MPP-2019-260

Vainshtein Screening in Bimetric Cosmology

Marvin Luben,1, ∗ Angnis Schmidt-May,1, † and Juri Smirnov2, 3, ‡

1Max-Planck-Institut fur Physik (Werner-Heisenberg-Institut), Fohringer Ring 6, 80805 Munich, Germany2Center for Cosmology and AstroParticle Physics (CCAPP),

The Ohio State University, Columbus, OH 43210, USA3Department of Physics, The Ohio State University, Columbus, OH 43210, USA

We demonstrate that the early universe in bimetric theory is screened by the Vainshtein mecha-nism on an FLRW background. The spin-2 mass serves as the cosmological Vainshtein scale in thiscase. This allows us to quantitatively address early universe cosmology. In particular, in a globalanalysis, we study data from the cosmic microwave background radiation and local measurementsof the Hubble flow. We show that bimetric cosmology resolves the discrepancy in the local andearly-time measurements of the Hubble scale via an effective phantom dark energy component.

I. INTRODUCTION

Modifications and extensions to the theory of generalrelativity (GR) are highly restricted [1, 2]. One possibledirection is the addition of new degrees of freedom tothe massless spin-2 field in a consistent manner. Ghost-free bimetric theory describes a massless and a massivespin-2 field with fully non-linear (self-)interactions [3–18].It contains both general relativity and massive gravityin certain parameter limits in which the massive or themassless field decouples, respectively. As such, bimetrictheory fills a gap in the list of consistent field theoriesfor massive and massless particles with spin up to 2 andrepresents an extended gravitational theory with a richphenomenology.

Bimetric theory can address several open questionsin modern cosmology. The theory gives rise to self-accelerating solutions where the interaction energy be-tween the massive and massless spin-2 field acts as dy-namical dark energy [19–26]. The modification of thegravitational potential (as compared to GR) affects therequired Dark Matter abundance from galactic to galaxycluster scales [27]. Also, the massive spin-2 field itselfserves as a candidate for Dark Matter [28–30]. Despitethese successes, the stability of perturbations around thecosmological background [25, 26, 31–35] still poses anopen problem1. At early times, certain perturbations be-come large, rendering linear perturbation theory invalid.

In static systems with spherical symmetry an analo-gous behavior occurs [7, 36–39]: Non-linear effects in theperturbations are relevant and render the linear approxi-mation invalid. This is the well-known Vainshtein screen-ing mechanism [7] that restores GR in spacetime regions

∗ mlueben@mpp.mpg.de† angnissm@mpp.mpg.de‡ smirnov.9@osu.edu1 Linear perturbation theory breaks down at a scale that can be

pushed to arbitrary early times [35]. This is precisely the GR-limit of the theory.

where the energy density is large (compared to the spin-2mass). In massive gravity, this behavior is crucial becauseit cures the so-called vDVZ discontinuity [5, 6].

In this paper, we give a physical argument for the cos-mological version of the Vainshtein mechanism. We findthat the spin-2 mass sets the energy scale at which Vain-shtein screening sets in. Above this energy scale, theenergy density coming from the interaction of the mas-sive and massless mode is suppressed. We show that thenon-linear massless and massive modes indeed decouplein this regime. It then becomes clear that the universetransitions from a Vainshtein screened period at earlytimes into a late time de Sitter phase. We use this insightto discuss the instability of linear perturbations aroundthe cosmological background in the light of existing re-sults.

The Vainshtein mechanism is also known to play a cru-cial role in scalar-tensor galileon cosmologies [40]. It hasbeen pointed out, that at early times the galileon dy-namics is governed by self-interactions, which results ina suppressed energy density relative to matter and radi-ation [41]. In this work we focus on an analogous effectin the framework of bimetric theory.

We use our result to address a highly debated prob-lem of the ΛCDM model, the H0-tension. As discussedin the literature [42–55], CMB observations tend to pre-dict a lower Hubble scale than local measurements. Wedemonstrate that this tension can be resolved in a min-imal realization of bimetric theory with only two moreparameters than the ΛCDM-model. These are the massof the spin-2 field and its coupling constant to ordinarymatter. We perform a fit to the CMB, Cepheid, super-nova Type Ia (SN Ia) and baryon-accoustic-oscillations(BAOs) observables and demonstrate that the tensionwith the current data is alleviated. Instead, all observ-ables are within 1σ error intervals. This is achieved viaan effective phantom phase of the dark energy componentin the redshift interval 1 . z . 10. Finally, we commenton the compatibility between the different observablesand argue that the ΛCDM and the β0β1β4 models aredistinguishable with future measurements.

arX

iv:1

912.

0944

9v2

[gr

-qc]

26

Nov

202

0

2

II. REVIEW OF BIMETRIC THEORY

In this section, we review some technical aspects of thebimetric theory. Readers familiar with bimetric theorymight want to jump directly to the next section. For areview on bimetric theory we refer to Ref. [56]. For anintroduction to the broad field of massive gravity we referto Refs. [16, 57].

A. Action and equations of motion

The action of bimetric theory with standard modelmatter minimally coupled to one metric is [9, 10, 15]

S =m2g

2

ˆd4x

(√−g R(g) + α2

√−f R(f)

)−m2

g

ˆd4x√−g V (g, f) +

ˆd4x√−gLm(g,Φ) , (1)

where R(g) and R(f) are the Ricci scalars of the met-ric tensors gµν and fµν , respectively. The parameter mg

is the Planck mass of gµν and α parametrizes the ratioto the Planck mass of fµν . The interaction part of theLagrangian has been derived prior to the bimetric for-mulation in [9, 10]. The potential, whose structure isentirely fixed by the absence of ghost instabilities, reads

V (g, f) =

4∑n=0

βnen

(√g−1f

)(2)

in terms of the elementary symmetric polynomials en.The interaction parameters βn have mass-dimension 2 inour parametrization. Matter fields, collectively denotedby Φ, minimally couple to the metric gµν via the genericmatter Lagrangian Lm.

Varying the action (1) with respect to the metric ten-sors yields two sets of modified Einstein equations,

Ggµν + V gµν =1

m2g

Tµν , Gfµν + V fµν = 0 , (3)

where Ggµν and Gfµν are the Einstein tensors of gµν andfµν , respectively. The variations of the potential yield

V gµν = gµλ

3∑n=0

(−1)nβnYλ(n)ν

(√g−1f

), (4)

V fµν = fµλ

3∑n=0

(−1)nβ4−nYλ(n)ν

(√f−1g

), (5)

where the matrices Y λ(n)ν(S) are sums of different powers

and contractions of the matrix√g−1f [11]. Since matter

minimally couples to gµν , the stress-energy tensor for gµνis given by

Tµν =−2√−g

δ√−gLmδgµν

. (6)

For usual matter (invariant under diffeomorphisms),stress-energy is conserved

∇µTµν = 0 , (7)

where∇µ is the covariant derivative compatible with gµν .Due to the Bianchi identity ∇µGgµν = 0, this implies anon-shell constraint on the potential,

∇µV gµν = 0 , (8)

referred to as Bianchi constraint.

