Modified gravity as an alternative to dark

energy

Lecture 2. Theory of modified gravity models

Theory of modified gravity

Requirements Must explain the late time accelerationMust recover GR on small scalesMust be free from pathologies

Two examples to see how difficult it is to satisfy these conditions!

f(R) gravity, DGP braneworld model

Example: f(R) gravityThe modification should act at low energies

Ricci curvature is smaller at low energies example

must be fine-tunedcf. high energy corrections

4 ( )S d x gF R= −∫4S d x gR= −∫

4

( )F R RRμ

= −

μ0Hμ

2( )F R R Rα= +

Late time accelerationFriedman equation now becomes 4th order differential equation

this analysis does not include matter/radiation

acceleration

H

H&

2( )a t t∝

Problemf(R) theory is equivalent to BD theoryLegendre transformation

This is BD theory withThe potential is of order for

( )4 ( )S d x R Vψ ψ= −∫

4 ( )S d x gF R= −∫( )4 ( ) ( ) '( )S d x g F R Fφ φ φ= − + −∫ R φ=

''( ) 0F R ≠

4( )V μΨ

0BDω =4

( )F R RRμ

= −

(Wands, Chiba)

Contradicts to solar system constraintsthe potential is of order and can be neglected

Cosmologyinclusion of radiation/ matterNariai 1969

BD theory yields for any kind of matter

MD era does not exist !

We expected to recover GR at early times where the correction is tiny...

40H

0BDω =1/ 2( )a t t∝ 1/ 3effw =

(Amendola et.al astro-ph/0603703)

Instability with matter

(mCDTT model has a singularity inthe late accelerating phase)

This instability was known by Dolgov & Kawasaki for a staticsource

The condition for the instability

4

( )F R RRμ

= mCDTT

mCDTT

GR solution

BD solution

''

RR

R

F HB RF H

= 0B < unstable

(Hu and Sawicki astro-ph/0702278)

2

Rμ

ln a

Engineering f(R) models

Can we avoid solar system constraints?Hu and Sawicki

Starobinsky

22 1

22

( / )( )( / ) 1

n

n

c R mF R R mc R m

= −+

22

/ 0lim ( ) n

m Rf R R C DR−

→= − +

2( ) (1 ( / ) ) 1nF R R R mλ −⎡ ⎤= + + −⎣ ⎦

0lim ( ) 0R f R→ =no ‘cosmological constant’

Quasi-static perturbationsLinear perturbations

Equations of motion in BD theory

2 2 2

2 2 2

(3 2 ) 8142

BD m

m

a R Ga

Ga

ω ϕ δ π δρ

π δρ ϕ

ϕ

+ ∇ = −

∇ Φ = − − ∇

Φ +Ψ = −

2 2 2 2(1 2 ) ( ) (1 2 )ds dt a t dx= − + Ψ + + Φ v

0ψ ψ ϕ= +

mδρ

ϕΦ

2 2 2

2 2 2

(3 2 ) 8142

BD m

m

a R Ga

Ga

ω ϕ δ π δρ

π δρ ϕ

ϕ

+ ∇ = −

∇ Φ = − − ∇

Φ +Ψ = −

2 2

8

40

m

m

R G

Ga

δ π δρ

π δρ

=

∇ Φ = −

Φ +Ψ =

2 2

2 2

1

(3 2 ) 8

2(1 )43 2

21

BD m

BDm

BD

BD

BD

Ga

Ga

ω ϕ π δρ

ωπ δρω

ω γω

−

+ ∇ = −

⎛ ⎞+∇ Φ = − ⎜ ⎟+⎝ ⎠

+Ψ = Φ ≡ Φ

+

GR and BD limit

0ϕ →0Rδ →

f(R) gravity

linearisation

The inverse of mass determines the length scales where the scalar propagateson large scales we recover GR on small scales we recover 0BDω =

23RR

dRR F mdF

δ δ ϕ⎛ ⎞

= ≡⎜ ⎟⎝ ⎠

0, ( )BD RF Rω ψ= =

2 2 2 23 3 8 mm a Gaϕ ϕ π δρ∇ = − −

1L m−>1L m−<

12

γ Φ≡ − =

Ψenhances Newtonian potential and growth rate of structure formation

1m−

BD

GR

Example

In the cosmological background

20

0( ) ( )1 R

RRf R F R R C fAR R

= − ∝ → −+

21 0

0 36 RRm fR

− =

1 00 63.2 Mpc

10Rfm−−

0Rf controls the deviation from LCDM

Fractional difference of linear growth rate compared to LCDM

kz

(Hu and Sawicki 0705.1158)

On small scales, we getthis contradicts with solar system constraints?

depends on curvature and becomes smaller for dense region

Chameleon mechanism mass of the filed becomes large for dense region and hides the scalar degree of freedom

In general linearization breaks down and need to solve the non-linear equation for

0BDω =

1m−

21 0

0 36 RRm fR

− =

0R R Rδ = −

RFψ =2 3

2

8 23 3mG dV

a dπ ψψ δρ

ψ∇

− = − +

30 3 24 30 10 / , 10 /galaxyg cm g cmρ ρ− −≈ ≈

Example of solutions

1ψ −

51 10γ −− <

(Hu and Sawicki 0705.1158)

Singularity problemBD scalar

potential

this can be reached in strong gravity

20

01 RRfR

ψ ⎛ ⎞= + ⎜ ⎟⎝ ⎠

1ψ = corresponds to curvature singularity

(Frolov, Kobayashi and Maeda)

f(R) gravity summary

Naïve models do not worklow curvature modification in action changes GR even in high curvature regime

Contrive models using Chameleon mechanism can give acceptable cosmology & weak gravity

O(1) modification of GR on cosmological scalesbut additional mechanisms would be needed for strong gravity

Complicated version of quintessence?

