#  $k-$Dependent Dark Matter

Parisa Arabameri1, Zahra Davari2 and Nima Khosravi1
1Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran;
2School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.
Box 19395-5531, Tehran, Iran

###### Abstract

With the emersion of precise cosmology and the emergence of cosmic tensions,
we are faced with the question of whether the simple model of cold dark matter
needs to be extended and whether doing so can alleviate the tensions and
improve our understanding of the properties of dark matter. In this study, we
investigate one of the generalized models of dark matter so that the behavior
of this dark matter changes according to the scale of $k$. In large scales
(small $k$’s), the dark matter is cold, while it becomes warm for small scales
(large $k$’s). This behavior is modeled phenomenologically for two different
scenarios. We show that the $S_{8}$ tension can be alleviated, but the $H_{0}$
tension becomes milder while not too much.

###### keywords:

Cold Dark Matter, Warm Dark Matter

##  1 Introduction

The cold dark matter (CDM) paradigm is an important feature in particle
physics and cosmology, assuming cold and collisionless dark matter particles
interact only gravitationally. This component is one of the main bases in the
standard $\Lambda$CDM model and it is responsible for about 26% of the energy
density of Universe (Scott, 2020; Scolnic et al., 2018). A wide range of
cosmological observations from many different epochs and at large and small
scales, including CMB missions, BAO data, observations of galaxy clusters, and
weak lensing experiments, supported this paradigm. However, the physical
nature of DM particles remains unclear and a mystery after decades of
research.
On the other hand, the CDM paradigm is remarkably successful in many aspects,
especially in explaining the observed properties of large-scale structures
(LSS) in the Universe (in the range $\sim$ 1 Gpc down to $\sim$ 10 Mpc);
however, it conflicts with observations on galactic and sub-galactic scales
($\leq$1 Mpc). For instance, we can point to:

  * •

The "missing satellites problem," which refers to the fact that there is an
overestimation of dwarf galaxies by the CDM model than observed in the
Universe (Rubin & Ford, 1970; Rubin et al., 1980; Moore et al., 1999).

  * •

The "cusp-core problem," which refers to the fact that the CDM model predicts
that dark matter halos should have a cuspy density profile at their centers,
while observations suggest that they have a more constant density profile
(Gentile et al., 2004).

  * •

The "too big to fail problem," which refers to the lack of observation of the
most massive halos, which are predicted to be luminous (Purcell & Zentner,
2012).

The small scale crisis motivated the study of scenarios that predict damped
matter fluctuations below a characteristic free-streaming scale through either
modification of the primordial power spectrum or non-cold dark matter models,
which modify (suppress) the power spectrum at late times.
Furthermore, the recent high-precision cosmological data has shown a
statistically significant discrepancy in the estimation of the current values
of the Hubble parameter ($H_{0}$) and the fluctuations amplitude of density
perturbations at 8 h-1Mpc scale ($\sigma_{8}$) between early-time and late-
time observations, which poses another challenge to the standard $\Lambda$CDM
model. Early universe measurements like CMB Planck collaboration (Aghanim et
al., 2020) estimate $H_{0}\sim(67.0-68.5)$ km/s/Mpc, while late-time distance
ladder measurements like SH0ES and H0LiCOW collaborations report
$H_{0}=(74.03\pm 1.42)$ (Riess et al., 2019).
The mentioned problems, together with the lack of understanding of the nature,
mass, and dynamics of dark matter particles, have sparked several extensions
and alternatives to standard dark matter models of particle physics, which are
theoretically well-motivated and inspire new search strategies. There are many
approaches in order to investigate dark matter, such as warm dark matter
(WDM), cannibal Dark Matter (Buen-Abad et al., 2018), decaying dark matter
(Davari & Khosravi, 2022), dynamical dark matter, fuzzy dark matter and
interacting dark matter (Loeb & Weiner, 2011; Archidiacono et al., 2019).
If dark matter particles decouple from the primordial plasma when still
relativistic and soon become non-relativistic, the particles are called “warm
dark matter". These WDM particles would have a smaller free-streaming length
than cold dark matter particles, preventing them from clustering on small
scales and potentially solving the missing satellite problem. Furthermore, the
WDM particles significantly affect the clustering of matter on large $k$ limit
and could flat the inner regions of most galaxies more than the CDM model,
reconciling these values with observation and alleviating the core-cusp
problem. At the large $k$ limit, DM behaves as WDM as it slightly reduces the
DM preferred mass range to a size that includes a moderate initial velocity
dispersion and free streaming, sufficient to erase some small scale
structures. The suppression in WDM models has a variety of observable
implications: abundances of galaxies at high redshift (Pacucci et al., 2013;
Menci et al., 2016), high-redshift gamma-ray bursts (GRBs) (de Souza et al.,
2013), strong gravitational lensing (Gilman et al., 2020; Hsueh et al., 2020).
One extension of WDM is to assume that DM comes in two components, a cold one
and a warm one, which can be produced via two co-existing mechanisms. These
models are called mixed dark matter (MDM) (Maccio et al., 2013; Diamanti et
al., 2017; Parimbelli et al., 2021).
In this paper, we decided to investigate the case that dark matter consists of
only one component, but its behavior depends on $k$-scale such that in small
$k$ it behaves like cold DM, and in large $k$ it shows the properties of warm
DM. This scale dependent transition in the behavior can have some motivations
in the physics of critical phenomena. The $k$-Dependent dark energy has been
studied in Farhang & Khosravi (2023) based on a phenomenological gravitational
phase transition model (Khosravi & Farhang, 2022; Farhang & Khosravi, 2021).

