AIDnon-localgravity,non-singularsolutionsand inﬂationeichhorn/Kumar.pdf · K. Sravan Kumar...

AID non-local gravity, non-singular solutions andinflation

K. Sravan Kumar

Van Swinderen Institute of Particle physics and gravity, University of Groningen,Groningen, Netherlands

Based on collaborations with A. S. Koshelev, S. Maheswari, A.Mazumdar and A. A. Starobinsky (arXiv:1711.08864 and

arXiv:1905.03227 )

K. Sravan KumarQuantum gravity and matter, IWH, Heidelberg, Germany

General relativity and beyond

I GR has been the most successful theory (perfect with solarsystem and also perfect in many astrophysical observations)and even today it surprises us with its utmost predictions(Recent LIGO achievements).

I But GR put us in singularities (big bang and black holes) or itjust direct us to modify it at high energies.

These strongly indicate GR modifications.


Cosmological singularitiesBig bang singularity (for FLRW) i.e., Ricci tensor (RHS) blows upand radiation EM tensor (blows up)

Gµν ≡ Rµν − gµνR2 = 1

M2p

Tµν

R = 12H2 + 6H = 0 H = 12t .

Anisotropic singularities (Kasner type with R = 0, Rµν = 0 butRiemann is singular as t → 0).

ds2 = −dt2 + t2p1dx21 + t2p2dx2

2 + t2p3dx23 .

Kasner solution is vacuum solution with R = 0, Rµν = 0 withp1 + p2 + p3 = p2

1 + p22 + p2

3 = 1.K. Sravan KumarQuantum gravity and matter, IWH, Heidelberg, Germany

Blackhole singularitiesSchwarzschild solution

ds2 = −(1− 2GM

r

)dt2 +

(1− 2GM

r

)−1dr2 + r2dΩ2 .

This metric is Ricci flat i.e., Rµν = R = 0 for r 6= 0.

What happens at r = 0? .

Using theory of distributions we see curvature tensors are delta functions(making use of the definition of generalized functions, GeneralizedFunctions, Volume 1 I. M. Gel’fand and G. E. Shilov, H. Balasin, H.Nachbagauer arXiv:gr-qc/9305009)

Gµν = Tµ

ν = diag−Mδ(r),−Mδ(r), 12Mδ(r), 12Mδ(r)


Beyond GR (local modifications)

Stelle’s (1977) theory of gravity is known to be renormalizable butcontains a tensor ghost which spoils the Unitarity

S =∫

d4x√−g

[M2

p2 R + αR2 + βR2

µν − Λcc

]

≡∫

d4x√−g

[M2

p2 R + αR2 + βW 2

µνρσ − Λcc

].

Equations of motion of Stelle’s gravity are

Gµν + 4βBµν + 2αRSµν + 2β (gµνR −∇µ∇νR) = 0=⇒ R = r1R + r2 .


Quadratic curvature gravity: UV complete but..

I It was long ago shown by K. S. Stelle (1977) that R2 gravity with aWeyl square term (RµνρσRµνρσ − 2RµνRµν + 1

3 R2) isrenormalizable but however a tensor ghost (around Minkowski)appear in the theory that spoils unitarity.

I We will see that non-local gravity solves this problem of unitarity.I Any finite derivative extension of Stelle’s gravity obviously

introduces a ghost.Q. How can we remove the tensor ghost ?

A1. Critical gravity i.e., quadratic curvature theory withcosmological constant as a particular function of Λ (α, β) H. Lu and

C.N.Pope (2011) .A2. Upcoming slides!


Non-singular solutions of R = r1R + r2 in cosmologyConsider FLRW metric dS2 = −dt2 + a2dx2

I R = constant = − r2r1

I a = cosh (Mst)I a = eγt2

I Starobinsky inflation (r2 = 0)

1. Cosine hyperbolic scale factor satisfies equations of motion of Stellegravity with a non-zero radiation

ρr = −272 α

1a4 .

2. For exponential bounce r1 < 0 =⇒ α < 0.


Anisotropic solutions of Stelle gravity

I Kasner metric is the vacuum solution of Stelle gravity. Therefore,there exist singularity

I It was studied D. Muller et al (arXiv:1710.08753) that Kasnersolution become the past attractor of Starobinsky inflation.

I Starobinsky theory is not free from singular solutions.

What modification of quadratic gravity can be ghost free andsingularity free?


