Nemanja Kaloper, UC Davis HAUNTINGS Nemanja Kaloper UC Davis This presentation will probably involve...

Nemanja Kaloper, UC DavisNemanja Kaloper, UC Davis

HAUNTINGSHAUNTINGSNemanja KaloperNemanja Kaloper

UC DavisUC Davis

Based on: C. Charmousis, R. Gregory, A. Padilla, hep-th/0604086; and work in preparation with D. Kiley.


OverviewOverview Who cares?Who cares? Chasing ghosts in DGPChasing ghosts in DGP

Codimension-1 caseCodimension-1 case Specteral analysis: diagnosticsSpecteral analysis: diagnostics Shock therapyShock therapy

Shocking codimension-2 Shocking codimension-2 Gravity of photons = electrostatics Gravity of photons = electrostatics on coneson cones

Gravitational See-SawGravitational See-Saw SummarySummary


We DO NOT understand 95% of the We DO NOT understand 95% of the contents of our Universe!contents of our Universe!

Splitting the cosmic Splitting the cosmic piepie


The Concert of Cosmos?The Concert of Cosmos?

Einstein’s GR: a beautiful theoretical Einstein’s GR: a beautiful theoretical framework for gravity and cosmology, framework for gravity and cosmology, consistent with numerous experiments and consistent with numerous experiments and observations:observations: Solar system tests of GR Solar system tests of GR

Sub-millimeter (non)deviations from Newton’s law Sub-millimeter (non)deviations from Newton’s law Concordance Cosmology! Concordance Cosmology!

How well do we How well do we REALLYREALLY know gravity? know gravity? Hands-on observational tests confirm GR at scales Hands-on observational tests confirm GR at scales between roughly between roughly 0.1 mm0.1 mm and - say - about and - say - about 100 MPc; 100 MPc; are are wewe certain certain that GR remains valid at that GR remains valid at shortershorter and and longerlonger distances? distances?

New tests?New tests?

Or, Dark Discords?Or, Dark Discords?

New tests?New tests?



Cosmological constant Cosmological constant failurefailure The situation with the cosmological constant is The situation with the cosmological constant is

desperatedesperate (by at least 60 orders of magnitude!) (by at least 60 orders of magnitude!) desperate measures required?desperate measures required?

Might changing gravity help? A (very!) heuristic Might changing gravity help? A (very!) heuristic argument:argument: Legendre transformsLegendre transforms:: adding adding ∫ dx∫ dx (x) J(x)(x) J(x) to to SS trades an independent variable trades an independent variable for another for another independent variable independent variable JJ..

Reconstruction of Reconstruction of from from W(J)W(J) yields a yields a family family of of effective actionseffective actions parameterized byparameterized by an arbitraryan arbitrary JJ; ; J=0J=0 is put in by hand! is put in by hand!

∫ ∫ dxdx √√det(g) det(g) isis a Legendre transform. a Legendre transform. In GR, general covariance In GR, general covariance det(gdet(g)) does not does not propagate! propagate!

So the Legendre transform So the Legendre transform ∫ dx∫ dx √√det(g) det(g) ‘loses’ ‘loses’ information about information about only ONE only ONE IR parameter - IR parameter - . . Thus Thus is not calculable, but is an input!is not calculable, but is an input!

Could changing gravity alter this, circumventing no-Could changing gravity alter this, circumventing no-go theorems?…go theorems?…


HeadachesHeadaches

Changing gravity Changing gravity →→ adding new DOFs in the IR adding new DOFs in the IR They can be problematic:They can be problematic:

Too light and too strongly coupled Too light and too strongly coupled →→ new long range new long range forces forces

Observations place bounds on these!Observations place bounds on these! Negative mass squaredNegative mass squared or negative residue of the pole in or negative residue of the pole in the propagator for the new DOFs: the propagator for the new DOFs: tachyonstachyons and/orand/or ghostsghosts

Instabilities can render the theory nonsensical!Instabilities can render the theory nonsensical!

