EFT of Gravity and Cosmologywork with:

Alessandro Codello

EFT of Gravity and Cosmology

work with:

J Joergensen, F Sannino, O Svendsen R Percacci, A Tonero, L Rachwal

R K Jain

Probing the fundamental

nature of spacetime with the RG

NORDITA, Stockholm 23-27 March 2015

•

Effectivity vs Universality

•

Example: EFT of Pions (CPT)

•

Covariant EFT of Gravity in three lines

•

Renormalization in different schemes

•

Phenomenological parameters and their estimation

•

Adding matter

•

LO quantum corrections

•

Marginally deformed Starobinsky

•

Effective Friedmann equations

Outline of the talk

Effectivity vs Universality

theory space Universality

describes the massless IR

Effective field theories

describe the massive IR

Two main reasons why mathematical modeling of nature actually works

Effectivity vs Universality

theory space Universality

describes the massless IR

Effective field theories

describe the massive IR

Massive IR lies in the broken phase (G to G/H) Characteristic large scale at which G is broken_M

The theory of pions (CPT)

•

Low energy QCD can be described by an EFT of pions

•

Symmetry braking pattern

•

is the pion decay constant

•

Phenomenological parameters to be fixed by experiments ( )

•

Renormalization and scale dependence

•

Low energy expansion of physical quantities A(E) = E²

M²

�

1 + C₁l₁ E²

M² + C₂l₂ E² M²

�

+ ...

l_i(µ₁) = l_i(µ₂) + γ_i

(4π)² log µ₁ µ₂ l_i

M F_π ∼ 10² MeV

i = 1, 2 SU (2)_L × SU(2)^R → SU(2)^V

•

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

g_µν → g^µν + √

16πGh_µν = g_µν + 1

M h_µν

M ≡ 1

√16πG = M_{P lanck}

√16π

M_{P lanck} = 1

√G = 1.2 × 10¹⁹ GeV

EFT of Gravity

S_{ef f}[g] =

�

d⁴x√ g

�

M⁴c₀ + M²(−R) + c^2,1R² + c_2,2Ric² + c_2,3Riem² + 1

M² c_3,1R³ + ...

�

EFT of Gravity

S_{ef f}[g] =

�

d⁴x√ g



M⁴ c₀

����

R⁰

+M² (−R)

� �� �

R

+ c_2,1R² + c_2,2Ric² + c_2,3Riem²

� �� �

R²

+ 1

M² c_3,1R³ + ...

� �� �

R³





EFT of Gravity

S_{ef f}[g] =

�

d⁴x√ g



M⁴ c₀

����

R⁰

+M² (−R)

� �� �

R

+ c_2,1R² + c_2,2Ric² + c_2,3Riem²

� �� �

R²

+ 1

M² c_3,1R³ + ...

� �� �

R³





≡ M²

�

I₁[g] + 1

M² I₂[g] + 1

M⁴ I₃[g] + ...

�

m² = −2Λ

Covariant EFT of Gravity

EFT: saddle point expansion in ¹ M²

e

^−Γ[g]

=

�

1P I

Dh

^µν

e

^{−Sef f[g+ M1 h]}

=

�

1P I

Dh

^µν

e

^−M2

{

^{I1[g+ M1 h]+} M 2¹ I₂[g+ _M¹ h]+...

}

Covariant EFT of Gravity

LO

NLO CT

NNLO Γ[g] = I₁[g]

+ 1 M²

�

I₂[g] + 1

2Tr log I₁⁽²⁾[g]

�

+ 1 M⁴

�

I₃[g] + 1

2Tr ��

I₁⁽²⁾[g]�₋₁

I₂⁽²⁾[g]

�

+ 2-loops with I₁[g]

�

+ . . .

EFT: saddle point expansion in ¹ M²

Covariant EFT of Gravity

Γ =

� + 1

+ 1 2 M

²

�

� + 1

2 − 1

12 + 1 + 1 8

M

⁴

�

+ . . .

LO

NLO CT

NNLO

I₁

I₂

I₃

Covariant EFT of Gravity

Γ =

� + 1

+ 1 2 M

²

�

� + 1

2 − 1

12 + 1 + 1 8

M

⁴

�

+ . . .

