SHOCK THERAPY

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SHOCK THERAPY

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• Nemanja Kaloper
• UC Davis

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Shock box
Modified Gravity
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Overview

• Two messages
• Changing gravity Why Bother?
• Exploring modified gravity Shocks
• DGP a toy arena
• DGP in shock
• Future directions
• Summary

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The Concert of Cosmos

• Einsteins GR a beautiful theoretical framework
  for gravity and cosmology, consistent with
  numerous experiments and observations
• Solar system tests of GR
• Sub-millimeter (non) deviations from Newtons law
• Principal cornerstone of Concordance Cosmology!
• How well do we REALLY know gravity?
• Hands-on observational tests confirm GR at scales
  between roughly 0.1 mm and - say - about 100 MPc
  why are we then so certain that the extrapolation
  of GR to shorter and longer distances is
  justified?

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The Concert of Cosmos?

• Einsteins GR a beautiful theoretical framework
  for gravity and cosmology, consistent with
  numerous experiments and observations
• Solar system tests of GR
  Pioneers ????...
• Sub-millimeter (non) deviations from Newtons law
  new tests ???
• Principal cornerstone of Concordance Cosmology!
  Things Dark ?!
• How well do we REALLY know gravity?
• Hands-on observational tests confirm GR at scales
  between roughly 0.1 mm and - say - about 100 MPc
  why are we then so certain that the extrapolation
  of GR to shorter and longer distances is
  justified?
• Discords in the Concordate? Are we pushing GR
  too far?

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Dark thoughts

• Can we change the nature of the cosmological
  constant problem by changing gravity?
• Can we explain galaxy rotation curves by changing
  gravity instead of using dark matter? (e.g.
  covariant MOND?... (Bekenstein, 2004))
• Even if this fails - exploring modifications of
  gravity could teach us just how robust the
  framework of GR is

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Cosmological constant failure

• Cosmological constant problem is desperate (by
  60 orders of magnitude!) ? desperate measures
  required?
• Might changing gravity help? A (very!) heuristic
  argument
• Legendre transforms adding ? dx ?(x) J(x) to S
  trades an independent variable F for another
  independent variable J.
• Reconstruction of G(F) from W(J) yields a family
  of effective actions parameterized by an
  arbitrary J, where J0 must be put in by hand!
• Cosmological constant term ? dx vdet(g) L is a
  Legendre transform.
• In GR, general covariance ? det(g) does not
  propagate!
• So the Legendre transform ? dx vdet(g) L loses
  information about only ONE IR parameter - L.
  Thus L is not calculable, but is an input!
• Could changing gravity alter this, circumventing
  no-go theorems?
• Even failure is success exploring ways of
  modifying gravity should teach us just how
  robust GR is

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Headaches

• Changing gravity ? adding new DOFs in the IR
• They can be problematic
• Too light and too strongly coupled ? new long
  range forces
• Observations place bounds on these!
• Negative mass squared or negative residue of the
  pole in the propagator for the new DOFs tachyons
  and/or ghosts
• Instabilities could render the theory
  nonsensical!

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DGP Braneworlds

• Use braneworlds as a playground to learn how to
  change gravity in the IR
• Brane-induced gravity (Dvali, Gabadadze, Porrati,
  2000)
• Ricci terms BOTH in the bulk and on the
  end-of-the-world brane, arising from e.g. wave
  function renormalization of the graviton by brane
  loops
• May appear in string theory (Kiritsis, Tetradis,
  Tomaras, 2001 Corley, Lowe, Ramgoolam, 2001)

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DGP Action

• Action
• Assume 8 bulk 4D gravity has to be mimicked by
  the exchange of bulk DOFs!
• How do we then hide the 5th dimension???
• Gravitational perturbations assume flat
  background perturb while perhaps dubious this
  is simple, builds up intuition

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DGP Action

• Action
• Assume 8 bulk 4D gravity has to be mimicked by
  the exchange of bulk DOFs!
• How do we then hide the 5th dimension???
• Gravitational perturbations assume flat
  background perturb while perhaps dubious this
  is simple, builds up intuition

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Masses and filters

• Propagator
• Gravitational filter
• Terms M5 in the denominator of the propagator
  dominate at LOW p, suppressing the momentum
  transfer as 1/p at distances r M42/2M53 ,
  making theory look 5D.
• Brane-localized terms M4 dominate at HIGH p and
  render theory 4D, suppressing the momentum
  transfer as 1/p2 at distances shorter than rc
  M42/2M53 .

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vDVZ

• Terms M5 like a mass term resonance composed
  of bulk modes, with 5 DOFs ? massive from the 4D
  point of view. So the resonance has extra
  longitudinal gravitons discontinuity when M5 ? 0
  similar to mg ? 0 (van Dam, Veltman Zakharov
  1970)
• Fourier expansion for the field of a source on
  the brane
• Take the limit M5 ? 0 and compare with 4D GR

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Strongly coupled scalar gravitons

• However naïve linear perturbation theory in
  massive gravity on a flat space breaks down ?
  nonlinearities yield continuous limit
  (Vainshtein, 1972).
• There exist examples of the absence of vDVZ
  discontinuity in curved backgrounds (Kogan et al
  Karch et al 2000).
• The reason the scalar graviton becomes strongly
  coupled at a scale much bigger than the
  gravitational radius. (Arkani-Hamed, Georgi,
  Schwartz, 2002)
• EFT analysis of DGP (Porrati, Rattazi Luty,
  2003) a naïve expansion around flat space
  suggests a breakdown of EFT at r 1000 km loss
  of predictivity at macroscopic scales! But
  inclusion of curvature pushes it down to 1 cm
  (Rattazi Nicolis, 2003) whats going on???

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Strong coupling in DGP

• EFT analysis of DGP (Luty, Porrati, Rattazzi,
  2003) finds similar behavior. The gist
  linearized expansion around masses in DGP breaks
  down when GNm/r r2/rc2, ( rc M42/2M53 is the
  scale where gravity is modified) ? at distances r
  (GNm rc2)1/3 gravity is strong. For small
  masses with Planck scale horizon size, r
  (rc2/MPl)1/3. There one expects strong QUANTUM
  effects as well!
• Borne out by perturbative analysis on a flat
  background integrating out the bulk dynamics
  LPR find EFT for the pullback of the metric on
  the brane. The Goldstone mode analysis finds the
  scalar graviton is strongly coupled at r
  (rc2/MPl)1/3 . Plugging in rc 1/H0 shows that
  rc 1000 km.
• The theory loses predictivity at macroscopic
  scales???
• LPR also find that the scalar mode becomes a
  ghost on the self-inflating branch (to be defined
  below), in the regime where the gravity
  modifications dominate cosmology.

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Curvature as a coupling controller?

• What if we include the curvature of the source
  itself? (Nicolis, Rattazzi, 2003)
• By including the effects of the source mass on
  the local geometry, via the local value of the
  extrinsic curvature KAB , NR find that the
  strongly coupled scalar may in fact receive large
  renormalization from the background fields
• L
  Zmn ? mf ? n f
• where
• Near a source Z (GNmrc2/r3)1/2 substituting
  r yields Z (m/MPl)1/2 , which is a HUGE
  suppression for big masses! This could restore
  EFT down to much shorter distances than 1000 km!
  NR for an Earth-based observer EFT remains
  valid down to scales 1 cm
• But why only these counterterms?

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Beyond naïve perturbation theory

• Difficulty both background and interactions have
  been treated perturbatively. Can we do better?
• Construct realistic backgrounds solve
• Look at the vacua first
• Symmetries require (see e.g. N.K, A. Linde,
  1998)
• where 4d metric is de Sitter in static
  patch

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Penrose diagram for a tensional brane in 5D
locally Minkowski bulk
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Normal and self-inflating branches

• The intrinsic curvature and the tension related
  by (N.K. Deffayet,2000)
• e 1 an integration constant e 1 normal
  branch,
• i.e. this reduces to the usual inflating
  brane in 5D!
• e -1 self-inflating branch
• inflates even if tension vanishes!

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Fields of small lumps of energy

• Trick using analyticity it is always possible to
  find a solution for compact ultra-relativistic
  sources!
• Consider the geometry of a mass point, which is a
  solution of some gravitational field equations,
  which obey
• Analyticity in m
• Principle of relativity
• Then pick an observer who moves VERY FAST
  relative to the mass source. In his frame the
  source is boosted relative to the observer. Take
  the limit of infinite boost.
• Only the first term in the expansion of the
  metric in m survives, since p m cosh b const.
  All other terms are mn cosh b, and so for n 1
  they vanish in the extreme relativistic limit!

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Shock waves

• Physically because of the Lorentz contraction
  in the direction of motion, the field lines get
  pushed towards the instantaneous plane which is
  orthogonal to V.
• The field lines of a massless charge are confined
  to this plane! (Bergmann, 1940s)
• The same intuition works for the gravitational
  field.

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Aichelburg-Sexl shockwave

• In flat 4D environment, the exact gravitational
  field of a photon found by boosting linearized
  Schwarzschild metric (Aichelburg, Sexl, 1971).
• Here u,v (x t)/v2 are null coordinates of the
  photon.
• For a particle with a momentum p , f is, up to a
  constant
• where R (y2 z2)1/2 is the transverse
  distance and l0 an arbitrary integration
  parameter.

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Dray-t Hooft trick

• Shock the geometry with a discontinuity in the
  null direction of motion v using orthogonal
  coordinate u , controlled by the photon momentum.
  Field equations linearize, yield a single field
  eq. for the wave profile ? the Israel junction
  condition on a null surface. The technique has
  been generalized by K. Sfetsos to general 4D GR
  (string) backgrounds. Extends to DGP, and other
  brany setups! (NK, 2005)
• Idea pick a spacetime and a set of null
  geodesics.
• Trick substitute
• 
  
• 
  change to
• discontinuity

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DGP in a state of shock

• The starting point for shocked DGP is (NK, 2005
  )
• Term f is the discontinuity in dv . Substitute
  this metric in the DGP field equations, where the
  new brane stress energy tensor includes photon
  momentum
• Turn the crank!

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Shockwave field equation

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

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Antipodal photons in the static patch on the
de Sitter brane
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Shockwave solutions

• Using the symmetries of the problem, this
  equation can be solved by the expansion (NK,
  2005)
• The solution is (using texp(-Hz), x cos q,
  g2M53/M42H1/rcH )
• 
  
  
  
• 
  

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Shockwave solutions II

• The series can be rewritten as an integral,
  analogous to the Poisson integral (NK, 2005),
• OK, but where is the physics???

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Arc lengths

• The horizon is at rH 1/H. So the distance
  between the photon at q0 and a point at a small
  q is
• R q/H

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Short distance properties I

• Consider first the limit g 0 on the brane at
  z0, the integral yields
• Identical to the 4D GR shockwave in de Sitter
  background, found by Hotta Tanaka in 1993.
  Using arc length R q/H, the 4D profile in dS
  reduces to the flat Aichelburg-Sexl at short
  distances (x1-H2R2/2 )
• What about the short distance properties when g
  ?0 ?

