# Super Interacting Dark Sector: An Improvement on Self-Interacting Dark
Matter via Scaling Relations of Galaxy Clusters

Mahdi Naseri<EMAIL_ADDRESS>Department of Physics, K.N. Toosi
University of Technology, P. O. Box 15875-4416, Tehran, Iran  Javad T.
Firouzjaee<EMAIL_ADDRESS>Department of Physics, K.N. Toosi University
of Technology, P. O. Box 15875-4416, Tehran, Iran  School of physics,
Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531,
Tehran, Iran

###### Abstract

Abstract: Self-interacting dark matter is known to be one of the most
appropriate candidates for dark matter. Due to its excellent success in
removing many astrophysical problems, particularly in small scale structure,
studying this model has taken on added significance. In this paper, we focus
on the results of two previously performed simulations of cluster size halos
with self-interacting dark matter and introduce a new function for the density
profile of galaxy clusters, which can perfectly describe the result of these
simulations. This density profile helps to find a velocity dispersion profile
and also a relation between cluster mass and concentration parameter. Using
these relations, we investigate two scaling relations of galaxy clusters,
namely mass-velocity dispersion and mass-temperature relations. As an
important discrepancy appears in the calculated masses for given temperatures
regarding the obtained mass-temperature relation, we propose a new interacting
model and show that this model can completely solve the problem. This model is
a combination of self-interacting dark matter idea with another model of the
dark sector in which dark matter particle mass is determined according to its
interaction with dark energy. We find that in this super interacting dark
sector (SIDS) model, the virial ratio changes to have higher values than
$\frac{1}{2}$. Finally, we provide quantitative expressions that can describe
the constants of this interacting model with the value of cross-section per
unit mass of dark matter particles.

###### Contents

  1. I Introduction
  2. II Halo Profiles
    1. II.1 Density Profile
    2. II.2 Mass-Concentration Relation
    3. II.3 Velocity Dispersion
  3. III Scaling Relations
    1. III.1 Mass-Velocity Dispersion Relation
    2. III.2 Mass-Temperature Relation
  4. IV Super Interacting Dark Sector
  5. V Conclusion

##  I Introduction

Dark matter (DM) is believed to be the second largest component of the total
mass-energy of the universe, accounting for about five times as much as the
contribution of baryonic matter Ade:2015xua . Studying velocity dispersion of
galaxies in the Coma cluster Zwicky:1937zza , rotation curves of the Andromeda
galaxy Babcock , and spiral galaxies Rubin  were the preliminary reasons that
led to the idea of DM. In more recent efforts, many other cosmological and
astrophysical tests have verified the existence of DM in the cosmos, including
X-ray observations of the Coma cluster Briel:1997hz , the Bullet cluster
observations Komatsu , gravitational lensing of galaxies
AdelmanMcCarthy:2005se  and cosmic microwave background (CMB) anisotropies
Challinor:2012ws .

Despite strong evidence for the existence of DM, its nature has been under
discussion for a prolonged period of time. There is a wide range of candidates
that have been proposed in particle physics to justify different cosmological
effects of DM, and no absolute consensus has been built yet among physicists.
Some of those candidates had been under study for a while but were finally
ruled out by further exploration. For example, once stellar remnants (such as
black holes and neutron stars) were considered as DM candidate, namely massive
compact halo objects (MACHOs), which can explain the missing mass of the
galaxies Mohapatra:1999ih . However, it was found that MACHOs can only contain
about $3\%$ of the Milky Way dark matter mass Bahcall:1994xu ; Freese:2000vi .

On the contrary, many candidates in particle physics can still attract
attention as a DM model, such as Axions and WIMPs, to name but a few. Axion
was postulated as a Peccei-Quinn solution to the strong-CP problem in quantum
chromodynamics Peccei:1977ur  and immediately considered as a DM candidate
Weinberg:1977ma . Axionic DM particles are neutral low-mass particles that can
interact via the weak and the strong forces and decay into microwave photons.
Thus, Axionic DM could theoretically be detected with the aid of these emitted
photons. The other introduced candidate, WIMPs (weakly interacting massive
particles), might be the most interesting suggestion. WIMPs merely interact
via the weak force (no interaction through the electromagnetic force brings
about no emission of light) and have a mass of between 2 GeV (Lee-Weinberg
bound) and 100 TeV Roszkowski:2017nbc . Given these properties, one can
theoretically obtain the relic density of WIMPs consistent with its right and
observed value in the universe. This amazing feature is known as the ’WIMP
miracle’ in literature.

The standard $\Lambda$CDM model with WIMPs have been very successful in
describing large scale structures. Nevertheless, there are a number of
problems with this model in small scales of the cosmos. The first problem
stems from the detection experiments of DM. These experiments try to detect DM
particles via thair scattering with atomic nuclei in underground laboratories.
While ’SuperCDMS’ Agnese:2017njq  and ’PICO’ Amole:2017dex  experiments found
the lower limit of DM-nucleon scattering cross-section to be somewhere in the
region of $10^{-42}-10^{-45}cm^{2}$ for DM particle with the mass of
$\approx$10 GeV, theoretical predictions attribute this value to WIMPs with
the mass of $\approx$100 GeV. This discrepancy remains with many orders of
magnitude different compared to theory in some other experiments. In addition,
cold DM (CDM) cosmological simulations predict the formation of $\approx$500
subhalos around the Milky Way Moore:1999nt , whereas solely about $\approx$50
satellites and dwarf spheroidal galaxies (dSphs) have been identified around
the Milky Way so far satellite . These simulations Springel:2008cc  also show
that hundreds of thousands of satellites should be formed within the Milky
Way, with masses of $\approx 10^{5}M_{\odot}$ or higher. Since the number of
predicted satellites for galaxy size halos is way higher than discovered
satellites, this contradiction has been known as the ’missing satellite
problem’. Eventually, the so-called ’core-cusp problem’ is another challenge
for collisionless DM. Although CDM simulations of the density profile of
halos, embracing Navarro-Frenk-White (NFW) density profile Navarro:1996gj ,
lead to a steep slope of density profile in the central region of halos (cuspy
profile), galaxies rotation curves do not indicate a steep slope and instead,
reveal rather cored profiles near the center cored .

All these mentioned cosmological problems can be solved considering a type of
collisional DM with a non-negligible cross-section. The idea of self-
interacting dark matter (SIDM) was initially proposed in Spergel:1999mh  in
order to resolve these problems and rapidly turned into an interesting
alternative for CDM. In this model, DM particles can interact with themselves
as they have a non-negligible cross-section, similar to that of the strong
force. It has been proved that such an assumption may result in a heat
transfer which reduces the density of the central region of halos and
consequently, eliminates the core-cusp problem. The decline of density in the
inner region of halos also leads to the formation of fewer satellites and
removes the missing satellite problem.

Particle physics has a number of candidates for SIDM, including Q-balls
Kusenko:2001vu , hidden sectors in the String theory Faraggi:2000pv  and
hidden charged dark matter Feng:2009mn . Basically, SIDM may comprise two
different possibilities with regard to its scattering cross-section per unit
mass i.e. $\frac{\sigma}{m}$. In the first possibility, self-interaction is
resulted from assuming hard-sphere scattering among DM particles, which
suggests a constant value of $\frac{\sigma}{m}$. On the other hand,
$\frac{\sigma}{m}$ could be velocity dependent considering a Yukawa potential
for SIDM scattering in the second possibility Loeb:2010gj .

