Dark Energy - Dark Matter Unification: Generalized Chaplygin Gas - PDF document

1 Dark Energy - Dark Matter Unification: Generalized Chaplygin Gas Model • Type Ia Supernovae and Accelerated Expansion • Quintessence • Quintessence and the Brane • Generalized Chaplygin Gas Model CMBR Constraints Supernovae and Gravitational Lensing Constraints Structure Formation M.C. Bento, O. B., A.A. Sen Phys. Rev. D66 (2002) 043507; D67 (2003) 063003; D70 (2004) 083519 Phys. Lett. B575 (2003) 172; Gen. Rel. Grav. 35 (2003) 2063 O. B., A.A. Sen, S. Sen, P.T. Silva Mon. Not. Roy. Astron. Soc. 353 (2004) 329 P.T. Silva, O. B. Astrophys. J. 599 (2003) 829 O. B. astro-ph/0403310; astro-ph/0504275 M.C. Bento, O. B., N.M. Santos, A.A. Sen Phys. Rev. D71 (2005) 063501 Orfeu Bertolami Instituto Superior T´ ecnico, Depto. F´ ısica, Lisbon KIAS-APCTP-DMRC Workshop on ”The Dark Side of the Universe” 24-26 May, 2005, KIAS, Seoul, Korea

2 Type Ia Supernovae and Accelerated Expansion Study of recently discovered Type Ia Supernovae with z ≥ 0 . 35 indicates that the deceleration parameter q 0 ≡ − ¨ a a a 2 , ˙ where a ( t ) is the scale factor, is negative − 1 < ∼ q 0 < 0 . [Permutter et al. 1998; Riess et al. 1999] For an homogeneous and isotropic expanding geometry driven by the vac- uum energy, Ω V and matter Ω M with Eqs. of state of the form p = ωρ − 1 ≤ ω ≤ 1 , it follows from the Friedmann and Raychaudhuri Eqs. q 0 = 1 2(3 ω + 1)Ω M − Ω Λ . A negative q 0 suggests that a dark energy, an “invisible” smoothly dis- tributed energy density, is the dominant component. This energy density can have its origin either on a non-vanishing cosmological constant, Λ , or on a dynamical vacuum energy, “quintessence”, Ω Q ( ω Q < − 1 / 3 ).

3 Cosmological Constraints Observational Parameters Ω M ≃ 0 . 30 Ω Λ ,Q ≃ 0 . 70 Ω k ≃ 0 H 0 = 100 h km s − 1 Mpc − 1 , h = 0 . 71 Big-Bang Nucleosynthesis (BBN) V ( φ ) = V 0 exp ( − λφ ) Ω φ < 0 . 045 (2 σ ) ⇒ λ > 9 [Bean, Hansen, Melchiorri 2001] Cosmic Microwave Background ω φ < − 0 . 6 (2 σ ) Flat models > 6 Ω φ < 0 . 39 (2 σ ) ⇒ λ ∼ [Efstathiou 1999]

4 Figure 1: Concordance Model (latest)

5 Some Ideas - V 0 exp ( − λφ ) (Troublesome on the brane) [Ratra, Peebles 1988; Wetterich 1988; Ferreira, Joyce 1998] - V 0 φ − α , α > 0 (Fine on the brane for 2 < α < 6 ) [Ratra, Peebles 1988] - V 0 φ − α exp ( λφ 2 ) , α > 0 [Brax, Martin 1999, 2000] - V 0 [exp ( M p /φ ) − 1] [Zlatev, Wang, Steinhardt 1999] - V 0 (cosh λφ − 1) p [Sahni, Wang 2000] - V 0 sinh − α ( λφ ) [Sahni, Starobinsky 2000; Ure˜ na-L´ opez, Matos 2000] - V 0 [exp ( βφ ) + exp ( γφ )] [Barreiro, Copeland, Nunes 2000] - Scalar-Tensor Theories of Gravity [Uzan 1999; Amendola 1999; O.B., Martins 2000; Fujii 2000; ...] - V 0 exp ( − λφ )[ A + ( φ − B ) 2 ] [Albrecht, Skordis 2000] - V 0 exp ( − λφ )[ a +( φ − φ 0 ) 2 + b ( ψ − ψ 0 ) 2 + c φ ( ψ − ψ 0 ) 2 + d ψ ( φ − φ 0 ) 2 ] [Bento, O.B., Santos 2002]