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Short distance properties II

• In general the solution is a Greens function
  for the two source problem and can only contain
  the physical short distance singularities. For
  ANY finite value of g those yield
• The only singular term is logarithmic just like
  in the 4D GR wave profile. Thus at short
  distances the shockwave looks precisely the same
  as in 4D! The corrections appear only as the
  terms linear in R, and are suppressed by 1/H g
  1/rc . (NK, 2005)

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Recovering 5th D

• We can take the limit g ? 8 (rc ? 0 ) on the
  normal branch while keeping positive tension we
  find 5D 4D contributions
• 
  
• 
  (NK,
  2005)
• The first term is the 5D A-S (Ferrari, Pendenza,
  Veneziano, 1987 de Vega, Sanchez, 1989)
• So only in the limit rc ? 0 will we find no
  filter whenever rc is finite, the filter will
  work preventing singularities worse than
  logarithms in the Greens function, and thus
  screening X-dims!

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Gravitational filter beyond perturbation theory

• How does the filter work? The key is that in the
  Greens function expanded as a sum over 5D modes,
  the coefficients are suppressed by l of P2l(x)
  their momentum is q l/H hence the effective
  coupling for momenta q 1/rc is
• Rewrite this as (NK, 2005)
• bulk Planck mass
  
• 
  
  filter
• 
  volume dilution

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4D Graviton resonance

• In the 4D language, the structure of the
  singularity of the Greens function shows that
  the sum of the bulk modes behaves exactly as a 4D
  resonance.
• At short distance its effective coupling to the
  brane matter is
• i.e. it mimics 4D gravity!

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Planckian scattering

• The cross section for shockwave scattering in DGP
  is another test of the filter. It controls the
  black hole formation rate in (very!) high energy
  particle collisions. For impact parameter b GN
  ECM , use eikonal approximation to compute the
  cross section. The cross section can be extracted
  from the shockwave profile (Amati, Ciafaloni,
  Veneziano t Hooft 1987). The eikonal and the
  profile are related by
• Plugging in the solution it looks 4D when b , 1/H ! (NK, 2005)

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Where is the scalar graviton?

• A very peculiar feature of the shockwave solution
  is that the scalar graviton has NOT been turned
  on if f is viewed as a perturbation, hmn f ,
  then hmm 0 .
• At first, that seems trivial ? hmm is sourced
  by Tmm , which vanishes in the ultrarelativistic
  limit. So it is OK to have ? 0
• as long as we are in a weak coupling limit
  where we can trust the perturbative effective
  action! However
• this survives for DGP sources with a lot of
  momentum in spite of the issues with strong
  coupling! This suggests that the nonlinearities
  may improve the theory.

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Paranormal phenomena?

• There are concerns that ghosts are present when
  gravity alterations drive cosmic acceleration
  (Luty, Porrati, Rattazzi, 2003 Rattazi, Nicolis,
  2003 Koyama, 2005 but some disagreements!) .
• Indeed we see a spectacular instability for e
  -1 when g ? 1
• The l0 mode diverges when it is perturbed by a
  particle of momentum p ! A possibility
  poltergeist !?
• Copious production of delocalized bulk gravitons!
  Deserves more attention.

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Chasing scalar gravitons

• A new perturbative expansion?
• Take a source at rest let a fast moving observer
  probe it.
• Let her move a little bit more slowly than c.
• In her rest frame the source is fast. So it can
  be approximated by a shockwave corrections
  controlled by m/p (1/v2-1)1/2.
• She can use m/p as a small expansion parameter
  and compute the field, then boost the result back
  to an observer at rest relative to the mass.
• Analyticity suggests that perturbation theory may
  be under control worth checking!

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Summary

• The cornerstone of the DGP gravitational filter
  - hides the extra dimension. But strongly
  coupled scalar graviton is dangerous!
• Shockwaves are the first example of exact DGP
  backgrounds for compact sources and a new arena
  to study perturbation theory.
• Shock therapy may thus yield new insights into
  the filter (but it wont rid us of ghosts on the
  self-inflating branch)
• More work we may reveal interesting new realms
  of gravity!
• Applicable elsewhere just to whet your appetite
  Locally Localized Gravity The Inside Story (NK
  L. Sorbo, see hep-th/0507191).