Adiabatic regularization of the graviton stress-energy tensor in de Sitter space-time F. Finelli,1,∗ G. Marozzi,2,† G. P. Vacca,2,‡ and G. Venturi2,§ 1CNR-INAF/IASF, Istituto di Astrofisica Spaziale e Fisica Cosmica Sezione di Bologna Via Gobetti 101, I-40129 Bologna – Italy 2Dipartimento di Fisica, Universit`a degli Studi di Bologna and I.N.F.N., via Irnerio, 46 – I-40126 Bologna – Italy We study the renormalized energy-momentum tensor of gravitons in a de Sitter space-time. Af- ter canonically quantizing only the physical degrees of freedom, we adopt the standard adiabatic subtractionusedformassless minimally coupled scalarfieldsasaregularization procedureandfind that the energy density of gravitons in the E(3) invariant vacuum is proportional to H4, where H is the Hubble parameter, but with a positive sign. According to this result the scalar expansion rate,which is gauge invariantin deSitterspace-time, is increased bythefluctuations. Thisimplies 5 that gravitons may then add to conformally coupled matter in driving the Starobinsky model of 0 inflation. 0 PACSnumbers: 98.80.Cq,04.62.+v 2 n a Interest in the back-reaction of particles produced by Since inflation is very close to de Sitter, this calculation J an external field has always been great in the last thirty mayalsobeusefulinthecontextofback-reactionduring 4 years. Thisissuehasbeenstudiedincosmologicalspace- inflation. For the case of inflation initiated by quantum 1 times following the seminal works by L. Parker [1]. The anomalies of test fields [5], this would lead to the com- back-reaction problem for quantized test fields of dif- plete evaluation of the gravitational sector. Throughout 3 v ferent spin has been examined in great detail over the units are chosen such that h¯ =c=1. 1 decades for a de Sitter space-time [2]. According to the action 0 Gravitationalback-reactiontreatedinaself-consistent 1 way is much more difficult. Taking into account metric S = 1 d4x√ g[R 2Λ] (1) 7 fluctuationstogetherwithmatterfluctuationstofirstor- 16πG − − 0 Z der is a textbook subject. To second order, the level of 4 the Einstein equations are: difficultyincreasesduetonon-linearityandtotheinvari- 0 / ance under coordinate transformations. 1 c Vacuumspace-timesarethesimplestarenainwhichto Rµν − 2gµνR+Λgµν =0 (2) q study the energy content carried by metric fluctuations, - InthedeSitterspace-timetheonlynon-vanishingmet- r since the latter just reduce to gravitational waves (i.e. g ricfluctuationsarethedynamicaldegreesoffreedom(the tensormodes)inthe absence ofdynamicalmatter. Even v: in such a simple setting, non-linear self-consistent solu- gravitons): i X tions are quite surprising: the gravitational geon [3] is ds2 = g(0)+δg dxµdxν one of the historic examples, which triggered more work µν µν r a in this area. = h dt2+a(t)2i[γ +h ]dxidxj ij ij Although gravitationalwaves are somewhat similar to − = a(η)2 dη2+(γ +h )dxidxj (3) massless minimally coupled test scalar fields, they are ij ij − not exactly the same and the former may reveal quan- wherea(t)=eHt isth(cid:2)e scalefactor(H2 =Λ/3)(cid:3),η is the tum gravity aspects which are not proper of the latter: conformaltime and γ is the spatially flat metric (greek here we shall show this relevant difference in the de Sit- ij indices go from 0 to 3, latin ones from 1 to 3 unless oth- ter case. In this paper we study the energy-momentum erwise stated). The gravitons are traceless, transverse tensor (EMT