Cosmology, Scalar Fields and Hydrodynamics

Transcription

1 Cosmology, Scalar Fields and Hydrodynamics Alexander Vikman (CERN)

2 THIS TALK IS BASED ON WORK IN PROGRESS AND Imperfect Dark Energy from Kinetic Gravity Braiding arxiv: [hep-th], JCAP 1010:026, 2010 The Imperfect Fluid behind Kinetic Gravity Braiding arxiv: [hep-th] IN COLLABORATION WITH Cédric Deffayet, Oriol Pujolàs & Ignacy Sawicki

3 Good news: now cosmologists have some understanding about the processes in the very early universe when it was seconds young! - Inflation Image Credit WMAP Science Team Bad news: we do not know what the universe is made of now % is Dark: Dark Matter (DM) & Dark Energy (DE) Image Credit: NASA Excellent news - a lot of work to do for physicists!

4 ISOTROPY AND HOMOGENEITY OF THE FRIEDMANN UNIVERSE T µν =(E + P ) u µ u ν Pg µν One can characterize the universe by energy density and pressure (and equation of state ) P w X = P/E A perfect fluid or a simple scalar field have this Energy - Momentum Tensor (EMT) on all backgrounds E

5 WHAT DO WE KNOW ABOUT DARK ENERGY? It is accelerating the universe and is Dark Energy scale 10 3 ev the same as for DM approximately 75% of the energy budget of the universe today P E equation of state up to ±5-10% Λ Is it just? Cosmological Constant Problem...

6 MAY BE DARK ENERGY IS NOT JUST Λ?

7 INDEED, THERE WAS INFLATION- ANOTHER STAGE OF THE ACCELERATED EXPANSION IN THE VERY EARLY UNIVERSE

8 WHAT DO WE KNOW ABOUT INFLATON? It accelerated the universe (how Dark is it???) Mass scale GeV dominated the universe when it was seconds young P E equation of state up to % lasted at least couple of dozens of cosmic times

9 MAIN RESULTS OF THE SIMPLEST INFLATION the universe is huge and spatially flat Image Credit WMAP Science Team the cosmological perturbations e.g. Newtonian potential are Gaussian with the power spectrum speed of sound c 2 s ( P E δ 2 Φ (k) ) φ δ Φ (k) E c s (1 + P/E) H=c s k : fluctuation of on length scale δ 2 h µν (k) E Φ Garriga & Mukhanov 99 l = k 1 for the Gravity Waves H=k n s 1= d ln δ2 Φ d ln k =

10 CAN ONE GO BEYOND THE PERFECT FLUID, BUT STILL KEEPING ONLY ONE SINGLE DEGREE OF FREEDOM?

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12 φ g µν BRAIDING METRIC WITH A SCALAR FIELD- Kinetic Gravity Braiding

13 WHAT IS KINETIC GRAVITY BRAIDING? S φ = d 4 x g [K (φ,x)+g (φ,x) φ] k-inflation/essence, Armendariz-Picon, Damour, Mukhanov, Steinhardt 1999/2000 where X 1 2 gµν µ φ ν φ Minimal coupling to gravity S tot = S φ + S EH However, derivatives of the metric are coupled to the derivatives of the scalar, provided G X 0 shift-symmetry: theory is not parity symmetric: φ φ + c φ φ

14 ACTION FOR KINETIC GRAVITY BRAIDING IS SIMILAR TO EINSTEIN-HILBERT ACTION The second derivatives (higher derivative -HD) enter the action but only linearly One can eliminate the HD(in time) only by breaking the Lorentz-invariant formulation of the theory. Boundary terms are required! Despite the HD in the action, the equations of motion are still of the 2nd order: NO new degrees of freedom - NO Ostrogradsky s ghosts order of equations of motion >2 implies Hamiltonian unbounded from below - ghosts

15 KINETIC GRAVITY BRAIDING IS SIMILAR TO GALILEON ( Nicolis, Rattazzi, Trincherini 2008) BUT Does not require the Galilean symmetry: φ φ + c General functions µ φ µ φ + c µ K (φ,x) Minimal coupling to gravity, NO NO higher order therms like φt µ µ & & G (φ,x) φ ;λ φ ;λ (( φ) 2 φ ;µν φ ;µν 1 4 φ ;µφ ;µ R General Galileon Indeed, manifestly healthy Galileons are NEVER Galilean symmetric!!! Deffayet, Esposito-Farese, AV, 2009 ) DGP in decoupling limit Kinetic Gravity Braiding K-Essence, DBI K (φ,x)

