the identical Hubble parameters shared by the $\Lambda$CDM and CIDER models
in the entire redshift range. Our results imply that the bound-zone velocity
slope can break a degeneracy even between the $\Lambda$CDM and CIDER models
with $\beta\leq 0.03$, which the standard cosmological diagnostics fail to
distinguish. We devise a simple analytic formula for the bound-zone slope as a
function of $\beta$, and prove its validity at all of the three redshifts. It
is concluded that the slope of the mean bound-zone peculiar velocity profile
should be in principle a powerful probe of dark sector interaction.

##  1 Introduction

It has been well known that as the peculiar velocity field contains an
independent dynamical information on the background cosmology [3], it is most
suitable for testing degenerate non-standard cosmologies like dynamical dark
energy and modified gravity scenarios [4, 5, 6, 7]. In the seminal work of
ref. [2] it was numerically proven that in the bound-zones around massive dark
matter (DM) halos where the gravitational attraction is greatly diluted by the
cosmic acceleration in the late-universe, the peculiar velocity behaves like a
linear variable and thus its radial profile can be analytically tractable.
Here, a bound-zone of a massive halo with virial radius, $r_{\rm v}$, refers
to the distance range, ($3-8$)$r_{\rm v}$ at which its neighbor lower-mass
halos recede from it at lower speeds than the global Hubble flow due to the
halo’s gravitational attraction [2]. It is located in-between the infall
($<3r_{\rm v}$) and Hubble ($>8r_{\rm v}$) zones where the recession speeds
become essentially zero and equal to the global value, respectively [4, 8].

Multiple numerical works have shown that the bound-zone velocity profile is
well approximated by the following universal formula with two adjustable
parameters $A$ and $n$ [9, 10, 2, 8, 7]:

$\frac{V(r)}{V_{\rm v}}=A\left(\frac{r}{r_{\rm v}}\right)^{-n}\,,$ (1.1)

where $V(r)$ is the mean radial component of the peculiar velocity field at a
bound-zone distance $r$ from a massive halo with mass $M$, while $V_{\rm v}$
is the radial component of virial velocity defined as $V_{\rm v}\equiv
GM/r_{\rm v}$. The universality of eq. (1.1) is manifested by the weak
dependences of $n$ and $A$ on $M$ [2] as well as on the initial conditions of
the concordance model [8] where the gravitational law is described by the
general relativity (GR), dark energy (DE) responsible for the cosmic
acceleration is the cosmological constant $\Lambda$, and the most dominant
source of gravity is the cold dark matter (CDM).

This universal form of the bound-zone velocity profile is believed to be
established by a delicate counter-balance between the gravity and accelerating
spacetime [8]. Its amplitude, $A$, measures the overall strength of
gravitational influence at the boundary between the infall and bound zones
where the global Hubble flow begins to be dominant. Meanwhile, its slope, $n$,
measures how rapidly the gravitational influence wanes in the bound zone due
to the cosmic acceleration. Thus, it is naturally expected that any deviation
of the background from the $\Lambda$CDM cosmology would affect the shape of
$V(r)$. Ref. [7] explored if and how the shape of $V(r)$ would change in the
presence of modified gravity (MG) and massive neutrinos ($\nu$). According
their results, a substantial difference exists in the best-fit values of $A$
and $n$ between the $\Lambda$CDM cosmology and a certain MG+$\nu$ model, which
cannot be discriminated by the conventional cosmological diagnostics [11].

The shape of the bound-zone velocity profile, $V(r)$, will be particularly
useful to test those non-standard cosmologies which affect only one of the two
parameters, $n$ and $A$ in eq. (1.1) and thus suffer from no extra degeneracy
between the two parameters. For example, if a non-standard cosmology differs
from the $\Lambda$CDM in its growth history but not in the expansion
counterpart, then, it would leave the amplitude, $A$, of $V(r)$ intact,
modifying only its slope, $n$, since $A$ depends most sensitively on the value
of the Hubble parameter at a given epoch [2, 8, 7]. The recently proposed
constrained interacting dark energy (CIDER) models [12] are such non-standard
models as could be powerfully tested by the slope of $V(r)$.

