Lorentz Center workshop Non-Linear Structure in the Modified Universe July 14 th Claudia de Rham

Transcription

1 Lorentz Center workshop Non-Linear Structure in the Modified Universe July 14 th 2014 Claudia de Rham

2 Modified Gravity Lorentz-Violating LV-Massive Gravity Ghost Condensate Horava- Lifshitz Cuscuton Extended HL Gravity as a spin-2 Lorentz-Invariant higher spin partially-massless spin-3 f(r) Chameleon Symmetron Massless spin-2 Non-minimally coupled (Scalar-Tensor) Brans-Dicke Galileon DBI-Galileon Massive Graviton-Galileon Multi-Galileon Massive Spin-2 DGP Cascading Gravity Quasidilaton Massive Gravity Bi/Multi Gravity

3 Modified Gravity Lorentz-Violating LV-Massive Gravity Ghost Condensate f(r) Chameleon Symmetron Horava- Lifshitz Cuscuton Extended HL Massless spin-2 Non-minimally coupled (Scalar-Tensor) Brans-Dicke Gravity as a spin-2 Galileon DBI-Galileon Massive Graviton-Galileon Multi-Galileon Lorentz-Invariant Cascading Gravity higher spin New dof usually couple weakly to matter (no direct screening) partially-massless spin-3 Massive Spin-2 DGP Quasidilaton Massive Gravity Bi/Multi Gravity

4 Modified Gravity Lorentz-Violating LV-Massive Gravity Ghost Condensate Horava- Lifshitz Cuscuton Extended HL Gravity as a spin-2 Lorentz-Invariant higher spin partially-massless spin-3 f(r) Chameleon Symmetron Massless spin-2 Non-minimally coupled (Scalar-Tensor) Brans-Dicke Galileon DBI-Galileon Massive Graviton-Galileon Multi-Galileon Massive Spin-2 DGP Cascading Gravity Quasidilaton Massive Gravity Bi/Multi Gravity

5 Modified Gravity Lorentz-Violating LV-Massive Gravity Ghost Condensate f(r) Chameleon Symmetron Horava- Lifshitz Cuscuton Extended HL Massless spin-2 Non-minimally coupled (Scalar-Tensor) Brans-Dicke Gravity as a spin-2 Galileon DBI-Galileon Massive Graviton-Galileon Multi-Galileon Lorentz-Invariant Cascading Gravity higher spin partially-massless spin-3 New dof screened via the Chameleon mechanism Massive Spin-2 DGP Quasidilaton Massive Gravity Bi/Multi Gravity

6 Modified Gravity Lorentz-Violating LV-Massive Gravity Ghost Condensate Horava- Lifshitz Cuscuton Extended HL Gravity as a spin-2 Lorentz-Invariant higher spin partially-massless spin-3 f(r) Chameleon Symmetron Massless spin-2 Non-minimally coupled (Scalar-Tensor) Brans-Dicke Galileon DBI-Galileon Massive Graviton-Galileon Multi-Galileon Massive Spin-2 DGP Cascading Gravity Quasidilaton Massive Gravity Bi/Multi Gravity

7 Modified Gravity Lorentz-Violating LV-Massive Gravity Ghost Condensate f(r) Chameleon Symmetron Horava- Lifshitz Cuscuton Extended HL Massless spin-2 Non-minimally coupled (Scalar-Tensor) Brans-Dicke Gravity as a spin-2 Galileon DBI-Galileon Massive Graviton-Galileon Multi-Galileon Lorentz-Invariant Cascading Gravity higher spin partially-massless spin-3 New dof screened via the Vainshtein mechanism Massive Spin-2 DGP Quasidilaton Massive Gravity Bi/Multi Gravity

8 Nicolis, Rattazzi, Trincherini, MG from Extra Dimensions Best way to obtain a sensible theory of Modified Gravity (with no ghost) is to start with GR 5d Gravity Soft massive Gravity (DGP) Galileons Hard massive Gravity Ghost-free mgr

9 Galileons Brane localized at y=p(x) CdR, Tolley,

10 Galileons (Cosmological Constant/tension on the brane) 4d curvature induced on the brane CdR, Tolley,

11 Galileons Induced Metric (disformal coupling) From From From From CdR, Tolley,

12 Galileons Finite Class of Interactions That enjoy a shift and Galileon symmetry Have no ghost (2 nd order eom) Enjoy a non-renormalization theorem Couples to matter with gravitational strength Nicolis, Rattazzi, Trincherini,

13 Galileons Finite Class of Interactions Galileon scalar fields can play an important role on cosmological scales (eg. Candidate for dark energy) That enjoy a shift and Galileon symmetry Yet remain frozen on short distances via a Have no ghost (2 nd order eom) Vainshtein mechanism Enjoy a non-renormalization theorem Couples to matter with gravitational strength Nicolis, Rattazzi, Trincherini,

