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A short introduction to massive gravity... or ... Can one give a mass to the graviton?

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Review talk by Prof. Cedric Deffayet at the SuperJEDI Conference, July 2013 …

Review talk by Prof. Cedric Deffayet at the SuperJEDI Conference, July 2013

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  • 1. SUPERJEDI Pointe aux Piments July, the 4th 2013 A short introduction to « massive gravity » … or … Can one give a mass to the graviton 1. The 3 sins of massive gravity (or why it is hard ?) 2. Cures (or why it may be possible ?) (to be discussed later) Cédric Deffayet (APC, CNRS Paris)
  • 2. Part 1. the 3 sins of massive gravity 1.1. Introduction: why « massive gravity » ? and some properties of « massless gravity » 1.2. The DGP model (as an invitation to take the trip) 1.3. Kaluza-Klein gravitons 1.4. Quadratic massive gravity: the Pauli-Fierz theory and the vDVZ discontinuity 1.5. Non linear Pauli-Fierz theory and the Vainshtein Mechanism 1.6 The Goldstone picture (and « decoupling limit ») of non linear massive gravity, and what can one get from it ?
  • 3. 1.1 Introduction: Why « massive gravity » ? • Dark matter, to explain in particular the dynamics of galaxies. • Dark energy (to explain in particular the observed acceleration of the expansion of the Universe) … in a form close to that of a cosmological constant ½ » constant Friedmann (Einstein) equation Standard model of cosmology requires the presence in the matter content of the Universe of
  • 4. Why « massive gravity » ? One way to modify gravity at « large distances » … and get rid of dark energy (or dark matter) ? Changing the dynamics of gravity ? Historical example the success/failure of both approaches: Le Verrier and • The discovery of Neptune • The non discovery of Vulcan… but that of General Relativity Dark matter dark energy ?
  • 5. for this idea to work… One obviously needs a very light graviton (of Compton length of order of the size of the Universe) I.e. to « replace » the cosmological constant by a non vanishing graviton mass… NB: It seems one of the Einstein’s motivations to introduce the cosmological constant was to try to « give a mass to the graviton » (see « Einstein’s mistake and the cosmological constant » by A. Harvey and E. Schucking, Am. J. of Phys. Vol. 68, Issue 8 (2000))
  • 6. Some properties of « massless gravity » (i.e. General Relativity – GR ) In GR, the field equations (Einstein equations) take the same form in all coordinate system (« general covariance ») Einstein tensor: Second order non linear differential operator on the metric g¹ º Energy momentum tensor: describes the sources Newton constant The Einstein tensor obeys the identities In agreement with the conservation relations
  • 7. Einstein equations can be obtained from the action With
  • 8. If one linearizes Einstein equations around e.g. a flat metric ´¹º one obtains the field equations for a « graviton » given by Kinetic operator of the graviton h¹ º : does not contain any mass term / (undifferentiated h¹ º) The masslessness of the graviton is guaranteed by the gauge invariance (general covariance) Which also results in the graviton having 2 = (10 – 4 £ 2) physical polarisations (cf. the « photon » A¹)
  • 9. 1.2 The DGP model (as an invitation to take the trip) Peculiar to DGP model Usual 5D brane world action • Brane localized kinetic term for the graviton • Will generically be induced by quantum corrections A special hierarchy between M(5) and MP is required to make the model phenomenologically interesting Dvali, Gabadadze, Porrati, 2000 Leads to the e.o.m.