Caveat emptor: this need not be a theory killer; it means Caveat emptor: this need not be a theory killer; it means that a naive perturbative description about some background that a naive perturbative description about some background is very bad. BUT: one is very bad. BUT: one *must**must* develop a meaningful develop a meaningful perturbative regime before surveying phenomenological perturbative regime before surveying phenomenological issues and applications.issues and applications.


Shock box

Modified Gravity


DGP BraneworldsDGP Braneworlds

Brane-induced gravityBrane-induced gravity (Dvali, Gabadadze, (Dvali, Gabadadze,

Porrati, 2000)Porrati, 2000):: Ricci terms BOTH in the bulk and on the Ricci terms BOTH in the bulk and on the end-of-the-world brane, arising from e.g. end-of-the-world brane, arising from e.g. wave function renormalization of the wave function renormalization of the graviton by brane loopsgraviton by brane loops

May appear in string theory May appear in string theory (Kiritsis, (Kiritsis, Tetradis, Tomaras, 2001; Corley, Lowe, Ramgoolam, 2001)Tetradis, Tomaras, 2001; Corley, Lowe, Ramgoolam, 2001)

Related works on exploration of brane-Related works on exploration of brane-localized radiative corrections localized radiative corrections (Collins, (Collins, Holdom, 2000) Holdom, 2000)


Codimension-1 Codimension-1 Action: for the case of codimension-1 Action: for the case of codimension-1 brane,brane,

Assume ∞ bulk: 4D gravity has to be Assume ∞ bulk: 4D gravity has to be mimicked by the exchange of bulk DOFs!mimicked by the exchange of bulk DOFs!

55thth dimension is concealed by the brane dimension is concealed by the brane curvature enforcing momentum transfer curvature enforcing momentum transfer 1/p1/p22 for for p > 1/rp > 1/rc c (DGP, 2000; Dvali, Gabadadze, (DGP, 2000; Dvali, Gabadadze, 2000):2000):


Strong coupling caveatsStrong coupling caveats In massive gravity, naïve linear perturbation theory in massive In massive gravity, naïve linear perturbation theory in massive

gravity on a flat space breaks down gravity on a flat space breaks down →→ idea: nonlinearities improve idea: nonlinearities improve the theory and yield continuous limitthe theory and yield continuous limit (Vainshtein, 1972)(Vainshtein, 1972)??

There are examples without IvDVZ discontinuity in curved There are examples without IvDVZ discontinuity in curved backgroundsbackgrounds (Kogan (Kogan et alet al; Karch ; Karch et alet al; Porrati; 2000); Porrati; 2000). (dS with a . (dS with a rockrock of of salt!)salt!)

Key: the scalar graviton is strongly coupled at a scale much bigger Key: the scalar graviton is strongly coupled at a scale much bigger than the gravitational radiusthan the gravitational radius (a long list of people… sorry, y’all!)(a long list of people… sorry, y’all!)..

In DGP a naïve expansion around flat space also breaks down at In DGP a naïve expansion around flat space also breaks down at macroscopic scalesmacroscopic scales (Deffayet, Dvali, Gabadadze, Vainshtein, 2001; Luty, (Deffayet, Dvali, Gabadadze, Vainshtein, 2001; Luty, Porrati, Rattazi, 2003; Rubakov, 2003).Porrati, Rattazi, 2003; Rubakov, 2003). Including curvature may push it Including curvature may push it down to about down to about ~~ 1 cm1 cm ( (Rattazi & Nicolis, 2004Rattazi & Nicolis, 2004). ).

LPR also claim a ghost in the scalar sector on the self-LPR also claim a ghost in the scalar sector on the self-accelerating branch; after some vacillation, people seem to - accelerating branch; after some vacillation, people seem to - mostly - agreemostly - agree (Koyama, 2005; Gorbunov, Koyama, Sibiryakov, 2005; (Koyama, 2005; Gorbunov, Koyama, Sibiryakov, 2005; Charmousis, Gregory, NK,, Padilla, 2006; Izumi, Koyama, Tanaka, Charmousis, Gregory, NK,, Padilla, 2006; Izumi, Koyama, Tanaka, 2006; Carena, Lykken, Park, Santiago, 2006 (two days ago); attempt 2006; Carena, Lykken, Park, Santiago, 2006 (two days ago); attempt to remove it by weird boundary conditions, by Deffayet, Gabadadze, to remove it by weird boundary conditions, by Deffayet, Gabadadze, Iglesias, 2006, fails to convince this speaker; ghost after all Iglesias, 2006, fails to convince this speaker; ghost after all means that the system leaks to infinity, so finding that the means that the system leaks to infinity, so finding that the system is sensitive to what happens faraway is an indicator of system is sensitive to what happens faraway is an indicator of occult phenomena)occult phenomena)


