The large scale structure formation in an expanding universe
Mir Hameeda, Behnam Pourhassan, Syed Masood, Mir Faizal, Li-Gang Wang, Shohaib Abass
aa r X i v : . [ phy s i c s . g e n - ph ] J un MNRAS , 000–000 (0000) Preprint 8 July 2020 Compiled using MNRAS L A TEX style file v3.0
The large scale structure formation in an expandinguniverse
M. Hameeda , ⋆ , B. Pourhassan , † , S. Masood ‡ , Mir Faizal , , § , L-G. Wang ¶ , S. Abass k Department of Physics, S.P. Collage, Srinagar, Kashmir, 190001 India Inter University Centre for Astronomy and Astrophysics , Pune India School of Physics, Damghan University, Damghan, 3671641167, Iran Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China Irving K. Barber School of Arts and Sciences, University of British Columbia, Kelowna, British Columbia, V1V 1V7, Canada Department of Physics and Astronomy, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada Department of Physics, Central University of Kashmir, Tulmulla Campus Ganderbal, 191131
ABSTRACT
In this paper, we will analyze the effects of expansion on the large scale structureformation in our universe. This will be done by incorporating a cosmological constantterm in the gravitational partition function. This gravitational partition function witha cosmological constant would be used for analyzing the thermodynamics for thissystem. We will analyze the viral expansion for this system, and obtain its equationof state. It is observed that the equation of state is the Van der Waals equation. Wealso analyze a gravitational phase transition in this system. This will be done usingthe mean field theory for this system. We construct the cosmic energy equation forthis system of galaxies, and compare it with observational data. We also analyze thedistribution function for this system, and compare it with the observational data.
Key words:
Dark energy, Thermodynamics and Statistics, Cluster of Galaxies.
The clustering of galaxies in our universe is responsiblefor the formation of larger scale structure in our universe(Peebles (1980); Peebles (1993); Voit (2005)). So, it is veryimportant to analyze the clustering of galaxies, and thiscan be done using a gravitation partition function (Saslaw(1986); Saslaw et al., (1990)). In this gravitational partitionfunction, the galaxies are approximated as point particles,as the size of galaxies is much smaller than the distancebetween them. It is possible to use this gravitational par-tition function for analyzing the clustering of galaxies. Asthe extended structure of the galaxies is approximated bya point particle, it is possible for this gravitational parti-tion function to diverge. However, these divergences in thegravitational partition function can be removed by usinga softening parameter (Ahmad and Hameeda (2010)). Thissoftening parameter incorporates the extended structure of ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ¶ E-mail: [email protected] k E-mail: [email protected] galaxies in the gravitational partition function. This soft-ening parameter modifies the thermodynamic fluctuationsin this system of galaxies. The thermodynamics for thissystem of galaxies can also be analyzed using this grav-itational partition function (Saslaw and Hamilton (1984)).The clustering of a system of galaxies has also been stud-ied using the grand canonical gravitational partition func-tions (Ahmad et al., (2002)). This has been done by ana-lyzing the distribution functions and moments of distribu-tions such as their skewness and kurtosis for such a sys-tem of galaxies. The distribution function for galaxies hasbeen studied for a wide range of samples, and it has beenobserved that galaxy clusters are surrounded by individualhalos (Sivakoff and Saslaw (2005); Rahmani et al., (2009)).The grand canonical partition function has also been usedto analyze the thermodynamics for a system of galaxies, andobtain the specific heats and isothermal compressibility forsuch a system (Ahmad et al., (2006)).It may be noted that the clustering occurs due to grav-itational interaction between different galaxies. However, asthe gravity pulls the galaxies towards each other (causingclustering), the expansion of the universe is also expected tomove the galaxies away from each other. It is now knownthat the universe is undergoing accelerated expansion. This c (cid:13) M. Hameeda et al. is based on observation of Type Ia Supernovae (SNeIa)(Riess et al., (1998); Perlmutter et al., (1998)). Thus, it isimportant to consider the effects of this expansion on thelarge scale structure formation in our universe. This can bedone by incorporating a cosmological constant term in thegravitational partition function (Hameeda et al., (2016a)).This gravitational partition function has been used to ob-tain the Helmholtz free energy for this system of galaxies.This Helmholtz free energy has in turn been used to ob-tain the entropy of this system. The thermodynamics of thissystem is used to obtain the the clustering parameter forthis system, and analyze the effect of the cosmological con-stant on the clustering of galaxies. As the internal energyof this system depends on the clustering parameter, whichdepends on the cosmological constant, the dependence ofthe internal energy