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February 2, 2008 Dynamical Mutation of Dark Energy L. R. Abramo1, R. C. Batista1, L. Liberato2 and R. Rosenfeld2 1 Instituto de F´ısica, Universidade de São Paulo CP 66318, 05315-970, São Paulo, Brazil and 2 Instituto de F´ısica Teórica, Universidade Estadual Paulista R. Pamplona 145, 01405-900, São Paulo, Brazil Wediscusstheintriguingpossibilitythatdarkenergymaychangeitsequationofstateinsituations where large dark energy fluctuations are present. We show indications of this dynamical mutation in some generic models of dark energy. 8 PACSnumbers: 98.80.Cq 0 0 2 INTRODUCTION darkenergycanmutateintoafluidwithclusteringprop- erties similarto those ofdark matter. We willshow that n a Thereisampleevidencethatsupportstheexistenceof this effect is a generic feature of dark energy, and that J darkmatter (DM) anddark energy(DE) in the universe it has a simple origin: when pressure perturbations are 3 [1]. Due to its charge neutrality and dust-like equation large, the effective equation of state inside a collapsed of state (i.e., negligible pressure), dark matter starts to region can be completely different from the equation of ] h cluster gravitationally very early in the history of the state of its homogeneous component. p Universe, and is crucial for the formation of large scale - structure. Darkenergy,ontheotherhand,becomesrele- o r vantonlymorerecently,andispresumedto be asmooth GRAVITATIONAL COLLAPSE WITH DARK st component with a negative equation of state in order to ENERGY a fuel the accelerated expansion of the Universe [2]. [ At the background level, dark energy (or any other In the following, in order to specify the properties 2 fluid) is completely determined by its equation of state of the dark energy perturbations we will neglect the v 8 w =pe/ρe, where pe is the pressure and ρe is the energy anisotropic stress. Moreover, we will characterize the density of dark energy. Already atthis leveldark energy pressure perturbation in a simplified manner, using the 6 3 can affect large scale structures [3] – see also [4] for the so-called effective (or non-adiabatic) sound velocity [14], 2 effect in different parameterizations of the equation of defined as c2 ≡δp /δρ , which we will assume to be a eff e e 0. state of dark energy. function of time only. Notice that this assumption lacks 1 However, if dark energy is in fact a manifestation of a a formal basis in perturbation theory, since δpe is a per- 7 dynamical mechanism such as a scalar field, then it will turbed variable whose time and spatial dependences can 0 also develop inhomogeneities due to its gravitational in- be, and often are, independent of the variations of δρ . e : v teractions with itself and with dark matter [5]. In linear Nevertheless, in a particular gauge (the so-called “rest i perturbation theory, besides the energy density pertur- frame” of the fluid, where Ti = 0), the effective sound X 0 bationδρ, we needtwoextra degreesoffreedomto char- speedcoincideswiththephasevelocityoflinearrelativis- ar acterizecosmologicalperturbations: thepressurepertur- tic perturbations, c2 [14, 15]. X bationδpandthescalaranisotropicstressπ[6,7]. Alter- Describingthe pressureperturbationasδp =c2 δρ e eff e natively,one canalsouse the velocity potentialθ =∇~ ·~v allows us to treat a wide variety of dark energy models and the anisotropic stress [8]. and, crucially, it also allows us to compute non-linear The inhomogeneities of dark energy are often quite structure formation using the Spherical Collapse model small, particularly in the case of ordinary (canonical) (SC) [16]. Furthermore, in this case the SC equations scalarfieldmodelswithalmostΛ-likebehaviour–thatis, (derivedinasimplifiedrelativisticframework)areidenti- whenw ≃−1[5,9,10]. Infact,asw →−1theperturba- caltotheequationsofpseudo-Newtoniancosmology[17]. tionsinalldarkenergymodelsaresuppressedinrelation This means that the physics