1. Spin-2 ghost in brane gravity
Takahiro Tanaka
Yukwa Institute for Theoretical Physics
Kyoto university
In collaboration with
Keisuke Izumi
Kazuya Koyama.
Oriol Pujolas
Prog.Theor.Phys.121: ,2009 K.I. and T.T.
Prog.Theor.Phys.121: ,2009 K.I. and T.T.
Phys.Rev.D76:104041,2007 K.I., K.K, O.P. and T.T.
JHEP 0704:053,2007 K.I. K.K. and T.T.

2. Dvali-Gabadadze-Porrati model
(Phys. Lett. B485, 208 (2000))
model action
Critical length scale
For r<rc , 4-D induced gravity term dominates?
Extension is infinite, but 4-D GR seems to be recovered for r < rc . ?? y
Brane
Minkowski
Bulk
y=constant

3. Cosmology in DGP model Flat Friedmann equation In early universe ,
cosmic expansion is normal.
Late time behavior for e = +1
figure taken from
Chamousis et.al.
hep-th/
in the limit r → 0
self-acceleration
prototype of alternative to dark energy.

4. Self-acceleration has a Ghost
normal
ghost
Spontaneous pair production of ghost and normal particles
unstable vacuum.
Once a channel opens, Lorentz invariance leads to divergence.
Maybe we need non-Lorentz invariant cutoff.

5. Ghost in self-accelerating branch
in DGP model
A massive graviton in de Sitter space with 0 < m2 < 2H2 contains a spin-2 ghost mode in general (Higuchi Nucl. Phys. B282, 397 (1987))
Spin2 ghost:
helicity decomposition (0, 1, 2)
(scalar, vector, tensor) in cosmological perturbation
This mode becomes a ghost
The mass of the lowest KK graviton in self-accelerating branch
m2 = 2H2 for r = 0, < m2 < 2H2 for r > 0.
Marginally there is a ghost mode in DGP. (Gorbunov, Koyama and Sibiryakov, Phys. Rev. D73, (2006))

6. Bulk is Rindler wedge of
Can we erase the ghost?
Can we erase the ghost simply by putting the second regulator brane in the bulk?
The idea is: if the distance between two branes becomes closer, the KK mass will increase.
m2 > 2H2
The ghost will disappear. t r
y=y+
y=y-
Bulk is Rindler wedge of
Minkowski space

7. self-acceleration H+=1/rc
In fact, there is no ghost in spin-2 sector once the second brane (or negative energy density) is introduced. t
r+ < 0: H+<1/rc
y=y+
y=y- r
self-acceleration H+=1/rc
spin-2 ghost exists
r+ > 0: H+>1/rc
far limit
close limit
However, at the point where the spin 2 ghost disappears, spin 0 (brane bending mode) ghost appears instead.
(K. Izumi, K. Koyama & T.T, 2007)
(single brane: Charmousis et al. 2006)

8. Do we really need to be afraid of spin 2 ghost in de Sitter space?
Here we discuss general massive gravity theory in de Sitter background.
Spontaneous pair production of ghost and normal particles
Vacuum becomes unstable.
particle production rate diverges due to UV contribution, but it is a bit strange that UV behavior is affected by the value of cosmological constant.
In H→0 limit there is no ghost.
If there is a non-covariant cutoff scale, the pair production rate becomes finite. Then, the model might be saved.

9. Strong coupling scale If a mode has a small quadratic kinetic term
may introduce a natural cutoff scale??
If a mode has a small quadratic kinetic term
a → 0 limit strong coupling. x x x x
strong coupling scale = L/a1/2
When spin-2 ghost marginally appears (m2= 2H2),
the model is necessarily in the strong coupling limit???

10. Strong coupling scale –continue-
However, DGP self-acceleration limit is special !
The quadratic kinetic terms for two modes simultaneously vanish.
spin2 ghost mode & spin0 brane bending mode.
Two contributions cancel with each other.
In this situation, can we really say that
the model is in the strong coupling limit?
RS-II model is also strongly coupled in a slightly different sense
4D effective interaction
z – z 2× z –(1/2)×3 =z 3/2

11. Can we justify 3-momentum cutoff?
(K. Izumi & T.T, 2007)
Spin2 ghost:
helicity decomposition (0, 1, 2)
(scalar, vector, tensor) in cosmological perturbation
ghost
Action for spin 2, helicity-0 mode
Action for helicity-0 mode, depending on 3-momentum k, is not covariant,
but spin 2 graviton in total is covariant classical mechanically.
≡background covariance
≡de Sitter invariance

12. However, quantum mechanical state will lose covariance.
Quantization of a ghost)
a ⇒ +ve (normal case)
&
a ⇒ -ve (ghost case)
To make the ground state wave function normalizable,
negative energy states
Quantization which avoids negative norm distinguishes ghost from normal case.

13. s Particle creation in de Sitter space < sk2 > is large
helicity 0 graviton s
⇒ particle production rate will be large.
conformally coupled scalar field
However, coupling to the matter field is also suppressed.
In the massless limit, hmn for helicity 0 mode is pure gauge.