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 E^n≥4

The EFT recipe in three lines

I₁

I₂

I₃

Covariant EFT of Gravity

Γ =

� + 1

+ 1 2 M

²

�

� + 1

2 − 1

12 + 1 + 1 8

M

⁴

�

+ . . .

LO

NLO CT

NNLO What do we already know?

UV divergencies and renormalization

G. ’t Hooft and M. J. G. Veltman, Annales Poincare Phys. Theor. A 20 (1974) 69 G. W. Gibbons, S. W. Hawking and M. J. Perry, Nucl. Phys. B 138 (1978) 141 S. M. Christensen and M. J. Duff, Nucl. Phys. B 170 (1980)

Covariant EFT of Gravity

Γ =

� + 1

+ 1 2 M

²

�

� + 1

2 − 1

12 + 1 + 1 8

M

⁴

�

+ . . .

LO

NLO CT

NNLO

Two loops UV divergencies

M.H. Goroff and A. Sagnotti, Nucl.Phys.B266, 709 (1986) A. E. M. van de Ven, Nucl. Phys. B378, 309 (1992)

What do we already know?

Covariant EFT of Gravity

Γ =

� + 1

+ 1 2 M

²

�

� + 1

2 − 1

12 + 1 + 1 8

M

⁴

�

+ . . .

LO

NLO CT

Finite LO terms NNLO

Leading logs

J.F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994)

A. C., J. Joergensen, F. Sannino and O. Svendsen, JHEP 1502 (2015) 050

Conformal anomaly

S. Deser, M. J. Duff and C. J. Isham, Nucl. Phys. B 111, 45 (1976) R.J. Riegert, Phys. Lett. B 134 (1984) 56

Four graviton vertex in Minkowski space

D. C. Dunbar and P. S. Norridge, Nucl. Phys. B 433, 181 (1995)

Curvature square terms

A. C. and R. K. Jain, in preparation

What do we already know?

Covariant EFT of Gravity

Γ =

� + 1

+ 1 2 M

²

�

� + 1

2 − 1

12 + 1 + 1 8

M

⁴

�

+ . . .

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

�

Λ⁴_{U V} − 10Λ²U V m² + 5m⁴ log Λ²_{U V} m² +

�

−23

3 Λ²_{U V} + 13

3 m² log Λ²_{U V} m²

� R +

� 7

20C² + 1

4R² + 149

180E − 19

15�R�

log Λ²_{U V} m²

� Cutoff regularization

m² = −2Λ

measured

+ phenomenological parameters

Renormalization

= 1 2

1 (4π)²

�

d⁴x√ g

�

Λ⁴_{U V} − 10Λ²U V m² + 5m⁴ log Λ²_{U V} m² +

�

−23

3 Λ²_{U V} + 13

3 m² log Λ²_{U V} m²

� R +

� 7

20C² + 1

4R² + 149

180E − 19

15�R�

log Λ²_{U V} m²

� Cutoff regularization

m² = −2Λ

Dimensional regularization

log Λ²_{U V} → 1

� Λ²_{U V} → 0 Λ⁴_{U V} → 0

measured

+ phenomenological parameters

Renormalization

= 1 2

1 (4π)²

�

d⁴x√ g

�

Λ⁴_{U V} − 10Λ²U V m² + 5m⁴ log Λ²_{U V} m² +

�

−23

3 Λ²_{U V} + 13

3 m² log Λ²_{U V} m²

� R +

� 7

20C² + 1

4R² + 149

180E − 19

15�R�

log Λ²_{U V} m²

� Cutoff regularization

m² = −2Λ

G runs if there is a mass scale involved also in dimensional regularization [Kirill’s talk]

Dimensional regularization

log Λ²_{U V} → 1

� Λ²_{U V} → 0 Λ⁴_{U V} → 0

c_i(µ₁) = c_i(µ₂) + γ_i

(4π)² log µ₁

µ₂ µ∂_µc_i = γ_i (4π)²

measured

+ phenomenological parameters

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 [Alfio’s talk]

Phenomenological parameters

ξ(k_∗) ∼ 10⁹ k_∗ ≡ 0.05 Mpc⁻¹ ∼ 10⁻⁴⁰GeV ξ ≡ cR²

Adding matter

Γ =

� + 1

2 + 1

M

²

� + 1

+ 2 + . . .

