1 / 62

Are Two Metrics Better than One ? The Cosmology of Massive Bigravity

Are Two Metrics Better than One ? The Cosmology of Massive Bigravity. Adam R. Solomon DAMTP, University of Cambridge Work in collaboration with: Yashar Akrami (Oslo), Tomi Koivisto (Nordita) Universit ä t Heidelberg, May 28 th , 2014. Why consider two metrics in the first place?.

giona
Download Presentation

Are Two Metrics Better than One ? The Cosmology of Massive Bigravity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Are Two Metrics Better than One?The Cosmology of Massive Bigravity Adam R. Solomon DAMTP, University of Cambridge Work in collaboration with:Yashar Akrami (Oslo), Tomi Koivisto (Nordita) Universität Heidelberg, May 28th, 2014

  2. Why consider two metrics in the first place? (Death to Occam’s razor?) Adam Solomon

  3. Why consider two metrics? • Field theoretic interest: how do we construct consistent interactions of multiple spin-2 fields? • NB “Consistent” crucially includes ghost-free • My motivation: modified gravity  massive graviton • 1) The next decade will see multiple precision tests of GR – we need to understand the alternatives • 2) The accelerating universe Adam Solomon

  4. Dark energy or modified gravity? Einstein’s equation + the Standard Model predict a decelerating universe, but this contradicts observations. The expansion of the Universe is accelerating! What went wrong?? Two possibilities: • Dark energy: Do we need to include new “stuff” on the RHS? • Modified gravity: Are we using the wrong equation to describe gravity at cosmological distances? Adam Solomon

  5. Cosmic acceleration has theoretical problems which modified gravity might solve • Technically natural self-acceleration: Certain theories of gravity may have late-time acceleration which does not get destabilized by quantum corrections. • This is THE major problem with a simple cosmological constant • Degravitation: Why do we not see a large CC from matter loops? Perhaps an IR modification of gravity makes a CC invisible to gravity • This is natural with a massive graviton due to short range Adam Solomon

  6. Why consider two metrics? • Take-home message: Massive bigravity is a natural, exciting, and still largely unexplored new direction in modifying GR. Adam Solomon

  7. How can we do gravity beyond GR?Some famous examples • Brans-Dicke (1961): make Newton’s constant dynamical:GN = 1/ϕ, gravity couples non-minimally to ϕ • f(R) (2000s): replace Einstein-Hilbert term with a general function f(R) of the Ricci scalar Adam Solomon

  8. How can we do gravity beyond GR? • These theories are generally not simple • Even f(R) looks elegant in the action, but from a degrees of freedom standpoint it is a theory of a scalar field non-minimally coupled to the metric, just like Brans-Dicke, Galileons, Horndeski, etc. Adam Solomon

  9. How can we do gravity beyond GR? • Most attempts at modifying GR are guided by Lovelock’s theorem (Lovelock, 1971): • The game of modifying gravity is played by breaking one or more of these assumptions GR is the unique theory of gravity which • Only involves a rank-2 tensor • Has second-order equations of motion • Is in 4D • Is local and Lorentz invariant Adam Solomon

  10. Another path: degrees of freedom(or, Lovelock or Weinberg?) • GR is unique. • But instead of thinking about that uniqueness through Lovelock’s theorem, we can also remember that (Weinberg, others, 1960s)… GR = massless spin-2 • A natural way to modify GR: give the graviton mass! Adam Solomon

  11. Non-linear massive gravity is a very recent development • At the linear level, the correct theory of a massive graviton has been known since 1939 (Fierz, Pauli) • But in the 1970s, several issues – most notably a dangerous ghost instability (mode with wrong-sign kinetic term) – were discovered Adam Solomon

  12. Non-linear massive gravity is a very recent development • Only in 2010 were these issues overcome when de Rham, Gabadadze, and Tolley (dRGT) wrote down the ghost-free, non-linear theory of massive gravity • See the reviews byde Rham arXiv:1401.4173, andHinterbichler arXiv:1105.3735 Adam Solomon

  13. dRGT Massive Gravity in a Nutshell • The unique non-linear action for a single massive spin-2 graviton iswhere fμν is an arbitrary reference metric which must be chosen at the start • βn are the free parameters; the graviton mass is ~m2βn • The en are elementary symmetric polynomials given by… Adam Solomon

  14. For a matrix X, the elementary symmetric polynomials are ([] = trace) Adam Solomon

  15. Much ado about a reference metric? • There is a simple (heuristic) reason that massive gravity needs a second metric: you can’t construct a non-trivial interaction term from one metric alone: • We need to introduce a second metric to construct interaction terms. • There are many dRGT massive gravity theories • What should this metric be? Adam Solomon

