Alessandro Codello

Covariant EFT of Gravityand Dark Energy

in collaboration with Rajeev K. JainarXiv:1507.06308arXiv:1507.07829

Jena6 Nov 2015

• Covariant EFT of gravity

• LO quantum corrections

• Curvature expansion

• Cosmological effective action

• Effective Friedmann equations

• Early times: Inflation

• Late times: Dark energy

Outline of the talk

• The theory of small fluctuations of the metric

• Planck’s scale is the characteristic scale of gravity

• Classical theory (CT) is successful over many orders of magnitude

EFT of Gravity

gµν → gµν +√16πGhµν = gµν +

1

Mhµν

M ≡ 1√16πG

=MPlanck√

16π

MPlanck =1√G

= 1.2× 1019 GeV

gµν

1

Mhµν

Seff [g] = M2

I1[g] +

1

M2I2[g] +

1

M4I3[g] + ...

I1[g] =

d4x

√gM2c0 − c1R

I2[g] =

d4x

√gc2,1R

2 + c2,2Ric2 + c2,3Riem

2

I3[g] =

d4x

√gc3,1RR+ c3,2RµνRµν + c3,3R

3 + ...

Derivative expansion of the UV action

EFT of Gravity

UV action contains all couplings expressed in terms of the scale M

EFT: saddle point expansion in 1

M2

e−Γ[g] =

1PI

Dhµν e−Seff [g+

1M h]

=

1PI

Dhµν e−M2I1[g+ 1

M h]+ 1M2 I2[g+ 1

M h]+...

Γ[g] = I1[g]

+1

M2

I2[g] +

1

2Tr log I(2)1 [g]

+1

M4

I3[g] +

1

2Tr

I(2)1 [g]

−1I(2)2 [g]

+ 2-loops with I1[g]

+ . . .

Covariant EFT of Gravity

Covariant EFT of Gravity

=Γ

+1

2+

1

M2

+

1

2− 1

12+

1

8+

1

M4

+ . . .

LO

NLO

CT

NNLO

I1

I2

I3

Covariant EFT of Gravity

=Γ

+1

2+

1

M2

+

1

2− 1

12+

1

8+

1

M4

+ . . .

LO

NLO

CT

NNLO

1) the general lagrangian of order is to be used both at tree level and in loop diagrams

2) the general lagrangian of order is to be used at tree level and as an insertion in loop diagrams

3) the renormalization program is carried out order by order

E2

En≥4

The EFT recipe in three lines

I1

I2

I3

Covariant EFT of Gravity

=Γ

+1

2+

1

M2

+

1

2− 1

12+

1

8+

1

M4

+ . . .

LO

NLO

CT

NNLO

LOQG: the only QG we will ever observe!

Even if we have a fundamental theory its is generally difficult to compute phenomenological parameters...

Renormalization

=1

2

1

(4π)2

d4x

√g

Λ4UV − 10Λ2

UV m2 + 5m4 log

Λ2UV

m2

+

−23

3Λ2UV +

13

3m2 log

Λ2UV

m2

R

+

7

20C2 +

1

4R2 +

149

180E − 19

15R

log

Λ2UV

m2

Cutoff regularization

measured+ phenomenological

parameters

Renormalization

=1

2

1

(4π)2

d4x

√g

Λ4UV − 10Λ2

UV m2 + 5m4 log

Λ2UV

m2

+

−23

3Λ2UV +

13

3m2 log

Λ2UV

m2

R

+

7

20C2 +

1

4R2 +

149

180E − 19

15R

log

Λ2UV

m2

Cutoff regularization

Dimensional regularization

logΛ2UV → 1

Λ2UV → 0 Λ4

UV → 0

measured+ phenomenological

parameters

Renormalization

=1

2

1

(4π)2

d4x

√g

Λ4UV − 10Λ2

UV m2 + 5m4 log

Λ2UV

m2

+

−23

3Λ2UV +

13

3m2 log

Λ2UV

m2

R

+

7

20C2 +

1

4R2 +

149

180E − 19

15R

log

Λ2UV

m2

Cutoff regularization

m2 = −2Λ

G runs if there is a mass scale involved alsoin dimensional regularization

measured+ phenomenological

parameters

Dimensional regularization

logΛ2UV → 1

Λ2UV → 0 Λ4

UV → 0

+1

2= − 1

2(4π)d/2

ddx

√g trR γi

−m2

R+ ...

