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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
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