of gravitational collapse of to those of dark matter. However, if that is not the case structuressuchasgalaxyclustersiswelldescribedwithin then the dark energy density contrast δ ≡ δρ /ρ can this framework. e e e be either small or large, depending on the pressure per- Consider then, in the spirit of the SC model, a spheri- turbations of dark energy [11]. cally symmetric region of constant dark energy overden- Here we present further evidence of an intriguing sity (the so-called “top-hat” density profile.) Let us call possibility which was first pointed out in supergravity- ρc =ρ +δρ andpc =p +δp theenergydensityandthe e e e e e e motivatedscalarfieldmodelsofdarkenergy[12,13]: that pressure of this region, which are modified with respect 2 to the corresponding background quantitites, ρ and p for the perturbed region for a fluid species j: e e by the perturbations δρ and δp . The equation of state e e ρ˙ ρ˙c w is defined as the ratio of the total pressure to the to- j =−3H(1+w ) , j =−3h(1+wc), (2) tal energy density, and hence it will be different for the ρj j ρcj j background and the interior of the collapsed region. A we obtain, for the density contrasts: simplecalculationshowsthattheequationofstateinside the collapsed region, wc, is given by: δ˙j =−3(1+δj)(cid:2)h(1+wjc)−H(1+wj)(cid:3). (3) p +δp δ wc = ρe+δρe =w+(c2eff −w)1+eδ . (1) Obviously, for matter we have wm =wmc =0. e e e Using Eq. (1) and that the local expansion rate is related to the velocity field in the perturbed region by For small density contrasts |δ | ≪ 1, the equation of e h=H +θ/3a, we arrive at: state inside the overdense region does not change ap- prergeicmiaeb,lyw.hHeroewδev&er,1i(fhca2elfofs)6=, thwe,rethceonulidnbtheeansounblsintaeanr- δ˙j+3H(c2jeff−wj)δj+aθ (cid:2)(1+wj)+(1+c2jeff)δj(cid:3)=0. tial modification in wc with respect to the background (4) equation of state. Even in underdense regions (voids), The equation that determines the evolution of θ comes where δ ≈ −1, there could be large modifications of the from the “acceleration” in the perturbed region: equation of state. Hence, in principle dark energy could evvoeidnse.ffectivelymutateintodarkmatterinsidehalosand θ˙+Hθ+3θa2+23H2a(cid:2)Ωmδm+(1+3c2eff)Ωeδe(cid:3)=0. (5) The above argument is completely general. What re- Notice that there is only one equation for the peculiar mains to be shown is whether there are models in which velocity, even in the case of 2 fluids. This is clearly nec- thisdramaticsituationisactuallyrealized. Thisrequires essary, because in the SC model it is a single spherically a non-linear analysis of the evolution of pertubations for symmetric region that detaches from the background, two gravitationally coupled fluids, dark energy and dark with a peculiar expansion rate given by h = H +θ/3a. matter (we will neglect radiation and baryons in what When there is pressure and pressure gradients,relativis- follows). Unfortunately, at present there are no totally tic corrections almost surely break this identity between rigorous methods for performing this analysis – except the velocities of the fluids, which means that the SC in the case of canonical scalar fields, but even then only model with a top-hat profile is inconsistent with a fully approximately [9, 10]. relativisticcalculation. Theassumptionthatc2 isonly eff Here we employ a generalization of the SC model for time dependent implies that δp is also only time depen- the case of a relativistic fluid with pressure. In the next dent–whichultimatelyguaranteesthevalidityofthetop section we present the relevant equations and mention hat SC model. under which conditions they are equivalent to a pseudo- The SC equations capture many features of the gravi- Newtonianapproach. Wewillthenanalysetheevolution tationalphysicsoneexpectstofindatthescaleofthecol- ofthe coupledsystemofperturbationsfora widevariety lapsedstructuresweseetoday. Thisisbecausethe exact of dark energy models