# Observational constraints of diffusive dark-fluid cosmology

Remudin Reshid<EMAIL_ADDRESS>and Amare
<EMAIL_ADDRESS>1 Faculty of Engineering and Computer
Science, Ala-too International University, Bishkek, Kyrgyzstan  2 Centre for
Space Research, North-West University, Potchefstroom, South Africa  3 National
Institute for Theoretical and Computational Sciences (NITheCS), South Africa

###### Abstract

In this work, we consider an interacting dark-fluid cosmological model in
which energy exchange between dark matter and dark energy occurs through
diffusion. After solving the background expansion history for a late-time
universe, we attempt to constrain the cosmological parameters by comparing
simulated values of the model against Supernovae Type 1A data. We consider
four different cases and compare them against the $\Lambda$CDM model as the
”true model”. Our results show that the diffusive model in which dark energy
flows to dark matter is the most likely alternative to $\Lambda$CDM model.
This model is not only in line with Planck 2018 observational results but can
also give a potential explanation to the so-called Hubble tension.

PACS numbers

04.50.Kd, 98.80.Jk, 98.80.-k, 95.36.+x, 98.80.Cq

###### pacs:

04.50.Kd, 98.80.Jk, 98.80.-k, 95.36.+x, 98.80.Cq

##  I Introduction

A lot has already been reported about the discrepancy between observational
findings riess1998observational ; perlmutter2003supernovae ; maity2019universe
; tegmark2004cosmological ; zwicky1937masses ; sofue2001rotation  and
theoretical predictions of the expansion history of the universe in standard
cosmology. The missing matter and energy in the universe, dubbed dark matter
(DM) and dark energy (DE), respectively, account for a whopping 95% of the
total content of the universe. The nature of these dark components of the
universe is not properly understood, but there are several candidates in the
literature, including unified dark-fluid models, proposed to describe them and
their effect on astrophysics and cosmology. On the DM side, most commonly
studied candidates include Weakly Interacting Massive Particles (WIMPS)
bergstrom2009dark ; colafrancesco2006multi ; Mekuria:2017TI ; mekuria2017multi
or some astrophysical modification of gravity such as the Modified Newtonian
Dynamics (MOND)milgrom1983modification  among many others, whereas on the DE
side, the cosmological constant $\Lambda$ carroll2001cosmological  is perhaps
the simplest addition to the standard cosmological model needed to explain
most of the observed data. There are some serious issues associated with the
cosmological constant, however, such as the eponymous cosmological constant
problem weinberg1989cosmological  and the coincidence problem
steinhardt2003quintessential ; velten2014aspects  which make the choice less
attractive. That is why there are currently a plethora of other alternatives
to explain current cosmological observations, such as modifications to the
gravitational theory itself (see, for example, saridakis2021modified ;
capozziello2011extended ; clifton2012modified ; debono2016general ), an
evolving $\Lambda$ peebles1999evolution ; peebles1988cosmology ;
reuter1987time , deviations from the standard homogeneous (see
bolejko2011inhomogeneous ; krasinski1997inhomogeneous  and references therein)
and isotropic universe (such as the various Bianchi cosmological models)
assumption, or some form of combination of these, among others.

Another aspect to consider, and one gaining much traction recently, is the
interaction of dark matter and dark energy bolotin2015cosmological ;
yang2014cosmological ; yang2015reconstructing ; maity2019universe ;
zimdahl2005interacting . Such an approach is interesting because it has the
potential to explain the cosmological and coincidence problems, the Hubble
tension and/or the $\sigma_{8}$ discrepancy vandark ; mvdw ; mahlatjidiffusive
.

Our current work pursues the last aspect, and studies the cosmological
viability of a model maity2019universe  of the dark-fluid interaction using
Supernovae Type 1A data. We organise the rest of the manuscript as follows: in
Sec. II we give a covariant thermodynamics description of, and derive the
field equations for, the background universe involving the diffusive dark-
fluid system. In Sec. III we give an observational-constraint analysis using
MCMC simulations of Supernovae Type 1A and Planck 2018 data and give some
predictions on the values of the defining parameters of our model. Finally in
Sec. IV we discuss the results and give conclusions.

##  II Background thermodynamics

The standard $\Lambda$CDM cosmology is a solution of the Einstein field
equations (EFEs) derived from the action (From here onwards, we will work with
units in which the speed of light $c=1$):

$S=\frac{c^{4}}{16\pi
G}\int{d^{4}x\sqrt{-g}\left[R+2\left(L_{m}-\Lambda\right)\right]}\;,$ (1)

where $R$, $L_{m}$ and $\Lambda$ are the Ricci scalar, the matter Lagrangian
density and the cosmological constant, respectively. The corresponding EFEs
read:

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}\;,$ (2)

with the first (geometric) term represented by the Einstein tensor, and the
RHS of the equation representing the total energy-momentum tensor (EMT) of
matter fluid forms. Both $G_{\mu\nu}$ and $T_{\mu\nu}$ are covariantly
conserved quantities. The EMT for perfect-fluid models is given by

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}\;,$ (3)

where $\rho$ and $p$ are the energy density and isotropic pressure of matter,
respectively, often related by the barotropic equation of state (EoS)
$p=w\rho$ for a constant EoS parameter $w$. The normalised vector $u_{\alpha}$
represents the four-velocity of fundamental observers comoving with the fluid.
The divergence-free EMT $T^{\mu\nu}{}_{;\mu}=0$ leads to the fluid
conservation equation

$\dot{\rho}+3\frac{\dot{a}}{a}(1+w)\rho=0\;,$ (4)

where $a(t)$ is the cosmological scale factor whose evolution is given by the
Friedmann equation

$\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi
G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^{2}}$ (5)

where $k$ is the normalised spatial curvature parameter with values
$-1\;,0\;,1$ depending on an open, flat or closed spatial geometry. In a
multi-component fluid system, it is usually assumed that the energy density of
each perfect-fluid component is assumed to evolve independently of the other
fluids of the system:

$\dot{\rho_{i}}+3\frac{\dot{a}}{a}(1+w)\rho_{i}=0\;,$ (6)

and in this case the EMT in Eq. (3) is the algebraic sum of the EMTS of each
fluid, so are the total energy density and total pressure terms of Eq. (5) the
algebraic sums of the individual components. However, if we relax this
assumption due to the presence of diffusion between the constituent components
of the fluid, the individual components do not obey the matter conservation
equation, but the total fluid still does. For the the $i$th component fluid,
the new conservation equation reads:

$T^{\mu\nu}_{i}{}_{;\mu}=N^{\nu}_{i}\;,$ (7)

where $N^{\nu}_{i}$ corresponds to the current of diffusion term for that
fluid. One can then write the non-conservation equation for the fluid as:

$\dot{\rho_{i}}+3\frac{\dot{a}}{a}(1+w)\rho_{i}=\frac{\gamma_{i}}{a^{3}}\;,$
(8)

where $\gamma_{i}$ is a constant for that fluid such that
$\sum_{i}\gamma_{i}=0$. Integrating this equation gives

$\rho_{i}=a^{-3(1+w_{i})}\left[\rho_{i0}+\gamma_{i}\int^{t}_{t_{0}}a^{3w_{i}}dt^{\prime}\right]\;,$
(9)

with $\rho_{i0}$ representing the present-day ($t=t_{0}$) value of the energy
density of the $i$th fluid. Using a late-time $t-t_{0}\ll t_{0}$ expansion and
expressing $a(t)=a_{0}\left[1-(t_{0}-t)H_{0})+\dots\right]$, we can write the
last term of the above integrand as

$\displaystyle\int^{t}_{t_{0}}a^{3w_{i}}dt$
$\displaystyle=\int^{t}_{t_{0}}a^{3w_{i}}\left[1-(t_{0}-t)H_{0})+\dots\right]^{3w_{i}}dt^{\prime}$
(10)
$\displaystyle=-\frac{1}{1+3w_{i}}\left[\left(1+(t_{0}-t)H_{0}\right)^{1+3w_{i}}-\left(1+(t_{0}-t)H_{0}\right)^{1+3w_{i}}+\dots\right]$
$\displaystyle\approx\frac{1}{1+3w_{i}}\left[1-\left(1+(t_{0}-t)H_{0}\right)^{1+3w_{i}}\right]$
$\displaystyle=\frac{1}{(1+3w_{i})H_{0}}\left[1-\left(2-a\right)^{1+3w_{i}}\right]\;,$

where in the last step, we have used normalised the scale factor to unity
today: $a_{0}=1$. Thus the energy density of each diffusive fluid component is
given according to the below relation:

$\rho_{i}=a^{-3(1+w_{i})}\left[\rho_{i0}+\frac{\gamma_{i}}{(1+3w_{i})H_{0}}\left[1-\left(2-a\right)^{1+3w_{i}}\right]\right]\;.$
(11)

Assuming the well-known component of radiation, dust-like matter (baryons and
dark matter) and vacuum energy, the above diffusive solution leads to:

$\displaystyle\rho_{\rm r}=a^{-4}\left[\rho_{\rm r0}+\frac{\gamma_{\rm
r}}{2H_{0}}\left[1-\left(2-a\right)^{2}\right]\right]\;$
$\displaystyle\rho_{\rm m}=a^{-3}\left[\rho_{\rm m0}+\frac{\gamma_{\rm
m}}{H_{0}}\left[1-\left(2-a\right)\right]\right]\;$
$\displaystyle\rho_{\Lambda}=\rho_{\Lambda
0}-\frac{\gamma_{\Lambda}}{2H_{0}}\left[1-\left(2-a\right)^{-2}\right]\;$

Let us now consider the Friedmann equation for the $\Lambda$CDM model,
assuming $k=0$, which can be given as:

$\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi G}{3}\left[\rho_{\rm r0}a^{-4}+\rho_{\rm
m0}+\frac{\gamma_{\rm
m}}{H_{0}}\left[1-\left(2-a\right)\right]a^{-3}+\rho_{\Lambda
0}-\frac{\gamma_{\Lambda}}{2H_{0}}\left[1-\left(2-a\right)^{-2}\right]\right]\;.$
(13)

We are going to assume the diffusive interaction is limited between dark
matter and dark energy for this work, and hence $\gamma_{\rm r}=0$ in the
above equation.

Let us now introduce the following dimensionless quantities:

$\Omega_{\rm i}\equiv\frac{8\pi G}{3H^{2}_{0}}\rho_{\rm
i}\;,\quad\quad\Delta_{\rm m}\equiv\frac{8\pi G}{3H^{3}_{0}}\gamma_{\rm
m}\;,\quad\quad\Delta_{\Lambda}\equiv\frac{8\pi
G}{3H^{3}_{0}}\gamma_{\Lambda}\;,\quad\quad 1+z\equiv a^{-1}\;,\qquad
h\equiv\frac{H}{H_{0}}\;.$ (14)

We can then show that the Friedmann equation can be recast as

$h^{2}=\Omega_{\rm r0}(1+z)^{4}+\Omega_{\rm m0}(1+z)^{3}+\Omega_{\Lambda
0}-\Delta_{\rm
m}z(1+z)^{2}-\Delta_{\Lambda}\left[\frac{1}{2}-\frac{1}{2}\left(\frac{1+2z}{1+z}\right)^{-2}\right]\;.$
(15)

Moreover, the deceleration parameter can be shown to be

$\displaystyle q$
$\displaystyle\equiv-\frac{\ddot{a}a}{\dot{a}^{2}}=\frac{4\pi
G}{3H^{2}}\sum_{i}\rho_{\rm i}(1+3w_{\rm i})$ (16)
$\displaystyle=\frac{1}{2}\left\\{\frac{2\Omega_{\rm r0}(1+z)^{4}+\Omega_{\rm
m0}(1+z)^{3}-2\Omega_{\Lambda 0}-\Delta_{\rm
m}z(1+z)^{2}+\Delta_{\Lambda}\left[1-\left(\frac{1+2z}{1+z}\right)^{-2}\right]}{\Omega_{\rm
r0}(1+z)^{4}+\Omega_{\rm m0}(1+z)^{3}+\Omega_{\Lambda 0}-\Delta_{\rm
m}z(1+z)^{2}-\Delta_{\Lambda}\left[\frac{1}{2}-\frac{1}{2}\left(\frac{1+2z}{1+z}\right)^{-2}\right]}\right\\}$

These equations reduce to their respective $\Lambda$CDM limits when
$\Delta_{\rm m}$ and $\Delta_{\Lambda}$ both vanish.

