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Cascading gravity and de gravitation

Cascading gravity and de gravitation. Claudia de Rham Perimeter Institute/McMaster. Miami 2008 Dec, 18 th 2008. Based on work on collaboration with. Stefan Hofmann, Nordita, Stockholm Justin Khoury, Perimeter, Waterloo Andrew Tolley, Perimeter, Waterloo Oriol Pujolas, CERN

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Cascading gravity and de gravitation

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  1. Cascading gravityanddegravitation Claudia de Rham Perimeter Institute/McMaster Miami 2008 Dec, 18th2008

  2. Based on work on collaboration with • Stefan Hofmann, Nordita, Stockholm • Justin Khoury, Perimeter, Waterloo • Andrew Tolley, Perimeter, Waterloo • Oriol Pujolas, CERN • Gia Dvali, NYU, New York &CERN • Michele Redi, EPFL, Lausanne “Cascading Gravity and Degravitation”, JCAP02(2008)011 “Cascading DGP”, PRL 100 (251603), 2008 “Tensing the ghost in 6D cascading gravity”, to appear “Towards Cosmology in theories of massive gravity”, to appear

  3. The c.c. problem • The current acceleration of the Universe is well described by a c.c., L=r/Mpl2, with r~(10-2 eV)4 whileme4/r ~ 1036 and Mpl4/r ~ 10120 • Why is the vacuum energy so small when quantum effects lead to much bigger corrections?

  4. The c.c. problem • The current acceleration of the Universe is well described by a c.c., L=r/Mpl2, with r~(10-2 eV)4 whileme4/r ~ 1036 and Mpl4/r ~ 10120 • Why is the vacuum energy so small when quantum effects lead to much bigger corrections? • Is the vacuum energy actually small or does it simply gravitate very little ? idea behinddegravitation Dvali, Hofmann&Khoury, hep-th/0703027

  5. Small c.c. / weakly gravitating • In GR, gravity is mediated by a massless spin-2 particle and gauge invariance makes both questions equivalent. (universality of graviton coupling) • If gravity was mediated by an effectively massive graviton, gravity would be weaker in the IR the vacuum energy (and other IR sources) would gravitate differently Dvali, Hofmann&Khoury, hep-th/0703027

  6. Filtering gravity • In Einstein’s gravity, the c.c. is bound to gravitate as any other source • The idea behind degravitation is to promote the Newton’s constant GN to a filter operator,

  7. Filtering gravity • At short wavelengths compared to L, if b>0 GN G0N there is no filter and sources gravitate normally, • While at long distances, GN 0, so sources with large wavelengths, (such as the c.c.) are filtered out and effectively gravitate very weakly. with

  8. Filtering and graviton mass • As such, the theory would not satisfy the Bianchi identity, • This cannot represent a consistent theory of massless spin-2 gravitons (with only 2 degrees of freedom) • Instead the theory should be understood as the limit of a theory of massive gravity, with mass ~1/L.

  9. Filtering and graviton mass • Any degravitating (filter) theory must reduce at the linearized level to a theory of massive gravity • Corresponding to the filter theory

  10. Filtering and graviton mass • To be a satisfying ghost-free degravitating theory, the mass should satisfywith 0 d a < 1. • a =1 corresponds to the effective 4d theory arising from the 5d DGP model. R(4) R(5) Dvali, Gabadadze & Porrati, hep-th/0005016

  11. DGP – eg. of massive gravity • Extra dof arise from 5d nature of theory. • We live in a (3+1)-brane embedded in an infinite flat extra dimension R(5) Dvali, Gabadadze & Porrati, hep-th/0005016

  12. DGP – eg. of massive gravity • Extra dof arise from 5d nature of theory. • We live in a (3+1)-brane embedded in an infinite flat extra dimension • In the UV, the 4d curvatureterm dominates, gravity looks 4d • In the IR, gravity is 5d. R(4) R(5) Dvali, Gabadadze & Porrati, hep-th/0005016

  13. DGP – eg. of massive gravity • Effective 4d propagator for DGP • This corresponds to a degravitating theory with a=1/2 with induced Friedmann eq. • a=1/2 is too large ! Is there an extension with a<1/2 ??? k: 4d momentum m5=M53/M42 Cf. Ghazal Geshnizjani ’s talk

  14. Gravity in higher dimensions • For a given spectral representation , we have the “Newtonian potential” • In a (4+n)-dimensional spacetime, the gravitational potential goes as ie. • If n=1 (DGP), in the IR G~p-1a=1/2 • If n=2, in the IR G~ log p a=0 • Any higher dim DGP model corresponds to a=0. r(s)~sn/2-1

  15. Higher-codimension sources • Cod-1 or pure tension cod-2 are the only meaningful distributional sources. (Geroch&Traschen) • Arbitrary matter on cod-2 and higher distributions lead to metric divergences on the defect. The defect should be regularized. Geroch & Traschen, 1987

  16. Cod-2 sources • Cod-1 example • Cod-2 divergences

  17. Regularizing Cod-2 sources • If we had insteadthe solution is regular (easier to see in momentum space) • The new kinetic term plays the role of a regulator. Effectively represents a brane localized kinetic term.

  18. Cascading gravity

  19. Cod-2 cascading • Consider the 6d actionwith couplings y L1 L2 z

  20. Momentum space • In momentum space, this corresponds to brane localized couplingsl1=-M53(q5+k2), and l2=-M42k2.with 2 mass scales m5=M53/M42 and m6= M64/M53 . 2 y L1 L2 z

  21. Cod-2 propagator • Including both couplings, the propagator on the brane is • As m6pk, the propagator behaves as in 6d (a=0) • As m5pkpm6 it takes a 5d behavior • At small scales, kpm5, we recover 4d. log k2G-1 log k

  22. Cascading Gravity:A Naïve approach • The generalization to gravity is straightforward • The tensor mode behaves precisely as the scalar field toy-model, • However one of the scalar modes propagates a ghost.

  23. Propagating modes • Working around flat space-time, • where the tensor mode behaves as expected • and the scalar field p is also regularized by the cod-1 brane source term

  24. 1 4 Ghost mode • p is finite on the cod-2 brane, • However in the UV, p ~ + T • While in the IR, p ~ - T. • The kinetic term changes sign, signaling the presence of a ghost. • In the UV, the gravitational amplitude is

  25. 1 4 Ghost mode • p is finite on the cod-2 brane, • However in the UV, p ~ + T • While in the IR, p ~ - T. • The kinetic term changes sign, signaling the presence of a ghost. • In the UV, the gravitational amplitude is = -1/3-1/6

  26. Ghost mode • This ghost is completely independent to the ghost present in the self-accelerating branch of DGP. • However, it is generic to any cod-2 and higher framework with localized kinetic terms. • In particular it is present when considering a pure cod-2 scenario(no cascading). L2 Gabadadze&Shifman hep-th/0312289

  27. Curing the ghost • There are two ways to cure the ghost:1. Adding a tension on the brane2. Regularizing the brane.

  28. Curing the ghost • There are two ways to cure the ghost:1. Adding a tension on the brane2. Regularizing the brane. • Both approaches lead to a well-defined 4d effective theory, with gravitational amplitude = 1/3-1/12 = 1/2-1/6-1/12

  29. Cosmology Cf. Ghazal Geshnizjani ’s talk

  30. de Sitter solutions • To find some de Sitter solution, can slice the 6d Minkowski bulk as • and take the cod-1 brane located atthe cod-2 at .

  31. R5 dS solutions in 6d • The Cod-1 is not flat • But the brane adapts its position to balance the extrinsic curvature and the Einstein tensor on the brane for y>0 for this configuration can only support a minimal H

  32. dS solutions in 6d • The Friedmann eq. on the brane is then from brane EH R4

  33. dS solutions in 6d • The Friedmann eq. on the brane is then • Solution only makes sense for minimal tension from brane EH R4

  34. dS solutions in 6d • The Friedmann eq. on the brane is then • Solution only makes sense for minimal tension • which is the same bound as the no-ghost condition in the deficit angle solution. from brane EH R4

  35. Properties of the solution • Away for the source, the cod-1 brane asymptotes to a constant position • The 6d bulk is Minkowski (in non trivial coordinates) volume of the extra dimensions is infinite, there are no separate massless zero mode. Asymptotically, the 5d brane is flat

  36. Properties of the Friedmann eq. • Does correspond to a IR modification of gravity • Could in principle have a large r with a small H • BUT still a local expression…

  37. Properties of the Friedmann eq. • Does correspond to a IR modification of gravity • Could in principle have a large r with a small H • BUT still a local expression… • In the absence of brane EH term, there is a self-accelerating solution ghost??

  38. Properties of the Friedmann eq. • Does correspond to a IR modification of gravity • Could in principle have a large r with a small H • BUT still a local expression… • In the absence of brane EH term, there is a self-accelerating solution ghost?? • although different from the “standard self-acceleration’’

  39. Properties of the Friedmann eq. • Does correspond to a IR modification of gravity • Could in principle have a large r with a small H • BUT still a local expression… • In the absence of brane EH term, there is a self-accelerating solution ghost?? • If the solution was unstable, would be interesting to see where it decays to…

  40. Conclusions • Models of massive gravity represent a novel framework to understand the c.c. problem • There is to date only one known ghost-freenon-perturbative theory capable of exhibiting a model of massive gravity that does not violate Lorentz invariance: that is DGP and its Cascading extension.

  41. Conclusion • In 6d cascading gravity, there are at least 2 kind of different solutions for a pure tension source: static, “wedge solution” L de Sittersolution L

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