# Some Classes of Interacting Two-Fluid Model of the Expanding Universe

Subhra Bhattacharya<EMAIL_ADDRESS>Department of Mathematics,
Presidency University, Kolkata-700073, India

###### Abstract

We consider interacting dark matter-dark energy models arising out of a
general interaction term
$Q=f(\rho_{m},\rho_{d},\dot{\rho}_{m},\dot{\rho}_{d}).$ Here $f$ is a
functional relation connecting the energy densities $\rho_{m}$ and $\rho_{d}$
and their derivatives w.r.t. time $t.$ In our model we consider two
interacting barotropic fluid with constant equation of state $\omega_{m}$ and
$\omega_{d}.$ By considering a dynamical interaction between them we trace out
the cosmological evolution dynamics of the universe. We analytically solve the
model by considering a constant ratio between the two fluids and then track
the corresponding analytical results using observational data from the baryon
acoustic oscillation measurements, Type Ia supernovae measurements and the
local Hubble constant measurements. From this general setting we introduce
three different models and nine different interaction function. Our final aim
is to set up a comparative analysis of the various class of models under the
different interaction function using common theoretical and numerical
analysis.

$\Lambda$CDM, Dark Matter, Dark Energy, Interaction

##  I Introduction

Observational data from distant supernovae and cosmic microwave background
have established that the universe is undergoing accelerated expansion sup1 ;
sup2 . Theoretical predictions suggest that the acceleration is driven by some
matter component with a negative pressure that is sufficient to counter the
attractive gravitational force. Considering a homogeneous and isotropic
universe, studies show that the primary matter component of our universe is
the dark energy (DE), characterized by negative pressure while most of the
remaining matter is made up of pressure-less dark matter (DM). The rest of the
universe comprises of negligible amount of baryonic matter and radiations.
Such a universe is modelled by the simple concordance model or the
$\Lambda$CDM model lcdm , that best fits the observational data. Infact
$\Lambda$CDM is so successful that a locally inhomogeneous universe can mimic
$\Lambda$CDM cosmology at large scales sb1 . In the $\Lambda$CDM model the
expansion of the universe is attributed to cosmological constant $\Lambda$
with negative pressure lam . $\Lambda$ corresponds to the vacuum energy
density, resulting from vacuum fluctuations of matter with equation of state
$\omega_{\Lambda}=\frac{p_{\Lambda}}{\rho_{\Lambda}}=-1.$

Despite the phenomenal success of the $\Lambda$CDM model, it is mired with
several theoretical limitations. Two important, yet unresolved puzzles are the
cosmological constant (CC) problem cc  and the coincidence problem coin .
Calculation reveal a 121 order of magnitude discrepancy in the numerical
estimates of the cosmological constant when compared with the actual estimate
of vacuum energy density. This difference between the numerical and
theoretical estimates is called the cosmological constant problem. The second
perplexing puzzle is the similar orders of magnitude of the DE and DM energy
density in the current epoch. The densities of different matter components in
the universe did not remain same throughout its evolution. The hot big bang
divides the evolution into three phases. An initial hot radiation dominated
epoch. This was followed by the DM dominated epoch, during which most of the
structures we observe in the universe were created. Curiously this changed at
a recent time around red-shift distance of $z\approx 0.5$ when densities of DM
and DE became equivalent. During the first two phases of evolution the density
of DE component was negligible in the universe, yet it is it only by
coincidence that in the current epoch it evolved to similar orders of
magnitude as the other matter components. This is the coincidence problem.

A considerable bulk of the literature has been devoted to tackle these
inconsistencies, yet the questions remain. Models like scalar field
description of DE, modified gravity theories like $f(R),f(T)$ and recently
$f(Q)$ mgt , the dynamical DE models with variable equation of state parameter
(also called $\omega CDM$), cosmic fluids etc. have been used time and again
to resolve these issues rev . One such theoretical model is the interacting DE
and DM description of the universe ide1 ; ide2 ; ide3 ; ide4 ; pwan ; am ; guo
; he ; mvg ; liu ; 1c2 ; 2c2 ; 3c2 ; 4c2 ; c51 ; c52 ; c53 ; sb2 ; c81 ; sb3 ;
c9 ; sp ; wyan ; iw . Most cosmological models assume independent evolution
for DE and DM. It was shown that scalar field models can couple with other
matter fields via energy transfer between the various matter sectors. Various
work showed that models depicting viable interaction between DE and DM could
help to resolve the cosmological constant problem idcc . Since the current
energy budget of the universe show predominance of the two dark components, it
is not unusual to assume non-gravitational interaction between them resulting
in their simultaneous evolution, which in turn can lead to successful
resolution of the coincidence problem idcoin . Several models of interacting
DE and DM has been suggested in the literature. For a complete review see
idrev . Recent studies have also predicted that interacting models can have
significant contribution in easing the present tension in Hubble constant
$H_{0}$ ht ; 1c2 ; 2c2 . Due to the variety of possibilities, the interacting
models are some of the extensively studied models in the literature rec2
-rec36 . Most models predict minimal phenomenological interaction, involving
the Hubble parameter, the fluid energy densities and sometimes the derivatives
of the dark fluid energy densities.

In this article we shall consider one such general interaction function
$Q=f(\rho_{m},\rho_{d},\dot{\rho}_{m},\dot{\rho}_{d},H).$ Here $\rho_{m}$ and
$\rho_{d}$ denote the two interacting dark fluids and
$\dot{\rho}_{m},\dot{\rho}_{d}$ denote their derivatives w.r.t time $t.$ These
two dark fluid interact via non gravitational coupling term $Q$. In the
limiting case of $Q\rightarrow 0$ we can get back the usual non-interacting
scenario. It may be noted that the choice of interaction function is based
purely on phenomenological considerations. Attempts to obtain suitable
interaction based on underlying physics has so far produced no successful
results. Since interacting DM-DE cosmologies track the background dynamics of
the universe, it is expected that interactions will have dependence on the
Hubble parameter and and the two dark densities affecting the cosmology. Hence
it is logical to assume a functional relation between them and their rate of
evolution. We therefore choose a general interaction having dependence on all
the factors that can affect interaction, namely $H,\rho_{m},\rho_{d}$ and
their rate of change. We solve the ensuing system by theoretical means with
the assumption of a constant ratio between DE and DM. We divide the resulting
solution into nine subclasses of solution, depending upon specific parameter
choices. These nine sub-classes are constrained using binned Supernovae data
from JLA light curves, the Hubble and BAO data in three separate models. The
various parameter constraints are used to set up a comparative analysis
between the several interaction functions. We also trace the evolution of the
equation of state the effective matter and the corresponding dynamical
deceleration parameter. Finally we do statistical fitness analysis among the
various classes of interaction function considered to understand which models
to accept and which ones to rule out.

The article is arranged as follows: in section 2 we provide details of the
relevant equations, the analytical results, the interaction functions and the
model formulations. In section 3 we provide details of the data sets used with
the details and discussion of the corresponding results obtained. In section 4
we discuss the effective equation of state, while in section 5 the
deceleration parameter for the model. In section 6 we provide statistical
model selection details and finally in section 7 we end with a brief
discussion.

##  II Modelling the Interacting Dark Sector

We consider a universe described by the flat
Friedmann–Lemaître–Robertson–Walker metric (FLRW) given by:

$dS^{2}=dt^{2}-a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right)$ (1)

where $t$ is the cosmic time and $a(t)$ is the scale factor. The stress energy
tensors consist of coupled barotropic fluids $\rho_{m}$ and $\rho_{d}$ with
matter distribution $T_{\mu\nu}$ following the general equation:

$T_{\mu\nu}=(p+\rho)u_{\mu}u_{\nu}+pg_{\mu\nu}$ (2)

with $u_{\mu}$ the four velocity of the fluid and $p$ being the fluid
pressure. The Einstein’s field equations are given by:

$\displaystyle 3H^{2}$ $\displaystyle=8\pi G(\rho_{m}+\rho_{d})$ (3)
$\displaystyle-2\dot{H}$ $\displaystyle=8\pi G(\rho_{m}+p_{m}+\rho_{d}+p_{d})$
(4)

where with the Hubble parameter $H=\frac{\dot{a}}{a}$ and _dot_ denoting
derivative w.r.t time $t$. We consider that the interacting dark sectors have
equation of state $p_{m}=\omega_{m}\rho_{m}$ and $p_{d}=\omega_{d}\rho_{d}$
with both $\omega_{m}$ and $\omega_{d}$ being constants. Since the total
energy density is conserved, the continuity equations evolve as:

$\displaystyle\dot{\rho_{m}}+3H\rho_{m}(1+\omega_{m})=Q$ (5)
$\displaystyle\dot{\rho_{d}}+3H\rho_{d}(1+\omega_{d})=-Q$ (6)

with $Q$ being the interaction function. Here a $Q>0$ term indicate an energy
transfer from DE to DM, while $Q<0$ indicates transfer of energy from DM to
DE. Let $\rho_{t}=\rho_{m}+\rho_{d}$ be the total energy density of the
combined dark sectors. Then from the above continuity equations we observe
that $\rho_{t}$ follows the conservation equation
$\dot{\rho_{t}}+3H\rho_{t}(1+\omega_{t})=0$ with
$\omega_{t}=\frac{\omega_{m}\rho_{m}+\rho_{d}\omega_{d}}{\rho_{m}+\rho_{d}}.$
The continuity equations are modified using $y=3\log a$ to give

$\displaystyle\rho_{m}^{\prime}+\rho_{m}(1+\omega_{m})=R$ (7)
$\displaystyle\rho_{d}^{\prime}+\rho_{d}(1+\omega_{d})=-R$ (8)

with dash denoting differentiation w.r.t the new variable $y$ and
$R=\frac{Q}{3H}.$ Here we consider the interaction as a linear combination of
the factors affecting the background dynamics and is given by:

$R=\xi\rho_{m}+\eta\rho_{d}+\alpha\rho^{\prime}_{m}+\beta\rho^{\prime}_{d}+\gamma\rho^{\delta}_{m}\rho^{1-\delta}_{d}$
(9)

where parameters $\xi,~{}\eta,~{}\alpha,~{}\beta,~{}\gamma$ and $\delta$ are
free constants that will be determined using observational data. With

$\rho_{d}=r\rho_{m}$ (10)

we solve (7) and (8) and obtain :

$\displaystyle\rho_{m}=\rho_{m0}(1+z)^{3\left(\frac{1+\omega_{m}-\xi-\eta
r-\gamma r^{1-\delta}}{1-\alpha-\beta r}\right)}$ (11)
$\displaystyle\rho_{d}=\rho_{d0}(1+z)^{3\left(\frac{1+\omega_{d}+\frac{\xi}{r}+\eta+\frac{\gamma}{r^{\delta}}}{1+\frac{\alpha}{r}+\beta}\right)}$
(12)

where $1+z=\frac{1}{a}$ is the red-shift distance. Using the Friedmann’s
equation (3) we obtain that the evolution dynamics of the universe in terms of
the Hubble’s parameter as:

$3H^{2}=8\pi G\left(\rho_{m0}(1+z)^{3\left(\frac{1+\omega_{m}-\xi-\eta
r-\gamma r^{1-\delta}}{1-\alpha-\beta
r}\right)}+\rho_{d0}(1+z)^{3\left(\frac{1+\omega_{d}+\frac{\xi}{r}+\eta+\frac{\gamma}{r^{\delta}}}{1+\frac{\alpha}{r}+\beta}\right)}\right)$
(13)

In terms of the density parameters the above expression can be rewritten as:

$E^{2}(z)=\Omega_{m}(1+z)^{3\left(\frac{1+\omega_{m}-\xi-\eta r-\gamma
r^{1-\delta}}{1-\alpha-\beta
r}\right)}+\Omega_{d}(1+z)^{3\left(\frac{1+\omega_{d}+\frac{\xi}{r}+\eta+\frac{\gamma}{r^{\delta}}}{1+\frac{\alpha}{r}+\beta}\right)}$
(14)

where $E(z)=\frac{H(z)}{H_{0}}$ is the dimensionless expansion coefficient
($H_{0}$ is the Hubble parameter at the present time),
$\Omega_{m}=\frac{\rho_{m0}}{\rho_{c0}},~{}\Omega_{d}=\frac{\rho_{d0}}{\rho_{c0}}$
with $\rho_{c0}=\frac{3H_{0}^{2}}{8\pi G}$ and $\Omega_{m}+\Omega_{d}=1.$ We
can obtain different interacting functions from (14) by setting the free
parameters to zero in order. Accordingly this general function has been split
up into nine different interacting functions as below:

  * •

C1: $R=\xi\rho_{m}$
Here $(\eta,~{}\alpha,~{}\beta,~{}\gamma,~{}\delta)=0$ with the expansion term
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3(1+\omega_{m}-\xi)}+\Omega_{d}(1+z)^{3(1+\omega_{d}+\frac{\xi}{r})}.$

  * •

C2: $R=\eta\rho_{d}$
Here we take $(\xi,~{}\alpha,~{}\beta,~{}\gamma,~{}\delta)=0$ with the
expansion factor $E^{2}(z)=(1-\Omega_{d})(1+z)^{3(1+\omega_{m}-\eta
r)}+\Omega_{d}(1+z)^{3(1+\omega_{d}+\eta)}.$

  * •

C3: $R=\xi\rho_{m}+\alpha\rho^{\prime}_{m}$
Here we take $(\eta,~{}\beta,~{}\gamma,~{}\delta)=0.$ This gives the expansion
coefficient as:
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3\left(\frac{1+\omega_{m}-\xi}{1-\alpha}\right)}+\Omega_{d}(1+z)^{3\left(\frac{1+\omega_{d}+\xi/r}{1+\alpha/r}\right)}.$

  * •

C4: $R=\eta\rho_{d}+\beta\rho^{\prime}_{d}$
In this case we take $(\xi,~{}\alpha,~{}\gamma,~{}\delta)=0$ with
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3\left(\frac{1+\omega_{m}-\eta r}{1-\beta
r}\right)}+\Omega_{d}(1+z)^{3\left(\frac{1+\omega_{d}+\eta}{1+\beta}\right)}.$

  * •

C5: $R=\xi\rho_{m}+\eta\rho_{d}$
Here we take $(\alpha,~{}\beta,~{}\gamma,~{}\delta)=0$ with
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3(1+\omega_{m}-\xi-\eta
r)}+\Omega_{d}(1+z)^{3(1+\omega_{d}+\xi/r+\eta)}.$

  * •

C6: $R=\eta\rho_{d}+\alpha\rho^{\prime}_{m}$
For this we assume $(\xi,~{}\beta,~{}\gamma,~{}\delta)=0$ and the expansion
given by: $E^{2}(z)=(1-\Omega_{d})(1+z)^{3\left(\frac{1+\omega_{m}-\eta
r}{1-\alpha}\right)}+\Omega_{d}(1+z)^{3\left(\frac{1+\omega_{d}+\eta}{1+\alpha/r}\right)}.$

  * •

C7: $R=\xi\rho_{m}+\beta\rho^{\prime}_{d}$
Here we take $(\eta,~{}\alpha,~{}\gamma,~{}\delta)=0$ and the expansion factor
reduces to:
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3\left(\frac{1+\omega_{m}-\xi}{1-\beta
r}\right)}+\Omega_{d}(1+z)^{3\left(\frac{1+\omega_{d}+\xi/r}{1+\beta}\right)}.$

  * •

C8: $R=\alpha\rho^{\prime}_{m}+\beta\rho^{\prime}_{d}$
Here $(\xi,~{}\eta,~{}\gamma,~{}\delta)=0$ and the expansion factor is:
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3\left(\frac{1+\omega_{m}}{1-\alpha-\beta
r}\right)}+\Omega_{d}(1+z)^{3\left(\frac{1+\omega_{d}}{1+\alpha/r+\beta}\right)}.$

  * •

C9: $R=\gamma\rho_{m}^{\delta}\rho_{d}^{1-\delta}$ Finally we take
$(\xi,~{}\eta,~{}\alpha,~{}\beta)=0$ which gives:
$E^{2}(z)=(1-\Omega_{d})(1+z)^{3\left(1+\omega_{m}-\gamma
r^{1-\delta}\right)}+\Omega_{d}(1+z)^{3\left(1+\omega_{d}+\gamma
r^{-\delta}\right)}.$

We shall consider these nine independent interaction functions within the
purview of three different cosmological models.

  * M1:

_Interacting Cold Dark Matter and Cosmological Constant_
We will consider $\rho_{m}$ as the pressure-less cold dark matter (CDM) with
$\omega_{m}=0$ and $\rho_{d}$ as the cosmological constant (CC) with
$\omega_{d}=-1$.

  * M2:

_Interacting Cold Dark Matter and Dark Energy_
Here although $\rho_{m}$ is considered to be CDM, $\rho_{d}$ is considered to
be a fluid with free equation of state parameter $\omega_{d}.$

  * M3:

_Interacting Two Fluid_
In this model we consider interaction between two fluid with constant equation
of state $\omega_{m}$ and $\omega_{d}$ which are which are free to be
determined using the observational data sets.

We constrain the free parameters in the nine interaction functions using the
models M1-M3. Based on the results we set up a comparison between the
different models and the interaction functions. Although one can find numerous
literature on coupled DE and DM cosmology, yet we cannot bring them under a
single umbrella due to diversity in solution approach and data sets. This work
thus aims to bring a fairly large class of coupled DE-DM models using a
unified theoretical and numerical approach for their better understanding and
classification. Further, analysing these scenarios in three different models
will help us underline the most convincing form of DE and DM that supports
coupled evolution of the two most prominent energy sources of the universe.