Difficulty: equations are hard, perturbative Difficulty: equations are hard, perturbative treatments of treatments of bothboth background and interactions background and interactions subtle... Can we be more precise?subtle... Can we be more precise?

An attempt: construct realistic backgrounds; solveAn attempt: construct realistic backgrounds; solve

Look at the vacua first: Look at the vacua first:

Symmetries requireSymmetries require (see e.g. N.K, A. Linde, 1998)(see e.g. N.K, A. Linde, 1998)::

where 4d metric is de Sitter.where 4d metric is de Sitter.

Perturbing cosmological Perturbing cosmological vacuavacua


Codimension-1 vacua Codimension-1 vacua


The intrinsic curvature and the tension related byThe intrinsic curvature and the tension related by ((N.K.; Deffayet,2000N.K.; Deffayet,2000))

= = ±1 ±1 an integration constant; an integration constant; = - = -1 1 normal normal branch,branch,

i.e. this reduces to the usual inflating brane i.e. this reduces to the usual inflating brane

in 5D!in 5D! ==1 1 self-inflating branch, inflates even if self-inflating branch, inflates even if

tension vanishes!tension vanishes!

Normal and self-inflating Normal and self-inflating branchesbranches


SpecteroscopySpecteroscopy Logic: start with the cosmological vacua and perturb the Logic: start with the cosmological vacua and perturb the

bulk & brane system, allowing for brane matter as well; bulk & brane system, allowing for brane matter as well; gravity sector isgravity sector is

But, there are still unbroken gauge invariances of the But, there are still unbroken gauge invariances of the bulk+brane system! Not all modes are physical.bulk+brane system! Not all modes are physical.

The analysis here is The analysis here is linearlinear - think of it as a - think of it as a diagnostic tool. But: it reflects problems with diagnostic tool. But: it reflects problems with perturbations at lengths > Vainshtein scale. perturbations at lengths > Vainshtein scale.


Gauge symmetry IGauge symmetry I Infinitesimal transformationsInfinitesimal transformations

The perturbations change asThe perturbations change as

Set e.g. and to zero; that leaves Set e.g. and to zero; that leaves us with and us with and


Gauge symmetry IIGauge symmetry II Decomposition theorem Decomposition theorem (see CGKP, 2006)(see CGKP, 2006) : :

Not all need be propagating modes!Not all need be propagating modes! To linear order, vectors decouple by gauge To linear order, vectors decouple by gauge

symmetry, and the only modes responding to brane symmetry, and the only modes responding to brane matter are TT-tensors and scalars.matter are TT-tensors and scalars.

Write down the TT-tensor and scalar Lagrangian.Write down the TT-tensor and scalar Lagrangian.


Gauge symmetry IIIGauge symmetry III Note: there still remain residual gauge Note: there still remain residual gauge

transformationstransformations

under whichunder which

so we can go to a brane-fixed gauge so we can go to a brane-fixed gauge F’=0 F’=0 andand


ForkingForking Direct substitution into field equations yields Direct substitution into field equations yields

the spectrum; use mode decompositionthe spectrum; use mode decomposition

Get the bulk eigenvalue problemGet the bulk eigenvalue problem

A constant potential with an attractive A constant potential with an attractive --function function well.well.

