Massive mimetic cosmology

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Massive

mimetic

cosmology

Adam

R. Solomon

a

,

∗

,

Valeri Vardanyan

b

,

c

,

Yashar Akrami

d

,

b

a_{DepartmentofPhysics&McWilliamsCenterforCosmology,CarnegieMellonUniversity,Pittsburgh,PA 15213,USA} b_{LorentzInstituteforTheoreticalPhysics,LeidenUniversity,P.O.Box9506,2300RALeiden,theNetherlands} c_{LeidenObservatory,LeidenUniversity,P.O.Box9513,2300RALeiden,theNetherlands}

d_{DépartementdePhysique,ÉcoleNormaleSupérieure,PSLResearchUniversity,CNRS,24rueLhomond,75005Paris,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received14March2019

Receivedinrevisedform24May2019 Accepted28May2019

Availableonline30May2019 Editor:H.Peiris

Westudythefirstcosmologicalimplicationsofthemimetictheoryofmassivegravityrecentlyproposed byChamseddineandMukhanov.Thisisanoveltheoryofghost-freemassivegravitywhichadditionally containsamimeticdarkmattercomponent.Inanechoofothermodifiedgravitytheories,thereare self-acceleratingsolutionswhichcontainaghostinstability.Intheghost-freeregionofparameterspace,the effectofthegravitonmassonthecosmicexpansionhistoryamountstoaneffectivenegativecosmological constant,aradiationcomponent,andanegativecurvatureterm.Thisallowsustoplaceconstraintsonthe model parameters—thegravitonmassandthe Stückelbergvacuumexpectationvalue—byinsisting that theeffective radiationandcurvaturetermsbewithinobservationalbounds.Thelate-timeacceleration must be accountedfor by aseparatepositive cosmological constantorother darkenergy sector. We imposefurtherconstraintsatthelevelofperturbationsbydemandinglinearstability.Wecommenton the possibilityof distinguishingthistheory fromCDM withcurrentand future large-scale structure surveys.

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Chamseddine and Mukhanov have recently proposed [1,2] a novel ghost-free theory of massive gravity in which one of the fourStückelbergscalars isconstrainedinthesamewayasinthe mimetictheoryofdarkmatter [3],spontaneouslybreakingLorentz invariance.InthisLetter,we studytheimmediateimplicationsof thismimeticmassivegravityforcosmologicaltheoryand observa-tion.

From a field-theoretic perspective, general relativity is the uniquetheory(in fourspacetimedimensions)ofamassless spin-2particle,orgraviton. Itis thereforenaturaltoask whetheritis possibleto endow thegraviton witha non-zero mass, andwhat sort of theoretical structures would result [4]. A closely related lineofinquiryaskswhetheritispossiblefortwo ormore gravi-tonsto interact[5]. Most nonlinear realizations of such theories sufferfromtheso-calledBoulware-Deserghostinstability[6].The pastdecadehasseentheconstructionofmodelswhichavoidthis instability, allowing forthe construction ofghost-free theoriesof

*

Correspondingauthor.

E-mailaddresses:adamsolo@andrew.cmu.edu(A.R. Solomon), vardanyan@lorentz.leidenuniv.nl(V. Vardanyan),akrami@ens.fr(Y. Akrami).

massive gravity [7–13] andbimetric andmultimetric gravity [12,

14,15].Wereferthereadertothereviews[16,17] onmassive grav-ityand[18,19] onbimetricgravity.Thetheoryofmimeticmassive gravityproposedinRefs. [1,2] takesanewandalternativepathto aghost-freenonlineartheoryofmassivegravity.

A generic theory ofmassive gravity propagatessix degreesof freedom,whichshould bethoughtofasthefivehelicitystatesof a massive graviton plus an additional, ghostly scalar. The easiest waytounderstandthedegrees-of-freedom countingisto observe that a graviton mass breaks diffeomorphism invariance.This isa gauge symmetry andso can be restored by the addition of four Stückelberg scalars



A, which propagate in addition to the two (nowpotentiallymassive)tensormodesofgeneralrelativity.

As an illustration, considera Lorentz-invariant theory of mas-sivegravity.Inordertoconstructnon-trivial, non-derivative inter-actions for the metric, one requires a second “reference” metric. The simplestchoiceforthis metricisthat offlat space,

η

μν , but

theadditionofthispriorgeometrybreaksdiffeomorphism invari-ance;forinstance,therearepreferredcoordinatesystemsinwhich

η

μν

=

diag(

−

1,1,1,1). But diffeomorphism invariance is simply

a redundancy in description, and can be restored by the addi-tion ofredundantvariables, i.e., replacing

η

μν

→

η

A B

∂

μ



A

∂

ν



B,

where

η

A B

=

diag(

−

1,1,1,1)andthefourfields



A transformas https://doi.org/10.1016/j.physletb.2019.05.045

spacetimescalars.Onecanalways,bymeansofadiffeomorphism, choosethe unitary gauge inwhich



A

=

xA,andwe recoverthe originaldescriptionofthetheoryintermsofasymmetry-breaking referencemetric. Generic interaction terms for the graviton, e.g., genericfunctionsof gμα

η

A B

∂

α



A

∂

ν



B,willleadtodynamicsfor

eachofthesefourscalars,inadditiontothetwomodesofgeneral relativity,foratotalofsixdegreesoffreedom.

At thelinear level,i.e., linearizing themetric aboutflat space inunitarygauge, gμν

=

η

μν

+

hμν and



A

=

xA,wefindthatone

of the sixdegrees of freedom leads to a ghost instability unless we specifically arrange the mass term into the Fierz-Pauli form,

L

mass

∼

h2μν

−

h2,inwhichcasethedynamicsoftheghostlymode

take the form of a total derivative. Continuing thisprocedure at higher orders in perturbation theory—i.e., continually packaging ghostlyoperatorsintototalderivativestructures—leadsuniquelyto thenon-linearmassivegravitytheoryofdeRham,Gabadadze,and Tolley(dRGT)[8,9].

The recent proposal of Chamseddine and Mukhanov takes a novel alternative approach to eliminating the dangerous ghostly mode [1,2]. Noticing that the ghost can be associated to the



0 _{Stückelberg mode, they propose imposing the constraint} gμν

∂

μ



0

∂

ν



0

= −

1. This is motivatedby a similar construction

known as mimetic gravity [3], in which the constrained scalar winds up behaving like dark matter.1 Mimetic massive gravity takesthisconstrainedscalartobeoneoftheStückelbergmodesof amassivegraviton,eliminatingtheghost.Theyproposethe follow-ing action,designedto ensurestability at thelinear level(notice thatthemasstermisnotoftheFierz-Pauliform),

S

=



d4x

√

−

g



_M2 Pl 2 R

+

m2_M2 Pl 8



1 2h

¯

2

_{− ¯}

_h2 A B



+ λ(

X

+

1

)



+

Smatter

,

(1) withX

≡

gμν

∂

μ



0

∂

ν



0,and

¯

hA B

≡

gμν

∂

μ



A

∂

ν



B

−

η

A B

.