The outline of this paper is as follows: in section 2, we derive Boltzmann
equations governing the evolution at the perturbation level. Then, we
implement the related equations in the publicly available numerical code
CLASS111https://github.com/lesgourg/class_public(the Cosmic Linear Anisotropy
Solving System) (Lesgourgues & Tram, 2011) and using the code
MONTEPYTHON-v3222https://github.com/baudren/montepython_public (Audren et al.,
2013; Brinckmann & Lesgourgues, 2019) to perform a Monte Carlo Markov chain
(MCMC) analysis with a Metropolis-Hasting algorithm against the high- CMB TT,
TE, EE +low- TT, EE+lensing data from Planck 2018 (Aghanim et al., 2020) in
combination with other probes such as the Baryon acoustic oscillations, BAO (
BOSS DR12 (Alam et al., 2017), eBOSS Ly-$\alpha$ combined correlations).

##  2 Phenomenology of $k$-Dependent DM model in perturbation level

In the framework of general relativity, let us consider the flat, homogeneous,
and isotropic universe with energy density $\rho(\tau)$ and pressure $P(\tau)$
that is described by the FLRW metric. Using the Einstein equations, we can
obtain the following evolution equations for the expansion factor $a(\tau)$.

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}a^{2}\rho,$
(1) $\displaystyle\frac{d}{d\tau}\left(\frac{\dot{a}}{a}\right)=-\frac{4\pi
G}{3}a^{2}(\rho+3P),$ (2)

where the dots denote derivatives with respect to conformal time, $\tau$. The
most convenient way to solve the linearized Einstein equations is in the two
gauges in the Fourier space $k$. In the synchronous gauge, the scalar
perturbations are characterized by $h(\vec{k},\tau)$ and $\eta(\vec{k},\tau)$.
The scalar mode of $h_{ij}$ is given as a Fourier integral

$h_{ij}(\vec{x},\tau)=\int
d^{3}k\left(h(\vec{k},\tau){\hat{k}_{i}\hat{k}_{j}}+6\eta(\vec{k},\tau)({\hat{k}_{i}\hat{k}_{j}}-\frac{1}{3}\delta_{ij})\right)e^{i{\vec{k}.\vec{x}}},$
(3)

where, h is used to denote the trace of $h_{ij}$ in both the real space and
the Fourier space (Aoyama et al., 2014).
The perturbations are characterized by two scalar potentials
$\psi(\tau,\vec{x})$ and $\phi(\tau,\vec{x})$ which appear in the line element
as

$ds^{2}=a^{2}(\tau)\bigg{(}-(1+2\psi)d\tau^{2}+(1-2\phi)d\vec{x}^{2}\bigg{)},$
(4)

and for a perfect fluid of energy density, $\rho$, and pressure, $P$, the
energy-momentum tensor has the form

$T^{\mu}_{\nu}=Pg^{\mu}_{\nu}+(\rho+P)u^{\mu}u_{\nu},$ (5)

where $u^{\mu}$ is the four-velocity of the fluid. The perturbed part of
energy-momentum conservation equations in $k$-space implies the synchronous
gauge as

$\displaystyle\dot{\delta}=-3{\cal{H}}(c_{s}^{2}-w)\delta-(1+w)(\theta+\frac{\dot{h}}{2}),$
(6)
$\displaystyle\dot{\theta}=-{\cal{H}}(1-3c_{g}^{2})\theta+\frac{c_{s}^{2}}{1+w}k^{2}\delta-k^{2}\sigma,$
(7)

and for the conformal Newtonian gauge as

$\displaystyle\dot{\delta}=-3{\cal{H}}(c_{s}^{2}-w)\delta-(1+w)(\theta-3\dot{\phi}),$
$\displaystyle\dot{\theta}=-{\cal{H}}(1-3c_{g}^{2})\theta+\frac{c_{s}^{2}}{1+w}k^{2}\delta-k^{2}\sigma+k^{2}\psi.$
(8)

The evolution equation for the shear can be obtained as

$\dot{\sigma}=-3\big{[}\frac{1}{\tau}+{\cal{H}}(\frac{2}{3}-c^{2}_{g}-\frac{1}{3}\frac{{\cal{P}}}{p})\big{]}\sigma+\frac{4}{3}\frac{c^{2}_{\rm
vis}}{1+w}(2\theta+\dot{h}).$ (9)

$c_{s}$ and $c_{g}$ in the above equations are the effective sound speed and
the adiabatic sound speed, respectively. In equation 9, $c^{2}_{\rm vis}$ is a
new parameter named viscosity speed, and in implementation of CLASS, it is
assumed as $c^{2}_{\rm vis}=3wc^{2}_{g}$ (Lesgourgues & Tram, 2011). The
adiabatic sound speed can be expressed as

$c^{2}_{g}=\frac{\dot{p}}{\dot{\rho}}=w-\frac{\dot{w}}{3{\cal{H}}(1+w)},$ (10)

or in another form
$c^{2}_{g}=\frac{\dot{p}}{\dot{\rho}}=-w\frac{\dot{p}}{p}(\frac{\dot{a}}{a})^{-1}\frac{1}{3(1+w)}$,
that it is stated in Tram et al. (2019),
$\frac{\dot{p}}{p}=(\frac{\dot{a}}{a})(5-\frac{{\cal{P}}}{p})$. So, the
adiabatic sound speed can be rewrote as