  • 10. Phenomenological interest A new way to modify gravity at large distance, with a new type of phenomenology … The first framework where cosmic acceleration was proposed to be linked to a large distance modification of gravity (C.D. 2001; C.D., Dvali, Gababadze 2002) (Important to have such models, if only to disentangle what does and does not depend on the large distance dynamics of gravity in what we know about the Universe) Theoretical interest Consistent (?) non linear massive gravity … DGP model Intellectual interest Lead to many subsequent developments (massive gravity, Galileons, …)
  • 11. Energy density of brane localized matter Homogeneous cosmology of DGP model One obtains the following modified Friedmann equations (C.D. 2001) • Analogous to standard (4D) Friedmann equations In the early Universe (small Hubble radii ) • Deviations at late time (self-acceleration) Two branches of solutions Light cone Brane Cosmic time Equal cosmic time Big Bang
  • 12. DGP Vs. CDM Maartens, Majerotto 2006 (see also Fairbairn, Goobar 2005; Rydbeck, Fairbairn, Goobar 2007)
  • 13. • Newtonian potential on the brane behaves as 4D behavior at small distances 5D behavior at large distances • The crossover distance between the two regimes is given by This enables to get a “4D looking” theory of gravity out of one which is not, without having to assume a compact (Kaluza-Klein) or “curved” (Randall-Sundrum) bulk. • But the tensorial structure of the graviton propagator is that of a massive graviton (gravity is mediated by a continuum of massive modes) Leads to the « van Dam-Veltman-Zakharov discontinuity » on Minkowski background (i.e. the fact that the linearized theory differs drastically – e.g. in light bending - from linearized GR at all scales)! In the DGP model : the vDVZ discontinuity, is believed to disappear via the « Vainshtein mechanism » (taking into account of non linearities) C.D.,Gabadadze, Dvali, Vainshtein, Gruzinov; Porrati; Lue; Lue & Starkman; Tanaka; Gabadadze, Iglesias;…
  • 14. Many (open) questions about DGP model… … but for the purpose of this talk, just take it as an example of a theory with some flavour of « massive gravity »… … and extra space dimensions.
  • 15. 1.3. Kaluza-Klein gravitons Massive gravitons (from the standpoint of a 4D observer) are ubiquitous in models with extra space-time dimensions in the form of « Kaluza-Klein » modes. Consider first a massless scalar-mediated force in 4D.  E = - /0  N = 4  GN m It is obtained from the Poisson equation (e.g. for an electrostatic or a gravitationnal potential) Yielding a force between two bodies / 1/r2
  • 16. A force mediated by a massive scalar would instead obey the modified Helmholtz equation   -  /C 2 / source Compton length C = ~ / m c And results in the finite range Yukawa potential (r) / exp (-r/C) / r This comes from a quadratic (m2 ©2) mass terms in the Lagrangian
  • 17. Inserting this into the 5D massless field equation: with mk = R k Field equation for a 4D scalar field of mass mk A 5D massless scalar appears as a collection of 4D massive scalars (Tower of “Kaluza-Klein” modes): m0 = 0 m1 = R 1 m2 = R 2 m3 = R 3 m4 = R 4 Experiments at energies much below m1 only see the massless mode Low energy effective theory is four- dimensional m 2 k k
  • 18. The same reasoning holds for the graviton: with g (x,y) =   + h (x,y) Metric describing the 4+1D space-time Flat metric describing the reference cylinder Small perturbation in the vicinity of a reference “cylinder” : Decomposed in terms of a Fourier serie : ² One massless graviton ² A tower of massive “Kaluza-Klein” graviton Can one consider consistently a single massive graviton ?