This is self-adjoint with respect to the normThis is self-adjoint with respect to the norm


Brane-localized modes: Brane-localized modes: TensorsTensors Gapped continuum:Gapped continuum:

Bound state:Bound state:


Bound state specificsBound state specifics On the normal branch, On the normal branch, =-1=-1, the bound state is , the bound state is

massless! This is the normalizable graviton zero mode, massless! This is the normalizable graviton zero mode, arising because the bulk volume ends on a horizon, a arising because the bulk volume ends on a horizon, a finite distance away. It has additional residual gauge finite distance away. It has additional residual gauge invariances, and so only 2 propagating modes, with invariances, and so only 2 propagating modes, with matter couplings matter couplings g ~ Hg ~ H. It decouples on a flat brane.. It decouples on a flat brane.

On the self-accelerating branch, On the self-accelerating branch, =1=1, the bound state , the bound state mass is not zero! Instead, it has Pauli-Fierz mass mass is not zero! Instead, it has Pauli-Fierz mass term and 5 components,term and 5 components,

Perturbative Perturbative ghostghost: : mm22<2H<2H22, , helicity-0 component has helicity-0 component has negative kinetic term negative kinetic term (Deser, Nepomechie, 1983; Higuchi, 1987; (Deser, Nepomechie, 1983; Higuchi, 1987; I. Bengtsson, 1994; Deser, Waldron 2001).I. Bengtsson, 1994; Deser, Waldron 2001).


Brane-localized modes: Brane-localized modes: ScalarsScalars Single mode, with Single mode, with mm2 2 = 2H= 2H22, obeying, obeying

with the brane boundary conditionwith the brane boundary condition

Subtlety: interplay between normalizability, brane Subtlety: interplay between normalizability, brane dynamics and gauge invariance. On the normal dynamics and gauge invariance. On the normal branch, the normalizable scalar can always be branch, the normalizable scalar can always be gauged away by residual gauge transformations; not gauged away by residual gauge transformations; not so on the self-accelerating branch. There one so on the self-accelerating branch. There one combination survives:combination survives:


Full perturbative Full perturbative solutionsolution

Full perturbative solution of the problem isFull perturbative solution of the problem is

On the normal branch, this solution has no scalar On the normal branch, this solution has no scalar contribution, and the bound state tensor is a zero contribution, and the bound state tensor is a zero mode. Hence there are no ghosts.mode. Hence there are no ghosts.

On the self-accelerating branch, the bound state On the self-accelerating branch, the bound state is massive, and when is massive, and when its helicity-0 mode is a its helicity-0 mode is a ghost; for ghost; for , the surviving scalar is a ghost , the surviving scalar is a ghost (its kinetic term is proportional to (its kinetic term is proportional to ).).

Zero tension is tricky.Zero tension is tricky.


Zeroing inZeroing in Zero tension corresponds to Zero tension corresponds to mm22 = 2H = 2H22 on SA branch. The on SA branch. The

lightest tensor and the scalar become completely lightest tensor and the scalar become completely degenerate. In Pauli-Fierz theory, there is an degenerate. In Pauli-Fierz theory, there is an accidental symmetry accidental symmetry (Deser, Nepomechie, 1983)(Deser, Nepomechie, 1983)

so that helicity-0 is pure gauge, and so it decouples so that helicity-0 is pure gauge, and so it decouples – ghost gone! – ghost gone!

With brane present, this symmetry is spontaneously With brane present, this symmetry is spontaneously broken! The brane Goldstone mode becomes the broken! The brane Goldstone mode becomes the Stuckelberg-like field, and as long as we demand Stuckelberg-like field, and as long as we demand normalizability the symmetry lifts to normalizability the symmetry lifts to

We can’t gauge away both helicity-0 and the scalar; the We can’t gauge away both helicity-0 and the scalar; the one which remains is a ghost one which remains is a ghost (see also Dubovsky, Koyama, (see also Dubovsky, Koyama, Sibiryakov, 2005).Sibiryakov, 2005).


(d)Effective action II(d)Effective action II By focusing on the helicity zero mode, we By focusing on the helicity zero mode, we can check that in the unitary gauge can check that in the unitary gauge (see (see

Deser, Waldron, 2001; CGKP, 2006)Deser, Waldron, 2001; CGKP, 2006) its Hamiltonian its Hamiltonian isis

where , and where , and therefore this mode is a ghost when therefore this mode is a ghost when mm22 < < 2H2H22; by mixing with the brane bending it ; by mixing with the brane bending it does not decouple even when does not decouple even when mm22 = 2H = 2H2 2 . .