B. Proportional background

The theory has a well-defined mass spectrum onlyaround proportional backgrounds where the two metricsare related as fµν = c2gµν for some non-vanishing con-stant c [15, 58]. The solution is an Einstein background,Rµν(f) = Rµν(g) = Λgµν , with a cosmological constantgiven by

Λ =β0 + 3β1c+ 3β2c2 + β3c

3

=1

α2

(β1

c+ 3β2 + 3β3c+ β4c

2

).

(9)

The equality of the first and second line is required forthe consistency of the vacuum solution. It is a quarticpolynomial in c and each root corresponds to a differentEinstein solution to the bimetric field equations.

The Fierz-Pauli mass of the massive spin-2 mode thatpropagates on the proportional background is given by

m2FP =

(1 +

1

α2c2

)c(β1 + 2β2c+ β3c

2) , (10)

in terms of the bimetric parameters. For later use, let usintroduce the short-hand notation α = αc. In the rest ofthe paper, we will refer to α, mFP, and Λ as the physicalparameters of a solution [59].

C. FLRW solutions

To describe the universe on large scales, we assumespacetime to be homogeneous and isotropic according tothe cosmological principle. Both metrics take on FLRWform [20, 21, 23],

ds2g = a(η)2

(−dη2 + d~x2

)ds2f = b(η)2

(−(1 + µ(η))2dη2 + d~x2

),

(11)

where we fixed the time-reparametrization invariancesuch that we work in conformal time η of the g-metric.The scale factors a(η) and b(η) of the metrics gµν andfµν are functions of time only. The f -metric lapse (1+µ)parameterizes the relative twist between the coordinatesystems of the two metrics. In this sense, it is similar

3

to a Stuckelberg field since it would be shifted by timereparametrizations of the metric fµν . From now on, wesuppress the η-dependence. For convenience, let us in-troduce the Hubble rate and the ratio of the scale factorsas

H =a

a, y =

b

a, (12)

where the dot represents derivative w.r.t. conformal time.The conformal and physical Hubble rates are related viaH = aH.

We assume the universe to be filled with a perfect fluidwith stress-energy tensor,

Tµν = (ρm + pm)uµuν + pmgµν , (13)

where ρm is the energy density and pm the pressure of thematter fluid. They are related via the linear equation ofstate wm = pm/ρm. uµ is the 4-velocity of the fluid. Theconservation eq. (7) leads to the continuity equation,

ρm + 3H(1 + wm)ρm = 0 , (14)

which is solved by ρm = ρm,0a−3(1+wm), where ρm,0 is an

integration constant. The physical scale factor is relatedto the redshift as a = (1 + z)−1.

The Bianchi constraint on the dynamical branch2 issolved by

µ =y

Hy=y′

y, (15)

where we have introduced the derivative w.r.t. e-folds,′ = d/d ln a. On the dynamical branch, the modifiedFriedmann equations for gµν and fµν read,

3H2 = a2

(ρDE

m2g

+ρm

m2g

), (16)

3α2H2 = a2

(β1

y+ 3β2 + 3β3y + β4y

2

). (17)

Here, we defined the energy density coming from the po-tential as

ρDE

m2g

= β0 + 3β1y + 3β2y2 + β3y

3 , (18)

which can be interpreted as dynamical dark energy.Combining both Friedmann equations yields a quarticpolynomial for y,

α2β3y4 + (3α2β2 − β4)y3 + 3(α2β1 − β3)y2

+

(α2β0 − 3β2 +

α2ρm

m2g

)y − β1 = 0 , (19)

2 Another solution to the Bianchi constraint is the algebraicbranch, where the ratio of the scale factors is fixed, y = const.This branch was studied in the literature and found to be patho-logical, see, e.g., Refs. [26, 31, 60].

that can be thought of determining y as a function ofρm. The polynomial has in general up to four real-valuedroots, of which only one (the so-called finite branch) isphysical [24, 26]. On this branch, the ratio of scale factorsevolves from y = 0 in the early universe (where ρm =∞),to a finite constant value y = c in the asymptotic future(where ρm = 0, i.e. de Sitter space).

Taking the derivative w.r.t. e-folds and using the con-servation eq. (14), we can express y′ as a function of yas [22]

y′ =3(1 + ωm)α2y2ρm/m

2g

β1 − 3β3y2 − 2β4y3 + 3α2y2(β1 + 2β2y + β3y2),

(20)

where ρm is a function of y via eq. (19). In terms of thedynamical mass parameter

m2eff =

1 + α2y2

α2y2y(β1 + 2β2y + β3y

2) , (21)

we can rewrite eq. (20) as

y′

y=

(1 + wm)ρm/m2g

m2eff − 2H2

. (22)

In the following discussion we always assume an ex-pansion history on the finite branch3 implying m2

eff >2H2 [61].

III. VAINSHTEIN SCREENING

The Vainshtein mechanism was discovered in systems,where due to a locally increasing gravitational field, GRis restored by non-linear interactions [7, 37]. We trans-late the analysis to the case where the gravitational fieldvaries in time. By studying the cosmological backgroundand linear perturbations we identify the energy scaleat which non-linearities become important. Due to ouranalogy this can be interpreted as a cosmological Vain-shtein mechanism.

A. The standard Vainshtein mechanism

On the technical level, the Vainshtein screening effect iscaused by the strong coupling of the longitudinal helicity-0 mode of the massive spin-2 field.

In the case of a point source in bimetric theory, thecritical radius below which the non-linearities of the fieldsbecome strong enough is given by [38],

rV =

(rS

m2FP

)1/3

. (23)

3 The finite branch is only well-defined for β1 > 0 [24]. Hence, inthis paper we exclude models with β1 = 0.

4

Here, the Schwarzschild radius for a source of mass M isgiven by

rS =M

4πm2g

. (24)

For a central, point-like source (or on scales muchlarger than the massive object itself), the induced gravi-tational potential is then given by [29, 38, 62, 63],

φ(r) =

−1m2g

1r r rV ,

− 1m2

Pl

(1r + 4α2

3e−mFPr

r

)r rV .

(25)

Inside the Vainshtein sphere, i.e. when r rV, the grav-itational potential is the same as in GR with Planckmass mg. Outside the Vainshtein sphere, the mas-sive spin-2 field propagates, contributing an attractiveYukawa term to the potential. On scales much largerthan the Compton wavelength, r m−1

FP, the Yukawaterm is suppressed and the potential coincides with theone in GR, however with a different Planck mass givenby mPl =

√1 + α2mg, compared to the one inside the

Vainshtein sphere. In this sense, there are two differ-ent scales on which GR is recovered, but with differentPlanck masses. Note that we neglect an asymptotic cos-mological constant in eq. (25).

Given a Schwarzschild geometry, on a technical level,the crucial indicator for the transition from the Vain-shtein screened regime to the massive gravity regime, isthe radial, relative metric twist µ(r) [37, 63]. This func-tion drops off quickly outside the Vainshtein regime andcan be used as a small parameter in perturbation theory.Inside the Vainshtein regime µ(r) becomes large and theperturbative expansion breaks down. On the non-linearlevel, however, in this regime, the GR solution is recov-ered. We will show in the following, that the same be-havior is present on the FLRW background and use thetemporal metric twist function µ(η), introduced above,to study when it becomes non-linear.

