On Nonlocal Modified Gravity and its Cosmological Solutions

0.15 MB·

by  Ivan Dimitrijevic

On Nonlocal Modified Gravity
and its Cosmological Solutions
IvanDimitrijevic,BrankoDragovich,JelenaStankovic,AlexeyS.Koshelev
andZoranRakic
7
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2

n
a
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AbstractDuringhundredyearsofGeneralRelativity(GR),manysignificantgrav-
9
itationalphenomenahavebeenpredictedanddiscovered.GeneralRelativityisstill

] thebesttheoryofgravity.Nevertheless,some(quantum)theoreticaland(astrophys-
h icalandcosmological)phenomenologicaldifficultiesofmoderngravityhavebeen
t
- motivation to search more general theory of gravity than GR. As a result, many
p
modificationsofGRhavebeenconsidered.Oneofpromisingrecentinvestigations
e
h isNonlocalModifiedGravity.Inthisarticlewepresentabriefreviewofsomenon-
[ localgravitymodelswiththeircosmologicalsolutions,inwhichnonlocalityisex-
pressedbyananalyticfunctionofthed’Alembert-Beltramioperator(cid:3).Somenew
1
resultsarealsopresented.
v
0
9
0
2
0
.
1 I.Dimitrijevic
0 FacultyofMathematics,UniversityofBelgrade,Studentskitrg16,Belgrade,Serbia
7 e-mail:[email protected]
1
B.Dragovich
:
v InstituteofPhysics, UniversityofBelgrade;MathematicalInstituteSANU,Belgrade, Serbia
i e-mail:[email protected]
X
J.Stankovic
r
TeacherEducationFaculty,UniversityofBelgrade,KraljiceNatalije43,Belgrade,Serbia
a
e-mail:[email protected]
A.S.Koshelev
DepartamentodeF´ısicaandCentrodeMatema´ticaeAplicac¸o˜es,UniversidadedaBeiraInterior,
6200Covilha˜,Portugal;Theoretische Natuurkunde, VrijeUniversiteitBrussel, andTheInterna-
tionalSolvayInstitutes,Pleinlaan2,B-1050Brussels,Belgium
e-mail:[email protected]
Z.Rakic
FacultyofMathematics,UniversityofBelgrade,Studentskitrg16,Belgrade,Serbia
e-mail:[email protected]
1
2 I.Dimitrijevicetal.
1 Introduction
Generalrelativity(GR)wasformulatedonehundredyearsagoandisalsoknownas
Einsteintheoryofgravity.GRisregardedasoneofthemostprofoundandbeautiful
physical theories with great phenomenologicalachievements and nice theoretical
properties. It has been tested and quite well confirmedin the Solar system, and it
hasbeenalsousedasatheoreticallaboratoryforgravitationalinvestigationsatother
spacetimescales.GRhasimportantastrophysicalimplicationspredictingexistence
ofblackholes,gravitationallensingandgravitationalwaves1.Incosmology,itpre-
dicts existence of about95% of additionalnew kind of matter, which makes dark
sideoftheuniverse.Namely,ifGRisthegravitytheoryfortheuniverseasawhole
andiftheuniverseishomogeneousandisotropicwiththeflatFriedmann-Lemaˆıtre-
Robertson-Walker(FLRW) metricat thecosmic scale, thenit containsabout68%
ofdarkenergy,27%ofdarkmatter,andonlyabout5%ofvisiblematter[2].
Despite of some significant phenomenological successes and many nice theo-
retical properties, GR is not complete theory of gravity. For example, attempts to
quantizeGRleadtothe problemofnonrenormalizability.GRalso containssingu-
larities like the Big Bang andblack holes. At the galactic andlarge cosmic scales
GR predicts new forms of matter, which are not verified in laboratory conditions
andhavenotsofarseeninparticlephysics.Hence,therearemanyattemptstomod-
ifyGeneralrelativity.Motivationsforitsmodificationusuallycomefromquantum
gravity,string theory,astrophysicsand cosmology(fora review,see [22, 60, 63]).