##  III Observational constraints

In the following we will provide the result for observational constraints for
the diffused dark fluid models we have introduced in this work. We have used
the distance modulus equation which can be obtained by combining the different
cosmological distance definitions to fit against the supernovae data in our
MCMC simulation, which is presented in the work of hough2021confronting .

As shown in Fig. 1 we run MCMC simulation for the diffusive model, by
combining Eqs. 14 and 15, we find on average the best-fitting parameter value
for each free parameter to be h = 0.6966 for the Hubble uncertainty parameter,
$\Omega_{m0}$ = 0.2678 for the matter density parameter, and $\Omega_{r0}$ =
0.00050 for the radiation density parameter, along with a newly introduced
parameters $\Delta_{\rm m}=0.00252$ and $\Delta_{\Lambda}=-0.00251$. We shall
henceforth refer to this diffusive model case as Case I.

Figure 1: The MCMC simulation results for the diffusive model’s which is given
in Eq. 15 for cosmological free parameters ($\Omega_{m0}$, $\bar{h}$,
$\Omega_{r0}$, $\Delta_{m}$ and $\Delta_{\Lambda}$), with the “true” values
for $\Omega_{m0}$ = $0.315$, $\bar{h}=0.674$, and $\Omega_{r0}=2.47\times
10^{-5}$ /$\bar{h}^{2}$ provided by the Planck2018 collaboration data release
aghanim2020planck . We used 100 random walkers and 10000 iterations.

Among some of our optimum results we have also, as Case II, obtained the
situation where $\Omega_{m0}$ result is much closer to the observational
result of Planck 2018 (= $0.315^{+0.555}_{-0.111}$) as shown below which are
also obtained with MCMC simulation for the diffusive model, by combining Eqs.
14 and 15, we find on average the best-fitting parameter value for each free
parameter to be h = 0.6955 for the Hubble uncertainty parameter, $\Omega_{m0}$
= 0.3134 for the matter density parameter, and $\Omega_{r0}$ = 0.00050 for the
radiation density parameter, along with a newly introduced parameters
$\Delta_{\rm m}=0.1246$ and $\Delta_{\Lambda}=-0.1244$.

Figure 2: The MCMC simulation results for the diffusive model’s which is given
in Eq. 15 for cosmological free parameters ($\Omega_{m0}$, $\bar{h}$,
$\Omega_{r0}$, $\Delta_{m}$ and $\Delta_{\Lambda}$), with the “true” values
for $\Omega_{m0}$ = $0.315$, $\bar{h}=0.674$, and $\Omega_{r0}=2.47\times
10^{-5}$ /$\bar{h}^{2}$ provided by the Planck2018 collaboration data release
aghanim2020planck . We used 100 random walkers and 10000 iterations.

In Figs. 4 and 4 the above discussed two diffusive cosmological cases were
given which clearly showsto fit the extremely well with the data. Even the
corresponding 1$\sigma$-deviation do not really have an impact on the full
range predicted by them.

Figure 3: The diffusive model’s Eq. 15 for best-fitting free parameters for
the Supernovae Type 1A data with cosmological parameter values as
$h=0.6966^{+0.0047}_{-0.0047}$, $\Omega_{m0}=0.2678^{+0.0248}_{-0.0237}$,
$\Omega_{r0}=0.0005^{+0.0003}_{-0.0003}$,
$\Delta_{m}=0.0025^{+0.0169}_{-0.0173}$ and
$\Delta_{\Lambda}=-0.0025^{+0.0172}_{-0.0169}$ of the MCMC simulation result
shown in Fig. 1.

Figure 4: The diffusive model’s Eq. 15 for best-fitting free parameters for
the Supernovae Type 1A data with cosmological parameter values as
$h=0.6955^{+0.0047}_{-0.0047}$, $\Omega_{m0}=0.3134^{+0.0531}_{-0.0478}$,
$\Omega_{r0}=0.0005^{+0.0003}_{-0.0003}$,
$\Delta_{m}=0.1246^{+0.1098}_{-0.0887}$ and
$\Delta_{\Lambda}=-0.1244^{+0.0877}_{-0.1096}$ of the MCMC simulation result
shown in Fig. 2.

Additionally, Figs. 6 and 6 display the residuals obtained in the above two
cases. It can clearly be seen that at no point do the models over-or under-
estimate the resulting distance modulus for each supernovae. We also note that
the average off-set the model has, compared to the data, is
$\bar{x}_{res}=-0.0374$ Mpc in both cases with a standard deviation of
$\sigma_{res}=0.2148$ and $\sigma_{res}=0.2152$, respectively. These results
show that these are very strong relationships between the models and the data
points.

Figure 5: This is the residuals distance in Mpc between the predicted model
values and the data points for the diffusive model’s Eq. 15 for best-fitting
free parameters shown in Fig. 4.

Figure 6: This is the residuals distance in Mpc between the predicted model
values and the data points for the diffusive model’s Eq. 15 for best-fitting
free parameters shown in Fig. 4.