14. After lengthy calculation,
helicity 0 graviton
Although we have to say that this is still a preliminary result, s
conformally coupled scalar field
total energy density of created scalar particles is
L: 3-momentum cutoff scale
Resulting energy density is divergent in L→∞ limit, but
if we set 　　　　⇒
Strong coupling scale L=(M4mn)1/(n+1) is much lower.
If we cut off the theory at this energy scale,
particle creation is extremely suppressed.

15. How is the fate of ghost instability?
(K.I., K.K, O.P. and T.T (2007))
De Sitter spin2 ghost might be milder, but spontaneous particle production is unavoidable.
Then, what is the consequence of ghost instability?
(+)-branch: unstable
(-)-branch: stable
Bubble nucleation like false vacuum decay may happen?

16. Is there such an instanton solution?
instanton connecting (+)-branch with (-)-branch.
Instanton: imaginary time (Euclidean) classical solution
connecting initial and final configurations.
Time derivative on the boundaries is 0.
(+)-branch: unstable
(-)-branch: stable
(-)-branch bubble
in (+)-branch sea

17. It looks possible to construct this type of instanton solution?
initial
final t t

18. The solution in the previous slide is an artifact.
Thick domain wall
However...
Euclidean EOM:
Bulk is Minkowski
by Birkov’s theorem
Friedmann eq.
in closed chart
scalar field eq. t
a is the radius from the axis.
conflicting
axis
The solution in the previous slide is an artifact.

19. Summary DGP model in self-acceleration branch has a ghost.
Minor modification of the model does not erase the ghost.
brane tension, two branes, brane stabilization by a bulk scalar.
Models with a ghost may survive, if we introduce 3-momentum cutoff
at the strong coupling scale.
3-momentum cutoff is not compatible with de Sitter invariance.
However, de Sitter invariance cannot be maintained in the situation where spin-2 ghost appears.
spin 2 ghost: Quantization of helicity 0 sector should be different from the others.
Instanton solution is found in the thin wall limit,
but such a solution turns out to be an artifact.
One can prove the absence of instanton under the Euclidean version of null energy condition.

20. mass spectrum in DGP brane world
tensionless brane
r = 0
positive tension brane
r > 0
normal
(-)-branch
self-accelerating
(+)-branch
discrete spectrum
m2 = 0
discrete spectrum
m2 = 0
continuum spectrum
m2 > 0
continuum spectrum
m2 > 9H2/4
discrete spectrum
m2 = 2H2
discrete spectrum
0 < m2 < 2H2
continuum spectrum
m2 > 9H2/4
brane bending mode
m2 = 2H2
massless graviton mode
normalizable only for compactified bulk
brane bending mode
pure gauge for compactified bulk

21. Transition rate Tunneling rate Bubbles with are suppressed.
Small bubbles are not suppressed?
Strong coupling scale
is much larger than this critical bubble size.

22. However, DGP self-acceleration limit is special !
Once the source of gravity Tmn is introduced,
perturbation will blow up.
However, DGP self-acceleration limit is special !
The quadratic kinetic terms for two modes simultaneously vanish.
spin2 ghost mode & spin0 brane bending mode.
Two contributions cancel with each other.
In this situation, can we really say that
the model is in the strong coupling limit?
RSII is also strongly coupled in a slightly different sense
4D effective interaction
z – z 2× z –(1/2)×3 =z 3/2

23. Then, when ghost is in Spin 0 sector, is de Sitter invariance maintained?
It is known that a scalar field with m2 < 0 does not have de Sitter invariant vacuum.
de Sitter invariant +ve frequency function
u(h) is Klein-Gordon normalized:
u(h) ~ real
Al diverges. l=m-3/2-j j=0,1,2,・・・
At this point wave fn becomes unnormalizable.
De Sitter invariant vacuum disappears.
Special cases
m2=0 : l = -j
l = 0 mode is special.
∃Kirsten-Garriga de Sitter invariant vacuum
l = 0 mode = shift of the origin of f.
m2=-4H2 : l = 1-j
l = 0,1 modes are special.
Violation of de Sitter invariance is subtle.
(Vachaspati-Vilenkin (1991))
l = 0,1 modes = translation = gauge in single brane case

24. Massive graviton + conformally coupled scalar field
Simple toy model
Massive graviton + conformally coupled scalar field
Action for spin 2, helicity-0 mode
For mi2 < 2H2, the signature of the action flips.
For large k, < sk2 > becomes large ⇒ strong coupling
Conformally coupled scalar field
Interaction term

25. Strong coupling scale Small quadratic kinetic term
a → 0 limit strong coupling. x x x x
strong coupling scale = L/a1/2
When spin-2 ghost marginally appears (m2= 2H2),
the model is in the strong coupling limit???
Once the source of gravity Tmn is introduced,
perturbation will blow up.

26. We can make an instanton by introducing a positive tension domain wall on the braneq1
H1-1 q2