##  III Data Sets Used

JLA Data: We used a sample of 31 binned data and their corresponding
covariance matrix data from the Joint Light-curve analysis jla . The
corresponding $\chi^{2}$ function that is used to constrain the theoretical
model is given by:

$\chi^{2}_{JLA}=(\bm{\mu^{obs}}-\bm{\mu})^{T}\bm{C}_{\mu}^{-1}(\bm{\mu^{*}}-\bm{\mu})$
(15)

with $\mu(z)$ being the distance modulus of type Ia supernova located at
redshift $z$ and $\bm{C}_{\mu}$ the covariance matrix for the binned JLA data
set.

OHD Data: We have used 35 sets of observed Hubble data corresponding to the
red shift range $0.07\leq z\leq 2.36.$ The data sets have been derived from
Cosmic Chronometers ccr , the baryon acoustic oscillation method al  and the
baryon acoustic oscillations in the Ly$\alpha$ forest of high red shift
quasars del . The corresponding $\chi^{2}$ is defined as:

$\chi^{2}_{OHD}=\sum_{i=1}^{35}\frac{\left(H^{obs}(z_{i})-H(z_{i}(\Theta))\right)^{2}}{\sigma_{i}^{2}}$
(16)

BAO Data: The BAO data used in the analysis were obtained from the following
sources:

  1. 1.

6dF Galaxy Survey at $z=0.106$ df ,

  2. 2.

the SDSS DR7 main galaxy sample at $z=0.15$ sd ,

  3. 3.

the BOSS LOWZ data at $z=0.32$ and

  4. 4.

BOSS CMASS at $z=0.57$ boss .

The corresponding $\chi^{2}$ is given by:

$\chi^{2}_{BAO}=\sum_{i=1}^{4}\frac{(\bm{(\eta}^{obs}(z_{i})-\bm{\eta}(z_{i}\Theta))^{2}}{\bm{\sigma}_{i}^{2}}$
(17)

where $\bm{\eta}(z_{i})$ is related to the ratio of $d_{V}$ the volume
dilation scale and $r_{s},$ the co-moving size of sound horizon at baryon drag
epoch. The volume dilation scale $d_{V}$ is related to the luminosity distance
$d_{L}$ by $d_{V}=\left(\frac{zd_{L}^{2}}{(1+z)^{2}H(z)}\right)^{1/3}.$

$\Theta$ in above three expressions indicate the various parameter set
constrained in the models.

###  III.1 Statistical Method Applied

We use the Markov Chain Monte Carlo (MCMC) technique of parameter estimation
using the $Python$ application of the MCMC that was devised by Goodman and
Weare gd  and Foreman-Mackey et al fr . Using the Bayes’ theorem we can relate
the posterior probability distribution of the parameter space to its prior
probability distribution and determine the corresponding likelihood function.
The minimum $\chi^{2}$ or the maximum likelihood function is given by:

$-\ln\mathfrak{L}(\Theta)=\frac{1}{2}\left(\chi^{2}_{JLA}+\chi^{2}_{OHD}+\chi^{2}_{BAO}\right)$
(18)

where $\Theta$ denotes the set of parameters.

###  III.2 Model Analysis

We have used one data set consisting of a combined JLA+OHD+BAO data to
constrain the free parameters
$r,\xi,\eta,\alpha,\beta,\gamma,\delta,\omega_{d},\omega_{m},\Omega_{d}.$ For
model M1 we have the ratio factor $r,$ the cosmological constant energy
density $\Omega_{d}$ and the interaction coefficients as parameters, while for
M2 we have the DE equation of state $\omega_{d}$ as an added parameter and
model M3 we have both DE and DM energy densities $\omega_{d},~{}\omega_{m}$ as
additional free parameters. In Table I-Table IX we give a tabular
representation of the constrained parameters, while in Figures 1-27 (provided
in Appendix A) we give the corresponding confidence contours on the 2
dimensional parameter space.

Table I, Fig.1 - Fig.3 shows the data for interaction function C1. For this
interaction all models predict marginal interaction between the two matter
sectors with $\xi=0.04$ for M1 and M2 and 0.08 for M3. Further in all the
models the $\Omega_{d}$ or the DE density matches with Planck Survey 2018
predictions plnk . Model M2 is found converging towards cosmological constant
DE candidate with $\omega_{d}$ fitted to a value of -1.00. However in M3 where
both DM and DE is left free it is seen that the model has tendency to predict
a marginally non-cold DM candidate with $\omega_{m}=0.04,$ and DE favouring
the phantom regime. The ratio $r$ of DE and DM, which can also be called the
coincidence factor when left free always has a tendency to hover around
$2.2-2.3.$ The interaction C1 with model M2 has been is one of the best
studied interaction in the literature pwan ; am ; guo ; he ; mvg ; liu .
Comparing the results of C1 in M2 with results in guo , we find our results
being completely compatible. The authors in guo  used WMAP CMB data, SDSS BAO
data and 71 Supernova Legacy Survey data. In the combined analysis they found
that the interaction varied as $(-0.4,0.3)$ while the DE density $\omega_{d}$
varied between (-1.16,-0.91). More recently in liu , the authors study C1 in
M2 in a different context using latest, CMB, Lensing, Pantheon and BAO data.
Again their combined analysis constrains the interaction to a slightly
positive value with $\omega_{d}$ constrained at -1.019. This clearly shows how
C1 compares in different models that we have used and with the existing data.

Table II, Fig.4 - Fig.6 show the corresponding data for interaction function
C2. All models predict marginal interaction with M3 favouring marginally more
interaction than M1 and M2 and the DE density $\Omega_{d}$ converge towards
Planck 2018 predicted value. Both M2 and M3 with C2 favours marginal phantom
DE while from M3 we see that marginal non-cold DM is favoured. The coincidence
factor for M1-M3 closely resembles, with $r$ value fitted same for M2 and M3.
Previously C2 has been studied in he ; mvg . Recently one can find renewed
interest in the interaction from the bulk of recent literature 1c2 ; 2c2 ; 3c2
; 4c2 . In 4c2  the authors use C2 in both M1 and M2. They constrain their
results using latest CMB, BAO and Pantheon data. In the combined analysis for
M1 they obtain a mean positive interaction value of 0.032, while in M2, they
use two different models one for $\omega_{d}<-1$ and other for
$\omega_{d}>-1.$ For the $\omega_{d}<-1$ case they obtain a slightly negative
interaction with constrained mean $\omega_{d}=-1.08.$ Again we see excellent
comparison with our results.

Table III, Fig.7 - Fig.9 shows the constrained parameter data for function C3.
Here C3 is driven via $\rho_{m}$ and $\rho^{\prime}_{m}.$ In all models the DE
density parameter $\Omega_{d}$ is fitted to values that closely mimics recent
Planck results. In both M1 and M2 we see DE converges to cosmological
constant, while the coincidence factor $r\simeq 2.3.$ The interaction
parameters all converge towards negative values with interaction coefficients
no longer marginal.

The results for C4 is shown in table IV and Fig.10 - Fig.12. Here interaction
is driven via $\rho_{d}$ and $\rho^{\prime}_{d}.$ Clearly this interaction
does not favour CDM and CC as coupled matter components as is evident from the
results in M1. Further results of M2 and M3 clearly shows preference to a
quintessence DE candidate. In M3 we again see that C4 also favouring a
marginal non-cold DM candidate. $r$ values in M2 and M3 are similar but away
from the $r$ value predicted by M1. Interaction is again non-marginal.

In C5 we find that interaction is driven via both $\rho_{m}$ and $\rho_{d}.$
The results are provided in table V, Fig.13 - Fig.15. From the results for M2
and M3 we observe that a marginal phantom matter is the favoured DE candidate.
M3 prefers a marginal non-cold DM candidate. The interaction is negligible for
all the models M1-M3, but interaction coefficients can change sign for the
models. This may be due to their predicted values constrained close to a zero
value. This interaction has been used previously in ide3 ; he ; c51 ; c52 ;
c53 ; sb2 . In sb2  C5 was used in model M2. The model was constrained using
194 Supernovae 1a data, which constrained $\omega_{d}$ to -1.01, with mild
positive interaction.

Table VI, and Fig.16 - Fig.18 gives the constrained parameters for function
C6. Here interaction is driven by $\rho_{d}$ and $\rho^{\prime}_{m}.$ From the
fitted values of the interaction coefficient we see that the interaction is
more significant for $\rho^{\prime}_{m}$ as compared to $\rho_{d}.$ In M2 we
see that fitted value of $\omega_{d}\sim-0.99$ whereas in M3 $\omega_{d}$ fits
to the cosmological constant. Given that both M1 and M3 prefers CC as the
preferred DE candidate we can say that C6 in general promotes interaction
driven via CC. As in the previous cases we observe that M3 always prefers a
marginal non-cold DM candidate.