$c_{g}^{2}=\frac{w}{3(1+w)}(5-\frac{{\cal{P}}}{p}),$ (11)

here $\cal{P}$ is the pseudo-pressure that for any pressureless species,
${\cal{P}}\simeq p\simeq 0$ and for relativistic species we have
${\cal{P}}\simeq p$ since in a higher moment pressure
$\frac{{\cal{P}}}{p}\simeq 1$.
Obtaining an analytical expression for $c^{2}_{s}$ is more complicated since
there is no dynamic equation for pressure perturbation, so in Abellán et al.
(2021) and Lesgourgues & Tram (2011), it is supposed that $c^{2}_{s}$ is
scale-independent and approximately equal to $c^{2}_{g}$. Nevertheless, the
full Boltzmann hierarchy calculations show that $c^{2}_{s}$ represents a
specific $k$-dependence and cannot be obtained with a background quantity such
as $c^{2}_{g}$, and it increases slightly on the scales $k$. We follow the
prescription in Abellán et al. (2021) for the synchronous sound speed as

$c^{2}_{s}(k)=c^{2}_{g}\left[1+\frac{1}{5}\sqrt{\frac{k}{k_{fs}}}\right],$
(12)

where $k_{fs}=\sqrt{\frac{3}{2}{{\cal{H}}(a)}/c_{g}(a)}$ is the free-streaming
length of the WDM particles.
Equations 6-9 are valid for a single uncoupled fluid or for the net (mass-
averaged) $\delta$ and $\theta$ for all fluids. They need to be modified for
individual components if the components interact with each other.
The CDM particles can be used to define the synchronous coordinates and
therefore have zero peculiar velocities in this gauge. Setting
$\theta=\sigma=0$ and $w=\dot{w}=0$ in equation 6 for synchronous gauge lead
to

$\dot{\delta}_{\rm CDM}=-\frac{1}{2}\dot{h}.$ (13)

Table 1: The best and mean values and 68% confidence limit (CL) constraints
for the free parameters of CDM and two $k$-DM models. They are given using
Planck and Planck+Other datasets described in the paper.

Model | Parameter | $Planck$ | $Planck+Other$
---|---|---|---
|  | best-fit | $mean\pm\sigma$ | best-fit | $mean\pm\sigma$
| $\Omega{}_{m}$ | $0.3113$ | $0.3148\pm 0.0067$ | $0.2903$ | $0.2888\pm 0.0044$
| $\log_{10}\alpha$ | $7.59$ | $7.88^{+0.15}_{-0.12}$ | $7.72$ | $7.14^{+0.61}_{-0.34}$
| $k_{0}$ | $1.68$ | $1.70^{+0.19}_{-0.12}$ | $0.988$ | $1.003^{+0.049}_{-0.065}$
$k$-DM(1) | $c_{g}^{2}$ | $0.052$ | $<0.014$ | $0.0116(10^{-4})$ | $<0.0373(10^{-4})$
| $w$ | $0.011$ | $<0.009$ | $0.181(10^{-4})$ | $0.126\pm 0.049(10^{-4})$
| $H_{0}$ | $67.67$ | $67.40\pm 0.49$ | $69.25$ | $69.36\pm 0.36$
| $S_{8}$ | $0.8280$ | $0.829\pm 0.012$ | $0.7651$ | $0.7649^{+0.0067}_{-0.0083}$
| $\Omega{}_{m}$ | $0.3107$ | $0.3129\pm 0.0073$ | $0.2954$ | $0.2969^{+0.0054}_{-0.0066}$
| $\log_{10}\alpha$ | $7.65$ | $7.7^{+1.4}_{-1.1}$ | $6.54$ | $6.49^{+0.77}_{-0.63}$
$k$-DM(2) | $k_{0}$ | $2.51$ | $12^{+15}_{-10}$ | $1.66$ | $1.93^{+0.11}_{-0.30}$
| $w$ | $0.570$ | $<0.532$ | $1.18(10^{-7})$ | $<5.48(10^{-5})$
| $H_{0}$ | $67.66$ | $67.54\pm 0.53$ | $68.84$ | $68.67^{+0.55}_{-0.48}$
| $S_{8}$ | $0.822$ | $0.826\pm 0.013$ | $0.7795$ | $0.7875\pm 0.0099$
| $\Omega{}_{m}$ | $0.313328$ | $0.3142\pm 0.0065$ | $0.2929$ | $0.2933\pm 0.0045$
CDM | $H_{0}$ | $67.52$ | $67.46\pm 0.47$ | $69.47$ | $69.02\pm 0.37$
| $S_{8}$ | $0.828789$ | $0.831\pm 0.0012$ | $0.7872$ | $0.7833\pm 0.0074$

However, the CDM fluid velocity in the conformal Newtonian gauge is not zero
in general. In $k$-space, equation 2 gives

$\dot{\delta}_{\rm CDM}=-\theta_{\rm CDM}+3\dot{\phi},\qquad\dot{\theta}_{\rm
CDM}=-\frac{\dot{a}}{a}\theta_{\rm CDM}+k^{2}\psi.$ (14)

As we mentioned, in this study, we intend to consider dark matter such that
its behavior changes in terms of scale, so it behaves as relativistic such as
warm dark matter particles in large $k$ scales, and as non- relativistic such
as cold dark matter in small $k$ scales. Therefore, we introduce a step
function, ${\cal{S}}(k)$, for switching between these two boundary conditions.
${\cal{S}}(k)$ could be any kind of step (switching) function; for example we
consider it as

${\cal{S}}(k)=\frac{1+\tanh[\alpha(k-k_{0})]}{2}.$ (15)

$\alpha$ and $k_{0}$ are free parameters that $\alpha$ control the smoothness
of the transition between cold and warm dark matter. We rewrite 6 by using
${\cal{S}}(k)$ as

