# Modified theory of gravity and clustering of multi-component system of galaxies

## Abstract

In this paper, we analyze the clustering of galaxies using a modified theory of gravity, in which the field content of general relativity has been be increased. This increasing in the field content of general relativity changes the large distance behavior of the theory, and in weak field approximation, it will also modify the large distance behavior of Newtonian potential. So, we will analyzing the clustering of multi-component system of galaxies interacting through this modified Newtonian potential. We will obtain the partition function for this multi-component system, and study the thermodynamics of this system. So, we will analyze the effects of the large distance modification to the Newtonian potential on Helmholtz free energy, internal energy, entropy, pressure and chemical potential of this system. We obtain also the modified distribution function and the modified clustering parameter for this system, and hence observe the effect of large distance modification of Newtonian potential on clustering of galaxies.

## 1 Introduction

Observations made on the dynamics of galaxies indicate a discrepancy between the observed mass of galaxy from dynamics of galaxies and the mass inferred from the existence of luminous matter [1, 2]. It appears that a large part of the mass of the galaxies and thus universe as a whole universe is not visible, this non-luminous missing mass of the universe is known as dark matter [3, 4]. Several models have been proposed for the dark matter, such as axion [5], black holes [6], neutrino [7] and gravitino [8]. However, none of these dark matter models has been verified, and this has led to the development of alternative approaches to explain this discrepancy between the observed and measured rotation of galaxies. These approaches are motivated by a large distance modification of dynamics, such that this modified dynamics can resolve this discrepancy. In fact, it has been demonstrated that by modifying the Newtonian dynamics at galactic scales, it is possible to resolve this discrepancy, and this large distance correction to the Newtonian dynamics is called Modified Newtonian Dynamics (MOND) [9]. Even though the MOND explains the dynamics at galactic scale, it does not correct describe the dynamics at intra-galactic scale, and hence it cannot be used to analyze the clustering of galaxies [10, 11, 12].

It has also been argued that it is possible to have other modifications to gravity, such as modified theory of gravity (MOG), which do not have this problem, and can explain the clustering of galaxies [13, 14]. In MOG, the field content of general relativity are increased to include scalar, and vector fields, apart from the tensor field [15]. The dynamics of a test particle in MOG are modified by the inclusion of these additional fields. This is because the coupling of the metric to both the scalar and the vector fields, modifies the usual solution to the field equations for a point mass [16]. In fact, the rotation curves of galaxies in MOG have also been analyzed using a static spherically symmetric point mass solution derived from the field equations [17, 18]. The same procedure has been applied to the dynamics of globular clusters [19], clusters of galaxies [20], and the bullet cluster [21]. It has been observed that MOG can explain the dynamics at intra-galactic scales, and hence it can be used to analyze clustering of galaxies. In the weak field approximation, MOG produces the Newtonian potential with and a large distance correction to the Newtonian potential. The weak field approximation of the MOG has been used for analyzing such systems [22, 23, 24], so in this paper, we will also use the weak field approximation to MOG.

As the intra-galactic distances are much larger than the diameter of individual galaxies, we can approximate the individual galaxies as point particle [25]. So, such a system of galaxies interacting through a potential can be studied using standard techniques of statistical mechanics. In fact, such a system of galaxies interacting thought a Newtonian potential has already been studied using such techniques from statistical mechanics [26, 27, 28]. It has also been observed that for this system the gravitational clustering can evolve through a sequence of quasi-equilibrium state [26, 29]. Thus, the cosmological many body partition function has been obtained by using an ensemble of co-moving cells containing galaxies interacting through the usual Newtonian potential [30]. It has also been observed that if the cells is smaller than the correlation length, then each member of this ensemble is correlated gravitationally with other cells. So, for such cells, the correlations within a cell is greater than correlations among cells, so that extensivity is a good approximation to study such a system [30, 31]. The techniques of statistical mechanics can also used to analyze the clustering of different types of galaxies. In fact, the clustering of different kind of galaxies has been studied using a multi-component system [32, 33, 34]. The galaxies in this multi-component system were again assumed to interact through a Newtonian potential.

