A fluid of diffusing particles and its cosmological behaviour - PowerPoint PPT Presentation

A fluid of diffusing particles and its cosmological behaviour Zbigniew Haba Institute of Theoretical Physics, University of Wroclaw

Content I. RHS of Einstein equations II. Relativistic diffusion on RHS III.Homogeneous metric IV. Non-homogeneous metric V. Temperature fluctuations

Classical equations The Einstein equations ( g µν is the metric ds 2 = g µν dx µ dx ν ) R µν − 1 2 g µν R = 8 π GT µν , (1) with ( T µν ) ; µ = 0 . (2)

RHS of Einstein equations ◮ phase space distribution of particles ◮ fields ◮ fluids

Phase space distribution The phase-space distribution satisfies Liouville equation (for geodesic motion, here Γ µ νρ are Christoffel symbols) ( p µ ∂ x µ − Γ k µν p µ p ν ∂ k )Φ( x , p ) = 0 . (3) The formula for the energy-momentum tensor is T µν = √ g � d p 1 ˜ p 0 p µ p ν Φ , (4) (2 π ) 3 ˜ T is conserved g is the determinant of the metric and p 0 is determined from the mass-shell condition p µ p µ = m 2 ( m is the particle’s mass, we set c = 1). Greek indices run from 0 to 3, Latin indices denoting spatial components have the range from 1 to 3, the covariant ∂ derivative is over the space-time, derivatives over the momenta ∂ p k are denoted ∂ k and ∂ x denotes a derivative over a space-time coordinate x .

Fluids Assuming we have a phase space distribution we can define v µ = � p µ � (5) Then, � p µ p ν � = � 1 � v µ v ν + � ( p µ − v µ )( p ν − v ν ) � (6) Let u µ be a normalized v µ ,i.e. g µν u µ u ν = 1 (7) Then, the identity for � p µ p ν � can be expressed as T µν = Eu µ u ν − π ( g µν − u µ u ν ) + π µν (8)

Fields If we have the action W then δ W T µν = (9) δ g µν ( x ) For the scalar field � dx √ g ( g µν ∂ µ φ∂ ν φ − V ( φ )) W = (10) Classical scalar fields are applied to generate inflation

What if the energy-momentum is not conserved? R µν − 1 2 g µν R = T µν = T µν D + ˜ T µν , (11) where T D is the energy-momentum of a certain (dark) matter and ˜ T is the energy-momentum of the system of diffusing particles. From the lhs it follows that ( T µν D ) ; µ = − ( ˜ T µν ) ; µ . (12) Knowing the rhs of we can determine the lhs up to a constant. We represent T D by a time-dependent cosmological term Λ. A dynamical relation of the cosmological term to the matter density seems to be unavoidable for an explanation of the coincidence problem.

Why diffusion? ◮ Diffusion equilibrates to a temperature (equilibrium) state washing out initial conditions The diffusion on the mass-shell g µν p µ p ν = m 2 (13) The diffusion is generated by the Laplace-Beltrami operator on H + ∂ j G jk √ 1 △ H = √ G ∂ k (14) G where G jk = m 2 g jk + p j p k (15) ∂ ∂ j = ∂ p j and G = det( G jk ) is the determinant of G jk .

The transport equation for the linear diffusion generated by △ H reads ( p µ ∂ x µ − Γ k µν p µ p ν ∂ k )Ω = κ 2 △ H Ω , (16) where κ 2 is the diffusion constant, ∂ x ∂ µ = ∂ x µ and x = ( t , x )

Quantum phase space distributions If the phase space distribution has the Bose-Einstein or Fermi-Dirac equilibrium limit which is a minimum of the relative entropy (related to the free energy ) then the diffusion equation must be non-linear. The proper generalization reads � � G jk p − 1 ( p µ ∂ x µ − Γ k µν p µ p ν ∂ k )Ω = κ 2 p 0 ∂ j 0 ∂ k Ω + β p j Ω(1 + ν Ω) , (17) where ν = 1 for bosons and ν = − 1 for fermions. The classical (Boltzmann) statistics can be described by ν = 0.

Solutions of linear and non-linear diffusion equations at finite temperature We have the time-dependent equilibrium � − 1 � Ω PL exp( β a 2 ( p + µ )) − ν E = (18) where µ is an arbitrary constant (the chemical potential). In the ultrarelativistic limit ( a large p ) the Planck distribution is the same as the J¨ uttner distribution. For the equilibrium solution we obtain standard Friedmann cosmology. There are other solutions of the diffusion equation whose energy momentum tensor gives different Friedmann equation for the scale factor a

We solve the conservation equation for Λ then with H = a − 1 da d τ the Friedmann equation reads � τ 8 π G H 2 ≡ d τ ) 2 = ˜ τ 0 dra − 4 ∂ r ( a 4 ˜ 3 8 π G ( a − 1 da 3 T 00 ( τ ) − T 00 ) + Λ 8 π G ( τ 0 ) � τ = ˜ τ 0 drH ( r ) ˜ T 00 ( τ 0 ) − 4 T 00 ( r ) + Λ 8 π G ( τ 0 ) (19) T µν energy-momentum of diffusing particles and any Here, ˜ conserved energy-momentum (e.g. scalar fields), T µν D = Λ g µν and ( τ is the cosmic time) ds 2 = d τ 2 − a 2 ( τ ) d x 2

