2015/02/10 seminar@APC

Breaking of Vainshtein screeningin scalar-tensor theories
beyond Horndeski
YAMAUCHI, Daisuke
Research Center for the Early Universe (RESCEU),
U. TokyoReference

Breaking of Vainshtein screening in scalar-tensor theories
beyond Horndeski
Tsutomu Kobayashi(Rikkyo), Yuki Watanabe(RESCEU) and DY,
arXiv:1411.4130.

Why modified gravity?
 Discovery of cosmic acceleration: Our understanding of the Universe
is incomplete!
 Need better understanding of gravity:-- Dark energy or modified gravity ?
• Small scale : consistent withexperiments in the solar-systemand on the Earth• Large scale : can be tested though
the cosmological observations[Planck2013]

Modified gravity theories
General relativity : massless spin-2

Modified gravity : A new d.o.f. is introduced toachieve the accelerating expansion
f(R), DGP, Galileon, massive gravity, bi-gravity, …

Scalar-tensor theories• Gravity is mediated by gμν and φ• Can capture essential modification of gravity

Modified gravity theories
Scalars mediates fifth forces strongly constrainedby precision tests of gravity at Solar system scale:

E.g. |Ψ/Φ -1| < O(10-4)
Scalar field must be screenedin the vicinity of sources.
• Need screening mechanism on short scalesCosmological test of gravity theories

VAINSHTEIN SCREENINGMECHANISMGeneral Relativity∇Φ = ∇Ψ = GNM/r2r
Scalar-tensor theories
Scalar d.o.f. can participate in gravity
dynamics : φ(x) = φ0 + π(x)∇Φ = ∇Ψ !!!

Any modified model involving scalar dof should
accommodate a mechanism to suppress the scalarinteraction on small scalesrVr∇Φ = ∇Ψ >> ∇π ∇Φ = ∇ΨScreened

Recover GR

Unscreened

Non-GRAn example : cubic Galileon
 Look for spherical symmetric solution : ρ=ρ(r), π = π(r)Algebraic equation for π’/Λ3r

Enclosed mass∝ρ

∝A(r)

A
Vainshtein
radius

Surface
Screened

Unscreened

1rS

rV~(M/MPl)1/3Λ-1r

A
Vainshtein
radius

Surface Outer region: linear regime

Screenedπ’ = O(A)<<1⇒ π’ ~ Φ’ ~= Ψ’ = O(A)Unscreened

1rS

rV~(M/MPl)1/3Λ-1r

Aπ’ = O(A )>>1
⇒ π’ << Φ’ = Ψ’ = O(A)
Scalar interactionrS canbe suppressed!

Vainshtein
radiusSurface
 Inner region
Screened: Quadratic term is dominated.
1
1/2

UnscreenedrV~(M/MPl)1/3Λ-1r

Lesson from this example
Naively, scalar-tensor theories predict∇π ~ ∇Φ =~ ∇ΨNonlinear derivative interaction of π can be large,
leading to self-screening in the vicinity of source(even if Φ<<1)(∇2π)n∇π~∇π<<∇Φ=∇Ψ
 2 derivatives acting on π plays central role for
implementing Vainshtein mech.

EFFECTIVE THEORYFOR VAINSHTEIN MECHANISM

[Horndeski (1974),Deffayet+Gao+Steer+Zahariade (2011),
Kobayashi+Yamaguchi+Yokoyama (2011)]

Horndeski theory: The most general scalar-tensor theory withsecond-order field equations.

have 4 arbitrary functions of φ and X=-1/2(∇φ)2.
is quite useful for a comprehensive study ofmodified gravity.

Effective theory for screening mech.
[Cosmological background :Kimura+Kobayashi+Yamamoto(2012), Kobayashi+Watanabe+DY (2014)][Minkowski background :Narikawa+Kobayashi+DY+Saito (2013), Koyama+Niz+Tasinato(2013)]

The general theory exhibiting Vainshtein screening
mechanism can be derived from Horndeski Lagrangian.
Step 1 : Expand the Lagrangian around FLRW backgroundEffective theory for screening mech.
[Cosmological background :Kimura+Kobayashi+Yamamoto(2012), Kobayashi+Watanabe+DY (2014)][Minkowski background :Narikawa+Kobayashi+DY+Saito (2013), Koyama+Niz+Tasinato(2013)]

Step 2 : Quasi-static approximation

Step 3 : Assume scaling of coefficients

etc…
Note : A concrete realization of the above scaling
can be found in massive gravity.

Effective theory for screening mech.
[Cosmological background :Kimura+Kobayashi+Yamamoto(2012), Kobayashi+Watanabe+DY (2014)][Minkowski background :Narikawa+Kobayashi+DY+Saito (2013), Koyama+Niz+Tasinato(2013)]

Step 4 : Keep (2+n)-th order terms with (2+2n) spatial derivatives• These can be larger than the quadratic terms below
a certain scale.
• Such nonlinear derivative interactions are crucial for
realizing the Vainshtein mechanism.

Effective theory from Horndeski[Kimura+Kobayashi+Yamamoto(2012), Kobayashi+Watanabe+DY (2014)]

Effective theory from Horndeski[Kimura+Kobayashi+Yamamoto(2012), Kobayashi+Watanabe+DY (2014)]

O(∇2ε2) terms
F, G, η,… are O(1) unless they vanish.
Matter overdensity δ is assumed tobe O(∇2Φ).