$\dot{\delta}=\frac{\dot{h}}{2}+{\cal{S}}(k)\big{[}-3{\cal{H}}(c_{s}^{2}-w)\delta-(1+w)\theta\big{]},$
(16)
$\dot{\theta}={\cal{S}}(k)\big{[}-{\cal{H}}(1-3c_{g}^{2})\theta+\frac{c_{s}^{2}}{1+w}k^{2}\delta-k^{2}\sigma\big{]},$
(17)
$\dot{\sigma}={\cal{S}}(k)\bigg{[}-3\big{(}\frac{1}{\tau}+{\cal{H}}(\frac{2}{3}-c^{2}_{g}-\frac{1}{3}\frac{{\cal{P}}}{p})\big{)}\sigma+\frac{4}{3}\frac{c^{2}_{\rm
vis}}{1+w}(2\theta+\dot{h})\bigg{]}.$ (18)

It is obvious that if ${\cal{S}}(k)$ vanishes, the above equations reduce to
the CDM. This case happens for $k<k_{0}$ and is more precise for larger
$\alpha$’s.
We implement the above equations in the public Boltzmann solver CLASS. Since
we expect this model to behave similarly to CDM in the cosmological
background, we only change the perturbation equations in module
perturbation.c. We analyzed this model in two cases: i) the sound speed
behaves independently of $k$ as a constant parameter ($k$-DM(1)) and ii) the
case where it changes depending on $k$ given to equation 12 ($k$-DM(2)).

##  3 $k$-Dependent DM model verse data

In this section, we present constraints on the $k$-Dependent dark matter model
we have introduced. For MCMC analysis, we use the Metropolis-Hastings
algorithm of the cosmological sampling package MONTEPYTHON-v3, connected to an
altered version of the Boltzmann Solver CLASS.
We use the following dataset combination to perform statistical inference:

Figure 1: 1D likelihoods and 2D contours in 68% and 95% CL marginalized joint
regions for chosen free parameters while they are constrained by using
$Planck+Other$ datasets. It seems the $k$-DM models predict lower $S_{8}$ to
alleviate this tension while the $H_{0}$ value is not affected too much.

  * •

CMB: We use the CMB temperature and polarization auto- and cross-correlation
measurements of the most recent Planck 2018 legacy release, including the full
temperature power spectrum at multipoles 2 $\leq$ l $\leq$ 2500 and the
polarization power spectra in the range 2 $\leq$ l $\leq$ 29 (lowP). We also
include information on the gravitational lensing power spectrum estimated from
the CMB trispectrum analysis. (Aghanim et al., 2020).

  * •

BAO: We use the BAO measurements from the Baryon Oscillation Spectroscopic
Survey Data Release 12 (BOSS DR12) (Alam et al., 2017) ,SS DR14-Ly-$\alpha$
combined correlations (de Sainte Agathe et al., 2019), Lyman-$\alpha$ forest
autocorrelation de Sainte Agathe et al. (2019), and the cross correlation of
Lyman-$\alpha$ and QSO (Blomqvist et al., 2019).

  * •

LSS: We use three different sets of LSS data in order to check whether $k$\-
Dependent dark matter model leads to a suppression in the matter power
spectrum relative to the CDM:
1- KiDS + Viking 450 (KV450) matter power spectrum shape data; this combined
analysis of data from the KiloDegree Survey (KiDS) and the VISTA Kilo-Degree
Infrared Galaxy Survey (VIKING) includes photometric redshift measurements
with cosmic shear/weak-lensing observations to measure the matter power
spectrum over a wide range of $k$-scales at redshifts between 0.1 and 1.2
(Hildebrandt et al., 2020).
2- Planck SZ (2013): Another independent LSS dataset is the Planck SZ which
studies the properties of galaxy clusters by measuring the Sunyaev-Zeldovich
effect. But we should note that the measurements of galaxy distribution from
the SZ effect depend on a mass bias factor $(1-b)$ that relates the observed
SZ signal to the true mass of galaxy clusters. In Planck SZ (2013), a
numerical simulation of the $S_{8}$ measurement is reported by fixing the mass
bias to its central value $(1-b)=0.8$. Later, the Planck SZ (2015) report
allowed $(1-b)$ to vary with a Gaussian prior centered at 0.79. The central
value of the resulting $S_{8}^{\rm SZ}$ becomes smaller but has a much larger
uncertainty, $S_{8}^{\rm SZ}=0.744\pm 0.034$, and less tension to CMB
measurements (Ade et al., 2016). For our analysis, we chose this data set
since the central value $\sigma_{8}$ of the SZ (2013) analysis is consistent
with many low-redshift measurements (Zu et al., 2023).
3-WiggleZ $P(k)$ data: Since dark energy has an effect on the expansion
history of the Universe and on the growth of cosmological structures, we also
use WiggleZ data in this study. The WiggleZ Dark Energy Survey is a survey to
measure the large scale structure of the Universe by mapping the distance-
redshift relation with baryon acoustic oscillations (Kazin et al., 2014).

