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 28 th, 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? 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? Einstein’s equation + the Standard Model predict a decelerating universe, but this contradicts observations. The expansion of the Universe is accelerating! 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? Ta ke-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: G N = 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

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. How can we do gravity beyond GR? How can we do gravity beyond GR? Adam Solomon

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 How can we do gravity beyond GR? How can we do gravity beyond GR? 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

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: , and Hinterbichler arXiv: Adam Solomon

dRGT Massive Gravity in a Nutshell The unique non-linear action for a single massive spin-2 graviton is where f μν is an arbitrary reference metric which must be chosen at the start β n are the free parameters; the graviton mass is ~m 2 β n The e n 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 ( f μν ) for a constant scalar (M f ) – 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 self- accelerating 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, g μν Adam Solomon

Massive bigravity effectively competes with ΛCDM A comprehensive comparison to background data was undertaken by Akrami, Koivisto, & Sandstad [arXiv: ] 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: 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 β 1 nonzero is a viable alternative to ΛCDM The “minimal” model with only β 1 nonzero is a viable alternative to ΛCDM Adam Solomon

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

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: See also Könnig and Amendola, arXiv: Linearize metrics around FRW backgrounds, restrict to scalar perturbations {E g,f, A g,f, F g,f, and B g,f }: Full linearized Einstein equations (in cosmic or conformal time) can be found in ARS, Akrami, and Koivisto, arXiv: Adam Solomon

Scalar perturbations in massive bigravity ARS, Y. Akrami, and T. Koivisto, arXiv: (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 ( E g,f, A g,f, and B f - B g ) 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) (GR and massive bigravity) In GR, there is no anisotropic stress so E g (time-time perturbation) is related to δ through Poisson’s equation, In bigravity, the relation beteen E g 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 Growth rate/index (f/ γ) : measures growth of structures Modification of Newton’s constant in Poisson eq. (Q): Anisotropic stress ( η ): GR: Adam Solomon

The “observables”: Modified gravity parameters We have analytic solutions (messy) for A g and E g as (stuff) x δ, so Can immediately read off analytic expressions for Q and η : (h i are non-trivial functions of time; see ARS, Akrami, and Koivisto arXiv: , App. B) Can solve numerically for δ using Q and η : Adam Solomon

The minimal model (B 1 only) SNe only: B 1 = ± SNe + CMB + BAO: B 1 = ± ARS, Y. Akrami, & T. Koivisto arXiv: Adam Solomon

The minimal model (B 1 only) k = 0.1 h/Mpc: γ = 0.46 for B 1 = 1.35 γ = 0.48 for B 1 = 1.45

The minimal model (B 1 only) will measure η within 10% Euclid: k = 0.1 h/Mpc: γ = 0.46 for B 1 = 1.35 γ = 0.48 for B 1 = 1.45

21 cm HI forecasts arXiv: 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 B i parameters are degenerate at background level when more than one are nonzero Simple analytic argument: see arXiv: Can structure formation break this degeneracy? Yes! Yes! Adam Solomon

Two-parameter models B 1, B 3 ≠ 0; Ω Λ eff ~ 0.7 Adam Solomon

Two-parameter models B 1, B 3 ≠ 0; Ω Λ eff ~ 0.7 Adam Solomon

Two-parameter models B 1, B 3 ≠ 0; Ω Λ eff ~ 0.7 Adam Solomon

Two-parameter models Recall: B i parameters are degenerate at background level when more than one are nonzero Simple analytic argument: see arXiv: Can structure formation break this degeneracy? Yes! Yes! See arXiv: for extensive analysis We repeated the previous analysis for all cosmologically viable two-parameter models Note: we found an instability in the B 1 -B 2 model – dangerous? Adam Solomon

Bimetric Cosmology: Summary The β 1 -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: See also Könnig and Amendola, arXiv: 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: ] for introduction and background cosmology Adam Solomon

Doubly-coupled cosmology Y. Akrami, T. Koivisto, D. Mota, and M. Sandstad [arXiv: ] 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 β 2 ≠ 0 or β 3 ≠ 0 are now viable Adam Solomon

Doubly-coupled cosmology Y. Akrami, T. Koivisto, D. Mota, and M. Sandstad [arXiv: ] Candidate partially massless theory has non-trivial dynamics β 0 = β 4 = 3 β 2, β 1 = β 3 = 0: has partially-massless symmetry around maximally symmetric (dS) solutions (arXiv: ) 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 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: ] 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 Need to step beyond the confines of metric geometry Adam Solomon

Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv: ] Consider a Maxwell field After all, we make observations by tracking photons! Imagine it is minimally coupled to an effective metric, h μν : 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: ] 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: ] Rewrite the action as where Rewrite the action as where Adam Solomon

Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv: ] Rewrite the action as where 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: ]  Point particles move in an effective geometry defined by  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: ] Finsler geometry: the most general line element that is homogeneous of degree 2 in the coordinate intervals dx μ Related to disformal couplings (cf. Bekenstein, gr-qc/ ) We can define a quasimetric: Adam Solomon

Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv: ] 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: ] “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 self- accelerating cosmologies arXiv: Plot: ratio of the two scale factors as a function of redshift (in a particular model) Adam Solomon

B i parameters can be degenerate in FRW background The mass term in the Friedmann equation acts as an effective CC density today: while y 0 obeys a quartic equation (with Ω m + Ω Λ eff = 1): The mass term in the Friedmann equation acts as an effective CC density today: while y 0 obeys a quartic equation (with Ω m + Ω Λ eff = 1): So if multiple B i parameters are nonzero, then fixing Ω Λ eff only restricts to a parameter subspace. E.g., if B 1, B 2 ≠ 0: Adam Solomon

B i 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 (B 1 only) k = 0.1 h/Mpc: γ = 0.46 for B 1 = 1.35 γ = 0.48 for B 1 = 1.45 Adam Solomon

The minimal model (B 1 only) k = 0.1 h/Mpc: γ = 0.46 for B 1 = 1.35 γ = 0.48 for B 1 = 1.45 will measure η within 10% Euclid: Adam Solomon

Two metrics or none? Y. Akrami, T. Koivisto, and ARS [arXiv: ] 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