There are also different constraints on the value of a constant
$\frac{\sigma}{m}$ which have been obtained so far. A lower limit of
approximately $\frac{\sigma}{m}>0.1cm^{2}g^{-1}$ is required for SIDM to be
distinguishable from CDM on small scales Zavala . The shape of clusters can be
studied to find upper limits on velocity independent cross-section per unit
mass of SIDM. In this sense, the value of nearly
$\frac{\sigma}{m}<1cm^{2}g^{-1}$ is obtained Peter:2012jh . In fact, most
physical and observational constraints have been derived in this range and it
would be a suitable choice to consider
$0.1cm^{2}g^{-1}<\frac{\sigma}{m}<1cm^{2}g^{-1}$ for acceptable cases.

Galaxy clusters could be as much helpful as small scale structures to conduct
research into SIDM. In this regard, simulations play a significant role, and
making a comparison between their results and observations provides a host of
opportunities to investigate cluster size SIDM halos. In this work, we use the
outcomes of previously performed simulations, in order to study halo profiles
and scaling relations of galaxy clusters, and ultimately introduce a novel
description of SIDM through connecting our results to another dark energy (DE)
model.

The paper is organized as follows: Section II is dedicated to finding
quantitative relations for halo profiles, embracing density profile and
velocity dispersion profile, according to the result of simulations. A mass-
concentration relation can also be derived regarding the modified density
profile. In Section III, two important scaling relations are obtained, namely
mass-velocity dispersion and mass-temperature relations. As the obtained mass-
temperature suggests too high masses for ordinary temperatures, a new model is
explained in Section IV to reform the SIDM model and remove this problem.
Finally, we conclude and summarize the main results in Section V.

##  II Halo Profiles

Characterizing a halo includes various profiles and variables, such as density
profile, total mass, concentration parameter, halo shape, and velocity
dispersion profile. Considering self-interaction among DM particles may
dramatically alter these characteristics in both large and small scale
structures, and the imprint of collisional DM can emerge in a variety of
astrophysical tests. For example, the effect of SIDM on galaxy wraps is
investigated in Pardo:2019wie .

In order to study SIDM cluster size halo profiles, Brinckmann et al. performed
a dark-matter-only simulation, containing 28 dynamically relaxed halos with
masses of $\approx 10^{15}M_{\odot}$ in CDM and SIDM models Brinckmann:2017uve
. In a more comprehensive and more recent work, Andrew Robertson et al.
performed another similar simulation, but with adding baryonic matter effects
and also another mass window of $\approx 10^{14}M_{\odot}$ Robertson:2018anx .
Both studies have been presented for two choices of $\frac{\sigma}{m}$, which
are $\frac{\sigma}{m}=0.1cm^{2}g^{-1}$ (hereafter SIDM0.1) and
$\frac{\sigma}{m}=1cm^{2}g^{-1}$ (hereafter SIDM1). Even though the latter
simulation has also been performed with velocity-dependent $\frac{\sigma}{m}$,
this model is not our objective in this paper. We concentrate on the results
of the previous simulations, mainly the latter, and try to convert those
results to mathematical expressions that can be used to determine halo
characteristics and to derive scaling relations in the next Section. As we
deduce our relations from Robertson:2018anx , we can solely obtain and use our
equations for $0.1cm^{2}g^{-1}<\frac{\sigma}{m}<1cm^{2}g^{-1}$ and
$10^{14}<\frac{M}{M_{\odot}}<10^{15}$, analogous to the upper and lower limits
of the given simulation. Meanwhile, all the results are found for low-redshift
clusters, due to the assumption of $z\approx 0$ in the simulation.

###  II.1 Density Profile

A well-known density profile according to a CDM simulation is the NFW density
profile Navarro:1996gj , which is expressed as

$\frac{\rho}{\rho_{0}}=\frac{1}{x(1+x)^{2}}\,,$ (1)

where $x=\frac{r}{r_{s}}$ and $r_{s}$ is the scale radius of halo. Moreover,
$\rho_{0}=\rho_{crit}\delta_{c}$ is calculated for each halo, where
$\rho_{crit}$ is the critical density of the universe at given redshift (z)
and $\delta_{c}$ depends on the concentration parameter (c) with the relation
of

$\delta_{c}=\frac{200}{3}\frac{c^{3}}{ln(1+c)-c/(1+c)}\,.$ (2)

The overdensity of a virialized halo is usually considered to be equal to
$\approx 200$ times the critical density of the universe and hence, $r_{200}$
is the radius of the virialized halo, and $M_{200}$ indicates the enclosed
mass by this radius. With this definition, the relation of
$c=\frac{r_{200}}{r_{s}}$ connects concentration parameter to the scale radius
and $r_{200}$ of a cluster.

It can easily be seen from Eq. (1) that in small radii ($x\to 0$), NFW density
profile reduces to $\rho\propto r^{-1}$, while in large radii ($x\to\infty$),
it changes with radius as $\rho\propto r^{-3}$. As it has been mentioned
before, observation of cored profiles ($\rho=constant$) gave rise to efforts
to reform CDM simulations. However, SIDM simulations managed to resolve this
problem in galaxy size halos. Even in larger scales e.g. clusters of galaxies,
SIDM makes differences. According to simulation (see the top plots of Fig. (2)
in Robertson:2018anx ), the slope of density profile slowly changes from
$\rho\propto r^{-1}$ (consistent with NFW profile) to $\rho=constant$ in small
radii with two parameters: the total mass of the cluster and the value of
$\frac{\sigma}{m}$. The more massive the cluster is, the less cuspy the
profile is. Likewise but more effectively, larger $\frac{\sigma}{m}$ leads to
a more apparent cored profile. With respect to this behavior, we propose a
mathematical function that describes this change and has the form of

$f(x)=\frac{\rho(x)}{\rho_{0}}=\frac{1+(1-\alpha)e^{-x/k}}{x^{\alpha}(1+x)^{3-\alpha}}\,.$
(3)

This density profile has two constants of $k$ and $\alpha$. $k$ is just a
mathematical parameter and is added to get a closer result to simulation
outcomes. Whereas, $\alpha$ is the primary parameter that characterizes the
density profile for each input of $\frac{M}{M_{\odot}}$ and
$\frac{\sigma}{m}$. Due to dependency of $\alpha$ on cluster mass and cross-
section per unit mass of SIDM, we suggest the following relation as an ansatz,

$\alpha=1-\sigma_{m}(\frac{M}{10^{15}M_{\odot}})^{A}\,,$ (4)

and introduce the dimensionless cross-section per unit mass of
$\sigma_{m}=\frac{\sigma/m}{1cm^{2}g^{-1}}$ to have a dimensionless parameter
of $\alpha$ (the value of $1cm^{2}g^{-1}$ is chosen in the denominator since
it is the upper limit of $\frac{\sigma}{m}$ in our study). Here, $A$ is a new
constant that determines how much the change in the slope of the profile
depends on cluster mass in small radii, in comparison with the role of
$\sigma_{m}$. A larger value of $A$ reveals a higher dependency on cluster
mass. Having mentioned that, $A$ should have a negligible positive value to be
in agreement with the simulation result.

Clearly, $\alpha=1$ in Eq. (3) results in NFW profile, while $\alpha=0$
describes a completely cored profile. In other words, we have designed the
above ansatz in a way that it reduces to NFW profile if $\frac{\sigma}{m}$ is
negligible, and shows a cored profile for the upper limit of
$\frac{\sigma}{m}$ and the highest mass in the given range.

