Download - Nemanja Kaloper, UC Davis SHOCK THERAPY Nemanja Kaloper UC Davis This presentation will probably involve audience discussion, which will create action.

Nemanja Kaloper, UC Davis

SHOCK SHOCK THERAPYTHERAPY

Nemanja KaloperNemanja Kaloper

UC DavisUC Davis


Shock box

Modified Gravity


OverviewOverview

Two messages:Two messages: Changing gravity: Changing gravity: Why BotherWhy Bother?? Exploring modified gravity: Exploring modified gravity: ShocksShocks

DGP: a toy arenaDGP: a toy arena DGP in shockDGP in shock Future directionsFuture directions SummarySummary


The Concert of CosmosThe Concert of Cosmos Einstein’s GR: a beautiful theoretical framework Einstein’s GR: a beautiful theoretical framework

for gravity and cosmology, consistent with for gravity and cosmology, consistent with numerous experiments and observations:numerous experiments and observations: Solar system tests of GRSolar system tests of GR Sub-millimeter (non) deviations from Newton’s lawSub-millimeter (non) deviations from Newton’s law Principal cornerstone of Concordance Cosmology!Principal cornerstone of 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; why are we then why are we then so certainso certain that the extrapolation of that the extrapolation of GR to GR to shortershorter and and longerlonger distances is justified? distances is justified?


The Concert of Cosmos?The Concert of Cosmos? Einstein’s GR: a beautiful theoretical framework for Einstein’s GR: a beautiful theoretical framework for

gravity and cosmology, consistent with numerous gravity and cosmology, consistent with numerous experiments and observations:experiments and observations: Solar system tests of GR Solar system tests of GR

Pioneers ????...Pioneers ????... Sub-millimeter (non) deviations from Newton’s law Sub-millimeter (non) deviations from Newton’s law new new

tests ???tests ??? Principal cornerstone of Concordance Cosmology! Principal cornerstone of Concordance Cosmology! Things Things

Dark ?!Dark ?! How well do we How well do we REALLYREALLY know gravity? know gravity?

Hands-on observational tests confirm GR at scales between Hands-on observational tests confirm GR at scales between roughly roughly 0.1 mm0.1 mm and - say - about and - say - about 100 MPc; 100 MPc; why are we then why are we then so so certaincertain that the extrapolation of GR to that the extrapolation of GR to shortershorter and and longerlonger distances is justified?distances is justified?

Discords in the Concordate?Discords in the Concordate? Are we pushing GR too far?… Are we pushing GR too far?…


Cosmological constant Cosmological constant failurefailure

Cosmological constant problem is Cosmological constant problem is desperatedesperate (by ≥ 60 (by ≥ 60 orders of magnitude!) orders of magnitude!) desperate measures desperate measures required?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 trades an

independent variable independent variable (x)(x) for another independent variable for another independent variable J(x)J(x).. Cosmological constant term Cosmological constant term ∫ dx∫ dx √√det(g) det(g) isis a Legendre a Legendre

transform. transform. In GR, general covariance In GR, general covariance det(gdet(g)) does not propagate! does not propagate! So the Legendre transform So the Legendre transform ∫ dx∫ dx √√det(g) det(g) ‘loses’ information ‘loses’ information

about about only ONE only ONE IR parameter - IR parameter - . . Thus Thus is not calculable, but is not calculable, but is an input!is an input!

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

Even failure isEven failure is success: exploring ways of modifying gravity should success: exploring ways of modifying gravity should teach us just how robust GR is… teach us just how robust GR is…


HeadachesHeadaches

Changing gravity → adding new DOFs in Changing gravity → adding new DOFs in the IRthe IR

They can be problematic:They can be problematic: Too light and too strongly coupled → new long Too light and too strongly coupled → new long

range forces range forces

Observations place bounds on these!Observations place bounds on these! Negative mass squaredNegative mass squared or negative residue of the or negative residue of the

pole in the propagator for the new DOFs: pole in the propagator for the new DOFs: tachyonstachyons and/orand/or ghostsghosts

Instabilities could render the theory Instabilities could render the theory nonsensical!nonsensical!


