Physics Letters B 685 (2010) Contents lists available at ScienceDirect. Physics Letters B.

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1 Physics Letters B 685 () Contents lists available at ScienceDirect Physics Letters B Spherical collapse model with and without curvature Seokcheon Lee a,b, a Institute of Physics, Academia Sinica, Taipei, Taiwan 59, ROC b Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei, Taiwan 67, ROC article info abstract Article history: Received November 9 Accepted January Available online 9 January Editor: M. Trodden We investigate a spherical collapse model with and without the spatial curvature. We obin the exact solutions of dynamical quantities such as the ratio of the scale factor to its value at the turnaround epoch and the ratio of the overdensity radius to its value at the turnaround time with general cosmological parameters. The exact solutions of the overdensity at the turnaround epoch for the different models are also obined. Thus, we are able to obin the nonlinear overdensity at any epoch for the given model. We obin that the nonlinear overdensity of the Einstein de Sitter (EdS) universe at the virial epoch is 8π ( π + ) 7 instead of the well-known value 8π 78. In the open universe, perturbations are virialized earlier than in the flat one and thus clusters are denser at the virial epoch. Also the critical density threshold of EdS universe from the linear theory at the virialized epoch is obined as (9π + 6).58 instead of (π).69. This value is same for the close and the open universes. We find that the observed quantities at high redshifts are less sensitive between different models. Even though the low redshift cluster shows the stronger model dependence than high redshift one, the differences between models might be still too small to be distinguished by observations if the curvature is small. rom these analytic forms of dynamical quantities, we are able to estimate the abundances of both virialized and non-virialized clusters and the temperature and luminosity functions at any epoch. The current concordance model prefers the almost flat universe and thus the above results might be restricted by the academic interests only. However, the mathematical structure of the evolution equations of physical quantities for the curved space is identical with that for the flat universe including the dark energy with the equation of ste ω de. Thus, we might be able to extend these analytic solutions to the general dark energy model and they will provide the useful tools for probing the properties of dark energy. Elsevier B.V. Open access under CC BY license.. Spherical collapse model Background evolution equations of the physical quantities in a RW universe with the matter are given by (ȧ ) H 8π G ρ m k a a 8π G ρ cr, (.) ä π G ρ m, (.) a (ȧ ) ρ m + ρ m, (.) a where a is the scale factor, ρ m is the energy density of the matter, ρ cr is the critical energy density, and k is chosen to be +, * Address for correspondence: Institute of Physics, Academia Sinica, Taipei, Taiwan 59, ROC. address: skylee@phys.sinica.edu.tw Elsevier B.V. Open access under CC BY license. doi:.6/j.physletb...58, or for spaces of consnt positive, zero, or negative spatial curvatures, respectively. In terms of the ratio of the matter density to the critical density Ω m, the above riedmann equation (.) becomes k H a Ω k Ω m, (.) which is valid for all times. We consider a spherical perturbation in the matter density. ρ cluster is the matter density within the spherical overdensity radius R. The flatness condition is not held because of the perturbation in the matter. Thus, we have another set of equations governing the dynamics of the spherical perturbation []: R π G ρ cluster, (.5) R ( ) Ṙ ρ cluster + ρ cluster, (.6) R