6 Quintessence and the Brane Brane-World Scenarios [L. Randall, R. Sundrum 1999, ...] - 5 -dim AdS spacetime in the bulk with matter confined on a 3 -brane ⇒ 4 -dimensional Einstein Eqs. � 8 π � 2 G µν = − Λ g µν + 8 π T µν + S µν − E µν . M 2 M 3 P 5 [Shiromizu, Maeda, Sasaki 2000] If T µν is the energy-momentum of a perfect fluid on the brane, then S µν = 1 2 ρ 2 u µ u ν + 1 12 ρ ( ρ + 2 p ) h µν , ρ , p are the energy density and isotropic pressure of a fluid with 4 -velocity u µ , h µν = g µν + u µ u ν , E µν = − 6 k 2 λ [ ǫ ( u µ u ν + 1 3 h µν ) + P µν + Q µ u ν + Q ν u µ ] , so that k 2 ≡ 8 π/M 2 P (GR limit λ − 1 → 0 ) and the tensors P µν and Q µ correspond to non-local contributions to pressure and flux of energy. For a perfect fluid P µν = Q µ = 0 and ǫ = ǫ 0 a − 4 . The 4 -dimensional cosmological constant is related to the 5 -dimensional one and the 3 -brane tension, λ : Λ = 4 π � Λ 5 + 4 π � λ 2 M 3 3 M 3 5 5 while the Planck scale is given by � M 3 3 5 √ M P = . 4 π λ

7 In a cosmological setting, where the 3 -brane resembles our Universe and the metric projected onto the brane is an homogeneous and isotropic flat Robertson-Walker metric, the generalized Friedmann Eq. reads � 8 π � 4 π H 2 = Λ � � ρ 2 + ǫ 0 3 + ρ + a 4 . 3 M 2 3 M 3 P 5 [Bin´ etruy, Deffayet, Ellwanger, Langlois 2000] [Flanagan, Tye, Wasserman 2000] 5 and dropping the term ǫ 0 a − 4 which quickly Choosing Λ 5 ≃ − 4 πλ 2 / 3 M 3 vanishes after inflation: 8 π 1 + ρ H 2 = � � ρ . 3 M 2 2 λ P Extra brane term: Beneficial for some quintessence models, but harmful for some others! [Mizuno, Maeda 2001]

8 Generalized Chaplygin Gas • Radical new idea: change of behaviour of the missing energy density might be controlled by the change in the equation of state of the back- ground fluid. • Interesting case: Chaplygin gas, described by the Eq. of state p = − A ρ α , with α = 1 and A a positive constant. • From the relativistic energy conservation Eq., within the framework of a Friedmann-Robertson-Walker cosmology, � A + B ρ = a 6 , where B is an integration constant. √ Ba − 3 , • Smooth interpolation between a dust dominated phase where, ρ ≃ and a De Sitter phase where p ≃ − ρ , through an intermediate regime de- scribed by the equation of state for “stiff” matter, p = ρ . [Kamenshchik, Moschella, Pasquier 2001] This setup admits a brane interpretation via a parametrization invariant Nambu-Goto d -brane action in a ( d + 1 , 1) spacetime. This action leads, in the light-cone parametrization, to the Poincar´ e-invariant Born-Infeld action in a ( d, 1) spacetime. The Chaplygin is the only known gas to admit a supersymmetric generalization. [Jackiw 2000]

9 • Bearing on the observed accelerated expansion of the Universe: Eq. of √ state is asymptotically dominated by a cosmological constant, 8 πG A . • Inhomogeneous generalization can be regarded as a dark energy - dark matter unification. [Bili´ c, Tupper, Viollier 2001] [Bento, O.B., Sen 2002]