DGP BraneworldsDGP Braneworlds

Use braneworlds as a playground to Use braneworlds as a playground to learn how to change gravity in the IRlearn how to change gravity in the IR

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

2000)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 loops graviton by brane loops

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


DGP ActionDGP Action Action:Action:

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!

How do we then hide the 5How do we then hide the 5thth dimension??? dimension??? Gravitational perturbations: assume flat Gravitational perturbations: assume flat

background & perturb; while perhaps background & perturb; while perhaps dubious this is simple, builds up dubious this is simple, builds up intuition…intuition…


Masses and filtersMasses and filters Propagator:Propagator:

Gravitational filterGravitational filter:: Terms Terms ~ M~ M55 in the denominator of the propagator in the denominator of the propagator

dominate at dominate at LOW LOW pp, suppressing the momentum , suppressing the momentum transfer as transfer as 1/p1/p at distances at distances r > Mr > M44

22/2M/2M5533 , making , making

theory look 5D.theory look 5D. Brane-localized terms Brane-localized terms ~ M~ M44 dominate at dominate at HIGH HIGH pp

and render theory 4D, suppressing the momentum and render theory 4D, suppressing the momentum transfer as transfer as 1/p1/p22 at distances shorter than at distances shorter than rrcc < < MM44

22/2M/2M5533 ..


vDVZvDVZ

Terms Terms ~ M~ M55 like a mass term; resonance like a mass term; resonance composed of bulk modes, with 5 DOFs → massive composed of bulk modes, with 5 DOFs → massive from the 4D point of view. So the resonance has from the 4D point of view. So the resonance has extra longitudinal gravitons; discontinuity when extra longitudinal gravitons; discontinuity when MM5 5 → 0 → 0 similar to similar to mmg g → 0→ 0 (van Dam, Veltman; (van Dam, Veltman;

Zakharov; 1970Zakharov; 1970):): Fourier expansion for the field of a source on the Fourier expansion for the field of a source on the

brane:brane:

Take the limit Take the limit MM5 5 → 0 → 0 and compare with 4D GR: and compare with 4D GR:


Strongly coupled scalar Strongly coupled scalar gravitonsgravitons

However: naïve linear perturbation theory in massive However: naïve linear perturbation theory in massive gravity on a flat space breaks down → nonlinearities gravity on a flat space breaks down → nonlinearities yield continuous limityield continuous limit (Vainshtein, 1972)(Vainshtein, 1972)..

There exist examples of the absence of vDVZ There exist examples of the absence of vDVZ discontinuity in curved backgroundsdiscontinuity in curved backgrounds (Kogan (Kogan et alet al; Karch ; Karch et et alal; 2000); 2000)..

The reason: the scalar graviton becomes strongly The reason: the scalar graviton becomes strongly coupled at a scale much bigger than the gravitational coupled at a scale much bigger than the gravitational radius.radius. (Arkani-Hamed, Georgi, Schwartz, 2002)(Arkani-Hamed, Georgi, Schwartz, 2002)::

EFT analysis of DGP (EFT analysis of DGP (Porrati, Rattazi & Luty, 2003Porrati, Rattazi & Luty, 2003): a naïve ): a naïve expansion around flat space suggests a breakdown of expansion around flat space suggests a breakdown of EFT at EFT at rr* * ~ 1000 km; ~ 1000 km; loss of predictivity loss of predictivity at at macroscopic scales!macroscopic scales! But inclusion of curvature pushes But inclusion of curvature pushes it down to it down to ~~ 1 cm1 cm ( (Rattazi & Nicolis, 2003Rattazi & Nicolis, 2003); what’s going ); what’s going on???on???


