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 S =

• d4x√−g
• M 2

PR

2

• , M 2

P =

1 8πGN

1/15

Renormalizable ghost-free gravity Set-up

The set-up

We start with GR, which must be the IR limit anyway S =

• d4x√−g
• M 2

PR

2

• , M 2

P =

1 8πGN 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 S =

• d4x√−g
• M 2

PR

2 +λ 2RF1()R

• 1/15

Renormalizable ghost-free gravity Set-up

The set-up

We start with GR, which must be the IR limit anyway S =

• d4x√−g
• M 2

PR

2

• , M 2

P =

1 8πGN 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 S =

• d4x√−g
• M 2

PR

2 +λ 2RF1()R

• and after some deeper thinking

S =

• d4x√−g
• M 2

PR

2 +λ 2

• RF1()R + RµνF2()Rµν + RµνλσF4()Rµνλσ
• 1/15

Renormalizable ghost-free gravity Set-up

The set-up

We start with GR, which must be the IR limit anyway S =

• d4x√−g
• M 2

PR

2

• , M 2

P =

1 8πGN 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 S =

• d4x√−g
• M 2

PR

2 +λ 2RF1()R

• and after some deeper thinking

S =

• d4x√−g
• M 2

PR

2 +λ 2

• RF1()R + RµνF2()Rµν + RµνλσF4()Rµνλσ
• WHY???

1/15

Renormalizable ghost-free gravity History

History

• Classical gravity and GR; also Ostrogradski 1850
• Stelle, 1977,1978, renormalizable R2 type gravity (containing ghosts)
• Starobinsky, 1980-s, R2 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 kα, such that kαkα = 0. Then Rµνkµkν < 0 for a non-singular space-time

• GR equations of motion are:

M 2

PGµν = Tµν

Assuming the matter is a perfect fluid T µ

ν = diag(−ρ, p, p, p) ⇒ Tµνkµkν = ρ + p

• Then

M 2

PRµνkµkν = ρ + p

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 kα, such that kαkα = 0. Then Rµνkµkν < 0 for a non-singular space-time

• GR equations of motion are:

M 2

PGµν = Tµν

Assuming the matter is a perfect fluid T µ

ν = diag(−ρ, p, p, p) ⇒ Tµνkµkν = ρ + p

• Then

0 >M 2

PRµνkµkν = ρ + p> 0

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∂µφ∂µφ + . . . ghost: L = +1 2∂µφ∂µφ + . . . 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

• f 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 − m2→G() = ( − m2)eγ() γ() must be an entire function. This guarantees that no extra degrees

• f 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 2φ( − m2)φ + λ 4

• e−βφ

4 ⇒ 1 2ϕ( − m2)e2βϕ + λ 4ϕ4 The Lagrangian to understand is S =

• dDx
• 1

2ϕG()ϕ − λv(ϕ) + . . .

• G() =

n≥0

gnn, i.e. it is an analytic function. Canonical physics has G() = − m2, i.e. L = 1

2ϕϕ − m2 2 ϕ2

Ghosty example G() = − m2 + g22

7/15

Renormalizable ghost-free gravity NLG

Coming back to the advertised setting

S =

• d4x√−g
• M 2

PR

2 +λ 2

• RF1()R + RµνF2()Rµν + RµνλσF4()Rµνλσ
• Linearization around Minkowski space-time, gµν = ηµν + hµν, h = hµ

µ:

δ(2)S = d4x 2

• 1

2hµνα()hµν + ∂σhσ

µα()∂νhµν + hγ()∂µ∂νhµν − 1

2hγ()h −∂α∂βhαβγ() − α()

• ∂µ∂νhµν
• α() = M 2

P − λ

2F2() − 2λF4(), γ() = M 2

P + 2λF1() + λ

2λF4() γ() − α() = 2λ[F1() + 1 2F2() + F4()] Notice: a generalized Gauss-Bonnet term F1() = −4F2() = F4() always gives γ() = α()

8/15

Renormalizable ghost-free gravity Propagator

Propagator

Projection operators (van Nieuwenhuizen, 1973): P2 = 1 2(θµρθνσ + θνρθµσ) − 1 3θµνθσρ, P0

s = 1

3θµνθσρ, θµν = ηµν − kµkν k2 Plus two more which are not relevant here. The propagator: Π = P2 α(−k2)k2 + P0

s

(α(−k2) − 3γ(−k2))k2 Recall the pure GR propagator: ΠGR = P2 k2 − P0

s

2k2

• 1. Absence of new degrees of freedom requires α(−k2) and α(−k2) − 3γ(−k2)

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:

F4() = 0, F1() = −1 2F2() F1() = eσ() − 1

• , σ() is an entire function and σ(0) = 0
• This leads to a manifestly asymptotically-free gravity:

σ() = − M , Φ ∼ −1 rerf Mr 2

• →

const as r → 0

1 r as r → ∞

• 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 S =

• d4x√−g
• M 2

PR

2 +λ 2

• R ˜

F1()R + Rµν ˜ F2()Rµν + Cµνλσ ˜ F4()Cµνλσ

• To satisfy the conditions obtained from the consideration of the propa-

gator one can set ˜ F2() = 0, ˜ F1() = −1 3 ˜ F4() Claim: any solution of the local R2 gravity is a solution here upon 3 algebraic conditions on the action parameters Accounting R = r1R + r2 (which is an EOM rather than a constraint in a local R2 gravity) and letting the cosmological term Λ to be in the action F(1)(r1) = 0, r2 r1 = −M 2

P − 6λF(r1)r1

2λ[F(r1) − F(0)], Λ = −r2M 2

P

4r1 ,

11/15

Renormalizable ghost-free gravity Bounce, Inflation

Examples of solutions

Let as usual a is the scale factor of the FRW metric Explicit non-singular bouncing solutions a = a0 cosh(σt)

Biswas, Muzumdar, Siegel, JCAP, 2006; AK, CQG, 2013

a = a0

• cosh(σt) and also a = a0e−σ

2t2

AK, CQG, 2013

Starobinsky solution a ≈ a0 √t∗ − teσ(t∗−t)2

Craps, De Jonckheere, AK, JCAP, 2014

Quantization of perturbations was studied and shown to reproduce the values close to observable in the cosmological experiments

12/15

Renormalizable ghost-free gravity Further

A wishful solution

In contrast with the local R2 gravity one can arrange such a parameter range that both bounce type and inflation type solutions coexist. We are however lack of an explicit construction of such a solution yet.

AK, work in progress

Renormalizability

The renormalizability must follow since the newly emerged momentum- dependent factors yield an exponential suppression in the UV. An example, for us a toy-model, we have in mind is the p-adic string theory proven to be manifestly finite. Its Lagrangian is L = −1 2φp−/m2

pφ +

1 p + 1φp+1

13/15

Renormalizable ghost-free gravity Outlook

Conclusions

• Non-local SFT motivated generalization of Einstein’s gravity is pre-

sented

• The ghost-free conditions on the propagator are clearly formulated
• There are exist exact analytic solutions including bounce and the

Starobinsky inflation in this framework

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Renormalizable ghost-free gravity Outlook

Open questioins

• Study of generalized models of the non-local gravity

I.Dimitrijevic, B.Dragovich, J.Grujic, AK, Z.Rakic, to appear next week

• There are preliminary results for the graviton action in the bosonic

closed SFT

AK, work in progress

• The next major step is to derive in some lowest approximation an

action for the massless states in a heterotic SFT

15/15

Thank you for listening!