WearemainlyinterestedincosmologicalreasonstomodifyEinsteintheoryofgrav-
ity,i.e.tofindsuchextensionofGRwhichwillnotcontaintheBigBangsingularity
andofferanotherpossibledescriptionoftheuniverseaccelerationandlargeveloci-
tiesingalaxiesinsteadofmysteriousdarkenergyanddarkmatter.Itisobviousthat
physical theory has to be modified when it contains a singularity. Even if it hap-
penedthatdarkenergyanddarkmatterreallyexistitisstillinterestingtoknowis
thereamodifiedgravitywhichcanimitatethesameorsimilareffects.Hence,ade-
quategravitymodificationcanreduceroleandrateofthedarkmatter/energyinthe
universe.
Anywell foundedmodificationof the Einstein theoryof gravityhas to contain
generalrelativityandtobeverifiedatleastonthedynamicsoftheSolarsystem.In
otherwords,ithastobeageneralizationofthegeneraltheoryofrelativity.Mathe-
matically,itshouldbeformulatedwithinthepseudo-Riemanniangeometryinterms
of covariantquantities and take into account equivalence of the inertial and grav-
itational mass. Consequently, the Ricci scalar R in gravity Lagrangian L of the
g
Einstein-Hilbertactionshouldbereplacedbyan adequatefunctionwhich,ingen-
eral, may contain not only R but also some scalar covariant constructions which
arepossibleinthepseudo-Riemanniangeometry.However,wedonotknowwhatis
hereadequatefunctionandthereareinfinitelymanypossibilitiesforitsconstruction.
Unfortunately,sofarthereisnoguidingtheoreticalprinciplewhichcouldmakeap-
propriatechoicebetweenallpossibilities.InthiscontexttheEinstein-Hilbertaction
1Whilewepreparedthiscontribution,thediscoveryofgravitationalwaveswasannounced[1].
OnNonlocalModifiedGravityanditsCosmologicalSolutions 3
isthesimplestone,i.e.itcanbeviewedasrealizationoftheprincipleofsimplicity
inconstructionofL .
g
Oneofpromisingmodernapproachestowardsmorecompletetheoryofgravityis
itsnonlocalmodification.Motivationfornonlocalmodificationofgeneralrelativity
can be found in string theory which is nonlocal theory and contains gravity. We
present here a brief review and some new results of nonlocalgravity with related
bouncecosmologicalsolutions.Inparticular,wepayspecialattentiontomodelsin
whichnonlocalityisexpressedbyananalyticfunctionofthed’Alembertoperator
(cid:3)= 1 ¶ m √ ggmn ¶ n like nonlocality in string theory. In these models, we are
√ g −
mainly−interestedinnonsingularbouncesolutionsforthecosmicscalefactora(t).
In Sect. 2 we mentiona few differentapproachesto nonlocalmodifiedgravity.
Section3 containsrathergeneralmodifiedactionwith an analyticnonlocalityand
with corresponding equations of motion. Cosmological equations for the FLRW
metricispresentedinSect.4.Cosmologicalsolutionsforconstantscalarcurvature
are considered separately in Sect. 5. Some new examplesof nonlocalmodels and
relatedAnsa¨tzeareintroducedinSect.6.Attheandafewremarksarealsonoticed.
2 Nonlocal Modified Gravity
We consider here nonlocal modified gravity. Usually a nonlocal modified gravity
modelcontainsaninfinitenumberofspacetimederivativesintheformofapower
seriesexpansionwithrespecttothed’Alembertoperator(cid:3)= 1 ¶ m √ ggmn ¶ n .In
√ g −
thisarticle,wearemainlyinterestedinnonlocalityexpressedin−theformofanana-
lyticfunctionF((cid:3))=(cid:229) ¥ f (cid:3)n,wherecoefficients f shouldbedeterminedfrom
n=0 n n
varioustheoreticaland phenomenologicalconditions.Some conditionsare related
totheabsenceoftachyonsandgosts.
Beforetoproceedwiththisanalyticnonlocalityitisworthtomentionsomeother
interestingnonlocalapproaches.Forapproachescontaining(cid:3) 1 onecansee,e.g.,
−
[27, 26, 66, 61, 45, 67, 42, 43, 46, 47] and referencestherein. For nonlocalgrav-
ity with(cid:3) 1 see also [8, 58].Manyaspectsofnonlocalgravitymodelshavebeen
−
considered,seee.g.[20,16,17,59,18,36]andreferencestherein.