The CIDER is a special type of interacting scalar field DE, $\phi$, coupled
with the CDM through energy-momentum exchange [13, 14, 15, 16, 17, 18, 19, 20,
21, 22]. In the classical interacting DE models, the DE-CDM coupling have two
prolonged effects on the dark sector. First, it drives the dynamical evolution
of the DE density and its equation of stage. Second, it promotes the density
growths by exerting an attractive fifth force on the CDM particles. By tracing
the cosmic expansion and growth histories, it is in principle possible to
constrain the strength of DE-CDM coupling, which enters as a coefficient in
the scalar field DE potential, $U(\phi)$ [13, 14, 15, 18]. A critical
difference between the CIDER and the classical interacting DE lies in the fact
that the shape of $U(\phi)$ is specified a priori for the latter but not for
the former, whose dynamical evolution instead determines $U(\phi)$ under the
condition of identical Hubble parameter to the $\Lambda$CDM case in the whole
redshift range. In other words, the $\Lambda$CDM and CIDER models can be
differentiated from each other only through their difference in the growth
histories, as the two share the same expansion history.

The CIDER scenario has recently garnered probing attention as it turned out to
be capable of alleviating the $\sigma_{8}$ tension that the local estimates of
the linear density power spectrum amplitude, $\sigma_{8}$, is significantly
lower than the best-fit value from the cosmic microwave background radiation
temperature spectrum (see ref. [23] and references therein for a detailed
review of the current status on the $\sigma_{8}$ tension). In contrast to the
majority of other interacting DE and MG effects, a lower $\sigma_{8}$ is
predicted by the CIDER models in which the dark sector coupling has an effect
of suppressing the density growths, the degree of which increases with the
coupling strength [12]. Yet, no large difference in the growth history can be
accommodated by the current observational results between the $\Lambda$CDM and
a viable CIDER model. In other words, a CIDER model is viable only provided
that it is almost degenerate with the $\Lambda$CDM, having very weak dark
sector couplings.

Given this distinct features of the CIDER models, we speculate that the CIDER
might affect only the slope, $n$, of the mean bound-zone velocity profile with
amplitude $A$ intact, and thus that $V(r)$ might be able to break the
degeneracy between the $\Lambda$CDM and a viable CIDER model. Here, we will
test our speculation against a large-volume simulation performed by ref. [1]
for three different CIDER models. The organization of this paper as follows.
In section 2.1, we will briefly review the datasets from the CIDER
simulations. In section 2.2, we will explain how to measure the mean bound-
zone velocity profiles from simulation data and present what effects the CIDER
has on the bound-zone velocity slopes. In section 2.3, we will present a
simple fitting formula for the dependence of bound-zone velocity slope on the
dark sector coupling. In section 3, we will summarize the results and draw a
conclusion.

##  2 Bound-zone velocity profiles in the CIDER models

###  2.1 The CIDER simulations: a brief review

Figure 1:  Numerically obtained bound-zone velocity profiles (black filled
circles) compared with the analytic single parameter (red solid line) and
double parameter (blue dashed line) formulae for three different CIDER models
as well as for the standard $\Lambda$CDM cosmology at $z=0$. Figure 2:  Same
as figure 1 but at $z=0.5$.  Figure 3:  Same as figure 1 but at $z=1$. Figure
4:  $68\%$, $95\%$ and $99\%$ contours of $\chi^{2}(A,n)$ between the
numerical and analytical results of the bound-zone velocity profiles in the 2D
plane spanned by $A$ and $n$ that denote two adjustable parameters of the
analytical formula given in eq. (1.1) for the four cosmologies at $z=0$.
Figure 5:  Same as figure 4 but at $z=0.5$. Figure 6:  Same as figure 4 but at
$z=1$.