16 EXPANSIONS IN GRADIENT TERMS K-Essence, DBI etc K (φ,x) X ( 1+c 1 (φ) X + c 2 (φ) X ) Kinetic Gravity Braiding integrate the canonical kinetic energy by parts G (φ,x) φ φ φ ( 1 + c 1 (φ) X + c 2 (φ) X )

17 EQUATION OF MOTION I L µν µ ν φ +( α β φ) Q αβµν ( µ ν φ)+ +Z G X R µν µ φ ν φ =0 Braiding EOM is of the second order: L µν, Q αβµν, Z constructed from field and it s first derivatives Q αβµν is such that EOM is a 4D Lorentzian generalization of the Monge-Ampère Equation, always linear in φ

18 EQUATION OF MOTION II Shift-Charge Current: J µ J µ =(L X 2G φ ) µ φ G X µ X New Equivalent Lagrangian: P P = K 2XG φ G X λ φ λ X Equation of motion is a conservation law : µ J µ = P φ

19 BRAIDING Einstein Equations (φ, φ, φ, g, g, g) =0 φeom (φ, φ, φ, g, g, g) =0 Cannot solve separately!!!! characteristics (cones of propagation ) depend on external matter

20 IMPERFECT FLUID FOR TIMELIKE GRADIENTS Four velocity : u µ µφ 2X φ is an internal clock projector: Time derivative: Expansion : µν = g µν u µ u ν ( ) d dτ uλ λ θ λ µ λ u µ = V /V Shift-symmetry φ φ + c violates φ φ and introduces arrow of time comoving volume

21 EFFECTIVE MASS & CHEMICAL POTENTIAL κ 2XG X charge density: n J µ u µ = n 0 + κθ Braiding energy density: E T µν u µ u ν = E 0 + θ φκ effective mass per shift-charge / chemical potential: ( E ) m n V,φ = 2X = φ

22 SHIFT-CURRENT AND DIFFUSION J µ = nu µ κ m λ µ λ m Diffusion 59, L&L, vol. 6 κ 2XG X Is a diffusivity / transport coefficient

23 IMPERFECT FLUID ENERGY-MOMENTUM TENSOR Pressure P 1 3 T µν µν = P 0 κṁ Energy Flow q µ µλ T λ ν u ν = m µν J ν q µ = κ ν µ ν m No Heat Flux! Energy Momentum Tensor T µν = Eu µ u ν µν P +2u (µ q ν) Solving for ṁ for small gradients or small κ one obtains bulk viscosity

24 DIFFUSION OF CHARGE For incompressible motion θ 0 equation of motion is: (D ) ṅ = µ µ n + Da µ µ n where the diffusion constant: D κ c.f. 59, L&L, vol. 6, p 232 n m m 4-acceleration: a µ u µ spatial gradient: µ ν µ ν

25 ENERGY CONSERVATION IN COMOVING VOLUME Energy conservation: u ν µ T µν =0 de = PdV + mdn dif Euler relation: E = mn P 0 Momentum conservation: µν λ T λν =0

26 VACUUM-ATTRACTORS Euler relation: E = mn P 0 for no particles: n =0 E = P κ ṁ almost ds!

27 COSMOLOGY q µ =0 and θ =3H Friedmann Equation: H 2 = κmh (E 0 + ρ ext ) r 1 c = κm crossover scale in DGP

28 CHARGE CONSERVATION ṅ +3Hn = P φ If there is shift-symmetry then P φ =0 n a 3

29 INFLATION BRINGS THE SCALAR TO ATTRACTOR n =0

30 EXAMPLE: SIMPLEST IMPERFECT DARK ENERGY Only one free parameter µ Lagrangian L = X ( 1+µ φ) shift-charge density n = m (3µHm 1)

31 NONTRIVIAL ATTRACTOR No Particles: n =0 m = (3µH) 1 m =0 H 2 = 1 6 ρ ext ( (µρ ext) 2 ) H 2 = 1 3 ρ ext STABLE GHOSTY

32 DARK ENERGY assume today 3 2 µρ ext 1 H µ 1 Λ µ 1 3ρ CDM 3 2 ρ CDMµ Mass Scale µ 1/3 (H 2 0 M Pl ) 1/ ev Length Scale: 1000 km In Quintessence - the size of the universe

33 HIGH FREQUENCY STABILITY Effective metric for perturbations No Ghosts G µν = Du µ u ν + Ω µν 2κ m K µν 2κ m a (µ u ν) D = E m κθ m κ2 Ω = n + λ Extrinsic curvature for φ = const ( κu λ ) In general propagation is anisotropic, but in cosmology: c 2 s = m Ωm 2κH md 1 2 κ2