  16. From massive gravity to massive bigravity • Simple idea (Hassan and Rosen, 2011): make the reference metric dynamical • Resulting theory: one massless graviton and one massive – massive bigravity Adam Solomon

  17. From massive gravity to massive bigravity • By moving from dRGT to bimetric massive gravity, we avoid the issue of choosing a reference metric (Minkowski? (A)dS? Other?) • Trading a constant matrix (fμν) for a constant scalar (Mf) – simplification! • Allows for stable, flat FRW cosmological solutions (do not exist in dRGT) • Bigravity is a very sensible theory to consider Adam Solomon

  18. Massive bigravity has self-accelerating cosmologies • Homogeneous and isotropic solution:the background dynamics are determined byAs ρ -> 0, y -> constant, so the mass term approaches a (positive) constant late-time acceleration • NB: We are choosing (for now) to only couple matter to one metric, gμν Adam Solomon

  19. Massive bigravity effectively competes with ΛCDM • A comprehensive comparison to background data was undertaken by Akrami, Koivisto, & Sandstad [arXiv:1209.0457] • Data sets: • Luminosity distances from Type Ia supernovae (Union 2.1) • Position of the first CMB peak – angular scale of sound horizon at recombination (WMAP7) • Baryon-acoustic oscillations (2dFGRS, 6dFGS, SDSS and WiggleZ) Adam Solomon

  20. Massive bigravity effectively competes with ΛCDM • A comprehensive comparison to background data was undertaken by Akrami, Koivisto, & Sandstad (2012), arXiv:1209.0457 • Take-home points: • No exact ΛCDM without explicit cosmological constant (vacuum energy) • Dynamical dark energy • Phantom behavior (w < -1) is common • The “minimal” model with only β1 nonzero is a viable alternative to ΛCDM Adam Solomon

  21. Massive bigravity effectively competes with ΛCDM Y. Akrami, T. Koivisto, and M. Sandstad[arXiv:1209.0457] See also F. Könnig, A. Patil, and L. Amendola [arXiv:1312.3208]; ARS, Y. Akrami, and T. Koivisto [arXiv:1404.4061]

  22. On to the perturbations… These models provide a good fit to the background data, but look similar to ΛCDM and can be degenerate with each other. Can we tease these models apart by looking beyond the background to structure formation? (Spoiler alert: yes.) Adam Solomon

  23. Scalar perturbations in massive bigravity • Extensive analysis of perturbations undertaken by ARS, Y. Akrami, and T. Koivisto, arXiv:1404.4061 • See also Könnig and Amendola, arXiv:1402.1988 • Linearize metrics around FRW backgrounds, restrict to scalar perturbations {Eg,f, Ag,f, Fg,f, and Bg,f}: • Full linearized Einstein equations (in cosmic or conformal time) can be found in ARS, Akrami, and Koivisto, arXiv:1404.4061 Adam Solomon

  24. Scalar perturbations in massive bigravity ARS, Y. Akrami, and T. Koivisto, arXiv:1404.4061 (gory details) • Most observations of cosmic structure are taken in the subhorizon limit: • Specializing to this limit, and assuming only matter is dust (P=0)… • Five perturbations (Eg,f, Ag,f, and Bf - Bg) are determined algebraically in terms of the density perturbation δ • Meanwhile, δis determined by the same evolution equation as in GR: Adam Solomon

  25. (GR and massive bigravity) • In GR, there is no anisotropic stress so Eg (time-time perturbation) is related to δ through Poisson’s equation, • In bigravity, the relation beteen Eg and δ is significantly more complicatedmodified structure growth Adam Solomon

  26. The “observables”:Modified gravity parameters We calculate three parameters which are commonly used to distinguish modified gravity from GR: • Growth rate/index (f/γ): measures growth of structures • Modification of Newton’s constant in Poisson eq. (Q): • Anisotropic stress(η): GR: Adam Solomon

  27. The “observables”:Modified gravity parameters • We have analytic solutions (messy) for Ag and Eg as (stuff) x δ, so • Can immediately read off analytic expressions for Q and η:(hi are non-trivial functions of time; see ARS, Akrami, and Koivisto arXiv:1404.4061, App. B) • Can solve numerically for δusing Q and η: Adam Solomon

  28. The minimal model (B1 only) SNe + CMB + BAO:B1 = 1.448 ± 0.0168 SNe only:B1 = 1.3527 ± 0.0497 ARS, Y. Akrami, & T. KoivistoarXiv:1404.4061 Adam Solomon

  29. The minimal model (B1 only) k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45

  30. The minimal model (B1 only) k = 0.1 h/Mpc: Euclid: will measure ηwithin 10% γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45

  31. 21 cm HI forecasts arXiv:1405.1452 Adam Solomon

  32. Euclid and SKA forecasts for bigravity in prep.[work with Yashar Akrami (Oslo), Tomi Koivisto (Nordita), and Domenico Sapone (Madrid)] Adam Solomon

