Galactic Clustering Under Power-law Modified Newtonian Potential

Since last decade, the substantial progresses have been made in the understandings of galaxy clusters, from their internal structure and evolution to their place in the large scale structure of the universe. All these progresses are due to the stupendous improvements in the theoretical modeling and numerical simulation, viz-a-viz abundance of new information provided by multi-wavelength surveys of the universe. Various theories of cosmological many-body distribution function have been developed from the thermodynamic point of view. Benkestien [1], Hawking [2] and Unruh [3] originated the relation between relativity and thermodynamics. Later, Jacobson [4] introduced the Einstein equation as thermodynamic equation of state. In fact, the Einstein equation is derived from the proportionality of entropy and the horizon area together with the fundamental relation $\delta Q=TdS$, where $\delta Q$ and $T$ are interpreted as the energy flux and Unruh temperature. Recently Verlinde [5] proposed an entropic origin of gravity and interpreted gravity as an entropic force. In this regard, it is argued that the central notion needed to derive gravity is information. Employing the holographic principle and equipartition law of energy, Newton’s law of gravitation, Poisson’s equation and Einstein’s field equations are successfully obtained. Verlinde had used area-entropy relation of black holes in Einstein’s gravity (i.e. $S=\frac{A}{4l_{p}^{2}}$) to get the Newtons law, where $S,A$ and $l_{P}$ refer the entropy of the black hole, the area of horizon and the Planck length, respectively.

The study of modified gravity on the cosmological scales is an active area of present research. Recently, the corrected gravitational potential plays a vital role in estimation of the total mass of a sample of 12 clusters of galaxies which provides a better fit to the mass of visible matter [6]. At large distances, the modification in Newtonian potential occurs due to the propagation of gravity into the bulk [7]. Recently, Sheykhi and Hendi studied the effect of power-law corrections in entropy to the Newton’s law [8]. These corrections appear due to entanglement of quantum field inside and outside of horizon [9]. It is also found that a viable source of black hole entropy is quantum entanglement of degrees of freedom inside and outside the horizon. Also, the black hole entropy is found directly proportional to the surface area of the sphere when the field is in the ground state. But when the field is in a superposition of ground and excited states, a correction term proportional to a fractional power of area appears. These corrections are negligible for large horizon areas. Another quantum correction to Newton’s law, a logarithmic correction, has also been studied which appears due to the result of thermal equilibrium fluctuations and the quantum fluctuations of loop quantum gravity [10, 11, 12]. Recently, the effect of this logarithmic correction on the galaxy clustering has also been studied  [13]. The clustering of galaxies has been studied under the modified gravity under various potentials [14, 15, 16, 17, 18, 19, 20, 21].

The power-law corrected entropy leads to modification in Newton’s gravitational potential by adopting the viewpoint of gravity as an entropic force. We consider a power-law corrected entropy of the following form  [9, 22]:

$$S=\dfrac{A}{4l_{p}^{2}}[1-K_{\alpha}A^{1-\frac{\alpha}{2}}],$$

(1)

where $\alpha$ is a dimensionless constant whose value is not confirmed yet and the parameter $K_{\alpha}$ that depends on the power $\alpha$ of the entropy correction as following:

$$K_{\alpha}=\frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{(4-\alpha)r_{c}^{2-\alpha}},$$

here $r_{c}$ denotes crossover scale. The Boltzmann constant is set unit here. The last term of equation (1) appears due to the superposition of ground state and excited state wave function of the field. Interestingly, Verlinde [5] proposed an entropic origin of gravity and interpreted gravity as an entropic force. The gravity derived here was from the notion of information associated with matter and its location measured in terms of entropy. The Newton’s law of gravitational force $F$ is related to the entropy $S$ of the system as [8]

$\displaystyle F=-4l_{p}^{2}\frac{GM^{2}}{r^{2}}\frac{\partial S}{\partial A}.$

(2)

For the power-law corrected entropy given in (1), this relation leads to the following modified Newton’s force [8]:

$$F=-\frac{GM^{2}}{r^{2}}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{r}\right)^{\alpha-2}\right],$$

(3)

which coincides with the original Newtons law when $\alpha$ is set to zero. Keeping in view the attractive nature of gravity, we should have $F<0$. This requires,

$$\alpha<2\left(\frac{r}{r_{c}}\right)^{\alpha-2}.$$

In this paper, we study the effect of power-law corrected Newtonian potential on the clustering of galaxies. We also consider the effect of dark matter on the formation of galaxy clusters via the incorporation of cosmological constant in the Newtonian potential. This is because of the role played by cosmological constant $\Lambda$ in the expansion of universe  [5]. In order to study the effects of all the corrections made in gravitational potential in galaxy clustering, we first derive the $N$-body partition function by evaluating configuration integrals recursively. The resulting partition function is employed to calculate the various thermodynamic quantities viz the Helmholtz free energy, entropy, pressure, internal energy, and chemical potential which possess deviations from their original values due to the incorporation of corrections. Remarkably, a modified clustering parameter emerges naturally from the corrected equations of state. Furthermore, we derive the probability distribution function assuming that the system is in a quasi equilibrium state as described by the grand canonical ensemble. The expression of distribution function embeds modified clustering parameter. A comparative analysis is made with the original distribution function to study the deviation due to corrections.