Beyond naïve Beyond naïve perturbation theoryperturbation theory

Difficulty: Difficulty: bothboth background and interactions background and interactions have been treated perturbatively. Can we do have been treated perturbatively. Can we do better?better?

Construct realistic backgrounds; solveConstruct 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; in static patch:where 4d metric is de Sitter; in static patch:


The intrinsic curvature and the tension related by The 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 branch,normal branch,

i.e. this reduces to the usual inflating brane in 5D!i.e. this reduces to the usual inflating brane in 5D! ==--1 1 self-inflating branch:self-inflating branch:

inflates even if tension vanishes!inflates even if tension vanishes!

Normal and self-inflating Normal and self-inflating branchesbranches


Fields of small lumps of Fields of small lumps of energyenergy

Trick: using analyticity it is Trick: using analyticity it is alwaysalways possible to find a possible to find a solution for compact ultra-relativistic sources!solution for compact ultra-relativistic sources!

Consider the geometry of a mass point, which is a Consider the geometry of a mass point, which is a solution of some gravitational field equations, which solution of some gravitational field equations, which obeyobey Analyticity in Analyticity in mm Principle of relativityPrinciple of relativity

Then pick an observer who moves Then pick an observer who moves VERY FASTVERY FAST relative relative to the mass source. In his frame the source is boosted to the mass source. In his frame the source is boosted relative to the observer. Take the limit of infinite boost.relative to the observer. Take the limit of infinite boost.

Only the first term in the expansion of the Only the first term in the expansion of the metric in metric in mm survives, since survives, since p = m cosh p = m cosh = = constconst. All other terms are ~. All other terms are ~mmnn cosh cosh , and so , and so for for n > 1 n > 1 they vanishthey vanish in the extreme in the extreme relativistic limit!relativistic limit!


Shock wavesShock waves Physically: because of Physically: because of

the Lorentz contraction the Lorentz contraction in the direction of in the direction of motion, the field lines get motion, the field lines get pushed towards the pushed towards the instantaneous plane instantaneous plane which is orthogonal to which is orthogonal 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! ((Bergmann, 1940’sBergmann, 1940’s))

The same intuition works The same intuition works for the gravitational field.for the gravitational field.


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.

Aichelburg-Sexl Aichelburg-Sexl shockwaveshockwave


Dray-’t Hooft trickDray-’t Hooft trick Shock the geometry with a Shock the geometry with a discontinuity in the null discontinuity in the null

directiondirection of motion of motion vv using orthogonal coordinate using orthogonal coordinate u u , , controlled by the photon momentum. Field equations controlled by the photon momentum. Field equations linearize, yield a single field eq. for the wave profile linearize, yield a single field eq. for the wave profile the the Israel junction condition on a null surface. The technique Israel junction condition on a null surface. The technique has been generalized by K. Sfetsos to general 4D GR has been generalized by K. Sfetsos to general 4D GR (string) backgrounds. Extends to DGP, and other brany (string) backgrounds. Extends to DGP, and other brany setups! setups! (NK, 2005)(NK, 2005)

Idea: pick a spacetime and a set of null geodesics. Idea: pick a spacetime and a set of null geodesics. Trick: substituteTrick: substitute

change to change to

discontinuitydiscontinuity


DGP in a state of shockDGP in a state of shock The starting point for ‘shocked’ DGP is The starting point for ‘shocked’ DGP is (NK, 2005 )(NK, 2005 )

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

Turn the crank!Turn the crank!