We employ the $\chi^{2}$ statistics to constrain our theoretical model as:

$\chi^{2}=\frac{({\cal{P}}_{\rm obs}-{\cal{P}}_{\rm
th})^{2}}{\sigma_{{\cal{P}}}^{2}},$ (19)

here ${\cal{P}}_{\rm obs}$, ${\cal{P}}_{\rm th}$ and $\sigma_{{\cal{P}}}^{2}$
indicate the observed values, the predicted values and the standard deviation,
respectively. Note that in addition to the six free parameters of the standard
model, i.e, $(\Omega_{b},\Omega_{DM},10\theta_{\rm MC},\ln
10^{10}A_{s},n_{s},\tau_{\rm reio})$, the $k$-Dependent dark matter model
introduced in the previous section includes for the first case:
$(\alpha,k_{0},w,c^{2}_{g})$ and the second case: $(\alpha,k_{0},w)$. To span
the $\alpha$ parameter’s space, we work with $\log_{10}\alpha$ instead of
$\alpha$. The flat priors we assumed for the parameters are given by
$k_{0}\in[0,{10}]$, $\alpha\in[0,10^{10}]$, $w\in[0,1]$, and
$c^{2}_{g}\in[0,1]$. The convergence of chains for each parameter is measured
by the Gelman-Rubin criterion, and one can obtain acceptable $R-1$ values
(i.e., below 0.01 for every parameter) with an iterative strategy (Gelman &
Rubin, 1992) and the average acceptance rate (acc) is around 0.2.
In order to check the cosmic tensions in these models, we added data step by
step in two MCMC scans as: Planck and then Planck+Other. This can provide us
with further intuition as a starting point, given that Planck’s data has
provided the most precise measurements of the early universe.
In the Table 1, we report the best and the mean values and 68% CL intervals
for the main parameters, including the total matter density parameter
($\Omega_{m}=\Omega_{B}+\Omega_{DM}$), the present-day expansion rate of the
Universe or the Hubble constant, $H_{0}$, and
$S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$ in different scenarios for two MCMC
analyzes. We also show posterior distributions (1$\sigma$ and 2$\sigma$
intervals) as dark and light-shaded contours for MCMC analysis, respectively,
in the plots of the Figure 1.
Some points in these plots need to be stressed. First, it is clear that by
considering all different data sets, the parameters are bound more tightly
than the analysis with Planck data333The only Planck-constrained parameters
are not shown in the figures, but it has checked that Planck and Planck+Other
are consistent. This means the contours for the latter are inside the only
Planck contours.. Second, we can see that the decrease of $H_{0}$ values are
associated with the increase of $\Omega_{m}$ values and vice versa in both
$k$-Dependent and cold dark matter scenarios.

Figure 2: The matter power spectrum for $k$-Dependent DM and CDM models, and
the fractional difference between them. The behavior of $k$-DM models mimics
the CDM for small $k$’s. However, we see a transition for large $k$’s in
$k$-DM models. However, there is no very precise data points at those scales.

As we see in Table 1, assuming $k$ dependence of dark matter behavior for the
Planck+Other analysis seems to improve the $S_{8}$ tension for the
$k$-Dependent dark matter scenario. However, we do not see any significant
improvement in addressing the $H_{0}$ tension. Note that the results show a
small deviation from CDM due to non-zero values for $w$ and $c_{g}^{2}$ when
we have Planck+Other datasets. Their values are at order ${\cal{O}}(10^{-5})$,
which are in agreement with generalized dark matter models (Ilić et al., 2021;
Kopp et al., 2016).

Next, to check whether the fit is good and also to choose the best and most
compatible model with the observational data, we employ the simplest method
that is usually used in cosmology, which is called the least squares method,
$\chi^{2}_{{}_{\rm tot}}$. In this case, the model with smaller
$\chi^{2}_{{}_{\rm tot}}$ is taken to be a better fit to the data (Davari &
Rahvar, 2021). Comparing $k$-Dependent model to the CDM scenario, we note that
$k$-Dependent DM model does better than the CDM model. However, one can have
the impression that the model with the lowest $\chi^{2}_{{}_{\rm tot}}$ is not
necessarily the best because adding more flexibility with extra parameters
will normally lead to a lower $\chi^{2}_{{}_{\rm tot}}$. In this work, the
$k$-Dependent model has three more parameters than the CDM scenario. In order
to deal with model selection, a standard approach is to compute the Akaike
Information Criterion (AIC). It is defined as

$AIC=\chi^{2}_{\rm min}+2M+\frac{2M(M+1)}{N-M+1},$ (20)

where $M$ is the number of free parameters in the model and $N$ is the number
of data points; thus, $\Delta AIC=\Delta\chi^{2}_{\rm min}+2\Delta M$. We
neglect the third term in the Equation 20 for large sample sizes, $N\geq M$.
We report the result of MCMC analysis for the best-fit $\chi^{2}_{min}$ for
observational Planck and total data sets and for both models in Table 2.
The results of this analysis can be interpreted with the Jeffreys’ scale as
follows: among all models, the one that minimizes the AIC is considered to be
the best one, and if the difference between the AIC of a given model and the
best model is smaller than 4, one concludes that the data equally support the
best fitted model and a given model. In the case of $4<|\Delta AIC|<10$,
observations still support the given model but less than the best one.
Finally, for $|\Delta AIC|>10$, observations basically do not support the
given model compared to the best model (Davari et al., 2018). According to
Table 2, the only Planck data prefers CDM with respect to $k$-DM models.
However, adding the other datasets make the situation in favor of $k$-DM
models. This may mean that $k$-DM models have more space to include all the
datasets altogether.