henceforth) of quantized gravitonsand the (hi = 0,∂ hij = 0) and therefore gauge-invariant with scalarexpansionratein de Sitter space-time ina pertur- i j respect to tensorial spatial transformations. First order bative way. We shall consider only half of the de Sit- scalar metric fluctuations vanish in the absence of dy- ter Carter-Penrose diagram, i. e. only the cosmological namical matter, as already mentioned. flat expanding branch. To our knowledge this case has In order to compute the graviton EMT we proceed as beenonlyapproachedbyFord[4],butnotfullyexplored. in textbooks [6]: τGW = 1 G(2) = M2 R(2) 1 g gαβR (2) µν −8πG µν − pl µν − 2 µν αβ ∗Electronicaddress: fi[email protected] (cid:20) (cid:21) †Electronicaddress: [email protected] (cid:0) (cid:1) (4) ‡Electronicaddress: [email protected] where we have set Mp2l = (8πG)−1. The above expres- §Electronicaddress: [email protected] sion will become, after using the first order equations of 2 motion: From Eqs. (9) and (10) we see that the amplitudes h s,k satisfy the same equation as massless minimally coupled 1 τGW = M2 R(2) g(0)g(0)αβR(2) . (5) scalar fields: µν − pl µν − 2 µν αβ (cid:18) (cid:19) k2 where by the superscript (2) we mean terms which are ¨h +3Hh˙ + h =0 (14) s,k s,k a2 s,k quadratic in the perturbation h . R(2) can be found in ij µν Eq. (35.58b) of [6]. and the solution for the Fourier mode which becomes a We obtain plane wave for short wavelengths is: 1 H 1 3 G(020) = −8h˙ijh˙ij + 2 h˙ijhij + 2hijh¨ij + 8∂khij∂khij hs,k = a3/21M 2πH 1/2H3(1/)2(−kη). (15) 1 pl ∂nhim∂ h (6) (cid:16) (cid:17) m in −4 This solution is valid for all values of k except for k = G(2) = 1h˙mn∂ h 1hmn∂ h˙ +hmn∂ h˙ (7)0 (corresponding to a zero measure) in which case the i0 4 i mn− 2 m in i mn solutionissimplyaspaceindependentpuregauge. When G(2) = a2δ 3h˙mnh˙ 3∂khmn∂ h averagedoverthevacuumstateannihilatedbyˆbtheEMT ij ij 8 mn− 8 k mn of gravitons takes a perfect fluid form: (cid:20) 1 1 a2 + ∂nhim∂ h + ∂ hmn∂ h + ∂khm∂ h τGW =diag(ǫ,a2p,a2p,a2p), (16) 4 m in 4 j i mn 2 j k mi h µν i (cid:21) 1 1 1 which is covariantly conserved in de Sitter space-time: hmn∂ ∂ h hmn∂ ∂ h + hmn∂ ∂ h −2 m j in− 2 m i jn 2 m n ij ǫ˙+3H(ǫ+p)=0 [8, 9]. 1 a2 1 For the vacuum expectation value of the effective en- + hmn∂ ∂ h h˙mh˙ ∂ hn∂ hm, (8) 2 i j mn− 2 i mj − 2 m i n j ergy and pressure we obtain the following value where ˙ denotes the derivative with respect to t. The d3k action can also be expanded as S = S(0) +S(2), where ǫ ≡ Mp2l (2π)3 ǫs,k S(0) is the background value. The second order piece s Z X S(2), omitting boundary terms, is: = M2 d3k 1 h˙ 2+ 1k2 h 2 pl (2π)3 4| s,k| 4a2| s,k| S(2) = M8p2l d4xa3"h˙mnh˙mn−∂khmn∂khmn#.(9) X+sH h˙sZ,kh∗s,k+(cid:20)hs,kh˙∗s,k (17) Z (cid:16) (cid:17)i LetusperformaFourierexpansionandconsideronlythe physical degrees of freedom (polarization states) h and + d3k h×: p ≡ Mp2l (2π)3 ps,k hij = (2π1)3 dkeik·x h+e+ij +h×e×ij , (10) = Xs M2 Z d3k 5 h˙ 2+ 7 k2 h 2(1,8) Z pl (2π)3 −12| s,k| 12a2| s,k| where e+ and e× are the pola(cid:2)rizationtensor(cid:3)s having the Xs Z (cid:20) (cid:21) following properties (s=+, ): where h are the solutions given in Eq. (15). The × s,k aboveexpressionsagreewiththe spaceaveragedEMTof e =e , kie =0, e =0, (11) ij ji ij ii gravitonsobtainedin[8]. Itisimportanttonotethatthe term Hhh˙ is reminiscent of a scalar field non-minimally e ( ~k,s)=e∗(~k,s), e∗ (~k,s)eij(~k,s)=4. (12) coupled to gravity: however, a