To numerically explore the effect of CIDER on the mean bound-zone velocity
profile, we utilize the halo catalogs extracted from the CIDER simulation [1]
that belongs to a second generation of the CoDECS 222It stands for COupled
Dark Energy Cosmological Simulations project [20], a suite of $N$-body
simulations for interacting DE models, called the CoDECS 2 project. As most of
the classical interacting DE models considered in the original project had
gone out of favor with current observational data [25, 26, 27], the CoDEC 2
has been launched to venture various new interacting DE scenarios including
the CIDER. The unique constraint that the Hubble parameter, $H(z)$, follows
the identical redshift evolution to that of the $\Lambda$CDM case
characterizes the CIDER models whose potential, $U(\phi)$, can be derived from
this constraint under the assumption that the DE-CDM coupling does not vary
with time [12]. Unlike in the majority of the interacting DE and MG models,
the matter density field in the CIDER models have suppressed powers [12], the
degree of which is determined by the value of its dark sector coupling
constant, $\beta$ [1].

The CIDER simulations were performed by ref. [1] on a periodic box of side
length $1\,h^{-1}$Gpc for four different cosmologies: the $\Lambda$CDM,
CIDER003, CIDER005, and CIDER008 corresponding to $\beta=0,\ 0.03,\ 0.05$ and
$0,08$, respectively. According to the results of ref. [1], the CIDER models
with $\beta\leq 0.03$ could not be discriminated from the $\Lambda$CDM by the
standard linear diagnostics such as linear density power spectrum and linear
growth fact, while the extreme CIDER008 model can be safely ruled out by the
current observational data [26]. Except for the density power spectrum
amplitude ($\sigma_{8}$), these four models share the identical initial
conditions of $h=0.677$, $\Omega_{\rm de}=0.689$, $\Omega_{m}=0.262$,
$\Omega_{b}=0.049$, and $n_{s}=0.9665$, with dimensionless Hubble parameter
($h$), DE density parameter ($\Omega_{\rm de}$), matter density parameter
($\Omega_{m}$) , baryon density parameter ($\Omega_{b}$) and spectral index
($n_{s}$). The values of $\sigma_{8}$ for these four models are listed in the
third columns of table 1.

Starting at redshift $z_{i}\sim 99$ with these initial conditions, the CIDER
simulations tracked down the dynamical evolution of $1024$ cDM particles and
equal number of baryonic particles as well, with the help of the C-Gadget code
[24], an adapted version of the Gadget 3 code [28] for the computation of dark
sector interaction. With the bound halos identified from the CIDER snapshots
via the standard friends-of-friends (FoF) algorithm with a linkage length
parameter of $0.2$, ref. [1] evaluated the halo mass functions, bias
functions, density profiles and concentration-mass relations for each model.
It was found that even these nonlinear statistics could not tell apart the
$\Lambda$CDM from the CIDER models if $\beta\leq 0.03$. For a comprehensive
description of the CIDER simulations, we refer the readers to ref. [1].

###  2.2 The effect of dark sector coupling on the slope of $V(r)$

For each model, we determine the mean bound-zone velocity profiles averaged
over the massive FoF halos at three different redshifts, $z=0,\ 0.5$ and $1$,
by taking the following steps [7]:

  1. 1.

Select those FoF halos with total mass $M\geq 3\times
10^{13}\,h^{-1}\,M_{\odot}$. Determine the virial radii of the selected halos
as $r_{\rm v}\equiv\left(3M/[4\pi\,\Delta\rho_{\rm crit}(z)]\right)^{1/3}$
where critical density of the universe $\rho_{\rm crit}(z)$ and halo
overdensity threshold $\Delta\equiv 100$ [2].

  2. 2.

Using each selected halo as a target, search for its bound-zone neighbor halos
located at radial separation distances, $r\in(3-8)r_{\rm v}$. If any of the
bound-zone neighbors turn out to be more massive than the target, then exclude
the selected halo from the target list. From here on, the neighbor lower-mass
halos located in the bound-zone of a target halo will be referred to as the
bound-zone neighbors.

