Report

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 Why consider two metrics in the first place? (Death to Occam’s razor?) Adam Solomon 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 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 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 Why consider two metrics? Take-home message: Massive bigravity is a natural, exciting, and still largely unexplored new direction in modifying GR. Adam Solomon 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 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 How can we do gravity beyond GR? Most attempts at modifying GR are guided by Lovelock’s theorem (Lovelock, 1971): 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 The game of modifying gravity is played by breaking one or more of these assumptions Adam Solomon Einstein-DilatonGauss-Bonnet Tessa Baker Cascading gravity ✓ Strings & Branes f DGP Randall-Sundrum Ⅰ& Ⅱ R ⇤ 2T gravity Higher dimensions Lorentz violation Hoř ava-Lifschitz ◆ Conformal gravity f (G) Some degravitation scenarios Non-local Higher-order f (R) General Rμν Rμν , ☐R,etc. Kaluza-Klein Generalisations of SEH Modiﬁed Gravity New degrees of freedom Gauss-Bonnet arXiv: 1310.1086 1209.2117 1107.0491 1110.3830 Einstein-Aether Lorentz violation TeVeS Lovelock gravity Vector Scalar-tensor & Brans-Dicke Ghost condensates Galileons the Fab Four Scalar Massive gravity Bigravity Chern-Simons Cuscuton Tensor EBI Chaplygin gases Bimetric MOND KGB Coupled Quintessence f(T) Einstein-Cartan-Sciama-Kibble Horndeski theories Torsion theories 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 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 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 by de Rham arXiv:1401.4173, and Hinterbichler arXiv:1105.3735 Adam Solomon dRGT Massive Gravity in a Nutshell The unique non-linear action for a single massive spin-2 graviton is where 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 For a matrix X, the elementary symmetric polynomials are ([] = trace) Adam Solomon 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 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 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 Massive bigravity has selfaccelerating cosmologies Homogeneous and isotropic solution: the background dynamics are determined by As 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 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 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 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] 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 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 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, δ is determined by the same evolution equation as in GR: Adam Solomon (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 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): GR: Anisotropic stress (η): Adam Solomon 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 δ using Q and η: Adam Solomon The minimal model (B1 only) SNe only: B1 = 1.3527 ± 0.0497 SNe + CMB + BAO: B1 = 1.448 ± 0.0168 1.2 Growth of Structures SNe Ia CMB/BAO Combined ARS, Y. Akrami, & T. Koivisto arXiv:1404.4061 1 Likelihood 0.8 0.6 0.4 0.2 0 1.1 Adam Solomon 1.2 1.3 1.4 B1 1.5 1.6 1.7 The minimal model (B1 only) k = 0.1 h/Mpc 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 f(z): B1 = 1.35 f(z): B1 = 1.45 Best-fit Wg: B1 = 1.35 g Best-fit W : B1 = 1.45 0.55 0.5 0 1 2 3 4 5 z k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45 The minimal model (B1 only) k = 0.1 h/Mpc 1 Q: B1 = 1.35 Q: B1 = 1.45 h: B1 = 1.35 h: B1 = 1.45 Euclid: 0.98 will measure η within 10% 0.96 0.94 0.92 0.9 0.88 0.86 0 1 2 3 4 5 z k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45 21 cm HI forecasts arXiv:1405.1452 Adam Solomon Euclid and SKA forecasts for bigravity in prep. [work with Yashar Akrami (Oslo), Tomi Koivisto (Nordita), and Domenico Sapone (Madrid)] Adam Solomon 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 Two-parameter models B1, B3 ≠ 0; ΩΛeff ~ 0.7 k = 0.1 h/Mpc 1.1 1 0.9 0.8 0.7 f(z): B 1 = 1.1 f(z): B1 = 1.45 f(z): B 1 = 2.5 f(z): B 1 = 5 g Best-fit W : B1 = 1.1 g Best-fit W : B1 = 1.45 Best-fit Wg: B1 = 2.5 g Best-fit W : B1 = 5 0.6 0.5 Adam Solomon 0 1 2 3 z 4 5 Two-parameter models B1, B3 ≠ 0; ΩΛeff ~ 0.7 k = 0.1 h/Mpc 0.65 0.6 GR Best-fit g 0.55 0.5 B1 = 1.45 0.45 0.4 Adam Solomon 0 2 4 6 B1 8 10 Two-parameter models B1, B3 ≠ 0; ΩΛeff ~ 0.7 k = 0.1h/Mpc 1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 