on the cosmological constant has alsobeen studied for this system. Finally, the distribution func-tion for a system of galaxies in an expanding universe isobtained using the grand canonical gravitational partition.The gravitational partition function with a time dependentcosmological constant has also been constructed, and it hasbeen used to study the effects dynamical dark energy onthe structure formation in our universe (Pourhassan et al.,(2017)). Thus, for such a dynamical dark energy model theHelmholtz free energy is used to obtain the entropy of thesystem, which in turn is used to obtain the dependence ofthe clustering parameter on the dynamical dark energy. Thecorrelation function between galaxies has also been calcu-lated and observed to be consistent with observations.It is possible to generalize general relativity by addinghigher powers of the curvature tensor to f ( R ) grav-ity, and this modification of general relativity also modi-fies the large distance behavior of gravitational potential(Sotiriou and Faraoni (2010)). It may be noted as f ( R ) grav-ity can also be used to study the expansion of the uni-verse (Sotiriou and Faraoni (2010)), the gravitational parti-tion function for f ( R ) gravity has been used to analyze theeffects of f ( R ) gravity on the large scale structure formation(Capozziello et al., (2018)). It was observed that this mod-ification the gravitational partition function is consistentwith observations. The thermodynamics of such a system ofgalaxies, interacting through the modified gravitational po-tential of f ( R ) gravity, has also been studied. It has also beendemonstrated that f ( R ) gravity can be constrained usingthe PLANCK data on galaxy clusters (De Martino et al.,(2014)). This was done by calculating the pressure profilesof different galaxy clusters. It was assumed that this gasof galaxies was in hydrostatic equilibrium within the f ( R )gravitational potential well. It was observed that the profileof this system of galaxies fits the observation data withoutrequiring dark matter. The thermodynamics of a system ofsystem of galaxies has also been analyzed using a MOND,and this was done by analyze the modifications to the grav-itational partition function from MOND (Upadhyay et al.,(2018)). It was observed that the modification of the grav-itational partition function from MOND, also modified thethermodynamic potential for this system, which in turnmodified the formation of large scale structure. The mod-ification to the gravitational partition function from MOGhas also been studied, and it was observed that the cluster-ing of galaxies depends on the large scale modifications tothe Newtonian potential in MOG (Hameeda et al., (2019)). This was done by analyzing the thermodynamics of a sys-tem of galaxies interacting through MOG Newtonian poten-tial (Hameeda et al., (2019)). The clustering in brane worldmodels has also been studied using a modification to grav-itational partition function (Hameeda et al., (2016b)). Thiswas done by analyzing the modification to the Newton’spotential from super-light brane world perturbative modes.These modified Newtonian potential modified the thermo-dynamics of the system, and this changes the large scalestructure formation. Thus, it was possible to analyze the ef-fects of super-light brane world perturbative modes on thelarge scale structure formation in our universe.It may be noted that it is also possible to study thegravitational phase transition for a system of galaxies usinggravitational partition function (Khan and Malik (2012)). Itwas observed that a first order phase transition occurs whenthis system of galaxies starts to cluster from an initial ho-mogeneous phase, to a phase with large scale structure. Thephase transition in a system of galaxies has also been an-alyzed using the gravitational partition function as a func-tion of complex fugacity (Khan and Malik (2013)). This wasdone by extending the Yang-Lee theory to the gravitationalphase transition. It was observed that masses of individualgalaxies can have an effect on the formation of large scalestructure in our universe. Now it is important to considerthe effects of the expansion of universe in the structure for-mation in our universe, we will analyze the gravitationalphase transition for a system of galaxies, with a cosmolog-ical constant term. We would like to point out that it ispossible to study the clustering in a system of galaxies usingcosmic energy equation (Wahid et al., (2011)). Even thoughthe cosmic energy equation is obtained by assuming galaxiesas point particles (Wahid et al., (2011)), the modification tothe cosmic energy equation from the extended structure ofgalaxies has also been studied (Ahmad et al., (2009)). Thecosmic energy equation can be used for analyzing the depen-dence of clustering on the gravitational potential. In fact, theeffects of a large distance modification to gravitational po-tential on clustering has also been analyzed using modifiedcosmic energy equation (Hameeda et al., (2018)). As it isimportant to consider the modification of the gravitationalpartition function from the cosmological constant, we willalso analyze the cosmic energy equation for a gravitationalpartition function with