  3. 3.

For each target halo, compute the relative radial velocities of its bound-zone
neighbors and rescale them as $\tilde{V}\equiv V/V_{\rm v}$. Also, rescaling
their separation distances as $\tilde{r}\equiv r/r_{\rm v}$, divide the range
of $3\leq\tilde{r}\leq 8$ into ten short intervals of equal length,
$\Delta\tilde{r}=0.5$.

  4. 4.

Find the bound-zone neighbors of all target halos that fall in each
$\tilde{r}$ bin. Compute the average of $\tilde{V}$ and its one standard
deviation error $\sigma(\tilde{V})$ over the stacked bound-zone neighbors at
each $\tilde{r}$ bin.

  5. 5.

Compare the numerically obtained $\tilde{V}(\tilde{r})$ with the formula,
$A\tilde{r}^{-n}$ over the whole range of $\tilde{r}$ to determine the best-
fit values of $A$ and $n$ via the minimization of $\chi^{2}(A,n)$:

$\chi^{2}(A,n)\equiv\frac{1}{(N_{\rm d.o.f})}\sum_{i=1}^{N_{\rm
bin}}\frac{\left[\log\tilde{V}(\tilde{r}_{i})-\log
A+n\log\tilde{r}_{i}\right]^{2}}{\sigma^{2}({\tilde{V}})}\,,$ (2.1)

where $N_{\rm bin}=10$ is the number of $\tilde{r}$ bins, and $N_{\rm d.o.f.}$
is the degree of freedom, equal to $N_{\rm bin}-2$ as the analytic formula has
two free parameters. Figures 1-3 plot the mean bound-zone velocity profiles
(filled black circles) obtained via the above steps for four cosmological
models, at $z=0,\ 0.5$ and $1$, respectively. The universal formula, eq.
(1.1), with the best-fit values of $A$ and $n$ is also shown (red solid line)
for comparison in each panel, confirming that this formula for the bound-zone
velocity profile is indeed valid even in the CIDER models. Note also that the
errors in $\tilde{V}(\tilde{r})$ appear to increase as $\beta$ increases,
which should be attributed to the smaller numbers of the selected massive
halos in the CIDER models with higher $\beta$ due to their more severely
suppressed density powers (see table 1).

Figures 4-6 plot the $68\%,\ 95\%$ and $99\%$ contours of $\chi^{2}(A,n)$
(blue, red and green curves) in the two-dimensional space spanned by $A$ and
$n$ for each case. Note again that as $\beta$ increases, the sizes of the
contours become larger due to the larger errors in $\tilde{V}$ caused by the
smaller numbers of the selected massive halos, as listed in table 1. As can be
seen, while the slope parameter, $n$, exhibits a sensitive variation with
$\beta$ at each redshift, the amplitude,$A$, appears to be almost independent
of $\beta$. The bottom panel of figure 7 plots the best-fit values of $A$
versus the scale factor, $a\equiv 1/(1+z)$, from the four cosmologies. For
this plot, the scale factor values of $A(a)$ are slightly offset for
visibility purposes. As $\beta$ increases, the best-fit slope, $n$, becomes
lower, which indicates that the stronger dark sector coupling causes the
gravitational influence in the bound zones to wane less rapidly.

Noting that the dark sector coupling does not affect the amplitude of the
bound-zone velocity profiles for the case of CIDER models, we regard eq. (1.1)
as a single parameter formula and repeat the procedure of fitting $V(r)$ to
eq. (1.1) by adjusting only $n$. The amplitude, $A$, is fixed at the best-fit
value of the $\Lambda$CDM case at each redshift, and the degree of freedom is
accordingly set at $N_{\rm bin}-1$ in this procedure. In figures 1-3, the
results of this one single-parameter fitting is shown (blue dashed lines) for
each case. As can be seen, despite that $A$ is fixed, the overall agreements
between the numerical and analytical results are excellent for every case. The
top-panel of figure 7 plots the resulting best-fit values of $n$ versus scale
factors, revealing that the four models yield significantly different values
of $n$ at each epoch. Even between the $\Lambda$CDM and CIDER003 models, the
difference in the best-fit value of $n$ is found to be as significant as $\geq
4\sigma$ at each redshift, which implies that the bound-zone velocity slope
could in principle break the degeneracy between the $\Lambda$CDM and CIDER
models with $\beta\leq 0.0.3$.