(2)

Internal indices (given by capital Roman letters) are raised and loweredwiththeMinkowskimetric.Thefieldequationsare2

Gμν

=

1 M2 Pl Tμν

−

2

λ

M2 Pl

∂

μ



0

∂

ν



0

+

m2 2



¯

hA B

−

1 2h

¯

η

A B

 

∂

μ



A

∂

ν



B

−

1 4h

¯

A B_g μν



,

(3) 0

= ∇

μ



2

λ

M2_Pl

∂

μ

_

0

_δ

0 A

−

m2 2



¯

hA B

−

1 2h

¯

η

A B



∂

μ



B



,

(4) X

= −

1

.

(5)

The last of thesealigns

˙

0 _{withthe lapse of gμν . An upshot of}

thisconstructionisthattheconstrainedmode behavesasa pres-sureless fluid, i.e., this theory provides a (mimetic) dark matter candidate[1,2].3

1 _{Foranearlierconstructioninwhichaconstrainedscalarmimicsdarkmatter}

anddarkenergy,andwhichcontainsmimeticdarkmatterasasubset,seeRef. [20].

2 _{Notethesigndifferencesbetweentheright-handsideoftheEinsteinequations}

andthe correspondingequationinRef. [1],whichisdue tothe mostlypositive metricconventionweemploy.

3 _{Oneshouldnotethatthephenomenologyofmimeticdarkmatterisstillinthe}

earlystagesofdevelopmentcomparedtotraditionalparticle darkmattermodels suchasweaklyinteractingmassiveparticles(WIMPs)oraxions,anditispremature toconsidermimetic gravity asaserious alternativetothosemodels. For

exam-We endthissection bymakinga connectionwiththeexisting literature on Lorentz-violatingmassivegravity anddemonstrating the absence of certain well-known features of Lorentz-invariant massive gravity, namely the van Dam-Veltman-Zakharov (vDVZ) discontinuity [22,23] andthe Higuchi bound [24]. The vDVZ dis-continuityreferstothefailureoflinearizedLorentz-invariant mas-sive gravity to reduce to general relativityin the massless limit; thisrequiresnonlineareffectsinordertorestoregeneralrelativity in the Newtonian limit [25,26]. The Higuchi bound is a stabil-itybound formassive gravityon de Sitterspace, placing a lower bound on thegraviton mass,m2

≥

2H2,with H the Hubble rate. Itiswell-knownthat breakingLorentzinvariancechangesbothof theseconclusionsdramatically[27,28].

At theleveloflinearperturbations aroundflatspace, the gen-eralSO(3)-invariantmassterminunitarygauge(A

=

xA) canbe writtenas[27]

L

mass

=

1 8M 2 Pl

m2₀h2₀₀

+

2m2₁h2_0i

−

m2₂h2_{i j}

+

m2₃h_ii2

−

2m2₄h00hii

.

(6)

Thelinearizedmasstermineq. (1) inunitarygaugeis(treating

λ

asfirst-order)

L

mass

=

m2M_Pl2 8



−

1 2h 2 00

+

2h20i

−

h2i j

+

1 2h 2 ii

−

h00hii



+ λ

h00

.

(7)

The

λ

equation ofmotion setsh00

=

0, whichwe can impose in theaction4_tofind

m2₀

=

m2₄

=

0

,

m₁2

=

m2₂

=

2m2₃

=

1

.

(8)

This allows us to easily make contact with the existing litera-tureonLorentz-violatingmassivegravity.TheanalysisofRef. [27] shows that for these mi parameters, the Newtonian limit is the usualone,whilethevDVZdiscontinuityisabsent.Theanalogueof theHiguchiboundinLorentz-violatingmassivegravitywasderived in Ref. [28], andfor ourvalues of the mi parameters, it reduces simplytoH2

_>

_{0,whichistriviallysatisfied.}

2. Flat-spaceperturbations

In this section, we briefly review the behavior of perturba-tions aboutflatspaceinmimeticmassivegravity, asdiscussedin Refs. [1,2].Thiswillplacestabilityconditionsonthetheorywhich willberelevantwhenwemovetocosmologicalsolutions.

Theequationsofmotion(3)–(5) invacuumaresolvedby5

gμν

=

η

μν

,



A

=

xA

,

λ

=

0

.

(9)

ple,sincethemimeticdarkmatteronlyinteractsgravitationallywiththeStandard Model,wedonotexpecttohaveathermalproductionmechanism,incontrastto manytraditionaldarkmatterscenariossuchasWIMPs.Indeed,whenthetheoryis shift-symmetricin0_{,theenergydensityofthiscomponentissetentirelybyan}

integrationconstantandsoisdeterminedbyinitialconditions.Itmayalsobe nec-essarytotunetheparametersofthemodelinordertoobtaintherightvaluesof thedarkmatterdensityovertheentirecosmichistory,andhigher-derivative effec-tivefieldtheorycorrectionsplayanimportantrole[21].Wereferthereaderto,e.g., Ref. [21] fordiscussionsoftheconstraintsthatearly-Universeconsiderationsplace onthepropertiesandevolutionofmimeticdarkmatterthroughoutcosmichistory.

4 _{Thisisjustifiedbecause,onshell,theh}

00equationofmotionsimplysetsthe

valueofλ,whileh00 dropsoutofthehi j equationsofmotion.Thedynamicsare

thereforeequivalent.

5 _{Thisistheonlysolutionthatismanifestlyinvariantunderrotations,i.e.,with} gμν=diag(−1,1,1,1)andA_{=ϕ(t), βx}i_{.Apriori itmaybepossibletohave}

flatsolutionswithinhomogeneousStückelbergsA_{,orequivalentlysolutionswith} A=xA_andg

Weexpandtheaction(1) toquadraticorderaroundtheMinkowski solution,focusingonscalarmodes,

g00

= −(

1

+

2

φ),

(10) g0i

= ∂

iB

,

(11) gi j

= (

1

−

2

ψ )δ

i j

+

2

∂

i

∂

jE

,

(12)



A

=

xA

+

π

0

_{, ∂}

i

_π



_,

₍₁₃₎

λ

= δλ.