  • 19. N.B., the PF mass term reads h00 enters linearly both in the kinetic part and the mass term, and is thus a Lagrange multiplier of the theory… … which equation of motion enables to eliminate one of the a priori 6 dynamical d.o.f. hij By contrast the h0i are not Lagrange multipliers 5 propagating d.o.f. in the quadratic PF h  is transverse traceless in vacuum. [cf. a massive photon (Proca field), which has 3 polarisation, vs a massless photon, which has 2]
  • 20. 1.5 Non linear Pauli-Fierz theory and the Vainshtein Mechanism Can be defined by an action of the form The interaction term is chosen such that • It is invariant under diffeomorphisms • It has flat space-time as a vacuum • When expanded around a flat metric (g  =   + h , f  =  ) It gives the Pauli-Fierz mass term Einstein-Hilbert action for the g metric Matter action (coupled to metric g) Interaction term coupling the metric g and the non dynamical metric f Matter energy-momentum tensor Leads to the e.o.m. M2 P G¹º = ¡ T¹º + Tg ¹º(f; g) ¢ Effective energy-momentum tensor ( f,g dependent) Isham, Salam, Strathdee, 1971
  • 21. Some working examples Look for static spherically symmetric solutions with H¹º = g¹º ¡ f¹º (infinite number of models with similar properties) Boulware Deser, 1972, BD in the following Arkani-Hamed, Georgi, Schwarz, 2003 AGS in the following (Damour, Kogan, 2003) S (2) int = ¡ 1 8 m2 M2 P Z d4 x p ¡f H¹ºH¾¿ (f¹¾ fº¿ ¡ f¹º f¾¿ ) S (3) int = ¡ 1 8 m2 M2 P Z d4 x p ¡g H¹ºH¾¿ (g¹¾ gº¿ ¡ g¹º g¾¿ )
  • 22. With the ansatz (not the most general one !) gABdxA dxB = ¡J(r)dt2 + K(r)dr2 + L(r)r2 d-2 fABdxA dxB = ¡dt2 + dr2 + r2 d-2 Gauge transformation g¹ºdx¹ dxº = ¡eº(R) dt2 + e¸(R) dR2 + R2 d-2 f¹ºdx¹ dxº = ¡dt2 + µ 1 ¡ R¹0 (R) 2 ¶2 e¡¹(R) dR2 + e¡¹(R) R2 d-2 Then look for an expansion in GN (or in RS / GN M) of the would be solution Interest: in this form the g metric can easily be compared to standard Schwarzschild form
  • 23. This coefficient equals +1 in Schwarzschild solution Wrong light bending! Vainshtein 1972 In « some kind » [Damour et al. 2003] of non linear PF … … O(1) ² O(1) ² Introduces a new length scale R in the problem below which the perturbation theory diverges! V with Rv = (RSmà 4 )1=5 For the sun: bigger than solar system!
  • 24. So, what is going on at smaller distances? Vainshtein’72 There exists an other perturbative expansion at smaller distances, defined around (ordinary) Schwarzschild and reading: with • This goes smoothly toward Schwarzschild as m goes to zero • This leads to corrections to Schwarzschild which are non analytic in the Newton constant ¸(R) = +RS R n 1 + O ³ R5=2 =R 5=2 v ´o º(R) = ¡RS R n 1 + O ³ R5=2 =R 5=2 v ´o R ¡5=2 v = m2 R ¡1=2 S
  • 25. To summarize: 2 regimes ÷(R) = à R RS (1 + 32 7 ï + ::: with ï = m4R5 RS Valid for R À Rv with Rv = (RSmà 4 )1=5 Valid for R ¿ Rv Expansion around Schwarzschild solution Crucial question: can one join the two regimes in a single existing non singular (asymptotically flat) solution? (Boulware Deser 72) Standard perturbation theory around flat space
  • 26. This was investigated (by numerical integration) by Damour, Kogan and Papazoglou (2003) No non-singular solution found matching the two behaviours (always singularities appearing at finite radius) (see also Jun, Kang 1986) In the 2nd part of this talk: A new look on this problem using in particular the « Goldstone picture » of massive gravity in the « Decoupling limit. » (in collaboration with E. Babichev and R.Ziour 2009-2010)
  • 27. 1.6 The Goldstone picture (and « decoupling limit ») of non linear massive gravity, and what can one get from it ? Originally proposed in the analysis of Arkani-Hamed, Georgi and Schwartz using « Stückelberg » fields … and leads to the cubic action in the scalar sector (helicity 0) of the model Other cubic terms omitted With  = (m4 MP)1/5 « Strong coupling scale » (hidden cutoff of the model ?)
  • 28. Analogous to the Stuckelberg « trick » used to introduce gauge invariance into the Proca Lagrangian (action for a massive photon) F¹ºF¹º + m2 A¹A¹ Unitary gauge The obtained theory has the gauge invariance A¹ ! A¹ + @¹® Á ! Á ¡ ® A¹ ! A¹ + @¹ÁDo then the replacement with the new field Á The Proca action is just the same theory written in the gauge, while gets a kinetic term via the Proca mass term ( ) Á = 0 Á m2 A¹A¹ ! m2 @¹Á@¹ Á g  f 
  • 29. Do the same for non linear massive gravity The theory considered has the usual diffeo invariance g¹º(x) = @¹x0¾ (x)@ºx0¿ (x)g0 ¾¿ (x0 (x)) f¹º(x) = @¹x0¾ (x)@ºx0¿ (x)f0 ¾¿ (x0 (x)) This can be used to go from a « unitary gauge » where fAB = ´AB To a « non unitary gauge » where some of the d.o.f. of the g metric are put into f thanks to a gauge transformation of the form f¹º(x) = @¹XA (x)@ºXB (x)´AB (X(x)) g¹º(x) = @¹XA (x)@ºXB (x)gAB (X(x)) g¹º(x)x ¹ ´AB X A f¹º(x) XA (x) A Á
  • 30. One (trivial) example: our spherically symmetric ansatz gABdxA dxB = ¡J(r)dt2 + K(r)dr2 + L(r)r2 d-2 fABdxA dxB = ¡dt2 + dr2 + r2 d-2 Gauge transformation g¹ºdx¹ dxº = ¡eº(R) dt2 + e¸(R) dR2 + R2 d-2 f¹ºdx¹ dxº = ¡dt2 + µ 1 ¡ R¹0 (R) 2 ¶2 e¡¹(R) dR2 + e¡¹(R) R2 d-2
  • 31. Expand the theory around the unitary gauge as XA (x) = ±A ¹ x¹ + ¼A (x) ¼A (x) = ±A ¹ (A¹ (x) + ´¹º @ºÁ) : Unitary gauge coordinates « pion » fields The interaction term expanded at quadratic order in the new fields A and  reads A gets a kinetic term via the mass term  only gets one via a mixing term M2 P m2 8 Z d4 x £ h2 ¡ h¹ºh¹º ¡ F¹ºF¹º ¡4(h@A ¡ h¹º@¹ Aº ) ¡ 4(h@¹ @¹Á ¡ h¹º@¹ @º Á)]
  • 32. One can demix  from h by defining h¹º = ^h¹º ¡ m2 ´¹ºÁ And the interaction term reads then at quadratic order S = M2 P m2 8 Z d4 x n ^h2 ¡ ^h¹º ^h¹º ¡ F¹ºF¹º ¡ 4(^h@A ¡ ^h¹º@¹ Aº ) +6m2 h Á(@¹@¹ + 2m2 )Á ¡ ^hÁ + 2Á@A io The canonically normalized  is given by ~Á = MP m2 Á Taking then the « Decoupling Limit » One is left with … MP ! 1 m ! 0 ¤ = (m4 MP )1=5 » const T¹º=MP » const;
  • 33. With  = (m4 MP)1/5 E.g. around a heavy source: of mass M + + …. Interaction M/M of the external source with þà P The cubic interaction above generates O(1) coorrection at R = Rv ñ (RSmà 4 )1=5 In the decoupling limit, the Vainshtein radius is kept fixed, and one can understand the Vainshtein mechanism as ® ( ~Á)3 + ¯ ( ~Á ~Á;¹º ~Á;¹º ) and ® and ¯ model dependent coefficients « Strong coupling scale » (hidden cutoff of the model ?)
  • 34. An other non trivial property of non-linear Pauli-Fierz: at non linear level, it propagates 6 instead of 5 degrees of freedom, the energy of the sixth d.o.f. having no lower bound! Using the usual ADM decomposition of the metric, the non-linear PF Lagrangian reads (for   flat) With Neither Ni , nor N are Lagrange multipliers 6 propagating d.o.f., corresponding to the gij The e.o.m. of Ni and N determine those as functions of the other variables Boulware, Deser ‘72
  • 35. Moreover, the reduced Lagrangian for those propagating d.o.f. read ) Unbounded from below Hamiltonian Boulware, Deser 1972 This can be understood in the « Goldstone » description C.D., Rombouts 2005 (See also Creminelli, Nicolis, Papucci, Trincherini 2005) Indeed the action for the scalar polarization Leads to order 4 E.O.M. ), it describes two scalars fields, one being ghost-like
  • 36. Summary of the first part: the 3 sins of massive gravity They can all be seen at the Decoupling Limit level • 1. vDVZ discontinuity Cured by the Vainshtein mechanism ? • 2. Boulware Deser ghost Can one get rid of it ? • 3. Low Strong Coupling scale Can one have a higher cutoff ?
  • 37. The end of part 1

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