The clustering has also been studied using the large distance modification of Newtonian potential. It is possible to obtain large distance correction to the Newtonian potential in brane world models [35], and the clustering of galaxies has been studied using such a large distance correction to Newtonian potential [36]. The effects of of cosmological constant on clustering of galaxies has been analyzed, and the thermodynamics for such a system of galaxies has been studied [37]. The cluster of galaxies under the effect of dynamical dark energy has also been studied, and the gravitational partition function for this system has been constructed [38]. This gravitational partition function has been used to analyze the thermodynamics of this system. As the dark energy is dynamical in this model, the time evolution of the clustering parameter is studied using the time dependence of this dynamical dark energy. So, it is both interesting and important to analyze the large distance modification to the clustering of galaxies using standard techniques of statistical mechanics. As MOG produces an phenomenologically important large distance modification of Newtonian potential, in this paper, we will use this MOG modified Newtonian potential to analyze the clustering of galaxies.

## 2 Modified Newtonian potential

*G*, a vector field coupling constant \(\omega \). The mass of the vector filed \(\mu \) acts as scalar fields, and so the mass of the scalar field is a dynamical function in space-time. This theory also contains the self interacting potentials for various field, which can be denoted by \(V_\phi (\phi _\mu \phi ^\mu )\), \(V_G(G)\), \(V(\omega )\) and \(V_\mu (\mu )\). Now the action for MOG can be written as [15, 24],

*G*is given by

*G*, we have

*v*is the internal velocity of the system. So, for clusters of galaxies, the deviation from the constant \(G_0\) is of the order of \(G_1/G_0\simeq 10^{-7}-10^{-5}\).

## 3 Gravitational partition function

*N*! takes the distinguish-ability of classical galaxies into account, and \(\Lambda \) is the normalization factor which results from integration over momentum space. Now, integrating the momentum space, we obtain the following expression

## 4 Multi-component system of galaxies

*F*. For \(N_{l} = 3\), the Helmholtz free energy is completely negative. For \(N_{l}=4\), we can see some positive values of

*F*, including a maximum, also a minimum for low temperature case (Fig. 1b). In the case of high temperature, we can find large value for the Helmholtz free energy. The value of the mentioned maximum of the Helmholtz free energy depends on number of components. Increasing number of components, increased value of the Helmholtz free energy at peak. There are some critical temperatures (\(T\approx 0.5\) and \(T\approx 5\) in the Fig. 1b), where the Helmholtz free energy of all multi-component systems are the same. Moreover, the Helmholtz free energy is zero at zero-temperature limit.

*N*is shown in Fig. 2. It is clear that the Helmholtz free energy is an increasing function of

*N*. As before, we can see that by increasing number of components, value of the Helmholtz free energy is increased. We also find that the Helmholtz free energy is a decreasing function of \(\alpha \).

*S*can now be calculated from the Helmholtz free energy,

*N*, and decreasing function of \(\alpha \). In the Fig. 4, we can see typical behavior of the internal energy with respect to the temperature. We can see a minimum of energy at low temperature, and large energy at high temperature. However, such minimum has negative internal energy, and negative entropy (Fig. 3b). Hence, we can see negative entropy and internal energy below a critical temperature (\(T_{c}\approx 1\) with fixed parameters as given by figures), which may be sign of thermodynamical instability.

*P*and chemical potential \(\mu \) as follows,

*T*,

*N*, \(\alpha \) and

*V*. In the Fig. 5a we can see that chemical potential is increasing function of the temperature. Also, increasing number of component increases value of the chemical potential. From the Fig. 5b we can see that chemical potential is decreasing function of

*N*. Figure 5c shows that chemical potential is linearly decreasing function of

*N*. Finally, Fig. 5d shows that chemical potential is decreasing function of

*V*.

*N*galaxies can be written as

*z*is the activity.

*x*, in terms of the number of galaxies as

*b*is given by

## 5 Conclusion and discussion

In this paper, we have studied the clustering of a system of galaxies interacting thought a MOG modified Newtonian potential. As it is possible for the system of galaxies to have different masses, we have analyzed this system using a multi-component systems. This MOG modified Newtonian potential can be obtained from the weak field approximation of MOG, and we have used it for calculating the partition function of this multi-component system. We compute the partition function, and studied the thermodynamics of this system using that partition function. We also analyzed the general clustering parameter for this multi-component system of galaxies interacting though MOG.