For a source of finite size, the Vainshtein mechanismcan be at work, once the radius within which the mat-ter is concentrated is smaller than the Vainshtein radiusitself. This indicates that there exists a minimal densityfor the Vainshtein mechanism. In the next section, wewill elaborate on this observation and apply its logic tocosmology.

B. The Vainshtein mechanism in cosmology

In this section, we will explore possible implicationsof the standard Vainshtein mechanism for the Universefilled with a homogeneous energy density. From analogywith the galileon cosmology [40, 41], we expect that a cos-mological Vainshtein regime would be present in bimetrictheory, and restore a GR-like solution at early times andlarge densities in the universe.

In particular, using properties of the solution for astatic, spherically symmetric system around a compactsource, we would like to answer the following question:What is the critical density ρc of a homogeneous massdistribution, for which the entire mass lies inside its ownVainshtein radius? We can only give a very rough esti-mate for this value since the standard expression for theVainshtein radius is derived assuming that its value islarger than the radius of the source. In the following, wesimply extend its definition to all possible configurations.

Let us thus consider a spherical, constant mass or en-ergy distribution ρ(r) = const. This distribution can beinfinitely extended with radius R =∞ or have a large butfinite radius R; for our argument below, this will makeno difference. The mass M(r) enclosed within a distancer < R from the center of the mass distribution is,

M(r) =

ˆ r

0

d3r′ ρ(r′) = ρ

ˆ r

0

d3r′ =4π

3r3ρ . (26)

The Schwarzschild radius corresponding to this enclosedmass is,

rS =M(r)

4πm2g

=r3ρ

3m2g

. (27)

The Vainshtein radius corresponding to the mass M(r)is,

rV =

(rS

m2FP

) 13

=

(ρ

3m2gm

2FP

) 13

r . (28)

The mass within the radius r fits precisely inside its ownVainshtein radius, when r = rV. This gives the value forthe critical density,

ρc = 3m2gm

2FP . (29)

For less dense systems with ρ < ρc, the Vainshtein radiusis smaller than the radius that encloses its correspondingmass. For denser systems with ρ > ρc, the mass enclosedby a radius r lies entirely inside its own Vainshtein radius.

We can now apply this result to cosmology, assumingthat time evolution will not invalidate the arguments.Our reasoning suggests that there exists a critical valueρc for the homogeneous energy density ρ(t) in the Uni-verse, given by (29). Since ρ(t) is time-dependent andincreases with redshift, there will be a point in timewhen it passes the value ρc. Using Friedmann’s equation,H2 = ρ/(3m2

g), with ρ being the total energy density ofthe universe, this translates into a critical value for theHubble rate,

Hc = mFP . (30)

Moving backward in time, we reach the point tc at whichH(t) passes the value Hc. At this time, the energy den-sity takes the value ρc, for which its corresponding masslies entirely inside its own Vainshtein radius.

5

Vainshtein rgime

H-1

rV

-30-20-1001020

-30

-20

-10

0

10

20

30

log10H [eV]

log 10r[cm]

Vainshtein vs. Hubble scale

FIG. 1. For a spin-2 mass of mFP = 1 GeV, the size H−1 ofthe Universe and its cosmological Vainshtein radius rV areplotted as functions of the Hubble rate H. Since theuniverse expands faster than its Vainshtein sphere, itunavoidably becomes larger than this sphere. That happensat the critical Hubble rate Hc = mFP.

For a homogenous matter distribution like in cosmol-ogy, the Vainshtein radius scales linearly with radius,rV ∼ r. Therefore, if the energy density of the uni-verse is below its critical value, ρ < ρc (or equivalentlyH < mFP), each Hubble patch is outside the Vainshteinregime. Analogously, for larger energy densities, ρ > ρc,and hence at early times, each Hubble patch lies withinits Vainshtein region. Even though we apply a conceptderived in a spherically symmetric setup to a homoge-nous energy distribution, the equations do not single outa preferred point in space. This is consistent with trans-lational invariance of FLRW; our analogy does not spoilthe cosmological principle.

Instead, this allows for another, rather curious inter-pretation. In the expanding universe, let us treat the BigBang singularity as a central source and measure the dis-tance to it by the inverse Hubble scale H−1. From thisperspective, the profile of the energy density distributionis ρ ∼ H2 ∼ r−2 according to Friedmann’s equation. Thecorresponding Schwarzschild radius is rS = H−1. Calcu-lating the Vainshtein radius in this setup yields

rV =(m2

FPH)−1/3

, (31)

which of course leads to the same critical Hubble rate Hc

derived above as this is just the special case of the aboveanalysis for r = H−1.

In fig. 1, we show the different scalings of the Vain-shtein radius and of the size of the observable universeas functions of the Hubble rate H. The cosmologicalVainshtein radius scales as rV ∼ H−1/3 while the sizeof the observable universe scales as H−1. Therefore, atearly times when the Hubble rate is large, the Vainshteinradius is larger than the size of a Hubble patch. At thecritical value set by the spin-2 mass, the former surpasses

H−1

rV

H−1

rV

H−1 rV H−1 rV

Vainshtein (early times) de Sitter (late times)

FIG. 2. Schematical depiction of the expansion of theuniverse. Since the Universe expands faster than itsVainshtein radius, it leaves the Vainshtein sphere at thecritical Hubble rate.

the latter, and at late times a Hubble patch is larger thanits Vainshtein sphere.

In fig. 2, the evolution of the universe is schemati-cally depicted, and can be interpreted as follows. TheBig-Bang singularity serves as an analog of the centralsource in spherically symmetric systems. Moving awayfrom the source, i.e. letting time pass, the evolution isexpected to be governed by equations of motion equiva-lent to GR because we are inside the Vainshtein sphere.When the Hubble rate becomes comparable to the crit-ical value, the universe exits the Vainshtein regime andenters a transition phase where the evolution is governedby the bimetric field equations. Asymptotically in thefuture, the universe approaches de Sitter space. In thissense, also at late times, GR is effectively recovered.

We emphasize again that we made simplifying assump-tions about the validity of the expression for the Vain-shtein radius4. Thus our arguments here can at mostgive very rough estimates for the critical values of cosmo-logical quantities. Nevertheless, these approximate val-ues are supported by results in cosmological perturbationtheory, see section IV B. Moreover, we observe interest-ing behavior of the solutions already at the backgroundlevel, as we demonstrate in the next section.

IV. THE SPIN-2 MASS IN COSMOLOGY

In the previous section we have identified Hc = mFP asthe critical Hubble rate, at which the universe transitionsfrom a Vainshtein screened to an unscreened phase. In

4 Note that the standard derivation of the Vainshtein mechanism isrestricted to scales r which are much smaller than the Comptonwavelength of the massive spin-2 field, r m−1

FP. When consid-

ering the entire Universe, the Vainshtein radius is rV,crit = m−1FP

and thus violates this condition.

6

this section, we analyze how cosmological solutions tobimetric theory behave in the two regimes. In order topresent explicit results, we use the β0β1β4-model as toymodel for which we provide details in appendix A.

A. Vainshtein screening of background cosmology

In this section, we demonstrate at the level of Fried-mann’s eq. (16) that the background cosmology indeedtransitions from the GR to the bimetric phase exactly atthe energy scale mFP.