Table 2: The result of MCMC analysis for the best-fit $\chi^{2}$, and AIC. It shows that $k$-DM models have more space to have Planck+Other datasets altogether consistently. Parameters | CDM | k-DM(1) | k-DM(2)
---|---|---|---
| $Planck$ | $Planck+Other$ | $Planck$ | $Planck+Other$ | $Planck$ | $Planck+Other$
$\chi^{2}_{\rm min}$ | $2780.9$ | $3824$ | $2781.02$ | $3812.2$ | $2780.44$ | $3816.7$
$\rm AIC_{k-DM(i)}-AIC_{CDM}$ | $0$ | $0$ | $8.12$ | $-3.8$ | $5.54$ | $-1.3$

In order to have a better understanding of the aspects of obtaining from MCMC
scans, in the following, we discuss the features of the $k$-Dependent model in
the CMB and the matter power spectrum. In Figure 2, we show the matter power
spectrum, $P(k)\equiv\langle\delta_{m}(k)\rangle^{2}$, in the $k$-Dependent
model relative to the CDM model for the best obtained values using
Planck+Other data. As we see, $k$-Dependent dark matter case mimics the CDM
scenario to $k\simeq 1.3$ for the $k$-DM(1) and $k\simeq 2.2hMpc^{-1}$ for the
$k$-DM(2), but then starts to deviate at larger $k$’s (i.e., small scales) and
suppresses the power spectrum of matter by a large difference compared to the
standard model. We include the information embedded in the Ly$\alpha$ forest
measured with the eBOSS-DR14 data release on scales of a few Mpc. One reason
for this difference could be the lack of observational data in this range. In
Figure 3, we notice that considering $k$ dependence for dark matter has the
influence of slowing down the evolution rate of the dark matter perturbations.
This means that structures cluster slower, as we predicted from the Figure 2
for $k>1$ with a slight difference $P_{k-DM(i)}(k)<P_{CDM}(k)$.

Figure 3: The growth rate of matter fluctuations for $k$-Dependent DM model
compared to CDM model. The observational constraints are taken from
(Kazantzidis & Perivolaropoulos, 2018).

Figure 4:  Temperature anisotropies in the CMB. The bottom part of the panel
displays the relative temperature differences between the $k$-DM and the CDM
model.

Since the Planck collaboration has measured the temperature and polarization
maps of the CMB very precisely, it has placed stringent limits on the
parameter space of the CDM model. This motivates us to study $k$-Dependent DM
signatures in CMB maps. In Figure 4, we show how $k$ dependency affects the
temperature power spectra, including the variation with respect to the CDM
model. We can see a suppression in the amplitude of the lower multipoles in
the temperature power spectrum. As we know, the integrated Sachs-Wolfe (ISW)
effect is important on such scales. Also, the small $l$’s of CMB (TT and, even
better, EE) give information on the reionization history. We obtain the
redshift of the reionization, $z_{\rm reio}$, using the best values of
parameters to be 6.02 , 6.41 and 7.38 for $k$-DM(1), $k$-DM(2) and CDM
respectively. The $z_{\rm reio}$ of the $k$-DM(1) model has the biggest
differences from the standard model.

A crucial quantity in determining the age and evolution of the universe in
cosmology is $H_{0}$. It represents the current rate of expansion of the
universe. Because of the impact of Hubble’s expansion on the growth of matter
perturbations, it is significant to survey the behavior of H(z) in various DM
cosmologies. We plot the evolution of H(z)/1+z in Figure 5.

Figure 5: The redshift evolution of H(z)/1+z using the best fit values of
cosmological parameters in Tables 1 by Planck+Other data for the proposed
$k$-DM scenarios and CDM model. Also, in the Figure, the observational data
points from the Riess et al. (Riess et al., 2019), Planck 2018 (Aghanim et
al., 2020), BOSS DR12 (Alam et al., 2017), DR14 quasars (Zarrouk et al.,
2018), and BOSS DR14 Ly-$\alpha$ (de Sainte Agathe et al., 2019) measurements
have been specified for comparison.

Our results in Tables 1 and Figure 6 show that the assumption of dependence
dark matter to $k$ scale can only reduce the $S_{8}$ tension and not the
$H_{0}$ tension. In general, $k$-DM(1) model, which considered the equation of
state, $w$, and adiabatic sound speed, $c^{2}_{g}$ independent of $k$ scale,
reduces $S_{8}$ tension more than other models.

Figure 6: Comparison between $H_{0}$ and $S_{8}$ tensions for all considered
models in this study using the Planck+Other data.

##  4 Discussion

The warm dark matter model has always been of interest mainly because of the
possible need to alleviate the small-scale problems of the $\Lambda$CDM. With
such insight and also motivated by the effect of adding this cosmological
component to reduce current cosmological tensions, in this work we considered
a scenario in which the behavior of dark matter depends on the scale. It
mimics CDM for small $k$’s and WDM for large $k$’s. A motivation for us was to
check if the trace of WDM, which can be seen in very small scales to address
e.g., can the core-cusp problem show itself in the (very short) cosmological
scales? Our results show that this transition can affect the amplitude of the
matter fluctuations, such that reducing the $S_{8}$ tension. However, it seems
the lack of cosmological data for very large $k$’s makes it hard to answer to
the above question. For future analysis, we can think of a more theoretical
framework and also find cosmological datas at very small scales which are
cleaned from the baryonic physics. One way can be tracing the effects of our
model in non-linear structure formation and the dark matter halo
distributions.

##  5 ACKNOWLEDGMENTS

NK would like to thank Marzieh Farhang for instructive discussions during
working on Farhang & Khosravi (2023). This work has been supported financially
by a grant from Basic Sciences Research Fund under grant number BSRF-
phys-399-06. ZD also acknowledges support from Iran Science Elites Federation
under grant number M401543.

##  6 DATA AVAILABILITY

No new data were generated or analysed in support of this research.