2 S. Lee / Physics Letters B 685 () ig.. t and τ for the different values of z.(left)h t versus Ωm for the different values of z.6,., and. (from top to bottom). (Right) τ versus Ωm for the same values of z as in the left panel. where ρ cluster is the energy density of the clustering matter. The radius of the overdensity R evolves slower than the scale factor a and reaches its maximum size R at the turnaround epoch z and then the system begins to collapse. Cosmological parameters and the curvature of the Universe can be constrained from the growth of large scale structure and the abundance of rich clusters of galaxies. There have been numerous works related to this [ ]. Most of them reach to the similar conclusions based on the conventional approximate solutions of the background scale factor and of the overdensity radius. It is natural to expect that the correct values for the virial radius and the nonlinear overdensity obined from the exact solutions might be different from those obined from the conventional approximate solutions. We investigate this. NowweadoptthenotionsinRef.[] to investigate the evolutions of a and R x a, (.7) a y R, R (.8) where a and R are the scale factor and the radius at z,respectively. Then Eqs. (.) and (.5) are rewritten as dx dτ x Q, (.9) d y dτ y, (.) where dτ H(x ) Ω m (x ) dt H Ωm dt, Ω m Ω z k Ω m Ω Ω m k Ω m Ω m Ω ( + z ), ρ cluster m ρ m z, and x x(z ) from Eq. (.7). Ωm and Ω represent the present values of the k energy density contrasts of the matter and the curvature term, respectively. Eqs. (.9) and (.) can be solved analytically. The analytic solution of Eq. (.9) is given by x dx τ dτ x Q [ x,, 5 ], x τ, (.) where is the hypergeometric function and we use the boundary condition x when τ (seeappendix A for deils). rom this equation, the exact turnaround time τ is given by τ [,, 5, Ω m ] Ωm ( + z ) H Ωm t H Ω m ( + z ) t, (.) where we use the fact that x, the relations Ω m Ω m ( + z ), and τ H Ωm t. This exact analytic form of the turnaround time will be used to investigate the other quantities. As expected, τ (t ) depends on Ωm (i.e. Ω k ) and z as given in Eq. (.). We show these properties of τ (t )inig.. Inthe left panel of ig., we show the dependence of t (normalized by multiplying with H )onωm for the different values of z models. The solid, dashed, and dot-dashed lines (from top to bottom) correspond to z.6,., and., respectively. Eq. (.) is the evolution of the background scale factor a and we can interpret it as the age of the Universe is a decreasing function of Ωm.LargerΩ m implies faster deceleration, which corresponds to a more rapidly expanding universe early on. Also larger z means the earlier formation of the structure and thus gives the smaller t. We also show the Ωm dependence of τ for the values of z in the right panel of ig.. Becauseτ H Ωm t, τ becomes larger for the larger values of Ωm. The exact analytic solution of y also can be obined as (see Appendix A) ArcSin[ y ] y( y) τ, when τ τ, (.) y( y) ArcSin[ y ]+ π (τ τ ), when τ τ, (.) where τ and τ are given in Eqs. (.) and (.). can be obined from this analytic solution (.) (or equally from Eq. (.)) by using the fact that y ( ) π τ ( π ) ( [,, 5, (Ω m ) ]) Ωm ( + z ), (.5) where we use Eq. (.). WhenΩ m, [,, 5, ] and thus ( π ). This factor ( π ) is the well-known value of for the Einstein de Sitter (EdS) universe (Ω m ) [,]. The general value of foropenorcloseduniverseisgivenbyeq.(.5). Weshow the behavior of in ig..

3 S. Lee / Physics Letters B 685 () ig.. versus Ω m when we choose z.6,., and.. As shown in Eq. (.5), is inversely proportional to τ.thus, decreases as Ωm increases. Because is the ratio of ρ cluster to ρ m at z z, it means that the smaller overdensity kes the longer time to turn around and collapse. In ig., the solid, dashed, and dot-dashed lines correspond to z.6,., and., respectively. After we obin the value of, the exact values of x and y at any τ are obined without any ambiguity. In order to better undersnd the above results, it is useful to investigate the virialized times for the different models. rom Eq. (.), we are able to obin the collapsing time of each model normalized to the turnaround time for the EdS universe t (y( y) ArcSin[ y ]+π ) t,eds π [,, 5 ], Q, (.6) where t,eds is the turnaround time for the EdS universe. We show this in ig.. Perturbations is virialized earlier in the EdS universe than in the close one, and even earlier in the open one. The dashed, solid, and dot-dashed lines represent t normalized to the t,eds of close, EdS, and open universes, respectively. The vertical dotted line depicts the ratio of the virialized radius to the turnaround one y vir. If only the matter virializes, then from the virial theorem and the energy conservation we have ( U cluster (z ) U cluster + R U cluster R ) U cluster(z vir ), (.7) GM where U cluster is the potential energy associated with 5 R the spherical