a cosmological constant term. fromthe cosmological constant, we will also analyze the cosmicenergy equation for a gravitational partition function witha cosmological constant term. We consider a large number galaxies distributed in an en-semble of cells, all of the same volume V , or radius R andaverage density ρ . Both the number of galaxies and their to-tal energies vary among these cells and hence can be appro-priately represented by a grand canonical ensemble. Thesegalaxies within the system have pairwise gravitational inter-action generated by the modified Newtonian potential. It isfurther assumed that the distribution is statistically homo-geneous over the large regions. Meanwhile, we mention thatwe perform our calculations in units of the Boltzmann andPlanck constants. MNRAS , 000–000 (0000) he large scale structure formation in an expanding universe The general partition function of a system of N galaxiesof mass m interacting through the modified gravitationalpotential energy Φ nl can be written as (Pourhassan et al.,(2017); Capozziello et al., (2018); Upadhyay et al., (2018);Hameeda et al., (2019)), Z ( T, V ) = 1Λ N N ! Z d N pd N r × exp (cid:18) − (cid:20)P Ni =1 p i m + Φ( r , r , r , t . . . , r N ) (cid:21) T (cid:19) , (1)where Λ ≡ √ πmT is the mean thermal wavelength, and p i is the momentum for different galaxies and T is the averagetemperature. Integrating the momentum space, we get Z N ( T, V ) = 1 N ! (cid:0) πmT Λ (cid:1) N/ Q N ( T, V ) , (2)where Q N ( T, V ) is the configurational integral of the system,given by Q N ( T, V ) = Z .... Z Y i A typical behavior of the potential energy (solid redline) and the corresponding Mayer function (dashed blue line)versus r ij for m = G = T = ǫ = 1 and ¨ a a − Λ6 = 1. In the weak interaction approximation, only first term isretained f ij = (cid:0) Gm T ( r ij + ǫ ) / + m Λ r ij T − m ¨ ar ij aT (cid:1) (9)In the Fig. 1 we can see typical behavior of potential energy(7) and Mayer function (8) in terms of r ij . It is illustrated bythe blue dashed line that f ij is bounded at all regions whichyields minus one at infinity and has finite positive value atorigin. It is observed that both Φ and f ij are zero at thesingle point as expected.for N = 2, we get Q ( T, V ) = 4 πV (cid:18)Z R r dr (cid:19) + 4 πV (cid:18) Gm T Z R r dr ( r + ǫ ) / (cid:19) + 4 πV (cid:18)(cid:0) Λ m T − m ¨ a a (cid:1) Z R r dr (cid:19) (10)Solving the above integral leads us to Q ( T, V ) = V (cid:0) αx (cid:1) , (11)where α = s ǫ R + ǫ R ln (cid:16) ǫR + p R + ǫ (cid:17) + 2 R Gm (cid:16) Λ6 − ¨ a a (cid:17) , (12)and 3 Gm R T = 3 Gm ρ − / T = 32 Gm ρ / T − (13)and using scale invariance, ρ → λ − ρ , T → λ − T and r → λr , we obtain32 ( Gm ) ρT − = βρT − = x, (14)where β = ( Gm ) . Following the same procedure, we getgeneral configurational integral as Q N ( T, V ) = V N (cid:0) αx (cid:1) ( N − , (15) MNRAS , 000–000 (0000) M. Hameeda et al. Figure 2. A typical behavior of the partition function per unitvolume versus a for m = G = T = R = Λ = ¨ a = 1 and N = 50. Using above equation, the gravitational partition functionfor a system of galaxies can be written as Z N ( T, V ) = 1 N ! (cid:0) πmT (cid:1) N V N (cid:0) αx (cid:1) N − (16)The term α shows the impact of dark matter and dark en-ergy, and the cosmological expansion on the partition func-tion. In the Fig. 2, we can see effect of scale factor on thepartition function. We note that presence of scale factor in-creases the value of the partition function density. The gravitational partition function gotten in the previoussection can help us to understand the thermodynamic be-havior of this system of interacting galaxies in an expandinguniverse. We can also derive general form of equation interms of the scale factor a followed by assuming a specificform of power law scale factor.Lets begin with the internal energy of this galaxy cluster,which is given by U = T d ln Z N dT , (17)and yields the following expression (assuming m = G = 1), U = 3 T (cid:2) aR − aU N (cid:3) aR − aU d , (18)where U N = 10 NR T + Λ R + 15 R q R + ǫ + 15 ǫ ln ( ǫR + p R + ǫ ) ,U d = 10 R T + Λ R + 15 R q R + ǫ + 15 ǫ ln ( ǫR + p R + ǫ ) . (19)We can also study Helmholtz free energy via the followinggeneral formula, F = − T ln Z N . (20) Figure 3. Typical behavior of the (a) internal energy and (b)Helmholtz free energy per unit volume versus a for m = G = T = R = Λ = ¨ a = 1 and N = 100. Interestingly, in the case of N = 1 we find U = 3 T whichis internal energy of 3-dimensional harmonic oscillator (inunits of Boltzmann constant) which indicates that it is inagreement with equipartition theorem. In the Fig. 3 (a) westudy behavior of the internal energy U against scale factor a . We can see a singular point for the small scale factor.The internal energy is initially negative which yields to apositive value after a phase transition. This phase transitionis corresponding to the maximum of Helmholtz free energy(see Fig. 3 (b)) which is indeed a divergent point at initialstage. We can see from the Fig. 3 (b) that the Helmholtzfree energy yields to its minimum at the late time which isa clear indication that the resulting configuration is stable.It means that increasing scale factor increases stability