Figs. (1) and (2) indicate Eq. (3) for two values of $A=0.05$ and $A=0.2$ and
also compare this function with NFW profile. In these two figures (and
following figures, too), blue lines show SIDM0.1 and red lines illustrate
SIDM1. As can be seen, the suggested function for the density profile is in
good agreement with the simulation of Robertson:2018anx . Both mass and
$\frac{\sigma}{m}$ make a contribution to the changes in the slope of density
profile at the inner region, in a way that is totally consistent with
simulation results, and therefore, this function can be used for the density
profile of SIDM galaxy clusters. Moreover, the exact value of $A$ should be
obtained via iterating simulations for different cluster masses and find the
fitted quantity of $A$. Owing to the fact that we have access to only two
windows of mass, we can just guess the value of this constant. We restrict our
study to these two mentioned values of $A$ but note that its precise value can
only be determined through performing simulation for a number of different
masses.

Figure 1: Density profile of SIDM cluster size halos, regarding the function
of Eq. (3) with $A=0.05$ and $k=4$. Black color shows the NFW profile, while
SIDM0.1 and SIDM1 are represented by blue and red lines, respectively. For
both cases, solid line illustrates $M=10^{15}M_{\odot}$ of each
$\frac{\sigma}{m}$ and dashed line of the same color stands for
$M=10^{14}M_{\odot}$. Figure 2: Density profile of SIDM cluster size halos
regarding the function of Eq. (3) with $A=0.2$ and $k=4$. Colors and types of
lines are chosen analogous to Fig. (1).

In Figs. (1) and (2), we used $k=4$ for all curves. This preference is kept
throughout this paper because it is an appropriate choice that leads the
profiles to the NFW profile in large radii, completely similar to simulation
results. To have a frame of reference, Fig. (3) reveals how different values
of $k$ may change the behavior of the profile. Among the three illustrated
choices of $k=0.5,4,15$, the best choice that would fit the behavior of the
function to the NFW profile is $k=4$.

Figure 3: Density profile of Eq. (3) with $\alpha=0.5$ and three values of
$k=0.5$ (purple line), $k=4$ (yellow line) and $k=15$ (green line). Black line
shows the NFW profile.

It should also be mentioned that there are many other cored density profiles
that have been yet proposed, such as Burkert Burkert:1995yz  and Hernquist
Hernquist . Nonetheless, we need a profile that slowly changes from a cuspy
form to a cored shape. To this purpose, previous density profiles cannot be
used as a suitable choice to describe galaxy clusters with SIDM, although they
would be used for galaxies.

###  II.2 Mass-Concentration Relation

The concentration parameter of a halo is defined with the relation of

$c=\frac{r_{200}}{r_{s}}\,,$ (5)

as a function of viral radius and scale radius in the NFW profile. The
relation between mass and concentration can be written with the form of

$c=a(\frac{M_{200}}{10^{12}h^{-1}M_{\odot}})^{b}\,,$ (6)

where Hubble parameter is given by $H=100h$ and the quantities of $a$ and $b$
vary in different studies. One of the most common forms of mass-concentration
(M-c) relation is given by Macciò et al. Maccio:2008pcd  with $a=8.3$ and
$b=-0.104$.

According to the definition of $r_{s}$, scale radius is the distance from the
center of the halo at which the logarithmic slope of the density profile
equates to $-2$, meaning that $\rho(x)\propto x^{-2}$. We may use this
definition to obtain a concentration parameter for the profile of Eq. (3).
Considering our density profile of $\rho(x)$, we can obtain $x_{s}$
(dimensionless radius where results in $\rho(x_{s})\propto x^{-2}$) via
calculating logarithmic derivative of density profile as

$\frac{d}{dx}(ln\rho(x))|_{x=x_{s}}=\frac{d}{dx}(ln(x^{-2}))|_{x=1}=-2\,.$ (7)

The bulk of cluster mass is accumulated in its central region. Due to the fact
that the SIDM density profile merely affects this inner region, not the
distant radii where is related to $r_{200}$, we assume that virial radius does
not see a dramatic change in SIDM halos compared with CDM halos. Therefore,
considering Eq. (5), it is possible to find concentration parameter in our new
profile as a function of concentration of NFW profile ($c_{NFW}$) and $x_{s}$
of the previous equation via

$\frac{c_{NFW}}{c}=\frac{r_{s}}{r_{s,NFW}}=x_{s}\,,$ (8)

and hence

$c=\frac{c_{NFW}}{x_{s}}\,.$ (9)

This method helps to calculate the concentration parameter for the new density
profile. The result is provided in Fig. (4) for M-c relation and it reveals
that SIDM leads to an upward shift in the M-c plot. The value of
$\frac{\sigma}{m}$ determines how remarkable this shift would be. In order to
obtain the relevant M-c relation, we fit these plots to Eq. (6). Assuming
$A=0.05$, using this equation results in the values of $a=8.423$ and
$b=-0.1029$ for SIDM0.1, and the values of $a=9.685$ and $b=-0.048$ is
obtained for SIDM1. For $A=0.2$, we can still fit SIDM0.1 to this equation,
which gives $a=8.272$ and $b=-0.1002$. However, a higher dependency of
$\alpha$ on mass ($A=0.2$) totally alters the graph for SIDM1 and we cannot
fit it to Eq. (6) anymore. Another relation which could be a good fitting
model for this case is

$c=a^{\prime}(\frac{M_{200}}{10^{12}h^{-1}M_{\odot}})^{b^{\prime}}+c^{\prime}(\frac{M_{200}}{10^{12}h^{-1}M_{\odot}})^{d^{\prime}}\,.$
(10)

With regard to this equation, one can obtain $a^{\prime}=7.903$,
$b^{\prime}=-0.04527$, $c^{\prime}=0.0004138$ and $d^{\prime}=1.205$.

Note that this upward change has also been obtained in several other studies
for SIDM1 Bondarenko:2017rfu  and also for a velocity-dependent cross-section
model of SIDM Sagunski:2020spe .

Figure 4: Mass-concentration relation of the introduced density profile. Black
line describes the M-c relation of Macciò et al. Maccio:2008pcd , while blue
and red lines stand for SIDM0.1 and SIDM1. Solid type of line denotes the case
of $A=0.5$ and the other case with $A=0.2$ is represented by dotted lines.

###  II.3 Velocity Dispersion

SIDM simulation Robertson:2018anx  also provides a velocity dispersion profile
of halos. In terms of a qualitative discussion, velocity dispersion in a CDM
halo increases with the radius until hitting its peak at $r_{max}\approx
2.1r_{s}$ and then declines until a very low value in high radii. Inside SIDM
halo, DM collisions give rise to heat transfer from the region with the
highest velocity dispersion ($r_{max}$) to the inner region of halo, resulting
in an increase in kinetic energy (and thus, velocity dispersion) in the
central part of the cluster. This phenomenon affects the velocity dispersion
profile and as can be seen in the third set of plots in Fig. (2) of
Robertson:2018anx , velocity dispersion approximately remains at its maximum
until $r_{max}$ of CDM and then falls, similar to CDM case.

In order to quantify such behavior, one can solve the Jeans equation
Sokolenko:2018noz  for 3-dimensional velocity dispersion ($\sigma_{v}$), which
is written as

$\frac{d}{dr}(\frac{\sigma_{v}^{2}}{3}\frac{r^{2}}{\rho}\frac{d\rho}{dr})=-4\pi
Gr^{2}\rho\,.$ (11)

To solve this first-order differential equation (using the density profile of
Eq. (3)), a boundary condition is required. In a very high radius
($r\to\infty$), velocity dispersion goes to zero. Nonetheless, it would not be
used as an appropriate boundary condition, owing to the fact that all halos
(with any value of $\alpha$) should satisfy this condition regarding
simulation results. We use some approximations to find a suitable initial
condition to solve this differential equation.

