CHAPLYGIN GAS COSMOLOGY unification of dark matter and dark energy

Transcription

1 CHAPLYGIN GAS COSMOLOGY unification of dark matter and dark energy Neven Bili Theoretical Physic Division Ruer Boškovi Institute IRGAC 006

2 Chaplygin gas An exotic fluid with an equation of state p = A ρ The first definite model for a dark matter/energy unification A. Kamenshchik, U. Moschella, V. Pasquier, PLB 511 (001) N.B., G.B. Tupper, R.D. Viollier, PLB 535 (00) J.C. Fabris, S.V.B. Goncalves, P.E. de Souza, GRG 34 (00)

3 The generalized Chaplygin gas p A = 0 α 1 ρ α M.C. Bento, O. Bertolami, and A.A. Sen, PRD 66 (00) The term quartessence was coined to describe unified dark matter/dark energy models M. Makler, S.Q. de Oliveira, and I. Waga, PLB 555 (003)

4 The Chaplygin gas model is equivalent to the (scalar) Dirac- Born-Infeld description of a D-brane R. Jackiw, Lectures on Fluid Mechanics (Springer, 00) String theory branes possess three features that are absent in the simple Nambu Goto p-dim membranes (i) support an Abelian gauge field A reflecting open strings with their ends stuck on the brane; (ii) couple to the dilaton (iii) couple to the (pull-back of) Kalb-Ramond field B A B C.V. Johnson, D-Branes, Cambridge University Press, 003.

5 D-branes and the Dirac-Born Born-Infeld (DBI) theory S DBI = p+ 1 Φ p ind A d x e ( 1) det( g + B ) the induced metric ( pull back ) of the bulk metric g ind µν = G ab X x the antisymetric tensor field B a µ X x πα µν µν ν b = B + F µν

6 Choose the coordinates such that X =x and let the p+1-th coordinate X p+1 be normal to the brane. From now on we set p=3. Then Gµν = gµν for µ = G µ 4 = 0 G44 = 1 X a x i x 0

7 Scalar Born-Infeld theory If we neglect the dilaton and the B -field, we find 4 ind 4 SBI A d x det ( g ) d x L T = = BI A 1 = θ θ L θ BI µν = θ, µ θ, ν L BIgµν θ L which is consistent with a perfect fluid description θ, µ Tµν = ( ρ + p) uµ uν pgµν with uµ = θ = BI g µν θ, µ θ, ν

8 and hence A ; 1 ρ = p = A θ 1 θ p A = ρ In a homogeneous model the conservation equation yields the Chaplygin gas density as a function of the scale factor a p = A + B a 6 where B is an integration constant.the Chaplygin gas thus interpolates between dust ( ~a -3 ) at large redshifts and a cosmological constant ( ~ A ½ ) today and hence yields a correct homogeneous cosmology

9 The inhomogeneous Chaplygin gas and structure formation The inhomogeneous Chaplygin gas based on a Zel'dovich type approximation has been proposed, and the picture has emerged that on caustics, where the density is high, the fluid behaves as cold dark matter, whereas in voids, w=p/ acceleration as dark energy. is driven to the lower bound -1 producing Soon it has been shown that the naive Chaplygin gas model does not reproduce the mass power spectrum H.B. Sandvik, M. Tegmark, M. Zaldarriaga, and I. Waga, PRD 69 (004) and the CMB D. Carturan and F. Finelli, PRD 68 (003); L. Amendola, F. Finelli, C. Burigana, and D. Carturan, JCAP 07 (003)

10 The physical reason is that although the adiabatic speed of sound c s p = ρ is small until a ~ 1, the accumulated comoving acoustic horizon c d = dt H a a s 1 7 / s 0 reaches MPc scales by redshifts of twenty, frustrating the structure formation even into a mildly nonlinear regime N. B., R.J. Lindebaum, G.B. Tupper, and R.D. Viollier, JCAP 11 (004) S

11 the density contrast described by a nonlinear evolution equation either grows as dust or undergoes damped oscillations depending on the initial conditions Evolution of the density contrast in the spherical model from a eq = for the comoving wavelength 1/k = 0.34 Mpc, k (a eq ) =0.004 (solid) k (a eq ) =0.005 (dashed).

12 The root of the structure formation problem is the last term, as may be understood if we solve the equation at linear order 3 cs δ + H δ H δ δ = 0 a with the solution δ k = 1/ 4 a J5/ 4 dsk ( ) behaving asymptotically as δ k a for d k1 cos ( dsk) δk for d sk1 a Hence damped oscillations at the scales below d s s

13 The suggested ways out to overcome the acoustic barrier: A k-essence type-model where the Lagrangian has a local minimum R.J. Scherrer, PRL 93 (004) Such a model is equivalent to the ghost condensate and hence shares its peculiarities N. Arkani-Hamed et al, JHEP 05 (004) A. Krause and S. P. Ng, IJMP A 1 (006) 1091 The generalized Chaplygin gas in a modified gravity approach, reminiscent of Cardassian models T. Barreiro and A.A. Sen, PRD 70 (004)

14 Another deformation of the Chaplygin gas M. Novello, M. Makler, L.S. Werneck and C.A. Romero, PRD 71 (005) If there are entropy perturbations such that the pressure perturbation p=0 and with it the acoustic horizon d s =0, no problem arises R.R.R. Reis, I. Waga, M.O. Calvão, and S.E. Joas, PRD 68 (003) The scenario with entropy perturbations, which is difficult to justify in the simple Chaplygin gas model, may be realized by introducing an extra degree of freedom, e.g., in terms of a quintessence-type scalar field

15 Nonadiabatic perturbations Suppose that the matter Lagrangian depends on two degrees of freedom, e.g., a Born-Infeld scalar field and one additional scalar field. In this case, instead of a simple barotropic form p=p( ), the equation of state involves the entropy density (entropy per particle) s and may be written in the parametric form p=p(, ) s=s(, ) The corresponding perturbations p p δ p = δρ + δφ ρ φ s s δ s = δρ + δφ ρ φ

16 The speed of sound is the sum of two nonadiabatic terms c s S 1 p p s s p = ρ ρ p φ φ Thus, even for a nonzero p/ ρ the speed of sound may vanish if the second term on the right-hand side cancels the first one. This cancellation will take place if in the course of an adiabatic expansion, the perturbation grows with a in the same way as ρ. In this case, it is only a matter of adjusting the initial conditions of with ρ to get c s =0.

