May the Force NOT be with you

Introduction
Results
May the force not be with you
Screening Modifications of Gravity and Disformal Couplings
Miguel Zumalacárregui
Instituto de F´ısica Teórica (IFT-UAM-CSIC) → ITP - Uni. Heidelberg
Refs: PRL 109 241102 (1205.3167) and PRD 87 083010 (1210.8016)
with Tomi S. Koivisto and David F. Mota
University of Geneva (May 2013)
Miguel Zumalacárregui May the force not be with you

Introduction
Results
Theories of Gravity
Screening Mechanisms
The Frontiers of Gravity
Ostrogradski’s Theorem (1850)
Theories with L ⊃
∂nq
∂tn
, n ≥ 2 are unstable∗
L(q(t), ˙q, ¨q) →
∂L
∂q
−
d
dt
∂L
∂ ˙q
+
d2
dt2
∂L
∂¨q
= 0
q, ˙q, ¨q,
...
q → Q1, Q2, P1,P2
H = P1Q2 + terms independent of P1

Introduction
Results
Theories of Gravity
The Frontiers of Gravity
Ostrogradski’s Theorem (1850)
Theories with L ⊃
∂nq
∂tn
, n ≥ 2 are unstable∗
L(q(t), ˙q, ¨q) →
∂L
∂q
−
d
dt
∂L
∂ ˙q
+
d2
dt2
∂L
∂¨q
= 0
q, ˙q, ¨q,
...
q → Q1, Q2, P1,P2
H = P1Q2 + terms independent of P1
∗ Loophole: Th’s with second order equations of motion

Introduction
Results
Theories of Gravity
Einstein’s Theory
Lovelock’s Theorem (1971)
gµν + Local + 4-D + Lorentz Theory with 2nd order Eqs.
√
−g
1
16πG
(R − 2Λ)
Ways out - Clifton et al. (Phys.Rept. 2012)
Additional fields −→
£
¢

¡
φ , Aµ, hµν...
“Higher derivatives” −→ f(R)...
Extra dimensions −→ DGP, Kaluza-Klein...
Weird stuff −→ Non-local, Lorentz violating...
Scalars fields: Simple + certain limits from other theories

Introduction
Results
Theories of Gravity
Scalar Tensor-Theories
Horndenski’s Theory (1974)
gµν +
£
¢

¡
φ + Local + 4-D + Lorentz Theory with 2nd order Eqs.
⇒ 4 free functions Gi(φ, X), X ≡ −1
2φ,µφ,µ
LH = G2 − G3 φ + G4R + G4,X ( φ)2
− φ;µνφ;µν
+ G5Gµνφ;µν
−
G5,X
6
( φ)3
− 3( φ)φ;µνφ;µν
+ 2φ ;ν
;µ φ ;λ
;ν φ ;µ
;λ
Jordan-Brans-Dicke: G4 = φ
16πG, G2 = X
ω(φ) − V (φ)

Introduction
Results
Theories of Gravity
gµν +
£
¢

¡
2φ,µφ,µ
LH = G2 − G3 φ + G4R + G4,X ( φ)2
− φ;µνφ;µν
+ G5Gµνφ;µν
−
G5,X
6
( φ)3
+ 2φ ;ν
;µ φ ;λ
;ν φ ;µ
;λ
16πG, G2 = X
ω(φ) − V (φ)
Kinetic Gravity Braiding - Deﬀayet et al. JCAP 2010

Introduction
Results
Theories of Gravity
gµν +
£
¢

¡
2φ,µφ,µ
LH = G2 − G3 φ + G4R + G4,X ( φ)2
− φ;µνφ;µν
+ G5Gµνφ;µν
−
G5,X
6
( φ)3
+ 2φ ;ν
;µ φ ;λ
;ν φ ;µ
;λ
16πG, G2 = X
ω(φ) − V (φ)
Deriv. couplings G4(X)

Introduction
Results
Theories of Gravity
gµν +
£
¢

¡
2φ,µφ,µ
LH = G2 − G3 φ + G4R + G4,X ( φ)2
− φ;µνφ;µν
+ G5Gµνφ;µν
−
G5,X
6
( φ)3
+ 2φ ;ν
;µ φ ;λ
;ν φ ;µ
;λ
16πG, G2 = X
ω(φ) − V (φ)
Deriv. couplings G4(X), G5 = 0

