Palatini cosmology in different frames Andrzej Borowiec joint work - PowerPoint PPT Presentation

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical Palatini cosmology in different frames Andrzej Borowiec joint work with Aleksander Stachowski Marek Szyd� lowski Aneta Wojnar Institute for Theoretical Physics, Wroclaw University, Poland 9th MATHEMATICAL PHYSICS MEETING: School and Conference on Modern Mathematical Physics 18 - 23 September 2017, Belgrade, Serbia

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical Einstein gravity is a beautiful theory which is very well tested in the Solar system scale. However it indicates some drawbacks in the other scales. The simplest way to generalize (/modify) it is by replacing Einstein-Hilbert Lagrangian n R → f ( R ) = R − 2Λ + γ R 2 + · · · = � γ i R i i =0 by an arbitrary function of the scalar R . Such modification might be helpful in solving dark matter and dark energy problems . Here we focus on some cosmological applications presented in arXiv:1707.01948; Eur.Phys.J. C77 (2017) no.9, 603 arXiv:1608.03196; Eur.Phys.J. C77 (2017) no.6, 406 arXiv:1512.04580; Eur.Phys.J. C76 (2016) no.10, 567, arXiv:1512.01199; JCAP01 (2016) 040

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical Introduction I In the Palatini f ( ˆ R ) gravity the action is dependent on a metric and a torsionless connection as independent variables � √− gf ( ˆ ρσ ) = S g + S m = 1 S ( g µν , Γ λ R ) d 4 x + S m ( g µν , ψ ) , (1) 2 R ( g , Γ) = g µν ˆ where ˆ R µν (Γ) is the generalized Ricci scalar and ˆ R µν (Γ) is the Ricci tensor of a torsionless connection Γ. EOM are R ( µν ) (Γ) − 1 f ′ ( ˆ R ) ˆ 2 f ( ˆ R ) g µν = T µν , (2) ∇ α ( √− gf ′ ( ˆ ˆ R ) g µν ) = 0 , (3) 2 δ L m where T µν = − δ g µν (e.g. PF = ( p + ρ ) u µ u ν + pg µν ) is EMT, √− g i.e. assuming that the matter couples minimally to the metric g µν .

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical Introduction II In order to solve equation (3) it is convenient to introduce a new metric √− ¯ g µν = √− gf ′ ( ˆ g ¯ R ) g µν (4) for which the connection Γ = Γ L − C (¯ g ) is a Levi-Civita connection. As a consequence in dim M = 4 one gets g µν = f ′ ( ˆ ¯ R ) g µν , (5) For this reason one should assume that the conformal factor f ′ ( ˆ R ) � = 0, so it has strictly positive or negative values. Taking the g − trace of (2), we obtain structural equation f ′ ( ˆ R ) ˆ R − 2 f ( ˆ R ) = T . (6) where T = g µν T µν (= 3 p − ρ ). Thus, the equation (2) can be treated both as determining the dynamics of the metric g or ¯ g (two frames !!)

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical The eq. (2) can be recast to the following form R µν − 1 1 ( T µν − 1 ¯ ¯ R ¯ g µν = 4 T g µν ) . (7) f ′ ( ˆ 4 R ) g µν ¯ R ) − 1 ˆ where ˆ R µν = ¯ R µν , ¯ R µν = f ′ ( ˆ g µν ¯ R = g µν ˆ R = ¯ R and ¯ R . 1. non-linear system of second order PDE. 2. for the linear Lagrangian ˆ R − 2Λ is fully equivalent to Einstein R − 2Λ, 3. any f ( ˆ R ) vacuum solutions ( T µν = 0) ⇒ Einstein vacuum solutions with cosmological constant; 4. PF: T µν = ( p + ρ ) u µ u ν + pg µν ⇒ T µν − 1 u µ u ν + 1 � � 4 Tg µν = ( p + ρ ) . Thus DE solutions ≡ 4 g µν vacuum solutions. Palatini gravity is the first cousin of Einstein theory (next of kin)!!

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical The action (1) is dynamically equivalent to the constraint system with first order Palatini gravitational Lagrangian with the ′′ ( ˆ additional scalar field χ , provided that f R ) � = 0 (This condition excludes the linear Einstein-Hilbert Lagrangian f ( ˆ R ) = ˆ R − 2Λ from our considerations.) d 4 x √− g ρσ ) = 1 � � � f ′ ( χ )( ˆ S ( χ, g µν , Γ λ R − χ ) + f ( χ ) + S m ( g µν , ψ ) , 2 κ (8) Introducing new scalar field Φ = f ′ ( χ ) and taking into account the constraint equation χ = ˆ R , one can rewrite the action in dynamically equivalent way as a Palatini action d 4 x √− g ρσ ) = 1 � � � Φ ˆ S (Φ , g µν , Γ λ R − U (Φ) + S m ( g µν , ψ ) , (9) 2 k where the potential U (Φ) encodes the information about the function f ( ˆ R ) is given by

