#  Statistical description of galaxy clusters in Finzi model of gravity

Abdul W. Khanday<EMAIL_ADDRESS>Department of Physics, National
Institute of Technology Srinagar, Jammu and Kashmir -190006, India.  Sudhaker
Upadhyay<EMAIL_ADDRESS>Department of Physics, K. L. S. College,
Nawada, Bihar 805110, India  Department of Physics, Magadh University, Bodh
Gaya, Bihar 824234, India  Inter-University Centre for Astronomy and
Astrophysics (IUCAA) Pune, Maharashtra-411007  School of Physics, Damghan
University, Damghan, 3671641167, Iran  Prince A. Ganai<EMAIL_ADDRESS>Department of Physics, National Institute of Technology Srinagar, Jammu and
Kashmir -190006, India.

###### Abstract

We exploit a new theory of gravity proposed by Finzi, which gives stronger
interaction at large scales, to study the thermodynamic description of galaxy
clusters. We employ a statistical model to deduce various thermodynamics
equations of state. In addition, we analyze the behavior of clustering
parameter in comparison to its standard (Newtonian) counterpart. The general
distribution function and its behavior with varying strength of clustering
parameter are also studied. The possibility of phase transition is
investigated and observed that a phase transition is possible though
hierarchically. We also analyze the model by comparing the results with data
available through SDSS-3, and obtain the parameters involved.

Finzi gravity; Distribution function of galaxies; Phase transition.

##  I Introduction

The large scale distribution of the matter is mainly described by the
gravitational interaction. It is believed that the large scale structure of
the universe is a gravitationally amplified descendant of a faint noise field
believed to be seeded by quantum fluctuations in the early universe. A linear
theory for initial perturbations to the present observed matter distribution
has been developed extensively 1 . The formation of first structures in the
universe took place at a red-shift of $10-30$ in the dark matter halos of
masses, $M>10^{5}-10^{8}M_{\odot}$ 2 . This structure formation took place on
the imprints of small matter density perturbations in the primordial matter
density field. Present day $N$-body simulations have confirmed that the
initial density perturbations have the potential to grow to the present day
observed structure 3 .

It is well-known that the velocity of galaxies in a cluster doesn’t agree well
with the total mass of the visible matter. The estimated mass was $200-400$
times less than the mass required to prevent the rupture of the galaxies from
the cluster 4 . This led to the concept of dark matter. Although there is a
strong support to the dark matter hypothesis but the identification of the
particles that compose this matter is yet under discussion. One explanation to
this could be a strong correlation between baryonic matter 5 and dark-matter.
The other intriguing idea could be a relevance of modified Newtonian dynamics
(MoND) at large scales.

The formation and distribution of galaxy clusters are crucial to understand
the evolution of the structure formation in the universe. Saslaw and Hamilton
developed a thermodynamic model for nonlinear regime of the clustering of
galaxies in an expanding universe 7 . The theoretical predictions of this
model involve the distribution of voids as well as galaxies corresponding to
under-dense and over-dense regions, respectively, in the initial density
field. The probability distribution function predicted by this theory for $N$
number of galaxies in volume $V$ is given by

$f(N)=e^{-\bar{N}(1-b)-Nb}\frac{\bar{N}(1-b)}{N!}\left[\bar{N}(1-b)+Nb\right]^{N-1},$
(1)

where $\bar{N}$ is the average number of particles found in volume $V$ and $b$
is clustering parameter.

It is established that the force accounting for the flat rotation curve of
galaxy clusters requires an enhancement at larger distances than predicted by
Newtonian gravity. In recent past, there have been a number of modifications
to general relativity 8  to account for this discrepancy. For instance, in
$f(R)$ gravity, an additional term having function $f(R)$ of Ricci curvature
is added to account for the enhanced gravitational force on large scales. The
force field that governs the large scale structure formation also governs the
distribution of mass at this scale. We can employ statistical methods to find
the statistical distribution of matter at largest possible scales 9 . Recently
there has been a lot of progress on the study of the effect of modified
gravity theories on the clustering of galaxies through statistical methods 10
; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19 . The effect of the modifications
incorporated has been anticipated in the strength of the clustering parameter
$b$. For instance, in 14 , the modified clustering parameter has been studied
as a function of the modification factor.

