On LongWavelength of Modifications of Gravity and LateTime Acceleration

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Degravitating the C.C.
Justin Khoury (Perimeter Institute)
Based on G. Dvali, S. Hofmann JK,
hep-th/0703027 C. de Rham, G. Dvali,
S. Hofmann, JK, O. Pujolas, M. Redi A.
Tolley, to appear
Earlier work Dvali, Gabadadze Shifman,
hep-th/0202174 Arkani-Hamed, Dimopoulos,
Dvali Gabadadze, hep-th/0209227
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Almost all of the effort in addressing the C.C.
problem has focused on the question
Why is ??so small?
But since we only infer vacuum energy through
gravity, perhaps the right question to ask is
Why does ??gravitate so weakly?

• In GR, equiv principle gt 2 questions are
  equivalent

• But in theories where graviton is massive or is a
  resonance, 2 questions can in principle be
  distinguished

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Phenomenological eqn
Arkani-Hamed, Dvali, Dimopoulos Gabadadze,
hep-th/0209227
where
i.e., gravity acts as high-pass filter
Vacuum energy is longest-wavelength source
gt
Therefore ??is filtered or degravitated
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Key point the filtering eqn
cannot be a consistent theory of massless spin-2!
Puzzle arises even at linearized level
which in de Donder gauge reduces to
Describes massive graviton with only 2
d.o.f.s, which is inconsistent!
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Indeed, only consistent theory of massive
graviton is Fiertz-Pauli
where h?? has 5 propagating degrees of freedom
2 helicity-2 2 helicity-1 1 helicity-0
Resolution is an effective eqn for helicity-2
(Einsteinian) part of metric, once other
helicities have been integrated out.
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EM analogue de-electrification
The analogue of the C.C. problem in EM is to
consider a uniform charge density
(like T?? ?g?? )

(like h00 t2 in de Sitter)

E-field grows unbounded.
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2 possible solns - give A? a mass
- introduce filter
These 2 are equivalent.
Start with Proca
Restore gauge inv. with Goldstone-Stuckelberg
field
with
Thus A? is gauge-invariant (physical observable)
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In terms of helicity-1 and helicity-0 fields
Take divergence of this gives eqn for Goldstone
Substitute for ? in the eom then gives
Hence, adding mass ltgt filtering
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Adding mass solves C.C. problem in EM
with solution
With mass term, vacuum becomes superconducting, an
d uniform sources get screened.
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Degravitation of c.c. is analogous to
screening of uniform charge density in the Higgs
vacuum of Maxwell theory
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Degravitation ltgt Massive or Resonance Graviton
where is the linearized Einstein tensor
Introduce Stuckelberg fields to restore diff inv
with
The physical metric h?? is gauge-inv. gt
observable
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As in spin-1 story, can integrate out the
Stuckelberg A?, and obtain eom for helicity-2
(Einsteinian) part
Any degravitating theory must reduce at linear
level to a theory of massive or resonance
graviton.
Extra degrees of freedom
2 helicity-2 2 helicity-1 1 helicity-0
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C.C. already a problem at linearized level
Soln in de Donder gauge is
Metric grows unbounded, indicating instability
of Minkowski vacuum
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Degravitating C.C. with graviton mass
Has (flat) solution
In massive gravity, metric is a physical
observable and hence cannot grow unbounded.
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Allowed modifications
In deep IR (p -gt 0), assume power-law form

• Clearly need ? lt 1 for it to be an IR modification

• Absence of ghosts requires ? gt 0

Hence, allowed range
Note Massive gravity corresponds to ? 0
DGP model corresponds ? 1/2
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Strong coupling r-effect
helicity-2
helicity-0 (couples to T??)
At linear level, have
helicity-0
helicity-2
(vDVZ discontinuity)
m 0
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Near source, however, non-linear interactions in
? are important

h??

h??

• For h is important when r rsch

h??
?
?

