Hui - modified gravity

Chameleon screening - Alberto Nicolis, Chris Stubbs - Phil Chang - Justin Khoury, Junpu Wang

Vainshtein screening - Alberto Nicolis

Lam Hui’s collaborators:

Cosmology School Lecture on Modified Gravity

Tuesday, January 24, 2012

Topics for discussions:

- Conformal transformation: Einstein vs Jordan frame

- Connection with self-acceleration

- Large scale tests

- Small scale tests

- Scalar-tensor theory as a framework for modifying gravityWeinberg’s theorem, quintessence


Weinberg/Deser theorem tells us that a Lorentz invariant theory of a massless spin 2 particle must be GR at low energies. Thus modified gravity oftenintroduce new d.o.f. such as scalars (e.g. DGP, f(R), massive graviton,degravitation, TeVeS, healthy extensions of Horava gravity ...)

Scalar-tensor theories

Let’s consider (Einstein frame):

S =�

d4x�

−1

2(∂ϕ)2 + Lint(ϕ) + αϕTm

�

+ ... + hµνTmµν

α = scalar-matter coupling = O(1)

dimensionless, ϕ MP = 1

gµν = ηµν + hµν

ϕ mediates a long range force, which must be screened to satisfy solarsystem tests. determines the screening mechanism - potential interactions give chameleon, derivative interactions give Vainshtein.

Lint(ϕ)

Absent symmetries, quintessence should be coupled to matter.





- Large scale tests

- Small scale tests


Einstein: extra (5th) force; Jordan: geodesic for test particle (only!)





S =�

d4x�

−1


�

+ ... + hµνTmµν





Lint(ϕ)






- Large scale tests

- Small scale tests



Self-acceleration versus acceleration by dark energy





- Large scale tests

- Small scale tests




Growth rate, Psi versus Phi, photons





S =�

d4x�

−1


�

+ ... + hµνTmµν





Lint(ϕ)






- Large scale tests

- Small scale tests





Screening mechanisms: chameleon versus Vainshtein

Violations of the equivalence principle: chameleon - non-relativistic; Vainshtein - relativistic.


Chameleon screening - environment dependent mass

Vainshtein screening - scale dependent interactions

(Einstein frame)

V (ϕ)

ϕ

(Tmµµ ∼ −ρm)

Khoury &

(Einstein frame)

ϕ ∝ 1

r

ϕ ∝√

r

large r

small r

e.g. DGP

point mass solution ϕ

r

r−1√r

α = scalar-matter coupling = O(1) generically

Sscalar ∼�

d4x�

−1

2(∂ϕ)2 − V (ϕ) + αϕTm

µµ

�

αρmϕ ✷ϕ ∼ [V + αρmϕ],ϕ

e.o.m.:

Sscalar ∼�

d4x�

−1

2(∂ϕ)2 − 1

m2 (∂ϕ)2✷ϕ + αϕTmµµ

�

✷ϕ +1

m2

�(✷ϕ)2 − ∂µ∂νϕ∂µ∂νϕ

�∼ αρm

e.o.m.:

graviton mass

MPϕ( dimensionless, = 1 )

Weltman

See also: galileon (Nicolis, Rattazzi, Trincherini)

See also: symmetron(Hinterbichler, Khoury)

rV ∼ (rSchwm−2)1/3


Constrasting chameleon and Vainshtein screening:Consider an object in the presence of a long wavelength external (i.e. ignoring tides).

ϕ

The object-scalar interaction is described by Sint ∼ −αQ�

dτϕ

where Q is the object’s scalar charge i.e. .

Chameleon: e.g.

sun earth

both have Q << M ,∇2ϕ = V,ϕ + αρm ∼ 0

Vainshtein: e.g. both have Q = M ,

because

because shift symmetry impliese.o.m. takes the Gauss-law form ∂ · J = αρm

A large scalar force on the earth is avoided by having the sun source a very suppressed scalar profile within the Vainshtein radius.

rV

F = −αQ∇ϕ


Observational tests of chameleon screeningAn object is chameleon screened (Q << M) if the -grav. potential(GM/R) is deeper than , and unscreened (Q=M) otherwise.Observationally, we know any object with -grav. pot. deeper than should be screened (from Milky way).

ϕext/α

A screened object does not experience scalar force, whilean unscreened object does. They therefore fall at rates that areO(1) different (violation of equivalence principle) i.e. objectdependent G.

10−6

Red giants would have a compact screened core, and a diffuse unscreened envelope. Thus, effectively Newton’s G changes value in the star. This affects the observed temperature at the 100 K level.

Note: a screened object does not move on Jordan frame geodesic!


Bulk motion tests:

1. Small galaxies should move faster than large galaxies (i.e. aneffective velocity bias - redshift distortion needs to be reworked)in unscreened environments. Beware: Yukawa suppression.

2. Small galaxies should stream out of voids faster than large galaxiescreating larger than expected voids defined by small galaxies(see Peebles; note: effect cares about sign of grav. pot.).

3. Diffuse gas (e.g. HI) should move faster than stars in small galaxieseven if they are on the same orbit. Beware: asymmetric drift.4. Gravitational lensing mass should agree with dynamical massfrom stars, but disagree with that from HI in small galaxies.

Internal motion tests:

Idea - unscreened small galaxies, screened large galaxies.

Idea - unscreened HI gas clouds, screened stars.

Key: avoid blanket screening.