ofclustering of galaxies.The negative value of the internal energy at the smallscale factor or singular point of initial stage is related tothe separation distance of galaxies. It means that there is aminimum value for R where the system will be stable, oth-erwise system has tendency to collapse due to gravitationalpull. Let’s substantiate this point by analyzing entropy S ofsystem obtained via the following relation, S = UT + ln Z N . (21)In the Fig. 4 we can see typical behavior of the clusterentropy in terms of R for various values of the scale factor a . It is pretty much clear that the entropy is positive as MNRAS , 000–000 (0000) he large scale structure formation in an expanding universe Figure 4. Typical behavior of the entropy per unit volume versus R for m = G = T = Λ = ¨ a = 1, ǫ = 0 . N = 100. Figure 5. Typical behavior of the specific heat per unit volumeversus a for m = G = T = R = Λ = ¨ a = 1 and N = 100. expected. However, for the smaller values of R , entropy isdecreasing function of time. We define a critical value of R as R c ≈ . 8, such that R > R c indicating that the entropy isan increasing function of time, in accordance with the secondlaw of thermodynamics. The system then goes to the stablephase which is attributed to the accelerating expansion ofthe universe. It is further confirmed by the heat capacityanalysis. The heat capacity in constant volume C V of thissystem reads as C V = (cid:18) dUdT (cid:19) V . (22)Behavior of the specific heat in constant volume given bythe Fig. 5. We can see that the specific heat is negative atthe early stage which confirms our previous point. Also, wecan see that the heat capacity rises to a maximum, which islike a Schottky anomaly (appears in some of the two-levelsystems). In what follows, we discuss the special cases bychoosing specific time-dependence of scale factor. We consider the special case of power law scale factorgiven by a = a t n , (23)where a is a constant and n is a real number. In aFriedmann-Lemaˆıtre-Robertson-Walker universe, the scalefactor a gives us the value of n , so that in the radiation-dominated universe, n = , while in a matter-dominateduniverse, n = . In those cases, one can obtain Hubble ex-pansion parameter as H ∝ t . On the other hand, holo-graphic dark energy models suggest that Hubble parameter H is proportional to the length, H ∝ R . Under this as-sumption, equation (23) changes to a = a R n , (24)where a is another constant. Hence we have,¨ a = n ( n − R a. (25)In that case the equation (12) reduced to α = s ǫ R + ǫ R ln ( ǫR + p R + ǫ )+ 2 R − n ( n − R ) . (26)where we assumed G = m = 1. In the plots of the Fig. 6 wecan see behavior of α for various values of n and R . Fromthe equation (26), we can find that dαdn = 0 yields n = .Hence, maximum value of α is obtained in the radiation-dominated universe as illustrated in Fig. 6 (a). Also, in theFig. 6 (b), we can see that α parameter is enhanced by in-creasing R .The maximum of α corresponds to minimum of theHelmholtz free energy. It is found by analyzing the equa-tion (20) which is illustrated in Fig. 7. It is clear from theFig. 7 (a) that minimum of the Helmholtz free energy corre-sponds to n = , while from the Fig. 7 (b), we can see thatmaximum of the Helmholtz free energy is obtained by thelarger R . It is possible to study virial expansion for this system ofgalaxies interacting through a gravitational potential in anexpanding universe. This viral expansion can be used to ob-tain the equation of state for this system. Thus, for the caseof large galaxy clustering (in the limit V → ∞ ) with fugac-ity z = e µT ( µ is the chemical potential of the given system),one can write PT = 1Λ ∞ X ν =1 I ν z ν , (27)and we can write N/V as NV = 1Λ ∞ X ν =1 νI ν z ν . (28)It may be noted that here I ν is given by I ν = 1 ν !Λ ν − V Z ν X ij = kl f ij f kl + ν Y i = j f ij d r · · · d r ν . (29) MNRAS , 000–000 (0000) M. Hameeda et al. Figure 6. Typical behavior of α for m = G = Λ = 1. (a) Interms of n for R = 1; (b) In terms of R for ǫ = 0 . Figure 7. Typical behavior of F for m = G = Λ = 1. (a) Interms of n for R = 1; (b) In terms of R for n = 0 . is the clustering integral which is a dimensionless parameter.It is easy to find that I = 1. Moreover, in the case of ν = 3we have, I = 16Λ V Z f d r d r d r , (30)where f ≡ f f + f f + f f + f f f (31)Also, in the simplest case of ν = 2, one can obtain I = m Z R (cid:18) GmT ( r + ǫ ) / + Λ r T − ¨ ar aT (cid:19) r dr = Gm T Λ (cid:18) R q R + ǫ − ǫ ln ( R + q R + ǫ ) + ǫ ln ǫ (cid:19) = m T Λ (cid:18) R − ¨ aa ) (cid:19) (32)Eliminating fugacity z between the equations (27) and (28),one can obtain the clustering equation of state. It yieldsfollowing virial expansion P VNT = ∞ X ν =1 c ν ( T ) (cid:18) Λ NV (cid:19) ν − , (33)where c ν ( T ) is called virial coefficient. In case of ν = 1, weobtain first virial coefficient c = I = 1, hence the equation(33) reduces to the ideal gas equation of state. Other virialcoefficients can also be expressed in terms of the clusteringintegral. For example, one can obtain c = − I ,c = 4 I − I ,c = − I + 18 I I − I . (34)In Fig. 8, we see the behavior of the second virial coefficient c for the model parameters. In Fig. 8 (a), effect of the soft-ening