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De Becker Arora<EMAIL_ADDRESS>
# Synchrotron radio emission as a proxy to identify long period massive
binaries

Michaël  Bharti  Space sciences, Technologies, and Astrophysics Research
(STAR) institute, University of Liège, Belgium

(30th April 2023)

###### Abstract

The multiplicity of massive stars is known to be significantly high. Even
though the majority of massive stars are located in binary systems, the census
of binaries is biased toward shorter periods as longer period systems are more
difficult to identify. Alternatively, the search for binary systems with
longer periods may proceed differently. As massive binary systems are
typically colliding-wind systems, hints for processes occurring in the
colliding-wind region could be used as a valuable proxy to identify likely
binary systems, and then organize dedicated spectroscopic or interferometric
campaigns on a short list of pre-selected targets. In this context, any hint
for synchrotron radio emission is seen as a promising indicator of long period
binaries, as short period systems undergo severe free-free absorption of the
synchrotron emission by the stellar wind material. Usual techniques to
identify synchrotron radio emitters constitute thus valid tools to explore
that poorly investigated part of the massive binary parameter space. In
addition, the identification of a synchrotron emission component in a short
period binary can be used as an indicator of the presence of a third companion
on a still unrevealed wider orbit in a triple system.

###### keywords:

Massive binaries, Particle acceleration, Non-thermal processes, Radio
astrophysics

##  1 The multiplicity of massive stars

A large fraction of massive stars are part of binary, or higher multiplicity
systems (Sana et al., 2012; Offner et al., 2022). However, the census of
binary systems is dependent on the specificities of the direct techniques used
to investigate their multiplicity (see blue boxes in Fig. 1),

  1. -

Shorter time scales are easier to monitor as the duration of the time series
is shorter, and observation campaigns are easier to organize.

  2. -

Photometry and spectroscopy are well-known techniques easy to implement, and
especially efficient at unveiling and characterizing short period systems
(typically a few days, even though spectroscopy can be efficient up to a few
year). However, these techniques are valid provided the inclination of the
system is appropriate. The measurement of eclipses through photometric time
series requires indeed the line-of-sight to be very close to the orbital
plane. An orbital plane seen under a not too low inclination angle is also a
requirement for the identification and study of spectroscopic binaries. The
Doppler shift of individual stellar spectral lines results from the periodic
motion along the line-of-sight of the components of the binary system (either
approaching or receding from the observer). A pole-on viewed orbit is thus not
able to display any radial velocity shifts along the orbital motion. In
addition, long period systems display typically lower amplitude radial
velocity curves, and their high eccentricity leads radial velocities not to
vary in a spectacular way over a large fraction of the orbit.

  3. -

Long baseline interferometry is a complementary technique very useful to
investigate periods of the order of one year, up to a few tens of years (Le
Bouquin et al., 2017; Lanthermann et al., 2023). However, only a couple of
facilities exist (e.g., the Very Large Telescope Interferometer at the
European Southern Observatory, Chile, or the Center for High Angular
Resolution Astronomy array on Mount Wilson, California), and this is not
enough to organize significant surveys.

  4. -

Non-interferometric high resolution imaging is relevant for much longer
periods, but such long time-scales are not easy to monitor (Sana et al.,
2014).

Given the constraints listed above, long period systems are still poorly
investigated, leading to a significant bias in the exploration of the massive
binary parameter space.

Figure 1: Diagram illustrating the main science drivers motivating the study
of massive binaries, along with the main techniques used to investigate their
multiplicity as a function of the orbital period.

##  2 Sciences drivers as a function of the orbital period

Depending on the orbital period of massive binaries, various science drivers
can be identified (see green boxes in Fig. 1). Shorter period systems are
privileged targets for modern stellar astrophysics for various reasons,
including in particular these ones:

  1. 1.

They constitute potential progenitors for gravitational wave (GW) events
triggered by black hole mergers or neutron star mergers (LIGO/VIRGO
Collaboration, 2019).

  2. 2.

Mass exchange between components of short period binary systems is a key
aspect of massive star evolution (Sana et al., 2012).

On the other hand, long period systems are relevant laboratories to
investigate shock physics. The stellar winds of both stars collide at high
speeds, typically their terminal velocity ($V_{\infty}$ = 2000 – 3000 km s-1),
leading to high Mach number adiabatic shocks. A specific topic of interest in
these systems is particle acceleration through the Diffusive Shock
Acceleration process (Drury, 1983). This process is known to accelerate
charged particles, including electrons, up to relativistic velocities. This
population of relativistic electrons is likely to participate in non-thermal
emission processes. In particular, in the presence of the magnetic field of
stars in the system, one can expect the synchrotron emission process to be
active in the radio domain (e.g. De Becker, 2007).

##  3 Synchrotron radiation as an tracer of long period massive binaries

Most particle accelerators among massive binaries, referred to as Particle-
Accelerating Colliding-Wind Binaries (PACWBs, De Becker and Raucq
2013111https://www.astro.uliege.be/~debecker/pacwb/) are identified through
the detection of synchrotron radio emission from the colliding-wind region
(e.g. Abbott et al., 1984; Benaglia et al., 2006; Dougherty et al., 2005;
Blomme et al., 2007; Benaglia et al., 2015, 2020; Marcote et al., 2021). Let’s
stress that single massive stars do not offer the opportunity to accelerate
particles, and thus to produce synchrotron radiation, except in the scarce
situation of a powerful wind interacting with a sufficiently dense
interstellar medium (the only known case to date is of WR102, Prajapati et al.
2019). Particle acceleration and synchrotron emission by massive stars is
strongly correlated to their multiplicity, as confirmed by the current census
of PACWBs. However, the synchrotron emission region can be significantly
embedded in the stellar wind plasma. Depending on the wind parameters, on the
orbital parameters and on the orbital phase, the synchrotron emission can be
significantly attenuated by free-free absorption (FFA). This process is
especially active in short period binaries where the wind collision occurs in
a denser plasma. PACWBs are thus more likely revealed in longer period systems
(typically, at least several weeks, but generally more than a few months).