For SIDM0.1, we can infer that there is only a negligible difference from CDM
outcome. We neglect this difference and consider that the SIDM0.1 velocity
dispersion profile is similar to that of CDM. The value of 1-dimensional
velocity dispersion ($\sigma_{1D}$) at virial radius can be obtained for CDM
halos via Evrard:2007py

$\sigma_{1D}=1082.9(\frac{M_{200}}{10^{15}h^{-1}M_{\odot}})^{0.3361}\,.$ (12)

Assuming that velocity dispersion profile is a symmetric function relative to
the vertical axis at $r_{max}$, we can claim that the obtained value of Eq.
(12) is also the value of $\sigma_{1D}$ at $r_{init}$, where we choose as the
initial condition to solve the Jeans equation. The radius of $r_{init}$ is
then determined by

$log(r_{max})-log(r_{init})=log(r_{200})-log(r_{max})\,,$ (13)

which reduces to

$log(x_{init})=2log(x_{max})-log(c)\,,$ (14)

where Eq. (5) is used. Finally, putting the rough value of $x_{max}\approx
2.1$ in above equation, $x_{init}$ is obtained via

$x_{init}=\frac{2.1^{2}}{c}\,.$ (15)

This procedure leads to finding a velocity dispersion profile, although
$x_{init}$ is quite large whereby we lose the substantial area in which the
SIDM effect differs from CDM. As a solution to this problem, one can fit a
Gaussian function of

$\sigma_{v}=ae^{-(\frac{t-b}{c})^{2}}\,,$ (16)

to the solution of the Jens equation, where $t=log(r/kpc)$ and the constants
of $a$, $b$ and $c$ vary for profiles with different values of $\alpha$. A
Gaussian function can be a very good fit for the solution of the Jeans
equation in SIDM0.1. Such an assertion could be proved in Fig. (5), which
illustrates both solution of the Jeans equation and its fitted Gaussian
function for SIDM0.1 in a halo with $\frac{M}{M_{\odot}}=10^{14}$. It is clear
that the fitted Gaussian function (dotted line) with $a=816.6$, $b=2.418$ and
$c=1.463$ is consistent with the solution of Eq. (11) with regard to the
mentioned initial condition, which is represented by blue line. Note that we
used $\sigma_{v}\approx\sqrt{3}\sigma_{1D}$ and neglected velocity anisotropy,
even though there could emerge different velocity anisotropy values in
different radii of the halo Brinckmann:2017uve .

Another result of Fig. (5) is a very negligible difference between two cases
of $A=0.05$ and $A=0.2$, thereby these two cases overlap in a plot.

Figure 5: The 3-dimensional velocity dispersion profile in a SIDM0.1 halo as a
function of radius. Solid blue line represents the solution of Jeans equation
for $A=0.05$. This solution for $A=0.2$ is provided by dashed blue line,
although it is not even distinguishable from the case of $A=0.05$ in this
plot. Fitted Gaussian curve is illustrated by dotted magenta line, which is in
a very good agreement with the solution.

For SIDM1, we assume that at $r<r_{max}$, velocity dispersion remains
unchanged, having its value at $r_{max}$ and then, overlaps with its relevant
Gaussian function in SIDM0.1. Figs. (6) and (7) show 3-dimensional velocity
dispersion profiles of SIDM0.1 (blue) and SIDM1 (red) for
$\frac{M}{M_{\odot}}=10^{14}$ and $\frac{M}{M_{\odot}}=10^{15}$, respectively,
in which the explained method is used. It is clear that these plots are in
good agreement with simulation and consequently, this procedure can be used in
Section III to derive scaling relations.

Figure 6: The 3-dimensional velocity dispersion as a function of radius in a
halo with the mass of $\frac{M}{M_{\odot}}=10^{14}$. SIDM0.1 is illustrated
with blue curve and red curve represents SIDM1. Figure 7: The 3-dimensional
velocity dispersion as a function of radius in a halo with the mass of
$\frac{M}{M_{\odot}}=10^{15}$. Colors are chosen the same as the previous
figure.

##  III Scaling Relations

In order to completely characterize a cluster with observational data, scaling
relations would be needed. Specifically, cluster masses are difficult to be
directly measured, and therefore, mass-temperature (M-T) and mass-velocity
dispersion (M-$\sigma_{v}$) relations make a notable contribution to
determining fundamental features of galaxy clusters. In this Section, we
investigate these two important relations in the SIDM model.

###  III.1 Mass-Velocity Dispersion Relation

Once we have the velocity dispersion profile, we can immediately find a
relation between mean velocity dispersion and mass of the clusters.
Considering $\bar{\sigma_{v}}$ to be the 3-dimensional mass-weighted mean
velocity dispersion inside the radius of $r_{200}$, this parameter can be
written as

$\bar{\sigma_{v}}=\frac{\int_{0}^{c}4\pi
x^{2}\rho(x)\sigma_{v}dx}{\int_{0}^{c}4\pi x^{2}\rho(x)dx}\,.$ (17)

Using this equation, it is possible to find $\bar{\sigma_{v}}$ as a function
of cluster mass. As can be seen in Fig. (8), a higher value of cross-section
leads to a higher $\bar{\sigma_{v}}$ for the same mass. This is not
surprising, as it can already be inferred from Figs. (6) and (7).

Figure 8: Mass-velocity dispersion relation for SIDM0.1 (blue) and SIDM1 (red)
models. Solid lines denote $A=0.05$ and dashed lines stand for $A=0.2$.
However, the difference between these two cases is very negligible.

The lines in Fig. (8) can also be fitted to suitable functions to have a
quantitive mass-velocity dispersion relation. To this purpose, we find the
relation of

$\bar{\sigma_{v}}=1634(\frac{M_{200}}{10^{15}h^{-1}M_{\odot}})^{0.3316}\,$
(18)

for SIDM0.1, and similarly the relation of

$\bar{\sigma_{v}}=1876(\frac{M_{200}}{10^{15}h^{-1}M_{\odot}})^{0.3247}\,$
(19)

is obtained for SIDM1. According to Fig. (8), plots of $A=0.05$ and $A=0.2$
approximately overlap with each other and as a result, the above relations can
be used for both these cases of $A$.

###  III.2 Mass-Temperature Relation

The mass-temperature relation plays a crucial role in determining cluster
masses. Mass of a cluster can be found via several methods, including
gravitational lensing. However, the M-T relation can pave the way for a more
straightforward way of mass measurement. In our previous work Naseri:2020uvn ,
we followed the method provided by Afshordi & Cen Afshordi:2001ze  and derived
the M-T relation with regard to the interacting dark sector model in which a
possible interaction is assumed between dark matter and dark energy. Here, we
pursue that method of obtaining M-T relation and then investigate its
modifications in the SIDM model.

