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SHOCK THERAPY

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SHOCK THERAPY

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  1. This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation • In Slide Show, click on the right mouse button • Select “Meeting Minder” • Select the “Action Items” tab • Type in action items as they come up • Click OK to dismiss this box • This will automatically create an Action Item slide at the end of your presentation with your points entered. SHOCK THERAPY Nemanja Kaloper UC Davis Nemanja Kaloper, UC Davis

  2. Shock box Modified Gravity Nemanja Kaloper, UC Davis

  3. Overview • Two messages: • Changing gravity: Why Bother? • Exploring modified gravity: Shocks • DGP: a toy arena • DGP in shock • Future directions • Summary Nemanja Kaloper, UC Davis

  4. The Concert of Cosmos • Einstein’s 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 Newton’s 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? Nemanja Kaloper, UC Davis

  5. The Concert of Cosmos? • Einstein’s 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 Newton’s 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?… Nemanja Kaloper, UC Davis

  6. 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 Strades an independent variable F for another independent variable J. • Reconstruction of G(F) from W(J) yields a family of effective actionsparameterized byan arbitraryJ,where J=0 must be put in by hand! • Cosmological constant term ∫ dx√det(g) L isa Legendre transform. • In GR, general covariance det(g) does not propagate! • So the Legendre transform ∫ dx√det(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… Nemanja Kaloper, UC Davis

  7. 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 squaredor negative residue of the pole in the propagator for the new DOFs: tachyonsand/orghosts Instabilities could render the theory nonsensical! Nemanja Kaloper, UC Davis

  8. 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) Nemanja Kaloper, UC Davis

  9. DGP Action • Action: • Assume ∞ 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… Nemanja Kaloper, UC Davis

  10. 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 pand render theory 4D, suppressing the momentum transfer as 1/p2 at distances shorter than rc < M42/2M53. Nemanja Kaloper, UC Davis

  11. 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: Nemanja Kaloper, UC Davis

  12. 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); what’s going on??? Nemanja Kaloper, UC Davis

  13. 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? Nemanja Kaloper, UC Davis

  14. 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: Nemanja Kaloper, UC Davis

  15. Penrose diagram for a tensional brane in 5D locally Minkowski bulk Nemanja Kaloper, UC Davis

  16. 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! Nemanja Kaloper, UC Davis

  17. 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 vanishin the extreme relativistic limit! Nemanja Kaloper, UC Davis

  18. 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, 1940’s) • The same intuition works for the gravitational field. Nemanja Kaloper, UC Davis

  19. 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)/√2 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. Nemanja Kaloper, UC Davis

  20. Dray-’t Hooft trick • Shock the geometry with a discontinuity in the null directionof 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 Nemanja Kaloper, UC Davis

  21. 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! Nemanja Kaloper, UC Davis

  22. 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) Nemanja Kaloper, UC Davis

  23. “Antipodal’’ photons in the static patch on the de Sitter brane Nemanja Kaloper, UC Davis

  24. Shockwave solutions • Using the symmetries of the problem, this equation can be solved by the expansion (NK, 2005) • The solution is (using t=exp(-H|z|), x = cos q, g=2M53/M42H=1/rcH ) Nemanja Kaloper, UC Davis

  25. Shockwave solutions II • The series can be rewritten as an integral, analogous to the Poisson integral (NK, 2005), • OK, but where is the physics??? Nemanja Kaloper, UC Davis

  26. Arc lengths • The horizon is at rH = 1/H. So the distance between the photon at q=0 and a point at a small q is R = q/H Nemanja Kaloper, UC Davis

  27. Short distance properties I • Consider first the limit g = 0; on the brane at z=0, 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 (x=1-H2R2/2 ): • What about the short distance properties when g ≠0 ? … Nemanja Kaloper, UC Davis

  28. Short distance properties II • In general: the solution is a Green’s 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) Nemanja Kaloper, UC Davis

  29. Recovering 5th D • We can take the limit g ∞ (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 Green’s function, and thus screening X-dims! Nemanja Kaloper, UC Davis

  30. Gravitational filter beyond perturbation theory • How does the filter work? The key is that in the Green’s 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 Nemanja Kaloper, UC Davis

  31. 4D Graviton resonance • In the 4D language, the structure of the singularity of the Green’s function shows that the sum of the bulk modes behaves exactly as a 4D resonance. • At short distance it’s effective coupling to the brane matter is i.e. it mimics 4D gravity! Nemanja Kaloper, UC Davis

  32. 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 < rc, 1/H !(NK, 2005) Nemanja Kaloper, UC Davis

  33. Where is the scalar graviton? • A very peculiar feature of the shockwave solution is that the scalar graviton hasNOT 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. Nemanja Kaloper, UC Davis

  34. Paranormal phenomena? • There are concerns that ghostsare 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 l=0 mode diverges when it is perturbed by a particle of momentum p ! A possibility: poltergeist !? • Copious production of delocalized bulk gravitons! Deserves more attention. Nemanja Kaloper, UC Davis

  35. 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! Nemanja Kaloper, UC Davis

  36. 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 won’t 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). Nemanja Kaloper, UC Davis

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