Ourmotivationtomodifygravityinananalyticnonlocalwaycomesmainlyfrom
string theory, in particular from string field theory (see the very original effort in
thisdirectionin[3])and p-adicstringtheory[15,38,39,40,65].Sincestringsare
one-dimensionalextendedobjects,theirfieldtheorydescriptioncontainsspacetime
nonlocalityexpressedbysomeexponentialfunctionsofd’Alembertoperator(cid:3).
Atclassicallevelanalyticnon-localgravityhasproventoalleviatethesingularity
oftheBlack-holetypebecausetheNewtonianpotentialappearsregular(tendingto
aconstant)onauniversalbasisattheorigin[41,11,9].Alsotherewassignificant
successinconstructingclassicallystablesolutionforthecosmologicalbounce[11,
13,48,51,55].
Analysisofperturbationsrevealedanaturalabilityofanalyticnon-localgravities
to accommodate inflationary models. In particular, the Starobinsky inflation was
4 I.Dimitrijevicetal.
studiedindetailsandnewpredictionsfortheobservableparametersweremade[24,
53]. Moreover, in the quantum sector infinite derivative gravity theories improve
renormalization,seee.g.whiletheunitarityisstillpreserved[56,57,53](notethat
justalocalquadraticcurvaturegravitywasproventoberenormalizablewhilebeing
non-unitary[64]).
3 Modified GR withAnalyticalNonlocality
Tobetterunderstandnonlocalmodifiedgravityitself,weinvestigateitherewithout
presence of matter. Models of nonlocal gravity which we mainly investigate are
givenbythefollowingaction
M2 l
S= d4x√ g PR L + P(R)F((cid:3))Q(R) , (1)
Z − (cid:18) 2 − 2 (cid:19)
¥
whereRisthescalarcurvature,L isthecosmologicalconstant,F((cid:3))= (cid:229) f (cid:3)n
n
n=0
isananalyticfunctionofthed’Alembert-Beltramioperator(cid:3)=(cid:209) m (cid:209) m where(cid:209) m is
thecovariantderivative.ThePlanckmassM isrelatedtotheNewtonianconstant
P
GasM2= 1 andP,Qarescalarfunctionsofthescalarcurvature.Thespacetime
P 8p G
dimensionality D=4 and our signature is ( ,+,+,+). l is a constant and can
be absorbedin the rescaling of F((cid:3)). Howe−ver,it is convenientto remain l and
recoverGRinthelimitl 0.
→
Note that to have physically meaningful expressions one should introduce the
scale of nonlocality using a new mass parameter M. Then the function F would
¥
be expanded in Taylor series as F((cid:3))= (cid:229) f¯(cid:3)n/M2n with all barred constants
n
n=0
dimensionless. For simplicity we shall keep M2 =1. We shall also see later that
analytic function F((cid:3))=(cid:229) ¥ f (cid:3)n, has to satisfy some conditions, in order to
n=0 n
escape unphysicaldegreesof freedomlike ghostsand tachyons,and to have good
behaviorinquantumsector(see[9,10,41]).
Varyingtheaction(1)bysubstituting
gmn gmn +hmn (2)
→
tothelinearorderinhmn ,removingthetotalderivativesandintegratingfromtime
totimebypartsonegets
mn
h
d S= d4x√ g Gmn , (3)
Z − 2 (cid:20)− (cid:21)
where
OnNonlocalModifiedGravityanditsCosmologicalSolutions 5
l l ¥
Gmn ≡MP2Gmn +gmn L − 2gmn PF((cid:3))Q+l (Rmn −Kmn )V− 2 (cid:229) fn
n=1
n 1
(cid:229)− Pm(l)Q(nn−l−1)+Pn(l)Q(mn−l−1) gmn (grs Pr(l)Q(sn−l−1)+P(l)Q(n−l)) =0
×l=0(cid:16) − (cid:17)
(4)
presentsequationsofmotionforgravitationalfieldgmn inthevacuum.In(4)Gmn =
Rmn 1gmn RistheEinsteintensor,
−2
Kmn =(cid:209) m (cid:209) n gmn (cid:3), V =PRF((cid:3))Q+QRF((cid:3))P,
−
where the subscript R indicates the derivative w.r.t. R (as many times as it is re-
peated)and
P(l)=(cid:3)lP, Pr(l)=¶ r (cid:3)lPwiththesameforQ, PR, ...