Figure 7: The Hubble parameter vs Redshift for the Model displayed in Fig. 1.
The blue curve represent the result obtained by considering diffusive fluid
and employing MCMC simulation, with 1-$\sigma$ deviation result displayed in
yellowish shaded region. The red curve represent $\Lambda$CDM cosmology result
using MCMC simulation where as the green curve represent one obtained directly
by using the Planck 2018 data for the purpose of comparison.

Figure 8: The Hubble parameter vs Redshift for the Model displayed in Fig. 2.
The blue curve represent the result obtained by considering diffusive fluid
and employing MCMC simulation, with 1-$\sigma$ deviation result displayed in
yellowish shaded region. The red curve represent $\Lambda$CDM cosmology result
using MCMC simulation where as the green curve represent one obtained directly
by using the Planck 2018 data for the purpose of comparison.

Figs. 8 and 8 shows the evolution of the Hubble parameter across the Redshift
for two cosmological model cases discussed above. The blue curves represent
the results obtained by considering diffusive fluid and employing MCMC
simulations, with 1-$\sigma$ deviation results displayed in yellowish shaded
regions. The red curves represent $\Lambda$CDM cosmology result using the
average values obtained from the MCMC simulations where as the green curves
represent those which are obtained directly by using the Planck 2018 data. In
Fig. 8 a complete overlap is observed between the two curves which are
obtained by using the MCMC simulation data in the $\Lambda$CDM model and the
average value of MCMC simulation data in diffusive model. In contrast in the
results of Fig. 8 we begin to notice a deviation between the two curves from
$\sim\rm{z}$ of 0.75 onward. Even though it is not expected to have a complete
overlap between the $\Lambda$CDM model using Planck 2018 values put directly
in the cosmological equations (green curves) and the MCMC results (blue curves
and yellowish shaded regions), the difference between them is observed to be
more prominent in the case of Fig. 8 than that of Fig. 8.

Figure 9: The graph of deceleration parameter vs redshift for the diffusive
model shown in Fig. 8.

Figure 10: The graph of deceleration parameter vs redshift for the diffusive
model shown in Fig. 8.

Fig. 10 and 10 shows the evolution of deceleration parameter across Redshift
for the two diffusive cosmological model cases discussed above. The blue
curves represent the results obtained by considering diffusive fluid and
employing MCMC simulations, with 1-$\sigma$ deviation results displayed in
yellowish shaded regions. The red curves represent $\Lambda$CDM cosmological
results by using the MCMC simulation data, where as the green curves represent
those which are obtained directly by substituting the Planck 2018 data values.
In Fig. 10 we observe a complete overlap between the two curves which are
obtained by using MCMC simulation data in the $\Lambda$CDM equation and the
average value of MCMC simulation data of the diffusive model, which is also
observed in the Hubble parameter vs Redshift plot of Fig. 8. Moreover, the
1-$\sigma$ deviation result which is indicated in the yellowish colored shaded
region is observed to encompass all the curves for about (z $\sim$ 0.5) of the
deceleration parameter values given in Fig. 10. The diffusive model in this
case has slightly larger values of deceleration parameter in the present
universe (z $\sim$ 0) compared to what is observed in the case of Fig. 10.

The above two cases (Case I and Case II) were obtained with a positive
$\Delta_{\rm m}$ and negative $\Delta_{\Lambda}$. In what follows, let us
provide the results corresponding to the cases of negative $\Delta_{m}$ which
can be explained as being the situation when energy flows from dark matter
sector to that of dark energy. As shown in Fig. 11 we run MCMC simulation for
the diffusive model, by combining Eqs. 14 and 15, we find on average the best-
fitting parameter value for each free parameter to be h = 0.6967 for the
Hubble uncertainty parameter, $\Omega_{m0}$ = 0.2655 for the matter density
parameter, $\Omega_{r0}$ = 0.00050 for the radiation density parameter, along
with a newly introduced parameters $\Delta_{\rm m}=-0.00251$ and
$\Delta_{\Lambda}=0.00246$. We will call this diffusive model case hereafter
Case III.

Figure 11: The MCMC simulation results for the diffusive model’s which is
given in Eq. 15 for cosmological free parameters ($\Omega_{m0}$, $\bar{h}$,
$\Omega_{r0}$, $\Delta_{m}$ and $\Delta_{\Lambda}$), with the “true” values
for $\Omega_{m0}$ = $0.315$, $\bar{h}=0.674$, and $\Omega_{r0}=2.47\times
10^{-5}$ /$\bar{h}^{2}$ provided by the Planck2018 collaboration data release
aghanim2020planck . We used 100 random walkers and 10000 iterations.

In the following we will also provide one interesting case in which the best-
fitting parameter value for each free parameter to be h = 0.6976 for the
Hubble uncertainty parameter, $\Omega_{m0}$ = 0.2283 for the matter density
parameter, and $\Omega_{r0}$ = 0.00050 for the radiation density parameter,
along with a newly introduced parameters $\Delta_{\rm m}=-0.10747$ and
$\Delta_{\Lambda}=0.10426$. We will refer to this diffusive model case as Case
IV.

Figure 12: The MCMC simulation results for the diffusive model’s which is
given in Eq. 15 for cosmological free parameters ($\Omega_{m0}$, $\bar{h}$,
$\Omega_{r0}$, $\Delta_{m}$ and $\Delta_{\Lambda}$), with the “true” values
for $\Omega_{m0}$ = $0.315$, $\bar{h}=0.674$, and $\Omega_{r0}=2.47\times
10^{-5}$ /$\bar{h}^{2}$ provided by the Planck2018 collaboration data release
aghanim2020planck . We used 100 random walkers and 10000 iterations.

In Fig. 14 and Fig. 14 the above discussed two diffusive cosmological model
cases were given which clearly shows to fit the extremely well with the data.
Even the corresponding 1$\sigma$-deviation do not really have an impact on the
full range predicted by them.