!610 !8

61/

10

α

ϕ/α

Ruled out by demanding screening in Milky way and sun

scalar-matter coupling

ϕ = scalar field value at mean density


Side remark: chameleon theories cannot support genuineself-acceleration.

g̃µν = eαϕgµν

Jordan frame metric Einstein frame metric

Want no acceleration in Einstein frame, but acceleration in Jordan frame i.e. do not want acceleration to be caused by some form of dark energy, but rather by the non-minimal scalar coupling itself.This suggests cannot be too small.αϕ

Since observations constrain for chameleon screening, it cannot support self-acceleration whateverthe actual model is (assuming ).

ϕ/α ∼< 10−6

α ∼ 1


Observational test of the Vainshtein mechanismIt would be nice if there are equivalence principle tests of thesort like those for chameleon.


Observational test of the Vainshtein mechanismIt would be nice if there are equivalence principle tests of thesort like those for chameleon. But we know already Q=M is respected by derivative interactions.Thus different objects fall at the same rate (i.e. “grav. charge/mass” = inertial mass).


Observational test of the Vainshtein mechanismIt would be nice if there are equivalence principle tests of thesort like those for chameleon. But we know already Q=M is respected by derivative interactions.Thus different objects fall at the same rate (i.e. “grav. charge/mass” = inertial mass).Wait! How about black holes, they have zero scalar charge right?Won’t they fall slower than stars? i.e. equivalence principle violationof the relativistic kind.


Observational test of the Vainshtein mechanismIt would be nice if there are equivalence principle tests of thesort like those for chameleon. But we know already Q=M is respected by derivative interactions.Thus different objects fall at the same rate (i.e. “grav. charge/mass” = inertial mass).Wait! How about black holes, they have zero scalar charge right?Won’t they fall slower than stars? i.e. equivalence principle violationof the relativistic kind.Issue 1: the existing derivations of no-scalar-hair theorem do notapply to galileons, but we can extend them to show black holeshave no galileon hair (at the moment for Schwarzchild).


Observational test of the Vainshtein mechanismIt would be nice if there are equivalence principle tests of thesort like those for chameleon. But we know already Q=M is respected by derivative interactions.Thus different objects fall at the same rate (i.e. “grav. charge/mass” = inertial mass).Wait! How about black holes, they have zero scalar charge right?Won’t they fall slower than stars? i.e. equivalence principle violationof the relativistic kind.Issue 1: the existing derivations of no-scalar-hair theorem do notapply to galileons, but we can extend them to show black holeshave no galileon hair (at the moment for Schwarzchild).Issue 2, a more serious problem: black holes and stars are generallyfound inside galaxies. Wouldn’t the fact that they are both insidethe Vainshtein radius of the galaxy mean the effect is very small?


9

FIG. 5: Dark matter power spectra from the nonlinear DGP model (nlDGP) , linear DGP (lDGP), and GR perturbations withthe same expansion history (GRH) at z = 1. The left panels show the power spectra, and the right panels shows ratios tobetter see the di!erences. Two sets of computational boxes are shown for each case, covering a di!erent range in k (see text).The solid line denotes the predictions from paper I for PnlDGP (left panel) and PGRH/PnlDGP (right panel).

FIG. 6: Same as Fig. 5 but for z = 0

than at z = 1 at large scales, so the Vainshtein e!ecthas to overcome a larger di!erence at z = 0. A Vain-shtein scale (analogous to r! in the Schwarzshild case)

may be defined by the scale at which PGRH/PnlDGP startsto decrease, this is about k! ! 2 h Mpc"1 at z = 1 andk! ! 1 h Mpc"1 at z = 0. Note that at intermediate

Chan & Scoccimarro 2009


The key is to recognize that there are regions in the universewhere the scalar is in the linear regime - in and around voids(see sim. by Chan & Scoccimarro). Rewriting the scalar e.o.m.:

ϕ

H−2∂2ϕ+ (H−2∂2ϕ)2 ∼ α

ρmρ̄m

in regions of sufficiently low density, the linear term dominatesover the nonlinear term i.e. is unsuppressed by interactions.

Consider a galaxy in such a region: ϕextthe linear galaxy

The galaxy (with its stars and dark matter) would fall underthis external scalar field. The black hole won’t. Both of coursestill respond in the same way to the Einstein part of gravity.

ϕ

falls


The key is to recognize that there are regions in the universewhere the scalar is in the linear regime - in and around voids(see sim. by Chan & Scoccimarro). Rewriting the scalar e.o.m.:

ϕ

H−2∂2ϕ+ (H−2∂2ϕ)2 ∼ α

ρmρ̄m

in regions of sufficiently low density, the linear term dominatesover the nonlinear term i.e. is unsuppressed by interactions.

Consider a galaxy in such a region: ϕextthe linear galaxy

The galaxy (with its stars and dark matter) would fall underthis external scalar field. The black hole won’t. Both of coursestill respond in the same way to the Einstein part of gravity.

Central massive black hole becomes off-centered!

ϕ

falls


The idea is to look for the offset of massive black holes from thecenters of galaxies which are streaming out of voids.The offset should be correlated with the direction of the streamingmotion. The massive black holes can take the form of quasars orlow luminosity galactic nuclei i.e. Seyferts.

The offset is estimated to be up to 0.1 kpc, for small galaxies.





- Large scale tests

- Small scale tests





Screening mechanisms: chameleon versus Vainshtein

Violations of the equivalence principle: chameleon - non-relativistic; Vainshtein - relativistic.


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Hui - modified gravity

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