Curvature expansion

The finite physical part of the effective action is covariantly encoded in the structure functionswhich can be computed using the non-local heat kernel expansion

Non-local heat kernel structure functions

Non-local heat kernelA. O. Barvinsky and G. A. Vilkovisky, Nucl. Phys. B 282 (1987) 163I. G. Avramidi, Lect. Notes Phys. M 64 (2000) 1A. Codello and O. Zanusso, J. Math. Phys. 54 (2013) 013513

γi

X

m2

≡ lim

ΛUV →∞

∞

1/Λ2UV

ds

ss−d/2+2 [fi(sX)− fi(0)]e

−sm2

= − 1

2(4π)2

d4x

√g tr

1RµνγRic

−m2

Rµν +

1

120RγR

−m2

R

−1

6RγRU

−m2

U+

1

2UγU

−m2

U+

1

12ΩµνγΩ

−m2

Ωµν

+

1

2

Curvature expansion

γRic(u) =1

40+

1

12u− 1

2

1

0dξ

1

u+ ξ(1− ξ)

2log [1 + u ξ(1− ξ)]

γR(u) = − 23

960− 1

96u+

1

32

1

0dξ

2

u2+

4

u[1 + ξ(1− ξ)]

− 1 + 2ξ(2− ξ)(1− ξ2)log [1 + u ξ(1− ξ)]

γRU (u) =1

12− 1

2

1

0dξ

1

u− 1

2+ ξ(1− ξ)

log [1 + u ξ(1− ξ)]

γU (u) = −1

2

1

0dξ log [1 + u ξ(1− ξ)]

γΩ(u) =1

12− 1

2

1

0dξ

1

u+ ξ(1− ξ)

log [1 + u ξ(1− ξ)]

Explicit form for the structure functions

u ≡ −m2

Curvature expansion

γRic(u) = − u

840+

u2

15120− u3

166320+O(u4)

γR(u) = − u

336+

11u2

30240− 19u3

332640+O(u4)

γRU (u) =u

30− u2

280+

u3

1890+O(u4)

γU (u) = − u

12+

u2

120− u3

840+O(u4)

γΩ(u) = − u

120+

u2

1680− u3

15120+O(u4)

= − 1

2(4π)2

d4x

√g tr

1RµνγRic

−m2

Rµν +

1

120RγR

−m2

R

−1

6RγRU

−m2

U+

1

2UγU

−m2

U+

1

12ΩµνγΩ

−m2

Ωµν

+

1

2

u ≡ −m2

High energy expansion u 1

Curvature expansion

γRic(u) =23

450− 1

60log u+

5

18u− log u

6u+

1

4u2− log u

2u2+O

1

u3

γR(u) =1

1800− 1

120log u− 2

9u+

log u

12u+

1

8u2+

log u

4u2+O

1

u3

γRU (u) = − 5

18+

1

6log u+

1

u− 1

2u2− log u

u2+O

1

u3

γU (u) = 1− 1

2log u− 1

u− log u

u− 1

2u2+

log u

u2+O

1

u3

γΩ(u) =2

9− 1

12log u+

1

2u− log u

2u− 3

4u2− log u

2u2+O

1

u3

= − 1

2(4π)2

d4x

√g tr

1RµνγRic

−m2

Rµν +

1

120RγR

−m2

R

−1

6RγRU

−m2

U+

1

2UγU

−m2

U+

1

12ΩµνγΩ

−m2

Ωµν

+

1

2

Low energy expansion u 1

u ≡ −m2

LO effective action to R2

Γ[g] =1

16πG

d4x√g (2Λ−R) +

1

2λ

d4x√g C

2 +1

ξ

d4x√g R

2

+

d4x√g Cµναβ G

−m2

C

µναβ +

d4x√g RF

−m2

R+O(R3)

G2(u) = − 1

2(4π)2

5γRic(u) + 3γU (u)− 12γΩ(u)

− 4γRic(u)− γU (u) + 4γΩ(u)

F2(u) = − 1

2(4π)2

10

3γRic(u) + 10γR(u) + 6γRU (u) + 4γU (u)− 2γΩ(u)

−8

3γRic(u)− 8γR(u) + 2γRU (u)−

2

3γU (u) +

2

3γΩ(u)

G0(u),F0(u),G1/2(u),F1/2(u),G1(u),F1(u)

Matter contributions:

Graviton contributions:

LO effective action to R2

Γ[g] =1

16πG

d4x√g (2Λ−R) +

1

2λ

d4x√g C

2 +1

ξ

d4x√g R

2

+

d4x√g Cµναβ G

−m2

C

µναβ +

d4x√g RF

−m2

R+O(R3)

0.01 0.1 1 10 100 10000.002

0.001

0.000

0.001

0.002

u

0.01 0.1 1 10 100 10000.04

0.02

0.00

0.02

0.04

u

u ≡ −m2

Adding matter

LO

NLO

CT=Γ

+1

2+

1

M2

+

+

1

2

+ . . .