In 1963, Finzi proposed a law of gravitation that gives a stronger interaction
at relatively larger distances than predicted by Newton’s law of gravity 20 .
This force form can explain the larger velocities of galaxies in clusters
without involving the concept of dark matter. The potential energy function
proposed by Finzi is given by

$\Phi(r)=G_{*}m^{2}\left(\frac{1}{r^{1/2}}\right),$ (2)

where $G_{*}=-2k/\rho^{1/2}$, $k$ is a constant (equivalent to gravitational
constant $G$) and $\rho$ is a characteristic length beyond which the potential
becomes significant. This potential reduces to the usual Newtonian potential
at distances $\rho\approx r$. The correlation and distribution function of
galactic cluster are not studied for Finzi approximation. Here, we try to
bridge this gap and also make a comparison of the effect of this model to the
already studied theories.

Our investigations are presented systematically as following. First, we
construct the partition function under Finzi potential for the system of
galaxies in section II. Here, in order to avoid divergence, we consider
extended nature of galaxies. In section III, we derive the various
thermodynamic equations of state along with their behavior under different
circumstances, e.g., Helmholtz free energy, pressure, internal energy, entropy
and chemical potential. In section IV, we compare the clustering parameters
with increasing radial distance corresponding to Newtonian and Finzi gravity.
In section V, we study the effects of this new force form on the statistical
distribution of the galaxy clusters. The power-law behavior for the
correlation function is presented in section VI. Finally we investigate the
possibility of phase transition in section VII. We conclude the results and
their importance in the last section.

##  II The partition function

In this section, we develop the partition function by taking into
consideration a system of particles interacting gravitationally through Finzi
gravity. We assume the clustering of galaxies in the expanding background to
be in quasi-equilibrium state forming an ensemble of co-moving cells. We
consider our system consists of a larger number of cells (ensemble of cells)
of equal volume $V$ with an average number density $\bar{N}$. The partition
function for such a system of pairwise interacting particles having
correlation energy $\Phi$ and average temperature $T$ is given by 9

$Z_{N}(T,V)=\frac{1}{N!}\left(\frac{2\pi
mT}{\lambda^{2}}\right)^{\frac{3N}{2}}Q_{N}(T,V).$ (3)

Here, $N!$ takes care of distinguishability of the system and $\lambda$ is a
normalization factor. Boltzmann’s constant is set to unity here. The
configuration part of the equation (3) can be written as

$Q_{N}(T,V)=\displaystyle\int...\int\exp\left[-T^{-1}\Phi(r_{1},r_{2},...r_{N})\right]d^{3N}r.$
(4)

The gravitational potential energy function $\Phi(r_{1},r_{2},...r_{N})$ is a
function of the relative positions $r=|r_{i}-r_{j}|$ and is summed over all
the pairs of particles. In our system of gravitating bodies, the potential
energy $\Phi(r_{1},r_{2},...r_{N})$ of the system is due to all pairs of
particles of the system, i.e.