Table VII, Fig.19 - Fig.21 gives data prescribed values for the free
parameters in C7. Interaction is driven via $\rho^{\prime}_{d}$ and
$\rho_{m}.$ From the prescribed interaction parameter values in the table one
can make out strong interaction for all models M1-M3. Further from the
constrained value of the DE density parameter we observe that CC is not the
preferred DE candidate. From the results of M3 we see that slightly phantom
regime is non-cold DM is the preferred Dark sectors of this interacting model.

The interaction function C8 is a combination of $\rho^{\prime}_{m}$ and
$\rho^{\prime}_{d}.$ From table VIII, Fig.22 - Fig.24 we see that the
interaction coefficients are marginal corresponding to both
$\rho^{\prime}_{m}$ and $\rho^{\prime}_{d}$ favouring only marginal
interaction for all models M1-M3. Further models M2 and M3 show a phantom
drift of DE candidate. DM candidate again favours a marginal non-cold
existence. Interaction C8 was first studied in c81  and later in sb3 . In sb3
C8 was studied using model M2. They used Hubble data, BAO data and 580 Union
2.1 Supernovae data to constrain their model. Their model showed marginal
negative interaction with $\omega_{d}$ constrained to around -1.095. This
again is in agreement with our data. In the above literature there was only
one interaction coefficient, that is the interactions $\alpha=\beta.$

From table IX, and Fig.25 - Fig.27 we see that interaction C9 closely follows
the data in table I. Non-linear IDE function like C9 has not been studied
closely in the literature so far. The results in table IX show that such
interactions needs to be analysed in greater details. In c9  a general
interaction term was used that included a product $\rho_{m},\rho_{d}$ and
$\rho_{m}+\rho_{d}$ raised to arbitrary powers such that the sum of the
exponents is one. C9 can be considered a special case of that interaction.
However this exact form has not been previously studied in the literature.

Interaction models C3, C4, C6 or C7 has not been previously studied as
considered here. In sp  a general interaction term was considered which was a
linear combination of $\rho_{m},\rho_{d}$ and $\rho^{\prime}_{m}.$ Both models
C3 and C6 can be considered to be subclasses of the interaction considered
there. In fact they study C6 in M1 as a subcategory. Their results with Cosmic
Chronometer Hubble data and binned JLA data constrains $\eta$ to a mean
negative value, while the coefficient $\alpha$ is constrained to a mean
positive value. Comparing the results we see that our model provides less
stringent constraints as compared to sp .

In wyan  an interaction involving only $\rho^{\prime}_{d}$ was studied.
Interactions C4 or C7 are more general cases of the above. Significantly in
both C4 or C7 we obtain negative mean interaction as has been predicted in
wyan  where the interaction was proportional to $\rho^{\prime}_{d}$ only, and
in model M2. The data sets consisted of CMB, Cosmic Chronometers, BAO, JLA,
Weak Lensing, Red shift space distortion and Local Hubble Data.

From the data in tables I-IX and figures 1-27, we observe that the coincidence
parameter $r$ is mostly constrained around 2.1-2.2 in all the models. Although
$r$ was defined as the ratio of DE to DM, yet observational data always
constrained it to a constant value irrespective of the dark sector matter
densities. Further from C3 and C4 we observe that interaction driven by only
$\rho_{m}$ and its derivative has a tendency to converge towards CC as the
preferred DE candidate, while interaction driven via $\rho_{d}$ and its
derivative favours a quintessence DE candidate. A constant feature in all the
interaction functions is that model M3 always converged to a marginal non-cold
DM candidate against the popular assumption of CDM in most of the literature
on interacting models. Usually DM is divided into three categories. The Hot DM
(HDM), Warm DM (WDM) and the Cold DM (CDM). In standard $\Lambda$CDM models,
the usual DM candidate is the CDM, which is pressure-less, that is
$p_{cdm}=0$. In fact CDM is associated with the large scale structure of the
universe at scales beyond 1 Mpc. However CDM cannot explain efficiently the
structures at small scales. Recent literature show that small scale
observations can be explained better by WDM. Free streaming velocity for WDM
candidates are intermediate of HDM and CDM, this helps to tackle some of the
problems associated with CDM structure formation. HDM are relativistic
candidates that were relevant only in the beginning of hot big bang, that is,
during the hot radiation dominated epoch. Our numerical results in model M3
prefers the existence of a non-cold DM candidate of the WDM type. Recently the
term interacting warm dark matter “IWDM” has been used in the context of
interacting non-cold dark matter in iw . The authors of iw  used the
interaction C2 between CC and non-cold DM. They analysed their model both at
the background and perturbation level. Further they constrained their model
using latest CMB, Lensing, BAO and Pantheon data. All of their results pointed
to the existence non-cold (warm)DM candidate. According to them using non-cold
DM candidate in interaction models “could herald the beginning of a new era in
cosmology.” We point that models like M3 has not been hitherto used in the
literature. Clearly our results in M3 are very much in agreement with the
results of iw  where we obtained slightly non-cold DM interacting with
marginally phantom DE or CC.