34 SOUND SPEED c 2 s = P m + 2 κ + κ (4H κm/2) E m 3κ (H κm/2) Ṗ E The relation between the equation of state, the sound speed and the presence of ghosts is very different from the k-essence & perfect fluid. w X < 1 A manifestly stable Phantom ( ) is possible even with a single degree of freedom and minimal coupling to gravity

35 Phantom Attractor Crossing Ghosty Total Energy negative 2µX Pressure singularity Phase portrait for scalar field & dust

36 Ω X w X c 2 s log a Evolution of dark energy properties in the Friedmann universe also containing dust and radiation. The scalar evolves on its attractor throughout the presented period. During matter domination w X = 2, while w X = 7/3 during radiation domination. The sound speed is superluminal when the scalar energy density is subdominant, becoming subluminal when Ω X 0.1 and w X 1.4

37 Ω X w X c 2 s log a Evolution of DE properties in the Friedmann universe which also contains dust and radiation. The energy density in the scalar is J-dominated (off attractor) until a transition during the matter domination epoch. This allows the scalar to increase its contribution to the total energy budget throughout radiation domination (w X =1/6) and provide an early DE peaked at matterradiation equality, from whence it begins to decline with w X =1/4. The transition to the attractor behaviour is rapid. The equation of state crosses w X = 1 and the scalar energy density begins to grow. The final stages of evolution are on the attractor and are similar to those presented in previous figure.

38 w X wx w X0 0.1 < Ω m < 0.5 and Ω Xeq < 0.1. The shading contours correspond to the energy density of DE today Ω X0. Two parameterisations of DE behaviour are shown: w X and w X evaluated today, and w X evaluated today and at z =1/2. The requirement that the energy density in DE at matter-radiation equality be small, Ω eq X < 0.1 forces the value of the shift charge to be small today Q 0 < This means that in the most recent history, the evolution has effectively been on attractor or very close to it and the permitted value of w X is very restricted and determined to all intents and purposes by Ω 0 X

39 w X wx w X The shading representing the contribution of DE to energy density at matterradiation equality. We choose to cut the parameters such that the contribution to this early DE at that time is no larger than 10%. It can clearly be seen that values of w X closer to 1 are obtained when the shift charge is larger, but this leads to more early DE, eventually disagreeing with current constraints

40 FURTHER DEVELOPMENT Kinetic Gravity Braiding with G X n arxiv: v2 [astro-ph.co], Rampei Kimura, Kazuhiro Yamamoto

41 CONSTRAINTS FROM CMB AND SN IA Kinetic Gravity Braiding with G X n arxiv: v2 [astro-ph.co], Rampei Kimura, Kazuhiro Yamamoto SN Ia CMB The SCP Union2 Compilation is a collection of 557 type Ia supernovae data whose range of the redshift is < z < 1.4 Thus n 3 mass scale 10 3 ev Length Scale: 1/10 mm

42 GROWTH FACTOR Kinetic Gravity Braiding with G X n arxiv: v2 [astro-ph.co], Rampei Kimura, Kazuhiro Yamamoto

43 EFFECTIVE NEWTON CONSTANT FOR PERTURBATIONS AND THE SOUND SPEED Kinetic Gravity Braiding with G X n arxiv: v2 [astro-ph.co], Rampei Kimura, Kazuhiro Yamamoto

44 FOR INFLATIONARY MODELS WITH BRAIDING SEE G-(& GENERALIZED G)-INFLATION T. Kobayashi, M. Yamaguchi, J. Yokoyama arxiv: [hep-th] arxiv: [hep-th]

45 CONCLUSIONS I φ Scalar field with a non-canonical action can behave like imperfect fluid: on general (not exactly isotropic and homogeneous FRW) background: T µν Eu µ u ν µν P Fluid elements (particles) are shift-charges: charges with respect to: φ φ + c φ kinetically mixes / braids with the metric: ( φ) 2 µ ( gg µν ν φ ) c.f. F (1) µν (A α ) F (2)µν (B β )

46 CONCLUSIONS II Manifestly stable (no ghosts and no gradient instabilities) and large violation of the Null Energy Condition (NEC) is possible even in theories minimally coupled to gravity: healthy Phantom with w< 1 n =0 Vanishing shift-charge,, corresponds to cosmological attractors similar to Ghost Condensate / bad k-inflation. But here these attractors can be manifestly stable (no ghosts and no gradient instabilities) and their exact properties depend on external matter. Through Euler relation these attractors generically E = mn P 0 evolve to de Sitter in late time asymptotic. Interesting for DE & Inflation!

47 THANKS A LOT FOR YOUR ATTENTION!