SHOCK THERAPY Nemanja Kaloper UC Davis Nemanja Kaloper

SHOCK THERAPY Nemanja Kaloper UC Davis Nemanja Kaloper, UC Davis

Shock box Modified Gravity Nemanja Kaloper, UC Davis

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

The Concert of Cosmos n Einstein’s GR: a beautiful theoretical framework for gravity and cosmology, consistent with numerous experiments and observations: n n 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? n 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

The Concert of Cosmos? n Einstein’s GR: a beautiful theoretical framework for gravity and cosmology, consistent with numerous experiments and observations: n n 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? n n 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

Cosmological constant failure n n Cosmological constant problem is desperate (by ≥ 60 orders of magnitude!) desperate measures required? Might changing gravity help? A (very!) heuristic argument: n n n Legendre transforms: adding ∫ dx (x) J(x) to S trades an independent variable (x) for another independent variable J(x). Cosmological constant term ∫ dx √det(g) L is a 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

Headaches Changing gravity → adding new DOFs in the IR n They can be problematic: n n n 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 theory nonsensical! Nemanja Kaloper, UC Davis

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

DGP Action n Action: n Assume ∞ bulk: 4 D gravity has to be mimicked by the exchange of bulk DOFs! How do we then hide the 5 th dimension? ? ? Gravitational perturbations: assume flat background & perturb; while perhaps dubious this is simple, builds up intuition… n n Nemanja Kaloper, UC Davis

Masses and filters § Propagator: § Gravitational filter: § Terms ~ M 5 in the denominator of the propagator dominate at LOW p, suppressing the momentum transfer as 1/p at distances r > M 42/2 M 53 , making theory look 5 D. § Brane-localized terms ~ M 4 dominate at HIGH p and render theory 4 D, suppressing the momentum transfer as 1/p 2 at distances shorter than rc < M 42/2 M 53. Nemanja Kaloper, UC Davis

v. DVZ n Terms ~ M 5 like a mass term; resonance composed of bulk modes, with 5 DOFs → massive from the 4 D point of view. So the resonance has extra longitudinal gravitons; discontinuity when M 5 → 0 similar to mg → 0 (van Dam, Veltman; Zakharov; 1970): n Fourier expansion for the field of a source on the brane: n Take the limit M 5 → 0 and compare with 4 D GR: Nemanja Kaloper, UC Davis

Strongly coupled scalar gravitons n n n 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 v. DVZ 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): n 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

Beyond naïve perturbation theory n n 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 4 d metric is de Sitter; in static patch: Nemanja Kaloper, UC Davis

Normal and self-inflating branches n The intrinsic curvature and the tension related by (N. K. ; Deffayet, 2000) n e = ± 1 an integration constant; e =1 normal branch, n i. e. this reduces to the usual inflating brane in 5 D! e =-1 self-inflating branch: inflates even if tension vanishes! Nemanja Kaloper, UC Davis

Fields of small lumps of energy n n n 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 n Analyticity in m n 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! Nemanja Kaloper, UC Davis

Shock waves n n n 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

Aichelburg-Sexl shockwave n In flat 4 D environment, the exact gravitational field of a photon found by boosting linearized Schwarzschild metric (Aichelburg, Sexl, 1971). n 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 n where R = (y 2 + z 2)1/2 is the transverse distance and l 0 an arbitrary integration parameter. Nemanja Kaloper, UC Davis

Dray-’t Hooft trick n n n 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 4 D 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

DGP in a state of shock n n n 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

Shockwave field equation n 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-d. S solution in 4 D (Hotta, Tanaka, 1993). The field equation is (NK, 2005) Nemanja Kaloper, UC Davis

Shockwave solutions n n 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=2 M 53/M 42 H=1/rc. H ) Nemanja Kaloper, UC Davis

Arc lengths n n OK, but where is the physics? ? ? Short distance expansion! The horizon is at r. H = 1/H. So the transverse distance between the photon at q=0 and a point at a small q is R = q/H Nemanja Kaloper, UC Davis

Short distance properties I n Consider first the limit g = 0; on the brane at z=0, the integral yields n Identical to the 4 D GR shockwave in de Sitter background, found by Hotta & Tanaka in 1993. Using arc length R = q/H, the 4 D profile in d. S reduces to the flat Aichelburg-Sexl at short distances (x=1 -H 2 R 2/2 ): n What about the short distance properties when g ≠ 0 ? … Nemanja Kaloper, UC Davis

Short distance properties II n 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 n The only singular term is logarithmic – just like in the 4 D GR wave profile. Thus at short distances the shockwave looks precisely the same as in 4 D! 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

Recovering 5 th D n We can take the limit g ∞ (rc 0 ) on the normal branch while keeping positive tension; we find 5 D + 4 D contributions: (NK, 2005) n The first term is the 5 D A-S (Ferrari, Pendenza, Veneziano, 1987; de Vega, Sanchez, 1989) n 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

Gravitational filter beyond perturbation theory n How does the filter work? The key is that in the Green’s function expanded as a sum over 5 D modes, the coefficients are suppressed by l of P 2 l(x) ; their momentum is q = l/H ; hence the effective coupling for momenta q > 1/rc is n Rewrite this as (NK, 2005) bulk Planck mass filter volume dilution Nemanja Kaloper, UC Davis

Where is the scalar graviton? n n 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 theory. Nemanja Kaloper, UC Davis

Paranormal phenomena? n n There are concerns that ghosts are present when gravity alterations drive cosmic acceleration (Luty, Porrati, Rattazzi, 2003; Rattazi, Nicolis, 2003; Koyama, 2005 – but they disagree with each other!). 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

Chasing scalar gravitons n A new perturbative expansion? n n n 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/v 2 -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

Summary n n n 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 yields 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: only a teaser here: “Locally Localized Gravity: The Inside Story” (NK & L. Sorbo, see yesterday’s hep-th/0507191). Nemanja Kaloper, UC Davis