$\Phi(r_{1},r_{2},...r_{N})=\sum_{1\leq i\leq j\leq N}\Phi_{ij}(r).$ (5)

With this simplification (5), equation (4) can now be written as

$Q_{N}(T,V)=\displaystyle\int...\int\displaystyle\prod_{1\leq i\leq j\leq
N}\exp\left[-T^{-1}\Phi_{ij}(r)\right]d^{3N}r,$ (6)

where $\Phi_{ij}$ is the two point interaction energy between the $i^{th}$ and
$j^{th}$ particle. The configuration integral can be written in terms of two-
point function $f_{ij}(=e^{-\frac{\Phi_{ij}}{T}}-1)$ defined as:

$Q_{N}(T,V)=\displaystyle\int...\int(1+f_{12})(1+f_{13})(1+f_{23})(1+f_{14})...(1+f_{N-1,N})d^{3}r_{1}d^{3}r_{2}...d^{3}r_{N}.$
(7)

Here, we note that the two-point function $f_{ij}$ is non-zero only when there
are interactions present. The Finzi potential (2) diverges for the point-
particle nature of galaxies. Therefore, we need to express potential (2) by
considering the extended nature of galaxies (galaxies with halos). This is
done by introducing the softening parameter $\epsilon$ $(0.01\leq\epsilon\leq
0.05)$ in the potential as

$\Phi(r)={\frac{G_{*}m^{2}}{(r+\epsilon)^{1/2}}}.$ (8)

By substituting the two-point function corresponding to potential (8) in
equation (7), the values of $Q_{N}$ for different values of $N$ can be
calculated. For $N=1$, we have

$Q_{1}(T,V)=V.$ (9)

For $N=2$, the integral is evaluated by fixing the position of $r_{1}$ and
evaluating over all the other particles. In this way the integral simplifies
to

$Q_{2}(T,V)=V^{2}[1+X\alpha],$

where $X=\frac{2G_{*}m^{2}}{Tr^{1/2}}$ and
$\alpha=\sqrt{1+\frac{\epsilon}{r}}\big{[}3-4(\epsilon/r)+8(\epsilon/r)^{2}-(\sqrt{\epsilon/r}8(\epsilon/r)^{2})\big{]}$.

Proceeding in similar fashion , the values of $Q_{N}$ for $N=3,4,5,...,N$ can
be obtained. E.g, for $N=3$, we get

$Q_{3}(T,V)=V^{3}\bigg{[}1+X\alpha\bigg{]}^{2}.$

In general the above equation for $N$ number of particles takes the form,

$Q_{N}(T,V)=V^{N}\bigg{[}1+X\alpha\bigg{]}^{N-1}.$ (10)

Finally, substituting equation (10) into equation (3), we obtain the general
partition function for a gravitating system of $N$ pairwise interacting
particles (galaxies) as

$Z_{N}(T,V)=\frac{1}{N!}\bigg{(}\frac{2\pi
mT}{\Lambda^{2}}\bigg{)}^{\frac{3N}{2}}V^{N}\bigg{[}1+\alpha X\bigg{]}^{N-1}.$
(11)

Equation (11) is the standard form of canonical partition function for a
system of $N$ particles interacting through the new gravity law (2). The
modification is inherent in the parameter $\alpha$.

##  III Thermodynamic equations of state

The partition function equation (11) contains all the necessary information
about the macroscopic variable (free energy, entropy, internal energy,
pressure, chemical potential ) of the system. The Helmholtz free energy for
the system of galaxies can be deduced from the partition function via the
general statistical relation $F=-T\ln Z_{N}(T,V)$. For our system of galaxies
the free energy takes the following form:

$\begin{split}F&=NT\ln\bigg{(}\frac{N}{V}T^{\frac{-3}{2}}\bigg{)}-NT-\frac{3}{2}NT\ln\bigg{(}\frac{2\pi
mT}{\lambda^{2}}\bigg{)}-NT\ln[1+X\alpha],\\\
&=NT\ln\bigg{(}\frac{N}{V}T^{\frac{-3}{2}}\bigg{)}-NT-\frac{3}{2}NT\ln\bigg{(}\frac{2\pi
mT}{\lambda^{2}}\bigg{)}+NT\ln[1-b_{n}].\end{split}$ (12)

Here, we approximated $N-1\approx N$. In equation (12), the parameter $b_{n}$
is defined as

$b_{n}=\frac{\alpha X}{1+\alpha X}.$ (13)

This is the new clustering parameter which estimates the strength of
correlation that governs the evolution of the galaxy cluster and can take
values between $0$ and $1$.