This procedure starts with the virial theorem and combines it with the
conservation of energy to find a relation between mass and temperature. While
in an interacting dark sector model, the virial ratio changes and results in
the main differences Naseri:2020uvn , it would not be remodeled in SIDM. In
fact, the possible interaction between DM particles has nothing to do with the
virial theorem, as a variety of interactions among baryons might not lead to a
different virial condition (for a review, you may follow Section II in
Naseri:2020uvn ). In a SIDM halo, none of the fundamental assumptions and
principles would change and thus, the M-T relation keeps its form in the
$\Lambda$CDM model, that is

$k_{B}T=(6.62keV)\tilde{Q}(\frac{M}{10^{15}h^{-1}M_{\odot}})^{2/3}\,,$ (20)

where $k_{B}$ is the Boltzmann constant and $\tilde{Q}$ is a dimensionless
factor. This factor contains every detail of SIDM inside itself and is written
as

$\tilde{Q}=(\frac{\tilde{\beta}_{spec}}{0.9})^{-1}(\frac{1+\nu}{1-\nu})y\,,$
(21)

with

$y(c,f(x))=\frac{\Delta^{1/3}(1-\nu)c\int_{0}^{c}f(x)g(x)xdx}{3\pi^{2/3}g^{2}(c)}\,.$
(22)

Here, $\Delta$ is the overdensity of the virialized halo (and thereby
$\Delta\approx 200$), and $\nu$ is a coefficient constant that has emerged
since the surface pressure term on the boundary of the cluster ($P_{ext}$) is
taken into account in virial condition. This constant relates $P_{ext}$ to the
potential energy of virialized sphere ($U$) with volume of $V$ via
Afshordi:2001ze

$\nu(c,f(x))=-\frac{3P_{ext}V}{U}=\frac{c^{3}\int_{c}^{\infty}f(x)g(x)x^{-2}dx}{\int_{0}^{c}f(x)g(x)xdx}\,.$
(23)

It is self-evident that Eqs. (22) and (23) are highly dependent on the
modified density profile ($f(x)$) and concentration parameter of SIDM halo.
Not only does the effect of density profile directly appears in the above
equations, but it also affects another function of $g(x)$, which has the
definition of

$g(x)=\int_{0}^{x}f(x)x^{2}dx\,.$ (24)

Another significant contributor to the M-T relation is $\tilde{\beta}_{spec}$,
which has the form of

$\tilde{\beta}_{spec}=\beta_{spec}[1+(f\beta_{spec}^{-1}-1)\frac{\Omega_{b}}{\Omega_{b}+\Omega_{DM}}]\,,$
(25)

where $f$ is the fraction of baryonic matter in hot gas and is considered to
be near one Afshordi:2001ze , $\Omega_{b}$ and $\Omega_{DM}$ are the relevant
density parameters of baryonic matter and DM, and the spectroscopic beta
parameter is defined as

$\beta_{spec}=\frac{\sigma_{1D}^{2}}{(k_{B}T/\mu m_{p})}\,.$ (26)

Here, $\mu=0.59$ is the mean molecular weight and $m_{p}$ denotes the proton
mass. The impact of velocity dispersion emerges in Eq. (26) and we should use
Eqs. (18) and (19) in that for SIDM0.1 and SIDM1, respectively. Given these
equations, it is plausible to have a relation between the mass and temperature
of a cluster. As a result, we obtain temperatures for given masses between
$10^{14}<\frac{M}{M_{\odot}}<10^{15}$ with the aid of the above equations.
Fitting the relation of

$\frac{M}{10^{14}h^{-1}M_{\odot}}=p(\frac{k_{B}T}{keV})^{q}\,,$ (27)

to the calculated T values for input M values give constants of $p$ and $q$
and consequently, we can find the M-T relation for the SIDM model. The result
is summarized in Table 1 and it can be seen that for SIDM1, the power-law
between mass and temperature faces a significant change, both in the power
index and normalization factor.

Using this method, the modified M-T relation is illustrated in Fig. (9). In
this figure, SIDM0.1 totally overlaps with the CDM result in the NFW profile.
However, it is clear that the obtained mass for each given temperature sharply
increases in SIDM1 halos. It is an important conclusion, as the cluster masses
are usually determined according to X-ray temperature. Therefore, clusters
should be way more massive than what has been considered if DM particles
extensively interact with each other.

In addition, high dependency on mass in density profile (or higher $A$,
equivalently) completely changes the behavior of M-T relation. Perhaps the
modified M-c relation plays the most profound role in this change, due to its
large difference from the other cases (take a look at Fig. (4)). Too strange
behavior of M-c and M-T relations for SIDM1 in $A=0.2$ raises doubts about the
high dependency of density profile on cluster mass. Overall, $A=0.05$ gives
rise to more familiar results in these two equations, although it is not
different than $A=0.2$ in the mass-velocity dispersion relation.

Table 1: The numerical values of constants in Eq. (27) which are obtained with fitting this equation to calculated temperatures for given masses in the range of $10^{14}<\frac{M}{M_{\odot}}<10^{15}$. |  | $p$ | $q$ |
---|---|---|---|---
A=0.05 | SIDM0.1 | 0.1654 | 1.552 |
SIDM1 | 4.952 | 1.254 |
A=0.2 | SIDM0.1 | 0.1623 | 1.559 |
SIDM1 | 5.96 | 0.7945 |
Figure 9: Mass-temperature relation for SIDM0.1 (blue line) and SIDM1 (red
line) models. Solid and dotted lines denote $A=0.05$ and $A=0.2$,
respectively. Moreover, the result of the NFW profile in provided by black
line, although it is not distinguishable from SIDM0.1 model.

##  IV Super Interacting Dark Sector

In the previous section, it was concluded that for SIDM1, relevant mass to
every input temperature is much higher than that of SIDM0.1 and CDM, even with
a difference of more than one order of magnitude. This is an extraordinary
result, owing to the fact that the other observational methods of mass
measurement do not reveal such a discrepancy with mass determination through
M-T relation. In this section, we show that the combination of a particular
cosmological model of DE with the self-interacting model of DM can completely
resolve the problem.

The interacting dark sector is a DE-DM model in which a possible interaction
is assumed between DE and DM. According to this model, DM particle mass is
determined with regard to its interaction with a scalar field with the energy
density of DE; similar to the attribution of quark and lepton masses to their
interaction with the Higgs field Farrar:2003uw . Given this assumption,
cosmological equations, including the energy balance and the Raychadhuri field
equations, change to the forms of

$\displaystyle\dot{\rho}_{b}=-3H\rho_{b}\,,$
$\displaystyle\dot{\rho}_{DM}=-3H\rho_{DM}+Q\,,$
$\displaystyle\dot{\rho}_{DE}=-3(1+w_{DE})H\rho_{DE}-Q\,,$
$\displaystyle\dot{H}=-4\pi
G\left[\rho_{b}+\rho_{DM}+(1+w_{DE})\rho_{DE}\right]\,.$ (28)

Here $\rho_{DE}$, $\rho_{DM}$, and $\rho_{b}$ denote energy density of DE, DM,
and baryonic matter, respectively, and $\omega_{DE}$ is constant of the
equation of state for DE. Furthermore, $Q$ is the new term that describes the
interaction between DE and DM. In other words, $Q$ indicates the rate of
energy transfer between these two components; while a positive $Q$ shows the
energy transfer from DE to DM, a negative $Q$ reveals this transfer from DM to
DE. A number of functions have been proposed and investigated for $Q$ so far.
One simple and useful choice is considering $Q$ to be proportional to the
Hubble parameter. For this model, some of the studied functions are
CalderaCabral:2008bx ; vonMarttens:2018iav

$\displaystyle\mbox{ Model I}:Q=3H(\alpha_{c}\rho_{c}+\alpha_{x}\rho_{x})\,,$
$\displaystyle\mbox{ Model
II}:Q=3H\xi_{1}\frac{\rho_{c}\rho_{x}}{\rho_{c}+\rho_{x}}\,,$
$\displaystyle\mbox{ Model
III}:Q=3H\xi_{2}\frac{\rho_{x}^{2}}{\rho_{c}+\rho_{x}}\,,$
$\displaystyle\mbox{ Model
IV}:Q=3H\xi_{3}\frac{\rho_{c}^{2}}{\rho_{c}+\rho_{x}}\,.$ (29)

The parameters of $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, $\alpha_{DE}$ and
$\alpha_{DM}$ are the so-called interacting constants that describe each
model, exclusively.