parameter ǫ on different values of the scale factor a isshown. It is observed that the c is an increasing function of ǫ with both positive and negative values allowed, dependingupon scale factor a . For an infinitesimal ǫ , c approximatelyconstant. Fig. 8 (b) depicts that c increases as the sepa-ration between galaxies increases. In this case also, we findthat infinitesimal R yields negative value for second virialcoefficient. Fig. 8 (c) shows variation of c in terms of scalefactor a . We observe that it is a decreasing function of a which approaches to a constant value for the larger a (latetime behavior). Finally, looking at Fig. 8 (d), we see that c is decreasing function of temperature T . It is observed thatlow T behavior is similar for various values of ǫ .For the first order approximation (reasonably for for V → ∞ ), we obtain P VNT = 1 − (cid:18) Λ NV (cid:19) I + O ( 1 V ) , (35)where I is given by the equation (32). We thus rewrite theequation (32) in the following simple form, I = f ( R )2Λ T , (36)where f ( R ) = Gm (cid:18) R q R + ǫ − ǫ ln ( R + q R + ǫ ) + ǫ ln ǫ (cid:19) + m (cid:18) R − ¨ aa ) (cid:19) . (37) MNRAS , 000–000 (0000) he large scale structure formation in an expanding universe Figure 8. A typical behavior of the second virial coefficient for m = G = Λ = ¨ a = 1. (a) In terms of ǫ for T = R = 1; (b) Interms of R for ǫ = 0 . a = 2; (c) In terms of the a for T = 1and ǫ = 0 . 8; (d) In terms of temperature for a = R = 1. With this new definition, we write the equation of state (35)as (cid:16) P + a v v (cid:17) ( v − b v ) = T (38)where v = VN is the volume per galaxies number, a v = f ( R )2 is strength of the interactions and b v denotes the size ofgalaxies (because we assumed the galaxies as point-like par-ticles, hence b v = 0). Interestingly, Eq. (38) is nothing butthe usual Van der Waals equation of state and is quite helpfulfor studying mean field theory of clustering phase transition. In the previous section, by using viral expansion, we derivedVan der Waals equation of state for a system of interactinggalaxies. In our case, we considered the modified Newtonianpotential taking into account the presence of dark matterand dark energy. Further, we are explored the dependenceof clustering on cosmological scale factor a and studied thepossibility of a phase transition. We now consider the equa-tion (38) for mean field theory of phase transition in thisclustering phenomenon (Huang (1987)).So here we extend previous case to a non-zero b v . We realizethat it is possible to find Landau free energy that can leadto the required equation upon minimizing with respect tothe order parameter. Here we choose v as the order param-eter and P as the conjugate field and we suppose that theequation (38) is obtained by minimizing Landau free energylabeled by ψ ( v, P, T ). Thus ∂ψ∂v = 0 (39) is the same as Eq. (38). Hence we get ψ ( v, P, T ) = P v − a v v − T ln( v − b v ) . (40)By putting Eq. (40) in Eq. (39) gives, P = T ( v − b v ) − a v v , (41)which is exactly same as Eq. (38). The isothermal compress-ibility is β T = − v (cid:18) ∂v∂P (cid:19) T = (1 − b v ) v ( T − a v (1 − b v ) ) . (42)Approaching the critical point from T > v = 1, wewrite the compressibility as β T = (1 − b v ) ( τ + 1 − a v (1 − b v ) ) , (43)where we denoted the critical temperature by T c and intro-duced a dimensionless quantity τ such that τ = T − T c T c . (44)Let us analyze the compressibility in the limit τ → β T as follows β T = (1 − b v ) − a v (1 − b v ) , (45)Thus in the limit τ → 0, this thermodynamic quantityshould have a singular part in addition to the regular part.For the phase transition to take place as τ → β T shouldbe infinite but the Eq. (44) shows that for cosmological con-stant and the extended galaxies, there is no phase transitionat T = T c i.e, τ → τ = 2 a v (1 − b v ) − , (46)therefore, T c ′ = T c (2 a v ((1 − b v ) ) (47)Thus, in our case, critical temperature changes from T c to T c ′ . Corresponding to a mean field theory in the region of afirst order phase transition, Landau free energy ψ must havetwo minima at volume v and v . The two conditions to besatisfied (cid:18) ∂ψ∂v (cid:19) v = v = (cid:18) ∂ψ∂v (cid:19) v = v (48) ψ ( v ) = ψ ( v ) . (49)which lead to the following two equations T (cid:18) v − e − v − e (cid:19) − a v (cid:18) v − v (cid:19) = 0 (50)and P ( v − v ) = Z v v P dv (51)These conditions combined together comprise the Maxwell’sConstruction.The Maxwell’s construction leads to coexisting volumes v MNRAS , 000–000 (0000) M. Hameeda et al. and v , symmetrically placed around the critical volume, say v . Let v = 1 + δ, v = 1 − δ, (52)where δ is a small parameter. Hence, by using the Maxwell’sconstruction and results obtained in the previous section, weget (1 + τ ) (cid:18) 11 + δ − b v − − δ − b v (cid:19) − a v (cid:18) δ ) − − δ ) (cid:19) = 0 (53)as a possible solution of the Eq. (53) is, δ ≈ s (1 + τ ) − a v (1 − b v ) τ − a v ) , (54)where we neglected the terms of O ( δ ). There is a coexistingvolume at phase transition when δ → τ = 2 a v (1 − b v ) − b v = 0, we find τ = 2 a v − f ( R ) − T I − . (56)Hence, under assumption T = m = 1, we get τ = I √ π − , (57)Therefore τ ∝ I . (58)In that case, behavior of τ fairly resembles that of the onerepresented by dotted green line of the Fig. 8 (b). It meansthat τ → R → In this section, we discuss cosmic energy equation which is anessential tool for analyzing galaxy clustering in an expandinguniverse. Originally, it was introduced to demonstrate, howin an expanding universe, a large assembly of pressurelessgalaxies interacting via a Newtonian gravity potential followthe energy conservation rules (Voit (2005); Peebles (1980);Peebles (1993)). Later, it has been studied for galaxies hav-ing extended structures (Ahmad et al., (2009)) and for non-point like masses (Wahid et al., (2011)). A large distance-inspired modified Newtonian potential based galaxy cluster-ing phenomenon has also been done Hameeda et al., (2018)and the corresponding cosmic energy equation has beenfound. In the present work, we extend this analysis for auniverse endowed with dark matter and dark energy, intro-duced in the modified Newtonian potential. For a system ofgalaxies with the internal energy U , pressure P and scalefactor a ( t ), the first law of thermodynamics can be writtenas d ( Ua ) dt + P da dt = 0 . (59) Writing the equations of energy U and pressure P in termsof the potential for cosmological constant, we have U = 32 NT + Nρ Z V Φ( r ) ξ ( r )4 πr dr, (60) P = NTV − ρ Z V r d Φ( r ) dr ξ ( r )4 πr dr, (61)where ρ is density number and ξ ( r ) is the correlation func-tion whose integral over a certain volume is obtained via themean square number fluctuation as Z ξdV = (2 − b ) b (1 − b ) , (62)where we used ∂b∂V = − xV dbdx = − b (1 − b ) V and the equation(14).By making use of Eq.(6), we can write the above parametersas U = 32 NT + W ǫ + W Λ , (63) P = 3 NT + W ǫ + ǫ W ′ ǫ − W M V (64)where W ǫ = − GNρm Z ξ ( r )( r + ǫ ) πr dr, (65) W M = Nρm a a − Λ6 ) Z r ξ ( r )4 πr dr, (66) W ′ ǫ = − GNρm Z ξ ( r )( r + ǫ ) πr dr. (67)Following Refs. (Hameeda et al., (2018); Hameeda et al.,(2016b)), the conservation law for the cosmological constantis d ( K + W ) dt + ˙ aa (2 K + W (1 + η )) = 0 , (68)where K is the kinetic energy and W is the correlation en-ergy. Also η = ǫ W ′ ǫ − W Λ W , (69)where W Λ is the total correlation energy in the presence ofcosmological constant Λ. This is the cosmic energy equationfor Λ . We use it to determine the critical value of the clus-tering parameter.The cosmic energy equation derived above is simpli-fied by using the definition of clustering parameter b Λ ,which is the ratio of gravitational correlation energy W to kinetic energy K for Λ. Thus we have b Λ = − W K Ahmad and Hameeda (2010). Then, following the Refs.(Saslaw (1986); Saslaw et al., (1990)), where the substitu-tion is made as y Λ ( t )= b Λ ( t ) , the cosmic equation becomes dy Λ ( t ) dt − − y Λ W dWdt − aa (1 − y Λ + η ) = 0 . (70)We use the power law form of the correlation energy W ( t ) ∝ t ω , (71)and power law form of the scale factor as given in Eq. (23)to solve Eq. (70), which yields y Λ ( t ) = y c + ( y Λ0 − y c )( aa ) − ( ωn + n ) , (72) MNRAS , 000–000 (0000) he large scale structure formation in an expanding universe where y c = 2 ω + 2 n + 2 nηω + 2 n , (73)with y c = b c ( b c is critical value of b where system is virial-ized). Further from the power law spectrum, ω ∼ − ¯ n in anEinstein-de Sitter model with n = , we have b c = 5 − ¯ n − n + 4 η . (74)Here η is extremely small and can be ignored. We note that b c turns out to be same for a galaxy with extendedness asthat of a point like. The inference from this is noteworthy.It indicates that b c is independent of any modification tothe Newtonian potential. Rather it only depends on thevalue of ¯ n , implying b c is model dependent only.The solutions of the cosmic energy equation for Λ andthe point-like approximation of a galaxy are respectivelywritten as y Λ ( t ) = y c + ( y Λ0 − y c )( aa ) j y ( t ) = y c + ( y − y c )( aa ) j (75)where j = − n ( 3 ω n ) . (76)Combining the above equations yields, y Λ = y + ( y Λ0 − y )( aa ) j (77)Here we have assumed that y Λ0 = y c , since y Λ = b Λ , y = b , y Λ0 = b Λ0 , and y = b . We further use the basic definitionof the form of b and use b = x x , b Λ0 = αx αx (78)where x = βρ T − (79)is the initial density fluctuation given by the equation (14).Hence we get b Λ ( t ) = αx bαx + b (1 − α ) (cid:0) aa (cid:1) j . (80)Further assuming x to be minimum fixed value using b = x x , we have b Λ ( t ) = αx xαx (1 + x ) + x (1 − α ) (cid:0) aa (cid:1) j . (81)Since x corresponds to the initial density fluctuations andfor any interaction to take place, the presence of initial fluc-tuations is important. Here gravity amplifies the fluctuationand which subsequently leads to the clustering and structureformation.Keeping x fixed at some minimum, we can compare theclustering of cosmological constant with that of the Newto-nian gravity from equation (81). This also provides us veryimportant clue that clustering depends on cosmological ex-pansion.Hence, we are able to find the impact of the redshift on clustering. For that purpose, we rewrite the equation (81)as1 b Λ = h b c + (cid:16) b − b c (cid:17)(cid:16) aa (cid:17) j i , (82)or b Λ = b c (cid:0) b c b − (cid:1)(cid:0) aa (cid:1) j . (83)Using the relation 1+ z = a a , where z is the red shift and a is current value of the scale factor, we can study the variationof b Λ with z for different models i.e, for different values of¯ n : 1 , , − , − 2, which correspond to different values of ω as ω is related to ¯ n as − ¯ n . For ¯ n = 2 , , , − , − , − 3, the cor-responding estimates of b c read as 1 . , , . , . , . , . n = 3, b c diverges. Fixing b atsome minimum, we can study variation of b Λ ( z ) verses z .In Fig. 9, we plot b Λ against z to understand its time-dependence by fixing b = 0 . G = m = 1.We see in Fig. 9 (a) the behavior of b Λ corresponding to¯ n = 2, for which we have ω = − and therefore b c = 1 . b Λ decreasessuddenly and soon becomes constant. Fig. 9 (b) shows thecase ¯ n = 1, hence ω = 0 and therefore b c = 1. It is clear herethat b Λ is decreasing function of z . However, it yields to aconstant at high redshift values. In all three plots, the be-havior of b Λ corresponding to the radiation-dominated uni-verse ( n = 1 / 2) is represented by solid red line and that ofa matter-dominated universe is represented by dashed blueline for n = 2 / n = . Then,we study the case of ¯ n = 0, for which ω = and therefore b c = 0 . 83 which is illustrated in Fig. 9 (c).For all other values mentioned above, we can see similar be-havior, i.e b Λ is a decreasing function of redshift for ω > ω c ,where ω c is a negative value which is independent of theredshift ( ω = ω c is a singular point). This result is differ-ent from that obtained in N-body simulation (Farieta et al.,(2019)), where b Λ is an increasing function of the redshift,which is possible only by choosing ω < ω c . This is illustratedin Fig. 10, which depicts behavior of b Λ in terms of ω . Forexample, the case n = yields ω c ≈ − . 34. In Fig. 10 (a) weshow that b Λ is increasing (decreasing) function of redshiftfor ω < ω c ( ω > ω c ), and there is singular point for ω = ω c .We see that b Λ at z = 0 the is constant and equal to initialvalue of b = 0 . 6. The value of ω c only depends on n whichis shown in Fig. 10 (b). We can see for the larger values of n , ω c is smaller.Thus, the value of b Λ varies from 0 h b Λ i 1, as ex-pected from physical considerations. However, in order tocheck validity of our method, we can use the clustering tem-perature and connect it to the observational data. For ex-ample, we know that T = 12 . ± . keV in units of Boltz-mann constant at redshift z ∼ . b Λ from Fig. 9, and use(74) to write b c in terms of the temperature. Then, we obtaina relation for the temperature in terms of the redshift. In or-der to do that we use the approximation, η = AT , where A is infinitesimal constant. We can see that ¯ n ≈ σ gal = 1434 ± kms − (Maurogordato et al., (2008)). It can be used to fix otherfree parameters in our model. MNRAS , 000–000 (0000) M. Hameeda et al. Figure 9. A typical behavior of b Λ in terms of the redshift for b = 0 . ω > ω c . We can also plot correlation function (62) and compareit with the observational data (Hong et al., (2012)). We canfind general behavior of the correlation function (see Fig. 12)is in agreement with the best-fit ΛCDM model (Hong et al.,(2012)), and coincide with 11103 clusters, with known red-shifts (Wen et al., (2009)). It is also possible to calculate the distribution function forthis system. This distribution function can then be used ki-netic energy fluctuations, and then we can use that to relatethis model to the observational data. Now as the galaxiesbehave as point particles, we can analyze this distributionfunction for this system using the standard methods used instatistical mechanics. Thus, we can write the probability of Figure 10. A typical behavior of b Λ in terms of the ω for b = 0 . Figure 11. Clustering temperature versus redshift for A = 0 . h b Λ i = and ω > ω c with different values of ¯ n . finding N -galaxies in grand canonical ensemble as F ( N ) = P i e NµT e − UiT Z G ( T, V, z ) = e NµT Z N ( V, T ) Z G ( T, V, z ) (84) MNRAS , 000–000 (0000) he large scale structure formation in an expanding universe Figure 12. Correlation function in unit volume. Tow-side arrowshow observational data. where Z G is the grand partition function defined by Huang(1987) as: Z G ( T, V, z ) = ∞ X N =0 z N Z N ( V, T ) (85)and z = e µT is the activity.Now this grand partition function can be expressed asln Z G = P VT = ¯ N (1 − b ) , (86)where ¯ N is the average number of galaxies in a cell. So, wecan express the probability of finding N -galaxies as F ( N ) = ¯ N (1 − b ) N ! (cid:0) ¯ N (1 − b ) + Nb (cid:1) N − e − [ ¯ N (1 − b )+ Nb ] . (87)This is in agreement with Ahmad et al., (2002) andSaslaw and Hamilton (1984).The basic assumption for a quasi-equilibrium, is the fluctu-ations in potential energy over a given volume are propor-tional to the fluctuations of local kinetic energy. Thus, for N galaxies and assuming N to be very large, we obtain Gm N ( N − < (cid:18) r ′ (cid:19) > = α Nm v (88)where h r ′ i is given by h r ′ i = h r + ǫ ) − Gm (cid:18) Λ6 − ¨ a a (cid:19) r ! i (89)Assuming h N i = 1, with G = m = R = 1, we obtain α = h (cid:18) r ′ (cid:19) ih v i − , (90)where v is the peculiar velocity.Now, we rescale F ( N ) from density fluctuations to kineticenergy fluctuations. This is done by replacing N with N h r ′ i and replacing the average number ¯ N with ¯ N h r ′ i . Further-more, we substitute α v for N h r ′ i and α h v i for ¯ N h r ′ i .Finally, using N ! = Γ( N + 1), we can express the kinetic en-ergy fluctuations in velocity fluctuations using the Jacobian Figure 13. Correlation function in unit volume. Tow-side arrowshow observational data. Unit value for the model parametersselected. α v , f ( v ) = 2 α h v i (1 − b )Γ( α v + 1) (cid:2) α h v i (1 − b ) + α bv (cid:3) α v − × exp (cid:0) − α h v i (1 − b ) − α bv (cid:1) v. (91)This result is in agreement with the earlier results ob-tained in Saslaw et al., (1990) and Saslaw and Yang (2009).This result demonstrates a Gaussian-like distribution (seeFig. 