In the action, the surviving combination In the action, the surviving combination isis


Shocking nonlocalitiesShocking nonlocalities What does this ghost imply? In the Lagrangian in What does this ghost imply? In the Lagrangian in

the bulk, there is no explicit negative norm the bulk, there is no explicit negative norm states; the ghost comes about from brane boundary states; the ghost comes about from brane boundary conditions - brane does not want to stay put.conditions - brane does not want to stay put.

Can it move and/or interact with the bulk and Can it move and/or interact with the bulk and eliminate the ghost? eliminate the ghost?

In shock wave analysis In shock wave analysis (NK, 2005) (NK, 2005) one finds a one finds a singularity in the gravitational wave field of a singularity in the gravitational wave field of a massless brane particle in the localized solution. massless brane particle in the localized solution. It can be smoothed out with a non-integrable mode.It can be smoothed out with a non-integrable mode.

But: this mode But: this mode GROWSGROWS far from the brane – it lives far from the brane – it lives at asymptotic infinity, and is sensitive to the at asymptotic infinity, and is sensitive to the boundary conditions there.boundary conditions there.

Can we say anything about what goes on there? Can we say anything about what goes on there?


Trick: shock wavesTrick: shock waves

Physically: because of the Physically: because of the Lorentz contraction in the Lorentz contraction in the direction of motion, the direction of motion, the field lines get pushed field lines get pushed towards the instantaneous towards the instantaneous plane which is orthogonal plane which is orthogonal to to V.V.

The field lines of a The field lines of a massless charge are massless charge are confined to this plane! confined to this plane! (P.G (P.G Bergmann, 1940’s)Bergmann, 1940’s)

The same intuition works The same intuition works for the gravitational for the gravitational field. field. (Pirani; Penrose; Dray, ‘t (Pirani; Penrose; Dray, ‘t Hooft; Ferrari, Pendenza, Hooft; Ferrari, Pendenza, Veneziano; Sfetsos; NK; …)Veneziano; Sfetsos; NK; …)


In flat 4D environment, the exact gravitational field of a In flat 4D environment, the exact gravitational field of a photon found by boosting linearized Schwarzschild metric photon found by boosting linearized Schwarzschild metric ((Aichelburg, Sexl, 1971).Aichelburg, Sexl, 1971).

Here Here u,v = (x ±t)/√2 u,v = (x ±t)/√2 are null coordinates of the photon.are null coordinates of the photon. For a particle with a momentum For a particle with a momentum p p ,, f f is, up to a constant is, up to a constant

where where RR = (y = (y22 + z + z22))1/21/2 is the transverse distance and is the transverse distance and

ll00 an arbitrary integration parameter. an arbitrary integration parameter. This will be our template…This will be our template…

4D4D: Aichelburg-Sexl : Aichelburg-Sexl shockwaveshockwave


The starting point for ‘shocked’ DGP is The starting point for ‘shocked’ DGP is (NK, 2005 )(NK, 2005 )

Term Term ~ ~ ff is the discontinuity in is the discontinuity in ddvv . Substitute . Substitute this metric in the DGP field equations, where the this metric in the DGP field equations, where the new brane stress energy tensor includes photon new brane stress energy tensor includes photon momentummomentum

Turn the crank!Turn the crank!

