Spherical collapse in DGP braneworld cosmology
aa r X i v : . [ a s t r o - ph . C O ] A ug Spherical collapse in DGP braneworld cosmology
Ankan Mukherjee Centre for Theoretical Physics,Jamia Millia Islamia, Jamia Nagar, New Delhi-110025, India
Abstract
The DGP (Dvali, Gabadaze and Porrati) braneworld cosmology gives an alternative todark energy. In DGP cosmology the alleged cosmic acceleration is generated by the modifi-cation of gravity theory. Nonlinear evolution of matter density contrast in DGP braneworldcosmology is studied in the present work. The semi-analytic approach of spherical collapseof matter overdensity is adopted in the present context to study the nonliner evolution.Further, the number count of galaxy clusters along the redshift is studied for the DGP cos-mology using Press-Schechter and Sheth-Tormen mass function formalisms. It is observedthat for same values of cosmological parameters, the DGP model enhances the number ofgalaxy cluster compared to the standard ΛCDM scenario.
Keywords: Cosmology, dark energy, modified gravity, spherical collapse, cluster number count.
The observed phenomenon of cosmic acceleration [1, 2, 3] and its further confirmations from latestcosmological observations [4, 5, 6, 7] have changed our understanding about the universe. Withinthe regime of general relativity (GR), the alleged accelerated expansion of the universe can beexplained by introducing an exotic component in the energy budget, dubbed as dark energy .Besides dark energy cosmology, the other way to look for a possible solution of the puzzle is tomodify the theory of gravity where the geometry itself be responsible of the cosmic acceleration.Higher dimensional gravity theories [8, 9], scalar-tensor theories [10, 11, 12, 13], Gauss-Bonnetgravity [14, 15, 16], f ( R ) gravity [17, 18, 19, 20] are amongs the popular modified gravity theories.Some of the f ( R ) gravity models are highly successful to explain the cosmic inflation [21, 22].On the other hand, some of the higher dimensional theories [23, 24, 25, 26, 27, 28, 29, 30], areconsistent in case of late time cosmology. Email: [email protected]
The present work is devoted to the study of the evolution of matter overdensity and clusteringof dark matter in the higher dimensional DPG (Dvali, Gabadaze and Porrati) braneworld cosmol-ogy [8]. The prime endeavour is to probe the effect of higher dimensional gravity scenario on thelinear and nonlinear evolution of matter density contrast and formation of large scale structure inthe universe. The semi-analytic approach of spherical collapse of matter over density is adoptedfor the present study. Further the number count of dark matter halos along redshift for the DGPcosmology is also studied. The results are compared with that of standard cosmological constant with cold dark matter (ΛCDM) model. It is indispensable for geometric dark energy model toproduce the appropriate dynamics of the growth of cosmological perturbations and the large scalestructure formation to be accepted as a viable cosmological model. In this regard the presentwork is important to study the viability of DGP cosmology. A study of structure formation inDGP cosmology was carried out by Koyama and Maartens [31]. Multamaki, Gaztanaga andManera studied the growth of large scale structure in braneworld cosmology [32]. Topology oflarge scale structure in non-standard cosmology was investigated by Watts et al. [33]. Large scalestructure formation on a normal branch of DGP brane was studied by Song [34]. Numerical studyof cosmological perturbation in DGP cosmology was carried out by Cardoso et al [35]. Structureformation in modified gravity scenario is studied by Brax and Valageas [36] and by Carrol etal. [37]. Aspects of spherical collapse in modified gravity theories are explored by Schafer andKoyama [38] and by Schmidt, Hu and Lima [39].As already mentioned that in the present analysis, the nonlinear evolution of dark matter over-density and its clustering is studied with the assumption of spherical collapse. Spherical collapseis a semi-analytic approach [40, 41, 42] to study the dynamics of matter overdensity. Evolutionof spherical homogeneous overdensity is studied using the fully nonlinear equation derived fromNewtonian hydrodynamics. The overdence region is assumed to be spherically symmetric with auniform density of dark matter which is higher than the background density. It is considered as aclosed sub-universe expanding with Hubble flow. But the expansion slows down and after reachinga maximum radius, it stars compression and eventually collapses due to gravitational attraction.Virialization of gravitational potential and thermal energy is introduced in the context of sphericalcollapse to explain the finite size of the collapsed object. The effective nature of dark energy canhave its signature on the collapse of dark matter overdensities and consequently on the formationof large scale structure. The spherical collapse approach is numerically easier technique to apply