LO

NLO CT

−

Adding matter

Γ =

� + 1

2 + 1

M

²

� + 1

+ 2 + . . .

LO

NLO CT

−

UV divergencies and renormalization with matter

Scalars

A. O. Barvinsky, A. Y. .Kamenshchik and I. P. Karmazin, Phys. Rev. D 48 (1993) 3677 ...

Gauge

S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 96, 231601 (2006) ...

Yukawa

A. Rodigast and T. Schuster, Phys. Rev. Lett. 104, 081301 (2010) ...

Adding matter

Γ =

� + 1

2 + 1

M

²

� + 1

+ 2 + . . .

LO

NLO CT

−

Finite LO terms with matter

Flat space corrections to Newton’s potential

J.F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994)

N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein (2003b) Phys. Rev. D 67 I.B. Khriplovich, G.G. Kirilin (2004) J. Exp. Theor. Phys. 98, 1063-1072

Covariant leading logs

A C, R Percacci, L Rachwal and A Tonero in preparation

Adding matter

Γ =

� + 1

2 + 1

M

²

� + 1

+ 2 + . . .

LO

NLO CT

−

Matter induced effective action

Integrating out matter

= −1 2

1 (4π)²

�

n=^d₂+1

1

m^2n−d B_2n 1

2

Matter induced effective action:

expandable in inverse powers of the (lightest) particle masse Local heat kernel coefficients

B6

Gilkey, PB

A. O. Barvinsky and G. A. Vilkovisky, Phys. Rept. 119 (1985) 1.

I. G. Avramidi, Lect. Notes Phys. M 64 (2000) 1.

Groh, Kai Saueressig, Frank Zanusso, Omar

B8

Amsterdamski, P Ven, AEM Van de

Integrating out matter

= −1 2

1 (4π)²

1

m² B₆ + ...

1 2

As in QED when we integrate out electrons

Integrating out matter

= −1 2

1 (4π)²

1 m²

�

d⁴x√ g

� 1

336R�R + 1

840R_µν�R^µν + 1

1296R³ − 1

1080RR_µνR^µν − 4

2835R_µ^νR^α_ν R^µ_α + 1

945R_µνR_αβR^µανβ + 1

1080RR_µναβR^µναβ + 1

7560R_µνR^µαβγR^ν_αβγ + 17

45360R_µν^αβR_αβ^γδR_γδ^µν − 1

1620R^{α β}_{µ ν}R^{µ ν}_{γ δ}R^{γ δ}_{α β}

�

+ . . . 1

2

1

M⁴ → 1 m²M²

Changes how cubic terms are suppressed:

Integrating out matter

= −1 2

1 (4π)²

1 m²

�

d⁴x√ g

� 1

336R�R + 1

840R_µν�R^µν + 1

1296R³ − 1

1080RR_µνR^µν − 4

2835R_µ^νR^α_ν R^µ_α + 1

945R_µνR_αβR^µανβ + 1

1080RR_µναβR^µναβ + 1

7560R_µνR^µαβγR^ν_αβγ + 17

45360R_µν^αβR_αβ^γδR_γδ^µν − 1

1620R^{α β}_{µ ν}R^{µ ν}_{γ δ}R^{γ δ}_{α β}

�

+ . . . 1

2

The same can be applied to and with some modifications to and thus find

Corrections to Newton’s interaction

iκ 2

�

P_αβ,γδ

�

k^µk^ν + (k − q)^µ(k − q)^ν + q^µq^ν − 3

2η^µνq²

�

+2q_λq_σ �

I^λσ,_αβI^µν,_γδ + I^λσ,_γδI^µν,_αβ − I^λµ,_αβI^σν,_γδ − I^σν,_αβI^λµ,_γδ� +�

q_λq^µ �

η_αβI^λν,_γδ + η_γδI^λν,_αβ�

+ q_λq^ν �

η_αβI^λµ,_γδ + η_γδI^λµ,_αβ�

−q² �

η_αβI^µν,_γδ + η_γδI^µν,_αβ�

− η^µνq^λq^σ (η_αβI_γδ,λσ + η_γδI_αβ,λσ)� +�

2q^λ �

I^σν,_αβI_γδ,λσ(k − q)^µ + I^σµ,_αβI_γδ,λσ(k − q)^ν

−I^σν,_γδI_αβ,λσk^µ − I^σµ,_γδI_αβ,λσk^ν� +q² �

I^σµ,_αβI_γδ,σ^ν + I_αβ,σ^νI^σµ,_αδ�

+ η^µνq^λq_σ �

I_αβ,λρI^ρσ,_γδ + I_γδ,λρI^ρσ,_αβ��

+

��

k² + (k − q)²��

I^σµ,_αβI_γδ,σ^ν + I^σν,_αβI_γδ,σ^µ − 1

2η^µνP_αβ,γδ

�

− �

k²η_γδI^µν,_αβ + (k − q)²η_αβI^µν,_γδ���

=

the truth behind Feynman diagrams...

Corrections to Newton’s interaction

V = − GM m r

� 1 + a G(M + m)

c

²

r + b G �

c

³

r

²

+ · · · �

Corrections to Newton’s interaction

V = − GM m r

� 1 + a G(M + m)

c

²

r + b G �

c

³

r

²

+ · · · � [G] = m

³

Kg s

²

[ �] = m

²

Kg

s [c] = m

s

Corrections to Newton’s interaction

Leading quantum corrections to Newton’s potential

J.F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994)

V = − GM m r

� 1 + 3 G(M + m)

c

²

r + 41 10π

G �

c

³

r

²

+ · · · �

Corrections to Newton’s interaction

GM

_⊙

c

²

r

_⊙

∼ 10

⁻⁶

G �

c

³

r

_⊙²

∼ 10

⁻⁸⁸

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

sun

Corrections to Newton’s interaction

Maybe there are some inside here...

Can we ever observe

quantum gravity effects?

LO quantum corrections

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R

�

α log R

µ² + β log −�

µ²

�

R + ...

Non-analytical vs non-local

LO quantum corrections

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R

�

α log R

µ² + β log −�

µ²

�

R + ...

Non-analytical vs non-local

Obtain finite part from beta functions (physical running)

2R ∂_R = −µ ∂^µ 2q² ∂_q² = −µ ∂^µ

LO quantum corrections

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R

�

α log R

µ² + β log −�

µ²

�

R + ...

Non-analytical vs non-local

Obtain finite part from beta functions (physical running)

2R ∂_R = −µ ∂^µ 2q² ∂_q² = −µ ∂^µ

As in QED

LO quantum corrections

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R

�

α log R

µ² + β log −�

µ²

�

R + ...

Non-analytical vs non-local

Obtain finite part from beta functions (physical running)

2R ∂_R = −µ ∂^µ 2q² ∂_q² = −µ ∂^µ

C = N_S

72 minimally coupled scalars C = 1

4 EFT gravity C = 5

36 HDG

α = C 4(4π)²

α = β A topology must be

chosen...

Marginally deformed Starobinsky

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R h

� R µ²

�

R + ...

Marginally deformed Starobinsky

R ∂_Rh = − C

2(4π)² h(R/µ²) = log R/µ² One loop flow equation

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R h

� R µ²

�

R + ...

Marginally deformed Starobinsky

R ∂_Rh = − C

2(4π)² h h(R) = h(R₀)

� R R₀

�₋ ^C

2(4π)2

+ 1

2

^{= −}

1 2(4π)²

�

d⁴x√

g R h

� R µ²

�

R + ...

C > 0 α = C α > 0 4(4π)²

RG improved flow equation R ∂_Rh = − C

2(4π)² h(R/µ²) = log R/µ² One loop flow equation

Marginally deformed Starobinsky (L)

Leading quantum corrections to tensor-to-scalar ratio

A. C, J. Joergensen, F. Sannino and O. Svendsen, JHEP 1502, 050 (2015)

γ

_i

� X m

²

�

≡ lim

Λ_{U V} →∞

�

_∞

1/Λ²_{U V}

ds

s s

^−d/2+2

[f

_i

(sX) − f

ⁱ

(0)]

+ 1

2

^{= −}

1

2(4π)^d/2

�

d^dx√

g tr R γⁱ

�−�

m²

�

R + ...

