1. Cosmology School Lecture on Modified Gravity
Lam Hui’s collaborators:
Chameleon screening - Alberto Nicolis, Chris Stubbs
- Phil Chang
- Justin Khoury, Junpu Wang
Vainshtein screening - Alberto Nicolis
Tuesday, January 24, 2012

2. Topics for discussions:
- Scalar-tensor theory as a framework for modifying gravity
Weinberg’s theorem, quintessence
- Conformal transformation: Einstein vs Jordan frame
- Connection with self-acceleration
- Large scale tests
- Small scale tests
Tuesday, January 24, 2012

3. Scalar-tensor theories
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 often
introduce new d.o.f. such as scalars (e.g. DGP, f(R), massive graviton,
degravitation, TeVeS, healthy extensions of Horava gravity ...)
Absent symmetries, quintessence should be coupled to matter.
Let’s consider (Einstein frame):
1
S= d x − (∂ϕ)2 + Lint (ϕ) + αϕTm + ... + hµν Tm µν
4
2
gµν = ηµν + hµν
ϕ dimensionless, MP = 1
α = scalar-matter coupling = O(1)
ϕ mediates a long range force, which must be screened to satisfy solar
system tests. Lint (ϕ) determines the screening mechanism - potential
interactions give chameleon, derivative interactions give Vainshtein.
Tuesday, January 24, 2012

4. Topics for discussions:
- Scalar-tensor theory as a framework for modifying gravity
Weinberg’s theorem, quintessence
- Conformal transformation: Einstein vs Jordan frame
Einstein: extra (5th) force; Jordan: geodesic for test particle (only!)
- Connection with self-acceleration
- Large scale tests
- Small scale tests
Tuesday, January 24, 2012

5. Scalar-tensor theories
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 often
introduce new d.o.f. such as scalars (e.g. DGP, f(R), massive graviton,
degravitation, TeVeS, healthy extensions of Horava gravity ...)
Absent symmetries, quintessence should be coupled to matter.
Let’s consider (Einstein frame):
1
S= d x − (∂ϕ)2 + Lint (ϕ) + αϕTm + ... + hµν Tm µν
4
2
gµν = ηµν + hµν
ϕ dimensionless, MP = 1
α = scalar-matter coupling = O(1)
ϕ mediates a long range force, which must be screened to satisfy solar
system tests. Lint (ϕ) determines the screening mechanism - potential
interactions give chameleon, derivative interactions give Vainshtein.
Tuesday, January 24, 2012

6. Topics for discussions:
- Scalar-tensor theory as a framework for modifying gravity
Weinberg’s theorem, quintessence
- Conformal transformation: Einstein vs Jordan frame
Einstein: extra (5th) force; Jordan: geodesic for test particle (only!)
- Connection with self-acceleration
Self-acceleration versus acceleration by dark energy
- Large scale tests
- Small scale tests
Tuesday, January 24, 2012

7. Topics for discussions:
- Scalar-tensor theory as a framework for modifying gravity
Weinberg’s theorem, quintessence
- Conformal transformation: Einstein vs Jordan frame
Einstein: extra (5th) force; Jordan: geodesic for test particle (only!)
- Connection with self-acceleration
Self-acceleration versus acceleration by dark energy
- Large scale tests
Growth rate, Psi versus Phi, photons
- Small scale tests
Tuesday, January 24, 2012

8. Scalar-tensor theories
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 often
introduce new d.o.f. such as scalars (e.g. DGP, f(R), massive graviton,
degravitation, TeVeS, healthy extensions of Horava gravity ...)
Absent symmetries, quintessence should be coupled to matter.
Let’s consider (Einstein frame):
1
S= d x − (∂ϕ)2 + Lint (ϕ) + αϕTm + ... + hµν Tm µν
4
2
gµν = ηµν + hµν
ϕ dimensionless, MP = 1
α = scalar-matter coupling = O(1)
ϕ mediates a long range force, which must be screened to satisfy solar
system tests. Lint (ϕ) determines the screening mechanism - potential
interactions give chameleon, derivative interactions give Vainshtein.
Tuesday, January 24, 2012

