CHAPLYGIN GAS CHAPLYGIN GAS COSMOLOGY COSMOLOGY ––unification of unification of dark matter and dark energydark matter and dark energy

Neven Bili�Theoretical Physic DivisionRu�er Boškovi� Institute

IRGAC 2006

Chaplygin gasChaplygin gas

An exotic fluid with an equation of state

The first definite model for a dark matter/energy unification

A. Kamenshchik, U. Moschella, V. Pasquier, PLB 511 (2001) N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002) J.C. Fabris, S.V.B. Goncalves, P.E. de Souza, GRG 34 (2002)

ρA

p −=

The generalized Chaplygin gas

M.C. Bento, O. Bertolami, and A.A. Sen, PRD 66 (2002)

The term “quartessence” was coined to describe unified dark matter/dark energy models

M. Makler, S.Q. de Oliveira, and I. Waga, PLB 555 (2003)

10 ≤≤−= αραA

p

• The Chaplygin gas model is equivalent to the (scalar) Dirac-Born-Infeld description of a D-braneR. Jackiw, Lectures on Fluid Mechanics (Springer, 2002)

• String theory branes possess three features that are absent in the simple Nambu Goto p-dimmembranes

• (i) support an Abelian gauge fieldA� 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, 2003.

DD--branesbranes and the and the DiracDirac--BornBorn--InfeldInfeld(DBI) theory(DBI) theory

the induced metric (“pull back”) of the bulk metric

νµµν x

X

x

XGg

ba

ab ∂∂

∂∂=ind

� +−−= Φ−+ )(det)1( ind1DBI BgexdAS pp

µνµνµν απ FB ′+= 2B

the antisymetric tensor field

xi

x0

Xa

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

10

3...0for

444 −==

==

GG

gG

µ

µνµν µ

Scalar BornScalar Born--InfeldInfeld theorytheory

If we neglect the dilaton and the B -field, we find

which is consistent with a perfect fluid description

4 ind 4BI BIdet ( )S A d x g d x= − − =� � L

2BI 1A θ= − −L

2BI

, , BI22T gµν µ ν µν

θ θ θθ

∂= −∂L

L

( )T p u u pgµν µ ν µνρ= + − ,

2u µ

µθθ

=with

2, ,g µνµ νθ θ θ=

and hence

In a homogeneous model the conservation equation yields theChaplygin gas density as a function of the scale factor a

where B is an integration constant.The Chaplygin gas thus interpolatesbetween dust ( ~a

-3 ) at large redshifts and a cosmological constant( ~ A½) today and hence yields a correct homogeneous cosmology

Ap

ρ= −

2

2; 1

1

Ap Aρ θ

θ= = − −

−

6Bp A

a= +

The inhomogeneous Chaplygin gas The inhomogeneous Chaplygin gas and structure formationand structure formation

• The inhomogeneous Chaplygin gas based ona 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, whereasin voids, w=p/ is driven to the lower bound -1 producingacceleration as dark energy.

• 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 (2004)

and the CMBD. Carturan and F. Finelli, PRD 68 (2003);

• L. Amendola, F. Finelli, C. Burigana, and D. Carturan, JCAP 07 (2003)

The physical reason is that although the adiabatic speed of sound

is small until a ~ 1, the accumulated comovingacoustic horizon

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 (2004)

2s

S

pc

ρ∂=∂

1 7 / 2ss 0

cd dt H a

a−= � �

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 aeq = 3 · 10-4 for the comovingwavelength 1/k = 0.34 Mpc, k (aeq ) =0.004 (solid)

k (aeq ) =0.005 (dashed).

The root of the structure formation problem is the last term, as may be understood if we solve the equation at linear order

with the solution

Hence damped oscillations at the scales below ds

22 s

2

32

2

cH H

aδ δ δ δ+ − − ∆ = 0�� �

1/ 45/ 4 s

s

ss2

( )

for 1

cos ( )for 1

k

k

k

a J d k

a d k

d kd k

a

δ

δ

δ

−=

� �

� �

behaving asymptotically as

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 (2004)

Such a model is equivalent to the ghost condensate and hence shares its peculiarities

N. Arkani-Hamed et al, JHEP 05 (2004) A. Krause and S. P. Ng, IJMP A 21 (2006) 1091

• The generalized Chaplygin gas in a modified gravity approach, reminiscent of Cardassian models

T. Barreiro and A.A. Sen, PRD 70 (2004)

• Another deformation of the Chaplygin gasM. Novello, M. Makler, L.S. Werneck and C.A. Romero, PRD 71 (2005)

• If there are entropy perturbations such that the pressure perturbation p=0 and with it the acoustic horizon ds=0, no problem arises

R.R.R. Reis, I. Waga, M.O. Calvão, and S.E. Jo�as, PRD 68 (2003)

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 �

NonadiabaticNonadiabatic perturbationsperturbationsSuppose 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 formp=p( , ) s=s( , )

The corresponding perturbations

p pp

s ss

δ δρ δφρ φ

δ δρ δφρ φ

∂ ∂= +∂ ∂∂ ∂= +∂ ∂

The speed of sound is the sum of two nonadiabaticterms

Thus, even for a nonzero p/ ρ the speed of soundmay 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 cs=0.