Curvature expansion

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

Non-local heat kernel structure functions Non-local heat kernel

A. O. Barvinsky and G. A. Vilkovisky, Nucl. Phys. B 282 (1987) 163 I. G. Avramidi, Lect. Notes Phys. M 64 (2000) 1

A. Codello and O. Zanusso, J. Math. Phys. 54 (2013) 013513

= − 1 2(4π)²

�

d⁴x√

g tr

�

1R_µνγ_Ric

�−�

m²

�

R^µν + 1

120Rγ_R

�−�

m²

� R

−1

6Rγ_RU

�−�

m²

�

U + 1

2Uγ_U

�−�

m²

�

U + 1

12Ω_µνγ_Ω

�−�

m²

�

Ω^µν

�

+ 1 2

Curvature expansion

γ_Ric(u) = 1

40 + 1

12u − 1 2

� 1 0

dξ

�1

u + ξ(1 − ξ)

�2

log [1 + u ξ(1 − ξ)]

γR(u) = − 23

960 − 1

96u + 1 32

� 1 0

dξ � 2

u² + 4

u [1 + ξ(1 − ξ)]

− 1 + 2ξ(2 − ξ)(1 − ξ²)�

log [1 + u ξ(1 − ξ)]

γ_RU(u) = 1

12 − 1 2

� 1 0

dξ

�1

u − 1

2 + ξ(1 − ξ)

�

log [1 + u ξ(1 − ξ)]

γU(u) = −1 2

� 1 0

dξ log [1 + u ξ(1 − ξ)]

γΩ(u) = 1

12 − 1 2

� 1 0

dξ

�1

u + ξ(1 − ξ)

�

log [1 + u ξ(1 − ξ)]

Explicit form for the structure functions

u ≡ −�

m²

Curvature expansion

γ_Ric(u) = − u

840 + u²

15120 − u³

166320 + O(u⁴) γ_R(u) = − u

336 + 11u²

30240 − 19u³

332640 + O(u⁴) γ_RU(u) = u

30 − u²

280 + u³

1890 + O(u⁴) γ_U(u) = − u

12 + u²

120 − u³

840 + O(u⁴) γ_Ω(u) = − u

120 + u²

1680 − u³

15120 + O(u⁴)

= − 1 2(4π)²

�

d⁴x√

g tr

�

1R_µνγ_Ric

�−�

m²

�

R^µν + 1

120Rγ_R

�−�

m²

� R

−1

6Rγ_RU

�−�

m²

�

U + 1

2Uγ_U

�−�

m²

�

U + 1

12Ω_µνγ_Ω

�−�

m²

�

Ω^µν

�

+ 1 2

u ≡ −�

m² Large energy expansion u � 1

Curvature expansion

γ_Ric(u) = − u

840 + u²

15120 − u³

166320 + O(u⁴) γ_R(u) = − u

336 + 11u²

30240 − 19u³

332640 + O(u⁴) γ_RU(u) = u

30 − u²

280 + u³

1890 + O(u⁴) γ_U(u) = − u

12 + u²

120 − u³

840 + O(u⁴) γ_Ω(u) = − u

120 + u²

1680 − u³

15120 + O(u⁴)

