Post on 24-Jun-2020
Modified Gravity 3
Radouane Gannouji
PUCV, Chile
UFES, December 18, 2015
3 ways to hide
S =
∫ √−gd4x
[
R − Zµν(φ, ∂φ, · · · )∂µφ∂νφ− V(φ)]
+ Sm[e2Qφgµν ,Ψm]
• Q ≪ 1
Unnatural like Brans-Dicke theories
• We use the potential V(φ) (Chameleon mechanism)
Like f (R)-gravity, chamelon gravity, Brans-Dicke with a potential ...
Nice properties, Simple predictions but the simplest models seem to be
ruled out
• We use the non-linearities Zµν (Vainshtein mechanism)
Brane models (DGP), massive gravity ...
• Examples
• K-inflation (C. Armendariz-Picon et al. 1999)
S =
∫
d4x[
R + K(φ, (∂φ)2)]
• K-essence (T. Chiba et al. 2000)
• Dirac-Born-Infeld (DBI) (D-branes in string theory) (T. Padmanabhan 2002)
S =
∫
d4x[
R − V(φ)√
1 − (∂φ)2
]
• Ghost condensate (N. Arkani-Hamed et al. 2004)
S =
∫
d4x[
R +1
2(∂φ)2 + a(∂φ)4
]
• Properties
• Non-gaussianity (D. Seery et al. 2005)
• Bouncing cosmology (because they violate the null energy condition) (P.
Creminelli et al. 2006)
• Supersymmetric version (J. Khoury et al. 2010)
• Study of large-scale structure (C. Armendariz-Picon et al. 2005)
• Study of topological defects, solitons ... (E. Babichev 2008)
• The simplest model is defined with a shift symmetry (φ(x) → φ(x) + a)
S =
∫ √−gd4x
[
R − 1
2(∂φ)2 +
α
4Λ4(∂φ)4 +
Q
MPl
φT]
φ− α
Λ4∂µ((∂φ)
2∂νφ) +Q
MPl
T = 0
• The simplest model is defined with a shift symmetry (φ(x) → φ(x) + a)
S =
∫ √−gd4x
[
R − 1
2(∂φ)2 +
α
4Λ4(∂φ)4 +
Q
MPl
φT]
φ− α
Λ4∂µ((∂φ)
2∂νφ) +Q
MPl
T = 0
We consider a point source T = −Mδ(3)(x)
div(
∇φ− α
Λ4(∂φ)2∇φ
)
=QM
MPl
δ(3)(x)
φ′ − α
Λ4φ′3 =
1
4πr2
gM
MPl
• The simplest model is defined with a shift symmetry (φ(x) → φ(x) + a)
S =
∫ √−gd4x
[
R − 1
2(∂φ)2 +
α
4Λ4(∂φ)4 +
Q
MPl
φT]
φ− α
Λ4∂µ((∂φ)
2∂νφ) +Q
MPl
T = 0
We consider a point source T = −Mδ(3)(x)
div(
∇φ− α
Λ4(∂φ)2∇φ
)
=QM
MPl
δ(3)(x)
φ′ − α
Λ4φ′3 =
1
4πr2
gM
MPl
φ′(r)
Λ2=
(
8π3
)1/3(r
rV
)2/3
(
−9 +
√
81 + 192π2(
rrV
)4)1/3− (72π)−1/3
(
−9 +
√
81 + 192π2
( r
rV
)4)1/3( rV
r
)2/3
rV =1
Λ
(QM
MPl
)1/2
(Gabadadze et al. 2012)
• In the asymptotic regime
• For r ≫ rV , φ′(r) ≃
Λ2
4π
(
rV
r
)2
• For r ≪ rV , φ′(r) ≃ (−α)1/3Λ2
(
rV
r
)2/3
• In the asymptotic regime
• For r ≫ rV , φ′(r) ≃
Λ2
4π
(
rV
r
)2
• For r ≪ rV , φ′(r) ≃ (−α)1/3Λ2
(
rV
r
)2/3
Fφ =Q
MPl
∇φ =Q
MPl
φ′(r)
FN =M
8πM2Pl
1
r2=
Λ2
8πMPl
(rV
r
)2
• Generalizations (Babichev et al. 2009)
S =
∫ √−gd4x
[
R − 1
2(∂φ)2 +
n∑
i=2
ci
Λ4i−4(∂φ)2i +
Q
MPl
φT]
• Generalizations (Babichev et al. 2009)
S =
∫ √−gd4x
[
R − 1
2(∂φ)2 +
n∑
i=2
ci
Λ4i−4(∂φ)2i +
Q
MPl
φT]
• For r ≫ rV , φ′(r) ≃
Λ2
4π
(
rV
r
)2
• For r ≪ rV , φ′(r) ≃ Λ2
(
rV
r
)2/(2n−1)
Fφ
FN
≃( r
rV
)
4n−42n−1
• Generalizations (Babichev et al. 2009)
S =
∫ √−gd4x
[
R − 1
2(∂φ)2 +
n∑
i=2
ci
Λ4i−4(∂φ)2i +
Q
MPl
φT]
• For r ≫ rV , φ′(r) ≃
Λ2
4π
(
rV
r
)2
• For r ≪ rV , φ′(r) ≃ Λ2
(
rV
r
)2/(2n−1)
Fφ
FN
≃( r
rV
)
4n−42n−1
If n → ∞, we have DBI Fφ/FN ≃ r2(de Rham et al. 2014)
• (D)BI (Burrage et al. 2014)
Negative tension brane embedded in a 5-dimensional space (with 2
time-like directions), at the lowest order, we have
S =
∫ √−gd4x
[
R + Λ4
√
1 − (∂φ)2
Λ4+
Q
MPl
φT]
φ′(r) =Λ2
√
1 + 16π2(r/rV)4
• Fφ ≃ 1/r2 (for r ≫ rV )
• Fφ/FN ≃ (r/rV)2 (for r ≪ rV )
General picture
S =
∫ √−gd4x
[
R − 1
2Zµν(φ)∂µφ∂νφ+
Q
Mpl
φT]
Zµν ≈ gµν +1
Λ3∂µ∂νφ+
1
Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energy Zµν ≈ gµν
• At high energy Zµν ≫ 1
General picture
S =
∫ √−gd4x
[
R − 1
2Zµν(φ)∂µφ∂νφ+
Q
Mpl
φT]
Zµν ≈ gµν +1
Λ3∂µ∂νφ+
1
Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energy Zµν ≈ gµν
• At high energy Zµν ≫ 1
General picture
S =
∫ √−gd4x
[
R − 1
2Zµν(φ)∂µφ∂νφ+
Q
Mpl
φT]
Zµν ≈ gµν +1
Λ3∂µ∂νφ+
1
Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energy Zµν ≈ gµν
• At high energy Zµν ≫ 1
General picture
S =
∫ √−gd4x
[
R − 1
2Zµν(φ)∂µφ∂νφ+
Q
Mpl
φT]
Zµν ≈ gµν +1
Λ3∂µ∂νφ+
1
Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energy Zµν ≈ gµν
• At high energy Zµν ≫ 1
−1
2Zµν(φ)∂µφ∂νφ = −1
2
(
∂φ)2
, φ =φ√Z
General picture
S =
∫ √−gd4x
[
R − 1
2Zµν(φ)∂µφ∂νφ+
Q
Mpl
φT]
Zµν ≈ gµν +1
Λ3∂µ∂νφ+
1
Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energy Zµν ≈ gµν
• At high energy Zµν ≫ 1
−1
2Zµν(φ)∂µφ∂νφ = −1
2
(
∂φ)2
, φ =φ√Z
Q
Mpl
φT =Q√ZMpl
φT ≪ 1
General picture
S =
∫ √−gd4x
[
R − 1
2Zµν(φ)∂µφ∂νφ+
Q
Mpl
φT]
Zµν ≈ gµν +1
Λ3∂µ∂νφ+
1
Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energy Zµν ≈ gµν
• At high energy Zµν ≫ 1
−1
2Zµν(φ)∂µφ∂νφ = −1
2
(
∂φ)2
, φ =φ√Z
Q
Mpl
φT =Q√ZMpl
φT ≪ 1
The field is strongly coupled to itself and becomes weakly coupled to
external sources
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
ds2 = gµνdxµdxν + eφ(xµ)dw2
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
ds2 = gµνdxµdxν + eφ(xµ)dw2
S =1
2
∫
d4x√−g
[
R − 1
2(∂φ)2 + αeφ/
√3(
RGB +4
3Gµνφ
;µφ;ν
− 1√3(∂φ)2
φ)]
+ Sm[e−φ/
√3gµν ;ψm]
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
ds2 = gµνdxµdxν + eφ(xµ)dw2
S =1
2
∫
d4x√−g
[
R − 1
2(∂φ)2 + αeφ/
√3(
RGB +4
3Gµνφ
;µφ;ν
− 1√3(∂φ)2
φ)]
+ Sm[e−φ/
√3gµν ;ψm]
10-6 1 106 1012
1
0.1
10-4
10-7
10-10
r HpcL
ÈFΠF
gÈ
Α
0
Α
10-20
Α
1
Α
1020
RV ≃ (αrS)1/3
α = H20α
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
ds2 = gµνdxµdxν + eφ(xµ)dw2
S =1
2
∫
d4x√−g
[
R − 1
2(∂φ)2 + αeφ/
√3(
RGB +4
3Gµνφ
;µφ;ν
− 1√3(∂φ)2
φ)]
+ Sm[e−φ/
√3gµν ;ψm]
10-6 1 106 10120.2
0.4
0.6
0.8
1.0
r HpcL
c W2
Α
10-20
Α
1
Α
1020
Considering perturbations around this solution
φ = φ+ δφ
S =
∫
d4xQ[
(∂tδφ)2 − c2
r (∂rδφ)2 − c2
Ω(∂Ωδφ)2]
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
ds2 = gµνdxµdxν + eφ(xµ)dw2
S =1
2
∫
d4x√−g
[
R − 1
2(∂φ)2 + αeφ/
√3(
RGB +4
3Gµνφ
;µφ;ν
− 1√3(∂φ)2
φ)]
+ Sm[e−φ/
√3gµν ;ψm]
10-6 1 106 10121.0
1.1
1.2
1.3
1.4
1.5
r HpcL
c r2
Α
10-20
Α
1
Α
1020
Considering perturbations around this solution
φ = φ+ δφ
S =
∫
d4xQ[
(∂tδφ)2 − c2
r (∂rδφ)2 − c2
Ω(∂Ωδφ)2]
• Gauss-Bonnet gravity (R.G., M. Sami 2011)
S =
∫
d5x√−g (R + αRGB)
ds2 = gµνdxµdxν + eφ(xµ)dw2
S =1
2
∫
d4x√−g
[
R − 1
2(∂φ)2 + αeφ/
√3(
RGB +4
3Gµνφ
;µφ;ν
− 1√3(∂φ)2
φ)]
+ Sm[e−φ/
√3gµν ;ψm]
There is always a direction in which the speed of propagation is superluminal
• Galileons
• Boundary effective theory on the DGP brane comes as (A. Nicolis, R. Rattazzi
2004)
S =
∫
d4x√−g
[
R −1
2(∂φ)2 + α(∂φ)2
φ+ φT]
• Generalized galileons (Horndeski) (G. W. Horndeski 1974)
• Most general action with a scalar field and second order differential
equation
• Galileons
• Boundary effective theory on the DGP brane comes as (A. Nicolis, R. Rattazzi
2004)
S =
∫
d4x√−g
[
R −1
2(∂φ)2 + α(∂φ)2
φ+ φT]
• Vainshtein mechanism• Self-accelerating solution• Superluminal behavior but causal• Self-accelerating solution is unstable (R.G., M. Sami 2010), we need
generalization
• Generalized galileons (Horndeski) (G. W. Horndeski 1974)
• Most general action with a scalar field and second order differential
equation
• Galileons
• Boundary effective theory on the DGP brane comes as (A. Nicolis, R. Rattazzi
2004)
S =
∫
d4x√−g
[
R −1
2(∂φ)2 + α(∂φ)2
φ+ φT]
• Vainshtein mechanism• Self-accelerating solution• Superluminal behavior but causal• Self-accelerating solution is unstable (R.G., M. Sami 2010), we need
generalization
• Generalized galileons (Horndeski) (G. W. Horndeski 1974)
• Most general action with a scalar field and second order differential
equation
• Galileons
• Boundary effective theory on the DGP brane comes as (A. Nicolis, R. Rattazzi
2004)
S =
∫
d4x√−g
[
R −1
2(∂φ)2 + α(∂φ)2
φ+ φT]
• Vainshtein mechanism• Self-accelerating solution• Superluminal behavior but causal• Self-accelerating solution is unstable (R.G., M. Sami 2010), we need
generalization
• Generalized galileons (Horndeski) (G. W. Horndeski 1974)
• Most general action with a scalar field and second order differential
equation
S =
∫
d4x√
−g[
K(φ, X)− G3(φ, X)φ+ G4(φ, X)R + G4,X(φ, X)[
(φ)2 − (∇µ∇νφ)
2]
+ G5(φ, X)Gµν∇µ∇νφ −1
6G5,X(φ, X)
[
(φ)3 − 3(φ)(∇µ∇ν)2 + 2∇µ∇αφ∇α∇βφ∇β∇µφ
]]
Origin of the Vainshtein mechanism
• Massive gravity
• Quadratic action for massive spin-2 particle (M. Fierz, W. Pauli 1939)
S =
∫
d4x[
−1
2∂αhµν∂
αhµν + ∂µhνα∂
νhµα − ∂µh
µν∂νh +
1
2∂αh∂
αh
−m2
2
(
hµνhµν − h
2)]
• After some algebra, it gives
Origin of the Vainshtein mechanism
• Massive gravity
• Quadratic action for massive spin-2 particle (M. Fierz, W. Pauli 1939)
S =
∫
d4x[
−1
2∂αhµν∂
αhµν + ∂µhνα∂
νhµα − ∂µh
µν∂νh +
1
2∂αh∂
αh
−m2
2
(
hµνhµν − h
2)]
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h
2)]
• After some algebra, it gives
Origin of the Vainshtein mechanism
• Massive gravity
• Quadratic action for massive spin-2 particle (M. Fierz, W. Pauli 1939)
S =
∫
d4x[
−1
2∂αhµν∂
αhµν + ∂µhνα∂
νhµα − ∂µh
µν∂νh +
1
2∂αh∂
αh
−m2
2
(
hµνhµν − h
2)]
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h
2)]
hµν + ∂µ∂νh − ∂µ∂αhαν − ∂ν∂αh
αµ + ηµν
(
∂α∂βhαβ − h
)
− m2 (hµν − ηµνh) = 0
• After some algebra, it gives
Origin of the Vainshtein mechanism
• Massive gravity
• Quadratic action for massive spin-2 particle (M. Fierz, W. Pauli 1939)
S =
∫
d4x[
−1
2∂αhµν∂
αhµν + ∂µhνα∂
νhµα − ∂µh
µν∂νh +
1
2∂αh∂
αh
−m2
2
(
hµνhµν − h
2)]
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h
2)]
hµν + ∂µ∂νh − ∂µ∂αhαν − ∂ν∂αh
αµ + ηµν
(
∂α∂βhαβ − h
)
− m2 (hµν − ηµνh) = 0
• After some algebra, it gives
(− m2)hµν = 0, ∂
µhµν = 0, h = 0
Origin of the Vainshtein mechanism
• Massive gravity
• Quadratic action for massive spin-2 particle (M. Fierz, W. Pauli 1939)
S =
∫
d4x[
−1
2∂αhµν∂
αhµν + ∂µhνα∂
νhµα − ∂µh
µν∂νh +
1
2∂αh∂
αh
−m2
2
(
hµνhµν − h
2)]
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h
2)]
hµν + ∂µ∂νh − ∂µ∂αhαν − ∂ν∂αh
αµ + ηµν
(
∂α∂βhαβ − h
)
− m2 (hµν − ηµνh) = 0
• After some algebra, it gives
(− m2)hµν = 0, ∂
µhµν = 0, h = 0
Wave equation for 5 propagating polarizations
• Solution for a point source
h00 =2M
3
1
4π
e−mr
r
hij =M
3
1
4π
e−mr
rδij
⇒ The PPN parameter is γ =h11
h00
=1
2like Brans-Dicke
• Even in the limit of zero mass, gives predictions which are order one
different from linearized GR
• This is the vDVZ (van Dam, Veltman, Zakharov) discontinuity (H. van Dam
and M. J. G. Veltman; V. I. Zakharov 1970)
• Solution for a point source
h00 =2M
3
1
4π
e−mr
r
hij =M
3
1
4π
e−mr
rδij
⇒ The PPN parameter is γ =h11
h00
=1
2like Brans-Dicke
• Even in the limit of zero mass, gives predictions which are order one
different from linearized GR
• This is the vDVZ (van Dam, Veltman, Zakharov) discontinuity (H. van Dam
and M. J. G. Veltman; V. I. Zakharov 1970)
• Solution for a point source
h00 =2M
3
1
4π
e−mr
r
hij =M
3
1
4π
e−mr
rδij
⇒ The PPN parameter is γ =h11
h00
=1
2like Brans-Dicke
• Even in the limit of zero mass, gives predictions which are order one
different from linearized GR
• This is the vDVZ (van Dam, Veltman, Zakharov) discontinuity (H. van Dam
and M. J. G. Veltman; V. I. Zakharov 1970)
• The Stückelberg trick
• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
• The Stückelberg trick
• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h2)
+ κhµνTµν]
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
• The Stückelberg trick
• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h2)
+ κhµνTµν]
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
• The Stückelberg trick
• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h2)
+ κhµνTµν]
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