non-minimal scalar field ij − ij ij s wouldalsohaveamasstermoforderH ( H2h2)which X ∼ instead is absent in Eqs. (17,18). Owing to the presence Theseshouldbesufficientforourone-loopcalculationon of the term Hhh˙, the EMT of gravitationalwaves is not shell. Thus we do not concern ourselves with unphysical invariant under the transformation h h + const. degrees of freedom and ghosts. This method of selecting ij ij → which has generated so much activity in the context of only the physical degrees of freedom was used in first massless minimally coupled scalar fields [10]. Of course, deriving the spectrum of gravitational waves [7] and is thisisnotsurprisingatall,sincezeromodesofgravitons used in computing the gravitational wave contribution are pure gauge, as already stated. to microwave anisotropies. In order to regularize bilinear quantities we proceed On quantizing we have in the following way: we subtract the fourth order term hˆs(t,x)= (2π1)3 dk hs,k(t)eik·x ˆbk+h∗s,k(t)e−ik·x ˆb†k . owfitthheaamdiaasbsatteicrmex[p1a1n,s1io2]n. fAorntEheMsToluwtiitohnatomEasqs. t(e1r4m) Z h (13)i m which regularizes the adiabatic expansion is needed 3 in order to proceed with the adiabatic subtraction. It which is only quadratically divergent in the ultraviolet is then necessary to add a term to the energy density (in contrast with the kinetic and gradient terms) and (ǫ ǫ (m)=ǫ +m2 h 2/4)andtothepressure does not lead to any bad infrared behaviour (in contrast s,k s,k s,k sk densit→y(p p (m)=p| +| 5m2 h 2/12)inorder to h 2). The term in Eq. (22) is therefore negative, s,k s,k s,k sk sk → | | | | to have a covariantly conserved EMT corresponding to with a sign which is opposite to that of the kinetic and (17,18) with a mass term. The covariantconservationof gradient terms: this difference of sign is also reflected in such an EMT is given by the renormalized values, leading to the positivity of the energy density for gravitons. ǫ˙ (m)+3H[ǫ (m)+p (m)]=0 (19) The result should be comparedwith that obtained for s,k s,k s,k a massless minimally coupled scalar field in the Allen- and is equivalent to the equation of motion (14) with Folacci (AF henceforth) vacuum [13, 17]: a mass term m2h . The regularized energy density is sk 119 then given by T AF =g H4, (23) h µνiREN µν960π2 d3k ǫ = lim lim M2 ǫ ǫ(4)(m) (,20)which corresponds to a contribution with a negative en- REN k∗→∞m→0Xs plZ|k|<k∗ (2π)3 h s,k− s,k i ergydensity. Onconsideringthe AFvacuumforthe cor- relator one obtains the following result [13, 17]: where ǫ(4)(m) is the fourth order term of the adiabatic s,k H3t expansion [2]. Let us note that, after performing the in- h2 = . (24) h siREN 4π2M2 tegrals, we first take the limit m 0 and finally remove pl → the ultraviolet cut-off: in this manner the correct adia- in which de Sitter invariance is broken in the standard baticsubtractionisimplemented. Indeedwhenthistech- way [20]. When the time derivative of Eq. (24) is con- nique is applied to a massless minimally coupled scalar sidered, one obtains fieldalltheresultsforanE(3)invariant-state[13]arere- produced. The above technique for computing integrals 1 H3 is therefore slightly different from the one we previously hhsh˙siREN = M2 8π2 , (25) employed [14, 15]. pl For the EMT we finally obtain which is the same result as given by our method. Thus ourapproachisconsistentwiththechoiceofthe AFvac- 361 τGW = g H4. (21) uum. The reason for the difference between (21) and h µν