Topics in f ( R ) THEORIES OF GRAVITY Nathalie Deruelle APC-CNRS, - PowerPoint PPT Presentation

1 Topics in f ( R ) THEORIES OF GRAVITY Nathalie Deruelle APC-CNRS, Paris ihes, April 22th 2010

2 INTRODUCTION The observed universe is well represented by a Friedmann-Lemaˆ ıtre spacetime the scale factor of which started to accelerate recently ds 2 = g ij dx i dx j = − dt 2 + a 2 ( t ) dσ 2 k D j T ij = 0 G ij = κ T total Supernova Cosmology Project ; 3 3 ij Knop et al. (2003) No Big Bang Spergel et al. (2003) Allen et al. (2002) H 2 + k 3 ρ total ; a 2 = κ H = 1 da 2 2 a dt Supernovae � 2 � 4 + Ω 0 � 3 + Ω 0 � 2 + Ω 0 � � a 0 � a 0 � a 0 H 1 = Ω 0 Ω Λ rad mat k Λ H 0 a a a CMB er r e v s f o a n d ex p ll y a rad ≈ 10 − 4 ; Ω 0 0 n t u ev e se s o l la p H 0 ≈ 70 ; Ω 0 mat ≈ 0 . 3 , Ω 0 r e c k ≈ 0 Clusters closed flat -1 open Ω 0 Λ ≈ 0 . 7 : Dark Energy 0 1 1 2 2 3 √ Ω Λ / 3 t , Λ = 3Ω 0 M κT Λ Λ H 2 = ⇒ ij = − Λ g ij , a ( t ) → e 0

3 Origin of this acceleration • An artefact of the averaging process ? G ij ( <g kl > ) = κ <T ij > instead of <G ij ( g kl ) > = κ <T ij > G ( <g kl > ) � = <G ( g kl ) > ) Ellis, 1971,..., see review Buchert, 2006 et seq. • Exotic matter ? ( ρ Λ + p Λ ≈ 0) – Chaplygin gas : pρ = − A , (Kamenshchik et al. , 2001) – Quintessence : R.G. plus ϕ with V ∝ 1 /ϕ n (Steinhardt et al. , 1997) • “Modified” gravity ? – Λ : the simplest explanation, (Bianchi and Rovelli, Feb 2010) – MOND, see, e.g., Navarro and Acoleyen, 2005 – “Branes” : DGP (2000), Deffayet (2001) – f ( R ) lagrangian, instead of Hilbert’s R C.D.T.T. (2003), Capozziello et al. (2003)

4 Outline of the talk 1. Introducing f ( R ) theories of gravity or : f ( R ) theories as scalar-tensor theories of gravity 2. f ( R ) cosmological models of dark energy or : the search for viable models 3. f ( R ) gravity and local tests or : how to hide the scalar d.o.f. of gravity 4. Back to cosmological models or : how to hide the scalar d.o.f. of gravity 5. Remarks on Black Holes in f ( R ) theories (uniqueness and thermodynamics)

5 1. Introducing f ( R ) theories of gravity • d.o.f. : gravity is described by a “graviton” and a “scalaron” d 4 x √− g f ( R ) + S m (Ψ ; g ij ) ¯ 1 � S [ g ij ] = 2 κ (Weyl 1918, Pauli 1919, Eddington, 1924) Metric variation yields a 4th order diff eqn for g ij : 2 ( Rf ′ − f ) g ij + g ij D 2 f ′ − D ij f ′ = κ T ij D j T ij = 0) f ′ ( R ) G ij + 1 ( ⇒ The trace : 3 D 2 f ′ + ( Rf ′ − 2 f ) = κ T is a (2nd order) eom for R (or f ′ ( R ) ), the “scalaron”(Starobinski, 1980) Remark : “Palatini” variations yield different eom (Vollick, 2003 et seq. )