###  2.3 Modeling the $\beta$-dependence of bound-zone velocity slope

Figure 7:  Best-fit values of the power-law index (top panel) and amplitude
(bottom panel) versus scale factor, compared with the linear fits (black
dashed lines) for the four cosmologies. To make the data points visible, we
place the off-set. Figure 8:  $\beta$ dependence of two linear-fit constants
(red filled circles) to the power-law index of the bound-zone velocity
profile, compared with the approximate formula (black dashed lines) where
$\beta$ denotes the dark sector coupling constant.

Now that the best-fit slope $n$ is found to vary almost linearly with the
scale factor $a$ for each case (figure 7), we model the differences in the
bound-zone slope between the $\Lambda$CDM and a CIDER model at a given epoch,
$n(a,\beta)$, as the following linear function: $\Delta n(a,\beta)$:

$\Delta n(a,\beta)\equiv C_{a}(1-a)+C_{\beta}\,,$ (2.2)

where $C_{a}$ represents the slope of this linear fit, while $C_{b}$ is the
value of $\Delta n$ at the present epoch ($a=1$). Through the
$\chi^{2}(C_{a},C_{\beta})$-minimization, we also determine the best-fit
values of $C_{a}$ and $C_{\beta}$, which are shown in the top and bottom
panels of figure 8, respectively. As can be seen, $C_{a}$ shows little
variation with $\beta$, while $C_{\beta}$ appears to be proportional to some
power of $\beta$.

We model the observed $\beta$ dependence of $C_{\beta}$ as a power-law
function:

$C_{\beta}\equiv\left(\frac{\beta}{\beta_{0}}\right)^{n_{\beta}}\,.$ (2.3)

Employing the $\chi^{2}$ statistics again, we determine the best-fit values of
$\beta_{0}$ and $n_{\beta}$ to be $0.301\pm 0.079$ and $1.781\pm 0.156$,
respectively, where the associated errors are the marginalized ones. The
bottom panel of figure 8 shows this best-fit power-law function (dashed line)
revealing that $\Delta n$ at the present epoch is indeed well described by our
power-law function, eq. (2.3).

Model | $\beta$ | $\sigma_{8}$ | $N_{h}(z=0)$ | $N_{h}(z=0.5)$ |  $N_{h}(z=1)$
---|---|---|---|---|---
$\Lambda$CDM | $0.00$ | $0.788$ | $151142$ | $92139$ |  $43462$
CIDER003 | $0.03$ | $0.763$ | $135339$ | $78806$ |  $35369$
CIDER005 | $0.05$ | $0.723$ | $109621$ | $58922$ |  $23777$
CIDER008 | $0.08$ | $0.642$ | $60748$ | $26093$ |  $8126$
Table 1:  Model, dark sector coupling constant ($\beta$) density power
spectrum amplitude ($\sigma_{8}$) and numbers of the selected massive halos
($N_{h}$) with $M\geq 3\times 10^{13}\,h^{-1}\,M_{\odot}$ at $z=0,\ 0.5$ and
$1$.

##  3 Summary and conclusion

We have numerically examined the usefulness of the mean bound-zone velocity
profiles as a test of the constrained interacting dark energy (CIDER) models,
which are believed to be capable of alleviating the $\sigma_{8}$ tension. For
this investigation, we have relied on the CIDER simulations [1] performed for
three CIDER models, which differ from one another in the value of dark sector
coupling constant ($\beta$) but share the same expansion history and key
cosmological parameters other than $\sigma_{8}$ as the $\Lambda$CDM case [12].