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical U f (Φ) ≡ U (Φ) = χ (Φ)Φ − f ( χ (Φ)) (10) and Φ = df ( χ ) R ≡ χ = dU (Φ) d χ . Thus one has ˆ d Φ . For a given f the potential U is a (singular) solution of the Clairaut’s differential equation: U (Φ) = Φ dU d Φ − f ( dU d Φ ). (One can observe that the trivial, i.e. constant, potential U (Φ) corresponds to the linear Lagrangian f ( ˆ R ) = ˆ R − 2Λ.) Palatini variation of this action provides � � R ( µν ) − 1 + 1 ˆ 2 g µν ˆ Φ R 2 g µν U (Φ) − κ T µν = 0 (11a) ∇ λ ( √− g Φ g µν ) = 0 ˆ (11b) ˆ R − U ′ (Φ) = 0 (11c) The last equation due to the constraint ˆ R = χ = U ′ ( φ ) is automatically satisfied. The middle equation (11b) implies that the connection ˆ Γ is a metric connection for the new metric g µν = Φ g µν . ¯

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical Now the equation (11a), (11c) can be written as a dynamical g µν ( ˆ R µν = ¯ R µν , ˆ R = Φ ¯ R , g µν ˆ g µν ¯ equation for the metric ¯ R = ¯ R ) R µν − 1 T µν − 1 ¯ g µν ¯ R = κ ¯ g µν ¯ 2 ¯ 2 ¯ U (Φ) (12a) R − (Φ 2 ¯ U ( Φ)) ′ = 0 Φ ¯ (12b) U ( φ ) = U ( φ ) / Φ 2 and ¯ where we have introduced ¯ T µν = Φ − 1 T µν . Thus the system (12a) - (12b) corresponds to a scalar-tensor action for the metric ¯ g µν and the (non-dynamical) scalar field Φ � ¯ d 4 x √− ¯ g µν , Φ) = 1 � R − ¯ + S m (Φ − 1 ¯ � S (¯ g U (Φ) g µν , ψ ) , (13) 2 κ non-minimally coupled to the matter ψ .

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical where 2 δ T µν = − u µ ¯ u ν + ¯ g µν = Φ − 3 T µν , (14) ¯ √− ¯ S m = (¯ ρ + ¯ p )¯ p ¯ δ ¯ g g µν u µ = Φ − 1 2 u µ , ¯ ρ = Φ − 2 ρ, ¯ p = Φ − 2 p , w = ¯ and ¯ w T µν = Φ − 1 T µν , ¯ ¯ T = Φ − 2 T . Further, the trace of (12a), provides R = 2 ¯ ¯ U (Φ) − κ ¯ T (15) The equation (12a), due to non-minimal coupling between the metric ¯ g µν and the matter, implies eneregy-momentum non-conservation T µν = − 1 T ∂ ν Φ ∇ µ ¯ ¯ ¯ (16) 2 Φ (however ∇ µ T µν = 0). In this, so-called Einstein frame case, the scalar field has no dynamics satisfying algebraic equation (12b).

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical By changing the frame (¯ g µν , Φ) → ( g µν , Φ) one gets that action for the original Palatini metric within scalar-tensor formulation S (Φ , g µν ) = 1 d 4 x √− g � Φ R + 3 � � 2Φ ∂ µ Φ ∂ µ Φ − U (Φ) , (17) 2 κ where U (Φ) is given as before by (10). In this case, a kinematical part of the scalar field does not vanish from the Lagrangian (17). We obtain Brans-Dicke action with the parameter ω BD = − 3 2 in the Jordan frame. In this case equations of motion take the form ( ∇ µ T µν = 0 ) � R µν − 1 � − 3 4Φ g µν ∇ σ Φ ∇ σ Φ + 3 Φ 2 g µν R 2Φ ∇ µ Φ ∇ ν Φ + g µν � Φ − ∇ µ ∇ ν Φ + 1 2 g µν U ( φ ) = κ T µν , (18a) R − 3 2Φ 2 ∇ µ Φ ∇ µ Φ − 1 3 2 U ′ (Φ) = 0 Φ � Φ + (18b)

Cosmological models based on Palatini f ( ˆ f ( ˆ Dynamics of Palatini gravity in different frames R )-gravity R ) Palatini cosmology Dynamical Cosmological applications I Assume that the metric g is a spatially flat FLRW metric ds 2 = dt 2 − a 2 ( t ) dr 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 ) � � , (19) where a ( t ) is the scale factor, t is the cosmic time. As a source of gravity we assume perfect fluid with the energy-momentum tensor T µ ν = diag( − ρ ( t ) , p ( t ) , p ( t ) , p ( t )) , (20) where p = w ρ , w = const is a form of the equation of state ( w = 0 for dust and w = 1 / 3 for radiation). Formally, effects of the spatial curvature can be also included to the model by introducing a curvature fluid ρ k = − k 2 a − 2 , with the barotropic factor w = − 1 3 ( p k = − 1 3 ρ k ). From the conservation condition T µ ν ; µ = 0 we obtain that ρ = ρ 0 a − 3(1+ w ) . Therefore, trace T reads as T = ρ 0 (3 w − 1) a ( t ) − 3(1+ w ) . (21)