SFT inspired Non-local gravity/AID gravity

Non-local gravity is a formulation aimed to construct UV complete(ghost free) theory of gravity. It has several motivations including Stringfield theory (String and branes are non-local objects and their interactioncarries vertex terms involving infinite derivatives e/M2). N. V.Krasnikov (1987), Y. V. Kuzmin (1989), Tomboulis, hep-th/9702146,Biswas, Mazumdar, Seigel hep-th/0508194, Biswas, Gerwik, Koivisto,Mazumdar (arXiv:1110.5249) and L. Modesto (arXiv:1107.2403 )

Non-locality in our scheme is introduced by inserting infinite derivativesin the action. The non-local extension of Stelle gravity is

S =∫

dDx√−g[

M2P2 R + λ

2 (RFR()R + WµνρσFW ()W µνρσ)− Λcc

].

Fi ’s are analytic. The theory is Analytic Infinite Derivative (AID) gravity.


Ghost free gravityI Infinite derivative non-locality may introduce more degrees of

freedom which include ghosts.I To avoid extra scalar dofs (ghosts) the function FR

(M2

)must be

analytic and should take the following form around MaximallySymmetric Backgrounds (MSS) (Biswas et al 1606.01250)

λ

M2pFR(/M2) = 1

61− eτ(/M2,R) (1−

M2

)ε(+ R

3

) ,

where H0 is an entire function of d’Alembertians.1. ε = 0 corresponds to a version of AID gravity with NO extrascalar degree of freedom. In this case, the theory is exactlyequivalent to GR.2. ε = 1 leads to AID gravity with an extra scalar of mass M2 > 0.In this case, the theory is equivalent to local R2.


I Tensor ghost that appears in the Stelle’s 4th derivative theory ofgravity can be consistently removed in its AID gravity extension byhaving the form factor

λ

M2pFW

(M2

)= 1

21− eα(− 2

3 R)(− 2

3 R)

I By demanding the above structure of form factors we indeed removeghosts in the theory and as such achieve UNITARITY. More recentdevelopments about it can be read from c.g., recent works fromAshoke Sen (in SFT), and A. Mazumdar et al (on Non-localgravitational theories)

I This is significant improvement of the Stelle’s 4th derivative theorywhich was proposed 40 years ago.

I It is natural to look for inflationary paradigm in this AID non-localgravity.


I Given we aimed to study inflation let us ask the question What isthe most general AID non-local theory of gravity around any MSS ?

I Recently, by Biswas et al in 1606.0125 derived a most generalnon-local gravity model that contributes to the study of linearperturbations around (A)dS or Minknowski backgrounds (i.e.,second order action)

S =∫

d4x√|g |[

M2P2 R + λ

2RFR

(M2

)R + λ

2RµνFS

(M2

)Rµν+

+ λ

2WµνρσFW

(M2

)W µνρσ − Λcc

].

In the local theory Ricci tensor term is absent, thanks to GaussBonnet invariance.


In our study arXiv: 1711.08864 we showed

I Using the Bianchi identity, Ricci tensor term can be eliminated fromthe theory (performing integration by parts and dropping the extraterms that contribute to beyond the second order action)

∇λRµνρσ +∇σRµνλρ +∇ρRµνσλ ≡ 0 .

As a result we obtain the most general theory of gravity relevant for thestudy of linear perturbations as

S =∫

d4x√|g |[

M2P2 R+λ

2RFR

(M2

)R+λ

2WµνρσFW

(M2

)W µνρσ−Λcc

].

The action exactly the AID non-local extension of Stelle’s theory.

We are interested in FLRW metric. This implies that the Weyl tensorvanishes and as such it does not manifest itself neither in the backgroundequations nor in linear perturbations.


EOM of AID theory (FLRW case)Equations of motion (EOM) of this theory are given by

Eµν ≡−[M2

p + 2λF(M2

)R]

Gµν − Λccδ

µν −

λ

2RF(M2

)Rδµν

+ 2λ (∇µ∂ν − δµν)F(M2

)R + λKµν −

λ

2 δµν

(Kσσ + K

)= −Tµ

ν ,

where

Kµν = 1M2

∞∑n=1

fnn−1∑l=0

∂µl

M2l R∂ν(M2

)n−l−1R , K =

∞∑n=1

fnn−1∑l=0

l

M2l R(M2

)n−lR .

The trace equation is

E = M2pR −

[4Λcc + 6λF

(M2

)R]−Kµµ − 2K = −Tµ

µ . (1)K. Sravan KumarQuantum gravity and matter, IWH, Heidelberg, Germany

I EOM of AID gravity even though looks complicated they can besolved by a simple equation which is exactly the trace equation oflocal R2 gravity (Biswas et al (2010), A. S. Koshelev (2017))

R = r1R + r2 =⇒ F(M2

)R = F1R + F2 .