O((∇2ε)n(∇ε)2) terms

Effective theory from Horndeski[Kimura+Kobayashi+Yamamoto(2012), Kobayashi+Watanabe+DY (2014)]

Effective theory from Horndeski[Kimura+Kobayashi+Yamamoto(2012),Kobayashi+Watanabe+DY
Hereafter
we focus on the subclassof β=γ=0, (2014)]
which amounts for switching off G5• β=0 → Vainshtein solutions are unstable [Koyama+(2013)]• γ=0 → Metric potentials do not show the correct
Newtonian behavior [Kobayashi+(2013)]

Spherical overdensity from Horndeski
In terms ofAlgebraic equation for x can be written as

Solving algebraic equation for x is essential
for constructing Vainshtein solution.

ArVr

rS

Vainshtein
radius

Surface1

A
Vainshtein
radius

Surface

Outer region → linear regime & A<<1
• There is always a decaying solution :x ~ O(A) <<1 → Φ’ =~ Ψ’ ~ O(A)
1rSrVrA
Vainshtein
radius

Surface

Inner region → nonlinear regime & A>>1
• Imposing the condition that two of three terms balance:1

 x ~ O(A1/2) case→ Φ’/a2MPl= Ψ’/a2MPl=GN(t)M/r2
GR can be reproduced inside the Vainshtein radiusrrV in r<r )S (other cases→ Φ’ = Ψ’ evenVr

EFFECTIVE THEORYFROM BEYOND-HORNDESKI

Horndeski theory
can be generalized w/o introducing any
propagating dof : beyond-Horndeski theory.
[Gleyzes+Langlois+Piazza+Vernizzi (2014)]

Inevitably has higher-order field equations.

Introduces a preferred time-slicing on which# of initial conditions remain the same.
 Lagrangian in a covariant form by reviving φ:Horndeski theory
can be generalized w/o introducing any
propagating dof : beyond-Horndeski theory.
[Gleyzes+Langlois+Piazza+Vernizzi (2014)]

Inevitably has higher-order field equations.

Introduces a preferred time-slicing on which# of initial conditions remain the same. 2
have 6 arbitrary functions of φ and X=-1/2(∇φ) .
 Lagrangian in a covariant form by reviving φ:

[Kobayashi+Watanabe+DY (2014)]

New terms from beyond-Horndeski
New terms

 Other terms generated by the beyond-Horndeski term can beabsorbed in the redefinition of the corresponding coefficients.
 Taking limit gμν→ημν, X→0, only the Honrdeski term survive
and there is no new term.

[Kobayashi+Watanabe+DY (2014)]

New terms from beyond-Horndeski
New terms

 Other terms generated by the beyond-Horndeski term can beabsorbed in the redefinition of the corresponding coefficients.
 Taking limit gμν→ημν, X→0, only the Honrdeski term survive
and there is no new term.

[Kobayashi+Watanabe+DY (2014)]

New terms from beyond-Horndeski
New termsEOMEOM[Kobayashi+Watanabe+DY (2014)]

Impact of: spherical overdensity

In terms ofAlgebraic equation for x can be written as

Inner solution is determined not only the enclosed mass(M∝A) but also from the local energy density (A’~ρ)!

ArSrVr
δ=0
δ=0
δ=0

Vainshtein
radius

Surface1

A
Vainshtein
radius

Surface

Outer region→ same as Horndeski
1
δ=0

δ=0

δ=0rSrVr

Vainshtein
radius

Surface

AInner region → nonlinear regime & A, A’, A’’>>1
Plugging the concrete expression of the coefficients,

1rS

New
r contribution
VrSurface

Outside the overdensity region

Vainshtein
radiusA

A=(rV/r)3 ⇔ M’=0
GR can be reproduced

1
δ=0

δ=0

δ=0rSrVr

A
Vainshtein
radius

SurfaceVainshtein
radius

Inside the overdensity region
1M’, M’’ = 0 ⇔ Φ’ = Ψ’
δ=0
δ=0Screening mech. fails to operate!

rS

δ=0rVrAn exampleA(r)=constant (r<rS=0.01rV)(rV/r)3

Breaking of screening mech.
Φ’/Λ3a2r
Φ’ = Ψ’32Ψ’/Λ a r
A=M/MPlΛ3r3(r>rS)

Φ’ = Ψ’

x=π’/Λ3a2rrS

Vainshtein
radiusSurface

Φ’ = Ψ’rV

F=2, G=1, ξ1=1.5, ξ2=1, η=0.5, μ=1, ν=0.5, α1=1.2, α2=1, α*=-0.25

Impact of: linear growth Cosmological Poisson equation• Time-dependent effective gravitational coupling• Additional friction from scalar field Evolution of density contrast+ Energy-momentum
conservation eq.
• Beyond-term modifies the expansion rate felt by
the density perturbations as well as GeffSummaryEffective Lagrangian on small scales is studied in thecontext of scalar-tensor theories beyond Horndeski
New terms generates 3rd spatial derivatives in field eqs
Vainshtein screening mechanism operates
outside the source and inside Vainshtein radiusΦ = Ψ inside the source even in r<rVLinear growth of density perturbations is modified Geff and additional friction

Future Observational consequence of breaking ofVainshtein screening• Motion of a galaxy inside a cluster?
• Modified structure of stars?

Thank you!