(14)

Threeof thesefields—φ, B,and

δλ—are

auxiliary,asthey appear withouttimederivativesintheaction,andsocanbeintegratedout usingtheirequationsofmotion.Notethattheauxiliarystructureis preciselythesameasingeneralrelativity,sincethemasstermand Lagrangemultiplierdonotintroduceanyderivativesofthemetric. Wecanusediffeomorphisminvariancetoremoveafurthertwo modes.Whengaugefixingattheleveloftheaction,onemusttake care to only eliminate variables whose equations of motion are containedin theequations ofmotion ofthe remaining variables, otherwisewe willloseinformationafterpicking agauge. Follow-ingtheprocedureofRefs. [29,30],weseethat wecansafelytake

π

0_andoneof

₍

_E

_,

_π

₎

_{tovanish.Pickingunitarygauge,}

_π

0

₌

_π

₌

_0,

weobtaintheflat-spacequadraticaction(inFourierspace),

δ

2S

=



dt M2_Pl

− ˙

X

T

_{K ˙}

_X

₊

_X

T

_k2

_G

₊

_m2

_M

_X

_,

₍₁₅₎

where

X ≡ (ψ,

k2_E

₎

_{andthematrices}

_K

_,

_G

_,and

_M

_aregivenby

K

=



3

+

4k2 m2 1 1 0



,

(16)

G

=



1 0 0 0



,

(17)

M

=

1 4



3 1 1

−

1



.

(18)

As described in Ref. [2], this system can be diagonalized by replacing

ψ

withtheLagrange multiplier

δλ,

which we had pre-viouslyintegratedoutusing

δλ

=

M 2 Pl 4



(

4k2

+

3m2

)ψ

+

k2m2E



,

(19) tofind

δ

2S

=



dt 1 4k2

₊

_3m2



k4m2M2_Pl

˙

E2

− (

k2

+

m2

)

E2

−

1 M_Pl2



16 m2

δλ

˙

2

−

4

δλ

2

 

.

(20)

Ifwetakem2

_>

_{0,wecancanonicallynormalize,}

δλ

c

≡

4 mMPl



2k2

₊

3 2m2

δλ,

(21) Ec

≡

mMPlk2



2k2

₊

3 2m2 E

,

(22)

toobtainthefinalaction,

δ

2S

=



dt



1 2E

˙

2 c

−

1 2

(

k 2

₊

_m2

₎

_E2 c

−

1 2

δλ

˙

2 c

+

1 8m 2

_δλ

2 c



.

(23)

The only dynamical degree of freedom here is Ec, which is healthyandhasmassm.Thefield

δλ

c hasthewrongsignonboth

itskineticandmassterms,butdoesnotpropagateduetothe ab-senceofagradientterm;itsequationofmotion,

¨

δλ

c

+

m2

4

δλ

c

=

0

,

(24)

leads toa dispersionrelation

ω

2

₌

_m2

_{/4 and}

_{issolved simplyby}

[2]

δλ

c

=

C

(

x

)

sin



mt 2



+

D

(

x

)

cos



mt 2



,

(25)

whereC andD are space-dependentconstantsofintegration.The authorsofRef. [2] identifythismodewiththemimeticdark mat-ter.6

When we discusscosmology inthenext section, we will find ourselvestemptedbythepossibilityoftakingm2

<

0.Apriori this

is merely a parameter choice, but the flat-space analysis shows whythiswouldbe apoordecision.Bylookingattheaction(20), we see that, fornegative m2,the overall signin front ofthe ac-tion flips depending onwhetherk2

>

3

|

m2

|/

4 or k2

<

3

|

m2

|/

4,a signofpathologicalbehavior.Inparticular,forscalesk2

_>

₃

_|

_m2

_|/

_4,

uponcanonicallynormalizingwefindtheaction(23) withan over-allminussign,sothatthedynamicalmode Ec isaghost.

3. Cosmologicalsolutions

In this section we investigate Friedmann-Lemaître-Robertson-Walker(FLRW) cosmologicalsolutionsofmimeticmassivegravity. Considerthehomogeneousandisotropicansatz

gμν

=

diag

(

−

1

,

a

(

t

)

2

δ

i j

),

(26)



A

=

ϕ

(

t

), β

xi



.

(27)

Inprincipleonecouldallow

β

todependontime,butthisbreaks homogeneityandisotropyasitinduces

x-dependenttermsinthe stress-energy tensorof the Stückelberg fields.Note that on-shell, theLagrangemultiplierenforces

ϕ

=

t (uptoaconstant). Wewill include a general matter sector with density

ρ

and pressure p.

Wewillfindthissector needstocontainacosmologicalconstant, muchlikeingeneralrelativity,butdoesnot needtoincludedark matter, as this role can be played by the mimetic dark matter (whichisanexactlypressurelessperfectfluid).

TheEinsteinandscalarequationsofmotionare

3H2

=

ρ

M2_Pl

−

2

λ

M2_Pl

−

3m2 16



β

4 a4

−

6

β

2 a2

+

5



,

(28) 2H

˙

+

3H2

= −

p M2 Pl

−

m2 16



3

−

β

4 a4

−

2

β

2 a2



,

(29) 0

=

d dt



a3



3m2 4



1

−

β

2 a2



+

2

λ

M2 Pl



.

(30)

Wecansolvefor

λ

byintegratingthe



0 equationofmotion(30), finding

6 _{SeeRef. [2] foranargumentforwhythismodeisnotaghost,despitehaving}

anoverallwrong-signaction.Inprinciple,onemightworrythatwhenquantizingor consideringnonlinearities,acouplingwillbeinducedbetweenδλcandotherfields

whichwillleadtoanOstrogradskiinstability.Ontheotherhand,duetothelack ofagradienttermthismodeisnotapropagatingdegreeoffreedomintheusual sense.Wewillremainagnosticaboutthisquestionandlimitourselvesto consid-erationsofclassical,linearstability,whichthissystemclearlysatisfiesform2_>0.

−

2

λ

M2_Pl

=

C

a3

+

3m2 4



1

−

β

2 a2



,

(31)

where

C

is an integration constant. Pluggingthis into the Fried-mannequation(28),weobtain

3H2

=

ρ

M2 Pl

+

C

a3

−

3m2 16



1

−

β

2 a2



2

.

(32)

Notethatthecontributionfrom

λ

exactlycancelsoutthatfromthe lasttermoftheEinsteinequation(3),sotheverysimpleformfor

ρ

ϕ

≡ −

3m2MPl2

(1

− β

2

_/

_a2

₎

2

_{/16 is}

_{entirelyduetotheterm}

propor-tionaltogμν inthestresstensor.Theintegrationconstantprovides adust-likecontributiontotheFriedmannequation,whichistobe expectedasthisisatheoryofmimeticdarkmatter.

We can geta better sense of the physicalpicture by expand-ing out the Friedmannequation andabsorbing the mimetic dark matter

C

into

ρ

,finding

3H2

=

ρ

M2 Pl

−

3m2 16



β

4 a4

−

2

β

2 a2

+

1



.

(33)

Form2

>

0 (m2

<

0),we seethatthe masstermgenerates an ef-fectivenegative (positive)cosmologicalconstant,an effective neg-ative (positive) curvature, and an effective radiation component withnegative (positive)energydensity.Notethat theseaddonto anycosmologicalconstant,radiation,andcurvaturealreadypresent cosmologically;forexample,whilewehaveassumedaflat cosmol-ogyasouransatz, observational boundsonspatial curvaturewill constrainthesumofanypre-existingcurvatureandthe curvature-liketermgeneratedbythegravitonmass.

Note that for m2

<

0 we have late-time acceleration, with



eff

=

3

|

m2

|/

16.However,asdiscussedintheprevioussection,we

needm2

>

0 inorder to avoida ghost around flat space. Thisis reminiscentofthesituationintheDvali-Gabadadze-Porrati(DGP) model [33], where one branch of solutions has self-accelerating cosmologicalexpansion[34,35] butisplaguedby aghost[36,37], whilethe other branch is healthybut cannot account forcosmic acceleration.

Letus assume that the energydensity

ρ

ineq. (33) contains dust(includingthe mimeticdarkmatter), radiation,anddark en-ergycomponents.Then,intermsofthedensityparameters,



i,0

=

ρ

i,0

3M_Pl2H₀2

,

(34)

thecomponentsoftheFriedmannequationwhicharemodifiedby mimeticmassivegravityare



,0

= ¯

,0

−

m2 16H2₀ (35)



K,0

=

m2 8H2₀

β

2

_,

₍₃₆₎



r,0

= ¯

r,0

−

m2 16H2₀

β

4

_,

₍₃₇₎

where

¯

,0 and

¯

r,0 arethe densities associatedto darkenergy

andStandardModelradiation.Usingobservationalboundsonthe curvatureandradiationdensities,wecanplaceconstraintsonthe model parameters m2 _and

_β.

_{We will not consider any bounds}

comingfrom thepresence ofthe effectivecosmologicalconstant, even though it contributes a negative and potentially large (if

m2

_{H0) amount to}

_

,0.Particle physics also predicts a large

(and potentially negative) vacuum energy, and since we are not worrying about that, it seems inconsistent to worry about the

contributionfrommimeticmassivegravity.Onemightexpectthat whateversolvestheformerproblemwillalsosolvethelatter.7

We will use observational constraints on



K,0 and



r,0 to

boundourtwofreeparameters,m2 and

β.

Planck 2018constrains



K,0

=

0.0007

±

0.0019, which we parametrize as

|

K,0

|

< δ

K, with

δ

K

∼

0.003 [39].Wewilltake thistobea constraintonthe contributionfrommimeticmassivegravityalone,

m2

8H₀2

β

2

_{< δ}

K

.

(38)

We remind the reader that what we are really bounding is the sum ofthe mimetic massive gravity contribution andany“bare” curvature,butunlessthereissignificanttuningbetweenthesetwo, we can simply take thisas a constrainton the mimetic massive gravitypiecealone.

Toboundthemimeticcontributiontotheradiationdensity,we will use constraintsfrom big bangnucleosynthesis (BBN). At the time of BBN,radiation dominates.The exact value ofthe Hubble rate at the time of nucleosynthesis, which dependson the radi-ation density, determines the freeze-out abundance of neutrons andthereforetheprimordialabundanceofhelium-4,whichis sub-jecttotightobservationalbounds.Theconstraintsareconveniently phrasedintermsofthe“speed-upfactor”

ζ

≡

H

/ ¯

H ,where H and

¯

H are theHubblerateandits expectedvalue, respectively,atthe time of BBN.The difference betweenthe observedandpredicted helium-4 abundance,

|

YP

|

, isrelated to the speed-upfactor by [40]



YP

=

0

.

08

(ζ

2

−

1

).

(39) Currentobservationalboundsimply[41]

|

YP

| 

0

.

01

.

(40)

Comparing the Friedmann equation (33) with and without the mimetic radiation contribution, and focusing on radiation domi-nation,wefind

ζ

2

−

1

= −

m

2

_β

4

16

¯

r,0H2₀

,

(41)

where the value for the present-day radiationdensity associated to photonsandneutrinos,

¯

r,0

∼

10−4,is determinedentirely by

theCMBtemperatureandtheeffectivenumberofneutrinospecies and is therefore not dependent on our modification of gravity.8 Combiningthiswitheq. (40) wearriveattheconstraint

m2 16H2₀

β

4

_{< δ}

r

,

(42) where

δ

r

≡

max

(

|Y

P

|) ¯

r,0 0

.

08

≈

O

(

10 −5

_).

₍₄₃₎

Wecanrewriteourconstraints(38) and(42) asinequalitiesfor

m

/

H0 and

β

aloneintwodifferentrégimes,

m H0

<

⎧

⎪

⎨

⎪

⎩

√ 8δK β

, β <



2δr δK 4√δr β2

,

β >



2δr δK

.

(44)

TheseareplottedinFig.1.

7 _{SeeRef. [38] foraproposedsolutiontothecosmologicalconstantproblemin}

thecontextofLorentz-violatingmassivegravity,whichiscloselyrelatedtomimetic massivegravity.

Fig. 1. Upper limits on m/H0andβfor(δK, δr)= (0.003,10−5). Finally,wenotethatthestrong-couplingscaleforthistheoryis oforder



2

=

√

mMPl[2].Ifm isoforderthepresent-dayHubble

scale,m

∼

10−33eV,thenthestrongcouplingscaleis



2

∼

meV,

i.e., the theory breaks down slightly below the millimeter scale. Aswe seefrom eq. (44), forsufficiently small

β

,m could poten-tiallybemuchlargerthan H0,leadingtoacorrespondinglylarger strong-couplingscale.

4. Cosmologicalperturbations

As we have seen, atthe background level, cosmological solu-tions in mimetic massive gravity do not differ appreciably from

CDM.

Wethereforeproceedtostudycosmologicalperturbations aroundthe FLRWbackground.This willtell ushowcosmological large-scalestructure(LSS)evolvesinthistheory incomparisonto

CDM.

Since mimetic massive gravity differs from general rela-tivity,we wouldexpectmodifications tothe gravitationalPoisson equationandthe sliprelation,whichcould inprinciple allow for observational tests of this alternative model against

CDM

and distinguishthetwousingthecurrentandfutureLSSsurveys. How-ever,aswewillsee,stabilityofcosmologicalperturbationsandthe bounds(44) placestrongconstraintsonthe modelwhichsuggest that thistheory should be observationally indistinguishablefrom GRinthelinearrégime.

4.1.Stabilitybound

We begin by studying the stability of cosmological pertur-bations using the second-order action formalism. Since, as dis-cussedin section 3,this theory doesnot possessghost-free self-accelerating solutions, we include a cosmological constant, al-thoughitwillnotaffectanyoftheresultsinthissection.Sincethe theory alreadycontains a pressurelessfluid, namely themimetic darkmatter,weneednotintroduceanadditionalmatterfield.Our analysisisthereforevalidforalltimesaftermatter-radiation equal-ity.

We define the linearized metric, Stückelberg fields, and La-grangemultiplieras ds2

= −(

1

+

2

φ)

dt2

+

2a

∂

iBdtdxi

+

a2



(

1

−

2

ψ )δ

i j

+

2

∂

i

∂

jE



dxidxj

,

(45)



0

=

t

+

π

0

,

(46)



i

= β

xi

+ ∂

i

π

,

(47)

λ

= ¯λ + δλ,

(48)

wherewe are restrictingourselves toscalar perturbations, and

¯λ

isthe background value given ineq. (31). The calculation of the

quadratic action proceeds analogously to the flat-space case dis-cussedinsection2.Expandingtheaction(1) (withacosmological constant)toquadraticorderinperturbations,wefindthatthe vari-ables

φ,

B,and

δλ

areauxiliary—thatis,theyappearwithouttime derivatives—andcanthereforebe integratedout usingtheir equa-tions ofmotion. To safelyfix a gauge at the level of the action, weagainfollowtheprocedureofRefs. [29,30],findingthatwecan eliminateoneeachof(ψ,

π

0_)and(E,

_π

_{).Wewillchoosetowork}

inunitarygauge,

π

0

₌

_π

₌

_0,sothat

_

A

_{= (}

_t

_,

_β

_xi

₎

_{isunperturbed.} Thefinalaction,inFourierspaceandafterintegrationsbyparts,is

δ

2S

=



dt M_Pl2a3



− ˙

X

T

_{K ˙}

_X

₊

_X

T



k2 a2

G

+

m 2

_M



X



,

(49)

where

X ≡ (ψ,

k2_E

₎

_{andthematrices}

_K

_,

_G

_,and

_M

_aregivenby

K

=



3

−

8a2 β2_−3a2 k 2 m2_β2 1 1 0



(50)

G

=



1 0 0 0



(51)

M

=

1 8

β

2 a2



1

+

β

2 a2

 

3 1 1

−

1



(52)

Sinceweareinterestedintheimplicationsofmimeticmassive gravityforthegrowthandpropertiesoflarge-scalestructureinthe late Universe, let usfocus on subhorizon scales (i.e., k2

a2H2) and assume the quasi-static (QS)approximation. In order to use thisapproximation,wefirstneedtoensurethatfluctuationsinthis régime are stable. Ignoring time variation in a

(

t

),

which will be subdominant in the limit k2

_a2_H2_{, and assuming solutions of}

theform

X =

X

0eiωt,theequationsofmotionfollowingfromthe action(49) are



−

ω

2

K

+

k 2 a2

G

+

m 2

_M



X

=

0

.

(53)

We canthen derive stability conditionsfromthe dispersion rela-tions,obtainedbysolving

0

=

det



−

ω

2

K

+

k 2 a2

G

+

m 2

_M



=

ω

4 a2

_{+ β}

2

−

ω

2_k2 a2

₍

_3a2

_{− β}

2

₎

−

5

ω

2_m2

_β

2 8a4

+

k2_m2

_β

2 8a6

+

m4

β

4

(

a2

+ β

2

)

16a8 (54) for

ω

2_.

The dispersionrelations arisingfromeq. (54) arecomplicated, butsimplifysignificantly in thelimit k

aH whenwe take into account theconstraints(44) onm

/

H0,whichwe obtainedby re-quiringthattheradiationandcurvaturedensitiesgeneratedbythe masstermnotexceedobservationalbounds.Consider replacingm

and

β

ineq. (54) withthefollowingtwoparameters,9



1

≡



m

β

k



2

,



2

≡



m

β

2 ka



2

.

(55)

Weproceedtoshowthatthebounds(44) implythateachofthese ismuchsmallerthanunityonsubhorizonscalesforalltimesafter matter-radiationequality.

9 _{Todothisreplacement, firstreplacem→}√_

1β/k,and thenreplaceany

Forboth



1 and



2 wecan putupperboundson the

numera-torsandlower boundsonthedenominators.Letusstartwiththe numerators. For



1, multiply each side of eq. (44) by

β

.We see

thereisastrictupperboundonthecombinationm

β

,

m

β

≤



8

δ

KH0

≈

0

.

15H0 (56)

wherewehavetaken

δ

K

∼

0.003 asarepresentativevalue.Wecan similarlyfindaboundonthenumeratorof



2 bymultiplyingboth

sidesofeq. (44) by

β

2,finding

m

β

2

≤

4



δ

rH0

≈

10−2H0 (57)

for

δ

r

∼

10−5.

Now we move on to the denominators. The subhorizon limit isgivenbyk

aH .Forthesake ofargumentlet usbe conserva-tiveandassumethatk isonlyslightlysubhorizon,k

/

a

≈

O(1)

H .10

At any giventime frommatter-radiation equality to the present, wherewe can trustour analysis, theHubblerate H is relatedto itspresent-dayvalue H0 byH

=

H0





,0

+ 

m,0a−3.Puttingthis

together with the boundswe have derived on m

β

andm

β

2, we find



1



0

.

02



,0a2

+ 

m,0a−1

1

,

(58)



2



10−4



,0a4

+ 

m,0a

1 for z



3000

.

(59)

Note that while the upper bound on



1 is always much smaller

thanunityfor0

<

a

≤

1,the upperboundon



2 infactgrows as a−1 atearly times.However, itgrows slowly and hasa factorof 10−4_{tocompetewith,sothat max(}

_

2

)

doesnotreachunityuntil z

∼

3000,rightaroundmatter-radiationequality.Thereforein prin-cipletheremightbeahandfulofmodes—rightaroundthehorizon scaleandattheearliestmomentsofmatterdomination—forwhich termsgoingas



2 affectthesubhorizondispersionrelation,if m

β

2

takes the largest value allowed by the constraints. We will con-tinuetotake



2

1,withtheunderstandingthatifthisparticular

situation is realized,then atthose very early times we are only consideringmodeswithk



10aH ,forwhich



2iscertainlysmaller

thanunity.

Droppingtermssubdominantin



1 and



2,thedispersion

rela-tion(54) becomes 0

≈

ω

4 a2

_{+ β}

2

−

ω

2_k2 a2

₍

_3a2

_{− β}

2

₎

+

k2_m2

_β

2 8a6

.

(60)

Solving for

ω

2_{, and again dropping terms subleading in}

_

1

=

(

m

β/

k

)

2_{,wefindthedispersionrelationsforourtwomodes,}

ω

2

≈

k 2 a2 a2

_{+ β}

2 3a2

_{− β}

2

,

(61)

ω

2

≈

m2

β

2 8a2



3

−

β

2 a2



.

(62)

Eachoftheseimpliesthesamestabilitycondition,

β

2

a2

<

3

.

(63)

Thistellsusthatnomatterwhatthevalueof

β

is,our cosmologi-calsolutionsareunstable atsufficientlyhighredshifts,

10 _{Ofcourse,the deeperinthesubhorizon régimek is,the smaller} 1 and2

become.

z

>

√

3

β

−1

−

1

.

(64)

This early time instability can howeverbe safely pushed back to unobservably early times by taking the parameter

β

to be suffi-ciently small.11 Becauseweareassuming matteranddarkenergy domination, we can trust our stability condition as far back as matter-radiation equalityat zeq

≈

3400.Demandingstabilityfrom

zeq onward,wefindaconstrainton

β

,12

β



5

×

10−4

.

(65)

4.2. Cosmologicaltensormass

Another possible cosmological bound on the parameters m

and

β

comes from constraints on the graviton mass. The tight-estboundscurrentlycomefromLIGO,mT

≤

7.7

×

10−23eV[44].13

To compute the mass of tensor fluctuations on a cosmological background,we linearizethe Einsteinequation (3) around gμν

=

¯

gμν

+

hμν ,withgμν

¯

=

diag(

−

1,a2

_δ

i j

),

h00

=

0,andhi j transverse andtraceless,i.e.,hii

= ∂

ihi j

=

0.TheEinsteinequationis

¨

hi j

+

3Hh

˙

i j

−

∇

2 a2hi j

+

m 2 Thi j

=

0 (66)

withthetensormassgivenby

m2_T

≡

m 2 2

β

2 a2



1

+

β

2 a2



(67)

The structureoftheEinstein equation issuchthat m2_T

/

m2 has to be a (quadratic) polynomial in

β

2

/

a2. What is non-trivial is that thedegree-zeroterminthat polynomialcancelsout, i.e.,the expressionform2

T

/

m2 startsatorder

β

2

/

a2.Thismeansthat

grav-itational waves propagating over cosmological distances (at low redshift,i.e.,a

∼

O(

1))donotdependonm alone;insteadthey in-volvethecombinationsm

β

andm

β

2 which,aswehaveseen,are stronglyconstrainedbythecosmologicalbackground.Inparticular, recalling that m2

_β

2

_

₁₀−2_H2

0 andm2

β

4



10−4H 2

0,we see that mT atthepresenteraisatmostoforder10−1H0

∼

10−34eV,far belowtheLIGObounds.Moreover,ourstabilitycondition (65) has no bearingon mT. No matter howtiny

β

is,the constraints(44) placeaconstantupperboundonm

β

,sothatthesmaller

β

is,the largerm isallowedtobe,leavingmT

≈

m

β/(

√

2a)fixed.Itis inter-esting to note that, without demanding that thismodelprovides cosmic acceleration, the tensormass isnevertheless forcedtobe smallerthan theHubblescale.Finally,we notethat arounda flat background,thetensormassissimplym, solocaltestsofgravity mightbeabletoplaceconstraintsonm thatarenotpossiblewith gravitationalwavesthatpropagateovercosmologicaldistances.

4.3. Quasistaticlimit

Finally, let uscomment on the testability ofmimetic massive gravity using near-future LSSsurveys. We willfind it convenient

11 _{Thisissimilartomassivebimetricgravity,whichpossessesanearly-time}

in-stabilitythatcanberenderedsafeinthelimitwheretheratioofthetwoPlanck massesbecomessmall [43].

12 _{Itisplausiblethattheresult(63) holds,atleastonanorder-of-magnitudebasis,}

throughradiationdominationaswell(see,again,theexampleofbigravity[43]).In thiscase,weshoulddemandthattheinstabilitybepushedbacktobeforebigbang nucleosynthesis,withzBBN≈3×108,whichwouldimplyastrongerconstraintof

β10−8_{.Wedonothavemuchobservationalhandleonthepresumably}

radiation-dominatederabeforeBBN,andthereforeshouldnotdemandthattheinstabilitybe absentthen;indeed,amildenoughinstabilitymighthaveinterestingconsequences, suchastheformationofprimordialblackholes.

13 _{SeeRef. [45] forahelpfulsummaryofboundsonthe gravitonmassfroma}

towork in Newtonian gauge, B

=

E

=

0.Linearizing the Einstein equations (3), and leaving in a generic stress-energy tensor Tμν

forcompleteness,weobtain

6H2

φ

−

2 a2

∂

i

_∂

i

ψ

+

6H

˙ψ

=

1 M_Pl2

δ

T 0 0

+

2

δλ

M2_Pl

−

m2 4

β

2 a2



3

−

β

2 a2



3

ψ

+ ∂

i

∂

i

π

,

(68)

−

2

∂

i

 ˙

ψ

+

H

φ



=

1 M_Pl2

δ

T 0 i

+

2

¯λ

M_Pl2

∂

i

π

0

+

m2 4



3

−

β

2 a2



∂

i

π

0

− β

2

∂

i

π

˙

,

(69) 6



¨ψ +

3H

˙ψ +

H

˙ + (

3H2

+

2H

˙

)φ



+

2 a2

∂

i

_∂

i

(φ

− ψ)

=

1 M2 Pl

δ

Ti_i

−

m 2 4

β

2 a2



1

+

β

2 a2



3

ψ

+ ∂

i

∂

i

π

,

(70) 1 a2

∂

i

_∂

j

(ψ

− φ) =

1 M2_Pl

δ

T i j

+

m2 2

β

2 a2



1

+

β

2 a2



∂

i

∂

j

π

,

i

=

j

.