1

2s

S

p p s s pc

pρ ρ φ φ

−� �∂ ∂ ∂ ∂ ∂≡ = − � �∂ ∂ ∂ ∂ ∂� �

NonadiabaticNonadiabatic perturbations in perturbations in the DBI theorythe 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

Abbreviations:

4DBI DBI

2 2 2 2DBI

det

(1 )(1 ) ( )

S d x g

A e θ θ θ−Φ

= −

= − − + + − ∗

� L

L B B B B

2, ,

2

2, ,

1

2

det

g

g

µνµ ν

µνµν

µνρσµν ρσ

µ νρµ ν ρ

θ θ θ

ε

θ θ θ θ

=

=

∗ =

=

1

8 -

B B B

B B B B

B B B

It is convenient to write everything in Einstein's frame using the transformation

2g e gµν µνΦ→

2, ,e eµ µ µν µνθ θΦ Φ→ →B B

and simultaneously rescaling

which gives

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 uand h . Hence

b DBIT T Tµν µν µν= +

;T u u ph

h g u u

µν µ ν µν µν

µν µν µ ν

ρ= − + Π

= −

1;

3T u u p T hµν µ ν µν µνρ = = −

Implications for linear perturbationsImplications for linear perturbationsA convenient choice is the temporal (synchronous) gauge

First-order perturbations

Homogeneity and isotropy imply

Hence, �,i , �,i and (B )2 count as first order

2 2 2 ( ) i jij ijds dt a f dx dxδ= − −

ρ ρδρ−= θδ δ θ θΦ = Φ − Φ = −

, ,0 0 0i i µνθΦ = = =B

With these assumptions we may neglect higher-order terms in T and we find simple expressions for the density and the pressure

associated with the DBI part TDBI of the energy momentum

2DBI 3

2

6DBI 2

1

1

1 11

3 3

u u T Ae

Aep h T

µ νµν

µνµν

ρθ

ρ

Φ

Φ

+≡ =−� �≡ − = − +� �� �

B

B

216

3p p

δρδ δρ

� �= − + Φ −� �� �

B

216 0

3

δρ δρ

+ Φ − =B

The pressure perturbation is given by

1 1/ 2c 0d dt a H a−= � �

and hence, the nonadiabatic perturbation cancellation scenario is realized by

as an initial condition outside the causal horizon

The The dilatondilatonRetaining only the dominant terms, the dilaton perturbationsatisfies

with the solution in k-space

Then, once the perturbations enter the causal horizon dc (but are still outside the acoustic horizon ds ), 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.

2

13H

aδ δ δΦ + Φ − ∆ Φ = 0�� �

3/ 43/ 2 c( )k a J d kδ −Φ =

The KalbThe Kalb--RamondRamond fieldfield

The temporal gauge ansatz

20 4

0i j

ki ij ijk

B BB

aε= = =B B B

Retaining the dominant terms, the field equation takes the form

2,

3

20

ji ijiBd B A B

dt a a a

κρ

� �− + =� �

� �

�

i i iB B B⊥= +� 0i

i B⊥∂ =

If we make the decomposition

the key point becomes evident: whereas the longitudinal part Bi suffers the same problemas , the transverse part Bi does not experience spatial gradients

The last term is of the order cs2 H2 compared with

the first term of the order H2 and may be neglected.The equation is further simplified to

,

30

jijiBd B

dt a a

� �− =� �

� �

�

( ) ( )2

30

B kd kB k

dt a a

� �− =� �� �

� �

�

�

�

The longitudinal equation (in k-space)

has the solution

( ) 5/ 45/ 2 c( )B k a J d k=

�

Hence, the longitudinal term B2/a4 oscillates withan amplitude decreasing as a

-2

43 0

i iB B

aδ⊥ ⊥ − =

0Bd

dt a⊥� �

=� �� �

�

4 20

consti iB Ba

a H⊥ ⊥ =

The transverse part satisfies

�

Hence, the transverse term grows linearly with the scale factor a. Owing to this we can arrange nonadiabatic perturbations such that

0pδ =�

We have achieved the cancellation of nonadiabatic perturbations with the assurance that this will hold independent of scale until a~1. With vanishing cs

2, the spatial gradient term is absent, and the density contrast satisfies

with the growing mode solution

which is our main result: the growing mode overdensities here donot display the damped oscillations of the simple Chaplygin gas below ds but grow as dust. We remark that it matters little that this applies only for > 0 since the Zel'dovich approximationimplies that 92% ends up in overdense regions.

232 ; 0

2H Hδ δ δ δ+ − = 0 >�� �

aδ �

ConclusionsConclusions• 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 field B 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, self-gravitating Chaplygin gas has a scaling solution ρ~r-2/3 far different from the ρ~r-2/3 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.

The DBI Lagrangian is invariant under

the Kalb-Ramond gauge transformations

• A. Kamenshchik, U.\ Moschella, and V.\ Pasquier, Phys.\ Lett.\ B{\bf 511} (2001) 265.•• N.\ Bili\'{c}, G.B.\ Tupper, and R.D.\ Viollier, Phys.\ Lett.\ B{\bf 535} (2002) 17.•• J.C.\ Fabris, S.V.B.\ Goncalves, P.E.\ de Souza, Gen.\ Rel.\ Grav.\ {\bf 34} (2002) 53; 2111.•• M.C. Bento, O. Bertolami, and A.A. Sen, Phys.\ Rev.\ D{\bf 66} (2002) 043507.

• M. Makler, S.Q. de Oliveira, and I. Waga, PLB 555 (2003) 1.

• C.V. Johnson, D-Branes, Cambridge University Press, Cambridge, 2003.

• H.B.\ Sandvik, M.\ Tegmark, M.\ Zaldarriaga, and I.\ Waga, Phys.\ Rev.\ D{\bf 69} (2004) 123524.

• D.\ Carturan and F.\ Finelli, Phys.\ Rev.\ D{\bf 68} (2003) 103501;

• R.\ Bean and O.\ Dor\'{e}, Phys.\ Rev.\ D{\bf 68} (2003) 023515.

• L.\ Amendola, F.\ Finelli, C.\ Burigana, and D.\ Carturan,• JCAP {\bf 07} (2003) 005.• %

• N. B., R.J. Lindebaum, G.B. Tupper, and R.D. Viollier, JCAP 11 (2004) 008.

• R.J.\ Scherrer, Phys.\ Rev.\ Lett.\ {\bf 93} (2004) 011301.

• N.\ Arkani-Hamed, H.C.\ Cheng, M.A.\ Luty, and S.\ Mukohyama, JHEP {\bf 05} (2004) 074.

• A.~Krause and S.~P.~Ng, Int.\ J.\ Mod.\ Phys.\ A {\bf 21} (2006) 1091,• hep-th/0409241.

• T.~Barreiro and A.~A.~Sen, Phys.\ Rev.\ D {\bf 70} (2004) 124013, • astro-ph/0408185.

• K.\ Freese and M.\ Lewis, Phys.\ Lett. B{\bf 540} (2002) 1.

• M.~Novello, M.~Makler, L.~S.~Werneck and C.~A.~Romero,• Phys.\ Rev.\ D {\bf 71} (2005) 043515,• astro-ph/0501643.

• R.R.R.\ Reis, I.\ Waga, M.O.\ Calv\~{a}o, and S.E.\ Jo\'{r}as, Phys.\ Rev.\ D{\bf 68} (2003) 061302;

• M.\ Kalb and P.\ Ramond, Phys.\ Rev.\ D{\bf 9} (1974) 2273.

• N.~Bili\'c, G.~B.~Tupper and R.~D.~Viollier, JCAP {\bf 0510} (2005) 003,

• E.J.\ Chun, H.B.\ Kim, and Y.\ Kim; hep-ph/0502051.

• M.\ Bordemann and J.\ Hoppe, Phys. Lett. B{\bf 325} (1994) 359

• N.\ Ogawa, Phys.\ Rev. D{\bf 62} (2000) 085023.

• R. Jackiw, Lectures on Fluid Mechanics (Springer, 2002).

• N. Bili\'{c}, R.J.\ Lindebaum, G.B.\ Tupper, and R.D.\ Viollier, proceedings of the PhysicalCosmology Workshop, 15 Rencontres de Blois, France, 2003, astro-ph/0310181.

• N.\ Seiberg and E.\ Witten, JHEP {\bf 09} (1999) 032;

• G.W. Gibbons and C.A.R. Herdeiro, Phys. Rev.\ D{\bf 63} (2001) 064006.

• T Damour, S.\ Deser, and J.\ McCarthy, Phys.\ Rev.\ D{\bf 47} (1993) 1541.

• E.J.\ Copeland, M.R.\ Garousi, M.\ Sami, and S.\ Tsujikawa, Phys.\ Rev.\ D{\bf 71} (2005) 043003.

• M.~R.~Garousi, M.~Sami and S.~Tsujikawa, Phys.\ Rev.\ D {\bf 71} (2005) 083005, hep-th/0412002.