Considering the interacting dark sector model, the virial theorem turns into a
different form. Most significantly, the virial ratio ($\lambda$) between
kinetic energy $K$ and potential energy $U$ is not $\frac{1}{2}$ anymore.
Writing the virial condition as

$K=-\lambda_{i}U\,,$ (30)

the modified virial ratio becomes Naseri:2020uvn

$\displaystyle\mbox{ Model
I}:\lambda_{I}=\frac{1-6\alpha_{DM}}{2+3\alpha_{DM}+3\alpha_{DE}/R}\,,$
$\displaystyle\mbox{ Model
II}:\lambda_{II}=\frac{1-\frac{6\xi_{1}}{R+1}}{2+\frac{3\xi_{1}}{R+1}}\,,$
$\displaystyle\mbox{ Model
III}:\lambda_{III}=\frac{1}{2+\frac{3\xi_{2}}{R(R+1)}}\,,$
$\displaystyle\mbox{ Model
IV}:\lambda_{IV}=\frac{1-\frac{6R\xi_{3}}{R+1}}{2+\frac{3R\xi_{3}}{R+1}}\,,$
(31)

where $i=I,II,III,IV$ and $R=\frac{\rho_{DM}}{\rho_{DE}}$. With this modified
virial theorem, the coefficient factor of Eq. (20) changes to

$\tilde{Q}=(\frac{\tilde{\beta}_{spec}}{0.9})^{-1}(\frac{2\lambda_{i}+\nu}{2-2\lambda_{i}-\nu})y\,,$
(32)

with

$y(c,f(x))=\frac{\Delta^{1/3}(2-2\lambda_{i}-\nu)c\int_{0}^{c}f(x)g(x)xdx}{3\pi^{2/3}g^{2}(c)}\,.$
(33)

Having these relations in mind, we now introduce a new model which we call
super interacting dark sector, or SIDS, in order to resolve the problem that
has been created in the previous Section in the M-T plot of SIDM1. This model
is founded on two fundamental assumptions: ($i$) DM particles have non-
negligible cross-section per unit mass of $\frac{\sigma}{m}$ whereby they can
interact with themselves (self-interacting DM) and ($ii$) DM particles
interact with DE and their mass stems from this interaction (interacting dark
sector). According to the second assumption, interacting constants ($\xi_{1}$,
$\xi_{2}$, $\xi_{3}$, $\alpha_{DE}$ and $\alpha_{DM}$) are functions of DM
mass and thereby, functions of $\frac{\sigma}{m}$.

SIDS is a simple combination of two different models of DE and DM. As
interacting dark sector can solve many cosmological problems, e.g. the
’coincidence problem’ CalderaCabral:2008bx , and SIDM can provide solutions to
some other astrophysical problems, such as core-cusp and missing satellite
problems, SIDS could even be more effective and justify both groups of
problems. Moreover, there is no contradiction between the two basic models in
theory.

To investigate M-T relation for the SIDS model, Eq. (20) should be considered,
again with density profile, concentration, and the velocity dispersion of
SIDM, but also with adding the new term of modified virial ratio, i.e. Eqs.
(32) and (33). Fig. (10) reveals how SIDS with $\lambda=0.65$ for
$\frac{\sigma}{m}=1cm^{2}g^{-1}$ omits the unusual and large upward shift of
the M-T plot. Interestingly, not only does this model correct the upward
shift, but it also changes the power index of $q=1.254$ to $q=1.513$, more
consistent with the previously known relation of $M\propto T^{\frac{3}{2}}$
for galaxy clusters. As can be seen, SIDS1 (SIDS model for
$\frac{\sigma}{m}=1cm^{2}g^{-1}$) gives much better result than SIDM1.

Figure 10: Mass-temperation relation for SIDS in comparison with SIDM. Similar
to the previous figure, SIDM1 and SIDM0.1 are shown with red and blue lines,
respectively. The dashed magenta line illustrates SIDS model with
$\frac{\sigma}{m}=1cm^{2}g^{-1}$ and $\lambda=0.65$. All the lines are
obtained for $A=0.05$.

For the given virial ratio of $\lambda=0.65$, it is possible to obtain
interacting constants of Eq. (IV) for specific models of Eq. (IV). In
addition, it was mentioned that the constants are functions of cross-section
per unit mass of DM particles, for example
$\xi_{1}=\xi_{1}(\frac{\sigma}{m})$. In order to have our result consistent
with M-T relation in a low $\frac{\sigma}{m}$, e.g. SIDM0.1, interacting
constants should reduce to zero for low values of $\frac{\sigma}{m}$. In other
words, a larger value of $\frac{\sigma}{m}$ indicates a more strong
interaction between DE and DM and therefore, results in larger values of
interacting constants. This would be the main effect of adding SIDM to the
theory of interacting dark sector.

To provide an opportunity for a more general conclusion, we iterated all steps
of our approach for three other values of $\frac{\sigma}{m}$; they are
$\frac{\sigma}{m}=0.4cm^{2}g^{-1}$ (or SIDS0.4),
$\frac{\sigma}{m}=0.6cm^{2}g^{-1}$ (or SIDS0.6) and
$\frac{\sigma}{m}=0.8cm^{2}g^{-1}$ (or SIDS0.8). The only stage at which we
lack information is the velocity dispersion profile since the SIDM simulations
have not been performed for these cases. On the other hand, looking at Eqs.
(18) and (19) may help to iron out this issue. In the mass-velocity dispersion
relation, the power index is approximately $\approx\frac{1}{3}$, which is in
agreement with the obtained relation in $\Lambda$CDM model. We fix the power
index to this value and assume that the coefficient factor linearly changes
from 1636 (for SIDM0.1) to 1876 (for SIDM1) with $\frac{\sigma}{m}$. Regarding
this assumption, it is plausible to calculate $\lambda$ in a way that the M-T
relation does not show any upward shift. As long as the quantity of $\lambda$
is determined, interacting constants for each model can immediately be found.
Table 2 shows these obtained values of $\lambda$ and interacting constants for
the given models. As there are two interacting constants for Model I, we
separate this model into two particular cases of $\alpha_{DE}=0$ and
$\alpha_{DM}=0$.