9. Topics for discussions:
- Scalar-tensor theory as a framework for modifying gravity
Weinberg’s theorem, quintessence
- Conformal transformation: Einstein vs Jordan frame
Einstein: extra (5th) force; Jordan: geodesic for test particle (only!)
- Connection with self-acceleration
Self-acceleration versus acceleration by dark energy
- Large scale tests
Growth rate, Psi versus Phi, photons
- Small scale tests
Screening mechanisms: chameleon versus Vainshtein
Violations of the equivalence principle:
chameleon - non-relativistic; Vainshtein - relativistic.
Tuesday, January 24, 2012

10. Khoury
Chameleon screening - environment dependent mass Weltman
1
Sscalar ∼ d x − (∂ϕ)2 − V (ϕ) + αϕTm µ µ
4 (Einstein frame)
2
e.o.m.:
V (ϕ)
αρm ϕ
✷ϕ ∼ [V + αρm ϕ],ϕ (Tm µ µ ∼ −ρm )
(ϕ dimensionless, MP = 1 )
ϕ See also: symmetron(Hinterbichler, Khoury)
Vainshtein screening - scale dependent interactions e.g. DGP
1 1 (Einstein frame)
Sscalar ∼ 4 2
d x − (∂ϕ) − 2 (∂ϕ)2 ✷ϕ + αϕTm µ µ
2 m
1
ϕ∝ large r
e.o.m.: ✷ϕ + 1 (✷ϕ)2 − ∂ µ∂ ν ϕ∂µ ∂ν ϕ ∼ αρm r
2 m √
ϕ ∝ r small r
ϕ graviton mass
√ point mass solution
r r−1
See also: galileon (Nicolis, Rattazzi, Trincherini)
r
rV ∼ (rSchw m−2 )1/3
α = scalar-matter coupling = O(1) generically
Tuesday, January 24, 2012

11. Constrasting chameleon and Vainshtein screening:
Consider an object in the presence of a long wavelength external ϕ
(i.e. ignoring tides).
The object-scalar interaction is described by Sint ∼ −αQ dτ ϕ
where Q is the object’s scalar charge i.e. F = −αQ∇ϕ .
Chameleon: e.g. both have Q M ,
because ∇2 ϕ = V,ϕ + αρm ∼ 0
earth
sun
Vainshtein: e.g.
both have Q = M ,
because shift symmetry implies
e.o.m. takes the Gauss-law
rV
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.
Tuesday, January 24, 2012

12. Observational tests of chameleon screening
An object is chameleon screened (Q M) if the -grav. potential
(GM/R) is deeper than ϕext /α , and unscreened (Q=M) otherwise.
Observationally, we know any object with -grav. pot. deeper than
10−6 should be screened (from Milky way).
A screened object does not experience scalar force, while
an unscreened object does. They therefore fall at rates that are
O(1) different (violation of equivalence principle) i.e. object
dependent G.
Note: a screened object does not move on Jordan frame geodesic!
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.
Tuesday, January 24, 2012

13. Bulk motion tests:
Idea - unscreened small galaxies, screened large galaxies.
1. Small galaxies should move faster than large galaxies (i.e. an
effective 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 galaxies
creating larger than expected voids defined by small galaxies
(see Peebles; note: effect cares about sign of grav. pot.).
Internal motion tests:
Idea - unscreened HI gas clouds, screened stars.
3. Diffuse gas (e.g. HI) should move faster than stars in small galaxies
even if they are on the same orbit. Beware: asymmetric drift.
4. Gravitational lensing mass should agree with dynamical mass
from stars, but disagree with that from HI in small galaxies.
Key: avoid blanket screening.
Tuesday, January 24, 2012

14. Ruled out by demanding
screening in Milky way and sun
α
scalar-matter
coupling
1/ 6
10 −8 10 −6
ϕ = scalar field value at mean density ϕ/α
Tuesday, January 24, 2012

15. Side remark: chameleon theories cannot support genuine
self-acceleration.
gµν = eαϕ gµν
˜
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 αϕ cannot be too small.
Since observations constrain ϕ/α 10−6 for chameleon
∼
screening, it cannot support self-acceleration whatever
the actual model is (assuming α ∼ 1).
Tuesday, January 24, 2012

16. Observational test of the Vainshtein mechanism
It would be nice if there are equivalence principle tests of the
sort like those for chameleon.
Tuesday, January 24, 2012