The paper is organized as following. In section 2, we consider a power-law and cosmological constant modified gravitational potential to calculated partition function of the system of galaxies and galaxy clusters. With the help of resulting partition function, the various thermodynamical equation of states are calculated in section 3. The study the distribution of galaxies and galaxy’s clusters under the modified gravitational potential, we estimate distribution function in section 4. Finally, we draw concluding remarks in the last section.

In this section, we derive a modified Newtonian potential due to power-law corrected force and estimate the partition function.

In this section, we derive various more exact equations of states for the system of galaxies interacting under modified potential. More precisely, we derive the Helmholtz free energy, entropy, internal energy, pressure and chemical potential. We also emphasize the effect of corrections in Newton’s potential on these equation of states.

In order to find the probability distribution function $F(N)$, which contains void distribution as well as statistics of counts of the number of galaxies in cells throughout the system. For the system of galaxies, wherein galaxies as well as energy can cross the cell boundary, one has to estimate the grand canonical partition function. The grand canonical partition function, a weighted sum of all canonical partition functions, is defined by

$$Z_{G}(T,V,z)=\sum_{N=0}^{\infty}e^{\frac{N\mu}{T}}Z_{N}(T,V),$$

(28)

where $z$ is the fugacity. The grand partition function for the system of galaxies is expressed in terms of thermodynamic variables as

$$\ln Z_{G}=\frac{PV}{T}=N(1-b_{\star}).$$

(29)

The probability distribution function $F(N)$ for finding $N$ galaxies in a cell of volume $V$ and energy $U(N,V)$ is defined by

$$F(N)=\frac{\sum_{i}e^{\frac{N\mu}{T}}e^{\frac{U_{i}}{T}}}{Z_{G}(T,V,z)}=\frac{e^{\frac{N\mu}{T}}Z_{N}(V,T)}{Z_{G}(T,V,z)}.$$

(30)

Making use of Eqs. (16), (27) and (29), the distribution function for the extended mass galaxies under modified potential is estimated as

$$F(N)=\frac{\bar{N}}{N!}(1-b_{\star})[\bar{N}(1-b_{\star})+Nb_{\star}]^{N-1}e^{-Nb_{\star}-\bar{N}(1-b_{\star})}.$$

(31)

This distribution function is structurally similar to those derived originally by Saslaw and Hamilton  [27] from thermodynamic point of view and by Ahmad and Saslaw  [24] from statistical point of view. Also the distribution function derived from logarithmic and volume corrected Newtonian potential  [13] has the same general structure.

The behavior of distribution function $F(N)$ versus $N$ can be seen from the comparative analysis as given in Fig. 9. The presence of correction term decreases the peak value of distribution function which occurs for system of small number of galaxies. However, for the system of large number of galaxies the correction terms increase the distribution function.

We have presented a study of galaxy clustering under the modified Newton’s law. These modifications to Newton’s law incorporate a power-law entropic corrections along with the inclusion of cosmological constant $\Lambda$ term, in order to take into account the effect of quantum entanglement (a possible source of black hole entropy) and dark energy respectively. Utilizing the modified Newtonian potential, we have derived the corresponding canonical partition function for the system of extended mass galaxies with the assumption that the system is made of $N$ equal volume cells and average particle density $\bar{\rho}$, which are statistically homogeneous over large regions. In this regard, we have used of softening parameter $\epsilon$ in the first term of Hamiltonian to get rid of the divergence of the Hamiltonian when galaxies are considered as point like. This justifies that the extended nature (finite size) of galaxies (or galaxies with halos). From the resulting partition function, we have calculated various important thermodynamic equations of state. Namely, these are free energy, entropy, internal energy, pressure and chemical potential. The exact expressions of equations of state contain a corrected correlation (clustering) parameter which emerges naturally for the clusters of the galaxies with halos. The new clustering parameter $b_{\star}$ reduces to the original parameter $b_{\epsilon}$ when $\gamma=0$ and $\beta=0$. Moreover, the distribution function is modified due to correction in potential but has the similar structure as the original one.