|  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\xi$ |
---|---|---|---|---|---|---|---|---
|  M1 |  0.789 |  $2.27^{+0.50}_{-0.51}$ |  – |  – |  $0.71^{+0.02}_{-0.02}$ |  $0.04^{+0.03}_{-0.03}$ |
|  M2 |  0.795 |  $2.24^{+0.52}_{-0.51}$ |  $-1.00^{+0.20}_{-0.14}$ |  – |  $0.70^{+0.12}_{-0.08}$ |  $0.04^{+0.06}_{-0.12}$ |
|  M3 |  0.81 |  $2.23^{+0.52}_{-0.49}$ |  $-1.01^{+0.21}_{-0.14}$ |  $0.04^{+0.04}_{-0.04}$ |  $0.71^{+0.12}_{-0.07}$ |  $0.08^{+0.07}_{-0.13}$ |
Table 1: The modelled parameter constraints for IDE function C1  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\eta$ |
---|---|---|---|---|---|---|---|---
|  M1 |  0.789 |  $2.17^{+0.53}_{-0.47}$ |  – |  – |  $0.71^{+0.02}_{-0.02}$ |  $0.02^{+0.01}_{-0.01}$ |
|  M2 |  0.795 |  $2.14^{+0.55}_{-0.46}$ |  $-1.01^{+0.19}_{-0.13}$ |  – |  $0.70^{+0.11}_{-0.07}$ |  $0.02^{+0.03}_{-0.05}$ |
|  M3 |  0.81 |  $2.14^{+0.56}_{-0.47}$ |  $-1.02^{+0.19}_{-0.13}$ |  $0.04^{+0.04}_{-0.04}$ |  $0.71^{+0.11}_{-0.07}$ |  $0.04^{+0.03}_{-0.05}$ |
Table 2: The modelled parameter constraints for IDE function C2  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\xi$ |  $\alpha$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.795 |  $2.27^{+0.51}_{-0.51}$ |  – |  – |  $0.67^{+0.08}_{-0.08}$ |  $-0.08^{+0.24}_{-0.17}$ |  $-0.16^{+0.31}_{-0.24}$ |
|  M2 |  0.807 |  $2.26^{+0.51}_{-0.51}$ |  $-1.00^{+0.19}_{-0.14}$ |  – |  $0.67^{+0.11}_{-0.09}$ |  $-0.10^{+0.27}_{-0.19}$ |  $-0.18^{+0.34}_{-0.22}$ |
|  M3 |  0.82 |  $2.26^{+0.50}_{-0.51}$ |  $-1.00^{+0.19}_{-0.13}$ |  $0.04^{+0.04}_{-0.04}$ |  $0.69^{+0.11}_{-0.10}$ |  $-0.07^{+0.28}_{-0.20}$ |  $-0.17^{+0.33}_{-0.28}$ |
Table 3: The modelled parameter constraints for IDE function C3  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\eta$ |  $\beta$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.795 |  $2.20^{+0.54}_{-0.51}$ |  – |  – |  $0.59^{+0.13}_{-0.10}$ |  $-0.13^{+0.18}_{-0.11}$ |  $-0.22^{+0.25}_{-0.19}$ |
|  M2 |  0.807 |  $2.11^{+0.58}_{-0.44}$ |  $-0.98^{+0.18}_{-0.15}$ |  – |  $0.65^{+0.13}_{-0.11}$ |  $-0.17^{+0.23}_{-0.16}$ |  $-0.23^{+0.28}_{-0.18}$ |
|  M3 |  0.82 |  $2.15^{+0.52}_{-0.45}$ |  $-0.92^{+0.23}_{-0.19}$ |  $0.05^{+0.04}_{-0.04}$ |  $0.64^{+0.14}_{-0.12}$ |  $-0.17^{+0.20}_{-0.13}$ |  $-0.26^{+0.25}_{-0.17}$ |
Table 4: The modelled parameter constraints for IDE function C4  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\xi$ |  $\eta$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.802 |  $2.09^{+0.59}_{-0.42}$ |  – |  – |  $0.71^{+0.02}_{-0.02}$ |  $-0.05^{+0.37}_{-0.32}$ |  $0.04^{+0.15}_{-0.17}$ |
|  M2 |  0.807 |  $2.15^{+0.53}_{-0.46}$ |  $-1.01^{+0.20}_{-0.13}$ |  – |  $0.70^{+0.12}_{-0.07}$ |  $0.03^{+0.31}_{-0.36}$ |  $-0.00^{+0.16}_{-0.15}$ |
|  M3 |  0.82 |  $2.14^{+0.57}_{-0.45}$ |  $-1.02^{+0.23}_{-0.19}$ |  $0.04^{+0.04}_{-0.04}$ |  $0.70^{+0.12}_{-0.07}$ |  $-0.01^{+0.33}_{-0.35}$ |  $-0.02^{+0.17}_{-0.16}$ |
Table 5: The modelled parameter constraints for IDE function C5  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\eta$ |  $\alpha$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.795 |  $2.17^{+0.52}_{-0.48}$ |  – |  – |  $0.67^{+0.08}_{-0.07}$ |  $-0.03^{+0.12}_{-0.08}$ |  $-0.15^{+0.34}_{-0.25}$ |
|  M2 |  0.807 |  $2.10^{+0.59}_{-0.43}$ |  $-0.99^{+0.21}_{-0.15}$ |  – |  $0.69^{+0.12}_{-0.10}$ |  $-0.05^{+0.14}_{-0.10}$ |  $-0.17^{+0.37}_{-0.24}$ |
|  M3 |  0.82 |  $2.17^{+0.55}_{-0.48}$ |  $-1.00^{+0.21}_{-0.14}$ |  $0.04^{+0.04}_{-0.04}$ |  $0.69^{+0.11}_{-0.10}$ |  $-0.03^{+0.13}_{-0.10}$ |  $-0.17^{+0.35}_{-0.24}$ |
Table 6: The modelled parameter constraints for IDE function C6  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\xi$ |  $\beta$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.795 |  $2.18^{+0.52}_{-0.47}$ |  – |  – |  $0.62^{+0.10}_{-0.08}$ |  $-0.21^{+0.30}_{-0.20}$ |  $-0.16^{+0.19}_{-0.15}$ |
|  M2 |  0.807 |  $2.14^{+0.55}_{-0.42}$ |  $-1.00^{+0.17}_{-0.14}$ |  – |  $0.63^{+0.13}_{-0.10}$ |  $-0.20^{+0.32}_{-0.21}$ |  $-0.14^{+0.19}_{-0.15}$ |
|  M3 |  0.82 |  $2.14^{+0.55}_{-0.46}$ |  $-1.01^{+0.18}_{-0.13}$ |  $0.05^{+0.04}_{-0.04}$ |  $0.64^{+0.18}_{-0.12}$ |  $-0.18^{+0.33}_{-0.22}$ |  $-0.15^{+0.19}_{-0.17}$ |
Table 7: The modelled parameter constraints for IDE function C7  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\alpha$ |  $\beta$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.801 |  $2.07^{+0.60}_{-0.41}$ |  – |  – |  $0.69^{+0.02}_{-0.03}$ |  $-0.05^{+0.36}_{-0.31}$ |  $-0.00^{+0.15}_{-0.17}$ |
|  M2 |  0.807 |  $2.18^{+0.54}_{-0.46}$ |  $-1.04^{+0.16}_{-0.11}$ |  – |  $0.66^{+0.12}_{-0.07}$ |  $-0.02^{+0.33}_{-0.35}$ |  $-0.04^{+0.16}_{-0.15}$ |
|  M3 |  0.82 |  $2.11^{+0.59}_{-0.43}$ |  $-1.03^{+0.18}_{-0.13}$ |  $0.05^{+0.03}_{-0.04}$ |  $0.66^{+0.14}_{-0.08}$ |  $-0.00^{+0.32}_{-0.34}$ |  $-0.04^{+0.15}_{-0.17}$ |
Table 8: The modelled parameter constraints for IDE function C8  |  Models |  $\chi^{2}_{min}$ |  $r$ |  $\omega_{d}$ |  $\omega_{m}$ |  $\Omega_{d}$ |  $\gamma$ |  $\delta$ |
---|---|---|---|---|---|---|---|---|---
|  M1 |  0.801 |  $2.14^{+0.54}_{-0.46}$ |  – |  – |  $0.70^{+0.02}_{-0.02}$ |  $0.02^{+0.02}_{-0.02}$ |  $0.09^{+0.30}_{-0.37}$ |
|  M2 |  0.808 |  $2.17^{+0.57}_{-0.47}$ |  $-1.00^{+0.20}_{-0.14}$ |  – |  $0.70^{+0.12}_{-0.07}$ |  $0.02^{+0.03}_{-0.06}$ |  $0.08^{+0.31}_{-0.36}$ |
|  M3 |  0.82 |  $2.17^{+0.55}_{-0.46}$ |  $-1.03^{+0.18}_{-0.12}$ |  $0.04^{+0.04}_{-0.04}$ |  $0.69^{+0.11}_{-0.07}$ |  $0.04^{+0.04}_{-0.05}$ |  $0.09^{+0.30}_{-0.38}$ |
Table 9: The modelled parameter constraints for IDE function C9

##  IV Effective Equation of State

Here we evaluate the effective equation of state
$\omega_{t}=\frac{\rho_{m}\omega_{m}+\rho_{d}\omega_{d}}{\rho_{m}+\rho_{d}}.$
Using equations (11) and (12) we write
$\omega_{t}=\frac{\omega_{m}\rho_{m0}(1+z)^{3m_{1}}+\omega_{d}\rho_{d0}(1+z)^{3m_{2}}}{\rho_{m0}(1+z)^{3m_{1}}+\rho_{d0}(1+z)^{3m_{2}}},$
where $m_{1}=\frac{1+\omega_{m}-\xi-\eta r-\gamma r^{1-\delta}}{1-\alpha-\beta
r}$ and
$m_{2}=\frac{1+\omega_{d}+\frac{\xi}{r}+\eta+\frac{\gamma}{r^{\delta}}}{1+\frac{\alpha}{r}+\beta}.$
From this we see that,

  * •

as
$z\rightarrow\infty,~{}\omega_{t}\rightarrow\begin{cases}\omega_{d},\text{if}~{}m_{2}-m_{1}>0,\\\
\omega_{m},\text{if}~{}m_{2}-m_{1}<0\end{cases},$

  * •

as
$z\rightarrow-1,~{}\omega_{t}\rightarrow\begin{cases}\omega_{m},\text{if}~{}m_{2}-m_{1}>0,\\\
\omega_{d},\text{if}~{}m_{2}-m_{1}<0\end{cases}.$

We see in that for all the interaction functions C1-C9 and Models M1-M3,
$\omega_{t}\rightarrow\omega_{m}$ as $z\rightarrow\infty$ while
$\omega_{t}\rightarrow\omega_{d}$ as $z\rightarrow-1.$ This shows that all the
models predict an early matter dominated epoch followed by a late time DE
dominated epoch. It may be noted that, it is only in the recent epoch that DE
and DM have evolved to similar order of magnitude. However such constant $r$
is not valid in an early universe. Yet, once we have deduced the model using
analytical as well as numerical means, it is imperative that we check the
model behaviour in the limiting scenarios of early and late epochs, where an
early epoch would be given by a large $z$ value or in the limit
$z\rightarrow\infty$ and a late or future epoch would mean $z\rightarrow-1.$
Accordingly the above limits on $z$ becomes important. Significantly the
numerical results obtained above, when combined with the limiting analytical
scenarios, our models provide evidence of expected early time as well as late
time cosmology.

##  V The Deceleration Factor

The deceleration factor is defined as $q=-1-\frac{\dot{H}}{H^{2}}.$ For the
given model using equation(14) we can rewrite the deceleration factor as
$q=-1+\frac{3}{2}\frac{m_{1}\rho_{m0}(1+z)^{3m_{1}}+m_{2}\rho_{d0}(1+z)^{3m_{2}}}{\rho_{m0}(1+z)^{3m_{1}}+\rho_{d0}(1+z)^{3m_{2}}}.$
Here also we can extrapolate the model to the limiting scenarios of
$z\rightarrow\infty$ and $z\rightarrow-1$ observe the model predictions at
early and late times.