Shockwave field equationShockwave field equation In fact it is convenient to work with two In fact it is convenient to work with two

‘antipodal’ photons, that zip along the past ‘antipodal’ photons, that zip along the past horizon (ie boundary of future light cone) in horizon (ie boundary of future light cone) in opposite directions. This avoids problems with opposite directions. This avoids problems with spurious singularities on compact spaces. It is spurious singularities on compact spaces. It is also the correct infinite boost limit of also the correct infinite boost limit of Schwarzschild-dS solution in 4D Schwarzschild-dS solution in 4D (Hotta, Tanaka, 1993) (Hotta, Tanaka, 1993) . . The field equation is The field equation is (NK, 2005)(NK, 2005)


Shockwave solutionsShockwave solutions Using the symmetries of the problem, this equation can be solved Using the symmetries of the problem, this equation can be solved

by the expansion by the expansion (NK, 2005)(NK, 2005)

The solution is (using The solution is (using =exp(-H=exp(-H||zz||), x = cos ), x = cos gg=2M=2M55

33/M/M4422H=1/rH=1/rccH H ))


Arc lengthsArc lengths OK, but where is the OK, but where is the

physics??? Short physics??? Short distance expansion!distance expansion!

The horizon is at The horizon is at rrHH = = 1/H.1/H. So the So the transverse distance transverse distance between the photon at between the photon at =0=0 and a point at a and a point at a small small is is

RR = = /H/H


Short distance properties Short distance properties II

Consider first the limit Consider first the limit g g = 0; = 0; on the brane aton the brane at z=0, z=0, the the integral yieldsintegral yields

Identical to the 4D GR shockwave in de Sitter background, Identical to the 4D GR shockwave in de Sitter background, found by Hotta & Tanaka in 1993.found by Hotta & Tanaka in 1993. Using arc length Using arc length RR = = /H/H, , the 4D profile in dS reduces to the flat Aichelburg-Sexl at the 4D profile in dS reduces to the flat Aichelburg-Sexl at short distances (short distances (x=1-Hx=1-H22RR22/2 /2 ):):

What about the short distance properties when What about the short distance properties when g ≠g ≠0 0 ? …? …


Short distance properties Short distance properties IIII

In general: the solution is a Green’s function for the In general: the solution is a Green’s function for the two source problem and can only contain the two source problem and can only contain the physical short distance singularities. For physical short distance singularities. For ANYANY finite finite value of value of gg those yield those yield

The only singular term is logarithmic – just like in the The only singular term is logarithmic – just like in the 4D GR wave profile. Thus at short distances the 4D GR wave profile. Thus at short distances the shockwave looks precisely the same as in 4D! The shockwave looks precisely the same as in 4D! The corrections appear only as the terms linear in corrections appear only as the terms linear in RR, and , and are suppressed by are suppressed by 1/H 1/H gg = 1/r = 1/rc c . . (NK, 2005)(NK, 2005)


Recovering 5Recovering 5thth D D We can take the limit We can take the limit gg ∞ ∞ ( (rrcc 0 0 ) on the normal ) on the normal

branch while keeping positive tension; we find 5D + 4D branch while keeping positive tension; we find 5D + 4D contributions:contributions:

(NK, (NK,

2005)2005)

The first term is the 5D A-S The first term is the 5D A-S ((Ferrari, Pendenza, Veneziano, 1987; Ferrari, Pendenza, Veneziano, 1987; de de Vega, Sanchez, 1989)Vega, Sanchez, 1989)

So only in the limit So only in the limit rrcc 0 0 will we find no filter; whenever will we find no filter; whenever rrcc is finite, the filter will work preventing singularities is finite, the filter will work preventing singularities worse than logarithms in the Green’s function, and thus worse than logarithms in the Green’s function, and thus screening X-dims!screening X-dims!


Gravitational filter beyond Gravitational filter beyond perturbation theoryperturbation theory

How does the filter work? The key is that in the Green’s How does the filter work? The key is that in the Green’s function expanded as a sum over 5D modes, the coefficients function expanded as a sum over 5D modes, the coefficients are suppressed by are suppressed by l l of of PP2l2l(x) (x) ; their momentum is ; their momentum is q = q = ll/H /H ; ; hence the effective coupling for momenta hence the effective coupling for momenta q > 1/rq > 1/rcc is is

Rewrite this as Rewrite this as (NK, 2005)(NK, 2005)

bulk Planck massbulk Planck mass

filterfilter volume dilutionvolume dilution


Where is the scalar Where is the scalar graviton?graviton?