As a result, one can claim that not all massive binaries are able to reveal
some synchrotron radio emission, but all synchrotron-emitting massive stars
are part of a binary or higher multiplicity system. This fact allows to
consider that the firm identification of a synchrotron component in the radio
emission from massive stars is telling us about their multiplicity.

##  4 General strategy

Even though large scale surveys of stellar populations can turn out to be
quite useful to investigate the multiplicity of massive stars, it will most of
the time require to gather long-term time series. This is quite demanding in
term of telescope time and, as stated in Sect. 1, observational biases are
strongly affecting most techniques implemented for such an endeavour.
Alternatively, direct methods such as spectroscopic and high angular
resolution imaging techniques can be used following a more efficient approach,
focusing on target lists made of stars whose multiplicity has been revealed by
indirect methods, such as identifying them as synchrotron emitters.

The radio measurement of many massive stars not known to be long period
binaries is thus expected to reveal synchrotron radiation from some of them,
according to well-established interpretation guidelines (De Becker and Raucq,
2013; De Becker et al., 2017): variability studies, high-angular resolution
measurements, spectral index measurements based on at last two spectral bands,
search for high brightness temperature radio emission. If synchrotron
radiation is identified, it points to a binary system with a period long
enough to warrant that FFA is not too severe. From this point, direct
techniques can be applied to pre-selected targets, aiming for the detection of
the companion, the identification of its spectral type and the determination
of the orbital parameters of the system.

This approach deserves also to be considered for known short period binaries,
as it is likely to reveal triple systems with a third companion on a much
wider orbit. This approach has already been quite useful to confirm the triple
system status of a few massive star systems whose short orbit was too small
(typically a few days) to explain the detection of synchrotron radio emission.
One may mention the case of HD 167971 whose wide orbit was revealed by long
baseline interferometry (De Becker et al., 2012), or that of HD 150136
investigated by using spectroscopic and interferometric techniques (Sana et
al., 2013). More recently, the careful analysis of the radio emission from the
112-d binary WR133 led to the conclusion that it could be a triple system,
with a still undetected third object on an orbit of potentially a few years
(De Becker et al., 2019).

##  5 Summary and concluding remarks

The key points addressed in this paper are the following:

  1. $\bullet$

Given the difficulty to reveal long period binaries with usual direct
techniques, the census of long period systems is underestimated. As a result,
complementary indirect approaches to reveal long period systems are welcome.

  2. $\bullet$

An important feature of massive binaries is their capability to allow their
stellar winds to collide. These wind collisions are appropriate to drive shock
physics, including particle acceleration.

  3. $\bullet$

PACWBs are mainly identified through their synchrotron radio emission. One can
thus consider this non-thermal radio emission signature as an indicator of
binarity.

  4. $\bullet$

Synchrotron radio emission is easily attenuated by free-free absorption that
is especially strong in short period systems. The synchrotron signature is
thus expected to be revealed provided the orbital period is not too short, as
the colliding-wind region is not too deeply embedded in the dense part of the
stellar winds. As a result, synchrotron radio emission and particle
acceleration can be considered as good indirect indicators of binary systems
with rather long periods.

  5. $\bullet$

Once a candidate binary system is identified thanks to its synchrotron radio
signature, a more focused use of direct multiplicity study techniques (such as
spectroscopy or long baseline interferometry) can be envisaged to characterize
the newly discovered systems. In particular, the identification of the
companion’s nature and the determination of orbital elements cannot be
achieved by the sole use of radio measurements.

It is important to stress that such an approach also turns out to be efficient
at revealing the existence of a third companion in already known binary
systems. Typically, a massive star system known as a very short period binary
and displaying significant synchrotron radio emission could actually be a
triple system, with the non-thermal radio emission arising from the collision
region produced by the interaction of the stellar wind of a third star with
that of the previously known close binary. Several members of the catalogue of
PACWBs were confirmed as binaries or higher multiplicity systems thanks to
specific multiplicity studies motivated by their particle accelerator status.
The use of their synchrotron radio emission is thus a relevant approach to
investigate the multiplicity of massive stars in general.

Beside the interest of this approach to unveil new particle accelerators to be
discussed in the context of the role played by PACWBs in the acceleration of
galactic cosmic rays (see De Becker et al., 2024, in this volume), a better
census of the multiplicity of massive stars is of high importance for the
question of their formation and dynamic evolution in stellar clusters.

###### Acknowledgements.

The authors would like to thank the organizers for the opportunity to present
this poster, along with the Belgo-Indian Network for Astronomy and
Astrophysics (BINA) for financial support. Bharti Arora acknowledges financial
support from Wallonia-Brussels International (WBI). This research has made use
of NASA’s Astrophysics Data System Bibliographic Services. This work is
supported by the Belgo-Indian Network for Astronomy and astrophysics (BINA),
approved by the International Division, Department of Science and Technology
(DST, Govt. of India; DST/INT/BELG/P-09/2017) and the Belgian Federal Science
Policy Office (BELSPO, Govt. of Belgium; BL/33/IN12).

0000-0002-1303-6534M.De Becker

Both co-authors actively contributed to this work.

The authors declare no conflict of interest.

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