Inthecaseofgravitywithmatter,thefullequationsofmotionareGmn =Tmn ,where
Tmn istheenergy-momentumtensor.Thankstotheintegrationbypartsthereisal-
waysthesymmetryofanexchangeP Q.
Whenl =0in(4)werecognizeth↔eEinstein’sGRequationwiththecosmolog-
icalconstantL .If f =0forn 1then(4)correspondstoequationsofmotionof
n
≥
an f(R)theory.
4 CosmologicalEquations forFLRWMetric
WeusetheFLRWmetric
dr2
ds2= dt2+a2(t) +r2dq 2+r2sin2q df 2
− (cid:18)1 kr2 (cid:19)
−
and look for some cosmological solutions. In the FLRW metric the Ricci scalar
curvatureis
a¨ a˙2 k
R=6 + +
(cid:18)a a2 a2(cid:19)
and
(cid:3)= ¶ 2 3H¶ ,
− t − t
whereH=a˙ istheHubbleparameter.Weusenaturalsystemofunitsinwhichspeed
a
oflightc=1.
Due to symmetries of the FLRW spacetime, in (4) there are only two linearly
independentequations.Theyare:traceand00,i.e.whenindicesm =n =0.
Thetraceequationand00-equation,respectively,are
6 I.Dimitrijevicetal.
M2R 4L +2l PF((cid:3))Q l (R+3(cid:3))V
P − −
¥ n 1 (5)
l (cid:229) fn (cid:229)− grs ¶ r (cid:3)lP¶ s (cid:3)n−l−1Q+2(cid:3)lP(cid:3)n−lQ =0,
− n=1 l=0(cid:16) (cid:17)
l l ¥
M2G L + PF((cid:3))Q+l (R (cid:209) (cid:209) (cid:3))V (cid:229) f
p 00− 2 00− 0 0− − 2 n
n=1
(6)
n 1
(cid:229)− 2¶ 0(cid:3)lP¶ 0(cid:3)n−l−1Q+grs ¶ r (cid:3)lP¶ s (cid:3)n−l−1Q+(cid:3)lP(cid:3)n−lQ =0.
×l=1(cid:16) (cid:17)
5 CosmologicalSolutions forConstantScalarCurvature R
WhenRisaconstantthenPandQarealsosomeconstantsandwehavethat(cid:3)R=0,
F((cid:3))= f .Thecorrespondingequationsofmotion(5)and(6)containsolutionsas
0
in the localcase. However,metricperturbationsatthe backgroundR=const. can
givenontrivialcosmicstructureduetononlocality.
LetR=R =constant=0.Then
0
6
a¨ a˙ k
2
6 + + =R . (7)
a a a2 0
(cid:16) (cid:0) (cid:1) (cid:17)
Thechangeofvariableb(t)=a2(t)transforms(7)intoequation
3b¨ R b= 6k. (8)
0
− −
DependingonthesignofR ,thefollowingsolutionsofequation(8)are
0
b(t)= 6k +s eqR30t+t e−qR30t, R0>0,
R
0
(9)
6k R R
b(t)= +s cos − 0t+t sin − 0t, R <0,
R r 3 r 3 0
0
wheres andt aresomeconstantcoefficients.
SubstitutionR=R intoequationsofmotion(5)and(6)yields,respectively,
0
M2R 4L +2l Pf Q l R V =0, (10)
p 0− 0 − 0 0
l
M2G L + Pf Q+l R V =0, (11)
p 00− 2 0 00 0
whereV = f (P Q+Q P) andG =R +R0.