13) in agreement with earlier observation obtained inRaychaudhury and Saslaw, (1996). In this paper, we have analyze the effects of expansion of theuniverse on the structure formation in the universe. This isdone using the gravitational partition function. As the dis-tance between galaxies is much larger than the size of galax-ies, it is possible to approximated galaxies as point particlesin this gravitational partition function. These point particlesinteract through a gravitational potential. The gravitationalforce pulls these galaxies towards each other, leading to theformation of large scale structure in our universe. However,the expansion of universe moves these galaxies away fromeach other. Thus, it is important to analyze the effects ofthe expansion of the universe on the structure formation inour universe. This can be done by incorporating a cosmo-logical constant term in this gravitation partition function.We have also used this gravitational partition function witha cosmological constant term, to study the thermodynam-ics for this system. Then we have used the viral expansionto obtain equation of state for this system. It was observedthat the equation of state for this system is the Van derWaals equation. We have also analyze a gravitational phasetransition in this system. This was done using the mean fieldtheory for this system of galaxies. We have also analyzed theeffect of cosmological construct the cosmic energy equation.This cosmic energy equation was also used for analyzing thetime evolution of the clustering parameter. We have alsoused compared our model with the observational data, and MNRAS , 000–000 (0000) M. Hameeda et al. used the observational data to constraint the free parame-ter in our model. This done using both the cosmic energyequation, and distribution function.It may be noted that the modification of gravitationalpartition function from f ( R ) gravity has also been analyzed.It was observed that this modified partition function wasconsistent with the observations. It would be interesting toanalyze the gravitational phase transition for this system.The details of such a gravitational phase transition woulddepend on the specific kind of f ( R ) gravity model. It wouldalso be interesting to obtain cosmic energy equation for thismodified gravitational partition function, and use it for ana-lyzing the effects of f ( R ) gravity on the time evolution of theclustering parameter. It would also be interesting to use adifferent model of f ( R ) gravity, and analyze its effect on clus-tering. It would be interesting to calculate the dependence ofclustering parameter on different models of f ( R ) gravity. Itwould also be interesting to calculate the correlation betweengalaxies using these different models of f ( R ) gravity. Thiscan then be compared with observation, and then used toconstraint the free parameters in f ( R ) gravity. As it is possi-ble to use the gravitational partition function for analyzingclustering in MOND (Upadhyay et al., (2018)), it would beinteresting to analyze gravitational phase transition usingMOND. It would also be interesting to perform such a cal-culation for MOG, as MOG can predict a large scale modi-fication of gravitational potential (Hameeda et al., (2019)).It would be interesting to obtain the distribution of differ-ent galaxies for such modified theories of galaxies, and thencompare it with observations. These observations can thenbe used to constraint certain free parameters in these mod-els of modified gravity. It may be noted that it is possible toobtain large scale correction to the gravitational potentialusing brane world models, and then use this modified gravi-tational potential to analyze gravitational partition functionfor brane world models (Hameeda et al., (2016b)). Thesegravitational partition functions can then be used to obtainthe dependence of the corrected gravitational potential onlarge extra dimensions. This effects can be observed in na-ture, and thus they can be used to constraint the size of suchlarge extra dimensions. These large scale corrections to thegravitational potential can be obtained from the super-lightbrane world perturbative modes. ACKNOWLEDGEMENT We would like to thank Mushtaq B Shah for useful discus-sions. LGW would like to acknowledge support from Zhe-jiang Provincial Natural Science Foundation of China un-der Grant No. LD18A040001 and National Natural ScienceFoundation of China under Grant Noˆa ˘A´Zs. 11674284 and11974309. 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