Crossover scale

4D Newtonian gravity5D Newtonian gravity

cr r<cr

Infinite extra-dimension

gravity leakage

5 4 4(5) (5)

1 132 16 m

c

S d x g R d x gR d x gLGr Gπ π

= − + − + −∫ ∫ ∫

cr r>

cr

(Dvali, Gabadadze,Porrati)

Example 2. braneworld model

Friedmann equation

early times 4D Friedmann

late times

As simple as LCDM model(and as fine-tuned as LCDM )

Cosmology in DGP model

2 83c

H GHr

π ρ= −

1cHr >>

1

c

Hr

→0ρ →

0cr H≈

(Deffayet)

Quasi-static perturbations

Non-linearity of brane bending mode

Solving bulk perturbations imposing regularity condition in the bulk

junction conditions on a brane

( ) ( ){ }

2 2 2

2 2 2 2

14 ,2

3 ( ) 8j i jc j j i

Ga

t r Ga

π ρδ ϕ ϕ

β ϕ ϕ ϕ ϕ ϕ π ρδ

−∇ Φ = + ∇ Φ +Ψ = −

∇ + ∂ ∂ ∇ −∂ ∂ ∂ ∂ =

ϕ( ) ( )2 2 2 2 2 2

,

1 2 1 2 (1 2 )

2 ic i

ds N dt A dx G dy

r dydxϕ

= − + Ψ + + Φ + +

+

r

21 2 13cHHrH

β⎛ ⎞

= − +⎜ ⎟⎝ ⎠

&

Silva and KK hep-th/0702169

Solutions for metric perturbations

Linear theory

2

2

2

2

14 1 ,3

14 1 ,3

k Ga

k Ga

π ρδβ

π ρδβ

⎛ ⎞Φ = −⎜ ⎟

⎝ ⎠⎛ ⎞

Ψ = − +⎜ ⎟⎝ ⎠

( ) ( )2 2 2 21 2 ( ) 1 2ds dt a t dx= − + Ψ + + Φr

Φ

−Ψ

a

21 2 13cHHrH

β⎛ ⎞

= − +⎜ ⎟⎝ ⎠

&

Lue et.al, KK and Maartens

( )3 1 (1)2BD Oω β= −

Non-linear evolution

Non-linearity of brane bending becomes important when

then

GR is recovered on non-linear scales

( ) ( ){ }

2 2 2

2 2 2 2

14 ,2

3 ( ) 8j i jc j j i

Ga

t r Ga

π ρδ ϕ ϕ

β ϕ ϕ ϕ ϕ ϕ π ρδ

−∇ Φ = + ∇ Φ + Ψ = −

∇ + ∂ ∂ ∇ − ∂ ∂ ∂ ∂ =

2 2( ) (1)cHr Oβ δ− <

2 2 2

c

H Hr

ϕ δ δ∇ ∇ Φ21 2 1

3cHHrH

β⎛ ⎞

= − +⎜ ⎟⎝ ⎠

&

Spherically symmetric solution

Vainstein radius

2

2

2

2

1 ,2 2

12 2

g g

c

g g

c

r r rr r

r r rr r

ββ

ββ

Φ = +

Ψ = − +

3 3*

2*

2( ), ( ) 1 13

gr rd rr rdr r r rϕ

β

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= Δ Δ = + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

12 3

* 42

8, 2

9c g

g

r rr r G M

β⎛ ⎞

= =⎜ ⎟⎜ ⎟⎝ ⎠

4D BD 5D

11 ,2 3

112 3

g

g

rrrr

β

β

⎛ ⎞Φ = −⎜ ⎟

⎝ ⎠⎛ ⎞

Ψ = − +⎜ ⎟⎝ ⎠

*r cr

4D Einstein

(Gruzinov; Middleton, Siopsis; Tanaka; Lue, Sccoccimarro, Starkman)

Solar system constraints are avoided if 1

0cr H −>

Problem

Ghost

32β−

Negative BD parameter

In Einstein frame, kinetic term for the scalarif the scalar becomes a ghost

ghost mediates anti-gravity and suppressed the growth of structure

( )3 12BDω β= − 21 2 1

3cHHrH

β⎛ ⎞

= − +⎜ ⎟⎝ ⎠

&

0β <

Strong coupling problemCovariant effective theory Minkowski background

We need quantum gravity below ! This is due to the fact disappears as

Strong gravityno known BH solution

( ) ( ) ( )132 2 4

2

1 1 ,2 c

MSrμ μϕ ϕ ϕ

⎛ ⎞= ∂ + ∂ Λ=⎜ ⎟Λ ⎝ ⎠

1 1000km−Λ =ϕ

cr →∞

DGP gravity summary

Classically, the model works very wellthere is only one parameter and we do not need to introduce additional mechanism to recover solar system constraints

Theory shows pathologies at quantum level there are debates on whether they are fatal or not

KK, Class.Quant.Grav.24:R231-R253,2007.arXiv:0709.2399 [hep-th]

Common features in f(R) and DGP

3 regime of gravity

Large scale structure

Scalar tensorGR

1*m −

Solar system

10m −

0cr H≈( )1

2 3* g cr r r≈

GR

5D

f(R)

DGP

Scalar-tensor

GRGR

GR