6 • Isolating the scalaron and coupling it to matter Introduce a “Helmholtz” lagrangian : d 4 x √− g [ f ′ ( s ) R − ( sf ′ ( s ) − f ( s )] + S m (Ψ ; ˜ ¯ 1 g ij = e 2 C ( s ) g ij ) � S [ g ij , s ] = 2 κ (No reason for the scalaron not to couple to matter.) hence TWO second order differential equations of motion : f ′ ( s ) G ij + 1 2 g ij ( sf ′ ( s ) − f ( s )) + g ij D 2 f ′ ( s ) − D ij f ′ ( s ) = κT ij ( ⇒ D j T j s = R − 2 κC ′ ( s ) T/f ′′ ( s ) i = TC ′ ( s ) ∂ i s ) C ( s ) = 0 : standard f ( R ) gravity ; s = R , same eom as before. C ( s ) � = 0 : “detuned” f ( R ) gravity (ND, Sasaki, Sendouda, 2007)

7 • Jordan vs Einstein frame description of f ( R ) gravity ˜ g ij = e 2 C g ij to – The “Jordan frame” is the spacetime, M , with metric ˜ T ij = 0 , e.g. ˜ which matter is minimally coupled (that is : ˜ D j ˜ a 3 ). ρ ∝ 1 / ˜ In this frame the action is a Brans-Dicke type action (up to a divergence) d 4 x √− ˜ � � ∂ Φ) 2 − 2 U (Φ) ˜ Φ ˜ R − ω (Φ) Φ (˜ 1 � S [˜ g ij , Φ] = g + S m [Ψ; ˜ g ij ] 2 κ Φ( s ) = f ′ ( s ) e − 2 C ( s ) U ( s ) = 1 2 ( sf ′ ( s ) − f ( s )) e − 2 C ( s ) where , ω ( s ) = − 3 K ( s )( K ( s ) − 2) d C d ln √ and with K ( s ) = f ′ . 2( K ( s ) − 1) 2 For standard f ( R ) gravity, C ( s ) = 0 ; the Jordan frame is the original one. And ω = 0 ; if U ≈ 0 , f ( R ) gravity is ruled out since ω > 40000 (Cassini) (see Damour Esposito-Farese, 1992, and below)

8 – The “Einstein frame” is the spacetime, M ∗ , the metric of which, g ∗ ij = e − 2 k ˜ g ij , makes the action for f ( R ) gravity look like Einstein’s : d 4 x √− g ∗ � R ∗ � 2 ( ∂ ∗ ϕ ) 2 − V ( ϕ ) S ∗ [ g ∗ ij , ϕ ] = 2 4 − 1 g ij = e 2 k ( ϕ ) g ∗ � + S m [Ψ; ˜ ij ] κ √ f ′ ( s ) , V ( s ) = sf ′ ( s ) − f ( s ) e 2 k ( s ) = e 2 C ( s ) � where ϕ ( s ) = 3 ln , f ′ ( s ) 4 f ′ 2 ( s ) M � = M ∗ unless ˜ ˜ g ij = g ∗ � f ′ ( s ) , – ij , that is, C ( s ) = ln (Magnano-Sokolewski, 1993, 2007) – hence : f ( R ) gravity is coupled quintessence Ellis et al. (1989), Damour-Nordvedt-Polyakov (1993), Wetterich (1995), Amendola (1999), Copeland et al. (2006),...

9 • Jordan vs Einstein frames : an endless debate – Einstein frame is the “physical” frame : Magnano-Sokolewski (93, 07), Gunzig-Faraoni (98) (but see Faraoni (06)) (“DEC does not hold in JF hence no positive energy theorem”) – Jordan frame is the “physical” frame : Damour Esposito-Farese (92) · · · “Jordan metric defines the lengths and times actually measured by laboratory rods and clocks (which are made of matter)” (Esposito-Farese Polarski, 2000) – Jordan and Einstein frames are equivalent (classically) : Flanagan (04); Makino-Sasaki (91), Kaiser (95) (CMB anisotropies) ; Catena et al (06) (cosmo)

10 • Jordan vs Einstein frames : an example – Capozziello et al, 10. FRW metric in JF ( ds 2 = − dt 2 + a 2 ( t ) dx 2 ). a , z ( t ) ≡ a 0 Define H ( z ) as : H ( t ) ≡ ˙ a a − 1 Define : H ∗ ( t ) ≡ H ∗ a ∗ da ∗ n dt ∗ , z ∗ ( t ) ≡ a ∗ ( t ) − 1 n a ∗ � √ f ′ dt , a ∗ = √ f ′ a . (where t ∗ = and a ∗ n and H ∗ n such that q ∗ ( t n ) = q 0 and a ∗ n ≡ a ∗ ( t n ) , H ∗ ( t n ) = H 0 .) The H ( z ) and H ∗ ( z ∗ ) are different. (Correct.) Since H ( z ) and H ∗ ( z ∗ ) are “Hubble laws”, “the Jordan and Einstein frames are physically inequivalent”. (Wrong.) Indeed :