0 0 R R |R=R0 00 00 2
Combiningequations(10)and(11)oneobtains
OnNonlocalModifiedGravityanditsCosmologicalSolutions 7
M2R 4L +2l Pf Q l R V =0, (12)
p 0− 0 − 0 0
4R +R =0. (13)
00 0
Equation (12) connects some parameters of the nonlocal model (1) in the al-
gebraicform with respectto R , while (13) implies a conditionon the parameters
0
s ,t ,kandR insolutions(9).Namely,R isrelatedtofunctionb(t)as
0 00
3a¨ 3(b˙)2 2bb¨
R = = − . (14)
00 − a 4 b2
ReplacingR in(13)by(14)andusingdifferentsolutionsforb(t)in(9)weobtain
00
9k2=R2st , R >0,
0 0
(15)
36k2=R2(s 2+t 2), R <0.
0 0
5.1 Case: R >0
0
Letk=0.From9k2=R2st followsthatatleastoneofs andt hastobezero.
• 0
Thusthereispossibilityforanexponentialsolutionfora(t)anda(t)=0.Taking
t =0ands =a2onehas
0
b(t)=a20eqR30t. (16)
If k=+1 one can find j such that s +t = 6 coshj and s t = 6 sinhj .
• R0 − R0
Moreover,weobtain
12 1 R
b(t)= cosh2 0t+j ,
R 2 r 3
0 (cid:16) (cid:17) (17)
12 1 R
a(t)= cosh 0t+j .
rR 2 r 3
0 (cid:16) (cid:17)
Ifk= 1onecantransformb(t)anda(t)to
• −
12 1 R
b(t)= sinh2 0t+j ,
R 2 r 3
0 (cid:16) (cid:17) (18)
12 1 R
a(t)= sinh 0t+j .
rR0(cid:12) 2(cid:16)r 3 (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 I.Dimitrijevicetal.
5.1.1 Case:R=12g 2
This is a special case of R , which simplifies the aboveexpressionsand yieldsde
0
Sitter-likecosmologicalsolutions.
k=0:
•
b(t)=a2e2gt, a(t)=a egt. (19)
0 0
k=+1:
• 1 j
b(t)= cosh2 g t+ ,
g 2 2
(cid:16) (cid:17)
(20)
1 j
a(t)= cosh g t+ .
g 2
(cid:16) (cid:17)
| |
k= 1:
• − 1 j
b(t)= sinh2 g t+ ,
g 2 2
(cid:16) (cid:17)
(21)
1 j
a(t)= sinh g t+ .
g 2
(cid:12) (cid:16) (cid:17)(cid:12)
| |(cid:12) (cid:12)
(cid:12) (cid:12)
5.2 Case: R <0
0
Whenk=0thens =t =0,andconsequentlyb(t)=0.
• Ifk= 1onecandefinej bys = 6cosj andt = 6sinj ,andrewriteb(t)
• − −R0 −R0
anda(t)as
12 1 R
b(t)= − cos2 0t j ,
R0 2(cid:16)r− 3 − (cid:17) (22)
12 1 R
a(t)= − cos ( 0t j ) .
r R0 (cid:12) 2 r− 3 − (cid:12)
(cid:12) (cid:12)
Inthelastcasek=+1,bythesame(cid:12)procedureasfork=(cid:12) 1,onecantransform
• −
b(t)toexpression
12 1 R
b(t)= sin2 0t j , (23)
R0 2(cid:16)r− 3 − (cid:17)
whichisnotpositiveandhenceyieldsnosolution.
5.3 Case: R =0
0
The case R =0 can be considered as limit of R 0 in both cases R >0 and
0 0 0
R <0.WhenR >0 thereis condition9k2 =R2st →in (15). Fromthiscondition,
0 0 0
OnNonlocalModifiedGravityanditsCosmologicalSolutions 9
R 0impliesk=0andarbitraryvaluesofconstantss andt .Thesameconclusion
0
obt→ainswhenR <0withcondition36k2=R2(s 2+t 2).Inboththesecasesthereis
0 0
Minkowskisolutionwithb(t)=constant>0andconsequentlya(t)=constant>0,
see(9).