Introduction
Results
Theories of Gravity
Frames and Forces
f(R) + Lm ,
vv
φ=f , V =R
((
φR + Lm + Lφ ,
uu
gµν ↔φ−1˜gµν
**
˜R + Lm[φ−1
˜gµν] + ˜Lφ
Jordan frame Einstein frame
√
−g
R
16πG
+
√
−γLm γµν[φ, gµν]
matter metric
, · · · +
√
−gLφ

Introduction
Results
Theories of Gravity
Frames and Forces
f(R) + Lm ,
vv
φ=f , V =R
((
φR + Lm + Lφ ,
uu
gµν ↔φ−1˜gµν
**
˜R + Lm[φ−1
˜gµν] + ˜Lφ
Jordan frame Einstein frame
√
−g
R
16πG
+
√
matter metric
, · · · +
√
−gLφ
Point Particle:
¨xα
= − Γα
µν + Kα
µν
γαλ
( (µγν)λ−1
2 λγµν )
˙xµ
˙xν
⇒ Fi
φ = Ki
00 +O(vi
/c) ≈ f[φ] i
φ

Introduction
Results
Theories of Gravity
Subtle the Force can be
Fi
φ ≈ f[φ] iφ
“You must feel the Force around you;
here, between you, me, the tree, the rock,
everywhere, yes”
Master Yoda

Introduction
Results
Theories of Gravity
Subtle the Force can be
Fi
φ ≈ f[φ] iφ
“You must feel the Force around you;
here, between you, me, the tree, the rock,
everywhere, yes”
Master Yoda
No Fφ observed in Solar System
Fφ
FG
1 when
ρ ρ0
r H−1
0

Introduction
Results
Theories of Gravity
§
¦
¤
¥
ρ ρ0 Chameleon Screening - Khoury & Veltman (PRL 2004)
Yukawa force: φ ∝ 1
r e−φ/mφ with mφ(ρ) increases with ρ
(cf. Symmetron - Hinterbichler & Khoury PRL 2010)

Introduction
Results
Theories of Gravity
§
¦
¤
¥
ρ ρ0 Chameleon Screening - Khoury & Veltman (PRL 2004)
Yukawa force: φ ∝ 1
r e−φ/mφ with mφ(ρ) increases with ρ
(cf. Symmetron - Hinterbichler & Khoury PRL 2010)
§
¦
¤
¥
r H−1
0 Vainshtein Screening - Vainshtein (PLB 1972)
L ⊃ (∂φ) + φX/m2
+ αφTm Non-linear derivative interactions
⇒ φ + m−2
( φ)2
− φ;µν φ;µν
= αMδ(r)
φ ∝
r−1 if r rV
√
r if r rV
Vainshtein radius rV ∝ (GM/m2)1/3

Introduction
Results
Theories of Gravity
Form of the Matter Metric
√
−g R
16πG +
√
matter metric
, · · · +
√
−gLφ
Disformal Relations - Bekenstein (PRD 1992)
γµν = C(φ)gµν
conformal
+ D(φ)φ,µφ,ν
disformal
d¯s2
γ = Cds2
g + D(φ,µdxµ
)2
C = 1 local rescaling, same causal structure

Introduction
Results
Theories of Gravity
√
−g R
16πG +
√
matter metric
, · · · +
√
−gLφ
γµν = C(φ)gµν
conformal
+ D(φ)φ,µφ,ν
disformal
d¯s2
γ = Cds2
g + D(φ,µdxµ
)2
D = 0 modiﬁed causal structure

Introduction
Results
Theories of Gravity
√
−g R
16πG +
√
matter metric
, · · · +
√
−gLφ
γµν = C(φ)gµν
conformal
+ D(φ)φ,µφ,ν
disformal
d¯s2
γ = Cds2
g + D(φ,µdxµ
)2
D = 0 modiﬁed causal structure
¯γ00 ∝ 1 − D
C
˙φ2 → Slow roll - MZ, Koivisto et al. (JCAP 2010)
+ relativistic MOND, VSL, Galileons...

Introduction
Results
Disformally Related Frames
Screening the Force
Disformally Related Theories - MZ, Koivisto, Mota (PRD 2013)
γµν = C(φ)gµν + D(φ)φ,µφ,ν
Einstein Frame: LEF =
√
−gR[gµν] +
√
−γLM (γµν, ψ)
Einstein
gµν → 1
C gµν − D
C φ,µφ,ν

Jordan
Jordan Frame: LJF =
√
−γR[γµν] +
√
−gLM (gµν, ψ)

Introduction
Results
Screening the Force
√
−gR[gµν] +
√
Einstein
gµν →C−1gµν
))
gµν →gµν − D
C
φ,µφ,ν
uu
gµν → 1
C gµν − D
C φ,µφ,ν