Figure 1: Variation of the free energy of the system with particle number for
different values of the $\alpha$ and unit value for the rest of parameters.

Once the free energy is known, other thermodynamic equations of state can be
estimated easily. For example, the entropy of the system of galaxies can be
obtained using the relation, $S=-\bigg{(}\frac{\partial F}{\partial
T}\bigg{)}_{V,N}$. Utilizing equation(12) the entropy of the system takes the
following form:

$S=N\ln\biggl{(}\frac{V}{N}T^{3/2}\biggl{)}+N\ln[1+X\alpha]-3N\frac{X\alpha}{1+X\alpha}+\frac{5}{2}N+\frac{3}{2}N\ln\bigg{(}\frac{2\pi
m}{\lambda^{2}}\bigg{)}.$ (14)

Entropy per particle of the system correspond to equation (13) can be written
as

$\frac{S}{N}=\ln\biggl{(}\frac{V}{N}T^{3/2}\biggl{)}-\ln[1-b_{n}]-3b_{n}+\frac{5}{2}+\frac{3}{2}\ln\frac{2\pi
m}{\lambda^{2}},$ (15)

Figure 2:  The variation of entropy ($S-S_{0}$) with an increasing particle
number for different values of clustering parameter $b_{n}$.

The total internal energy of the system of galaxies can be calculated using
the basic definition, $U=F+TS$. Substituting the values of free energy (12)
and entropy (13), the relation for internal energy in terms of the new
clustering parameter takes the form

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{3}{2}NT\bigg{[}1-2\frac{X\alpha}{1+X\alpha}\bigg{]}$ (16)
$\displaystyle=$ $\displaystyle\frac{3}{2}NT\left[1-2b_{n}\right].$

Figure 3:  The variation of internal energy $U$ as a function of particle
number for different values of correlation parameter $b_{n}$.

The graphical representation of the effect of the New clustering parameter on
the internal energy function for the system of galaxies can be visualized in
Fig. 3.

Similarly we can calculate the pressure caused by the particles within the
system utilizing the fundamental definitions $P=-\bigg{(}\frac{\partial
F}{\partial V}\bigg{)}_{T,N}$. The pressure of the system takes the form:

$\displaystyle P$
$\displaystyle=\frac{NT}{V}\left[1-\frac{X\alpha}{1+X\alpha}\right],$
$\displaystyle=\frac{NT}{V}\left[1-b_{n}\right].$ (17)

Figure 4:  The variation of the pressure with increasing particle number for
different values of clustering parameter $b_{n}$.

Finally, we can deduce the relation for the chemical potential of the system
using, $\mu=\bigg{(}\frac{\partial F}{\partial N}\bigg{)}_{T,V}$ as

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
T\bigg{(}\ln\frac{N}{V}T^{-\frac{3}{2}}\bigg{)}+T\ln\bigg{[}1-\frac{X}{1+X}\bigg{]}-T\frac{X}{1+X}-\frac{3}{2}T\ln\bigg{(}\frac{2\pi
m}{\lambda^{2}}\bigg{)},$ (18) $\displaystyle=$ $\displaystyle
T\left(\ln\frac{N}{V}T^{-3/2}\right)+T\ln\left[1-b_{n}\right]-Tb_{n}-\frac{3}{2}T\ln\left(\frac{2\pi
M}{\lambda^{2}}\right).$

The graphical representation of the variation of the chemical potential with
an increase in the particle number can be seen in figure (5)

Figure 5:  The graph shows the behavior of the chemical potential $\mu(N)$ of
the system as a function of particle number for different values of clustering
parameter $b_{n}$.