  33. Two-parameter models • Bi parameters are degenerate at background level when more than one are nonzero • Simple analytic argument: see arXiv:1404.4061 • Can structure formation break this degeneracy? • Yes! Adam Solomon

  34. Two-parameter models B1, B3 ≠ 0; ΩΛeff~ 0.7 Adam Solomon

  35. Two-parameter models B1, B3 ≠ 0; ΩΛeff~ 0.7 Adam Solomon

  36. Two-parameter models B1, B3 ≠ 0; ΩΛeff~ 0.7 Adam Solomon

  37. Two-parameter models • Recall: Bi parameters are degenerate at background level when more than one are nonzero • Simple analytic argument: see arXiv:1404.4061 • Can structure formation break this degeneracy? • Yes! • See arXiv:1404.4061 for extensive analysis • We repeated the previous analysis for all cosmologically viable two-parameter models • Note: we found an instability in the B1-B2 model – dangerous? Adam Solomon

  38. Bimetric Cosmology: Summary • The β1-only model is a strong competitor to ΛCDM and is gaining increasing attention • Same number of free parameters • Technically natural? • This model – as well as extensions to other interaction terms – deviate from ΛCDM at background level and in structure formation. Euclid (2020s) should settle the issue. • Extensive analysis of perturbations undertaken by ARS, Akrami, & Koivisto in arXiv:1404.4061 • See also Könnig and Amendola, arXiv:1402.1988 Adam Solomon

  39. Generalization:Doubly-coupled bigravity • Question: Does the dRGT/Hassan-Rosen bigravity action privilege either metric? • No: The vacuum action (kinetic and potential terms) is symmetric under exchange of the two metrics:Symmetry: Adam Solomon

  40. Generalization:Doubly-coupled bigravity • The matter coupling breaks this symmetry. We can restore it (i.e., promote it from the vacuum theory to the full theory) by double coupling the matter: • New symmetry for full action:See Y. Akrami, T. Koivisto, D. Mota, and M. Sandstad [arXiv:1306.0004] for introduction and background cosmology Adam Solomon

  41. Doubly-coupled cosmology Y. Akrami, T. Koivisto, D. Mota, and M. Sandstad [arXiv:1306.0004] • Novel features (compared to singly-coupled): • Can have conformally-related solutions, • These solutions can mimic exact ΛCDM (no dynamical DE) • Only for special parameter choices • Models with onlyβ2 ≠ 0 or β3 ≠ 0 are now viable Adam Solomon

  42. Doubly-coupled cosmology Y. Akrami, T. Koivisto, D. Mota, and M. Sandstad [arXiv:1306.0004] • Candidate partially massless theory has non-trivial dynamics • β0 = β4 = 3β2, β1 = β3 = 0: has partially-massless symmetry around maximally symmetric (dS) solutions (arXiv:1208.1797) • This is a new gauge symmetry which eliminates the helicity-0 mode (no fifth force, no vDVZ discontinuity) and fixes and protects the value of the CC/vacuum energy • Attractive solution to the CC problems! • However the singly-coupled version does not have non-trivial cosmologies • Our doubly-coupled bimetric theory results in a natural candidate PM gravity with viable cosmology • Remains to be seen: is this really partially massless? • All backgrounds? Fully non-linear symmetry? Adam Solomon

  43. Problem: how do we relate theory to observation? • There is no single “physical” metric • See Akrami, Koivisto, and ARS [arXiv:1404.0006] • So how do we connect observables (e.g., luminosity distance, redshift) to theory parameters? • GR: derivations of these observables rely heavily on knowing what the metric of spacetime is • Doubly-coupled theory possess mathematically two metrics, but physically none • Need to step beyond the confines of metric geometry Adam Solomon

  44. Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] • Consider a Maxwell field • After all, we make observations by tracking photons! • Imagine it is minimally coupled to an effective metric, hμν: • This implies • But the 00-00, 00-ii, and ii-ii components of this cannot be satisfied simultaneously! • No effective metric for photons Adam Solomon

  45. Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] • The point particle of mass m:has the “geodesic” equation:Is this the geodesic equation for any metric?NO. Adam Solomon

  46. Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] • Rewrite the actionaswhere Adam Solomon

  47. Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] • Rewrite the actionaswhere • It is easy to see that the point particle action can be written Adam Solomon

  48. Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] • Point particles move in an effective geometry defined by • This is not a metric spacetime. Rather, it is the line element of a Finsler geometry. Adam Solomon

  49. Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] • Finsler geometry: the most general line element that is homogeneous of degree 2 in the coordinate intervals dxμ • Related to disformal couplings (cf. Bekenstein, gr-qc/9211017) • We can define a quasimetric: Adam Solomon

More Related