We start by discussing the scale factor ratio y. Asdiscussed in section II, on the finite branch the scale fac-tor ratio evolves from zero at early times to a constantdetermined by eq. (9) in the asymptotic future. Moreprecisely, we can expand eq. (19) for ρm/m

2g βn to

find

y = β1

(ρm

m2g

)−1

+O(ρm

m2g

)−2

. (32)

Here we introduced the parameters y = αy and βn =α−nβn for convenience [59]. Plugging this result into theexpression for the dark energy density (18) yields

ρDE

m2g

= β0 +3β2

1m2g

ρm+O

(ρm

m2g

)−2

. (33)

Since β0 is a constant we thus find that ρm ρDE inthe early universe, as pointed out already in [20]. Hence,the energy density coming from the interaction betweenthe spin-2 fields does not contribute to the Hubble rate atearly times. This effect represents cosmological screeningin close analogy to galileon cosmology [41].

Next, we want to study when this screening sets in. Assuch, we determine the energy scale at which y′ is at itsmaximum. We therefore solve the equation y′′ = 0 andinfer the corresponding Hubble rate. However, analyticalsolutions could not be found for this polynomial in theβ0β1β4-model due to its high degree. Instead, we solvey′′ = 0 for the two submodels with β0 = 0 and β4 = 0,respectively. The results in both cases are lengthy, butin the limit α 1 we get the remarkably simple result

H∗ ' mFP , (34)

as the critical Hubble rate at which y′′ = 0 for bothsubmodels, up to an O(1) factor5.

5 Note that the limit α 1 has very different meaning in bothsubmodels [59]. Since we fix one of the interaction parameters,one of the physical parameters is not free, but depends on theother parameters. For the submodel where β0 = 0 we get α2 =Λ/(3m2

FP − Λ) while for the submodel with β4 = 0 the relationis α2 = m2

FP/Λ − 1, cf. eq. (A1). Hence our limit in the formercase implies m2

FP Λ, while in the latter it implies m2FP ' Λ.

Consequently, for the β0β1-model there is only one energy scaleinvolved. Expanding around α 1 (i.e. m2

FP Λ) instead

results in H∗ ' (mFPΛ)1/3 as critical Hubble rate at whichy′′ = 0. Only the β0β1-model gives rise to this behavior.

Let us provide a physical interpretation for the param-eter y = αy. The non-linear massless field Gµν is givenby [58]

Gµν = gµν + α2fµν , (35)

while the non-linear massive field is not unique. OnFLRW (11), the spatial components read Gij = a2(1 +y2)δij . Hence, in the limit y 1 the non-linear masslessmode and the physical metric are aligned. This impliesthat the non-linear massless and massive modes decou-ple. This observation is consistent with the decouplingof the linear massless and massive mode in the limitα 1 [35, 64]. We conclude that y parametrizes themixing of the massless and massive field on an FLRWbackground.

However, we also have to consider the temporal com-ponent of the non-linear massless field that is given byG00 = −a2(1 + y2(1 + µ)2). The decoupling is not onlycontrolled by y, but also the Stuckelberg field µ. Hence,let us study the evolution of µ. At late times, the matter-energy density ρm vanishes implying that y′ vanishes andthe Stuckelberg field is small,

µ −→ 0 for η → +∞ , (36)

cf. eq. (15). At early times y approaches zero and the en-ergy density diverges as ρm/m

2g ∼ β1/y. The Stuckelberg

field approaches the constant value,

µ −→ 3(1 + wm) for η → −∞ , (37)

which is O(1). The Stuckelberg field µ hence transi-tions from a large value at early times to a small valuein the asymptotic future. Although this behavior doesnot spoil the decoupling, it is analogous to the effect inthe Schwarzschild geometry. Here, µ asymptotes to aconstant value when approaching the source [38].

In fig. 3, we plot the Stuckelberg field as a function ofredshift z in the β0β1β4-model for three exemplary caseswith different mixing angles α and spin-2 masses mFP

as displayed in the caption. The vertical lines representthe critical redshift at which H = mFP. Qualitatively wefind that the Stuckelberg field µ indeed starts to deviatefrom its asymptotic value as soon as the Hubble rate isof the order of mFP. We have explicitly checked this forvarious examples and find that the behavior is completelygeneric6. Below the energy scale, mFP the Stuckelbergfield becomes small and enters its linear regime. We

6 Note that for the case represented by the red line µ develops apeak. This happens only for models with β0 < 0 as we checkedexplicitly. For submodels with β0 = 0, µ does not develop apeak. Instead, the Stuckelberg field for these submodels alwaysdecreases monotonically in time. Since this behavior is alreadycaptured by the blue and green examples, we do not demonstratethat explicitly here. Let us only note that for all these submodels,µ starts to deviate from its asymptotic value 3(1 + wm) as soonas the Hubble rate falls below the spin-2 mass.

7

0 5 10 15 20 25 300

1

2

3

4

5

6

7

z

μβ0β1β4-model: Evolution of Stuckelberg field

FIG. 3. The time evolution of the Stuckelberg field µ as afunction of redshift z. For all lines we set Λ = 0.7H2

0 andρm,0/m

2g = 0.3H2

0 as exemplary values. The bimetricparameters are mFP = H0 and α = 0.5 (blue line),m2

FP = 10H20 and α = 2 (red line), and m2

FP = 20H20 and

α = 0.01 (green line). The vertical lines represent theredshift at which H = mFP, respectively In each case, µtransitions from the early time asymptotic value to 0 aroundthe scale mFP.

have already estimated the energy scale at which theStuckelberg field transitions from the linear to the non-linear regime in eq. (34).

In the limit of small α we find the same expressionfor the critical Hubble rate as we found with the Vain-shtein analogy. Away from the limit, the critical Hubblerate does not only depend on mFP but also on α andΛ. This can be seen in fig. 3 because the inflection pointy′′ = 0 and the critical redshift are not exactly aligned.Our analogy with the local Vainshtein mechanism, hence,gives a rough estimate of the critical Hubble rate. The re-markable feature however is, that in the limit α 1, thevalue of the critical Hubble rate at which the Stuckelbergfield becomes non-linear and background cosmology isscreened is set only by mFP and becomes independent ofα and Λ. In fig. 5, we show a schematic overview of thedifferent regimes in bimetric cosmology.

Next, we study the Hubble rate. In fig. 4 we show theHubble rate as function of redshift, for several exemplaryvalues. Again, the Hubble rate starts to deviate from theenergy scale set by ρm/m

2g when H ∼ mFP, as indicated

by the vertical lines. However, the deviations are sup-pressed by α. Therefore, the green (α = 0.01) and blue(α = 0.5) line almost coincide. For the red line, α = 2is not small and the transition between the two phasescan be seen directly from the plot. Since the parame-ters that lead to the red line imply β0 < 0, the energycontribution from the bimetric potential is negative forsufficiently large redshift. This implies that the Hubblerate is smaller than the energy scale set by ρm/m

2g.

