may the force not be with you

IntroductionResults

May the force not be with youScreening Modifications of Gravity and Disformal Couplings

Miguel Zumalacarregui

Instituto de Fısica Teorica (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 Zumalacarregui May the force not be with you

IntroductionResults

Theories of GravityScreening 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

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

IntroductionResults

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

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

IntroductionResults

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

IntroductionResults

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 ≡ −12φ,µφ

LH = G2 −G3�φ+G4R+G4,X

[(�φ)2 − φ;µνφ

;µν]

+G5Gµνφ;µν −

[(�φ)3 − 3(�φ)φ;µνφ

;µν + 2φ ;ν;µ φ ;λ

;ν φ ;µ;λ

]Jordan-Brans-Dicke: G4 = φ

16πG , G2 = Xω(φ) − V (φ)

Kinetic Gravity Braiding - Deffayet et al. JCAP 2010

Deriv. couplings G4(X)

, G5 6= 0

IntroductionResults

[(�φ)2 − φ;µνφ

;µν]

+G5Gµνφ;µν −

[(�φ)3 − 3(�φ)φ;µνφ

;µν + 2φ ;ν;µ φ ;λ

;ν φ ;µ;λ

16πG , G2 = Xω(φ) − V (φ)

, G5 6= 0

IntroductionResults

[(�φ)2 − φ;µνφ

;µν]

+G5Gµνφ;µν −

[(�φ)3 − 3(�φ)φ;µνφ

;µν + 2φ ;ν;µ φ ;λ

;ν φ ;µ;λ

16πG , G2 = Xω(φ) − V (φ)

, G5 6= 0

IntroductionResults

[(�φ)2 − φ;µνφ

;µν]

+G5Gµνφ;µν −

[(�φ)3 − 3(�φ)φ;µνφ

;µν + 2φ ;ν;µ φ ;λ

;ν φ ;µ;λ

16πG , G2 = Xω(φ) − V (φ)

Deriv. couplings G4(X), G5 6= 0

IntroductionResults

Frames and Forces

f(R) + Lm ,vvφ=f ′, V ′=R

φR + Lm + Lφ ,uu

gµν↔φ−1gµν**

R+ Lm[φ−1gµν ] + Lφ

Jordan frame Einstein frame

√−g R

16πG+√−γLm

[γµν [φ, gµν ]︸︷︷︸matter metric

, · · ·]

+√−gLφ

? Point Particle:

xα = −(Γαµν + Kαµν︸︷︷︸

γαλ(∇(µγν)λ− 12∇λγµν)

)xµxν

⇒ F iφ = Ki00 +O(vi/c) ≈ f [φ]∇iφ

IntroductionResults

Frames and Forces

f(R) + Lm ,vvφ=f ′, V ′=R

φR + Lm + Lφ ,uu

gµν↔φ−1gµν**

R+ Lm[φ−1gµν ] + Lφ

Jordan frame Einstein frame

√−g R

16πG+√−γLm

, · · ·]

+√−gLφ

? Point Particle:

xα = −(Γαµν + Kαµν︸︷︷︸

γαλ(∇(µγν)λ− 12∇λγµν)

)xµxν

⇒ F iφ = Ki00 +O(vi/c) ≈ f [φ]∇iφ

IntroductionResults

Subtle the Force can be

F iφ ≈ 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

Screening Mechanisms∣∣∣∣FφFG∣∣∣∣� 1 when

{ρ� ρ0

r � H−10

IntroductionResults

Subtle the Force can be

F iφ ≈ 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

Screening Mechanisms∣∣∣∣FφFG∣∣∣∣� 1 when

{ρ� ρ0

r � H−10

IntroductionResults

Screening Mechanisms

�� ρ� ρ0 Chameleon Screening - Khoury & Veltman (PRL 2004)

Yukawa force: φ ∝ 1re−φ/mφ with mφ(ρ) increases with ρ

(cf. Symmetron - Hinterbichler & Khoury PRL 2010)

�� r � H−10 Vainshtein Screening - Vainshtein (PLB 1972)