iREN − µν960π2 (23)isthepresenceoftheterm2Hhh˙ forgravitons. Our mainresult(21)canbe easily verifiedby notingthat the This is our main result, which is in contrast with others renormalized energy density of gravitons given by Eqs. claiming that gravitons decrease the effective cosmologi- (17) is calconstantattheone-looporderindeSitterspace-time [4, 16]. ǫ = ǫAF +M2 2H h h˙ We stressthatthe result(21)isde Sitter invariant,al- REN REN pl h s siREN thoughthevacuumchosenwasE(3)invariant. Thesame Xs 119 H4 361 happensformasslessminimallycoupledscalarfields[17]. = H4+ = H4. (26) The interesting terms which break de Sitter invariance −960π2 2π2 960π2 in the regularized value of EMT found in [10, 18] for Thecontributionoftheterm2Hhh˙ ispositiveandlarger an O(4) state in the case of massless minimally coupled than the (negative) energy density of a massless mini- scalar fields are due to the use of closed spatial sections mally coupled scalar field. and are quickly redshifted after few e-folds of exponen- For the case of conformally invariant fields the EMT tial expansion [27]. We also note that the result in Eq. is independent of the vacuum state chosen and is fully (21) is obtained from the integration of the finite terms given by the trace anomaly T: of the adiabatic expansion: this is also whathappens for the averaged EMT of massless minimally coupled scalar gµν T = T (27) fields. h µνiREN 4 It is important to note that the renormalized EMT of with T given by [21]: gravitons in Eq. (21) has p = ρ as equation of state. − The relation p= ρ/3 is the correspondingunrenormal- β ized one for long-−wavelengthmodes [8]. T = α2R RαβγδRαβγδ 4RαβRαβ +R2 − 2 − Letusagainfocusontheterm2Hhh˙: itsFouriercom- +γC C(cid:0)αβγδ, (cid:1)(28) αβγδ ponent is where C is the Weyl tensor, which is zero for a met- αβγδ h˙ h∗ +h h˙∗ = H3η2 , (22) ric which is conformal to Minkowsky as is Robertson- sk sk sk sk −M2k Walker(andthereforedeSitter). Thecoefficientsαandβ pl 4 obtained by dimensional regularizations for scalar, four- backgrounds: the inflationary era in scalar field driven component spinors, and gauge fields, are respectively universes is a transient state (local attractor), while de [22]: Sitter is a global solution. To conclude we have computed the regularized gravi- 1 1 ton EMT in de Sitter space-time by quantizing only the 1 1 α= 6 , β = 11 . physical degrees of freedom. We have found that the 2880π2  2880π2  12 62 (one-loop)gravitoncontributiontothecosmologicalcon-   stantintheE(3)invariant-vacuumispositive,incontrast For (massless) conformally coupled scalar fields one with that of a massless minimally coupled scalar field. has: This effect also appears in a second order perturbative analysis of the geometrical quantity Θ, which shows an H4 T = g , (29) increasedexpansionrate. Accordingtothisresult,gravi- h µνiREN − µν960π2 tons may then add to the trace anomaly of conformally coupledmatterindrivingtheStarobinskymodelofinfla- and therefore gravitons contribute, for example, as 361 tion [5]. The contribution of gravitons to the cosmologi- conformally coupled scalar fields! cal constant is not negligible and corresponds to a large It is interesting to also investigate the effect of gravi- number (361) of conformally coupled scalar fields. This tons on the background space-time within the frame- contribution may also alter the inflationary phase of the workofsecondorderperturbationtheoryfortheEinstein Starobinsky