invarious dark energy models compared to the fully numerical simulations. There are several studiesin this direction in the literature [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]where spherical collapse of matter overdensity is investigated. Though the numerically sophisti-cated approach to study the nonlinear evolution of cosmological perturbation and formation oflarge scale structure in the universe is the N-body simulation [60, 61, 62, 63]. In case of geomet-ric dark energy the spherical collapse approach is also efficiently utilized to study the evolutiondynamics of matter perturbation [38, 39]. Besides the evolution of matter density contrast, inthe present work the number count of the collapsed objects in the DGP braneworld scenario isalso emphasized. To obtain the number count of collapsed objects or the dark matter halos, twodifferent mass function formalism has been adopted. Due to gravitational attraction the baryonicmatter follows the distribution dark matter. Hence the galaxy clusters are embedded in the darkmatter halos. Thus the observed distribution of galaxy clusters provides the information aboutthe distribution of dark matter halos in the universe. The cluster number count is an importantobservable of future observations of cosmic large scale structure. Study of number count of darkmatter halos or galaxy clusters along redshift would be useful to test any model with upcomingcosmological observations.This paper is arranged as the following. In the next section (section 2), the DGP braneworldcosmology in the FRW (Friedmann-Robertson-Walker) universe is discussed briefly. In section 3,the linear and nonlinear evolution of matter overdensity and its gravitational collapse is discussed.In section 4 the number count of dark matter halos or the galaxy clusters are studied for the DGPcosmology and the results are compared with that of ΛCDM. Finally in section 5, it is concludedwith a summarization of the results.
The DGP braneworld model is based on the idea that our four dimensional universe lives ona five dimensional manifold [8]. The gravitational interaction is modified due the presence ofthe five dimensional manifold. The modification in the gravitational interaction generates cosmicacceleration itself without introducing any exotic component in the energy budget of the universe.In this framework, the first Friedmann equation is written as, H + ka − M M H + ka ! = 8 πG ρ m + ρ r ) . (1)As the gravitational constant is modified for 5D manifold, the Planck mass is also get modified.Here M (5) and M (4) are the 5D and 4D Planck masses. Finally, the scaled Hubble parameter forthis model is obtained as, h ( a ) = H ( a ) H = (cid:20)q Ω r a − + Ω m a − + Ω rc + q Ω rc (cid:21) + Ω k a − , (2)where r c = M / M called the crossover scale, Ω rc = r c H . Here the density parameters aredefined as, Ω m = ρ m H / πG , Ω r = ρ r H / πG , Ω k = − k/H . The normalized condition, h ( z = 0) = 1provides Ω rc = (1 − Ω r − Ω m − Ω k ) − Ω k ) . (3)The dark energy density like contribution in the DGP braneworld model can be written as,∆ H DE = 2Ω rc + 2 q Ω rc q Ω rc + Ω m a − + Ω r a − , (4)where h ( a ) = Ω m a − + Ω r a − + Ω k a − + ∆ H DE . At low redshift, neglecting the radiationcontribution, the effective equation of state parameter for the geometric dark energy model is - - - - - - w e ff Figure 1:
The effective equation of state parameter w eff ( z ) plot of the geometric dark energy in DGPbraneworld cosmology. The parameters are fixed as Ω k = 0 . m = 0 . .
25 (dashedcurve), 0 .
35 (dotted curve). given as, w eff ( a ) = − − d ln ∆ H DE d ln a , (5)In figure 1, the plots of w eff ( z ) for the DGP braneworld cosmology are shown. The effectivegeometric dark energy equation of state evolves to lower values at recent time generating theeffective negative pressure required for the cosmic acceleration. Matter density contrast is defined as δ = ∆ ρ m /ρ m where ∆ ρ m is the deviation from homogeneousmatter density ρ m . The overdense region initially grows in size with Hubble expansion. Due togravitational attraction, it gathers mass from the surrounding. After certain amount of massaccumulation, it starts collapsing. It is the fundamental process to form the large scale structurein the universe. The nonlinear evolution of the matter overdensities is important to understandthe dynamics of gravitational collapse. The semi-analytic approach, namely the spherical collapsemodel [40, 41, 42] is the simplest one to probe the nonlinear evolution of matter over density. Itassumes that the overdense regions are spherically symmetric. The nonlinear differential equation,that governs the time evolution of the matter density contrast is,¨ δ + 2 H ˙ δ − πG eff ρ m δ (1 + δ ) −
43 ˙ δ (1 + δ ) = 0 . (6)The linear version of equation (6) is given as,¨ δ + 2 H ˙ δ − πG eff ρ m δ = 0 . (7) a δ a δ Figure 2:
Linear (gray curves) and nonlinear (black curves) evolution of δ ( a ). In the left panel, theparameter values are Ω k = 0 and Ω m = 0 . .