I Using the above trace equation the trace equation for AID non-localgravity become

A1R + A2(∂µR∂µR + 2r1R2) + A3 = 0

I It was showed that the only solution of the above equation forR 6= 0 (See Appendix C of 1711.08864) is

F (1)R (r1) = 0, −M2

P2λ +3r1F1 = F2, Λcc = −

M2p r2

4r1, where F1 ≡ FR

( r1M2

)K. Sravan KumarQuantum gravity and matter, IWH, Heidelberg, Germany

Singularity free AID gravityI Cosine hyperbolic bounce a = cosh(Mst) found to be a good

bouncing solution in IDG which is ghost free (see pprs of Biswas etal (2005), A. S. Koshelev et al (2010)). This bouncing solutionsatisfies R = r1R + r2

I Schwarzschild solution is shown to be NOT a solution in IDGgravity due to the fact that infinite derivatives acting on a deltafunction smearout the singularity c.f., recent pprs of L. Buoninfanteet al. (2018) eα∂2

x δ(x) = 1√2αe− x2

4α

I Having infinite derivative quadratic in Weyl term in the action,kasner type singularity cannot be solution IDG action (c.f., A. S.Koshelev, J. Marto, A. Mazumdar (2018))

I Starobinsky inflation can be exactly realized and is the onlyinflationary solution in IDG gravity (without Ricci tensor term) (c.f.,A. S. Koshelev, KSK, A. A. Starobinsky (2017) )


Predictions of AID non-local R2 model: Scalar part

I We compute the second order action for the scalar perturbations asδ2S = δ2Slocal + δ2Snon−local

δ2Snon−local =∫

d4x√|g |ζZ2()ζ ,

where ζ = δ(R −M2R). By solving the trace equation of the AIDgravity we proved that ζ = 0 is the only solution, therefore thepower spectrum and the scalar spectral index are the same as in thelocal model.

PR|k=aH = H2

16π2ε21

3λF1R, ns ≡

d lnPRd ln k

∣∣∣∣k=aH

≈ 1− 2N ,


Predictions of AID non-local R2 inflation: Tensor partI In the action we have Weyl square term with AID operators in it.

Therefore, it is natural to expect the tensor power spectrum getsmodified compared to local model. This is exactly what happens,the second order action for tensor perturbations become

δ2Stensor =∫

d4x√−g λF1R

4 hµν(− R

6

)P()hµν ,

where hµν is transverse and traceless and

P() = 1 + 1F1R

(− R

3

)FW

(+ R

3

)= e−2ω() .

I Computing the tensor power spectrum in the leading order inslow-roll we obtain

PT |k=aH = H2

π2λF1Re−2ω( R

6M2 ) .

Note that function ω() must be evaluated at the position of thepole which is R/6.K. Sravan Kumar

Quantum gravity and matter, IWH, Heidelberg, Germany

Tensor to scalar ratio

I The ratio of tensor to scalar power spectrum becomes

r = PTPR

∣∣∣∣k=aH

= 12N2 e

−2ω(

R6M2

)

which can be any value r < 0.07.I Computing the tensor tilt which gets modified as

nt ≡d lnPTd ln k

∣∣∣∣k=aH

≈ −d lnPTdN

(1 + 1

2N

)≈ − 3

2N2 +( 8

N + 6N2

) R6M2ω

′(

R6M2

),


Modified consistency relationI It is often said the standard consistency relation r = −8nt is

said to be the smoking gun to falsify single field (canonical)models of inflation.

I The relation is obeyed in local R2 inflation as well.I In the non-local case, due to the addition of Weyl square term

we modify consistency relation as follows

rnt

= −8 e−2ω

(R

6M2

)1− 16

3 N R6M2ω′

(R

6M2

) .All these predictions of non-local R2 inflation are the natural targetfor future CMB probes. Validating this theory also gives us greaterunderstanding of this UV completion.


Peturbations beyond MSSSecond order perturbations of AID gravity around non-MSS backgroundinvestigated in arXiv:1905.03227

Figure: Venn diagram with our specific conformally-flat, non-maximallysymmetric background. ∂µR = 0 follows from ∇αSµν = 0. dS/AdS limitis achieved by putting Sµν = 0.


Conclusions

I AID non-local gravity gives a interesting generalization of 4thorder gravity which is ghost free and singularity free.

I Although EoM are complicated we can exactly solve by usingthe trace equation of local theory. Cosmological perturbationsaround the backgrounds satisfying local theory trace equationcan be done exactly.

I Starobinsky inflation can be exactly embedded in AID gravitywith spectral index remains unmodified but tensor to scalarratio and tensor tilt gets corrections. This can be interestingin the light of future CMB probes.


Thank you for your attention


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