To conclude, background cosmology of bimetric theoryresembles GR when the massless and massive mode de-

0 5 10 15 20 25 300.1

1

10

100

1000

104

z

H2/H02

β0β1β4-model: Evolution of Hubble rate

FIG. 4. The Hubble rate (normalized to H0) as a function ofredshift z is shown. For all lines we set Λ = 0.7H2

0 andρm,0/m

2g = 0.3H2

0 as exemplary values. The bimetricparameters are mFP = H0 and α = 0.5 (blue line),m2

FP = 10H20 and α = 2 (red line), and m2

FP = 20H20 and

α = 0.01 (green line). The vertical lines represent theredshift at which H = mFP, respectively The horizontaldashed lines indicate (mFP/H0)2 (blue, red, green) andΛ/3H2

0 (black).

couple. They decouple when y 1. This can be achievedeither by adjusting α 1, i.e. by going to the GR-limitof the theory. Then, deviations from GR are suppressedat all redshifts. Alternatively, y 1 for H mFP aswe have just shown. In this case, the energy density dueto the non-linear spin-2 interaction is screened and hencenegligible above the same energy scale that we encoun-tered in section III B.

B. Linear scalar perturbations and Vainshtein

In this section, we connect our previous results to cos-mological perturbations around the FLRW backgroundin bimetric theory. Their analysis received a lot of atten-tion in the literature [25, 26, 32–35, 65] and here we willsummarize and rephrase the conclusions. It was foundthat the linear scalar perturbations in the WKB approx-imation are unstable on subhorizon scales during earlytimes (for an expansion history on the finite branch).More precisely, they are stable as long as the dynamicalbound [26],

y′′ <y′

2y

2Hy′(y′ − 3wmy)− 3a2ρmy2(wm + 1)(2wm + 1)

a2ρmy(w + 1) +H2y′,

(38)

is satisfied. For models with β2 = β3 = 0 this is equiva-lent to y′′ < 0 [26].

Hence, the linear perturbations are unstable exactlywhen the background is screened. In other words, ex-actly when the Stuckelberg field becomes non-linear, lin-

8

Vainshtein/GR

H2 ' ρm/3m2g

Transition

H2 ' m2FP

de Sitter

H2 ' Λ/3

t

FIG. 5. Schematic summary of the cosmic evolution inbimetric theory.

ear perturbation theory breaks down. For the β0β1-and β1β4-model we already identified the energy scaleat which y′′ = 0. For completeness, we also compute atwhich energy scale the dynamical bound in eq. (38) isviolated for the remaining two parameter models, β1β2

and β1β3. The expression for the critical Hubble rate istoo long to display it here, but in the limit α 1 we findH∗ ' mFP (up to an O(1) factor)7 as well.

The expansion history and the scalar perturbations aresensitive to the energy scale set by mFP as we have ex-plicitly demonstrated for all the two-parameter models.When the Hubble rate is of the order of the spin-2 mass,H ∼ mFP, the Stuckelberg field µ becomes non-linearand the linear perturbations start to grow exponentially.At the same energy scale, the universe becomes smallerthan its own Vainshtein radius. Combining these resultssuggests that the Vainshtein mechanism is active also ona time-dependent background like the FLRW and withspatially extended matter sources. Our result suggeststhat the scalar perturbations are cured by the Vainshteinmechanism.

Indeed, the stability of scalar perturbations aroundFLRW background was studied in the literature. Theauthors of Ref. [65] solved the perturbation equationsnon-linearly for early times8. Their analysis identifiesthe spin-2 mass mFP as the scale at which the Vain-shtein mechanism kicks in and restores GR (for a genericmodel). In Ref. [66], on the other hand, the equationsof motion were solved for an inhomogeneous mass distri-bution non-linearly. Again, no instabilities were found.Both these results show that the instabilities are in-deed an artifact of the linear approximation and that theVainshtein mechanism is active also on a time-dependentbackground with a spatially extended source. In a dif-ferent setting, the traditional Vainshtein mechanism wasused to investigate the effect of early time instabilities onstructure formation, see Ref. [67].

7 For the β1β2-model, H∗ was already computed in Ref. [35] in thelimit α 1, but not in terms of the physical parameters. Theydid not interpret their result as the spin-2 mass.

8 In [65], spherically symmetric perturbations on subhorizon scalesand on length scales smaller than the Compton wavelength m−1

FPwere studied. Furthermore, the background was assumed to beproportional, gµν = y2fµν , with a time-dependent conformalfactor y. This is only a solution to the equations of motionwhen both metrics couple to their own matter sector, which areproportional on-shell.

V. IMPACT ON THE H0 TENSION

Despite the huge success of General Relativity de-scribing gravitational systems on many different scalesto enormous precision and in particular of the ΛCDMmodel, latest data challenge the Standard Model of Cos-mology. Local observations of the Hubble flow are ingood agreement with each other and constrain the valueof the Hubble rate today to h = 0.7324± 0.0174 [42, 52]where

h =H0

100km/s/Mpc(39)

is the normalized Hubble rate today. Recently, a newmeasurement of the local Hubble rate was performed us-ing Megamasers [69]. This observation relies on indepen-dent distance determinations and thus has no overlap insystematic errors with previous studies. A Hubble ratevalue of h = 0.739± 0.030 was found, in agreement withprevious local measurements.

In the ΛCDM model, CMB data from the Planck ex-periment, however, favor a value h = 0.6781±0.0092 [46].This causes a ∼ 3.4σ tension with the Hubble rate valuefrom local observations. While local measurements arequite sensitive to systematics [54, 55, 70], the constraintsfrom CMB measurements highly depend on the gravita-tional model [46]. Indeed, CMB data alone favors a DarkEnergy component with a phantom equation of state,wDE < −1 [71].

A. Phantom and negative dark energy

Several mechanisms were proposed to alleviate the dis-crepancy in the value of H0 inferred from late-time andCMB measurements. One possible direction is to lowerthe Hubble rate via late-time modifications of the ΛCDMmodel. This can be achieved by a dynamical dark energycomponent that grows as the universe expands (phantomdark energy) or that changes sign at a certain redshift, i.e.with a negative energy density at sufficiently early times.Bimetric theory features both mechanisms depending onthe parameters.

Models with a phantom equation of state should betreated with caution. Phantom energy violates the dom-inant energy condition [72] and causes a future spacetimesingularity (Big Rip) [73–76]. For simple models thatbuild on a single field (such as quintessence) a phantomequation of state implies the presence of a low-energyghost [77]. There are ways to get around these issues.The Big Rip can be avoided if the equation of statevaries in time and approaches −1 sufficiently fast [78–80]. Ghost condensation can stabilize the vacuum [81].

In contrast to these simple realizations, bimetric theoryincorporates an effective phantom equation of state nat-urally. The dark energy density ρDE contained in Fried-

9

-

-

-

-

-

~ -

~ -

FIG. 6. The equation of state of the effective dark energycomponent. Two different scales for the spin-2 mass arechosen. The lighter corresponds to the Hubble scale today(red line) and the heaver to the Hubble scale at redshiftz ' 100 (blue line).

mann’s eq. (16) has the equation of state

wDE = −1− α2y2

1 + α2y2

(1 + wm)ρm

ρDE

m2eff

m2eff − 2H2

. (40)

A cosmic expansion history on the finite branch impliesm2

eff > 2H2 and hence for ρDE > 0 the equation of state isphantom, wDE < −1 [26]. At early times the asymptoticvalue of wDE is either −1 (for models with β0 6= 0) or−2 − wm (for models with β0 = 0). In the asymptoticfuture (when ρm → 0) the equation of state approacheswDE → −1 and the effect of the dynamical Dark Energyreduces to a cosmological constant. Hence, the Big Ripis avoided in bimetric theory.