Table 2: The calculated values of the virial ratio and interacting constants for SIDS with different values of $\frac{\sigma}{m}$. | $\lambda$ |  | $\alpha_{DM}$
---
($\alpha_{DE}=0$)
| $\alpha_{DE}$
---
($\alpha_{DM}=0$)
$\xi_{1}$ | $\xi_{2}$ | $\xi_{3}$
SIDS0.1 | 0.5 | 0 | 0 | 0 | 0 | 0
SIDS0.4 | 0.56 | $-0.0156$ | $-0.0265$ | $-0.0214$ | $-0.0364$ | $-0.0577$
SIDS0.6 | 0.60 | $-0.0256$ | $-0.0413$ | $-0.0352$ | $-0.0566$ | $-0.0947$
SIDS0.8 | 0.63 | $-0.0330$ | $-0.0511$ | $-0.0452$ | $-0.0701$ | $-0.1217$
SIDS1 | 0.65 | $-0.0377$ | $-0.0571$ | $-0.0518$ | $-0.0784$ | $-0.1393$

According to the Table 2, the virial ratio grows with $\frac{\sigma}{m}$ (or
$\sigma_{m}$). Thus, $\lambda$ can be expressed as a function of $\sigma_{m}$
via the first terms of a Fourier series with the form of

$\lambda=a_{0}+a_{1}cos(\omega\sigma_{m})+b_{1}sin(\omega\sigma_{m})\,,$ (34)

where $a_{0}=0.5456$, $a_{1}=-0.06348$, $b_{1}=0.08871$ and $\omega=1.892$.
Fig. (11) indicates this function and values of $\lambda$ from Table 2. It is
clear that Eq. (34) properly describes how the virial ratio changes with
$\sigma_{m}$.

Figure 11: The virial ratio as a function of $\sigma_{m}$; the squares are
selected from Table 2 and the curve is drawn with respect to Eq. (34) with
given constants.

Finding interacting constants for various $\sigma_{m}$ values also results in
a useful piece of information, which is the quantitive behaviour of
interacting constant of each model as a function of $\sigma_{m}$. Considering
the relation of

$\chi_{interaction}=A_{0}sin(\omega_{0}\sigma_{m}+\phi)\,,$ (35)

three constants of $A_{0}$, $\omega_{0}$ and $\phi$ can be determined via
fitting this relation to the obtained constants in Table 2 for each model. In
this expression,
$\chi_{interaction}=\alpha_{DM},\alpha_{DE},\xi_{1},\xi_{2},\xi_{3}$ denotes
the relevant interacting constant of each model. The result is collected in
Table 3. For two particular cases of Model I and for three other models, this
result is shown in Figs. (12) and (13), respectively. These plots prove that
Eq. (35) can be a true description for interacting constants as a function of
$\sigma_{m}$.

The last conclusion has specific importance. For interacting dark sector
models, $Q$ contains the quality of energy transfer between DE and DM. It is a
function of DM and DE densities, in addition to several constants. While in
the interacting dark sector, these constants do not necessarily have a
physical meaning, they directly describe DM particles characteristics in the
SIDS model. DM particles with higher $\frac{\sigma}{m}$ can transfer a more
substantial amount of energy to DE. In fact, the essence of interacting
constants is specified in SIDS.

Table 3: The obtained constants of Eq. (35) with regard to the result of Table 2 for five models of SIDS. | $A_{0}$ | $\omega_{0}$ | $\phi$
---|---|---|---
| Model I
---
($\alpha_{DE}=0$)
0.03991 | 1.387 | 2.999
| Model I
---
($\alpha_{DM}=0$)
0.05756 | 1.583 | 2.985
Model II | 0.05473 | 1.387 | 2.999
Model III | 0.07894 | 1.583 | 2.985
Model IV | 0.1473 | 1.387 | 2.999
Figure 12: Interacting constants of Model I as a function of $\sigma_{m}$ for
two specific cases of $\alpha_{DE}=0$ (magenta) and $\alpha_{DM}=0$ (orange).
In this plot, vertical axis reveals $\alpha_{DM}$ for magenta line, and
$\alpha_{DE}$ for orange line. The points are shown according to Table 2 and
lines are indicated regarding Eq. (35) with the matched values of constants
from Table 3. Figure 13: The same plot as Fig. 12, but with regard to Model II
(blue), Model III (green) and Model IV (purple).

##  V Conclusion

We studied halo profiles and scaling relations of galaxy clusters regarding
SIDM halos. Based on the result of previously performed simulations, we
proposed a function which can perfectly describe the simulated density profile
for the cross-section per unit mass of DM particles limited to
$0.1cm^{2}g^{-1}<\frac{\sigma}{m}<1cm^{2}g^{-1}$ and cluster masses in the
range of $10^{14}<\frac{M}{M_{\odot}}<10^{15}$. Unlike many other density
profiles, this profile is not totally cored and is flexible to change with
$\frac{\sigma}{m}$ and cluster mass, just as SIDM simulations predict. For
this profile, a relation between cluster mass and the concentration parameter
is found, which reveals higher values of concentration parameter in SIDM than
the NFW profile for the same masses.

Solving the Jeans equation with several assumptions led to a straightforward
but rather precise form of velocity dispersion profile. We also found the
scaling relation between cluster mass and mean velocity dispersion inside the
virial radius for SIDM0.1 and SIDM1. These results were used to find the
impact of the self-interaction of DM particles on the mass-temperature
relation in galaxy clusters. It demonstrates that any value of observed X-ray
temperature is attributed to much more massive clusters in the SIDM model in
comparison with CDM.

Due to the fact that observations do not show such an upward shift in
determined masses from the M-T relation compared with the other methods of
mass measurement, we proposed a new interacting model, namely super
interacting dark sector, which can effectively remove the problem. In this
model, there is a plausible interaction between DM and DE, in addition to
self-interaction among DM particles. Such a combination of two models of DM
and DE can justify a number of cosmological problems and also resolve the
problem that SIDM mass-temperature relation produced.

Studying the M-T relation suggested that constants of interacting models
should change with $\frac{\sigma}{m}$. The higher the cross-section per unit
mass of DM particles is, the stronger the interaction between DE and DM should
be. The virial ratio is obtained to be higher than $\frac{1}{2}$ for SIDS
models and accordingly, the interacting constants are found to be negative,
which indicates that energy flows from DM to DE.

We found many quantitive expressions for the virial ratio and interacting
constants as functions of $\sigma_{m}$. It indicates that in the SIDS model,
interacting constants have a physical meaning, which is related to DM
particles mass and cross-section of self-interaction. A higher value of
$\frac{\sigma}{m}$ reveals a more considerable amount of energy transfer from
DM particles to DE.

The approach of this research is founded on the outcomes of previous SIDM
simulations. Any development in these simulations can help to improve our
method and find better assumptions to more precisely studying of both the SIDS
model and scaling relations of SIDM. As a result, the provided results will
improve if the more intricate and comprehensive simulations are performed.