32 (black dashed). The linear δ ( a ) in curvein left panel is degenerate for Ω m = 0 .
3; 0 .
32. In the right panel, the parameter values Ω m = 0 . k = − .
01 (black solid), 0 .
01 (black dashed). The linear δ ( a ) curve in right panel is degenerate forΩ k = − .
01; 0 . The G eff is the effective gravitational constant for the braneworld model, given as, [64], G eff ( a ) G = 4Ω m ( a ) − − Ω k ( a )) + 2 p − Ω k ( a )(33Ω m ( a ) − − Ω k ( a )) + 2 p − Ω k ( a )(3 − K ( a ) + 2Ω m ( a )Ω k ( a ) + Ω k ( a )) − K ( a ) + 2Ω m ( a )Ω k ( a ) + Ω k ( a )) , or G eff = N ( a ) G , where G is the Newton’s gravitational constant. Using scale factor ‘a’ as theargument of differentiation in equation (6) yield, δ ′′ + h ′ h + 3 a ! δ ′ − m N ( a )2 a h δ (1 + δ ) − δ ′ (1 + δ ) = 0 . (8)Similarly the liner equation (eq. 7) is given as, δ ′′ + h ′ h + 3 a ! δ ′ − m N ( a )2 a h δ = 0 . (9)Equation (8) and (9) are studied numerically with the background given by equation (2).In figure 2 the linear and nonlinear evolution of δ are shown with varying values of parameters(Ω m , Ω k ). The initial conditions are fixed at a i = 0 .
001 and the initial values are fixed as δ i = 0 . δ ′ i = 0 . δ is found to be very less sensitive to thechange in parameter values, but the nonlinear evolution differs with variation in parameter values.The δ ( a ) curves are found to be indifferent with the change of the initial value δ ′ i .Another important quantity to study the spherical collapse of dark matter overdensity isthe critical density contrast ( δ c ). It is defined as the value of the linear density contrast at z c δ c Figure 3:
Critical density at collapse δ c as a function of redshift of collapse z c for the DGP braneworldcosmology (solid curve), and the LCDM cosmology (dotted curve). the redshift where the nonlinear density contrast diverges. Changing the initial condition δ i inequation (equation (8)) the redshift of nonlinear collapse can be changed. Thus the critical densitycontrast δ c is obtained as a function of redshift at the collapse ( z c ). The curves of δ c ( z c ) for theDGP braneworld cosmology is shown in figure 3. For a comparison, the δ c for ΛCDM cosmologyis also shown. The critical density at collapse δ c is important to study the distribution of galaxycluster number or the number dark matter halos along redshift. The δ c curves remain unchangedwith small variation of the cosmological parameters. It is clearly due the less sensitivity of linear δ to the change of parameter values. The gravitational collapse of the matter overdensity is the basic process of large scale structureformation in the universe. The objects, formed by the collapse, are called the dark matter halos.The baryonic matter follows the distribution of dark matter. The idea is that the galaxy clustersare embedded in the dark matter halos. Thus the observed distribution of galaxy clusters isthe probe of dark matter halos in the universe. In this section, the number distribution of darkmatter halos or the cluster number count is studied for the DGP braneworld cosmology using thespherical collapse model. In the semi-analytic approach, two different mathematical formulationof halo mass is used to evaluate the number count of collapsed objects or the galaxy clusters alongthe redshift. The first one is the Press-Schechter mass function [65] and the other one, which is ageneralization of the first one, is called the Sheth-Tormen mass function [66]. The mathematicalformulations of halo mass function are based on the assumption of a Gaussian distribution ofthe matter density field. The comoving number density of collapsed object (galaxy clusters) at acertain redshift z having mass range M to M + dM can be expressed as, dn ( M, z ) dM = − ρ m M d ln σ ( M, z ) dM f ( σ ( M, z )) , (10)where f ( σ ) is called the mass function. The mass function formula, proposed by Press andSchechter [65], is given as f P S ( σ ) = s π δ c ( z ) σ ( M, z ) exp " − δ c ( z )2 σ ( M, z ) . (11)The σ ( M, z ) is the corresponding rms density fluctuation in a sphere of radius r enclosing a massM. The σ ( M, z ) can be expressed in terms of the linerised growth factor g ( z ) = δ ( z ) /δ (0), andthe rms of density fluctuation at a fixed length r = 8 h − Mpc as, σ ( z, M ) = σ (0 , M ) (cid:18) MM (cid:19) − γ/ g ( z ) , (12)where M = 6 × Ω m h − M ⊙ , the mass within a sphere of radius r and the M ⊙ is the solarmass. The γ is given as γ = (0 . m h + 0 . (cid:20) .