17 Nonadiabatic perturbations in the DBI theory The DBI theory which we started from possesses extra degrees of freedom in terms of the dilaton and the tensor field B. The full action contains the bulk terms in addition to the DBI Lagrangian

18 4 SDBI = d x det g L DBI L DBI B B B B Φ = A e (1 θ )(1 + ) + θ θ ( ) Abbreviations: θ = g µν 1 θ θ, µ, ν µν B = B µνb B B = 8 - det g B = B B ε µνρσ µ νρ θ θ θ, µ ν θ, ρ 1 B B µν ρσ

19 It is convenient to write everything in Einstein's frame using the transformation and simultaneously rescaling which gives θ Φ gµν e gµν e θ B e B Φ Φ, µ, µ µν µν

20 The two parts of the energy momentum tensor are not in the form of a perfect fluid. To define ρ and p, we make the decomposition is a traceless anisotropic stress orthogonal to u u and h. Hence b DBI Tµν = Tµν + Tµν T = ρu u ph + Π µν µ ν µν µν h = g u u µν µν µ ν 1 ρ = T u u ; p = T h 3 µν µ ν µν µν ;

21 Implications for linear perturbations A convenient choice is the temporal (synchronous) gauge i ds = dt a ( δ f ) dx dx ij ij j First-order perturbations δ ρ ρ ρ = θ Homogeneity and isotropy imply δ = Φ Φ Φ δ = θ θ B Φ = 0 θ = 0 = 0, i, i µν Hence,,i,,i and (B ) count as first order

22 With these assumptions we may neglect higherorder terms in T and we find simple expressions for the density and the pressure µ ν ρ u u T = Ae DBI 3Φ µν B 1+ 1 θ 1 µν Ae 1 p h T = + 3 ρ 3 6Φ DBI µν 1 B associated with the DBI part T DBI momentum of the energy

23 The pressure perturbation is given by δ p = p δρ + 6δΦ ρ 1 3 B and hence, the nonadiabatic perturbation cancellation scenario is realized by δρ 1 δ ρ + Φ 3 B = 6 0 as an initial condition outside the causal horizon = d dt a H a 1 1/ c 0

24 The dilaton Retaining only the dominant terms, the dilaton perturbation satisfies 1 δφ + 3HδΦ δφ = 0 a with the solution in k-space δφ = 3/ ( c ) 3/ 4 k a J d k Then, once the perturbations enter the causal horizon d c (but are still outside the acoustic horizon d s ), undergoes rapid damped oscillations, so that the nonadiabatic perturbation associated with is destroyed. This means that the nonadiabatic perturbations are not automatically preserved except at long, i.e., superhorizon, wavelengths where the simple Chaplygin gas has no problem anyway.

25 The Kalb-Ramond field The temporal gauge ansatz B B a i k 0i = 0 ij = εijk B B = 4 B B j Retaining the dominant terms, the field equation takes the form + = i j i d B B, ji κ A B 3 dt a a ρ a 0

26 The last term is of the order c s H compared with the first term of the order H and may be neglected. The equation is further simplified to d dt a a i j B B, ji 0 3 = If we make the decomposition i i i B = B + B i = 0 i B the key point becomes evident: whereas the longitudinal part B i suffers the same problem as, the transverse part B i does not experience spatial gradients

27 The longitudinal equation (in k-space) d k B k dt a a ( ) B k ( ) = has the solution 0 3 ( ) 5/ 4 B k = a J5/ ( dck) Hence, the longitudinal term B /a 4 oscillates with an amplitude decreasing as a -

28 The transverse part satisfies d dt B = a 0 B B const i i = 4 a H0 a Hence, the transverse term grows linearly with the scale factor a. Owing to this we can arrange nonadiabatic perturbations such that i i B B 3δ = 0 4 a δ p = 0

29 We have achieved the cancellation of nonadiabatic perturbations with the assurance that this will hold independent of scale until a~1. With vanishing c s, the spatial gradient term is absent, and the density contrast satisfies 3 δ + δ δ = 0 δ > H H ; 0 with the growing mode solution δ a which is our main result: the growing mode overdensities here do not display the damped oscillations of the simple Chaplygin gas below d s but grow as dust. We remark that it matters little that this applies only for > 0 since the Zel'dovich approximation implies that 9% ends up in overdense regions.

30 Conclusions The estimate presented here can be taken as a starting point for investigating questions beyond linear theory, in the complete general-relativistic perturbation analysis including the electric-type fieldb 0i. One might hope that the acoustic horizon does resurrect at very small scales to provide the constant density cores seen in galaxies dominated by dark matter. The ``magnetic'' nature of the Kalb-Ramond field could generate rotation - it is pertinent to note that the simple, nonrotating, selfgravitating Chaplygin gas has a scaling solution ρ~r -/3 far different from the ρ~r -/3 wanted for flat rotation curves. Adding new degrees of freedom to some extent spoils the simplicity of quartessence unification. Ultimately, the model must be confronted with large-scale structure and the CMB.

31 The DBI Lagrangian is invariant under the Kalb-Ramond gauge transformations

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