Indeed we have thermodynamical study of clustering of the multi-component systems of galaxies in modified gravity to see how MOG (also number of components) affect thermodynamics quantities. We have shown that clustering parameter decreased value of most important thermodynamics quantities, while number of components increase value of thermodynamics variables. Helmholtz free energy for multi-component system of galaxies is evaluated and the variation Helmholtz free energy *F* with temperature depends on the value of \(N_l\). When \(N_l = 3\), the *F* is negative for all values of temperature. For \(N_l = 5\) the Helmholtz free has high positive values and keeps increasing for higher temperatures. For \(N_l = 4\), *F* has positive values with a minima and a maxima, we notice that changing the values of *l* the maxima or peak shifts upwards with increasing values of *l*. We also study the variation of free energy as a function of *N*, which increases with the value of *N*, and by increasing the number of components the free energy curve shifts upwards. The entropy of multi-component system of galaxy is also studied, it is seen that it has negative values for low values of temperature, and further increasing the temperatures the entropy becomes positive and keeps increasing. The variation of entropy with *N* shows that the value of entropy increases initially as a function of *N*, and then decreases on further increasing the value of *N*. We also see that in entropy versus *N* plot, increasing the value of *l* from 3 to 6, and the over all entropy curve is shifted upwards. To check the dependency of entropy on the parameters \(\alpha \), we plot *S* as a function of \(\alpha \) and notice that *S* decreases linearly with increasing value of \(\alpha \). The study of the internal energy of multi-component system of galaxies shows that it depends on the the multi-component clustering parameter, \(B_l\). The behavior of internal energy with respect to temperature is studied, and it is seen that at low temperature the internal energy has a minima. However, as the temperature increases further it increases and takes large values. Chemical potential plays very important role in clustering of galaxies, and it depends on the temperature, \(\alpha \), *N* and volume, *V*. The variation of chemical potential with respect to temperature shows that as temperature increases the chemical potential increases and for higher values *l*, the chemical potential increases rapidly. With *N* the chemical potential initially drops rapidly for small values of *N*, and remains almost constant for higher values of *N*. Furthermore, we notice that, the rate at which \(\mu \) changes for large values of *l* is slower in comparison to small values of *l*. The chemical potential decreases linearly as the value of \(\alpha \) increases. The chemical potential decreases logarithmically as the volume of the multi-component system increases. We found that distribution function is increasing function of clustering parameter as well as temperature, while is decreasing function of numbers.