17. Observational test of the Vainshtein mechanism
It would be nice if there are equivalence principle tests of the
sort 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).
Tuesday, January 24, 2012

18. Observational test of the Vainshtein mechanism
It would be nice if there are equivalence principle tests of the
sort 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 violation
of the relativistic kind.
Tuesday, January 24, 2012

19. Observational test of the Vainshtein mechanism
It would be nice if there are equivalence principle tests of the
sort 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 violation
of the relativistic kind.
Issue 1: the existing derivations of no-scalar-hair theorem do not
apply to galileons, but we can extend them to show black holes
have no galileon hair (at the moment for Schwarzchild).
Tuesday, January 24, 2012

20. Observational test of the Vainshtein mechanism
It would be nice if there are equivalence principle tests of the
sort 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 violation
of the relativistic kind.
Issue 1: the existing derivations of no-scalar-hair theorem do not
apply to galileons, but we can extend them to show black holes
have no galileon hair (at the moment for Schwarzchild).
Issue 2, a more serious problem: black holes and stars are generally
found inside galaxies. Wouldn’t the fact that they are both inside
the Vainshtein radius of the galaxy mean the effect is very small?
Tuesday, January 24, 2012

21. Chan Scoccimarro 2009 9
FIG. 5: Dark matter power spectra from the nonlinear DGP model (nlDGP) , linear DGP (lDGP), and GR perturbations with
the same expansion history (GRH) at z = 1. The left panels show the power spectra, and the right panels shows ratios to
better see the differences. Two sets of computational boxes are shown for each case, covering a different range in k (see text).
The solid line denotes the predictions from paper I for PnlDGP (left panel) and PGRH /PnlDGP (right panel).
Tuesday, January 24, 2012

22. The key is to recognize that there are regions in the universe
where the scalar ϕ is in the linear regime - in and around voids
(see sim. by Chan Scoccimarro). Rewriting the scalar e.o.m.:
−2 2 −2 2 2 ρm
H ∂ ϕ + (H ∂ ϕ) ∼ α
ρm
¯
in regions of sufficiently low density, the linear term dominates
over the nonlinear term i.e. ϕ is unsuppressed by interactions.
Consider a galaxy in such a region: the linear ϕext
galaxy
falls
The galaxy (with its stars and dark matter) would fall under
this external scalar field. The black hole won’t. Both of course
still respond in the same way to the Einstein part of gravity.
Tuesday, January 24, 2012

23. The key is to recognize that there are regions in the universe
where the scalar ϕ is in the linear regime - in and around voids
(see sim. by Chan Scoccimarro). Rewriting the scalar e.o.m.:
−2 2 −2 2 2 ρm
H ∂ ϕ + (H ∂ ϕ) ∼ α
ρm
¯
in regions of sufficiently low density, the linear term dominates
over the nonlinear term i.e. ϕ is unsuppressed by interactions.
Consider a galaxy in such a region: the linear ϕext
galaxy
Central massive black hole
becomes off-centered!
falls
The galaxy (with its stars and dark matter) would fall under
this external scalar field. The black hole won’t. Both of course
still respond in the same way to the Einstein part of gravity.
Tuesday, January 24, 2012

24. The idea is to look for the offset of massive black holes from the
centers of galaxies which are streaming out of voids.
The offset should be correlated with the direction of the streaming
motion. The massive black holes can take the form of quasars or
low luminosity galactic nuclei i.e. Seyferts.
The offset is estimated to be up to 0.1 kpc, for small galaxies.
Tuesday, January 24, 2012

25. Topics for discussions:
- Scalar-tensor theory as a framework for modifying gravity
Weinberg’s theorem, quintessence
- Conformal transformation: Einstein vs Jordan frame
Einstein: extra (5th) force; Jordan: geodesic for test particle (only!)
- Connection with self-acceleration
Self-acceleration versus acceleration by dark energy
- Large scale tests
Growth rate, Psi versus Phi, photons
- Small scale tests
Screening mechanisms: chameleon versus Vainshtein
Violations of the equivalence principle:
chameleon - non-relativistic; Vainshtein - relativistic.
Tuesday, January 24, 2012