Q: z = 0.5 Q: z = 1.5 h: z = 0.5 h: z = 1.5 B1 = 1.45 0.84 0.82 Adam Solomon 0 2 4 6 B1 8 10 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 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 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 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 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 only β2 ≠ 0 or β3 ≠ 0 are now viable Adam Solomon 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 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 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 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 Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] Rewrite the action as where Adam Solomon Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] Rewrite the action as where It is easy to see that the point particle action can be written Adam Solomon 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 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 Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] We can define proper time the usual way (dτ = -ds): Massive point particles move on timelike geodesics of unit norm Can now extend to massless particles as in GR (einbein) Crucial question: Does this describe photons?? Can be straightforwardly extended to multimetric theories Adam Solomon Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] “The geometry that emerges for an observer in a bimetric spacetime depends quite nontrivially upon, in addition to the two metric structures, the observer's four-velocity. This means she is disformally coupled to her own four-velocity, and thus effectively lives in a Finslerian spacetime.” Adam Solomon Summary 1. Massive gravitons can exist 2. Interacting spin-2 fields can exist 3. Late-time acceleration can be addressed (self-acceleration) 4. Dynamical dark energy – serious competitor to ΛCDM! 5. Clear non-GR signatures in large-scale structure: Euclid 6. Can couple both metrics to matter: truly bimetric gravity 7. Exciting cosmological implications: exact ΛCDM, PM, etc. 8. Conceptual challenges: mathematically two metrics, physically none. Finsler geometry? “Paradoxically, once we have doubled the geometry, we lose the ability to use its familiar methods. This is a call to go back to the basics, and rediscover the justifications for results which we have taken for granted for the better part of the last century.” Adam Solomon What’s next? Singly-coupled bigravity: Forecasts for Euclid Superhorizon scales: CMB (Boltzmann + ISW), inflation, tensor modes Nonlinear regime (N-body simulations) Inflation from bigravity Instabilities? Do they exist, are they dangerous? Doubly-coupled bigravity: Light propagation (connection to observables) Cosmological constraints (subhorizon, superhorizon, nonlinear) Inflationary constraints Local constraints Dark matter Adam Solomon Massive bigravity has selfaccelerating cosmologies arXiv:1209.0457 Plot: ratio of the two scale factors as a function of redshift (in a particular model) Adam Solomon Bi parameters can be degenerate in FRW background The mass term in the Friedmann equation acts as an effective CC density today: while y0 obeys a quartic equation (with Ωm + ΩΛeff= 1): So if multiple Bi parameters are nonzero, then fixing ΩΛeff only restricts to a parameter subspace. E.g., if B1, B2 ≠ 0: Adam Solomon Bi parameters can be degenerate in FRW background Adam Solomon Subhorizon evolution equations g metric Energy constraint (0-0 Einstein equation): Trace i-j Einstein equation: Off-diagonal (traceless) i-j Einstein equation: Adam Solomon Subhorizon evolution equations f metric Energy constraint (0-0 Einstein equation): Trace i-j Einstein equation: Off-diagonal (traceless) i-j Einstein equation: Adam Solomon The minimal model (B1 only) k = 0.1 h/Mpc 0.56 GR 0.54 0.52 Best-fit g 0.5 B1 = 1.35 0.48 0.46 B1 = 1.45 0.44 0.42 0.4 1.1 Solomon k Adam = 0.1 h/Mpc: 1.2 1.3 1.4 B1 γ = 0.46 for B1 = 1.35 1.5 1.6 1.7 γ = 0.48 for B1 = 1.45 The minimal model (B1 only) k = 0.1h/Mpc 1 Q: z = 0.5 Q: z = 1.5 h: z = 0.5 h: z = 1.5 Euclid: 0.98 will measure η within 10% B1 = 1.45 0.96 0.94 0.92 0.9 0.88 B1 = 1.35 0.86 1.1 Solomon k Adam = 0.1 h/Mpc: 1.2 1.3 1.4 B1 γ = 0.46 for B1 = 1.35 1.5 1.6 1.7 γ = 0.48 for B1 = 1.45 Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv:1404.0004] We can play the same game for other fields, e.g., a canonical scalar. This generally leads to a spacetime-varying mass for the scalar However, a massless scalar does have an effective metric, Adam Solomon