Moreover, the effective dark energy component ρDE

can change its sign. While in the infinite future thedynamical dark energy component approaches the valueof the asymptotic cosmological constant, at early timesρDE/m

2g → β0 because y → 0 on the finite branch.

Hence, if β0 < 0 the dark energy component is negativeabove a certain redshift. For the simple β0β1β4-modelthis redshift is determined by y = −β0/(3β1) as followsfrom eq. (18).

In Fig. 6, we show the time evolution of the Dark En-ergy equation of state for two different graviton masses.The phantom era of the cosmic evolution takes placewhen the size of the universe is comparable to the wave-length of the graviton, i.e. H(z) ' mFP and is relatedto the transition from the Vainshtein to the de Sitterregime.

The effective Dark Energy component violates the nullenergy condition (NEC) [82] allowing for a phantomequation of state while the sum of potential energy andmatter stress-energy satisfies the NEC. In bimetric the-ory, this does not imply the presence of a ghost modedue to the nontrivial interactions between the differentmodes that give rise to the effective phantom dark en-ergy. On the other hand, the NEC violation manifests

itself in linear perturbation theory as the gradient insta-bility. However, as we argued earlier, it appears to be anartifact of the calculation and higher-order terms have tobe taken into account due to the Vainshtein mechanism.It would be interesting to study the connection to ghostcondensation.

In the following section we perform a fit to data. We re-strict ourselves to the study of the β0β1β4-model. It is thesimplest minimal bimetric model where all three physicalparameters (mixing parameter α, spin-2 mass mFP, andcosmological constant Λ) are independent of each other.Already the authors of Ref. [52] studied phantom darkenergy as a possible resolution to the H0-tension9. Inparticular, the bimetric β0β1- and β1β2-model were usedas concrete models. They conclude that these models aredriven into their GR-limits and hence do not resolve theH0-tension. However, for these two-parameter-models ofbimetric theory, not all the physical parameters are in-dependent and the setup is too restricted. In particular,the fact that at small redshifts the equation of state isforced to be wDE ' −1 enforces a small mixing param-eter α 1. This is the GR-limit of these models andimplies an equation of state close to −1 at all redshifts.This is not the case for models where the three physicalparameters are free as we will demonstrate in this sec-tion with the β0β1β4-model. We find that solutions existwhich feature wDE ' −1 at small redshifts but deviatefrom that value at intermediate redshifts. In appendix Awe collect the equations that we need for the analysis.

B. Parametrization and scanning strategy

To treat cosmological observables of late and earlytimes on somewhat equal footings, we do the follow-ing. We approximate Friedmann’s equation at late timesby the Hubble law including the deceleration parameterq = −(1 + H/H2) as

H(z) = H0 + (1 + q)H0 z . (41)

Observations of Cepheid variables constrain the Hubblerate parameter today to be h = 0.7324 ± 0.0174 [42].The quoted value is derived from the combination of fourindependent Cepheid observables, which we will use toconstrain the local Hubble rate.

Furthermore, we use observations of Type Ia super-novae to constrain the deceleration parameter. We con-sider supernovae with redshifts z < 0.5 such that thedescription in terms of the deceleration parameter is stillvalid. We find the best-fit value q = −0.55 ± 0.1 whichis consistent with the results of Refs. [84–86]. This ap-proach projects a large number of SN-Ia observables on

9 Additionally in Ref. [83], the inverse distance ladder method isdiscussed with the goal of testing the parameter region in bimet-ric theory, relevant for the H0 tension.

10

one coarse-grained parameter. We choose this descrip-tion, to have a similar number of local and global ob-servables. We restrict our analysis to the supernovaewith moderate redshifts, as high redshift supernova ob-servations are subject to substantial luminosity uncer-tainties [87], and might be affected stronger by anisotropyeffects [88].

One crucial point of the present analysis is the follow-ing. We assume that the physics that controls the inho-mogeneities of the CMB in bimetric theory is identicalto GR. This assumption is justified since the Vainshteinmechanism ensures that GR is restored at early timesat the background level. For our analysis, we furtherassume, that small perturbations around this GR back-ground are well described by the standard CMB pertur-bation theory.

The essence of the CMB physics, can be well capturedby the following coarse-graining method, suggested inRef. [52]. At the core of the analysis, there are onlythree physical observables. Two of which are the shiftparameters based on the comoving angular distance tothe last scattering surface

DA(z∗) =

ˆ z∗

0

dz

H(z)given that ΩK = 0 , (42)

where z∗ is the redshift at which decoupling happens,and the sound horizon

rs(z∗) =

ˆ a∗

0

csda

a2H(a)=

1√3

ˆ a∗

0

da

a2H(a)√

1 + 3Ωb

4Ωγa,

(43)

where cs is the speed of sound that depends on the energydensity of baryons Ωb and of photons Ωγ . The physicallyconstrained combinations are

• the angular distance normalized to the Hubble hori-zon at decoupling R =

√Ωm,0H0DA(z∗),

• and the principle multipole number lA = πDA(z∗)rs(z∗) .

Here, Ωm,0 = ρm/(3m2gH

20 ) is the energy density to-

day of non-relativistic matter. The third parameteris the energy density of baryons at decoupling Ωbh

2.We use the CMB compressed likelihood [46] with val-ues (R, lA,Ωbh

2) = (1.7382, 301.63, 0.02262), errors(0.0088, 0.15, 0.00029) and the covariance matrix

DCMB =

1.0 0.64 −0.750.64 1.0 −0.55−0.75 −0.55 1.0

. (44)

Note that in our analysis, we modify the analysis ofRef. [52] by adding a weight factor in the χ2 function.The weight factor wp multiplies the contribution of lAto the χ2 funciton, and thus takes into account the peakmultiplicity in the CMB spectrum. We take wp = 7, as itcorresponds to the number of peaks, well resolved by the

[(/)/]

χ

Λ %

βββ %

FIG. 7. The χ2 functions and 95% confidence intervals ofH0, around the best-fit points of the ΛCDM model and theβ0β1β4-realization of bimetric theory. The fit improvementis substantial, with ∆χ2/dof ≈ 5.

Planck experiment. This approach approximately mod-els the statistical weight of the Planck data in comparisonto the local observables.

The same physical scale of the CMB perturbations isimprinted in the matter power spectrum as the baryon-accoustic-oscillations (BAOs), and is accessible to us inthe data of several surveys, measuring galaxy distribu-tions at different redshifts. The useful oblique parameterrelevant for the computation of the matter distributionobservable is the ratio of the sound horizon rs(zd) andthe spherical average of the angular scale and the red-shift separation dz = rs(zd)/DV (z), where

DV (z) =

(DA(z)2 z

H(z)

)1/3

, (45)

and zd is the drag epoch [89], the redshift at which thebaryons are released from the Compton drag of the pho-tons. We consider four experimental values of dz at dif-ferent effective redshifts, reported by 6dFGS [90]: dz =0.34±0.02 at zeff = 0.106, SDSS [91]: dz = 0.22±0.01 atzeff = 0.15, BOSS [92]: dz = 0.118± 0.002 at zeff = 0.32and dz = 0.072± 0.001 at zeff = 0.57.