Explicit solution We can find an explicit power-like solution of the integro-differential equation by a fine tuning of parameters showing that the exponential behaviour is not a necessity even if Λ( τ 0 ) > 0. Let us assume a ( τ ) = ν ( τ − q ) γ (20) with the initial condition a ( τ 0 ) = ν ( τ 0 − q ) γ . Inserting we determine the parameters γ = 1 , (21) 1 3 , ν = σ (22) ( τ 0 − q ) 2 = 2 θ (23) ν

Then Λ( τ 0 ) = 3 2( τ 0 − q ) − 2 . (24) We obtain E = 3 Λ = 8 π G ˜ 2( τ − q ) − 2 (25)

Non-homogeneous metric:Fluctuation spectrum ∞ � δ T T ( n ) δ T � T ( n ′ ) � = (2 l + 1) C l P l ( nn ′ ) (26) l =0 n direction in the sky Experimental result: COBE, WMAP l ( l + 1) C l ≃ const (27) ordinary Sachs-Wolfe effect

Einstein-Liouville-Vlasov equations We decompose g µν = h µν + h µν (28) where h µν describes homogenous metric in the conformal time ds 2 = h µν dx µ dx ν = a 2 ( dt 2 − d x 2 ) (29) and ds 2 = g µν dx µ dx ν = a 2 � (1 + 2 φ ) dt 2 − (1 + 2 ψ ) d x 2 − γ ij dx i dx j � (30)

For massless particles ( m = 0) and in the homogeneous metric h µν = 0 the J¨ uttner distribution Ω E = exp( − a 2 β | p | ) (31) with p 2 = � p j p j j is the solution of Liouville eq.

Non-homogeneous metric The solution can be expressed as Ω = Ω g E + β p 0 ΘΩ g (32) E where Ω g E = exp( − β p 0 ) (33) and p 0 is determined from g µν p µ p ν = 0

Θ is the solution of the equation ∂ t Θ + n k ∂ x k Θ = − 2 ∂ t ψ − 1 2 n j n k ∂ t γ jk (34) where n k = p k | p | − 1

We have p 0 Ω = exp( − T + δ T ) (35) Let δ T T = Θ (36) Then � t � Θ( t , x ) = Θ 0 ( x − n t ) − 2 ∂ s ψ ( s , x − ( t − s ) n ) 0 (37) 2 ∂ s γ jk ( s , x − ( t − s ) n ) n j n k � + 1 ds with the initial condition Θ 0 ( x ).

Diffusive space-time temperature variation We look for solutions in the form p 0 Ω = Ω g E + β p 0 ΘΩ g E + ra 2 Ω g E = (1 + ra 2 ) exp( − T + T Θ) (38) Inserting this formula in the diffusion equation we obtain equations for the temperature variation Θ and r k Θ + κ 2 β a 2 Θ = − 2 ∂ t ψ − 1 ∂ t Θ + n k ∂ x 2 n j n k ∂ t γ jk (39) ∂ t r + n k ∂ x k r = 3 κ 2 Θ (40) where n k = p k | p | − 1 (41)

The solution reads Θ t ( x ) = exp( − βκ 2 � t 0 a 2 ( s ) ds )Θ 0 ( x − t n ) � t 0 ds exp( − βκ 2 � t s a 2 ( r ) dr )(2 ∂ s ψ ( s , x − ( t − s ) n ) (42) − + 1 2 ∂ s γ jk n j n k ( s , x − ( t − s ) n )

Temperature fluctuations We restrict ourselves to tensor perturbations � Θ( t , n )Θ( t , n ′ ) � = 4 (2 π ) − 3 � t � t � t � t = 1 0 ds ′ � d q F ( s , s ′ , q ) exp( − βκ 2 ( s ′ ) dra 2 ( r )) s + 0 ds (2( n ∆( q ) n ′ ) 2 − ( n ∆( q ) n )( n ′ ∆( q ) n ′ )) exp( − i ( t − s ) nq + i ( t − s ′ ) n ′ q ) (43)

where n ∆( q ) n ′ = nn ′ − q − 2 ( qn )( qn ′ ) ≡ ∆( nn ′ , en , en ′ ) (44) n ∆( q ) n = 1 − q − 2 ( qn ) 2 ≡ ∆( en ) (45) where we write q = q e and F ( s , s ′ , q ) = ∂ s ∂ s ′ P ( s , s ′ , q ) (46) P is the expectation value of tensor perturbations.

If the power spectrum F is known then there remains to perform the integrals over s and q in order to obtain l =0 (2 l + 1) ˜ � Θ( t , n )Θ( t , n ′ ) � == � ∞ D l ( t , nn ′ ) P l ( nn ′ ) (47) = � ∞ l =0 (2 l + 1) C l ( t ) P l ( nn ′ ) where P l are the Legendre polynomials and ˜ D l P l still must be expanded in Legendre polynomials if the coefficients C l are to be independent of the angles. We have � t � t � t � t ˜ 1 d q F ( s , s ′ , q ) exp( − βκ 2 ( s ′ ) dra 2 ( r )) 0 ds ′ � D l == 0 ds s + 16 π 2 � 2∆( nn ′ , − i ∂ s , i ∂ s ′ ) 2 − ∆( − i ∂ s )∆( i ∂ s ′ ) � j l ( q ( t − s )) j l ( q ( t − s ′ )) (48)

Conclusions 1.Diffusion gives a damping factor for temperature variation 2. The damping factor for temperature fluctuations can give a finite result even if perturbation is acting for an infinite time 3. The diffusion can change the l behavior of multipoles