mass overdensity [,,5,,5]. rom the above equation, we are able to obin y vir R vir for any model R y vir. (.8) After replacing y vir into Eq. (.), weobin ( τ vir + ) ( τ + ) [ π π,, 5 ],. (.9) x vir is also obined from Eq. (.) with Eq. (.9) [ xvir,, 5, x ] ( vir + ) [ π,, 5 ],. (.) z vir ig.. t normalized to the turnaround time for the EdS universe versus y for close, EdS, and open universes (from top to bottom). Even though we obin the analytic expression of x vir in Eq. (.), generally this equation can be solved in a non-algebraic way. Thus, it might be useful to have the approximate analytic form of x vir for the wide ranges of the cosmological parameters (Ωm and z ) if one analyze the form by interpolating x vir values. We use the exact form in Eq. (.) though. Before we move further for the nonlinear overdensity, we investigate the common miske for Δ vir in the EdS universe. Eqs. (.), (.), and (.5) can be simplified in EdS universe because Q when Ωm. Then the above equations become x τ, (.) y( y) ArcSin[ y ]+ π ( τ ) π ( ) x, (.) ( ) π, (.) where we use τ. The commonly used assumption to obin the nonlinear overdensity Δ c in the EdS universe is that τ c τ which is the collapsing time for y c eventhough one uses y vir to obin Δ c. With this assumption, x c is obined from Eq. (.). Thus, one obins ( ) ( ) Δc EdS xc π ( ) 8π 78. (.) y vir However, we know the exact relation between x and y and thus we do not need to use the above assumption. If we insert y vir in Eq. (.), then we obin the correct virial time τ vir + π < τ c. With this correct value of τ vir, we obin the correct value of x vir ( + π ) by using Eq. (.). Thus, the correct value of the nonlinear overdensity Δ vir for the EdS universe is ( ) ( Δ EdS vir xvir π y vir 8π ( + π ) ( ( + π ) ) ) 7. (.5) Thus, the minimum overdensity for the flat universe is about 7 instead of 78. We show this in Table. Asz increases Δ c approaches to 7 instead of 78. Also for the close universe Δ vir is smaller than this value. rom Eq. (.7), we are able to obin the virial epoch from the given z

4 S. Lee / Physics Letters B 685 () Table, x vir, Δ vir,andz vir with the given values of cosmological parameters for the open, flat, and close universes. Perturbations reach turnaround and virialization earlier in the flat universe than in the close one, and even earlier in the open one. Thus, clusters are denser at virial epoch in flat universe than in the close one and even denserinthe open one. Ω m z.6 z. z. x vir Δ vir z vir x vir Δ vir z vir x vir Δ vir z vir z vir + z x vir. (.6) The ratio of cluster to background density at the virialized epoch z vir for open or close universes becomes Δ vir ρ cluster ρ m zvir ( ) xvir y vir 8π ( + π ) [,,, x vir ] 7 [,,, x vir ], (.7) where we use Eqs. (.), (.6), and (.5). We also use the fact that y vir independent of. rom Eq. (.7), we are able to investigate the linear perturbation at early epoch Δ τ + δ lin (see Appendix A for deils) τ Δ + δ NL + δ lin + ( ) τ 5 + ( ) t Ωm. (.8) 5 t This is equal to the famous result for EdS universe ( π and Ω m ), δ lin 5 ( π t t ) []. Again, there is a common miske for the value of the critical density threshold δ lin (z vir ).InEdSuniverse, τ c and one obins δ lin(z c ) from Eq. (.8) δ lin (z c ) 5 ( π ) ( ) (π).69. (.9) However, we show that the correct virialized epoch is given by τ vir ( + π )τ from Eq. (.9). Thus, the correct δ lin (z vir ) value is δ lin (z vir ) 5 ( ) (( + 9π 8 ) ) (6 + 9π).58, (.) where we use τ π. The above result given in Eq. (.) is true for with and without curvature and thus it is valid for the close, flat, and open universes. rom this critical density threshold δ lin (z vir ), we are able to obin δ lin at any epoch by using the relation δ lin (z) D g(z) D g (z vir ) δ lin(z vir ), (.) where D g is the linear growth factor. There is the exact analytic form of D g for the dark energy with the equation of ste ω de [6]. Mathematically, this form of D g is identical with that of curved space and thus we can adopt this form of D g in these models. rom the analytic forms of dynamical quantities x, y, and, it is straightforward to estimate the abundances of both virialized and non-virialized clusters at any epoch. Also the temperature and luminosity functions are able to be computed at any epoch [7]. Thus, these analytic forms provide an