model [5, 23] since gravitons are not con- equations,byevaluatingthegaugeinvariant(fordeSitter formally coupled, thus the back-reaction may alter the space) geometric quantity Θ associated with the expan- evolution of the gravitons themselves. sionrateoftheuniverse(seeforexample[15]). Hencewe consider the following second order metric fluctuations Since gravitons are not conformally coupled [24] the for the gauge fixed metric given in Eq. (3): averaged EMT may be state dependent. One may also worry about the problems concerning zero modes a δg(2) = 2α(2) , δg(2) = β(2), which plagued massless minimally coupled scalar fields 00 − 0i −2 ,i [10, 13, 17, 18]. Gravitational waves do not have zero δg(2) = a2 ∂ χ(2)+∂ χ(2)+h(2) , (30) modes (insofar as these correspond to a pure gauge), i. ij 2 i j j i ij e. k>0. Further all the contributions from the infrared (cid:16) (cid:17) to renormalized bilinear quantities which we compute in where in the second line the vector χ(i2) is divergenceless this paper are finite (we may even include the contribu- and the tensor h(2) is transverse and traceless. The ex- tionfromthek =0mode sincethe measureofthis point ij pansion scalar is defined by Θ = uµ (where uµ is a in Fourier space is zero). µ ∇ normalized vector field, uµuµ = 1, defining the comov- For a massless minimally coupled test scalar field in − ing frame) and simplifies to de Sitter space-time, it has been shown that the EMT evaluated in the AF vacuum is an asymptotic attractor 1 1 Θ=3H h h˙ij 3Hα(2)+ 2β(2). (31) amongallpossible vacua[25]. If this weretrue forgravi- ij − 2 − a∇ tationalwavesalso,theresultobtainedfortheE(3)state would be completely general and lead to an asymptotic On using the Einstein equations one obtains the second value for the generalized anomaly [25] Q2 =361/180 for order fluctuations as functions of the physical gravitons. gravitationalwaves. In particular we find for the expansion scalar averaged over the vacuum: Fromthe theoreticalpoint of view it wouldalso be in- terestingtoseeifourresult(classicalandquantumgrav- 121 H2 ity in de Sitter space) can be related to conformal field Θ =3H 1+ . (32) h i 2880π2Mp2l! theory,assuggestedbythedS/CFTcorrespondence[26]. Last, but not least, should the same result persists Thus we see that the choice of vacuum for the physical for cosmologies with H˙ = 0 and non-vacuum states for gravitons, which led to the result Eq. (21) correspond- 6 modes on large scales, it would be interesting to com- ingtoapositivecosmologicalconstantcontribution,also pute the contribution of cosmological perturbations to leads to a contribution of the same sign to the scalar the present energy density. expansion rate Θ. Acknowledgments Our result is only in apparent contradiction with the possibilitythatscalarfluctuationsactagainsttheacceler- WewouldliketothankR.Abramo,B.Allen,L.Ford,B. atedexpansioninchaoticinflation[15]. Apossibleexpla- Losic, K. Kirsten, L. Parker,R. Woodard and S. Zerbini nation of the difference is the stability of the space-time for useful comments and communications. 5 [1] L. Parker, Phys. Rev. 183, 1057 (1969). discussed by Tsamis and Woodard, Nucl. Phys. B 474, [2] N. D. Birrell and P. C. W. Davies, Quantum Fields in 235 (1996). Curved Space (Cambridge University Press, Cambridge, [17] D.BernardandA.Folacci,Phys.Rev.D34,2286(1986). 1982). [18] A.Folacci,J.Math.Phys.32,2828(1991)[Erratum-ibid. 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