(71)

MovingtoFourierspace, specializingtoa pressurelessfluid with-outanisotropic stress,andtakingthequasistaticlimit, X

¨

∼

HX

˙

∼

H2_X

_k2_{X for any perturbation X , eqs. (68), (70) and (71)}

be-come 2k2 a2

ψ

=

1 M_Pl2

(

2

δλ

− ¯

ρ

δ)

−

m2 4

β

2 a2



3

−

β

2 a2



3

ψ

−

k2

π

,

(72) 2k2 a2

(φ

− ψ) =

m2 4

β

2 a2



1

+

β

2 a2



3

ψ

−

k2

π

,

(73) 1 a2

(φ

− ψ) = −

m2 2

β

2 a2



1

+

β

2 a2



π

,

(74)

where

ρ

¯

and

δ

arethebackgrounddensityandoverdensity ofthe dustcomponent.Notethatthesearedegeneratewiththemimetic darkmatter,asexpected.

Combining these equations, we obtain the modified Poisson equationandthesliprelation,

−

k2

ψ

=

4

π

G

μ

(

a

,

k

)

a2

(δ

ρ

−

2

δλ),

(75)

ψ

=

η

(

a

,

k

)φ,

(76)

wherethemodified-gravityparameters

μ

and

η

aregivenby

μ

(

a

,

k

)

=

1 1

+

1₂m2β2 k2

3

−

β2 a2

,

(77)

η

(

a

,

k

)

=

1 1

+

1₂m2β2 k2

1

+

β2 a2

.

(78)

Theseparametrizeobservabledeviationsfromgeneralrelativity,in which

μ

=

η

=

1.

Theconstraintswehavealreadyderived onm and

β

preclude

μ

and

η

fromdeviating fromunity ata levelaccessible to near-futureobservations.Thestabilityconstraint(63) requirestheterms inparenthesesto be

O(

1),while the backgroundconstraint(56) setsm2

_β

2

_

_0.02H2 0,sothat

μ

−

1

∼

η

−

1

∼

O



m2

_β

2 k2





10−2



H0 k



2

.

(79)

It istherefore highly unlikely that cosmologicalobservations will be able to test this model against

CDM

in the linear and sub-horizonrégime.

5. Conclusions

In this Letter we have studied the first cosmological implica-tions of the recently-proposed theory of mimetic massive grav-ity. We find that the theory is unable to self-accelerate without introducinga ghost.Itseffects on Friedmann-Lemaître-Robertson-Walker cosmological backgrounds are tointroduce effective radi-ation, curvature, and cosmological constant terms, as well as a dust-like mimetic dark matter component. We place constraints (44) on the theory parameters by demanding that the effective radiationandcurvatureterms bewithin observational bounds.In theghost-freeregionofparameterspace,m2

_>

_{0,theeffective}

cos-mologicalconstant isnegative-definite, soa separate darkenergy sector,whichwetaketobeapositivecosmologicalconstant,is re-quiredtoexplainthelate-timeaccelerationoftheUniverse.

Wefurther studiedthebehaviorofcosmologicalperturbations inthe subhorizon,quasistaticlimit.The modelgenerically suffers fromaninstabilityatearlytimes.However,sinceouranalysisonly includeda pressurelessdustcomponent (inaddition toa cosmo-logicalconstant),thecalculationcanonlybetrustedasfarbackas matter-radiation equality.This allowedusto place afurther con-straintonthetheoryparametersbyinsistingthattheinstabilitybe absentthroughoutmatterdomination.Withtheseconstraints,the deviationsfrom

CDM

inthelinear,subhorizonrégimeare likely toosmalltobeobservable.

Notsurprisingly,sincethisisatheoryofmassivegravity,it pre-dicts massive tensormodes. We have calculatedthe tensormass around cosmologicalbackgrounds andfoundthat, takinginto ac-count the constraints imposed by the cosmological background, thismassmustbeatleastanorderofmagnitudebelowthe Hub-ble scale, far outside the currently-available constraints on the graviton mass. Unlikeother theoriesof massivegravity, in which the graviton mass is comparableto the Hubblescale in order to providelate-timeacceleration,thisbound onthegraviton massis solelyduetotherequirementthattheeffectiveradiationand cur-vaturetermsintheFriedmannequationnotbetoolarge.

What are the remaining prospects for cosmological tests of mimetic massive gravity? We emphasize that our analysis does notapply intwoimportantrégimes:horizon-sizescalesand non-linearscales. Oneor bothofthesemaypossesssignatures which couldbeusedtodistinguishmimeticmassivegravityfrom

CDM,

or otherwise to rule out additional regions of parameter space. OneexpectsthatnonlinearscaleswillrequireN-bodysimulations, while at horizon-size scales we cannot apply the quasistatic ap-proximation andwouldneed to solvethe perturbation equations numerically,asinother theoriesofmodified gravity[46]. Forthe latter, we note that themass scales appearing in theaction (49) forcosmologicalperturbationsarenotsimplym,whichcanbe ar-bitrarily large (in the limit of small

β

), butrather m

β

andm

β

2, which we haveshownmust bothbe at leastan order of magni-tudesmallerthantheHubblescale.Itthereforemightbe difficult forthistheory toproduceeffectsathorizonscalesthatarelarger than cosmic variance. Note that scales k

∼

m

β

and k

∼

m

β

2 are super-horizonandthereforenotobservable.

Acknowledgements

helpful discussions. We thank the referee for useful comments on the draft. A.R.S. is supported by DOE HEP grants DOE DE-FG02-04ER41338 and FG02-06ER41449 and by the McWilliams Center for Cosmology, Carnegie Mellon University. V.V. is sup-ported by a de Sitter PhD fellowship of the Netherlands Organi-zation for Scientific Research (NWO). Y.A. is supported by LabEx ENS-ICFP:ANR-10-LABX-0010/ANR-10-IDEX-0001-02PSL*.Y.A.also acknowledges support fromthe NWO and the Dutch Ministryof Education, Culture and Science (OCW), and also from the D-ITP consortium,aprogramoftheNWOthatisfundedbytheOCW.

References

[1]A.H.Chamseddine,V.Mukhanov,J.HighEnergy Phys.06(2018)060,arXiv: 1805.06283 [hep-th].

[2]A.H.Chamseddine,V.Mukhanov,J.HighEnergy Phys.06(2018)062,arXiv: 1805.06598 [hep-th].