## References

- 1.V.C. Rubin, E.M. Burbidge, G.R. Burbidge, K.H. Prendergast, Astrophys. J
**141**, 885 (1965)ADSCrossRefGoogle Scholar - 2.V.C. Rubin, W.K. Ford Jr., Astrophys. J
**159**, 379 (1970)ADSCrossRefGoogle Scholar - 3.D. Hooper, Phys. Dark Univ.
**15**, 53 (2017)CrossRefGoogle Scholar - 4.J. Sadeghi, H. Saadat, B. Pourhassan, Chaos Solitons Fractals
**42**, 1080 (2009)ADSCrossRefGoogle Scholar - 5.G.-C. Liua, K.-W. Ng, Phys. Dark Univ.
**16**, 22 (2017)CrossRefGoogle Scholar - 6.S. Clesse, J.G. Bellido, Phys. Dark Univ.
**15**, 142 (2017)CrossRefGoogle Scholar - 7.T. Asaka, S. Blanchet, M. Shaposhnikov, Phys. Lett. B
**631**, 151 (2005)ADSCrossRefGoogle Scholar - 8.E. Carquin, M.A. Diaz, G.A. Gomez-Vargas, B. Panes, N. Viaux, Phys. Dark Univ.
**11**, 1 (2016)CrossRefGoogle Scholar - 9.M. Milgrom, Astrophys. J.
**270**, 365 (1983)ADSCrossRefGoogle Scholar - 10.S. Dodelson, Int. J. Mod. Phys. D
**20**, 2749 (2011)ADSCrossRefGoogle Scholar - 11.L.E. Strigari, Phys. Rep.
**531**, 1 (2013)ADSCrossRefGoogle Scholar - 12.M.H. Chan, Phys. Rev. D
**88**(10), 103501 (2013)ADSCrossRefGoogle Scholar - 13.J.W. Moffat, V.T. Toth, arXiv:1112.4386 [astro-ph.CO]
- 14.A.O. Hodson, H. Zhao, arXiv:1703.10219 [astro-ph.GA]
- 15.J.W. Moffat, JCAP
**0603**, 004 (2006)ADSCrossRefGoogle Scholar - 16.J.W. Moffat, V.T. Toth, Class. Quantum Gravity
**26**, 085002 (2009)ADSCrossRefGoogle Scholar - 17.J.R. Brownstein, J.W. Moffat, Astrophys. J
**636**, 721 (2006)ADSCrossRefGoogle Scholar - 18.J. R. Brownstein, Ph.D. Thesis, University of Waterloo (2009)Google Scholar
- 19.J.W. Moffat, V.T. Toth, Astrophys. J
**680**, 1158 (2008)ADSCrossRefGoogle Scholar - 20.J.R. Brownstein, J.W. Moffat, Mon. Not. R. Astron. Soc.
**367**, 527 (2006)ADSCrossRefGoogle Scholar - 21.J.R. Brownstein, J.W. Moffat, Mon. Not. R. Astron. Soc.
**382**, 29 (2007)ADSCrossRefGoogle Scholar - 22.J.W. Moffat, S. Rahvar, Mon. Not. R. Astron. Soc
**441**, 3724 (2014)ADSCrossRefGoogle Scholar - 23.J.W. Moffat, S. Rahvar, Mon. Not. R. Astron. Soc.
**436**, 1439 (2013)ADSCrossRefGoogle Scholar - 24.J.R. Mureika, J.W. Moffat, M. Faizal, Phys. Lett. B
**757**, 528 (2016)ADSCrossRefGoogle Scholar - 25.W.C. Saslaw,
*Gravitational Physics of Stellar and Galactic Systems*(Cambridge University Press, Cambridge, 1985)CrossRefGoogle Scholar - 26.F. Ahmad, M. Hameeda, Astrophys. Sp. Sci.
**330**, 227 (2010)ADSCrossRefGoogle Scholar - 27.W.C. Saslaw, A.J.S. Hamilton, Astrophys. J
**276**, 13 (1984)ADSCrossRefGoogle Scholar - 28.F. Ahmad, W.C. Saslaw, M.A. Malik, Astrophys. J
**645**, 940 (2006)ADSCrossRefGoogle Scholar - 29.F. Ahmad, M.A. Malik, M. Hameeda, Astrophys. Sp. Sci.
**343**, 763 (2013)ADSCrossRefGoogle Scholar - 30.F. Ahmad, W.C. Saslaw, N.I. Bhat, Astrophys. J
**571**, 576 (2002)ADSCrossRefGoogle Scholar - 31.W.C. Saslaw,
*The Distribution of the Galaxies Gravitational Clustering in Cosmology*(Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar - 32.F. Ahmad, M.A. Malik, S. Masood, Int. J. Mod. Phys. D
**15**, 1267 (2006)ADSCrossRefGoogle Scholar - 33.M.A. Malik, R.N. Ali, F. Ahmad, Astrophys. Sp. Sci.
**336**, 447 (2011)ADSCrossRefGoogle Scholar - 34.M.A. Malik, F. Ahmad, S. Ahmad, S. Masood, Int. J. Mod. Phys. D
**18**, 959 (2009)ADSCrossRefGoogle Scholar - 35.L. Randall, R. Sundrum, Phys. Rev. Lett.
**83**, 3370 (1999)ADSMathSciNetCrossRefGoogle Scholar - 36.M. Hameeda, M. Faizal, A.F. Ali, Gen. Rel. Grav.
**48**, 47 (2016)ADSCrossRefGoogle Scholar - 37.M. Hameeda, S. Upadhyay, M. Faizal, A.F. Ali, Mon. Not. R. Astron. Soc.
**463**, 3699 (2016)ADSCrossRefGoogle Scholar - 38.B. Pourhassan, S. Upadhyay, M. Hameeda, M. Faizal, Mon. Not. R. Astron. Soc.
**468**, 3166 (2017)ADSCrossRefGoogle Scholar

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