L ⊃ (∂φ) +�φX/m2 + αφTm Non-linear derivative interactions

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

φ ∝{r−1 if r � rV√r if r � rV

Vainshtein radius rV ∝ (GM/m2)1/3

IntroductionResults

(cf. Symmetron - Hinterbichler & Khoury PRL 2010)�� r � H−10 Vainshtein Screening - Vainshtein (PLB 1972)

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

IntroductionResults

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

IntroductionResults

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

IntroductionResults

Form of the Matter Metric√−g R

16πG +√−γLm

, · · ·]

+√−gLφ

Disformal Relations - Bekenstein (PRD 1992)

γµν = C(φ)gµν︸︷︷︸conformal

+ D(φ)φ,µφ,ν︸︷︷︸disformal

ds2γ = Cds2

g +D(φ,µdxµ)2

? C 6= 1 local rescaling, same causal structure

? D 6= 0 modified causal structure

γ00 ∝ 1− DC φ

2 → Slow roll - MZ, Koivisto et al. (JCAP 2010)

+ relativistic MOND, VSL, Galileons...

IntroductionResults

16πG +√−γLm

, · · ·]

+√−gLφ

ds2γ = Cds2

g +D(φ,µdxµ)2

γ00 ∝ 1− DC φ

IntroductionResults

16πG +√−γLm

, · · ·]

+√−gLφ

ds2γ = Cds2

g +D(φ,µdxµ)2

γ00 ∝ 1− DC φ

IntroductionResults

Disformally Related FramesScreening 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 → 1C gµν −

DC φ,µφ,ν

��

Jordan

Jordan Frame: LJF =√−γR[γµν ] +

√−gLM (gµν , ψ)

IntroductionResults

Einstein

DC φ,µφ,ν

��

Jordan

IntroductionResults

Einstein

gµν→C−1gµν))

gµν→gµν−DC φ,µφ,νuu

DC φ,µφ,ν

��

D ⊂ matteroo

�� Galileon

Disformal

uuD ⊂ gravity //

Jordan

IntroductionResults

The Galileon Frame

Compute√−γR[γµν ] for

�� γµν = gµν + π,µπ,ν (π ≡∫ √

D(φ)dφ)

Disformal Curvature

√−γR[γµν ] =

√−g[

γR[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)

IntroductionResults

The Galileon Frame

�� γµν = gµν + π,µπ,ν (π ≡∫ √

D(φ)dφ)

Disformal Curvature

√−g[

γR[gµν ]− γ

((�π)2 − π;µνπ

;µν)

+∇µξµ]

√1 + π,µπ,µ

IntroductionResults

The Galileon Frame

�� γµν = gµν + π,µπ,ν (π ≡∫ √

D(φ)dφ)

Disformal Curvature

√−g[

γR[gµν ]− γ

((�π)2 − π;µνπ

;µν)

+∇µξµ]

√1 + π,µπ,µ

IntroductionResults

�� Einstein

gµν→C−1gµν))

gµν→gµν−DC φ,µφ,νuu

DC φ,µφ,ν

��

D ⊂ matteroo

��

Galileon

Disformal

uuD ⊂ gravity //

Jordan

IntroductionResults

The Einstein Frame

LEF =√−g(

16πG+ Lφ

)+√−γLM (γµν , ψ)

Gµν = 8πG(Tµνm + Tµνφ )

Matter-field interaction: ∇µTµνm = −Qφ,ν

C∇µ (Tµνm φ,ν)− C ′

2CTm +

2C− DC ′

)φ,µφ,νT

Kinetic mixing Conformal Disformal

IntroductionResults

The Einstein Frame

LEF =√−g(

16πG+ Lφ

2CTm +

2C− DC ′

)φ,µφ,νT

IntroductionResults

The Einstein Frame

LEF =√−g(

16πG+ Lφ

2CTm +

2C− DC ′

)φ,µφ,νT

IntroductionResults

Consequences of Kinetic Mixing - Koivisto, Mota, MZ (PRL 2012)

Static matter ρ(~x) + non-rel. p = 0[1 +

C − 2DX

]φ+ F(~∇φ,µ, φ,µ, ρ) = 0

Disformal Screening Mechanism

φ ≈ −D′

2Dφ2 + C ′

C− 1

)(If Dρ→∞)

φ(~x, t)�� independent of ρ(~x) and ∂iφ

⇒ No ~∇φ between M1,M2 ⇒�� No fifth force!