  * •

as
$z\rightarrow\infty,~{}q\rightarrow\begin{cases}-1+\frac{3}{2}m_{2},\text{if}~{}m_{2}-m_{1}>0,\\\
-1+\frac{3}{2}m_{1},\text{if}~{}m_{2}-m_{1}<0\end{cases},$

  * •

as
$z\rightarrow-1,~{}q\rightarrow\begin{cases}-1+\frac{3}{2}m_{1},\text{if}~{}m_{2}-m_{1}>0,\\\
-1+\frac{3}{2}m_{2},\text{if}~{}m_{2}-m_{1}<0\end{cases}.$

Analysing this for all our models we find that $q>0$ as $z\rightarrow\infty$
while $q<0$ as $z\rightarrow-1.$ This is expected with models predicting an
early time matter dominated decelerating universe followed by a late time
accelerating universe.

##  VI Model Selection based on information criteria

With the introduction of data based research in cosmology, models have taken a
centre stage in the efforts to understand the universe. Accordingly
prioritising one model over the other has become an important aspect in
cosmology. The most commonly used model selection criterion are the Akaike
Information Criteria (AIC) ak  and the Bayesian Information Criteria (BIC)
(sometimes called the Schwarz information criteria) bk . Both these criteria
penalizes a model based on the complexity introduced by the extra set of
parameters. A model with more parameters is likely to have an improved
likelihood estimation as compared with one with less number of parameters.
However this does not always give us a complete picture. AIC and BIC both
compensates for improved likelihood by correcting on the number of parameters.
The AIC directly compensates on the number of parameters by
$-2\ln\mathfrak{L}+2d$ where $d$ is the total number of free parameters, while
BIC is defined as $-2\ln\mathfrak{L}+d\ln N$ where $N$ is the total number of
data points used. The usual way to use these two information criteria is not
by actual estimation but by comparing it with a base model with minimum number
of parameters. AIC is said to be the criteria that defines how well a model is
favoured as compared to the base model, while BIC defines how well a model is
rejected as compared to the base model. $0<\Delta AIC<2$ can be interpreted as
a strongly favoured model, while $4<\Delta AIC<7$ means some evidence in
favour of the model. $\Delta AIC>10$ stands for no evidence in favour.
$0<\Delta BIC<2$ can be interpreted as not much strong evidence to reject the
model, $2<\Delta BIC<6$ can be interpreted as some evidence against the model,
while $\Delta BIC>10$ means good evidence to reject the model.

Since all our interacting coefficients were treated under three different
models, with M1 containing the least number of parameters, we consider the AIC
and BIC values for M1 in the corresponding interactions C1-C9 as the base
model and compare it with M2 and M3. Our results as listed in table X, show
that for all the interactions C1-C9, model M2 is highly favoured by AIC
although by BIC we can get some evidence against it. AIC is less harsh with M3
as is expected, which says that the model somewhat favoured in comparison with
M1. Whereas BIC being more stringent is more harsh with M3. However none of
the models can be ruled out in comparison to M1.

If $\Lambda$CDM is chosen as a base model for M1 and $\omega$CDM as a base
model for M2 and M3, then both M1 and M2 perform well, with IC values for M1
in the models C1-C9 ranging between $1.4\leq\Delta AIC_{\Lambda}<3,~{}3<\Delta
BIC_{\Lambda}<8,$ and for M2 the IC values for all the functions range between
$2<\Delta AIC_{\omega}<3,~{}3<\Delta BIC_{\omega}<8.$ However in case of M3,
when models C1-C9 is compared with $\omega$CDM we obtain the following range
for IC vales: $3<\Delta AIC_{\omega}<5.5,~{}7.5<\Delta BIC_{\omega}<12.1$ This
shows that BIC hints rejection of the model M3 in comparison to $\omega$CDM as
base model.

Interaction |  Model |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---
C1-C2 |  M1 |  $\simeq 58.91$ |  $\simeq 65.65$ |  - |  -
|  M2 |  $\simeq 60.51$ |  $\simeq 69.50$ |  1.6 |  3.85
|  M3 |  $\simeq 62.5$ |  $\simeq 73.75$ |  3.6 |  8.1
C3-C8 |  M1 |  $\simeq 60.5$ |  $\simeq 69.49$ |  - |  -
|  M2 |  $\simeq 62.49$ |  $\simeq 73.73$ |  1.98 |  4.24
|  M3 |  $\simeq 64.49$ |  $\simeq 77.98$ |  3.99 |  8.49
C9 |  M1 |  60.87 |  69.86 |  - |  -
|  M2 |  62.52 |  73.76 |  1.65 |  3.9
|  M3 |  64.51 |  78 |  3.64 |  8.14
Table 10: The AIC and BIC value with corresponding $\Delta$IC value of models
M2 and M3 when compared with M1 as a base models

##  VII Discussion

In this article we have considered a scenario of two interacting fluids.
Usually such models have been analysed under the purview of interacting CDM
and vacuum energy or the CDM and DE. Here we have broadened our perspective to
generalise DM and have considered a third model of interacting “two fluids”.
These three models were then analysed using nine interacting functions, all
obtained from a general interaction term.

In this article the main purpose was to bring forth a comparative study of
several competing interacting models and interacting functions under one
umbrella. Here we have also introduced the coincidence factor $r$ as a
parameter of the model. We have further computed the effective equation of
state as well as the deceleration parameter for the models considered.
Statistical model selection analysis shows relatively better acceptance for
models M1 and M2 as compared to M3. This is however expected given that, the
AIC and BIC tests are based on comparison w.r.t. already existing simple
models. Here M3 had no base model to compare unlike M1 which has $\Lambda$CDM
and M2 which had the $\omega$CDM as their base model for comparison. However
based on comparison with M1 as base model M3 could not be completely rejected.
The numerical results showed that all models favoured mostly vacuum DE or
marginally phantom DE, with M3 additionally preferring a slightly non-cold DM
candidate. Interaction function C4, which is driven via $\rho_{d}$ and
$\rho^{\prime}_{d}$ however preferred a quintessence DE candidate in both M3
and M2. This slightly difference in choice of C4 needs further analysis.
Clearly the results of C4 from M1 shows that CC is not a favoured DE candidate
for this interaction. This result along with the fact that when DM equation of
state is left free to evolve has a tendency to develop “non-cold” nature is
something that needs further analysis.

The current work offers only a glimpse and overview of the possibilities of
information that the models can offer. The interaction functions and the
models studied in this article needs greater subjective analysis using a
comprehensive pool of data sets, which we aim to do in near future. Further
one also needs to understand the stability of the above interaction functions
via a perturbation analysis, which will help us understand the underlying
mechanics of the factors driving the interactions. Preliminary analysis raises
interesting possibilities and show that the above interacting functions are
viable choices for interacting two fluid scenario of the universe. One may
note note that models like M3 which have not been extensively studied in the
literature can lead to interesting and otherwise unexplored possibilities. In
this context, results of iw  increases curiosity in the context of M3. In iw
C2 was treated from the perspective of interacting vacuum energy with a
perfect fluid equation of state, which the authors called interacting warm
dark matter and obtained similar results for the choice of DM candidate. This
clearly indicates that there is a need to understand and study models as M3
with greater depth and understanding.

##  VIII  Acknowledgemensts:

SB acknowledges the Department of Science and Technology, SERB, India for
financial help through ECR project (File No. ECR/2017/000569). SB also
acknowledges IUCAA, Pune, India, where the project was initiated.

The author acknowledges the anonymous reviewers whose feedback helped to
improve the article.

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##  Appendix A Figures

The following figures show the confidence contour on a two dimensional
parameter space with marginalized posterior distributions corresponding to the
interaction C1-C9 and models M1-M3. The figures were obtained by a combined
analysis of binned-LA+OHD+BAO data.The median values of the constrained
parameters with corresponding $1\sigma$ errors are given on top of the
posterior probability distribution panels.

Figure 1: The above depicts the confidence contour for function C1 and model
M1. Figure 2: The above depicts the confidence contour for function C1 and
model M2. Figure 3: The above depicts the confidence contour for function C1
and model M3. Figure 4: The above depicts the confidence contour for function
C2 and model M1. Figure 5: The above depicts the confidence contour for
function C2 and model M2. Figure 6: The above depicts the confidence contour
for function C2 and model M3. Figure 7: The above depicts the confidence
contour for function C3 and model M1. Figure 8: The above depicts the
confidence contour for function C3 and model M2. Figure 9: The above depicts
the confidence contour for function C3 and model M3. Figure 10: The above
depicts the confidence contour for function C4 and model M1. Figure 11: The
above depicts the confidence contour for function C4 and model M2. Figure 12:
The above depicts the confidence contour for function C4 and model M3. Figure
13: The above depicts the confidence contour for function C5 and model M1.
Figure 14: The above depicts the confidence contour for function C5 and model
M2. Figure 15: The above depicts the confidence contour for function C5 and
model M3. Figure 16: The above depicts the confidence contour for function C6
and model M1. Figure 17: The above depicts the confidence contour for function
C6 and model M2. Figure 18: The above depicts the confidence contour for
function C6 and model M3. Figure 19: The above depicts the confidence contour
for function C7 and model M1. Figure 20: The above depicts the confidence
contour for function C7 and model M2. Figure 21: The above depicts the
confidence contour for function C7 and model M3. Figure 22: The above depicts
the confidence contour for function C8 and model M1. Figure 23: The confidence
contour for function C8 and model M2. Figure 24: The confidence contour for
function C8 and model M3. Figure 25: The confidence contour for function C9
and model M1. Figure 26: The confidence contour for function C9 and model M2.
Figure 27: The confidence contour for function C9 and model M3.