accurate tool for probing the effect of the curvature on the clustering. As we mentioned, the mathematical structure of the evolution equations of the physical quantities for the curved space is identical with that of the dark energy with the equation of ste ω de in the flat Universe. Thus, these analytic solutions give the guideline for the extension of them to those of the general dark energy model and they will provide the useful tools for measuring the properties of dark energy [8]. Acknowledgement We thank K. Umetsu for useful comments. Appendix A irst, we derive the exact solution of τ as a function of x given in Eq. (.). After we replace the variables Z x and T x x, Eq. (.) becomes [9] x dx x x Q x T ( ZT) dt [,, 5 ], x, (A.) where is the hypergeometric function and we use the gamma function relation Γ [ + b]bγ [b]. We also derive the exact analytic solution of τ as a function of y givenineq.(.). This equation is solved as [] dy dτ y f (y) dy c + f (y ) dy c ± τ. (A.) We use f (y ) dy y to obin dy c + y c ± τ. (A.) After replacing Z c y and T y,thelhsofeq.(a.) becomes y

5 S. Lee / Physics Letters B 685 () y ( y ) dy y T ( c ) yt dt [ y,, 5, c ] y. (A.) We use the relation c y[,, 5, c y] [,,, c y] ( c y) [,,, c y], [,,, y ] y ArcSin[y], and [a, b, b, y]( y) a to obin y c c ( [ ] c c y ArcSin y c ) y c l + τ when τ τ, (A.5) ( [ ] y c c y ArcSin y c ) y c g τ when τ τ, (A.6) where ArcSin represents arcsine function. We choose the positive (negative) sign in τ to be consistent with the fact that R (i.e. y) increases (decreases) as τ increases before (after) it reaches to the maximum radius R ( y ) at τ. Integral consnts c, c l, and c g are obined from the boundary conditions dτ dy, y(τ ), and y dy dτ ± c + y c, (A.7) c l, (A.8) π c g τ, (A.9) where we use the fact that ArcSin[] π. τ is obined from Eq. (.) τ [,, 5, ]. (A.) rom Eqs. (A.8), (A.9), and (A.), we obin the exact analytic solution of y(τ ) ArcSin[ y ] y( y) τ, when τ τ, (A.) y( y) ArcSin[ y ]+ π (τ τ ), when τ τ. (A.) If we use the relationships ArcSin[ y ]+ π y ArcTan[ y( y) ] y [,, 5, y] y( y) + π, then we are able to rewrite the above equations (A.) and (A.) as y( y) [ ] y ArcTan + π y( y) [ y,, 5 ], y τ, when τ τ, (A.) y( y) + [ ] y ArcTan + π y( y) π [ y,, 5 ], y (τ τ ), when τ τ. (A.) Thus, all of these three representions are equal to each other. We also check the perturbation is linear at early epoch Δ τ + δ lin. We use Eqs. (A.) and (A.) because τ τ in this case. With the limit τ, we obin x τ and y + 5 y 5 τ. Thus we obin ( ) ( ( x τ Δ + δ NL + )) y y + 5 y δ NL δ lin ( ) 5 ( ) τ. (A.5) If we use τ H Ωm t and H t in the early matter dominated epoch, then we obin δ lin 5 ( Ω m t ).InEdS universe ( π and Ω m ), this gives the famous result δ lin 5 ( π t t ). or the close and open universes, δlin (z vir ) 5 ( τ vir) (6 + 9π) which is equal to that of the EdS one. References [] P.J.E. Peebles, Large-Scale Structure of the Universe, Princeton University Press, 98. [] J.E. Gunn, J.R. Gott III, Astrophys. J. 76 (97). [] P.J.E. Peebles, Astrophys. J. 8 (98) 9. [] O. Lahav, P.B. Lilje, J.R. Primack, M.J. Rees, Mon. Not. R. Astron. Soc. 5 (99) 8. [5] P.B. Lilje, Astrophys. J. 86 (99) L. [6] C. Lacey, S. Cole, Mon. Not. R. Astron. Soc. 6 (99) 67. [7] J.D. Barrow, P. Saich, Mon. Not. R. Astron. Soc. 6 (99) 77. [8] P.T.P. Viana, A.R. Liddle, Mon. Not. R. Astron. Soc. 8 (996), arxiv:astroph/957. [9] V.R. Eke, S. Cole, C.S. renk, Mon. Not. R. Astron. Soc. 8 (996) 6, arxiv: astro-ph/9688. [] T. Kiyama, Y. Suto, Astrophys. J. 69 (996) 8, arxiv:astro-ph/96. [] L. Wang, P.J. Steinhardt, Astrophys. J. 58 (998) 8, arxiv:astro-ph/985. [] C. Horellou, J. Berge, Mon. Not. R. Astron. Soc. 6 (5) 9, arxiv:astroph/565. [] T. Kihara, Publ. Astron. Soc. Jpn. (968). [] D.. Mo, C. van de Bruck, Astron. Astrophys. () 7, arxiv:astro-ph/ 5. [5] I. Maor, O. Lahav, JCAP 57 (5), arxiv:astro-ph/558. [6] S. Lee, K.-W. Ng, arxiv:95.5; S. Lee, K.-W. Ng, arxiv:96.6; S. Lee, K.-W. Ng, arxiv:97.8; S. Lee, arxiv:95.7. [7] S. Lee, K.-W. Ng, in preparation. [8] S. Lee, K.-W. Ng, arxiv:9.6. [9] M. Abramowitz, I.A. Stegun, Handbook of Mathematical unctions with ormulas, Graphs, and Mathematical Tables, Dover Publications, New York, 97. [] A.D. Polyanin, V.. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, nd edition, Chapman and Hall/CRC, Boca Raton,. t