Figure 13: The diffusive model’s Eq. 15 for best-fitting free parameters for
the Supernovae Type 1A data with cosmological parameter values as
$h=0.6967^{+0.0047}_{-0.0047}$, $\Omega_{m0}=0.2655^{+0.0248}_{-0.0237}$,
$\Omega_{r0}=0.0005^{+0.0003}_{-0.0003}$,
$\Delta_{m}=-0.0025^{+0.0169}_{-0.0170}$ and
$\Delta_{\Lambda}=0.0024^{+0.0172}_{-0.0167}$ of the MCMC simulation result
shown in Fig. 11.

Figure 14: The diffusive model’s Eq. 15 for best-fitting free parameters for
the Supernovae Type 1A data with cosmological parameter values as
$h=0.6976^{+0.0047}_{-0.0047}$, $\Omega_{m0}=0.2283^{+0.0387}_{-0.0374}$,
$\Omega_{r0}=0.0005^{+0.0003}_{-0.0003}$,
$\Delta_{m}=-0.1074^{+0.0716}_{-0.0641}$ and $\Delta_{\Lambda}$
$=0.1042^{+0.0660}_{-0.0697}$ of the MCMC simulation result shown in Fig. 12.

Additionally, Fig. 16 and Fig. 16 displays the residuals obtained in the above
two cases. It can clear be seen that at no point does the models over-or
under-estimate the resulting distance modulus for each supernovae. We also
note that the average off-set the model has, compared to the data, is
$\bar{x}_{res}=-0.0374$ Mpc and $\bar{x}_{res}=-0.0386$ Mpc, with a standard
deviation of $\sigma_{res}=0.2147$ and $\sigma_{res}=0.2145$, respectively.
These results show that these are very strong relationships between the models
and data points.

Figure 15: This is the residuals distance in Mpc between the predicted model
values and the data points for the diffusive model’s Eq. 15 for best-fitting
free parameters shown in Fig. 14.

Figure 16: This is the residuals distance in Mpc between the predicted model
values and the data points for the diffusive model’s Eq. 15 for best-fitting
free parameters shown in Fig. 14.

Figure 17: The Hubble parameter vs Redshift for the Model displayed in Fig. 1.
The blue curve represent the result obtained by considering diffusive fluid
and employing MCMC simulation, with 1-$\sigma$ deviation result displayed in
yellowish shaded region. The red curve represent $\Lambda$CDM cosmology result
using MCMC simulation where as the green curve represent one obtained directly
by using the Planck 2018 data for the purpose of comparison.

Figure 18: The Hubble parameter vs Redshift for the Model displayed in Fig. 2.
The blue curve represent the result obtained by considering diffusive fluid
and employing MCMC simulation, with 1-$\sigma$ deviation result displayed in
yellowish shaded region. The red curve represent $\Lambda$CDM cosmology result
using MCMC simulation where as the green curve represent one obtained directly
by using the Planck 2018 data for the purpose of comparison.

Fig. 18 and 18 shows the evolution of the Hubble parameter across the Redshift
for the two cosmological model cases discussed above. The blue curves
represent the result obtained by considering diffusive fluid and employing
MCMC simulation, with 1-$\sigma$ deviation results displayed in yellowish
shaded regions. The red curves represent $\Lambda$CDM cosmology result using
MCMC simulation where as the green curves represent those obtained directly by
using the Planck 2018 data. An overlap is observed in the result displayed in
Fig. 18 between the two curves which are obtained by using the MCMC simulation
data in the $\Lambda$CDM model and that of the average value of MCMC
simulation data of the diffusive model. In contrast, a noticeable deviation
emerges to be observed between the the Hubble parameter values of the
diffusive model result (the blue curve) and that of the $\Lambda$CDM result
based on MCMC simulation data (the red curve) in Fig. 18 from $\sim\rm{z}$ of
0.75 onward. Even though it is not expected to have a complete overlap between
the $\Lambda$CDM model using Planck 2018 values put directly in the
cosmological equations (green curves) and the MCMC results (blue curves and
yellowish shaded regions), the difference between them is observed to be more
prominent in the case of Fig. 18 than that of Fig. 18.

Figure 19: The graph of deceleration parameter vs redshift for the diffusive
model shown in Fig. 18.

Figure 20: The graph of deceleration parameter vs redshift for the diffusive
model shown in Fig. 18.

Fig. 20 and 20 shows the evolution of the deceleration parameter across
Redshift for the two cosmological model cases discussed above. The blue curves
represent the result obtained by considering diffusive fluid and employing
MCMC simulation, with 1-$\sigma$ deviation results displayed in yellowish
shaded regions. The red curves represent $\Lambda$CDM cosmology result using
MCMC simulation where as the green curves represent those obtained directly by
using the Planck 2018 data. In Fig. 20 we observe a complete overlap while
using MCMC simulation data in the $\Lambda$CDM model and the average value of
MCMC simulation data of the diffusive model, which is also observed in the
Hubble parameter vs Redshift plot of Fig. 18.

As redshift increases, the Hubble parameter values of the diffusive model
result (the blue curve) and that of the $\Lambda$CDM result based on MCMC
simulation data (the red curve), begin to acquire similar values as can be
seen in Fig. 20. The 1-$\sigma$ deviation results indicated in yellow colored
shaded region is observed to encompass both diffusive (blue curve) and non-
diffusive (red curve) cases of the deceleration parameter values given in Fig.
20. In the current universe, the diffusive model has slightly lower values of
deceleration parameter compared to what is observed in the case of Fig. 20.