11 – First, relate observable variables : � redshift ( Z = ν L 4 πl with L = Nhν 2 ) ν 0 − 1 ) vs luminosity ( D = where ν is the frequency of some atomic transition “there and then” ; where ν 0 and l are the observed frequency and apparent luminosity. In the JF, matter is minimally coupled, the EEP holds and ν is the same as � Z in the lab now. Hence, as in GR : Z = a 0 dZ a − 1 , D = (1 + Z ) 0 H – Second, recall that matter is not minimally coupled in EF : In the EF, the interaction of φ with matter implies m ∗ = m/ √ f ′ (Damour Gef, 92). Now ν ∗ ∝ m ∗ and the frequency “there and then” ( ν ∗ ) is NOT the frequency measured in the lab now ( ν ). Hence find : Z ∗ ≡ ν 0 − 1 = Z and D ∗ = D . (Catena et al,. ND Sasaki) : ν ∗ Relationships between observables do not depend on the frame.

12 • Hamiltonian structure of f ( R ) gravity In a nutshell : – Extra dof : either K ( ˙ g µν ) : “Odstrogradsky formulation”, (Buchbinder- Lyahovich 87, Querella 99, Esawa et al 99-09) or R ( ¨ g µν ), Boulware 84. – the action can be written in the Jordan or the Einstein frame ALL variables are related by (non-linear) canonical transformations. (N.D., Sendouda,Youssef 09, N.D., Sasaki, Sendouda, Yamauchi 09) – Equivalence at the quantum level ? At linear order, yes (CMB), otherwise ? • Junction conditions in f ( R ) gravity In a nutshell : – Do not impose the continuity of 1st, 2nd and 3rd order derivatives of JF ˜ g ij – Impose continuity of 1st and 2nd order derivatives of ˜ g ij and of R and its 1st derivative (Teyssandier-Tourrenc 83, ND Sasaki, Sendouda, 07)

13 Reminder : outline of the talk 1. Introducing f ( R ) theories of gravity or : f ( R ) theories as scalar-tensor theories of gravity 2. f ( R ) cosmological models of dark energy or : the search for viable models 3. f ( R ) gravity and local tests or : how to hide the scalar d.o.f. of gravity 4. Back to cosmological models or : how to hide the scalar d.o.f. of gravity 5. Remarks on Black Holes in f ( R ) theories (uniqueness and thermodynamics)

14 2. f ( R ) cosmological models of Dark Energy • The Carroll-Duvvuri-Trodden-Turner and Capozziello-Carloni-Troisi proposal (2003) f ( R ) = R − µ 2(1+ n ) µ 2 ∼ 10 − 33 eV µ 2 = 1 ℓ 2 with ℓ ∼ H − 1 ( n > 0) ; or R n 0 Late time Einstein frame Friedmann equations when matter has become negligible ( ϕ large, > 2 , say) : − ( n +2) ϕ √ ϕ 2 − 2 V ( ϕ ) ≈ 0 , ϕ + dV 3 H 2 ϕ + 3 H 2 ∗ − ˙ ¨ ∗ ˙ dϕ ≈ 0 with V ( ϕ ) ∝ e 2 3( n +1) √ Solution : a ∗ ( t ) ∝ t q ( q → 3 , w ∗ DE → − 0 . 77 for large n ) , ϕ ∼ 3 p ln t s 2 = t − 2 p ds 2 t 2 + ˜ ∗ = − d ˜ a 2 (˜ t ) dx 2 Jordan frame scale factor : d ˜ 2 a (˜ t ) ∝ ˜ hence : ˜ t 3(1+ ˜ w DE) 2( n +2) with w DE = − 1 + ˜ 3( n +1)(2 n +1) → − 1 for large n (2,3,4 is enough)