D ⊂ matteroo
§
¦
¤
¥Galileon
))
Disformal
uu
D ⊂ gravity //
OO
Jordan
Jordan Frame: LJF =
√
−γR[γµν] +
√
−gLM (gµν, ψ)

Introduction
Results
Screening the Force
The Galileon Frame
Compute
√
−γ ¯R[γµν] for
§
¦
¤
¥
γµν = gµν + π,µπ,ν (π ≡ D(φ)dφ)

Introduction
Results
Screening the Force
The Galileon Frame
Compute
√
§
¦
¤
¥
Disformal Curvature
√
−γR[γµν] =
√
−g
1
˜γ
R[gµν] − ˜γ ( π)2
− π;µνπ;µν
+ µξµ
with ˜γ−1 ≡ g/γ = 1 + π,µπ,µ

Introduction
Results
Screening the Force
The Galileon Frame
Compute
√
§
¦
¤
¥
Disformal Curvature
√
−γR[γµν] =
√
−g
1
˜γ
R[gµν] − ˜γ ( π)2
− π;µνπ;µν
+ µξµ
with ˜γ−1 ≡ g/γ = 1 + π,µπ,µ
Quartic DBI Galileon
π = brane coordinate in 5th dim.
- De Rham Tolley (JCAP 2010)

Introduction
Results
Screening the Force
√
−gR[gµν] +
√
§
¦
¤
¥Einstein
gµν →C−1gµν
))
gµν →gµν − D
C
φ,µφ,ν
uu
gµν → 1
C gµν − D
C φ,µφ,ν

D ⊂ matteroo

Galileon
))
Disformal
uu
D ⊂ gravity //
OO
Jordan
Jordan Frame: LJF =
√
−γR[γµν] +
√
−gLM (gµν, ψ)

Introduction
Results
Screening the Force
The Einstein Frame
LEF =
√
−g
R
16πG
+ Lφ +
√
Gµν = 8πG(Tµν
m + Tµν
φ )
Matter-ﬁeld interaction: µTµν
m = −Qφ,ν

Introduction
Results
Screening the Force
The Einstein Frame
LEF =
√
−g
R
16πG
+ Lφ +
√
Gµν = 8πG(Tµν
m + Tµν
φ )
m = −Qφ,ν
Q =
D
C
µ (Tµν
m φ,ν) −
C
2C
Tm +
D
2C
−
DC
C2
φ,µφ,νTµν
m

Introduction
Results
Screening the Force
The Einstein Frame
LEF =
√
−g
R
16πG
+ Lφ +
√
Gµν = 8πG(Tµν
m + Tµν
φ )
m = −Qφ,ν
Q =
D
C
µ (Tµν
m φ,ν) −
C
2C
Tm +
D
2C
−
DC
C2
φ,µφ,νTµν
m
Kinetic mixing Conformal Disformal

Introduction
Results
Screening the Force
Consequences of Kinetic Mixing - Koivisto, Mota, MZ (PRL 2012)
Static matter ρ(x) + non-rel. p = 0
1 +
Dρ
C − 2DX
¨φ + F( φ,µ, φ,µ, ρ) = 0

Introduction
Results
Screening the Force
1 +
Dρ
C − 2DX
¨φ + F( φ,µ, φ,µ, ρ) = 0
Disformal Screening Mechanism
¨φ ≈ −
D
2D
˙φ2
+ C
˙φ2
C
−
1
2D
(If Dρ → ∞)

Introduction
Results
Screening the Force
1 +
Dρ
C − 2DX
¨φ + F( φ,µ, φ,µ, ρ) = 0
Disformal Screening Mechanism
¨φ ≈ −
D
2D
˙φ2
+ C
˙φ2
C
−
1
2D
(If Dρ → ∞)
φ(x, t)
§
¦
¤
¥
independent of ρ(x) and ∂iφ
⇒ No φ between M1, M2 ⇒
§
¦
¤
¥No ﬁfth force!

Introduction
Results
Screening the Force
Disformal Screening: Potential Signatures
Assumptions:
Static ∂tρ = 0
Pressureless Dp X ≡ C − 2DX
Neglect p
ρ , p
ρ
∂φ
∂tφ
2
, X
Dρ , X
DρV /¨φ , Γµ
00φ,µ/¨φ ∼ 0

Introduction
Results
Screening the Force
Assumptions:
Static ∂tρ = 0
Neglect p
ρ , p
ρ
∂φ
∂tφ
2
, X
Dρ , X
DρV /¨φ , Γµ
00φ,µ/¨φ ∼ 0
Potential Signatures
Matter velocity ﬂows: T0i → Terms ∝ φ;0i and ˙φ φ,i
Suppressed by v/c → Binary pulsars?