DGP in a state of shockDGP in a state of shock


Chasing shocksChasing shocks Best to work with two ‘antipodal’ photons, that Best to work with two ‘antipodal’ photons, that

zip along the past horizon (ie boundary of future zip along the past horizon (ie boundary of future light cone) in opposite directions. This avoids light cone) in opposite directions. This avoids problems with spurious singularities on compact problems with spurious singularities on compact spaces. It is also the correct infinite boost spaces. It is also the correct infinite boost limit of Schwarzschild-dS solution in 4Dlimit of Schwarzschild-dS solution in 4D (Hotta, (Hotta,

Tanaka, 1993) Tanaka, 1993) . The field equation is. The field equation is (NK, 2005)(NK, 2005)


““Antipodal’’ photons in Antipodal’’ photons in the staticthe static

patch on de Sitter patch on de Sitter branebrane


Thanks to the symmetries of the problem, we Thanks to the symmetries of the problem, we can solve the equations by mode expansion:can solve the equations by mode expansion:

where the radial wavefunctions are where the radial wavefunctions are

Here is normalizable: it describes Here is normalizable: it describes gravitons localized on the brane. The mode gravitons localized on the brane. The mode is not normalizable. Its amplitude is not normalizable. Its amplitude diverges at infinity. This mode lives far diverges at infinity. This mode lives far from the brane, and is sensitive to boundary from the brane, and is sensitive to boundary conditions conditions therethere. .

Shocking solutions IShocking solutions I


Shocking solutions IIShocking solutions II Defining , Defining ,

using the spherical harmonic addition theorem, using the spherical harmonic addition theorem,

and changing normalization to and changing normalization to we can finally write the solution we can finally write the solution down as:down as:

The parameter controls the contribution from the The parameter controls the contribution from the nonintegrable modes. This is like choosing the vacuum of nonintegrable modes. This is like choosing the vacuum of a QFT in curved space. a QFT in curved space.

At short distances: the solution is well approximated by At short distances: the solution is well approximated by the Aichelburg-Sexl 4D shockwave - so the theory the Aichelburg-Sexl 4D shockwave - so the theory doesdoes look 4D!look 4D!

But at large distances, one finds that low-l (large But at large distances, one finds that low-l (large wavelength) are wavelength) are repulsive - repulsive - they resemble ghosts, from they resemble ghosts, from 4D point of view.4D point of view.


More on shocks…More on shocks… For integer For integer g g there are poles similar to the pole there are poles similar to the pole

encountered on the SA branch in the tensionless limit encountered on the SA branch in the tensionless limit g=1g=1 for the lightest brane mode. for the lightest brane mode.

This suggests that the general problem has more resonant This suggests that the general problem has more resonant channnels for energy losses into the bulk, once the door is channnels for energy losses into the bulk, once the door is opened to non-integrable modes. opened to non-integrable modes.

Once a single non-integrable mode is allowed, one cannot Once a single non-integrable mode is allowed, one cannot stop all of them from coming in without breaking bulk stop all of them from coming in without breaking bulk general covariance.general covariance.

In contrast, normal branch solutions are In contrast, normal branch solutions are completely well behaved. One may be able to use completely well behaved. One may be able to use them as a benchmark for looking for cosmological them as a benchmark for looking for cosmological signatures of modified gravity. Once a small signatures of modified gravity. Once a small cosmological term is put in by hand, cosmological term is put in by hand,

it simulates it simulates w<-1w<-1 (Sahni, Shtanov, 2002; Lue, Starkman, 2004) (Sahni, Shtanov, 2002; Lue, Starkman, 2004)

it changes cosmological structure formationit changes cosmological structure formation


Codimension-2 DGPCodimension-2 DGP Higher codimension models are different. A lump of energy Higher codimension models are different. A lump of energy

of codimension greater than unity gravitates. This lends of codimension greater than unity gravitates. This lends to gravitational short distance singularities which must to gravitational short distance singularities which must be regulated. be regulated.

The DGP gravitational filter may still work, confining The DGP gravitational filter may still work, confining gravity to the defect. However the crossover from 4D to gravity to the defect. However the crossover from 4D to higher-D depends on the short distance cutoff. higher-D depends on the short distance cutoff. (Dvali, (Dvali, Gabadadze, Hou, Sefusatti, 2001)Gabadadze, Hou, Sefusatti, 2001)

There were concerns about ghosts, and/or nonlocal There were concerns about ghosts, and/or nonlocal effects. effects. (Dubovsky, Rubakov; Kolanovic, Porrati, Rombouts; Gabadadze, (Dubovsky, Rubakov; Kolanovic, Porrati, Rombouts; Gabadadze, Veinshtein)Veinshtein)