for which the pressure perturba- sameequationscanbederivedfromapseudo-Newtonian tions are characterized by some homogeneous effective treatment of perturbations when gradients of pressure sound speed. can be neglected – as is the present case, of a top hat profile. When linearized, the ensuing equations corre- spond to the linear equations from General Relativity in NON-LINEAR EVOLUTION the case c2 = 0 for sub-horizon scales [18]. Further- eff more, even when c2 6= 0, although the equations are eff We define H = a˙/a and h = r˙/r as the expansion not equivalent anymore, still the growing modes of the rates for the background (a is the scale factor) and for linearized pseudo-Newtonian perturbations are identical the perturbed region (r is the size of the collapsing re- to the growing modes of the linearized relativistic per- gion), respectively. In what follows we work within two turbations [17]. assumptions, namely, that there is no non-gravitational interaction between DE and DM, and that the total energy of both DE and DM contained in the collapsed re- NUMERICAL SOLUTIONS gionis constant. The possibility ofincluding DM-DE in- teractionsinthestudyofstructureformationwasstudied We solvedthe coupleddifferential equationsfor δ , δ e m in [13, 19] and a discussion of possible outflow of energy andθ, fora few representativemodels ofdarkenergy. In from the collapsed region can be found in [12, 20]. ourapproximation,adarkenergymodelisdeterminedby Usingthecontinuityequationsforthebackgroundand itsbackgroundequationofstateandtheeffectivespeedof 3 0.4 0 0.2 -0.2 0 -0.4 -0.2 w c -0.4 wc -0.6 -0.6 -0.8 -0.8 -1 -1 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 z z FIG. 1: Clustered dark energy equation of state for different FIG. 2: Clustered dark energy equation of state for different cases of effective speed of sound for background w = −0.8: cases of effective speed of sound for background w = −0.99. c2 = w (solid line), c2 = 1 (dashed line), c2 = 0 (dot- LinesarethesameasinFigure1. Thecasesofc2 =w and eff eff eff eff dashed line),c2 =−w (dottedline) andc2 =−1(double c2 = −1, as well as c2 = 1 and c2 = −w, lie on top of eff eff eff eff eff dot-dashedline). Theinstantofturnaround(h=0)isz=0.3 each other. Turnaround occurs at approximately z ≃ 0.6 in for c2 =w, z =0.5 for both c2 =−w and c2 =1, and all cases. eff eff eff z=0.6 for c2 =0(for c2 =−1there is noturnaround.) eff eff However, substantial modifications are found for other possibilities of c2 . The largest modifications arise for sound. Inapreviouspaperweinvestigatedtheparticular eff casewherethe soundspeed ofdarkenergyis equalto its c2eff =1, which in our example is very similar to c2eff = equation of state, c2 = w, and therefore wc = w. In −w. In these two cases, even a complete metamorphosis eff that case, one can see clearly from Eq. (1) that there is of dark energy into a fluid which clusters as strongly as no mutation [11]. darkmatterispossibleatrecentepochs,due tothe large perturbationsinsidethecollapsedregion. Thebehaviour In this letter we expand our previous analysis to the followingcases: c2 =1,c2 =0,c2 =−1andc2 = for c2eff = −1 is also easily understood from Eq. (1), eff eff eff eff 2 −w. Thefirstcaseismotivatedbythecommonsituation since this is the only case where c −w <1. eff whendarkenergyismodelledbyacanonicalscalarfield, As expected, the effect of mutation is greatly reduced since in the gauge correspondingto the restframe of the for a backgroundequationof state close to that of a cos- scalar field the effective sound speed is c2 = 1 [14]. mological constant, independent of c2 . We illustrate eff eff The second case (null pressure perturbations) can occur that fact in Fig. 2, where we show the equation of state in so-called “silent quartessence” models [21]. The third inside the collapsing region for different models of clus- case represents a perturbation with behaviour close to a tered dark energy, in the case