Analyzing the FoF halos from the CIDER simulations, we have numerically
determined the mean bound-zone velocity profiles, $V(r)$, of massive halos
with $M\geq 3\times 10^{13}\,h^{-1}\,M_{\odot}$ at $z=0,0.5$ and $1$ for the
$\Lambda$CDM and three CIDER models. It has been found that the universal
power-law formula given by ref. [2] excellently describes $V(r)$ even in the
CIDER models and that the CIDER affects only the slope of $V(r)$ with its
amplitude intact. This $\beta$-independence of the bound-zone velocity
amplitude has been interpreted as a consequence of the same expansion history.

Noting that the bound-zone velocity slope has a progressively lower value as
$\beta$ increases, we have modeled $V(r)$ by a single-parameter power-law
formula with constant amplitude, and found excellent agreements with the
numerically obtained $V(r)$. Showing the high statistical significances of the
differences in the bound-zone velocity slope between the $\Lambda$CDM and
CIDER models, we have proven that the single parameter, i.e., the bound-zone
velocity slope, can in principle break the degeneracy between the $\Lambda$CDM
and those CIDER models with very low dark sector coupling constant, $\beta\leq
0.03$, which cannot be distinguished by the standard cosmological diagnostics
[1]. It has also been shown that the differences in the bound-zone velocity
slope between the $\Lambda$CDM and CIDER models behave like some powers of
$\beta$ at the present epoch, and evolve almost linearly with the scale
factor, $a$. Given this result, we have devised a simple power-law formula for
this slope difference as a function of $(a,\beta)$ and confirmed its validity.

Regarding the observational feasibility of using $V(r)$ as a cosmological
probe, the major obstacle is nothing but the noisy measurements of the
peculiar velocities of real galaxies (e.g., see ref. [30] for a review on this
issue). Although successful reconstructions of the large-scale peculiar
velocity field from the observational galaxy surveys have been reported in
several literatures [31, 32, 33, 34], the velocity reconstruction itself
required a priori assumption on the background cosmology. To test the CIDER
models with bound-zone velocity slope in practice, it is highly desirable to
make a cosmology independent estimation of $V(r)$ around massive clusters.

As shown in the previous works [2, 29, 35, 36, 37, 38], it is possible to
estimate the peculiar velocities of bound-zone galaxies around massive target
clusters from observations with information only on their redshifts if the
target clusters satisfy the following two conditions. First, their masses have
already been estimated with high accuracy. Second, the bound-zone galaxies are
located in anisotropic web environments like filaments and sheets whose axes
are not aligned with the line of sight directions [2, 35]. As most of the
galaxies in the universe are known to be embedded in filaments [39], the
second condition does not impose any severe observational limitation on the
section of targets. Whereas, the first condition would restrict the target
selection to those well relaxed low-$z$ clusters ($z\ll 1$) since the mass
measurements of high-$z$ clusters in the middle of merging often suffer from
large uncertainties [40]. Nevertheless, given that the upcoming next-
generation weak lensing surveys like the Legacy Survey of Space and Time5 [41]
are expected to achieve the unprecedentedly precise measurements of the masses
of high-$z$ clusters, we conclude that the bound-zone velocity slope holds a
good future prospect as a powerful complimentary probes of those non-standard
cosmologies whose expansion histories are the same as the $\Lambda$CDM model.

## Acknowledgments

JL acknowledges the support by Basic Science Research Program through the NRF
of Korea funded by the Ministry of Education (No.2019R1A2C1083855). MB
acknowledges support by the project “Combining Cosmic Microwave Background and
Large Scale Structure data: an Integrated Approach for Addressing Fun-
damental Questions in Cosmology”, funded by the MIUR Progetti di Ricerca di
Rilevante Interesse Nazionale (PRIN) Bando 2017 - grant 2017YJYZAH.

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