14 Nicolis, Rattazzi, Trincherini, MG from Extra Dimensions Best way to obtain a sensible theory of Modified Gravity (with no ghost) is to start with GR 5d Gravity Soft massive Gravity (DGP) Galileons Hard massive Gravity Ghost-free mgr

15 4d gravity 5d Gravity Brane bending mode behaves as a cubic Galileon in the DL Infinite extra dimension Confine Gravity on a brane DGP model Dvali Gabadadze Porrati, 2000 Deffayet, PLB 502 (2001) 199 Luty, Porrati, Rattazzi, 2003

16 4d gravity 5d Gravity Infinite extra dimension Confine Gravity on a brane DGP model

17 Positive Tension Corresponds to deficit angle in the 2 extra dimensions More than One Extra Dimension

18 4d gravity 5d Gravity 5d Gravity Infinite extra dimension Confine Gravity on a brane DGP model Finite-size extra dimension KK or deconstruction Hard massive Gravity (or multi-gravity)

19 Ghost-free Massive Gravity Structure of mass term is essential to avoid BD ghost f: reference metric not gauge invariant Boulware & Deser, PRD 6, 3368 (1972) CdR & Gabadadze, PRD 82, (2010) CdR, Gabadadze & Tolley, PRL 106, (2011)

20 Stückelberg Fields Diff. invariance can always be restored By adding four Stückelberg Scalar Fields 5 dofs: Helicity-2: behaves as GR with small mass Heliciy-1: decouples from matter Helicity-0: behaves as a Galileon scalar field

21 Stückelberg Fields Helicity-0 mode behaves as a `Galileon in some limit Where the Galilean symmetry is manifest And the equations of motion for p are second order. Screened via the Vainshtein Mechanism! CdR & Gabadadze, 2010

22 Vainshtein Mechanism Helicity-0 mode p couples to matter with the same strength as gravity Vainshtein, PLB39, 393 (1972)

23 Vainshtein Mechanism The non-linearities (kinetic interactions) are essential to screen the helicity-0 mode Vainshtein, PLB39, 393 (1972)

24 Vainshtein Mechanism The interactions for the helicity-0 mode are important at a very low energy scale, Can we trust these irrelevant interactions?

25 Vainshtein mechanism Relatively well understood for cubic Galileon for Static & Spherically Symmetric configurations

26 Vainshtein mechanism Static & Spherically Symmetric Cubic Galileon Background is analytic for

27 Vainshtein mechanism Static & Spherically Symmetric Cubic Galileon Background is analytic for Vainshtein radius:

28 Vainshtein mechanism Static & Spherically Symmetric Cubic Galileon 1

29 Vainshtein mechanism Static & Spherically Symmetric Cubic Galileon For Sun-Earth System

30 Vainshtein mechanism Static & Spherically Symmetric Cubic Galileon For Sun-Earth System For Earth-Moon System For Hulse-Taylor Pulsar

31 Vainshtein mechanism Static & Spherically Symmetric Cubic Galileon For Sun-Earth System For Earth-Moon System For Hulse-Taylor Pulsar More careful analysis gives instead 10-10!

32 Beyond the SSS approximation Static & Spherically Symmetric configuration not always a good indicator! - important for structure formation! - important for other models beyond the cubic Galileon! One of the main challenges for the Vainshtein mechanism

33 Beyond the SSS approximation Two-body system - numerical solutions of joint system match isolated system + small corrections - deviation at the order of 4% (on already a very suppressed effect) From T. Hiramatsu, W. Hu, K. Koyama & F. Schmidt,

34 Effects on structure formation For cubic Galileon (or DGP), (mainly spherical collapse) Lue, Scoccimarro & Starkman, astro-ph/ Lue, Scoccimarro & Starkman, astro-ph/ Lue, astro-ph/ Afshordi, Geshnizjani & Khoury, Scoccimarro, (linear level) Khoury & Wyman, Chan & Scoccimarro, Schmidt, Schmidt, Hu & Lima

35 Effects on structure formation For cubic Galileon From Khoury & Wyman,

36 Vainshtein Mechanism Close to a source the interactions are important Perturbations end up being weakly coupled to matter,

37 Vainshtein Mechanism Close to a source the interactions are important Perturbations end up being weakly coupled to matter,

38 Vainshtein mechanism Relies on irrelevant operators that arise at a (VERY!) low energy scale Can we rely on irrelevant operators and not be dominated by Quantum Corrections?

39 Vainshtein mechanism Relies on irrelevant operators that arise at a (VERY!) low energy scale Can we rely on irrelevant operators and not be dominated by Quantum Corrections? Relies on a redressing of the Kinetic Term Perturbations about a background

40 Vainshtein mechanism Relies on irrelevant operators that arise at a (VERY!) low energy scale Can we rely on irrelevant operators and not be dominated by Quantum Corrections? Relies on a redressing of the Kinetic Term Does it change the causal structure? Is it acausal???