##  IV A Comparison of New and old clustering parameter

In this section the clustering parameter $b_{n}$ developed in this work based
on the Finzi model of gravity is compared with the standard parameter $b$
defined in 9 . The clustering parameter gives the strength of correlation and
will in turn decide the time-length of clustering.

Figure 6: A comparison of the standard clustering parameter $b$ with the new
parameter $b_{n}$

From the graph (Fig.(6)) it can been seen the new clustering parameter is more
stronger than the standard one. This is due to the effect of the increased
strength of the potential energy function at large distances as defined in the
Finzi model.

##  V General distribution function

The general form of the distribution function $f(N)$, which characterizes the
galaxy clustering, contains the distribution of voids as well as the number of
galaxies in fixed volume cells distributed through out the system. Here we let
the system particles to cross the cell boundaries which in turn can change the
particle number in each cell. Thus we derive the grand canonical partition
function defined as

$Z_{G}(T,V,z)=\sum_{N=0}^{\infty}\exp\left(\frac{N\mu}{T}Z_{N}(T,V)\right).$
(19)

The probability distribution function of finding $N$ particles contained in a
cell of volume $V$ in a grand canonical ensemble is given by

$\displaystyle F(N)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{N}\frac{\exp\frac{N\mu}{T}\exp\frac{-U}{T}}{Z_{G}(T,V,z)},$
(20) $\displaystyle=$
$\displaystyle\frac{\exp\frac{N\mu}{T}Z_{N}(T,V)}{Z_{G}(T,V,z)}.$

Here, the factor $z=\exp\frac{\mu}{T}$ (fugacity) determines the activity of
the system. From this basic relation equation (20), we can determine the
general form of distribution function for our system of galaxies. Using the
partition function (19) with chemical potential equation (11) the distribution
function takes the following form:

$F(N)=\frac{\bar{N}}{N!}(1-b_{n})\bigg{[}\bar{N}(1-b_{n})+Nb_{n}\bigg{]}^{N-1}\exp-
Nb_{n}-\bar{N}(1-b_{n}).$ (21)

Till here
From equation (21) we confirm the basic structure of the distribution function
is same for Finzi model of gravity as has been derived by Saslaw and Hamilton
for Newtonian gravity 7 ; 13 . The graphical representation of the probability
distribution function as a function of particle number $N$ in is shown in
figure 7. From the figure, it is clearly visible that the peak of the
distribution function has flattens as the value of the clustering parameter
shifts to higher values.

Figure 7:  Graphical representation of the distribution function $F(N)$ with
increasing particle number, $N$ for different values of $b_{n}$.

##  VI Power law for two-point correlation function

Here we study the behavior of the two-point correlation function in the new
model of gravity. We write the clustering parameter in the form 21

$b_{n}=\frac{G_{*}\bar{\rho}}{6T}\int\left[\frac{1}{(r+\epsilon)^{1/2}}\right]\xi_{2}(\rho,r,T)d{V},$
(22)

where $\bar{\rho}$ is the number density and $\xi_{2}$ is the two-point
correlation function.

Differentiating with respect to $V$, equation (22) yields

$\begin{split}\frac{\partial b_{g}}{\partial
V}=\frac{G_{*}\bar{\rho}}{6T}\frac{\partial}{\partial
V}\int\left[\frac{1}{(r+\epsilon)^{1/2}}\right]\xi_{2}(\rho,r,T)d{V}\\\
+\frac{G_{*}}{6T}\left(-\frac{\partial\bar{\rho}}{\partial
V}\right)\int\left[\frac{1}{(r+\epsilon)^{1/2}}\right]\xi_{2}(\rho,r,T)d{V},\end{split}$
(23)

where we have used $\frac{\partial
V}{\partial{\bar{\rho}}}{\bar{\rho}}=-\frac{\bar{\rho}}{V}$.