92 + 13 log (cid:18) MM (cid:19)(cid:21) . (13)Here h is the Hubble constant H scaled by 100 km s − M pc − . The number of collapsed objectsor cluster number count having mass range M i < M < M s per redshift and square degree yieldas, N ( z ) = Z deg d Ω cH ( z ) "Z z cH ( x ) dx Z M s M i dndM dM. (14)A general nature of cluster count is successfully depicted by the Press-Schechter fromalism,but it suffers from the prediction of higher abundance of galaxy cluster at low redshift and lowerabundance of clusters at high redshift compared to the result obtained in simulation of darkmatter halo formation [67]. To alleviate this issue, Sheth and Tormen [66] proposed a modifiedversion of the mass function formula, which is given as, f ST ( σ ) = A s π " σ ( M, z ) aδ c ( z ) ! p δ c ( z ) σ ( M, z ) exp " − aδ c ( z )2 σ ( M, z ) . (15)Three new parameters ( a, p, A ) are introduced by the Sheth-Tormen mass function formula, givenin equation (15). For the values of the parameters ( a, p, A ) as (1 , , ) the Sheth-Torman massfuntion mimics the Press-Schechter mass function. In the present work, while studying the dis-tribution of cluster number count using Sheth-Tormen mass function formula, the values of theparameter ( a, p, A ) are fixed at (0 . , . , . z z × × × × z Figure 4:
Cluster number count N ( z ) plots using Press-Schechter mass function for DGP braneworldcosmology. In the left panel, the parameters are fixed at Ω m = 0 . σ = 0 . k = − .
01 (solidcurve), Ω k = 0 .
01 (dashed curve). In the middle panel, the parameters are fixed at Ω k = 0 . σ = 0 . m = 0 .
30 (solid curve), Ω m = 0 .
25 (dashed curve). In the right panel, the parameters are fixedat Ω m = 0 .
3, Ω k = 0 . σ = 0 . σ = 0 . σ = 0 . halo formation [67]. In the present work, the cluster number count is studied for the both themass function formulas.The number count of dark matter halos or galaxy cluster for the Press-Schechter mass functionare shown in figure 4. The variation of cluster count with changing values of parameters Ω m ,Ω k and σ are studied in figure 4. In the left panel of figure 4, N ( z ) plots are shown varyingthe value of Ω k . The middle panel of figure 4 shows the N ( z ) plots varying the values of Ω m .The right panel of figure 4 shows the N ( z ) plots for different values of the parameter σ . In thepresent study, the value of Hubble constant H is fixed at 70 km s − M pc − . Similarly figure 5presents the cluster number count count plots for the Sheth-Tormen mass function with similarvariation of the parameters. It is observed that the N ( z ) is less effected by the change of spatialcurvature. On the other hand, it varies significantly with changing Ω m . A higher value of Ω m produces higher number of dark matter halos. The cluster number count is also highly effected bythe rms fluctuation of matter density. Number count increases with the increase in the value ofparameter σ . The effect of changing parameter values is very similar in case of Press-Schechterand Sheth-Tormen mass function.For a comparison with ΛCDM cosmology, the cluster number count N ( z ) plots for DGPbraneworld model and ΛCDM model with same values of parameters are shown figure 6. Pa-rameters are fixed as Ω m = 0 .
3, Ω k = 0 . σ = 0 .
