Renormalizable ghost-free gravity Alexey Koshelev Universidade da - PowerPoint PPT Presentation

Renormalizable ghost-free gravity Alexey Koshelev Universidade da Beira Interior & Vrije Universiteit Brussel p-adics 2015 and Branko’s fest Belgrade, September 11, 2015

Happy Birthday, Branko!

Renormalizable ghost-free gravity Set-up The set-up We start with GR, which must be the IR limit anyway � � M 2 d 4 x √− g P R 1 � , M 2 S = P = 2 8 πG N 1/15

Renormalizable ghost-free gravity Set-up The set-up We start with GR, which must be the IR limit anyway � � M 2 d 4 x √− g � P R 1 , M 2 S = P = 2 8 πG N We proceed by modifying it in a covariant way, containing higher deriva- tives in a form of � operator and (as a zero try) focus on terms contribut- ing to the propagator on the Minkowski background. Thus � � M 2 d 4 x √− g P R + λ � S = 2 R F 1 ( � ) R 2 1/15

Renormalizable ghost-free gravity Set-up The set-up We start with GR, which must be the IR limit anyway � � M 2 d 4 x √− g � P R 1 , M 2 S = P = 2 8 πG N We proceed by modifying it in a covariant way, containing higher deriva- tives in a form of � operator and (as a zero try) focus on terms contribut- ing to the propagator on the Minkowski background. Thus � � M 2 d 4 x √− g P R + λ � S = 2 R F 1 ( � ) R 2 and after some deeper thinking � � M 2 d 4 x √− g � P R + λ R F 1 ( � ) R + R µν F 2 ( � ) R µν + R µνλσ F 4 ( � ) R µνλσ � � S = 2 2 1/15

Renormalizable ghost-free gravity Set-up The set-up We start with GR, which must be the IR limit anyway � � M 2 d 4 x √− g � P R 1 , M 2 S = P = 2 8 πG N We proceed by modifying it in a covariant way, containing higher deriva- tives in a form of � operator and (as a zero try) focus on terms contribut- ing to the propagator on the Minkowski background. Thus � � M 2 d 4 x √− g P R + λ � S = 2 R F 1 ( � ) R 2 and after some deeper thinking � � M 2 d 4 x √− g � P R + λ R F 1 ( � ) R + R µν F 2 ( � ) R µν + R µνλσ F 4 ( � ) R µνλσ � � S = 2 2 WHY??? 1/15

Renormalizable ghost-free gravity History History • Classical gravity and GR; also Ostrogradski 1850 • Stelle, 1977,1978, renormalizable R 2 type gravity (containing ghosts) • Starobinsky, 1980-s, R 2 inflation • Witten, 1986, String Field Theory (SFT) which by construction con- tains non-local vertexes • Vladimirov, Volovich, Zelenov; Dragovich, Khrennikov; Brekke, Fre- und, Olson, Witten, . . . , 1987+, p -adic strings, again non-local • Aref’eva, AK, 2004, models of non-local stringy inspired scalar fields coupled to gravity • Biswas, Mazumdar, Siegel, 2005, first explicit non-local gravity modi- fication • Recent activity by Biswas, Conroy, Dragovich, Koivisto, Mazumdar, Modesto, Pozdeeva, Rachwal, Vernov, AK and others 2/15

Renormalizable ghost-free gravity GR Problems of GR • GR does not explain the Dark Energy Or why the sky is dark in the night? Zel’dovich • GR is not renormalizable Obvious due to the dimensionful coupling M 2 P • GR is not geodesically complete 3/15

Renormalizable ghost-free gravity GR Problems of GR • GR does not explain the Dark Energy Or why the sky is dark in the night? Zel’dovich • GR is not renormalizable Obvious due to the dimensionful coupling M 2 P • GR is not geodesically complete • Either of the above has been overcome sacrificing the uni- tarity Ghosts appeared in the theory 3/15

Renormalizable ghost-free gravity GR Why Einstein’s GR is not enough? • Raychaudhuri equation: Consider a congruence of null geodesics characterized by a null vector k α , such that k α k α = 0 . Then R µν k µ k ν < 0 for a non-singular space-time • GR equations of motion are: M 2 P G µν = T µν Assuming the matter is a perfect fluid ν = diag( − ρ, p, p, p ) ⇒ T µν k µ k ν = ρ + p T µ • Then P R µν k µ k ν = ρ + p M 2 4/15

Renormalizable ghost-free gravity GR Why Einstein’s GR is not enough? • Raychaudhuri equation: Consider a congruence of null geodesics characterized by a null vector k α , such that k α k α = 0 . Then R µν k µ k ν < 0 for a non-singular space-time • GR equations of motion are: M 2 P G µν = T µν Assuming the matter is a perfect fluid ν = diag( − ρ, p, p, p ) ⇒ T µν k µ k ν = ρ + p T µ • Then P R µν k µ k ν = ρ + p> 0 0 >M 2 Either the space-time is singular or the NEC is violated. 4/15

Renormalizable ghost-free gravity Ghosts Who are ghosts? Terminology in ( − , + , + , +) signature: good: L = − 1 2 ∂ µ φ∂ µ φ + . . . ghost: L = +1 2 ∂ µ φ∂ µ φ + . . . Ghosts lead to a very rapid vacuum decay. Ostrogradski statement says that higher ( > 2 ) derivatives in a Lagrangian are equivalent to the presence of ghosts. This statement is not absolutely rigorous. There are systems with higher derivatives which have no ghosts. 5/15