Given this experimental input (i.e. from Cepheids,SNIa, BAOs, and CMB), we perform a χ2 analysis. Asa reference, we scan the ΛCDM model in the regionh = 0.65 − 0.75, ΩΛ = 0.6 − 0.8, Ωb,0 = 0 − 0.1, andΩγ,0 = 0 − 10−2. The value of Ωm,0 is fixed by theflatness condition. For the β0β1β4-model we scan overthe same region and additionally have the free param-eters α = 0 − 1 and ΩmFP

= m2FP/(3H

20 ) in the range

ΩmFP= 0 − 50. We perform at first a linear grid scan

and refine the χ2 fit by a Metropolis-Hastings method.

11

ΛCDM β0β1β4

h× 100 67.9+0.4−0.4 72.3+0.4

−0.4

α – 0.13+0.02−0.03

mFP/H0 – 0.59+0.17−0.13

ΩΛ 0.686+0.001−0.001 0.707+0.001

−0.002

Ωb,0 0.0488+0.0004−0.0004 0.0430+0.0004

−0.0004

TABLE I. The best-fit parameter values of the ΛCDM andthe β0β1β4 models. In both cases Ωγ is irrelevant for latetime observations.

C. Numerical results

In Tab. I, we show the best-fit values and one sigmaintervals of the ΛCDM and the β0β1β4 model. As ex-pected, the ΛCDM model fit is poor. The best-fit valueof χ2 ≈ 14 with seven degrees of freedom suggests a ∼ 3σtension of the global fit. The error intervals are derivedfrom projections on the one dimensional subspaces of thelikelihood function.

In contrast to this, when the fit is performed in theβ0β1β4 model, the fit is improved and χ2 ≈ 3. Given thatwe have only two additional fit-parameters, resulting infive degrees of freedom, this corresponds to an excellentfit value, indicating that all observables are within the 1σerror range. Another measure of the improvement is the∆χ2/dof, which in this case is ∼ 5. We conclude thatthe given data set favors the β0β1β4 model.

In Fig. 7, we show the χ2 as a function of H0 forthe ΛCDM and the β0β1β4-model. We find, that thealternative time evolution of the Hubble rate in bimet-ric theory can accommodate the CMB observables, anda larger H0 value today, than the ΛCDM scenario, whilebeing consistent with the BAO observations. The fa-vored Fierz-Pauli mass in the considered best-fit intervalis mFP ≈

(4 · 10−33 − 7 · 10−33

)eV, which is consistent

with cluster lensing [27] and other constraints [64].

D. Discussion of results

Now we discuss the results of the statistical analysisand underlying data sets. First note that the parametersat the best fit point imply β0 > 0 and hence ρDE > 0,cf. eq. (A1a). Thus dark energy is always phantom andpositive. In fig. 8, we show the equation of state of theeffective dark energy component in the β0β1β4-model atthe minimum of the χ2 function (solid line) and the 1σintervals (dashed lines). The values are in agreementwith current experimental bounds [93].

The phantom behavior is most pronounced in the red-shifts interval between z ∼ 1 and z ∼ 10. This is whenthe Hubble rate is of the order of the spin-2 mass. Thegeneral behavior is the following. The spin-2 mass con-trols at which redshifts the equation of state significantlydeviates from −1, while the mixing parameter α controls

-

-

-

-

-

-

-

~ -

FIG. 8. The equation of state of the effective dark energycomponent. The minimum of the χ2 function in the bimetricmodel corresponds to the red line, see table I. The dashedlines indicate the best-fit parameter intervals.

the magnitude of the deviation. To be precise, it is thevalue of β0 that controls the deviation. If its value isclose to zero, the equation of state significantly deviatesfrom−1. On the other hand, if β0 is positive and far awayfrom zero, the equation of state is close to −1 at all times.Note that in order to achieve a value β0 close to zero re-quires a non-zero mixing parameter α, cf. eq. (A1a). Inthe GR-limit α 1 the phantom era is absent. Thefreedom to allow for large spin-2 masses and thus shift-ing the phantom behavior to larger z while keeping αfinite to yield a significant phantom era, is not possiblein the more restricted two parameter models [52].

The phantom behavior of the dynamical dark energycomponent lowers the Hubble rate for redshifts where it isstill non-negligible, i.e. most importantly for z ∼ O(1),as compared to a cosmological constant. The loweredHubble rate increases the value of the integral in DA(z∗)which is compensated by a larger value of H0 to arriveat the same value of DA(z∗) as desired. Since the valueof rs(z

∗) remains unchanged by late-time modifications,the value of lA remains unchanged. However, within Rthe larger value of H0 must be compensated by a smallervalue of Ωm,0 and consequently a larger value of ΩΛ. Thisis indeed the case (see table I)10.

For the ΛCDM-model a smaller value of Ωm,0 is disfa-vored by local measurements as well as the CMB becausethe matter-dark energy equality is shifted to higher red-shifts, also affecting the value of lA. To see that bimetriccosmology with a smaller Ωm,0 is compatible with localexperiments, we compare the prediction of the β0β1β4-model (red solid line) to the four measurements of BAOobservables in fig. 9. For a better resolution, we normal-ized the plot to the ΛCDM prediction (dashed line). Wecan also see that, given the current experimental preci-sion, BAOs can not distinguish the predictions of the bi-

10 We are thankful to E. Mortsell for making this point clear.

12

βββ

Λ

/Λ

FIG. 9. The ratio of the baryon-acoustic-oscillationparameters normalized to the ΛCDM prediction dz/dz,ΛCDM

in the β0β1β4-model and the ΛCDM as a function of thecosmological redshift. Superposed are four measurementpoints at different redshifts.

metric β0β1β4-model from predictions of ΛCDM at theirbest-fit points.

However, baryon-acoustic-oscillations might provide atest of bimetric cosmology at the best-fit point found inthis section. In near future the DESI instrument [94, 95]will provide a new dataset of BAO observations at mul-tiple redshifts and even more advanced experiments willpush to larger redshifts [96]. A big advantage of DESIwill be, that BAO data will be obtained at multiple red-shifts with the same instrument, thus avoiding the prob-lem of different systematic errors among the instruments.With the newly collected data, this question should bere-examined. In particular with an estimated accuracyimprovement by a factor of two, as predicted in Ref. [97],the β0β1β4-model at its best-fit point will be distinguish-able from ΛCDM.

Moreover, measurements of supernovae at high red-shifts might allow to distinguish the bimetric β0β1β4-model from the ΛCDM model if the H0-tension shouldbe alleviated within bimetric theory solely due to thephantom phase. Although we only included super-novae with redshift z < 0.5 in our analysis, our best-fitpoint is consistent also with supernovae measurementsat higher redshifts [59] given the current experimentalprecision [87, 88].

When discussing the significance of the Hubble tension,a word of caution is in order. So far we have taken thelocal determinations of the Hubble rate at phase valuewith the quoted uncertainties, which are reported to beat the 2−3% level. However, the distance determinationwith the Cepheid observables is known to be subject tosystematic errors, which are hard to control, see for ex-ample Ref. [98].