8. The number count is found to besubstantially enhanced in case of DGP. As already mentioned, the lower value of σ effectivelydecreases the number count, the value of σ can be tuned for DGP model to produce numbercount of galaxy clusters similar to that of ΛCDM keeping other parameters same for both themodels. × × z × × z × × × × z Figure 5:
Cluster number count N ( z ) plots using Sheth-Tormen mass function for DGP braneworldcosmology. In the left panel, the parameters are fixed at Ω m = 0 . σ = 0 . k = − .
01 (solidcurve), Ω k = 0 .
01 (dashed curve). In the middle panel, the parameters are fixed at Ω k = 0 . σ = 0 . m = 0 .
30 (solid curve), Ω m = 0 .
25 (dashed curve). In the right panel, the parameters are fixedat Ω m = 0 .
3, Ω k = 0 . σ = 0 . σ = 0 . σ = 0 . The present study deals with the cosmological implication of a higher dimensional gravity the-ory, namely the DGP braneworld model. The sherical collapse of matter overdensity has beeninvestigated in the present context. The DGP braneworld is a 5-dimensional theory of gravityand at a length scale smaller than then the crossover length ( r c ), the gravity is restricted onthe 4-dimensional brane. The modification of gravity theory has its signature on the evolution ofmatter overdensity. The signature of higher dimensional gravity theory on the evolution dynamicsof matter density contrast comes through the modification of Friedmann equation governing thebackground as well as through the modification gravitational constant in the higher dimensionaltheory. The linear and nonlinear evolutions (equation (6) and (7)) of matter density contrast arestudied numerically in the present work. It is totally based on the assumption of an isotropiccollapse of the overdense region due to gravitational attraction. The curves of linear and nonlinearevolution of matter density contrast δ are shown in figure 2. The effect of variation of cosmolog-ical parameters on the δ ( a ) curve are emphasized. The linear evolution is very less effected bythe changes in the parameter values. The nonlinear evolution is non degenerate with the changeof parameter values. Increase in the value of background matter density increases the clusteringof dark matter. On the other hand, positive spatial curvature (Ω k <
0) slightly increases thedark matter clustering compared to the negative spatial curvature (Ω k > a i = 0 . z × × z Figure 6:
Cluster number count N ( z ) plots for DGP cosmology (solid curves) and for ΛCDM (dottedcurve) with parameter values Ω m = 0 .
3, Ω k = 0 . σ = 0 .
8. The left panel is obtained for thePress-Schechter mass function and right panel is obtained for Sheth-Tormen mass function. count of dark matter halos or galaxy clusters varies with the gravity theory. One prime endeavourof the present analysis was to check galaxy cluster number count in case of DGP cosmology. Infigure 4, the cluster count plots are shown for the Press-Schechter mass function formula. Theeffect of variation of the parameters are also investigated. It is observed that the positive spatialcurvature has a slightly lower cluster number at high redshift compared to that in case of negativespatial curvature. Lower matter density decreases the cluster number. The variation of clusternumber with the change in the rms fluctuation of matter density is also studied. The curvesare highly effected by the change in the value of σ . In figure 5, cluster number count for theSheth-Tormen mass function is shown. The effects to variation of parameters in this case aresame as in the case of Press-Schechter mass function.Further the cluster count distribution in the present scenario is compared to that of ΛCDMin figure 6. The DGP brameworld model produces much higher cluster number at high redshiftif the parameters are fixed at same values for DGP and ΛCDM. The redshift at which the clusternumber is maximum is also different in this two cases. A slightly lower value of the rms ofmatter density fluctuation subsequently decreases the number of galaxy cluster and also dragthe redshift of maximum cluster number to a lower value. Thus in DGP cosmology, the clusternumber count consistent with cosmological standard model can be achieved with a lower rms ofmatter fluctuation. Upcoming cosmological observations, like the South Pole Telescope (SPT),eROSITA, would be a effective to check the viability of various dark energy models including thegeometric dark energy scenarios through the precise observations of galaxy clusters number alongthe redshift.1 Acknowledgment
The author acknowledge the financial support from the Science and Engineering Research Board(SERB), Department of Science and Technology, Government of India through National Post-Doctoral Fellowship (NPDF, File no. PDF/2018/001859). The author would also like to thankProf. Anjan A. Sen and Dr. Supriya Pan for useful discussions.
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