of a background equation cosmologicalconstantandthelastcasereproducestheso- of state w =−0.99,and with the same initial conditions called generalized Chaplygin gas in a certain limit (α = that were used in Fig. 1. Nevertheless, even in this case 1) [22]. large density contrasts can still arise in the dark energy We use adiabatic initial conditions for the perturba- component, which lead to the mutation of dark energy. tions [δi = (1+w)δi ] at a redshift z = 1000, an initial e m velocity field coincident with the Hubble flow (θi = 0), H0 = 72 km s−1 Mpc−1, Ωm = 0.25 and Ωe = 0.75. CONCLUSIONS In the examples shown below we fixed the background equation of state of dark energy at w =−0.8. We have shown that it is possible to change radically In Fig. 1 we show the values of the clustered equation the clustering properties of dark energy in collapsed re- of state wc as a function of the redshift. The initial con- gions (halos and voids.) We exemplified this behaviour ditionδi waschosensuchthatthedarkenergyperturba- withafewmodelsforthedarkenergyperturbations,and m tion δe ∼ O(1) today. The perturbation in dark matter showed that it happens not only in scalar field models, is typically one to two ordersof magnitude larger,which but also in generic models of dark energy – in particular is consistent with the typical density contrast in galaxy the Generalized Chaplygin Gas and Silent Quartessence clusters, for which δm ∼O(103). models. Asdiscussedabove,forthecasec2 =wweobtainno Since the physics of most observed collapsed struc- eff mutationintheequationofstateintheperturbedregion. tures, such as galaxy clusters, is well approximated 4 by quasi-Newtonian physics, this dynamical mutation [4] L. Liberato and R. Rosenfeld, JCAP 0607, 009 (2006). should be a general phenomenom. Clearly, this is a cru- [5] K. Coble, S. Dodelson and J. A. Frieman, Phys. Rev. cial issue for all attempts to compute the influence of D55: 1851 (1997). [6] J. M. Bardeen, Phys.Rev. D 22, 1882 (1980). dark energy on the formation of large scale structures. [7] H. Kodama and M. Sasaki, Progr. Theor. Phys. Suppl. More detailed studies, including a relativistic approach 78: 1 (1984). andusingdifferentrealisticparameterizationsofthedark [8] C.-P. Ma and E. Bertschinger, Ap. J. 455: 7 (1995). energy equation of state are currently under way [17]. [9] S. Duttaand I. Maor, Phys. Rev.D 75, 063507 (2007). [10] D.F.Mota,D.J.ShawandJ.Silk,[arXiv:0709.2227]. [11] L. R.Abramo, R.C. Batista, L.Liberatoand R.Rosen- Acknowledgments feld, JCAP 0711:12 (2007). [12] D. F. Mota and C. van de Bruck, Astron. Astrophys. 421:71 (2004). We would like to thank Ioav Waga for many fruitful [13] N.J.NunesandD.F.Mota,Mon.Not.R.Ast.Soc.368, discussions. This work has been supported by FAPESP 751 (2006) grants 04/13668-0 (L.R.A. and R.R.) and 05/00554-0 [14] W. Hu,Astrophys. J. 506, 485 (1998). (R.C.B.), a CNPq grant 309158/2006-0 (R.R.) and a [15] J. Garriga and V. Mukhanov, Phys. Lett. B 458, 219 CAPES grant (L.L.). (1999). [16] J. E. Gunn and J. R.Gott III,Ap. J. 176: 1 (1972). [17] L. R.Abramo, R.C. Batista, L.Liberatoand R.Rosen- feld, in preparation. [18] R.R.R.Reis,Phys.Rev.D67,087301(2003),Erratum- ibid. D 68, 089901 (2003). [1] See, e.g., U. Seljak, A. Slozar and P. McDonald, JCAP [19] M.ManeraandD.F.Mota,Mon.Not.R.Ast.Soc.371, 0610, 014 (2006). 1373 (2006). [2] Forreviews, see e.g.T. Padmanabhan, Phys. Rept. 380: [20] N. J. Nunes, A. C. da Silva and N. Aghanim, Astron. 325 (2003); P. J. E. Peebles and B. Ratra, Rev. Mod. Astrophys. 450:899 (2006). Phys. 75: 559 (2003); V. Sahni and A. A. Statobinsky, [21] L. Amendola, I. Waga and F. Finelli, JCAP 0511:009 Int. J. Mod. Phys. D 15, 2105 (2006). (2005). [3] See, e.g., L. M. Wang and P. J. Steinhardt, Ap. J. 508: [22] L. Amendola, F. Fineli, C. Burigana and D. Carturan, 483(1998);R.A.BattyeandJ.Weller,Phys.Rev. D68: JCAP 0307:005 (2003). 083506 (2003); E. V. Linder, Phys. Rev. D72: 043529 (2005).