A very peculiar feature of the shockwave solution is A very peculiar feature of the shockwave solution is that the scalar graviton hasthat the scalar graviton has NOTNOT been turned on: if been turned on: if ff is viewed as a perturbation, is viewed as a perturbation, hh ~ f ~ f , then , then hh

= 0 = 0 .. At first, that seems trivial; At first, that seems trivial; = h= h

is sourced by is sourced by TT, ,

which vanishes in the ultrarelativistic limit. So it is OK which vanishes in the ultrarelativistic limit. So it is OK to have to have = 0= 0……

… … as long as we are in a weak coupling limit where we as long as we are in a weak coupling limit where we can trust the perturbative effective action! However…can trust the perturbative effective action! However…

… … this survives for DGP sources with a lot of this survives for DGP sources with a lot of momentum in spite of the issues with strong coupling! momentum in spite of the issues with strong coupling! This suggests that the nonlinearities may improve the This suggests that the nonlinearities may improve the theory.theory.


Paranormal phenomena?Paranormal phenomena? There are concerns that There are concerns that ghostsghosts are present when are present when

gravity alterations drive cosmic acceleration gravity alterations drive cosmic acceleration (Luty, (Luty, Porrati, Rattazzi, 2003; Rattazi, Nicolis, 2003; Koyama, 2005 – but Porrati, Rattazzi, 2003; Rattazi, Nicolis, 2003; Koyama, 2005 – but they disagree with each other!)they disagree with each other!) . .

Indeed: we see a spectacular instability for Indeed: we see a spectacular instability for = -1 = -1 when when gg 1 1 : :

The The l=0l=0 mode diverges when it is perturbed by a mode diverges when it is perturbed by a particle of momentum particle of momentum p p ! A possibility: ! A possibility: poltergeistpoltergeist !? !?

Copious production of delocalized bulk gravitons! Copious production of delocalized bulk gravitons! Deserves more attention. Deserves more attention.


Chasing scalar gravitonsChasing scalar gravitons

A new perturbative expansion?A new perturbative expansion? Take a source at rest; let a fast moving observer Take a source at rest; let a fast moving observer

probe it.probe it. Let her move a little bit more slowly than Let her move a little bit more slowly than cc.. In her rest frame the source is fast. So it can be In her rest frame the source is fast. So it can be

approximated by a shockwave; corrections approximated by a shockwave; corrections controlled by controlled by m/p = (1/vm/p = (1/v22-1)-1)1/21/2. .

She can use She can use m/pm/p as a small expansion parameter as a small expansion parameter and compute the field, then boost the result back and compute the field, then boost the result back to an observer at rest relative to the mass.to an observer at rest relative to the mass.

Analyticity suggests that perturbation theory Analyticity suggests that perturbation theory may be under control; worth checking!may be under control; worth checking!


SummarySummary The cornerstone of the DGP : The cornerstone of the DGP : gravitational filtergravitational filter - -

hides the extra dimension. But: strongly coupled hides the extra dimension. But: strongly coupled scalar graviton is scalar graviton is dangerousdangerous!!

Shockwaves are the first example of exact DGP Shockwaves are the first example of exact DGP backgrounds for compact sources and a new arena to backgrounds for compact sources and a new arena to study perturbation theory. study perturbation theory.

Shock therapyShock therapy yields new insights into the filter yields new insights into the filter (but (but it won’t rid us of ghosts on the self-inflating it won’t rid us of ghosts on the self-inflating branch…)branch…)

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

Applicable elsewhere: only a teaser here: “Locally Applicable elsewhere: only a teaser here: “Locally Localized Gravity: The Inside Story” (NK & L. Localized Gravity: The Inside Story” (NK & L. Sorbo, see yesterday’s hep-th/0507191).Sorbo, see yesterday’s hep-th/0507191).