Table 1: The best-fit for each tested model, including the $\Lambda$CDM model. The models are listed in the order from the largest likelihood function value $\textit{L}(\hat{\theta}|\rm{data})$ to the smallest likelihood of being viable. The reduced $\chi^{2}$ -values are given as an indication of the goodness of fit for a particular model. The AIC and BIC values are shown, as well as the $\Delta$AIC and $\Delta$BIC for each information criterion. The $\Lambda$CDM model is chosen as the ”true model”. Models | $\Delta_{\rm{m}}$ | $\Delta_{\Lambda}$ | $\textit{L}(\hat{\theta}|\rm{data})$ | $\chi^{2}$ | $\rm{Red.}\chi^{2}$ | $AIC$ | $|\Delta AIC|$ | $BIC$ | $|\Delta BIC|$
---|---|---|---|---|---|---|---|---|---
Diffusive Case II | +ve | -ve | -121.1677 | 242.3355 | 0.6845 | 252.3355 | 4.9405 | 271.7521 | 12.7072
Diffusive Case I | +ve | -ve | -120.7059 | 241.4118 | 0.6819 | 251.4118 | 4.0168 | 270.8285 | 11.7835
$\Lambda$CDM | 0 | 0 | -120.6975 | 241.3950 | 0.6780 | 247.3950 | 0 | 259.0449 | 0
Diffusive Case III | -ve | +ve | -120.6890 | 241.3781 | 0.6818 | 251.3781 | 3.9831 | 270.7947 | 11.7497
Diffusive Case IV | -ve | +ve | -120.3936 | 240.7872 | 0.6801 | 250.7872 | 3.3922 | 270.2039 | 11.1589

In Table 1 we provide some statistical result which allows us to determine the
best diffusive model case in comparisons to the $\Lambda$CDM model. The
statistical analysis test that we have used is the Akaike information
criterion (AIC) and Bayesian$/$Schwarz information criterion (BIC) selections
which were used in a similar work in hough2021confronting . These information
criteria evaluate the plausibility of an alternative model explaining the data
compared to an “accepted$/$true model”. In our case the $\Lambda$CDM model
will be considered as the “true model”. Following the suggestion made in
hough2021confronting  as the calculated values for the AIC and BIC can by very
random, we will also use the difference in AIC (i.e., $\Delta$AIC) and BIC
(i.e., $\Delta$BIC) values of each model compared to the “true model’s AIC and
BIC values, and we use the Jeffrey’s scale in order to make conclusions about
the viability of the various Diffusive model cases. Moreover, the reduced
$\chi^{2}$ -values are used as an indication of the goodness of fit for each
model on the supernovae data. It is observed that, the first two Diffusive
model cases (shown in Fig. 1 and 2) have obtained better likelihood function
values than the $\Lambda$CDM model based on a Gaussian probability
distribution, with Case II obtaining the larger likelihood function value.
However, in the reduced $\chi^{2}$ -values in which the number of parameters
are taken into account when determining the goodness of fit, the $\Lambda$CDM
model has the best value with the Diffusive model Case I (shown in Fig. 1)
managing to have a closer value to this accuracy. In order to find the better
fitting model among these two cases, we use AIC test, according to which the
Diffusive model Cases I and II have obtained more observational and less
observational support, respectively. Case I is seen to have a value just
missing out on the substantial observational support category, but is still
with a closer value to the boundary for less observational support. Therefore,
it can be concluded that Case I has some observational support according to
the AIC criterion, while Case II has less observational support. In terms of
the BIC criteria, we did not obtain one model to have some observational
support category, but Case I was the closer of being in one of the categories.
Therefore, statistically, based on the likelihood, the goodness of fit, the
AIC and BIC criteria, Case I is the most likely to be an alternative model to
the $\Lambda$CDM model, with Case II not being ruled out, but will have to be
tested on other datasets before being accepted or rejected.

##  IV Conclusions

In this manuscript we considered diffusive cosmological models where dark
matter and dark energy interact by exchanging energy. The background
cosmological parameters in particular the thermodynamics parameters have been
studied and compared against supernova cosmological data for different
Diffusive model cases using MCMC simulation results presented in the previous
section.

For the two new parameters which arise in our Diffusive cosmological model,
namely $\Delta_{m}$ and $\Delta_{\Lambda}$, we have examined the Hubble and
deceleration parameters results of Figs. 8 to 10 and that of Figs. 18 to 20.
Recalling the requirement that the sum of these two parameters need to be
zero, the magnitude of $\Delta_{m}$ and $\Delta_{\Lambda}$ of $\ \approx$
0.0025 fit the parameter space very well. Following which we investigated this
deeply based on the statistical analysis made in the above section which is
given in Table 1. From our analysis we observed that cases having positive
values of $\Delta_{\rm m}$ were showing the largest values of likelihood
function. Based on the analysis of likelihood, goodness of fit, AIC and BIC
criteria, one can conclude that overall Case I is the most likely to be an
alternative to the $\Lambda$CDM model.

As we have highlighted in the discussion part our current work is to provide a
viability test of the different cases considered, but to reject or accept any
of them more data and rigorous testing method is needed. Moreover, our initial
result such as the one shown in Figs. 8 and 18 suggest that one can look for a
potential explanation of the Hubble Tension in such models.

## Acknowledgements

AA acknowledges that this work is based on the research supported in part by
the National Research Foundation (NRF) of South Africa (grant number 112131).
This work was part of the research programme “New Insights into Astrophysics
and Cosmology with Theoretical Models confronting Observational Data” of the
National Institute for Theoretical and Computational Sciences of South Africa.

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# THE CHARACTERISTICS AND PROPERTIES OF GROWING NETWORK BASED ON HEXAHEDRON

Li Haijun111
Information Technology Centre
Hexi University
Zhangye 734000, P. R. China
<EMAIL_ADDRESS>
&Lu Qingping
College of Foreign Languages and Literature
Hexi University
Zhangye 734000, P. R. China
<EMAIL_ADDRESS>

###### Abstract

We constructs a new network by superposition of hexahedron , which are scale-
free, highly sparse, disassortative ,and maximal planar graphs. The network
degree distribution, agglomeration coefficient and degree of correlation are
computed separately using the iterative method, and these characteristics are
found to be very rich. The method of network characteristic analysis can be
applied to some actual systems, so as to study the complexity of real network
system under the framework of complex network theory.

Keywords. Study;Characteristics and Properties;Growing Network;Hexahedron. ††∗
Corresponding author.