• For ? is important when r r

?
where
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The picture
? (r/r) ?


h 1/r


h, ? 1/r

r
L
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In other words

• For r ltlt r, ??is non-linear and decouples
• have GR small O(r/r) corrections

Testable in LLR ??gt 1/2 ruled out ??lt 1/2
beyond expected accuracy.
Dvali, Gruzinov Zaldarriaga, hep-ph/0212069

• For r gtgt r, ??is linear and goes like 1/r
• have scalar-tensor theory

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Implications of r-effect for degravitation

• For H-1 ltlt L, then universe is within its own r,
• which means ??is suppressed, and thus

(Here h is full, physical metric)

• Once H-1 L, then ??no longer decouples, and
  presumably degravitation stops
• (leaves effective c.c. of order L-2?)

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Degravitation in the decoupling limit
with
fixed
Focuses on the non-linearity of helicity-0 mode.
Ignoring non-linear terms for a moment, soln is
is flat!
full metric
Clearly, non-linear terms leave soln intact if
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Summary thus far

• Absence of ghosts

• Non-linear degrav. (in decoupling limit)

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Realizing degravitation in brane-world
DGP model
z

• Because extra dimension is infinite, no massless
  graviton.
• Rather, graviton is a resonance with width
  L-1 M53/M42 .

• Linearized eom has degravitation form with ?
  1/2

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DGP model (contd)

• However, at non-linear level, degravitation
  fails.

• Israel junction condition fixes (local) Friedmann
  
• eqn on brane

Deffayet, hep-th/0010186
Consistent with earlier conclusion that ? 1/2
is borderline.
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Cod-N DGP All higher-codimension models have ?
0.


Proof. Potential is given by
where ?(s) is spectral density
At large distances, we have
for small s
Hence,

• Limit is finite for Ngt2 gt ? 0.

Q.E.D.

• For N2, get log divergence gt ? 0.

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Codimension-2 DGP
R4
R6
2 Issues

• Brane-to-brane propagator diverges in thin-brane
  limit

• Naïve DGP action

leads to a ghost. (More on this later.)
Gabadadze Shifman, hep-th/0312289 Dubovsky and
Rubakov, hep-th/0212222
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Cascading DGP
C. de Rham, G. Dvali, S. Hofmann, JK, O.
Pujolas, M. Redi A. Tolley, to appear
R5
R6
R4

• Brane-to-brane propagator is completely regular
  even in thin-brane limit

• When carefully treated, model does not suffer
  from ghost-like instabilities

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Brane-to-brane propagator is regular
where

• At long-wavelengths (p-gt0) this reduces to

which is the log(p) or ? 0 behavior of cod-2.

• Compare with pure cod-2 case

finite L6 plays the role of regulator here!
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Ghost stories

• The naïve action

leads to the exchange amplitude
where f goes through zero (i.e. no
positive-definite spectral representation)

• Moreover the trace of the metric still diverges
  on the brane must still regulate the cod-2
  brane.

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Curing the ghost
1. Regulate the brane
2. Add DGP term Natural to include Ricci term
appropriate for dimensionality of regularized
defect, in this case R6
Cf. Gabadadze Shifman, hep-th/0312289
Kolanovic, Porrati Rombouts, hep-th/0304148
3. Take thin-brane limit can do KK reduction
within worldvolume of branes.
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4. R6 terms give rise to scalar-tensor theories
on branes
5. These non-minimal couplings of ? and ? yield
additional contributions to TT part of amplitude
which is ghost-free (-1/4 -1/3 1/12).
6. Furthermore, trace of metric is now regular,
hence thin-brane limit is well-defined.
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Work in progress

• To get correct tensor structure, scalars ? and
  ??must get strongly coupled (r-effect).

Expect 1/4 -gt 1/3 -gt 1/2

• Study the cosmology of cascading DGP
• Is degravitation realized?

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Matryoshka brane-world
RD-1
RD
RD-2
all the way down to R4 on cod-N brane
RD-3

• Propagator is regular has tensor structure of
  graviton in D dimns (gt ghost-free).

• For codimension Ngt2, get behavior of massive
  gravity (? 0) at long wavelengths.

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Put another way our filtering eqn
gives exchange amplitude btwn conserved T T
which recovers GR result as m-gt0.
Whereas the answer in FP gravity is
which displays famous vDVZ discontinuity as m-gt0.