# Quantitative determination of anisotropic magnetoelectric coupling in
BiFeO3-CoFe2O4 nanostructures

Yoon Seok Oh  CeNSCMR, Department of Physics and Astronomy, Seoul National
University, Seoul 151-747, Republic of Korea  S. Crane  Department of
Materials Science and Engineering, University of California, Berkeley, CA
94720, USA  H. Zheng  Department of Materials Science and Engineering,
University of California, Berkeley, CA 94720, USA  Y. H. Chu  Department of
Materials Science and Engineering, National Chiao Tung University, HsinChu
30010, Taiwan  R. Ramesh  Department of Materials Science and Engineering,
University of California, Berkeley, CA 94720, USA  Kee Hoon Kim  CeNSCMR,
Department of Physics and Astronomy, Seoul National University, Seoul 151-747,
Republic of Korea

###### Abstract

The transverse and longitudinal magnetoelectric susceptibilities (MES) were
quantitatively determined for (001) heteroepitaxial BiFeO3-CoFe2O4
nanostructures. Both of these MES values were sharply enhanced at magnetic
fields below 6 kOe and revealed asymmetric lineshapes with respect to the dc
magnetic field, demonstrating the strain-induced magnetoelectric effect. The
maximum transverse MES, which reached as high as $\sim$60 mV/cm Oe, was about
five times larger than the longitudinal MES. This observation signifies that
transverse magnetostriction of the CoFe2O4 nanopillars is enhanced more than
the bulk value due to preferred magnetic domain alignment along the [001]
direction coming from compressive, heteroepitaxial strain.

The magnetoelectric (ME) effect is a physical phenomenon in which the electric
polarization $P$ (magnetization $M$) is modulated by the magnetic field $H$
(electric field $E$). There is a growing interest in the application of ME
effects toward various devices, including magnetic sensors nanjap  and energy
harvesters. dong  As such, numerous efforts have been made to obtain strong ME
couplings in novel ME composites made of ferroelectric (or piezoelectric) and
ferromagnetic materials. boomgaard ; jryu  In these types of ME composites,
$P$ is varied with $M$ via the strain ($u$)-coupling at the interface between
the piezoelectric and magnetostrictive phases. Thus, the configuration of this
interface is a significant control parameter for determining the extent of ME
coupling. nanjap  In this respect, a layered sandwich structure [i.e., (2-2)
structures] produces a larger ME coupling than for particulates that are
dispersed in a matrix [i.e., (0-3) structures] because the former has a larger
interface area. In contrast, this common practice cannot be applied to
multilayered thin films, where clamping of the nonmagnetic substrate can
prevent strain-coupling between the layers. nanprl

In 2004, an epitaxial thin film composed of CoFe2O4 (CFO) nanopillars embedded
in a BaTiO3 (BTO) matrix [i.e., (1-3) structures] was grown and suggested to
be an alternative to circumvent the substrate clamping effect. On the other
hand, it has been quite difficult to determine quantitatively the ME coupling
of such nanostructures and, more generally, numerous ME films, except
observing the existence of nontrivial ME coupling. zhengscience  One dominant
reason for this difficulty is that the ME voltage signal is proportional to
the film thickness; it typically becomes smaller than 1 $\mu$V for film
thicknesses less than 1 $\mu$m. To overcome this difficulty, a large ac
magnetic field ($H_{\rm{ac}}$) of up to $\sim$1 kOe was recently used to
obtain the effective MES as a function of $H_{\rm{ac}}$. yanapl ; yanjap
However, direct measurements of the MES based on a conventional scheme that
employs small $H_{\rm{ac}}$ values with variations in the dc magnetic field
($H_{\rm{dc}}$) would be useful for understanding the ME coupling of numerous
multiferroic films or nanostructures at the quantitative level. In this work,
by use of the conventional scheme, we provide experimental evidences that a
300 nm thick BiFeO3-CoFe2O4 (BFO-CFO) nanostructure has a peculiar MES
anisotropy that is not expected in bulk forms of those materials.

The self-assembled epitaxial BFO-CFO film with thickness of 300 nm was grown
on a (001) SrRuO3/SrTiO3 substrate by pulsed laser deposition. zhengadvm  The
film had the CFO nanopillars embedded in a BFO matrix with a volume fraction
of 1:1. Top electrodes Pt/SrRuO3 were deposited on the film surface (Fig.
1(a)). For magnetic hysteresis measurements, a vibrating sample magnetometer
was utilized. For ferroelectric hysteresis loops, the displacement current was
measured using a fast digitizer and a high voltage amplifier.

Figure 1:  (a) Schematic picture, (b) AFM image, and (c) XRD pattern of the
BFO-CFO film with nanopillar structures.

To investigate the MES (${\alpha}{\equiv}{\delta}P/{\delta}H_{\rm{ac}}$),
especially for thin films, we developed a highly sensitive ME susceptometer
that operates inside the PPMS (Quantum Design). A pair of solenoids was
designed to induce a $H_{\rm{ac}}$ of $\sim$4 Oe inside the solenoid pair and
a voltage pick-up coil was used to determine the phase of $H_{\rm{ac}}$. In
particular, modulated charges, instead of voltages, were measured using a
high-impedance charge amplifier with a gain factor of $10^{12}$ V/C. This
makes the amplified signal independent of film thickness but proportional to
electrode area so that the signal-to-noise ratio was improved. Based on this
scheme, we were able to detect small ME charges in a thin film with a
thickness $\sim$40 nm and a circular electrode of diameter $\sim$100 $\mu$m.
The lowest charge noise ($\Delta Q$) was ${\sim}10^{-17}$ C, which
corresponded to a voltage noise of $\Delta Q/C_{\rm{s}}$ ($C_{\rm{s}}$ =
sample capacitance). The determined $\alpha$ could be converted into the MES
expressed as a voltage unit $\alpha_{\rm{E}}{=}{\delta}E/{\delta}H_{\rm{ac}}$
using the relationship ${\alpha}{=}{\varepsilon}{\alpha}_{\rm{E}}$, where
$\varepsilon$ is the absolute permittivity of a specimen.

Figure 2: (a) Magnetic hysteresis loops for the in-plane and out-of-plane
directions measured at 300 K. $M$ is normalized to the volume fraction of CFO.
(b) $J_{\rm{PE}}(E)$ (solid circle) was integrated to estimate the $P$-$E$
hysteresis loop (solid line).

Fig. 1(b) shows an atomic force microscopy (AFM) image of the BFO-CFO film, in
which the CFO phase appears as rectangles embedded in the BFO matrix.
zhengadvm  Fig. 1(c) further shows an x-ray diffraction (XRD) pattern of the
film obtained through a $\theta-2\theta$ scan around the (002) SrTiO3 peak.
The distinct (00$l$) peaks of CFO, BFO, and SrRuO3 are consistent with the
previous result zhengadvm  that each phase was epitaxially grown on the SrTiO3
substrate. The $d$-spacing of the CFO nanopillars was estimated as 2.0867
$\rm{{\AA}}$ from the (004) peak. This indicates a compressive strain along
the [001] direction, $u_{001}{=}-0.33\%$, compared with the bulk CFO. Fig.
2(a) shows the magnetic hysteresis loops measured along the in-plane (i.e.,
$H{\parallel}[100]$) and out-of-plane (i.e., $H{\parallel}[001]$) directions.
The saturated moment of $\sim$3.4 $\mu_{B}$/f.u. for both directions was in
good agreement with the reported CFO value. pauthenet  Moreover, there existed
a large uniaxial magnetic anisotropy with an easy axis along the [001]
direction. A linear extrapolation of the in-plane loop yielded a magnetic
anisotropy field of $\sim$25 kOe. In a previous study on BTO-CFO
nanostructures, the compressive strain of CFO caused by heteroepitaxial growth
was found to be a primary contribution to the uniaxial magnetic anisotropy;
zhengapl  a large magnetic anisotropy field of 51 kOe was observed for
$u_{001}{=}-1.1\%$, while the anisotropy field decreased for smaller
$u_{001}$. Therefore, even in the BFO-CFO film studied here, the magnetic
anisotropy field of $\sim$25 kOe seems to originate from the presence of a
compressive, heteroepitaxial strain of $u_{001}{=}-0.33\%$ inside the CFO
nanopillars.