One important effect is the so-called blending effectand is based on the fact that the spatial resolution of ourinstruments gets worse with distance. Thus observationsof Cepheids, that are further away are more likely to

pick up light from unresolved background sources. Thisleads to systematically larger luminosities for more dis-tant objects. This effect tends to increase the recon-structed local Hubble rate, which is consistent with thesign of the observed discrepancy. Taking the blendingeffect into account would increase the uncertainty to the∼ 5% level [99, 100]. Even though this would not resolvethe Hubble tension, without proper control of the sys-tematic error, we can not make strong statements aboutthe true statistical significance of this anomaly.

VI. SUMMARY AND DISCUSSION

In the first part of this paper, we discussed the signa-tures of the Vainshtein mechanism in cosmology. We findthat the universe becomes larger than its own Vainshteinradius at the critical Hubble rate

Hc = mFP . (46)

For earlier times, each Hubble patch lies within its ownVainshtein regime and hence is screened. We find thesame scale appearing on the level of Friedmann’s equa-tion and linear perturbations. In particular, non-linearmassless and massive modes decouple at early times, i.e.for H mFP. Further, the Stuckelberg field becomeslarge and hence non-linear above the same scale. Finally,linear perturbation around the FLRW background areunstable and grow exponentially when H mFP. Weconclude that the early universe is screened by the cos-mological analogue of the Vainshtein mechanism. Whenstudying perturbative phenomena in the early universe,non-linearities have to be taken into account.

We assumed that these non-linearities are such thatthe massive mode becomes strongly coupled and can beintegrated out to restore GR, as demonstrated in [65].This allows us to study perturbative phenomena, such asthe formation of the CMB.

Under this assumption (that CMB physics is the sameas in GR due to the Vainshtein mechanism), we perform aglobal fit to the CMB observables and the local measure-ments on Cepheid variables, SNIa observations of the cos-mic acceleration and baryon-accoustic-oscillations. Wefind that due to the phantom equation of state of theeffective dark energy component in the redshift intervalbetween z ∼ 1 and z ∼ 10 the reported tension betweenthe local and CMB determination of the Hubble scale canbe resolved.

We discussed some potential shortcomings of our sta-tistical analysis. In particular the blending effect inCepheid measurements and the high-redshift supernovaeshould be taken into account. However, we do not ex-pect these to dramatically change our result as we dis-cussed. Instead, if the H0-tension is real and statisticallysignificant, bimetric theory appears as an experimentallyfavored, consistent, and theoretically well-motivated al-ternative to GR. Whether data eventually favors bimetric

13

theory also over other gravitational theories, remains anopen question.

We show that current data from BAOs can not dis-tinguish bimetric cosmology from ΛCDM. However, inthe near future, a strong improvement in sensitivity andsystematic uncertainty will likely change this situation.Under the requirement that the H0-tension is alleviatedsolely due to the bimetric phantom dark energy, theβ0β1β4-model will be distinguishable from the ΛCDM-model in the near future.

A next step would be a global fit to all current data.In particular, instead of the coarse-grained methods usedin our analysis the whole data sets should be used forstudying the statistical significance of the tension and itsalleviation within bimetric theory. Another direction isa non-linear study of the cosmological perturbations andof the Vainshtein mechanism. Within an entirely bimet-ric framework, the constraints from the CMB should bederived to explicitly check our assumption a posteriori.Also, cosmic structure formation should be addressed en-tirely within bimetric theory as it probes redshifts wherethe phantom era occurs.

Acknowledgments

We would like to thank Christopher Hirata, FlorianNiedermann, Moritz Platscher, Anna Porredon, FiorenzoVincenzo, Bei Zhou and especially Edvard Mortsellfor valuable discussions and helpful comments on themanuscript. This work is supported by a grant fromthe Max-Planck-Society. J.S. is largely supported by aFeodor Lynen Fellowship from the Alexander von Hum-boldt foundation. The work of M.L. is supported by agrant from the Max Planck Society.

APPENDIX

Appendix A: Explicit expressions for theβ0β1β4-model

Throughout the paper we used the β0β1β4-model thatis defined by setting β2 = β3 = 0 to provide an explicitexample. In this appendix, we report the exact expres-sions and discuss some features of the model. For details,we refer to [59].

First, let us find the relation between the interactionand physical parameters. The background eq. (9) is acubic polynomial in c and gives rise to up the threereal-valued roots. Each root describes a vacuum of theβ0β1β4-model with different spin-2 mass, mixing angleand cosmological constant. However as discussed inRef. [59], only one of the vacua is physical. Therefore, thevacuum eqs. (9) and (10) imply the following unique rela-tion between the interaction parameters and the physical

parameters,

β0 =−3α2m2

FP + (1 + α2)Λ

1 + α2(A1a)

α−1β1 =α

1 + α2m2

FP (A1b)

α−4β4 =−m2

FP + (1 + α2)Λ

α2(1 + α2). (A1c)

The physical parameters are not completely free but haveto satisfy the Higuchi bound, m2

FP > 3Λ/2, to ensureunitarity [68]. In the following, we still express all equa-tions in terms of the interaction parameters βn for brevitybut they should be understood as being functions of thephysical parameters. Furthermore, we rescale y = αy forbrevity.

Setting β2 = β3 = 0, eq. (19) reduces to

β4y3 − 3β1y

2 −(β0 +

ρm

m2g

)y + β1 = 0 , (A2)

where βn = α−nβn for brevity. This polynomial has upto three real-valued roots that yield y as a function of ρm.For a given set of parameters, only one of these solutionscorresponds to the finite branch. Since the expressionsare quite lengthy and not enlightening, we do not showthem explicitly here. The finite branch solution mustsatisfy 0 ≤ y ≤ α which allows picking the finite branchsolution numerically. Hence, we can express

µ =y′

y(A3)

either analytically (but lengthy) or numerically as a func-tion of matter-energy density ρm and consequently as afunction of redshift z only. We used that for producingthe exemplary plots in fig. 3.

Next, we want to find the Hubble rate as a function ofredshift z only. For the β0β1β4-model the Hubble ratereads

3H2 = β0 + 3β1y +ρm

m2g

=β1

y+ β4y

2 , (A4)

where the interaction parameters βn are understood asfunctions of the physical parameters and y as the finitebranch solution to eq. (A2) (either analytically or numer-ically). We used the result for drawing the plot in fig. 4and for the data analysis in section V.

Let us collect some more details for the data analysiswith the β0β1β4-model. Evaluating Friedmann’s equa-tion today yields a relation among the parameters of themodel,

3H20 = β0 + 3β1y0 +

ρm,0

m2g

, (A5)

where the subscript 0 indicates the value of the quantityat present time. We used this relation to eliminate Ωm,0in terms of the other parameters.

14

In the data analysis, we constrained the decelerationparameter q that is derived from Friedmann’s equation.We can express the definition more explicit in terms ofbimetric parameters as

1 + q = − H

H2=

1

2H2

dH2

dyy′ , (A6)

where y′ = ˙y/H is given by eq. (20) and

dH2

dy= − β1

3y2+

2

3β4y . (A7)

Again, with y understood as the finite branch solution,q is a function of redshift only. For the data analy-sis, we used the constraints on q0 (i.e. at z = 0) tofind the favored values of the physical parameters. Notethat eq. (A6) holds for any bimetric (sub)model.

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