= − 1 2(4π)²

�

d⁴x√

g tr

�

1R_µνγ_Ric

�−�

m²

�

R^µν + 1

120Rγ_R

�−�

m²

� R

−1

6Rγ_RU

�−�

m²

�

U + 1

2Uγ_U

�−�

m²

�

U + 1

12Ω_µνγ_Ω

�−�

m²

�

Ω^µν

�

+ 1 2

match with local heat

kernel

Large energy expansion u � 1

u ≡ −�

m²

Curvature expansion

γ_Ric(u) = 23

450 − 1

60 log u + 5

18u − log u

6u + 1

4u² − log u

2u² + O

� 1 u³

�

γ_R(u) = 1

1800 − 1

120 log u − 2

9u + log u

12u + 1

8u² + log u

4u² + O

� 1 u³

�

γ_RU(u) = − 5

18 + 1

6 log u + 1

u − 1

2u² − log u

u² + O

� 1 u³

�

γ_U(u) = 1 − 1

2 log u − 1

u − log u

u − 1

2u² + log u

u² + O

� 1 u³

�

γ_Ω(u) = 2

9 − 1

12 log u + 1

2u − log u

2u − 3

4u² − log u

2u² + O

� 1 u³

�

= − 1 2(4π)²

�

d⁴x√

g tr

�

1R_µνγ_Ric

�−�

m²

�

R^µν + 1

120Rγ_R

�−�

m²

� R

−1

6Rγ_RU

�−�

m²

�

U + 1

2Uγ_U

�−�

m²

�

U + 1

12Ω_µνγ_Ω

�−�

m²

�

Ω^µν

�

+ 1 2

Low energy expansion u � 1

u ≡ −�

m²

Cosmological effective action (L)

Γ[g] = 1 16πG

�

d⁴x√

−g (R − 2Λ) + ξ

�

d⁴x√

−g R² +

�

d⁴x√

−g RF (�)R

Cosmological effective action (L)

Γ[g] = 1 16πG

�

d⁴x√

−g (R − 2Λ) + ξ

�

d⁴x√

−g R² +

�

d⁴x√

−g RF (�)R

F (�) = α log −�

m²

+ β m²

−�

+ γ m²

−� log −�

m²

+ δ m⁴ (−�)²

+ ...

α, β, γ, δ

are calculable constants depending on matter

content

Cosmological effective action (L)

Γ[g] = 1 16πG

�

d⁴x√

−g (R − 2Λ) + ξ

�

d⁴x√

−g R² +

�

d⁴x√

−g RF (�)R

Leading logs

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

Non-local cosmology

S. Deser and R. P. Woodard, Phys. Rev. Lett. 99, 111301 (2007)

Non-local gravity and dark energy

M. Maggiore and M. Mancarella, Phys. Rev. D 90, 023005 (2014).

F (�) = α log −�

m²

+ β m²

−�

+ γ m²

−� log −�

m²

+ δ m⁴ (−�)²

+ ...

α, β, γ, δ

are calculable constants depending on matter

content

Effective non-local cosmology

A. C. and K. J. Jain in preparation

Effective Friedmann equations (L)

Local gravity @ LO

H² + 96πξG �

2H ¨H + 6H²H˙ − ˙H²�

= 1

3Λ + 8πG 3 ρ

Effective Friedmann equations (L)

Local gravity @ LO

H² + 96πξG �

2H ¨H + 6H²H˙ − ˙H²�

= 1

3Λ + 8πG 3 ρ

2H ¨H + 6H²H˙ − ˙H² = 0

t + C₂ =

� dH

C₁√

H − 2H² H = 1

2t H = 0

Pure Starobinsky gravity is exactly solvable

Effective Friedmann equations (L)

Local gravity @ LO

H² + 96πξG �

2H ¨H + 6H²H˙ − ˙H²�

= 1

3Λ + 8πG 3 ρ

2H ¨H + 6H²H˙ − ˙H² = 0

t + C₂ =

� dH

C₁√

H − 2H² H = 1

2t H = 0

T = 0 ⇒ R = 0 R + Rⁿ⁻¹δR = −8πG T

Pure Starobinsky gravity is exactly solvable

Effective Friedmann equations (L)

R + R² = 0 R + R² = Λ

R + R² = ρ_γ R + R² = ρ_γ + ρ_m

Effective Friedmann equations (L)

H²(t) + 16πGα

�

−

� a(t)¨ a²(t)�

a(t) + 2 H²(t) a(t)�

a(t)

� �

dt^� a³² (t^�)R(t^�) L₋(t − t^�)

+ 2 H(t) a(t)�

a(t)

�

dt^� L₋(t − t^�) d dt^�

�a³² (t^�)R(t^�)��

= 8πG

3 ρ(t) How to interpret non-local terms?