Integrating out matter

= −1

2

1

(4π)2

n= d2+1

1

m2n−dB2n

1

2

Matter induced effective action:expandable in inverse powers of the (lightest) particle mass

Integrating out matter

= −1

2

1

(4π)21

m2B6 + ...

1

2

As in QED when we integrate out electrons

Integrating out matter

= −1

2

1

(4π)21

m2

d4x

√g

1

336RR+

1

840RµνRµν

+1

1296R3 − 1

1080RRµνR

µν − 4

2835Rν

µRανR

µα

+1

945RµνRαβR

µανβ +1

1080RRµναβR

µναβ +1

7560RµνR

µαβγRναβγ

+17

45360R αβ

µν R γδαβ R µν

γδ − 1

1620Rα β

µ νRµ νγ δR

γ δα β

+ . . .

1

2

1

M4→ 1

m2M2

Changes how cubic terms are suppressed

Corrections to Newton’s potential

iκ

2

Pαβ,γδ

kµkν + (k − q)µ(k − q)ν + qµqν − 3

2ηµνq2

+2qλqσIλσ,αβI

µν,γδ + Iλσ,γδI

µν,αβ − Iλµ,αβI

σν,γδ − Iσν,αβI

λµ,γδ

+qλq

µηαβI

λν,γδ + ηγδI

λν,αβ

+ qλq

νηαβI

λµ,γδ + ηγδI

λµ,αβ

−q2ηαβI

µν,γδ + ηγδI

µν,αβ

− ηµνqλqσ (ηαβIγδ,λσ + ηγδIαβ,λσ)

+2qλ

Iσν,αβIγδ,λσ(k − q)µ + Iσµ,αβIγδ,λσ(k − q)ν

−Iσν,γδIαβ,λσkµ − Iσµ,γδIαβ,λσk

ν

+q2Iσµ,αβI

νγδ,σ + I ν

αβ,σ Iσµ,αδ

+ ηµνqλqσ

Iαβ,λρI

ρσ,γδ + Iγδ,λρI

ρσ,αβ

+

k2 + (k − q)2

Iσµ,αβI

νγδ,σ + Iσν,αβI

µγδ,σ − 1

2ηµνPαβ,γδ

−k2ηγδI

µν,αβ + (k − q)2ηαβI

µν,γδ

=

the truth behind Feynman diagrams...

Corrections to Newton’s interaction

Leading quantum corrections to Newton’s potentialJ.F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994)

V = −GMm

r

1 + 3

G(M +m)

c2r+

41

10π

Gc3r2

+ · · ·

Corrections to Newton’s interaction

GM⊙c2r⊙

∼ 10−6

Gc3r2⊙

∼ 10−88

Leading quantum corrections to Newton’s law are incredibly small!

sun

Corrections to Newton’s interaction

Look for physical situations whereLO corrections are enhanced

Can we ever observequantum gravity effects?

Cosmological effective action

Γ[g] =1

16πG

d4x

√−g (R− 2Λ) + ξ

d4x

√−g R2 +

d4x

√−g RF ()R

ds2 = −dt2 + a2(t)dx2

Cαβγδ = 0

FRW metric

Weyl tensor vanishes

Euclidean to Lorentzian

A predictive framework for cosmology

Phenomenological parameters

Cavendish 1797 (1% off best value!)

G = 6.67428× 10−11m3kg−1s−2

Phenomenological parameters

Supernova Cosmology Project

Λ = 10−47 GeV4

Planck mission

Phenomenological parameters

ξ(k∗) ∼ 109 k∗ ≡ 0.05Mpc−1 ∼ 10−40GeVξ ≡ cR2

Cosmological effective action

Γ[g] =1

16πG

d4x

√−g (R− 2Λ) + ξ

d4x

√−g R2 +

d4x

√−g RF ()R

Cαβγδ = 0

Cosmological effective action

Γ[g] =1

16πG

d4x

√−g (R− 2Λ) + ξ

d4x

√−g R2 +

d4x

√−g RF ()R

F () = α log−m2

+ βm2

−

+ γm2

− log−m2

+ δm4

(−)2

+ ...

