Are Two Metrics Better than One?

The Cosmology of Massive Bigravity

Adam R. SolomonDAMTP, University of Cambridge

Work in collaboration with:Yashar Akrami (Oslo), Tomi Koivisto (Nordita)

Universität Heidelberg, May 28th, 2014

Adam Solomon

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

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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

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Why consider two metrics?

Take-home message: Massive bigravity is a natural, exciting, and still largely unexplored new direction in modifying GR.

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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

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These theories are generally not simpleEven 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?

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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?

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

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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

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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

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…

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

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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?

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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

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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 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, 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

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]

Massive bigravity effectively competes with ΛCDM

Adam Solomon

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:

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(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

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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:

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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

ARS, Y. Akrami, & T. KoivistoarXiv:1404.4061

The minimal model (B1 only)

k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45

The minimal model (B1 only)

will measureη within 10%

Euclid:

k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45

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21 cm HI forecastsarXiv:1405.1452

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Euclid and SKA forecasts for bigravity in prep.

[work with Yashar Akrami (Oslo), Tomi Koivisto

(Nordita), and Domenico Sapone (Madrid)]

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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!

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Two-parameter modelsB1, B3 ≠ 0; ΩΛ

eff ~ 0.7

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Two-parameter modelsB1, B3 ≠ 0; ΩΛ

eff ~ 0.7

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Two-parameter modelsB1, B3 ≠ 0; ΩΛ

eff ~ 0.7

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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 analysisWe 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

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Generalization:Doubly-coupled bigravityQuestion: 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:

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Generalization:Doubly-coupled bigravityThe 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” metricSee 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 fieldAfter 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:

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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

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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

Summary1. 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: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 backgroundThe 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 ΩΛ

eff only restricts to a parameter subspace. E.g., if B1, B2 ≠ 0:

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Bi parameters can be degenerate in FRW background

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Subhorizon evolution equationsg metric

Energy constraint (0-0 Einstein equation):

Trace i-j Einstein equation:

Off-diagonal (traceless) i-j Einstein equation:

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Subhorizon evolution equationsf metric

Energy constraint (0-0 Einstein equation):

Trace i-j Einstein equation:

Off-diagonal (traceless) i-j Einstein equation:

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The minimal model (B1 only)

k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45

Adam Solomon

The minimal model (B1 only)

k = 0.1 h/Mpc: γ = 0.46 for B1 = 1.35 γ = 0.48 for B1 = 1.45

will measureη within 10%

Euclid:

Adam Solomon

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,