We find a very precise and simple description of the cod-We find a very precise and simple description of the cod-2 case. The shocks show both the short distance 2 case. The shocks show both the short distance singularities and see-saw of the cross-over scale by the singularities and see-saw of the cross-over scale by the UV cutoff. UV cutoff. (NK, D. Kiley, in preparation)(NK, D. Kiley, in preparation)

We suspect: no ghosts (very preliminary - but we almost We suspect: no ghosts (very preliminary - but we almost have the proof)! HOWEVER: There are light gravitationally have the proof)! HOWEVER: There are light gravitationally coupled modes so that the theory is coupled modes so that the theory is Brans-DickeBrans-Dicke below the below the crossover. Can the BD field be stabilized? crossover. Can the BD field be stabilized?


Unresolved codimension-Unresolved codimension-22 Look for the vacua; field equations:Look for the vacua; field equations:

Select 4D Minkowski vacuum x 2D cone:Select 4D Minkowski vacuum x 2D cone:

This is IDENTICAL to the codimension-2 flat brane in This is IDENTICAL to the codimension-2 flat brane in conventional conventional 6D6D gravity. (Sundrum, 1998) gravity. (Sundrum, 1998)

HereHere b b measures deficit angle: far from the core, measures deficit angle: far from the core, gg ~ ~ BB22 22 d d22, ,

The tension (The tension (a.k.a. brane-localized vacuum energya.k.a. brane-localized vacuum energy) ) dumped in the bulk! dumped in the bulk!

But to have static solution, one NEEDS But to have static solution, one NEEDS B>0 B>0 ! And, one ! And, one must have appearance of 4D to Hubble scales… How is must have appearance of 4D to Hubble scales… How is rrcc ~H~H00

-1-1 generated from generated from MM66 ≥ TeV ≥ TeV, and , and MM44 ~ 10 ~ 101919 GeV GeV ? ?


Unresolved vacuumUnresolved vacuum A conical singularity in two infinite extra dimensions:


Gravity on unresolved Gravity on unresolved conecone Put a photon on the brane:Put a photon on the brane:

Field equation, using Field equation, using l l = M= M44/M/M6622: conical : conical

electrostatics!electrostatics!

““Solution”: note that Solution”: note that D(k, D(k, ) ~ I(0) K(k) ~ I(0) K(k))

where where rr is the longitudinal and is the longitudinal and transverse distance. transverse distance. Both Both II and and KK are divergent at small argument; but on the are divergent at small argument; but on the brane brane ((=0)=0) divergences cancel, and for divergences cancel, and for r < r < l l /(1-b)/(1-b) one one finds the leading behavior of 4D A-S shockwave!finds the leading behavior of 4D A-S shockwave!

However for any However for any ‡ 0 ‡ 0 the divergence in the denominator the divergence in the denominator fixes fixes f=0 f=0 - very singular! We must regulate this…- very singular! We must regulate this…


Resolving the coneResolving the coneAn example of an ill-defined exterior boundary value An example of an ill-defined exterior boundary value problem in electrostatics! Resolution: replace the point problem in electrostatics! Resolution: replace the point charge with a ring source and solve by imposing regular charge with a ring source and solve by imposing regular boundary conditions in and out! This can be done by taking boundary conditions in and out! This can be done by taking a 4-brane with a massless scalar and wrapping it on a a 4-brane with a massless scalar and wrapping it on a circle of a fixed radius circle of a fixed radius rr00..