α, β, γ, δ

are calculable constants

depending on effective gravitons

and matter content

Cαβγδ = 0

Cosmological effective action

Γ[g] =1

16πG

d4x

√−g (R− 2Λ) + ξ

d4x

√−g R2 +

d4x

√−g RF ()R

Leading logsJ. F. Donoghue and B. K. El-Menoufi, Phys. Rev. D 89, 104062 (2014)

Non-local cosmologyS. Deser and R. P. Woodard, Phys. Rev. Lett. 99, 111301 (2007)

Non-local gravity and dark energyM. Maggiore and M. Mancarella, Phys. Rev. D 90, 023005 (2014).

F () = α log−m2

+ βm2

−

+ γm2

− log−m2

+ δm4

(−)2

+ ...

Cαβγδ = 0

α, β, γ, δ

are calculable constants

depending on effective gravitons

and matter content

Effective non-local cosmologyA. C. and K. J. Jain

Effective Friedmann equations (local)

H2 + 96πξG

2HH + 6H2

H − H2=

1

3Λ+

8πG

3ρ

Gµν +∆Gµν = 8πGTµν

− ξ

16πG∆Gµν = 2

Rµν − 1

4gµνR

R− 2 (∇µ∇ν − gµν)RR2

Modified Einstein’s equations

Local correction (covariant form)

Local correction (FRW form)

Effective Friedmann equations (local)

Pure Starobinsky gravity is exactly solvable

2HH + 6H2H − H

2 = 0

t+ C2 =

dH

C1

√H − 2H2

H =1

2t

H = 0 inflation

radiation

H2 + 96πξG

2HH + 6H2

H − H2=

1

3Λ+

8πG

3ρ

R R2 Λ ρ

Jordan frame Starobinsky gravity

Early times: Inflation

R+R2 = Λ

R+R2 = 0

Early times: Inflation

R+R2 = ργ

Early times: Inflation

R+R2 = ργ + ρm

Early times: Inflation

Effective Friedmann equations (non-local)

H2 − 16πGβm2

1

6U

2 − 2HU − 2H2U

=

8πG

3ρ

U + 3HU = 62H2 + H

1

16πGβm2∆G

R 1−R

µν = −2GµνU + 2gµνR+ 2∇µ∇νU +∇µU∇νU

− 1

2gµν∇αU∇αU

−U = R

Gµν +∆Gµν = 8πGTµν

Modified Einstein’s equations

Non local correction (covariant form)

Non local correction (FRW form)

Effective Friedmann equations (non-local)

H2 − 32πGδm4

2H2

S +HS +1

2HS − 1

6U S

=

8πG

3ρ

U + 3HU = 62H2 + H

S + 3HS = U

1

16πGδm4∆G

R 12 R

µν = − 2GµνS + 2gµνU + 2∇µ∇νS

+ 2∇µU∇νS − gµν∇αU∇αS +1

2gµνU

2

−U = R

−S = U .

Effective Friedmann equations (non-local)

H2 − 32πGδm4

2H2

S +HS +1

2HS − 1

6U S

=

8πG

3ρ

U + 3HU = 62H2 + H

S + 3HS = U

1

16πGδm4∆G

R 12 R

µν = − 2GµνS + 2gµνU + 2∇µ∇νS

+ 2∇µU∇νS − gµν∇αU∇αS +1

2gµνU

2

−U = R

−S = U .

H2 =

8πG

3(ρ+ ρDE )

0.001 0.1 10 1000 1051.0

0.8

0.6

0.4

0.2

0.0

0.2

Τ

w

R1

−R

R1

2R

Λ

matter

radiation

ρtot + 3H(1 + w)ρtot = 0 w = −1− 1

3H

ρtotρtot

Late times: Dark energy

0.001 0.1 10 1000 1051056

1046

1036

1026

1016

106

Τ

Ρ

ρr

ρm

ρde

R+R1

2R = ρr + ρm

Late times: Dark energy

Ωde

Ωm

Ωr

0.001 0.1 10 10000.0

0.2

0.4

0.6

0.8

1.0

Τ

Late times: Dark energy

Conclusions and Outlook

• Consistent cosmological history

• Compute all LO terms

• The role of the Conformal anomaly

• Apply to stars/black holes

• Connection with high energy quantum gravity

• Falsify!

Thank you