Using the relation $\frac{\partial
b_{n}}{\partial\bar{\rho}}=\frac{b_{n}(1-b_{n})}{\bar{\rho}}$ and equation
(23), we obtain the following expression for the power law

$\xi_{2}(r)=\frac{9Tb_{n}^{2}}{2\pi
G_{*}\bar{\rho}}\left(\frac{1}{(r+\epsilon)^{1/2}}\right).$ (24)

The graphical representation of the two point correlation function can be
visualized from the figure (8). We can see the strength of the correlation
function decreases with increasing radius. This leads us to an important
result about the correlation of system points in different cells.The galaxies
within a cell are more correlated than the galaxies in the adjacent cells.
Thus the transfer of system points from cell to cell are less likely although
not negligible.

Figure 8:  Graphical representation of the two point correlation function
$\xi(r)$ with increasing radial distance, $r$, for different values of
clustering parameter $b_{n}$.

##  VII Possibility of phase transition

If an interaction is introduced in a system characterized by Poisson
distribution the system has a high chance of changing phase from less
correlated to highly correlated. The possibility of phase transition in case
of our system of galaxies interacting gravitationally can not be ignored. Here
we will try to find out if the Finzi interaction can cause a phase transition
in the system of gravitationally interacting point particles. The result is
important as the possible phase transition can break the homogeneity of the
system and cause lumpiness in the structure. Among many indicators of phase
transition, specific heat is an important candidate to track.

The specific heat (at constant volume) $C_{V}$ is defined as

$C_{V}=\frac{1}{N}\left(\frac{\partial U}{\partial T}\right)_{N,V}.$ (25)

Using the relation for internal energy (16), the specific heat of the system
takes the following form

$C_{V}=\frac{3}{2}\left[\frac{1+6\alpha X-4\alpha^{2}X^{2}}{\left(1+\alpha
X\right)^{2}}\right].$ (26)

As $b_{n}\rightarrow 0,C_{V}\rightarrow 3/2$, this corresponds to no
interaction among the system particles. As $b_{n}\rightarrow
1,C_{V}\rightarrow-3/2$, which means the system is fully virialized. Between
these two extreme values lies the maximum values of specific heat at some
critical value of temperature. This extreme value of specific heat indicates a
possible phase transition at $T=T_{C}$.

$\frac{\partial C_{V}}{\partial T}\biggr{\rvert}_{T=T_{C}}=0.$

This gives an expression for the critical temperature as

$T_{C}=\left[3\frac{\bar{N}}{V}\left(GM^{2}\right)^{3}\alpha\right]^{1/3}.$
(27)

In terms of critical temperature the specific heat $C_{V}$ given in (26) can
be expressed as

$C_{V}=\frac{3}{2}\left[1-2\frac{1-4\left(T/T_{C}\right)^{3}}{\left\\{1+2\left(T/T_{C}\right)^{3}\right\\}^{2}}\right].$
(28)

At $T=T_{C},C_{V}=5/2$, a property of a diatomic gas. This is an indication of
system symmetry breaking due to the formation of binaries. The phase
transition is hierarchical and not spontaneous as observed in many other
physical systems. The graphical behavior of the specific heat can be
visualized in figure (9).

Figure 9: The behavior of specific heat $C_{V}$ with changing system
temperature $T/T_{C}$. We observe that the system symmetry breaks around the
critical temperature, $T=T_{C}$.

##  VIII observational data

In this section we test our model with the observational data obtained through
Sloan Digital Sky Survey 3 (SDSS-3) through its newest Data Release (DR12).
SDSS-3 contains additional sky coverage and better galaxy estimates than
SDSS-1 and SDSS-2. The data is present in catalog 22 . In figure (11) the sky
distribution of clusters in RA and DEC coordinates is presented from this
catalog. This catalog contains an information (RA, DEC, z, N etc) of almost
132,684 clusters. The catalog gives a parameter $r_{200}$ which is the
distance up to which mean density is $\approx$ 200 and also the number of
clusters in it i,e $N_{200}$. Figure (10,(a)-(i)) shows the model fitted to
the data.