• The Stückelberg trick
• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h2)
+ κhµνTµν]
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
• The Stückelberg trick
• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h2)
+ κhµνTµν]
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h
2)
−1
2FµνF
µν −1
2(∂φ)2 −
1
2m
2φ2 + κhµνTµν + κφT
]
• The Stückelberg trick• Formalism to expose the origin of this discontinuity
• Taking m → 0 in the equations does not yield to a smooth limit, because
degrees of freedom are lost
• To find the correct limit, the trick is to introduce new fields into the theory
in a way that does not alter the theory and then the limit can be found in
which no degrees of freedom are gained or lost
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h2)
+ κhµνTµν]
• The first part of the action is invariant under linear diffeomorphismhµν → hµν + ∂µξν + ∂νξµ
• So let us do the following transformation hµν → hµν + ∂µAν + ∂νAµ
• It is not sufficient to decouple all degrees of freedom, so we do
Aµ → Aµ + ∂µφ
S =
∫
d4x[
Lm=0 −m2
2
(
hµνhµν − h
2)
−1
2FµνF
µν −1
2(∂φ)2 −
1
2m
2φ2 + κhµνTµν + κφT
]
S =
∫
d4x
[
Lm=0 −1
2FµνFµν −
1
2(∂φ)2 + κhµνTµν + κφT
]
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
R = 0
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
R = 0
Mass terms as filters and degravitation
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
R = 0
Mass terms as filters and degravitation
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
R = 0
Mass terms as filters and degravitation
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
R = 0
Mass terms as filters and degravitation
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
hµν − ∂λ∂µhλν − ∂λ∂νh
λµ + ηµν∂λ∂σh
λσ + ∂µ∂νh − ηµνh − m2(hµν − ηµνh) = −κTµν
• Let us consider a constant source Tµν = Ληµν
• Taking double divergence of the equations gives
∂ν∂µhµν −h = 0
R = 0
Mass terms as filters and degravitation
• Maybe massive gravity is a path to a solution to the CCP
• But it has this discontinuity which makes it not viable theory
• Vainshtein (1972) proposed that non-linear terms can solve the
discontinuity
• Which means a non-linear theory of massive gravity
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Massive general relativity hµν = gµν − g(0)µν
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2g(0)µαg(0)νβ (hµνhαβ − hµαhνβ)
]
S =
∫
d4x
[
(√−gR)−
√
−g0m2
2U(g(0), h)
]
But the theory has now 6 degrees of freedom and the additional one is a
ghost (Boulware-Deser ghost)
• Fortunately, there exist one potential U without a ghost, dRGT model(de Rham, Gabadadze, Tolley 2011)
• It has Vainshtein screening
• Doesn’t admit flat FLRW solution (D’Amico et al. 2011)
• Exist open FLRW solution (Gumrukcuoglu et al. 2011)