PRELIMINARY RESULTS

Effective Friedmann equations (L)

H²(t) + 16πGα

�

−

� a(t)¨ a²(t)�

a(t) + 2 H²(t) a(t)�

a(t)

� �

dt^� a³² (t^�)R(t^�) L₋(t − t^�)

+ 2 H(t) a(t)�

a(t)

�

dt^� L₋(t − t^�) d dt^�

�a³² (t^�)R(t^�)��

= 8πG

3 ρ(t)

L(t − t^�) =

� _∞

0

ds

� 1

µ² + s − G(t − t^�; √ s)

�

G₋(t − t^�; m) = θ(t − t^�)

�a³(t)a³(t^�)

sin m(t − t^�)

m + O( ˙a) L ≡ log −�

µ²

L₋(t − t^�) = −2 lim

�→0

�θ(t − t^� − �)

t − t^� + δ(t − t^�) log µ �

�

How to interpret non-local terms?

PRELIMINARY RESULTS

Effective Friedmann equations (L)

a(t) = a₀e√_Λ

3 t

a(t) = (t/t₀)^1/2

Try an iterative solution...

a(t) = (t/t₀)^2/3 H²(t) + 16πGα

�

−

� a(t)¨ a²(t)�

a(t) + 2 H²(t) a(t)�

a(t)

� �

dt^� a³² (t^�)R(t^�) L₋(t − t^�)

+ 2 H(t) a(t)�

a(t)

�

dt^� L₋(t − t^�) d dt^�

�a³² (t^�)R(t^�)��

= 8πG

3 ρ(t)

PRELIMINARY RESULTS

Effective Friedmann equations (L)

a(t) = a₀e√_Λ

3 t

a(t) = (t/t₀)^1/2

Try an iterative solution...

a(t) = (t/t₀)^2/3

no effect H²(t) + 16πGα

�

−

� a(t)¨ a²(t)�

a(t) + 2 H²(t) a(t)�

a(t)

� �

dt^� a³² (t^�)R(t^�) L₋(t − t^�)

+ 2 H(t) a(t)�

a(t)

�

dt^� L₋(t − t^�) d dt^�

�a³² (t^�)R(t^�)��

= 8πG

3 ρ(t)

PRELIMINARY RESULTS

Effective Friedmann equations (L)

a(t) = a₀e√_Λ

3 t

a(t) = (t/t₀)^1/2

Try an iterative solution...

a(t) = (t/t₀)^2/3

no effect no effect H²(t) + 16πGα

�

−

� a(t)¨ a²(t)�

a(t) + 2 H²(t) a(t)�

a(t)

� �

dt^� a³² (t^�)R(t^�) L₋(t − t^�)

+ 2 H(t) a(t)�

a(t)

�

dt^� L₋(t − t^�) d dt^�

�a³² (t^�)R(t^�)��

= 8πG

3 ρ(t)

PRELIMINARY RESULTS

Effective Friedmann equations (L)

H²(t) + 256πGα 3t⁴

�

log µt + log

� t

t₀ − 1

�

+ 2 3

� t

t₀ − 1

��

= 8πG

3 ρ_m(t₀)

�t₀ t

�2

a(t) = a₀e√_Λ

3 t

a(t) = (t/t₀)^1/2

Try an iterative solution...

a(t) = (t/t₀)^2/3

... log R dominant a early times while log box at late times some effect

no effect no effect H²(t) + 16πGα

�

−

� a(t)¨ a²(t)�

a(t) + 2 H²(t) a(t)�

a(t)

� �

dt^� a³² (t^�)R(t^�) L₋(t − t^�)

+ 2 H(t) a(t)�

a(t)

�

dt^� L₋(t − t^�) d dt^�

�a³² (t^�)R(t^�)��

= 8πG

3 ρ(t)

PRELIMINARY RESULTS

Conclusions and Outlook

•

Compute all LO terms

•

Renormalization of NLO

•

Conformal anomaly contribution

•

Apply to cosmology

•

Apply to stars/black holes

•

Add the SM and constrain BSM

•

Connection with high energy quantum gravity

•

^Falsify!

Thank you