Resolved vacuumResolved vacuum Replace the 3-brane with a 4-brane and wrap it on a Replace the 3-brane with a 4-brane and wrap it on a

cylinder! To do this, put in the matter action an àxion’ cylinder! To do this, put in the matter action an àxion’ , so the vacuum action is, so the vacuum action is

Look for Look for 4D4D Minkowski vacuum x Minkowski vacuum x 2D2D truncated cone, with truncated cone, with qq; with one tuning condition, . ; with one tuning condition, . Then we can dimensionally reduce on the angle, viewing Then we can dimensionally reduce on the angle, viewing the matter as the matter as 4D4D with a with a KKKK tower of states moving around tower of states moving around the cylinder, with a mass gap ~ (the cylinder, with a mass gap ~ (r r Then we can think Then we can think of the cylindrical brane as a thin 3-brane at large of the cylindrical brane as a thin 3-brane at large distances, with the effective distances, with the effective 4D4D tension tension . The solution is precisely the conical `mesa’, with the . The solution is precisely the conical `mesa’, with the metric metric

The only difference that b is twice as big as for a naïve The only difference that b is twice as big as for a naïve thin brane, due to the àxion’ contribution: thin brane, due to the àxion’ contribution:


Shocking resolved Shocking resolved vacuumvacuum Now, put a photon (a massless ring) on the brane, a la Now, put a photon (a massless ring) on the brane, a la

Dray-’t HooftDray-’t Hooft

Field equation, using Field equation, using l l = M= M44/M/M662 2 and and R=R=+br+br00/(1-b),/(1-b), with with

rr00 the 4-brane radius: the 4-brane radius:

Solution!Solution!

everywhere regular! Explicitly taking the limits, at everywhere regular! Explicitly taking the limits, at distances distances r < rr < rcc one finds the 4D Aichelburg-Sexl shock one finds the 4D Aichelburg-Sexl shock wave! At wave! At r > rr > rcc changes to 6D changes to 6D (of Ferrari, Pendenza,Veneziano, (of Ferrari, Pendenza,Veneziano, 1988).1988).


See-SawSee-Saw Thus this theory simulates Thus this theory simulates 4D4D gravity at least in the gravity at least in the

helicity-2 sector, at distances shorter than the cross-helicity-2 sector, at distances shorter than the cross-over scale. over scale.

The cross-over scale The cross-over scale rrcc is not the naïve ratio of is not the naïve ratio of MM44 and and MM66, but it depends on the short ditsnace regularization , but it depends on the short ditsnace regularization scale scale rr0 0 . It is exactly the see-saw scale of DGHS:. It is exactly the see-saw scale of DGHS:

But now it is easy to see why. Recall that we really But now it is easy to see why. Recall that we really have brane graviton term on a 4-brane, with the brane have brane graviton term on a 4-brane, with the brane Planck scale Planck scale MM55. But then brane is wrapped on a cylinder . But then brane is wrapped on a cylinder truncation of the action to only the zero modes truncation of the action to only the zero modes yields effective graviton term on a 3-brane, with the yields effective graviton term on a 3-brane, with the normalisation given bynormalisation given by

Substituting in the cross-over scale formula, we find Substituting in the cross-over scale formula, we find exactlyexactly the codimension-1 result, but for the codimension-1 result, but for 5D5D : :


SummarySummary The keystone: The keystone: gravitational filtergravitational filter - hides the - hides the extra dimension. But: longitudinal scalar is extra dimension. But: longitudinal scalar is trickytricky!!

On SA brane, the localized mode is a perturbative On SA brane, the localized mode is a perturbative ghost. Cosmology with it running loose is ghost. Cosmology with it running loose is unreliable. unreliable.

What does the ghost do? What does the ghost do? Can it catalyze transition from SA to normal branch?Can it catalyze transition from SA to normal branch? Can it `condense’?Can it `condense’? What do strong couplings do? At short scales? At long What do strong couplings do? At short scales? At long scales?…scales?…

Cod-2: the simple wrapped 4-brane resolution Cod-2: the simple wrapped 4-brane resolution looks ghost-free. But: the tuning of the axion looks ghost-free. But: the tuning of the axion generates a “multiverse” of vacua. Can those generates a “multiverse” of vacua. Can those contain long deep “gulches” insensitive to the SM contain long deep “gulches” insensitive to the SM contributions?contributions?

More work: we may reveal interesting new realms More work: we may reveal interesting new realms of gravity! of gravity!

Date post:	18-Dec-2015
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Nemanja Kaloper, UC Davis HAUNTINGS Nemanja Kaloper UC Davis This presentation will probably involve...

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