## References

  * (1) P. A. R. Ade et al. [Planck], Astron. Astrophys. 594, A13 (2016) doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]].
  * (2) F. Zwicky, Astrophys. J. 86, 217-246 (1937) doi:10.1086/143864
  * (3) Babcock H. W., 1939, LicOB, 498, 41. doi:10.5479/ADS/bib/1939LicOB.19.41B
  * (4) Rubin V. C., Ford W. K., Thonnard N., 1980, ApJ, 238, 471. doi:10.1086/158003
  * (5) U. G. Briel and J. P. Henry, [arXiv:astro-ph/9711237 [astro-ph]].
  * (6) Lee J., Komatsu E., 2010, ApJ, 718, 60. doi:10.1088/0004-637X/718/1/60
  * (7) J. K. Adelman-McCarthy et al. [SDSS], Astrophys. J. Suppl. 162, 38-48 (2006) doi:10.1086/497917 [arXiv:astro-ph/0507711 [astro-ph]].
  * (8) A. Challinor, IAU Symp. 288, 42 (2013) doi:10.1017/S1743921312016663 [arXiv:1210.6008 [astro-ph.CO]].
  * (9) R. N. Mohapatra and V. L. Teplitz, Phys. Lett. B 462, 302-309 (1999) doi:10.1016/S0370-2693(99)00789-3 [arXiv:astro-ph/9902085 [astro-ph]].
  * (10) J. N. Bahcall, C. Flynn, A. Gould and S. Kirhakos, Astrophys. J. Lett. 435, L51-L54 (1994) doi:10.1086/187592 [arXiv:astro-ph/9406019 [astro-ph]].
  * (11) K. Freese, B. Fields and D. Graff, [arXiv:astro-ph/0007444 [astro-ph]].
  * (12) R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791-1797 (1977) doi:10.1103/PhysRevD.16.1791
  * (13) S. Weinberg, Phys. Rev. Lett. 40, 223-226 (1978) doi:10.1103/PhysRevLett.40.223
  * (14) L. Roszkowski, E. M. Sessolo and S. Trojanowski, Rept. Prog. Phys. 81, no.6, 066201 (2018) doi:10.1088/1361-6633/aab913 [arXiv:1707.06277 [hep-ph]].
  * (15) R. Agnese et al. [SuperCDMS], Phys. Rev. Lett. 120, no.6, 061802 (2018) doi:10.1103/PhysRevLett.120.061802 [arXiv:1708.08869 [hep-ex]].
  * (16) C. Amole et al. [PICO], Phys. Rev. Lett. 118, no.25, 251301 (2017) doi:10.1103/PhysRevLett.118.251301 [arXiv:1702.07666 [astro-ph.CO]].
  * (17) B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel and P. Tozzi, Astrophys. J. Lett. 524, L19-L22 (1999) doi:10.1086/312287 [arXiv:astro-ph/9907411 [astro-ph]].
  * (18) Bullock, J.A.M.E.S., 2013. Notes on the missing satellites problem. Local Group Cosmology, 20, p.95.
  * (19) V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, A. Jenkins, A. Helmi, J. F. Navarro, C. S. Frenk and S. D. M. White, Mon. Not. Roy. Astron. Soc. 391, 1685-1711 (2008) doi:10.1111/j.1365-2966.2008.14066.x [arXiv:0809.0898 [astro-ph]].
  * (20) J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. 490, 493-508 (1997) doi:10.1086/304888 [arXiv:astro-ph/9611107 [astro-ph]].
  * (21) Oh, S.H., De Blok, W.J.G., Brinks, E., Walter, F. and Kennicutt Jr, R.C., 2011. Dark and luminous matter in THINGS dwarf galaxies. The Astronomical Journal, 141(6), p.193.
  * (22) D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760-3763 (2000) doi:10.1103/PhysRevLett.84.3760 [arXiv:astro-ph/9909386 [astro-ph]].
  * (23) A. Kusenko and P. J. Steinhardt, Phys. Rev. Lett. 87, 141301 (2001) doi:10.1103/PhysRevLett.87.141301 [arXiv:astro-ph/0106008 [astro-ph]].
  * (24) A. E. Faraggi and M. Pospelov, Astropart. Phys. 16, 451-461 (2002) doi:10.1016/S0927-6505(01)00121-9 [arXiv:hep-ph/0008223 [hep-ph]].
  * (25) J. L. Feng, M. Kaplinghat, H. Tu and H. B. Yu, JCAP 07, 004 (2009) doi:10.1088/1475-7516/2009/07/004 [arXiv:0905.3039 [hep-ph]].
  * (26) A. Loeb and N. Weiner, Phys. Rev. Lett. 106, 171302 (2011) doi:10.1103/PhysRevLett.106.171302 [arXiv:1011.6374 [astro-ph.CO]].
  * (27) Zavala, J., Vogelsberger, M. and Walker, M.G., 2013. Constraining self-interacting dark matter with the Milky Way’s dwarf spheroidals. Monthly Notices of the Royal Astronomical Society: Letters, 431(1), pp.L20-L24.
  * (28) A. H. G. Peter, M. Rocha, J. S. Bullock and M. Kaplinghat, Mon. Not. Roy. Astron. Soc. 430, 105 (2013) doi:10.1093/mnras/sts535 [arXiv:1208.3026 [astro-ph.CO]].
  * (29) K. Pardo, H. Desmond and P. G. Ferreira, Phys. Rev. D 100, no.12, 123006 (2019) doi:10.1103/PhysRevD.100.123006 [arXiv:1911.04497 [astro-ph.CO]].
  * (30) T. Brinckmann, J. Zavala, D. Rapetti, S. H. Hansen and M. Vogelsberger, Mon. Not. Roy. Astron. Soc. 474, no.1, 746-759 (2018) doi:10.1093/mnras/stx2782 [arXiv:1705.00623 [astro-ph.CO]].
  * (31) A. Robertson, D. Harvey, R. Massey, V. Eke, I. G. McCarthy, M. Jauzac, B. Li and J. Schaye, Mon. Not. Roy. Astron. Soc. 488, no.3, 3646-3662 (2019) doi:10.1093/mnras/stz1815 [arXiv:1810.05649 [astro-ph.CO]].
  * (32) A. Burkert, IAU Symp. 171, 175 (1996) doi:10.1086/309560 [arXiv:astro-ph/9504041 [astro-ph]].
  * (33) Hernquist L., 1993, ApJS, 86, 389
  * (34) A. V. Maccio’, A. A. Dutton and F. C. v. d. Bosch, Mon. Not. Roy. Astron. Soc. 391, 1940-1954 (2008) doi:10.1111/j.1365-2966.2008.14029.x [arXiv:0805.1926 [astro-ph]].
  * (35) K. Bondarenko, A. Boyarsky, T. Bringmann and A. Sokolenko, JCAP 04, 049 (2018) doi:10.1088/1475-7516/2018/04/049 [arXiv:1712.06602 [astro-ph.CO]].
  * (36) L. Sagunski, S. Gad-Nasr, B. Colquhoun, A. Robertson and S. Tulin, [arXiv:2006.12515 [astro-ph.CO]].
  * (37) A. Sokolenko, K. Bondarenko, T. Brinckmann, J. Zavala, M. Vogelsberger, T. Bringmann and A. Boyarsky, JCAP 12, 038 (2018) doi:10.1088/1475-7516/2018/12/038 [arXiv:1806.11539 [astro-ph.CO]].
  * (38) A. E. Evrard, J. Bialek, M. Busha, M. White, S. Habib, K. Heitmann, M. Warren, E. Rasia, G. Tormen and L. Moscardini, et al. Astrophys. J. 672, 122-137 (2008) doi:10.1086/521616 [arXiv:astro-ph/0702241 [astro-ph]].
  * (39) M. Naseri and J. T. Firouzjaee, Phys. Rev. D , no. , (2020) [arXiv:2009.09956 [astro-ph.CO]].
  * (40) N. Afshordi and R. Cen, Astrophys. J. 564, 669-682 (2002) doi:10.1086/324282 [arXiv:astro-ph/0105020 [astro-ph]].
  * (41) G. R. Farrar and P. J. E. Peebles, Astrophys. J. 604, 1-11 (2004) doi:10.1086/381728.
  * (42) G. Caldera-Cabral, R. Maartens and L. A. Urena-Lopez, Phys. Rev. D 79, 063518 (2009) doi:10.1103/PhysRevD.79.063518 [arXiv:0812.1827 [gr-qc]].
  * (43) R. von Marttens, L. Casarini, D. F. Mota and W. Zimdahl, Phys. Dark Univ. 23, 100248 (2019) doi:10.1016/j.dark.2018.10.007 [arXiv:1807.11380 [astro-ph.CO]].