IntroductionResults

C − 2DX

]φ+ F(~∇φ,µ, φ,µ, ρ) = 0

φ ≈ −D′

2Dφ2 + C ′

C− 1

)(If Dρ→∞)

IntroductionResults

C − 2DX

]φ+ F(~∇φ,µ, φ,µ, ρ) = 0

φ ≈ −D′

2Dφ2 + C ′

C− 1

)(If Dρ→∞)

IntroductionResults

Disformal Screening: Potential Signatures

Assumptions:

Static ∂tρ = 0

Pressureless Dp < X ≡ C − 2DX

Neglect pρ ,

(~∂φ∂tφ

)2, XDρ ,

′/φ , Γµ00φ,µ/φ ∼ 0

Potential Signatures

Matter velocity flows: T 0i → Terms ∝ φ;0i and φ φ,iSuppressed by v/c → Binary pulsars?

Pressure: Instability and effects on radiation

Strong gravitational fields: Γµ00φ,µ not suppressed by DρΓr00 = GM

r3(r − 2GM) → Black holes?

Spatial Field Gradients: Evolution independent of ∂iφ

IntroductionResults

Assumptions:

Static ∂tρ = 0

Neglect pρ ,

(~∂φ∂tφ

)2, XDρ ,

′/φ , Γµ00φ,µ/φ ∼ 0

IntroductionResults

Assumptions:

Static ∂tρ = 0

Neglect pρ ,

(~∂φ∂tφ

)2, XDρ ,

′/φ , Γµ00φ,µ/φ ∼ 0

IntroductionResults

Assumptions:

Static ∂tρ = 0

Neglect pρ ,

(~∂φ∂tφ

)2, XDρ ,

′/φ , Γµ00φ,µ/φ ∼ 0

IntroductionResults

Assumptions:

Static ∂tρ = 0

Neglect pρ ,

(~∂φ∂tφ

)2, XDρ ,

′/φ , Γµ00φ,µ/φ ∼ 0

IntroductionResults

The Vainshtein Radius - Vainshtein (PLB 1972)

? Non-linear derivative interactions L ⊃ +�φX/m2 + αφTm

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

? Disformal coupling: L ⊃ −γ((�φ)2 − φ;µνφ

;µν)

(Jordan Fr.)

�φ = 0 ⇒ φ =S

rEinstein Fr.

D < 0 ⇒ asymptotic φ0 = Sr breaks down at rV =

IntroductionResults

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

;µν)

(Jordan Fr.)

�φ =−QµνδTµνmC +D(φ,r)2

⇒ φ =S

r(if r →∞)

IntroductionResults

⇒ �φ+m−2((�φ)2 − φ;µνφ

;µν)= αMδ(r)

;µν)

(Jordan Fr.)

�φ =−QµνδTµνmC +D(φ,r)2

⇒ φ =S

r(if r →∞)

IntroductionResults

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!

IntroductionResults

Backup Slides

IntroductionResults

Properties of the Field Equation

Canonical scalar field Lφ = X − V , solve for ∇∇φ

Mµν∇µ∇νφ+C

C − 2DXQµνTµνm − V = 0

Mµν ≡ gµν− DTµνmC − 2DX

, Qµν ≡C ′

2Cgµν+

(C ′D

C2− D′

)φ,µφ,ν

Coupling to (Einstein F) perfect fluid Tµν = diag(ρ, p, p, p)

M00 = 1 + Dρ

C−2DX , D, ρ > 0⇒ no ghosts

Mii = 1− Dp

C−2DX , ⇒ potential instability if p > C/D −X

- Does it occur dynamically?

- Consider non-relativistic coupled species Mii > 0

IntroductionResults

(Some) Cosmology

ρ+ 3Hρ = Q0φ ,

- Pure Conformal Q(c)0 = C′

- Pure Disformal Q(d)0 ≈ ρφ

2D (1 + wφ)− V ′

2V (1− wφ))

Simple models give

good background expansion with Λ = 0

too much growth:

G− 1 =

4πGρ2

But much room for viable models

may the force not be with you

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