##  1 Introduction

Scientists have studied all aspects of the graph, which is a very interesting
job.The graph of discrete objects has been studied extensively for more than
100 years.The analysis of the classification and nature of the graph and the
development of graph algorithm are important problems in this direction.In the
field of application, discrete graphics can represent physical, biological or
sociological objects, such as crystal or protein structure networks, community
network, etc.

The way to show that the real world’s complex systems use networks associated
with vertices and edges has been developed.[1-3]. In particular, some of the
characteristics of complex network properties such as high sparseness, scale-
free distribution has shown general attributes that plays an important role in
determining related special properties of the complex systems and complex
phenomena.Inspired by the two pioneering models[4,5],a large number of models
have been proposed in the study of network topology, and these models have
made great progress[6-11].Thanks to both the advances in network theory and
the increasing amount of network data available,the researchers began to
reveal a large number of different progress that could lead to some of the
aforementioned general characteristics.The research of universal structural
properties in the field of complex network modeling is very interesting[1-4].

This paper studies related properties based on the superimposed hexahedron of
the network, we built a new network using iterative superimposed methods.The
network generated after iteration $n$ times is random or determined,which are
maximal planar graphs, show scale-free degree distribution and assortative
degree-degree correlations.

##  2 Network construction

We will explore using hexahedron to generate a new network.We extended the
idea in Ref.[12] to deal with this problem. This literature introduces the
maximal planar graphs and generates a kind of planar network through regular
convex hexahedron.Therefore, the framework of bubble topology is readily
applicable to the hexahedron. This is illustrated in Fig.1.

Figure 1: (Color online) This Figure describes how to join two hexahedron by
merging the square faces

We generate a planar network in a determinate way, which iteratively merge the
newly introduced bubbles into the square faces of the graph $G(t)$.The plane
network generated by deterministic processing is called “Deterministic Bubble
Networks” (DBNs).These processes are illustrated in Fig.2.

Figure 2: Process of merging newly introduced bubbles to all polygonal faces
of a growing graph

We note that these processes can be seen as a generalization of the Apollonian
network construction process[13,14] and the Sierpinski gasket network of
trisecting sides [14].The essence of the Apollo network is the deterministic
network,it is generated by tetrahedral bubbles.The difference that
distinguishes the hexahedron DBN from Apollonian network is that the
Apollonian network considers only three triangular faces of the initial
tetrahedron,whereas the bubble approach considers all six faces of initial
hexahedron.

##  3 Iterative algorithm of the network

In the construction process of the hexahedron network,each face of the
original hexahedron is joined by a quadrilateral at the first time, and the
vertices of the new quadrilateral are connected to the vertices of the
original quadrilateral.The second generation of the new quadrangle vertex is
connected to the last generation of quadrilateral vertices, so that the
network is generated over and over again.When building a network, each group
has four new nodes that are added, and five new quadrilaterals are generated,
each of which can create four nodes in the next generation.In this idea we can
introduce an iterative algorithm to create the corresponding network,denoted
by $H(t)$ after $t$ generation evolutions.

The iterative algorithm for the network is shown below. For $t=0$, $H(0)$
consists of eight nodes forming a hexahedron. Then, we add four nodes into one
face of the hexahedron. These four new nodes are linked to each other shaping
a new quadrangle, a node of the new quadrilateral is connected to a node in
the original quadrilateral. Thus we get $H(1)$, see Figure 3.

Figure 3: (Color online) Iterative construction method for the network

Now we compute the order and size (number of all edges) of the network $H(t)$.
Let $L_{v}(t)$, $L_{e}(t)$ and $L_{\square}(t)$ be the number of vertices,
edges and active triangles created at step $t$, respectively. By construction
(see also Fig. 3),each active quadrangle in $H(t-1)$ will be added one
quadrangle in $H(t)$. So we can find the following relationship:
$L_{\square}(t)=L_{\square}(t-1)\times 5$. Since $L_{\square}(0)=6$,we have
$L_{\square}(t)=6\times 5^{t}$.

Each active quadrilateral of $H(t-1)$ in the iteration will result in adding
four new nodes and eight new edges at $t$,and then we get the following
relationship:$L_{v}(t)=4\times L_{\square}(t)=24\times 5^{t}$, and
$L_{e}(t)=8\times L_{\square}(t)=48\times 5^{t}$ for arbitrary $t>0$. From
these results, we can compute the order and size of the network. The total
number of vertices $N_{t}$ and edges $E_{t}$ present at step $t$ is

$N_{t}=\sum_{t_{i}=0}^{t}L_{v}(t_{i})+8=6\times 5^{t+1}+2$ (1)

and

$E_{t}=\sum_{t_{i}=0}^{t}L_{e}(t_{i})+12=12\times 5^{t+1},$ (2)

respectively. So for large t, the average degree
$\overline{k}_{t}=\frac{2E_{t}}{N_{t}}$ is approximately 4,which shows the
network is sparse as most real systems.

from equations(1) and (2),we have

$E_{t}=12\times 5^{t+1}=2\times(6\times 5^{t+1}+2)-4$ (3)

and

$E_{t}=2N_{t}-4,$ (4)

On the other hand, any two edges that obtained by each iteration during
network construction do not intersect each other.Thus this network is a
maximal planar graph[15], which is similar to some previously studied networks
[16-19].

##  4 Topological properties of the network

In this section, we will study some relevant characteristics of network
$H(t)$, which are degree distribution, clustering coefficient and degree
correlation.