Renormalizable ghost-free gravity Ghosts Exorcising ghosts • In some cases ghosts do not appear, like in f ( R ) gravity for special parameters. This is because the system is constrained. • There are special field theories which have higher derivatives in the La- grangian but no more than 2 derivatives act on a field in the equations of motion. For example KGB models or galileons. The fine-tuning is required. • Propagators can be modified and be non-local without changing the physical excitations � − m 2 →G ( � ) = ( � − m 2 ) e γ ( � ) γ ( � ) must be an entire function. This guarantees that no extra degrees of freedom appear. Let γ (0) = 0 to preserve the normalization. 6/15

Renormalizable ghost-free gravity G ( � ) G ( � ) physics, SFT motivation Low level example action from SFT: L ∼ 1 2 φ ( � − m 2 ) φ + λ � 4 ⇒ 1 2 ϕ ( � − m 2 ) e 2 β � ϕ + λ e − β � φ 4 ϕ 4 � 4 The Lagrangian to understand is � � 1 � d D x S = 2 ϕ G ( � ) ϕ − λv ( ϕ ) + . . . g n � n , i.e. it is an analytic function. G ( � ) = � n ≥ 0 2 ϕ � ϕ − m 2 Canonical physics has G ( � ) = � − m 2 , i.e. L = 1 2 ϕ 2 Ghosty example G ( � ) = � − m 2 + g 2 � 2 7/15

Renormalizable ghost-free gravity NLG Coming back to the advertised setting � � M 2 d 4 x √− g P R + λ � R F 1 ( � ) R + R µν F 2 ( � ) R µν + R µνλσ F 4 ( � ) R µνλσ � � S = 2 2 Linearization around Minkowski space-time, g µν = η µν + h µν , h = h µ µ : � d 4 x � 1 µ α ( � ) ∂ ν h µν + hγ ( � ) ∂ µ ∂ ν h µν − 1 2 h µν � α ( � ) h µν + ∂ σ h σ δ (2) S = 2 h � γ ( � ) h 2 � − ∂ α ∂ β h αβ γ ( � ) − α ( � ) ∂ µ ∂ ν h µν � P − λ P + 2 λ � F 1 ( � ) + λ α ( � ) = M 2 2 � F 2 ( � ) − 2 λ � F 4 ( � ) , γ ( � ) = M 2 2 λ � F 4 ( � ) γ ( � ) − α ( � ) = 2 λ � [ F 1 ( � ) + 1 2 F 2 ( � ) + F 4 ( � )] Notice: a generalized Gauss-Bonnet term F 1 ( � ) = − 4 F 2 ( � ) = F 4 ( � ) always gives γ ( � ) = α ( � ) 8/15

Renormalizable ghost-free gravity Propagator Propagator Projection operators ( van Nieuwenhuizen, 1973 ): P 2 = 1 2( θ µρ θ νσ + θ νρ θ µσ ) − 1 s = 1 3 θ µν θ σρ , θ µν = η µν − k µ k ν 3 θ µν θ σρ , P 0 k 2 Plus two more which are not relevant here. The propagator: P 2 P 0 s Π = α ( − k 2 ) k 2 + ( α ( − k 2 ) − 3 γ ( − k 2 )) k 2 Recall the pure GR propagator: Π GR = P 2 k 2 − P 0 s 2 k 2 1. Absence of new degrees of freedom requires α ( − k 2 ) and α ( − k 2 ) − 3 γ ( − k 2 ) have no roots 2. Presence of a GR limit requires α (0) = γ (0) = 1 9/15

Renormalizable ghost-free gravity Example Example • Non-local terms can be chosen as: F 4 ( � ) = 0 , F 1 ( � ) = − 1 2 F 2 ( � ) F 1 ( � ) = e σ ( � ) − 1 , σ ( � ) is an entire function and σ (0) = 0 � • This leads to a manifestly asymptotically-free gravity: � const as r → 0 � Mr � M , Φ ∼ − 1 σ ( � ) = − � r erf → 1 r as r → ∞ 2 • It is also singularity-free following the Raychaudhuri equation analysis Conroy, AK, Mazumdar, PRD, 2014 and a joint work in progress 10/15

Renormalizable ghost-free gravity Solutions Solutions of FRW type First we reshuffle the non-local terms by using the Weyl tensor � � M 2 d 4 x √− g � P R + λ F 2 ( � ) R µν + C µνλσ ˜ R ˜ F 1 ( � ) R + R µν ˜ F 4 ( � ) C µνλσ � � S = 2 2 To satisfy the conditions obtained from the consideration of the propa- gator one can set F 1 ( � ) = − 1 ˜ ˜ ˜ F 2 ( � ) = 0 , F 4 ( � ) 3 any solution of the local R 2 gravity is a solution here upon 3 Claim: algebraic conditions on the action parameters Accounting � R = r 1 R + r 2 (which is an EOM rather than a constraint in a local R 2 gravity) and letting the cosmological term Λ to be in the action = − M 2 2 λ [ F ( r 1 ) − F (0)] , Λ = − r 2 M 2 F (1) ( r 1 ) = 0 , r 2 P − 6 λ F ( r 1 ) r 1 P , r 1 4 r 1 11/15