• Exist anisotropic solutions (Gratia et al. 2012)
• But all these solutions contain a ghost (Kuhnel et al. 2013)
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
To date, there is no (extended) massive gravity model which possesses a
completely satisfactory self-acceleration solution
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
To date, there is no (extended) massive gravity model which possesses a
completely satisfactory self-acceleration solution
• But it opens the door to degravitation (N. Arkani-Hamed et al. 2002): a large
cosmological constant does not strongly curve space-time
• Because of these problems, massive gravity has been extended
• Background metric g(0)µν is dynamical, bi-metric theory (Hassan et al. 2012)
• Multi-metric theory (Hinterbichler et al. 2012)
• The mass is not constant but it’s a scalar field, quasi-dilaton massivegravity (D’Amico et al. 2013)
• Cosmology studied in (R.G. et al. 2013). The model has bouncing, turnaround
or cyclic solutions
To date, there is no (extended) massive gravity model which possesses a
completely satisfactory self-acceleration solution
• But it opens the door to degravitation (N. Arkani-Hamed et al. 2002): a large
cosmological constant does not strongly curve space-time
Any theory that exhibits degravitation must reduce to a theory of
massive/resonance gravity (continuum of massive gravitons) at linearized
level (G. Dvali et al. 2007)
• Vainshtein mechanism seems stronger than chameleon
• In time-dependent situations, Vainshtein mechanism has been found to
be less efficient than around static sources (Y. Chu, M. Trodden 2012)
• Possible way to find a degravitation model
• No satisfactory model for the moment
• Vainshtein mechanism seems stronger than chameleon
• In time-dependent situations, Vainshtein mechanism has been found to
be less efficient than around static sources (Y. Chu, M. Trodden 2012)
• Possible way to find a degravitation model
• No satisfactory model for the moment
• Vainshtein mechanism seems stronger than chameleon
• In time-dependent situations, Vainshtein mechanism has been found to
be less efficient than around static sources (Y. Chu, M. Trodden 2012)
• Possible way to find a degravitation model
• No satisfactory model for the moment
• Vainshtein mechanism seems stronger than chameleon
• In time-dependent situations, Vainshtein mechanism has been found to
be less efficient than around static sources (Y. Chu, M. Trodden 2012)
• Possible way to find a degravitation model
• No satisfactory model for the moment
• Vainshtein mechanism seems stronger than chameleon
• In time-dependent situations, Vainshtein mechanism has been found to
be less efficient than around static sources (Y. Chu, M. Trodden 2012)
• Possible way to find a degravitation model
• No satisfactory model for the moment