Introduction
Results
Screening the Force
Assumptions:
Static ∂tρ = 0
Neglect p
ρ , p
ρ
∂φ
∂tφ
2
, X
Dρ , X
DρV /¨φ , Γµ
00φ,µ/¨φ ∼ 0
Pressure: Instability and eﬀects on radiation

Introduction
Results
Screening the Force
Assumptions:
Static ∂tρ = 0
Neglect p
ρ , p
ρ
∂φ
∂tφ
2
, X
Dρ , X
DρV /¨φ , Γµ
00φ,µ/¨φ ∼ 0
Strong gravitational ﬁelds: Γµ
00φ,µ not suppressed by Dρ
Γr
00 = GM
r3 (r − 2GM) → Black holes?

Introduction
Results
Screening the Force
Assumptions:
Static ∂tρ = 0
Neglect p
ρ , p
ρ
∂φ
∂tφ
2
, X
Dρ , X
DρV /¨φ , Γµ
00φ,µ/¨φ ∼ 0
Strong gravitational ﬁelds: Γµ
00φ,µ not suppressed by Dρ
Γr
00 = GM
r3 (r − 2GM) → Black holes?
Spatial Field Gradients: Evolution independent of ∂iφ

Introduction
Results
Screening the Force
The Vainshtein Radius - Vainshtein (PLB 1972)
Non-linear derivative interactions L ⊃ + φX/m2
+ αφTm
⇒ φ + m−2
( φ)2
− φ;µν φ;µν
= αMδ(r)
φ ∝
r−1 if r rV
√
r if r rV
Disformal coupling: L ⊃ −˜γ ( φ)2
− φ;µν φ;µν
(Jordan Fr.)
φ = 0 ⇒ φ =
S
r
Einstein Fr.

Introduction
Results
Screening the Force
+ αφTm
⇒ φ + m−2
( φ)2
− φ;µν φ;µν
= αMδ(r)
φ ∝
r−1 if r rV
√
r if r rV
− φ;µν φ;µν
(Jordan Fr.)
φ =
−QµνδTµν
m
C + D(φ,r)2
⇒ φ =
S
r
(if r → ∞)

Introduction
Results
Screening the Force
+ αφTm
⇒ φ + m−2
( φ)2
− φ;µν φ;µν
= αMδ(r)
φ ∝
r−1 if r rV
√
r if r rV
− φ;µν φ;µν
(Jordan Fr.)
φ =
−QµνδTµν
m
C + D(φ,r)2
⇒ φ =
S
r
(if r → ∞)
D 0 ⇒ asymptotic φ0 = S
r breaks down at ˜rV = DS2
C
1/4

Introduction
Results
Conclusions
Life beyond the conformal coupling: Dφ,µφ,ν
→ don’t be a conformist!
New frames: Galileon Disformal
Einstein Frame: simpler Eqs. physical insight
Screening mechanisms:
Disformal: Dρ → ∞ ﬁeld eq. independent of ρ
Vainshtein: r rV ⇒ Fφ FG
Open questions potential applications (e.g. Cosmology)
May the force not be with you!

Introduction
Results
Backup Slides

Introduction
Results
Properties of the Field Equation
Canonical scalar ﬁeld Lφ = X − V , solve for φ
Mµν
µ νφ +
C
C − 2DX
QµνTµν
m − V = 0
Mµν
≡ gµν
−
D Tµν
m
C − 2DX
, Qµν ≡
C
2C
gµν +
C D
C2
−
D
2C
φ,µφ,ν
Coupling to (Einstein F) perfect ﬂuid Tµ
ν = diag(ρ, p, p, p)
M0
0 = 1 + Dρ
C−2DX , D, ρ 0 ⇒ no ghosts
Mi
i = 1 − Dp
C−2DX , ⇒ potential instability if p C/D − X
- Does it occur dynamically?
- Consider non-relativistic coupled species Mi
i 0

Introduction
Results
(Some) Cosmology
˙ρ + 3Hρ = Q0
˙φ ,
- Pure Conformal Q
(c)
0 = C
2C ρ
- Pure Disformal Q
(d)
0 ≈ ρφ
D
2D (1 + wφ) − V
2V (1 − wφ)
Simple models give
good background expansion with Λ = 0
too much growth:
Geﬀ
G
− 1 =
Q2
0
4πGρ2
But much room for viable models

May the Force NOT be with you

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