41 Vainshtein mechanism Relies on irrelevant operators that arise at a (VERY!) low energy scale No Wilsonian UV Can we rely on irrelevant operators and not be dominated by Quantum Corrections? completion Relies on a redressing of the Kinetic Term eg. see Eden, Landsho, Olive and Polkinghorne, The analytic S-matrix, 1966 Does it change the causal structure? Is it acausal???

42 Vainshtein in trouble Vainshtein mechanism relies on: Irrelevant operators: standard Cut-off reg. analysis would show the theory is not technically natural Exact Renormalization Group helps Redressing of the kinetic structure: standard analysis shows classical SL group velocities S-matrix not analytic. Duality may help

43 Galileon duality Gauge freedom in how introduce the Stückelberg field A Galileon Maps to another Galileon theory, with different coefficients

44 Eg.2 Dual to a Free theory A Free theory Maps to a specific quintic Galileon theory

45 Dual to a free theory The dual theory admits superluminal propagation in the vacuum! (ie. no matter or other sources)

46 Dual to a free theory The dual theory admits superluminal propagation in the vacuum! (ie. no matter or other sources) plane wave solutions Speed of fluctuations: Superluminal propagation for

47 Dual to a free theory The dual theory admits superluminal propagation Yet it maps to a free theory Trivially causal, unitary, UV complete,

48 Group vs front velocity No Paradox here! Group velocity is: - Not invariant - Has been observed to be SL in the real world - Was computed here classically: Valid till the strong coupling scale Classical phase velocity in dual theory

49 Group vs front velocity No Paradox here! Group velocity is: - Not invariant - Has been observed to be SL in the real world - Was computed here classically: Valid till the strong coupling scale Classical phase velocity in dual theory Classical and Quantum phase velocity in free theory

50 Group vs front velocity No Paradox here! Group velocity is: - Not invariant - Has been observed to be SL in the real world - Was computed here classically: Valid till the strong coupling scale Full phase velocity in dual theory Classical phase velocity in dual theory Classical and Quantum phase velocity in free theory

51 Group vs front velocity No Paradox here! Group velocity is: - Not invariant - Has been observed to be SL in the real world - Was computed here classically: Valid till the strong coupling scale Full phase velocity in dual theory Classical phase velocity in dual theory Classical and Quantum phase velocity in free theory

52 Group vs front velocity The classical group and phase velocities may depend on the field representation and may be SL The Causal structure is dictated by the front velocity The front velocity (and therefore causality) cannot be inferred by a simple classical calculation If the duality was going through at the quantum level one could compute the front velocity in the free theory. Since it is luminal we would infer that the quintic Galileon is actually causal

53 Classical Causal Structure Galileon coupled to a scalar field c with standard kinetic term The field c propagates luminaly independently of the configuration There are configurations where the Galileon is superluminal c is subluminal compared to the Galileon

54 Classical Causal Structure In the dual picture, has a standard kinetic term propagates luminaly acquires a non-standard kinetic term and propagates subluminaly

55

56 The Classical Causal structure remains preserved Shown for any field configuration, does not rely on plane waves does not rely on vacuum

57 If a UV completion exists then we should have

58 Vainshtein in trouble Vainshtein mechanism relies on: Irrelevant operators: standard Cut-off reg. analysis would show the theory is not technically natural Exact Renormalization Group helps Redressing of the kinetic structure: standard analysis shows classical SL group velocities S-matrix computation shows it is not analytic. Duality may help

59 Ridding on Irrelevant Operators Consider an arbitrary theory The theory exhibits the Vainshtein mechanism if

60 Ridding on Irrelevant Operators Consider an arbitrary theory The theory exhibits the Vainshtein mechanism if Coupling to heavier fields with naively detune the theory at least perturbatively would

61 Ridding on Irrelevant Operators Consider the exact Renormalization Group equation : regulator operator : effective average action : IR regulator Wetterich, 1993

62 Ridding on Irrelevant Operators Consider the exact Renormalization Group equation : regulator operator : effective average action : IR regulator Wetterich, 1993

63 Ridding on Irrelevant Operators Consider the exact Renormalization Group equation Deep in the Vainshtein Region, Fully Non-perturbatively CdR & Raquel Ribeiro,

64 Ridding on Irrelevant Operators Suppressing the loops Quantum corrections become irrelevant deep in the Vainshtein regime Fully Non-perturbatively CdR & Raquel Ribeiro,

65 Outlook Vainshtein mechanism is a highly non-linear process Need more insight - beyond the SSS -beyond the simplest cubic Galileon Strong Coupling, SL, S-matrix non-analyticity, no Wilsonian UV completion Need a better understanding of the cosmology in models exhibiting the Vainshtein mechanism If there is a resolution behind these issues, the Vainshtein mechanism should play a crucial role

66 Thank you!