To determine the $P$-$E$ loop, displacement current $J(E)$ was measured while
negative, positive, and zero biases were applied successively in sequence
(i.e., $-E$ to $E$ to 0). The $J(E)$ curve showed two extremes at $-500$ kV/cm
and 200 kV/cm, at which a reversal of $P$ occurred. In addition to these
extremes, a nonlinear background was found in the $J(E)$ curve. This could
have been from either a Schottky barrier at the interface or a ferroelectric
diode effect. choi ; yang  After subtracting the nonlinear background, the
remaining current density $J_{\rm{PE}}(E)$ was integrated. In the obtained
$P$-$E$ loop (Fig. 2(b)), the polarization value was normalized by the volume
fraction of BFO. The saturated $P$ ($P_{\rm{s}}$) of $\sim$62 $\mu$C/cm2 is
comparable to the previously obtained value in an epitaxial (001) BFO film.
wang  After fully poling the sample along the positive $P$ direction, i.e. top
electrode direction as indicated in the inset of Fig. 2(b), all the MES
measurements were subsequently performed.

Figure 3: (a) Transverse ($\alpha_{31}$) and (b) longitudinal MES
($\alpha_{33}$) of the BFO-CFO nanostructure at 300 K. (c) Dashed (solid)
lines represent the transverse ($\lambda_{31}$) and longitudinal
($\lambda_{33}$) magnetostriction curves of a Co0.8Fe2.2O4 crystal with the
demagnetized (single) magnetic domain. (d) Demagnetized, (e) single, and (f)
preferred magnetic domain patterns of CFO are schematically drawn.

Fig. 3 summarizes transverse ($\alpha_{31}$) and longitudinal ($\alpha_{33}$)
MES curves as a function of $H_{\rm{dc}}$. We note that the similar
$\alpha_{31}$ and $\alpha_{33}$ curves were obtained at many different
electrode spots of the same film and at those of the same kind of
nanostructured film with a thickness $\sim$40 nm. The $\alpha_{31}$ clearly
shows a sign reversal with the direction of $H_{\rm{dc}}$ and develops extreme
points at $H_{\rm{dc}}{=}{\mp}6$ kOe. This is the archetypal lineshape
expected in the strain-coupled ME media composed of piezoelectric and
magnetostrictive materials, supporting that the measured MES data are
reliable. In addition, the $\alpha_{33}$ curve exhibited an asymmetric
lineshape with $H_{\rm{dc}}$, which was similar to the case of $\alpha_{31}$.
However, in this $\alpha_{33}$ curve, a small but non-negligible offset at
$H_{\rm{dc}}{=}0$ was observed, of which value was proportional to the
electrode area. Thus, this offset was attributed to a small contribution from
eddy currents generated inside the electrode due to ${\delta}H_{\rm{ac}}$.
Except for this offset, the $\alpha_{33}$ curve was almost an odd function of
$H_{\rm{dc}}$. This further supports that the strain-coupling is a dominant
source of the longitudinal ME effect as well.

As shown in Fig. 3, $\alpha_{31}{>}0$ for $H_{\rm{dc}}{>}0$ while
$\alpha_{33}{<}0$ for $H_{\rm{dc}}{>}0$. This experimental result reflects
that $P_{3}$ (i.e., $P{\parallel}[001]$) increases under
$H_{\rm{dc}}{\parallel}[100]$, while $P_{3}$ decreases under
$H_{\rm{dc}}{\parallel}[001]$. When the $H_{\rm{dc}}{\parallel}[100]$ was
applied to the CFO crystal, the transverse magnetostriction ($\lambda_{31}$)
was positive (dashed lines in Fig. 3(c)). Thus, the CFO nanopillars and the
BFO matrix, via strain-coupling, were expected to be elongated along the [001]
direction. bozorth  It is known from an earlier study that $P_{3}$ increases
due to the rotation of $P$ with increasing length along the [001] direction.
ederer ; jang  Therefore, the observation of ${\alpha}_{31}{>}0$ for
$H_{\rm{dc}}{>}0$ can be qualitatively understood as the result of an
elongation of CFO/BFO along the [001] direction under
$H_{\rm{dc}}{\parallel}[100]$ and the subsequent increase of $P_{3}$. The case
of decreasing $P_{3}$ under $H_{\rm{dc}}{\parallel}[001]$ can also be
understood in a similar way because the longitudinal magnetostriction
($\lambda_{33}$) in the CFO crystal was negative.

However, our MES data are seemingly inconsistent with the magnetostriction
behavior of a bulk CFO at the quantitative level. The maximum to minimum value
of $\alpha_{31}$ ($\Delta\alpha_{31}$) in Fig. 3(a), amounting to $\sim$260
ps/m ($\sim$120 mV/cm Oe), is about five times larger than that of
$\alpha_{33}$ ($\Delta\alpha_{33}$). According to the magnetostriction of a
Co0.8Fe2.2O4 crystal with demagnetized domains (Fig. 3(c)), a slope of the
$\lambda_{33}$ vs. $H$ curve is at least twice that of the $\lambda_{31}$ vs.
$H$ curve. bozorth  Upon assuming that the magnetostriction of CFO nanopillars
follows a bulk behavior, these magnetostriction data predict that
$\Delta\alpha_{33}$ should be at least twice of $\Delta\alpha_{31}$, which is
in sharp contrast with the results in Fig. 3 (a) and (b).

Although there might exist several mechanisms to induce enhanced $\alpha_{31}$
as discussed in a recent anisotropic MES study using large $H_{\rm{ac}}$,
yanjap  one most decisive factor could be the preferred magnetic domains
existing in the CFO nanopillars. The solid lines in Fig. 3(c) reproduce
published $\lambda^{\rm{s}}_{31}$ and $\lambda^{\rm{s}}_{33}$ for a
Co0.8Fe2.2O4 crystal with a single magnetic domain along the [001] direction.
The single magnetic domain was obtained through the magnetic annealing
process, i.e. cooling under $H$ from high to room temperature. bozorth  In
this situation, applied $H{\parallel}[001]$ gave rise to the 180∘ domain wall
motion, which resulted in negligible $\lambda^{\rm{s}}_{33}$. In contrast,
$H{\parallel}[100]$ resulted in a 90∘ domain wall motion so that it produced
quite large $\lambda^{\rm{s}}_{31}$. As we discussed above, the compressive,
heteroepitaxial strain was a main source of enhanced magnetic anisotropy along
the [001] direction in the CFO nanopillars. It is thus likely that the
magnetic domains inside the CFO nanopillars have preferred alignment along the
[001] direction, as illustrated in Fig. 3(f). If so, similar to the case of
single magnetic domains, the CFO nanopillars are expected to have enhanced
$\lambda_{31}$ and suppressed $\lambda_{33}$. As a result, as observed in Fig.
3 (a) and (b), the BFO-CFO nanostructure will give rise to a bigger
$\alpha_{31}$ (smaller $\alpha_{33}$) than that expected based on the behavior
of bulk CFO magnetostriction.

These results point to the possibility that the strain-induced ME coupling in
the nanostructured film can be quite different from the macroscopic bulk
composite. Application of a compressive, heteroepitaxial strain to the CFO
nanopillars enables to achieve increased $\alpha_{31}$ to as high as $\sim$130
ps/m ($\sim$60 mV/cm Oe) at 6 kOe. Upon increasing $|u_{001}|$, as done in the
BTO-CFO nanostructures with growth temperatures, zhengapl  the $\alpha_{31}$
can be further optimized. In comparison, our previous study on a thin film
made of NiFe2O4 nanoparticulates embedded in a PbZr0.52Ti0.48O3 matrix [i.e.,
the (0-3) structure] showed maximum $|\alpha_{31}|{=}4$ mV/cm Oe ($\sim$4
ps/m) and $|\alpha_{33}|{=}16$ mV/cm Oe ($\sim$14 ps/m) hryu , which was
clearly smaller than the maximum $\alpha_{31}{\sim}130$ ps/m found here.
Therefore, our results strongly support that thin films with the (1-3)
nanostructure have larger ME couplings than the other nanostructures [e.g.,
(0-3) structure].

In conclusion, we have determined the anisotropic MES of a 300 nm thick
BiFeO3-CoFe2O4 nanostructure. An enhancement was observed in the transverse
configuration, which can be explained by the preferred alignment of magnetic
domain resulting from the heteroepitaxial strain that is unique to the present
(1-3) nanostructure. This investigation offers quantitative evidences that the
nanoscale engineering of strain coupling is useful for the design of ME
devices.

We appreciate discussions with T. W. Noh. This study was supported by the NRF,
Korea through National Creative Research Initiatives, NRL (M10600000238) and
Basic Science Research (2009-0083512) programs and by MOKE through the
Fundamental R${\&}$D Program for Core Technology of Materials. YSO is
supported by Seoul R${\&}$BD (10543). Y.H.C. is supported by the National
Science Council, R.O.C. (NSC 98-2119-M-009-M016).

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