We bin the data first on the basis of radius $(R)$ and then on the basis of
Redshift $(z)$. The bin size is chosen to be $\Delta R\approx 0.35Mpc$ and
$\Delta z\approx 20$. We divide cells by physical boundaries i.e., $R$. The
probability distribution of each cluster is determined by substituting
arbitrary values for the fitting parameter $b_{n}$ and $\rho$. The model is
then fit through Application Programming Interface (API)
Scipy.optimize.curve_fit of SciPy python Library. We obtained the optimized
values for the parameters involved in equation(21), listed in table(1).

The value of the charaecteristic length, $\rho$, is obtained through the
relation,

$\sqrt{\rho}=\frac{1-b_{n}}{b_{n}}.$ (29)

we have set all other parameters to unity in (29). The characteristic length
$\rho$ is measured in the same units as $r$. For instance, the value $\rho=5$
means for 1 unit of $r$, the characteristic length is 5 units.

Figure 10: The probability distribution of galaxy clusters in various red-
shift and radius bins

Figure 11: Figure shows the sky distribution of galaxies in various RA(deg)
and DEC(deg) coordinates Table 1:  The table shows the various parameters
(given/fitting) after fitting model to data points. Data points are divided
first in radius bins and then in Redshift bins. $N_{t}$ is the number of
clusters in each bin.

| $0.40<R<0.75Mpc$ | $0.75<R<1.10Mpc$ | $1.10<R<1.45Mpc$
---|---|---|---
$z$ | $N_{G}$ | $\bar{N}$ | $b$ | $\rho$ | $N_{G}$ | $\bar{N}$ | $b$ | $\rho$ | $N_{G}$ | $\bar{N}$ | $b$ | $\rho$
$[0.05,0.250]$ | $8259$ | $21.123$ | $0.30$ | $5.44$ | $368936$ | $16.876$ | $0.38$ | $2.66$ | $91826$ | $15.498$ | $0.0.37$ | $2.89$
$[0.251,0.450]$ | $19488$ | $20.406$ | $0.35$ | $3.45$ | $789028$ | $15.227$ | $0.31$ | $4.95$ | $146297$ | $14.591$ | $0.32$ | $4.51$
$[0.450,0.650]$ | $4995$ | $16.594$ | $0.40$ | $2.25$ | $387334$ | $12.536$ | $0.50$ | $1$ | $62609$ | $12.317$ | $0.33$ | $4.12$

From the plots (10), we observe that the models fits very closely to data in
redshift ranges $0.250<z<0.450$ and radius ranges $0.40<R<0.45,1.0<R1.45$,
fig(10(b,e)). In bins $0.450<z<0.650$;$0.75<R<1.10$, (fig10 (f)) the models
does not fit very closely to the data.

##  IX Conclusion

The velocities of galaxies in a cluster are much higher than depicted by the
visible matter that brings in the concept of gravitating non-visible matter
called dark matter. While the hunt for the exact description is underway, it
is plausible to reconsider Newtonian dynamics and modify it to fit the
observed data. A proposition of a new gravity law applicable at higher scales
is also not vague.

In this work, we have considered a gravitational potential proposed by Finzi
and, by using statistical methods, we deduced various thermodynamic quantities
along with correlation function. The graphical analysis of various
thermodynamic equations of state was also done. A new clustering parameter was
deduced based on this theory which shows a greater strength as compared to the
standard one. We have also made a graphical comparison of the the clustering
parameters for Finzi and Newtonian gravity. From the behavior of the
clustering parameter, it is obvious that the Finzi model of gravity at larger
scale is more significant. The power-law of correlation function for Finzi
model is also discussed. We also studied possible phase transition within the
system. The comparison of our results with the data was also studied and it
could be seen that the model fits the data very closely in some Redshift $z$
and radius ranges $R$ , while in some regions the fit is not too appropriate
although the trend follows.

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