10 A Model ( 0 < α ≤ 1 ) • Lagrangian density for a massive complex scalar field, Φ : L = g µν Φ ∗ ,µ Φ ,ν − V ( | Φ | 2 ) . φ Writing Φ = ( 2 m ) exp( − imθ ) in terms of its mass, m : √ L = 1 � φ 2 θ ,µ θ ,ν + 1 � 2 g µν − V ( φ 2 / 2) . m 2 φ ,µ φ ,ν Scale of inhomogeneities arises from the assumption: φ ,µ << mφ . • Lagrangian density in this “Thomas-Fermi” approximation: L TF = φ 2 2 g µν θ ,µ θ ,ν − V ( φ 2 / 2) . • Equations of motion: g µν θ ,µ θ ,ν = V ′ ( φ 2 / 2) , ( φ 2 √− gg µν θ ,ν ) ,µ = 0 , where V ′ ( x ) ≡ dV/dx . Phase θ can be regarded as a velocity field whether V ′ > 0 , that is U µ = g µν θ ,ν √ , V ′ so that, on the mass shell, U µ U µ = 1 . • Energy-momentum tensor takes the form of a perfect fluid: ρ = φ 2 2 V ′ + V , p = φ 2 2 V ′ − V .

11 • Covariant conservation of the energy-momentum tensor ρ + 3 H ( p + ρ ) = 0 , ˙ where H = ˙ a/a , leads for the generalized Chaplygin gas 1 � B � 1+ α ρ = A + . a 3(1+ α ) Furthermore d ln φ 2 = d ( ρ − p ) , ρ + p which, together with the Eq. of state implies that: φ 2 ( ρ ) = ρ α ( ρ 1+ α − A ) 1 − α 1+ α . • Generalized Born-Infeld theory: � α 1+ α , � 1 1+ α 1 − ( g µν θ ,µ θ ,ν ) L GBI = − A 1+ α 2 α which reproduces the Born-Infeld Lagrangian density for α = 1 . • L GBI can be regarded as a d -brane plus soft correcting terms as can be seen from the expansion around α = 1 : � α √ 1 − X + X log( X ) + (1 − X ) log(1 − X ) 1+ α = � 1+ α √ 1 − X (1 − α ) 2 α 4 1 − X + E + F + G 32(1 − X ) 3 / 2 (1 − α ) 2 + O ((1 − α ) 3 ) , where X ≡ g µν θ ,µ θ ,ν and E = X ( X − 2) log 2 ( X ) , F = − 2 X ( X − 1) log( X )[log(1 − X ) − 2] , G = ( X − 1) 2 [log(1 − X ) − 4] log(1 − X ) .

12 • Potential arising from the model V = ρ 1+ α + A � � = 1 Ψ 2 /α + A , 2 ρ α Ψ 2 2 where Ψ ≡ B − (1 − α/ 1+ α ) a 3(1 − α ) φ 2 , which reduces to the duality invariant, φ 2 → A/φ 2 , and scale-factor independent potential for the Chaplygin gas. • Intermediate regime between the dust dominated phase and the De Sitter phase: � � 1 B 1+ α a − 3(1+ α ) , 1 1+ α + ρ ≃ A α 1 + α A � � α B 1+ α a − 3(1+ α ) , 1 1+ α + p ≃ − A α 1 + α A 1 1+ α and matter which corresponds to a mixture of vacuum energy density A described by the “soft” equation of state: p = αρ .

13 Generalized d−brane α α+1 ( ) µν α+1 ) 2 α L = 1 − ( g Θ Θ ,µ ,ν ( α = 1: d−brane) Generalized Chaplygin gas A p = − α ρ ( α = 1: Chaplygin gas) p << ρ p = α ρ p = − ρ Dust De Sitter ( =1: stiff matter) α Figure 2: Cosmological evolution of a universe described by a generalized Chaplygin gas equation of state. Treatment of the Inhomogeneities • Second equation of motion admits as first integral a position dependent function B ( � r ) , after a convenient choice of comoving coordinates where 0 / √ g 00 . An induced induced 3 -metric the velocity field is given by U µ = δ µ γ ij = g i 0 g j 0 − g ij g 00 with determinant γ ≡ − g/g 00 can be built after choosing the proper time, dτ = √ g 00 dx 0 . For the relevant scales, function B ( � r ) can be regarded as approximately constant, hence