###  4.1 Degree distribution

When a new node $i$ is added to the network at step $t_{i}(t_{i}\geq 1)$, it
has a degree of 3.Let $L_{\square}(i,t)$ be the number of active quadrangle at
step $t$ that will create new nodes connected to the node $i$ at step $t+1$.
Then at step $t_{i}$,$L_{\square}(i,t_{i})=3$. From the iterative generation
process of the network, one can see that at any subsequent step each a new
neighbors of $i$ generate two active quadrangle involving $i$, and one of its
existing active quadrangle is deactivated simultaneously. We define $k_{i}(t)$
as the degree of node $i$ at time $t$, then the relation between $k_{i}(t)$
and $L_{\square}(i,t)$ satisfies:

$L_{\square}(i,t)=k_{i}(t)$ (5)

Now we compute $L_{\square}(i,t)$. By construction,
$L_{\square}(i,t)=L_{\square}(i,t-1)$. Considering the initial condition
$L_{\square}(i,t_{i})=3$, we can derive $L_{\square}(i,t)=3\times
2^{(t-t_{i})}$. Then at time $t$, the degree of vertex $i$ becomes

$k_{i}(t)=3\times 2^{(t-t_{i})}$ (6)

It should be mentioned that the initial eight vertices created at step 0 have
a little different evolution process from other ones. We can easily obtain
that at step t, the degree of one of the initial eight vertices and the number
of active quadrangle are $3\times 2^{t}$ .

Equation(6) shows that the degree spectrum of the network is discrete.It
follows that the cumulative degree distribution[2] is given by

$P_{\rm cum}(k)=\sum_{\tau\leqslant
t_{i}}\frac{L_{v}(\tau)}{N_{t}}=\frac{6\times 5^{t_{i}+1}+2}{6\times
5^{t+1}+2}$ (7)

Substituting for $t_{i}$ in this expression using $t_{i}=t+\frac{\rm
ln3-lnk}{\rm ln2}$ gives

$P_{\rm cum}(k)=\frac{6\times 5^{t+\frac{\rm ln3-lnk}{\rm ln2}}+2}{6\times
5^{t}+2}$ (8)

When $t$ is large enough,one can obtain

$P_{\rm cum}(k)=k^{(-\rm ln5/ln2)}$ (9)

So the degree distribution follows a power law form with the exponent
$\gamma=1+\frac{\rm ln5}{\rm ln2}$.Note that the same degree exponent has been
obtained in some other deterministic models such as Apollonian
networks[16-17,19] and pseudofractal scale-free web[20-22].

###  4.2 Clutering coefficient

The clustering coefficient [4] of a node i with degree $k_{i}$ is given by
$C_{i}=2e_{i}/[k_{i}(k_{i}-1)]$, where $e_{i}$ is the number of existing edges
among the $k_{i}$ neighbors. Using the connection rules, it is straightforward
to calculate analytically the clustering coefficient $C(k)$ for a single node
with degree $k$. When a node is added into the network,$k_{i}$ is 3, $e_{i}$
is zero. At each subsequent discrete time step, each of its active quadrangles
increases both $k_{i}$ and $e_{i}$ by 3 and 0,respectively. therefore
$C_{i}=0$,so this network is highly sparse.

###  4.3 Degree correlations

Degree correlation is a particularly interesting subject in the field of
network science [23-27],because it can give rise to some interesting network
structure effects.An interesting quantity related to degree correlations is
the average degree of the nearest neighbors for nodes with degree $k$,denoted
as $k_{nn}(k)$,which is a function of node degree $k$[24,25].When $k_{nn}(k)$
increases with $k$,it means that nodes have a tendency to connect to nodes
with a similar or larger degree .In this case the network is defined as
assortative [26,27].In contrast,if $k_{nn}(k)$ is decreasing with $k$, which
implies that nodes of large degree are likely to have near neighbors with
small degree ,then the network is said to be disassortative.If correlations
are absent,$k_{nn}(k)=const$.

We can exactly calculate $k_{nn}(k)$ for the networks using equations (3)
and(4) to work out how many links are made at a particular step to nodes with
a particular degree.By construction,we have the following expression[28,29]

$k_{\rm
nn}(k)=\frac{1}{L_{v}(t_{i})k(t_{i},t)}\Bigg{(}\sum_{t^{{}^{\prime}}_{i}=0}^{t_{i}^{{}^{\prime}}=t_{i}-1}L_{v}(t_{i}^{{}^{\prime}})L_{\square}(t_{i}^{{}^{\prime}},t_{i}-1)k(t_{i}^{{}^{\prime}},t)$

$+\sum_{t_{i}^{{}^{\prime}}=t_{i}+1}^{t_{i}^{{}^{\prime}}=t}L_{v}(t_{i})L_{\square}(t_{i},t_{i}^{{}^{\prime}}-1)k(t_{i}^{{}^{\prime}},t)\Bigg{)}$
(10)

for $k=3\times 2^{t-t_{i}}$ and where $k(t_{i},t)$ is the degree of a node $i$
at time $t$ that was born at step $t_{i}$.Here the first sum on the right-hand
side accounts for the links made to nodes with larger degree(i.e.
$t_{i}^{{}^{\prime}}<t_{i}$)when the node was generated at $t_{i}$.The second
sum describes the links made to the current smallest degree nodes at each step
$t_{i}^{{}^{\prime}}>t_{i}$.After some algebraic manipulations,we can rewrite
equation(19) in term of $k$ to obtain

$k_{\rm nn}(k)=\frac{690\times 5^{t}-36\times 2^{t}\times 5^{\frac{\rm lnk-
ln3}{\rm ln2}-t}-218\times 5^{\frac{\rm ln3-lnk}{\rm ln2}+t}\times k}{23k}$
(11)

Here,$k_{nn}(k)$ increases with $k$ in equation (11),it means that network
nodes have a tendency to connect to nodes with similar or larger degree.
Therefore,Network is assortative.

###  4.4 Conclusion

In conclusion, based on the deterministic bubble networks(DBNs), we
constructed the plane network by superimposing regular hexahedron, and
analyzed the topology properties of the network.The analytic expressions of
the main topological properties have been obtained through the iterative
method, such as degree distribution, clustering coefficient and degree
correlation.Research shows that this network shows some important features of
real systems: power-law degree distribution, highly sparse and positive degree
correlation.

This work opens up new interesting topics for future research and research on
complex networks.In this paper, the method of iterative loop superposition of
hexahedron can produce new planar graphs, which is also applicable to
polyhedra and they are therefore of general applicability.

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