[3]A.H.Chamseddine,V.Mukhanov,J.HighEnergy Phys.11(2013)135,arXiv: 1308.5410 [astro-ph.CO].

[4]M.Fierz,W.Pauli,Proc.R.Soc.Lond.A173(1939)211. [5]C.Isham,A.Salam,J.Strathdee,Phys.Rev.D3(1971)867. [6]D.Boulware,S.Deser,Phys.Rev.D6(1972)3368.

[7]P.Creminelli,A.Nicolis,M.Papucci,E.Trincherini,J.HighEnergyPhys.0509 (2005)003,arXiv:hep-th/0505147 [hep-th].

[8]C.deRham,G.Gabadadze, Phys.Rev.D82(2010) 044020,arXiv:1007.0443 [hep-th].

[9]C.deRham,G.Gabadadze,A.J.Tolley,Phys.Rev.Lett.106(2011)231101,arXiv: 1011.1232 [hep-th].

[10]S. Hassan, R.A. Rosen, Phys. Rev.Lett. 108(2012) 041101, arXiv:1106.3344 [hep-th].

[11]S.Hassan,R.A.Rosen,A.Schmidt-May,J.HighEnergyPhys.1202(2012)026, arXiv:1109.3230 [hep-th].

[12]S.Hassan,R.A.Rosen,J.HighEnergyPhys.1204(2012)123,arXiv:1111.2070 [hep-th].

[13]S.Hassan,A.Schmidt-May,M.vonStrauss,Phys.Lett.B715(2012)335,arXiv: 1203.5283 [hep-th].

[14]S.Hassan,R.A.Rosen,J.HighEnergyPhys.1202(2012)126,arXiv:1109.3515 [hep-th].

[15]K.Hinterbichler, R.A.Rosen,J.HighEnergyPhys.07(2012)047,arXiv:1203. 5783 [hep-th].

[16]K.Hinterbichler,Rev.Mod.Phys.84(2012)671,arXiv:1105.3735 [hep-th]. [17]C.deRham,LivingRev.Relativ.17(2014)7,arXiv:1401.4173 [hep-th]. [18]A.R.Solomon,CosmologyBeyondEinstein,Ph.D.thesis,CambridgeU.,Cham,

2015,arXiv:1508.06859 [gr-qc].

[19]A.Schmidt-May,M.vonStrauss,J.Phys.A49(2016)183001,arXiv:1512.00021 [hep-th].

[20]E.A.Lim,I.Sawicki,A.Vikman,J.Cosmol.Astropart.Phys.1005(2010)012, arXiv:1003.5751 [astro-ph.CO].

[21]L.Mirzagholi,A.Vikman,J.Cosmol.Astropart.Phys.1506(2015)028,arXiv: 1412.7136 [gr-qc].

[22]H.vanDam,M.J.G.Veltman,Nucl.Phys.B22(1970)397.

[23]V.I.Zakharov,JETPLett.12(1970)312;Pis’maZh.Eksp.Teor.Fiz.12(1970) 447.

[24]A.Higuchi,Nucl.Phys.B282(1987)397. [25]A.Vainshtein,Phys.Lett.B39(1972)393.

[26]E.Babichev,C.Deffayet,Class.QuantumGravity30(2013)184001,arXiv:1304. 7240 [gr-qc].

[27]S.L.Dubovsky,J.HighEnergyPhys.10(2004)076,arXiv:hep-th/0409124 [hep -th].

[28]D.Blas,D.Comelli,F.Nesti,L.Pilo,Phys.Rev.D80(2009)044025,arXiv:0905. 1699 [hep-th].

[29]M.Lagos,M.Bañados,P.G.Ferreira,S.García-Sáenz,Phys.Rev.D89(2014) 024034,arXiv:1311.3828 [gr-qc].

[30]H.Motohashi,T.Suyama,K.Takahashi,Phys.Rev.D94(2016)124021,arXiv: 1608.00071 [gr-qc].

[31]S.Dubovsky,T.Gregoire,A.Nicolis,R.Rattazzi,J.HighEnergyPhys.03(2006) 025,arXiv:hep-th/0512260 [hep-th].

[32]S.Endlich,A.Nicolis,R.Rattazzi,J.Wang,J.HighEnergyPhys.04(2011)102, arXiv:1011.6396 [hep-th].

[33]G.R.Dvali,G.Gabadadze,M.Porrati,Phys.Lett.B485(2000)208,arXiv:hep -th/0005016 [hep-th].

[34]C.Deffayet,Phys.Lett.B502(2001)199,arXiv:hep-th/0010186 [hep-th]. [35]C.Deffayet,G.R.Dvali,G.Gabadadze, Phys.Rev.D65(2002)044023,arXiv:

astro-ph/0105068 [astro-ph].

[36]D.Gorbunov,K.Koyama,S.Sibiryakov,Phys.Rev.D73(2006)044016,arXiv: hep-th/0512097 [hep-th].

[37]C.Charmousis,R.Gregory,N.Kaloper,A.Padilla,J.HighEnergyPhys.10(2006) 066,arXiv:hep-th/0604086 [hep-th].

[38]J.Khoury,J.Sakstein,A.R.Solomon,J.Cosmol.Astropart.Phys.1808(2018)024, arXiv:1805.05937 [hep-th].

[39]N.Aghanim,etal.,Planck,arXiv:1807.06209 [astro-ph.CO],2018.

[40]X.-l.Chen,R.J.Scherrer,G.Steigman,Phys.Rev.D63(2001)123504, arXiv: astro-ph/0011531 [astro-ph].

[41]R.H.Cyburt,B.D.Fields,K.A.Olive,T.-H.Yeh,Rev.Mod.Phys.88(2016)015004, arXiv:1505.01076 [astro-ph.CO].

[42]D.J.Fixsen,Astrophys.J.707(2009)916,arXiv:0911.1955 [astro-ph.CO]. [43]Y.Akrami,S.F.Hassan,F.Könnig,A.Schmidt-May,A.R.Solomon,Phys.Lett.B

748(2015)37,arXiv:1503.07521 [gr-qc].

[44]B.P.Abbott,etal.,LIGOScientific,VIRGO,Phys.Rev.Lett.118(2017)221101, arXiv:1706.01812 [gr-qc];Erratum:Phys.Rev.Lett.121 (12)(2018)129901. [45]C.de Rham,J.T. Deskins, A.J.Tolley, S.-Y.Zhou,Rev.Mod. Phys. 89 (2017)

025004,arXiv:1606.08462 [astro-ph.CO].