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.121:427-436,2009 K.I. and T.T. Prog.Theor.Phys.121:419-426,2009 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.

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

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/0604086 in the limit r → 0 self-acceleration prototype of alternative to dark energy.

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.

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 1 2 2 This mode becomes a ghost The mass of the lowest KK graviton in self-accelerating branch m2 = 2H2 for r = 0, 0 < m2 < 2H2 for r > 0. Marginally there is a ghost mode in DGP. (Gorbunov, Koyama and Sibiryakov, Phys. Rev. D73, 044016 (2006))

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

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)

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.

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???

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 –5 z 2×4 z –(1/2)×3 =z 3/2

Can we justify 3-momentum cutoff? (K. Izumi & T.T, 2007) Spin2 ghost: helicity decomposition (0, 1, 2) (scalar, vector, tensor) in cosmological perturbation 1 2 2 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

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.

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.

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.

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?

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

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

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.

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.

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

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

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 –5 z 2×4 z –(1/2)×3 =z 3/2

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

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

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.

We can make an instanton by introducing a positive tension domain wall on the brane q1 H1-1 q2