6 Some Models andRelatedAnsa¨tze forCosmologicalSolutions
6.1 NonlocalGravityModelQuadraticinR
NonlocalgravitymodelwhichisquadraticinRwasgivenbytheaction[11,12]
R 2L
S= d4x√ g − +RF((cid:3))R . (24)
Z − (cid:16) 16p G (cid:17)
Thismodelisimportantbecauseitisghostfreeandhassomenonsingularbounce
solutions,whichcanberegardedasasolutionoftheBigBangcosmologicalsingu-
larityproblem.
Thecorrespondingequationsofmotioncanbeeasilyobtainedfrom(5)and(6).
Toevaluaterelatedequationsofmotion,thefollowingAnsa¨tzewereused:
LinearAnsatz:(cid:3)R=rR+s,whererandsareconstants.
• QuadraticAnsatz:(cid:3)R=qR2,whereqisaconstant.
• QubicAnsatz:(cid:3)R=CR3,whereCisaconstant.
• Ansatz(cid:3)nR=c Rn+1, n 1,wherec areconstants.
n n
• ≥
TheseAnsa¨tzemakesomeconstraintsonpossiblesolutions,butsimplifyformalism
tofindaparticularsolution(see[29]andreferencestherein).
6.1.1 LinearAnsatzandNonsingularBounceCosmologicalSolutions
UsingAnsatz(cid:3)R=rR+safewnonsingularbouncesolutionsforthescalefactor
L 1 L t2
are found:a(t)=a cosh t (see [11, 12]), a(t)=a e2 3 (see [48, 49])
0 (cid:18)q3 (cid:19) 0 q
anda(t)=a (s el t+t e l t)[30].ThefirsttwoconsequencesofthisAnsatzare
0 −
s s
(cid:3)nR=rn(R+ ), n 1, F((cid:3))R=F(r)R+ (F(r) f ), (25)
0
r ≥ r −
whichconsiderablysimplifynonlocalterm.
Generalization of the above quadratic model in the form of nonlocal term
RpF((cid:3))Rq, where p andq are some naturalnumbers,wasrecentlyconsideredin
[28].Herecosmologicalsolutionforthescalefactorhastheforma(t)=a e gt2.
o −
10 I.Dimitrijevicetal.
6.2 GravityModelwith NonlocalTermR 1F((cid:3))R
−
Thismodelwasintroducedin[31]anditsactionmaybewrittenintheform
R
S= d4x√ g +R 1F((cid:3))R , (26)
Z − (cid:16)16p G − (cid:17)
whereF((cid:3))=(cid:229) ¥ f (cid:3)nand f = L playsroleofthecosmologicalconstant.
n=0 n 0 −8p G
ThenonlocaltermR 1F((cid:3))Rin(26)isinvariantundertransformationR CR.
−
→
ThisnonlocaltermdoesnotdependonthemagnitudeofscalarcurvatureR,buton
itsspacetimedependence,andintheFLRWcaseisrelevantonlydependenceofR
L
ontimet.Term f = iscompletelydeterminedbythecosmologicalconstant
L , which accordi0ng t−o8Lp CGDM model is small and positive energy density of the
vacuum. Coefficients f, i N can be estimated from other conditions, including
i
∈
agreementwith dynamicsthe Solar system. In comparisonto the modelquadratic
inR(24),completeLagrangianofthismodelremainstobelinearinRandinsuch
senseissimplernonlocalmodificationthan(24).
In this model are also used the above Ansa¨tze. Especially quadratic Ansatz
(cid:3)R=qR2, whereq is a constant,is effectiveto considerpower-lawcosmological
solutions,see[31,32,37,33].
6.3 SomeNewModelsand Ansa¨tze
Itisworthtoconsidersomeparticularexamplesofaction(1)whenP=Q=(R+
R )m,i.e.
0
1 l
S= R L + (R+R )mF((cid:3))(R+R )m √ gd4x, (27)
Z (cid:16)16p G − 2 0 0 (cid:17) −
whereR R,m Q,andwhichhavescalefactorsolutionas
0
∈ ∈
a(t)=Atnegt2, g R. (28)
∈
TothisendweconsidertheAnsatz
(cid:3)(R+R )m=p(R+R )m, (29)
0 0
where pisaconstantand(cid:3)isthed’AlembertoperatorinFLRWmetric.
¿FromAnsatz(29)andscalarcurvatureRfork=0,wegetthefollowingsystem
ofequations: