Nonlocal effective gravitational field equations and the running of Newton s constant G

Transcription

1 PHYSICAL REVIEW D 7, (005) Nonlocal effective gravitational field equations and the running of Newton s constant G H. W. Hamber* Deartment of Physics and Astronomy, University of California, Irvine, California , USA R. M. Williams Girton College, Cambridge CB3 0JG, United Kingdom and Deartment of Alied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom (Received 4 June 005; ublished 5 August 005) Nonerturbative studies of quantum gravity have recently suggested the ossibility that the strength of gravitational interactions might slowly increase with distance. Here a set of generally covariant effective field equations are roosed, which are intended to incororate the gravitational, vacuum-olarization induced, running of Newton s constant G. One attractive feature of this aroach is that, from an underlying quantum gravity ersective, the resulting long-distance (or large time) effective gravitational action inherits only one adjustable arameter, having the units of a length, arising from dimensional transmutation in the gravitational sector. Assuming the above scenario to be correct, some simle redictions for the long-distance corrections to the classical standard model Robertson-Walker metric are worked out in detail, with the results formulated as much as ossible in a model-indeendent framework. It is found that the theory, even in the limit of vanishing renormalized cosmological constant, generally redicts an accelerated ower-law exansion at later times t 1=H. DOI: /PhysRevD PACS numbers: m, Gw, Qc I. INTRODUCTION Nonerturbative studies of quantum gravity have recently suggested the ossibility that gravitational coulings might be weakly scale deendent due to nontrivial renormalization-grou effects. This would introduce a new gravitational scale, unrelated to Newton s constant, required in order to arametrize the gravitational running in the infrared region. If one is willing to accet such a scenario, then it seems difficult to find a comelling theoretical argument for why the nonerturbative scale entering the couling evolution equations should be very small, comarable to the Planck length. One ossibility ut forward recently is that the relevant nonerturbative scale is related to the curvature and therefore macroscoic in size, which could have observable consequences. One key ingredient in this argument is the relationshi, to some extent suorted by Euclidean lattice results combined with renormalization-grou arguments, between the scaling violation arameter and the scale of the average curvature. Irresective of the secific details of a gravitational theory at very short distances, such results would bring gravitation more in line with the rest of the standard model, where all gauge coulings are in fact known to run. In this aer we investigate the effects of a running gravitational couling G at large distances, with as few assumtions as ossible about the ultimate behavior of the theory at extremely short distances, where several ossible scenarios include a string cutoff at length scales S 0 1= [1], the aearance of higher derivative terms * address : HHamber@uci.edu address : R.M.Williams@damt.cam.ac.uk (either as direct contributions or as radiative corrections), or erhas a somewhat less aealing exlicit ultraviolet cutoff at the Planck scale. The running of the gravitational couling will generally be assumed to be driven by graviton vacuum-olarization effects, which roduce an antiscreening effect some distance away from the rimary source, and therefore tend to increase the strength of the gravitational couling. The above scenario is quite different from what one would exect, for examle, in suergravity theories, where significant cancellations arise in erturbation theory between graviton and matter loos [], and in contrast to ordinary gravity where in weak field erturbation theory L loos contribute L 1 owers of the curvature tensor to the effective action [3]. Instead, the running of Newton s constant is thought to arise due to the resence of a nontrivial, genuinely nonerturbative, ultraviolet fixed oint [4 6] (a hase transition in statistical mechanics arlance [4]). In this aer a ower-law (as oosed to a logarithmic) running of G will be imlemented via manifestly covariant nonlocal terms in the effective gravitational action and field equations. It ultimately will involve the inverse of the covariant d Alembertian raised to some fractional ower 1=, which in the framework of the resent aer remains largely unsecified, although nonerturbative models for quantum gravity have recently ut forward some rather secific redictions. Let us recall here, to rovide some degree of motivation, the recent discussions of [7,8] as a ossible theoretical framework for the running of Newton s G. The above results suggest that the gravitational constant G cannot be regarded a constant as in the classical theory, but instead changes slowly with scale due to the resence of weak =005=7(4)=04406(16)$ The American Physical Society

2 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) gravitational vacuum-olarization effects, in a way described by Gr G01 c r= 1= Or= = : (1.1) The exonent, generally related to the derivative of the beta function for ure gravity evaluated at the nontrivial ultraviolet fixed oint via the relation 0 G c 1=, is suosed to universally characterize the long-distance roerties of quantum gravitation, and is therefore exected to be indeendent of the secifics related to the nature of the ultraviolet regulator, or other detailed shortdistance features of the theory. 1 Recent estimates for the value of the universal scaling dimension 1 0 G c derived from nonerturbative studies of gravity vary from 1 3:0 [8] in the Euclidean Regge lattice case, to 1 3:8 in the exansion [10] about two dimensions carried to two loos [11 14], to 1 :7 [15] and 1 1:7 [16] in an aroximate renormalization-grou treatment a la Wilson based on an Einstein-Hilbert truncation, with some significant uncertainties in all three aroaches. More details, as well as a systematic comarison of the various methods and estimates, can be found in [17], where we argued, based on geometric arguments, in favor of the exact value of the exonent 1=3 for ure gravity in four dimensions, and O1=d 1 for large d. It is erhas a testament to how far these calculations have rogressed that actual numbers have emerged which can meaningfully be comared between different (lattice and continuum) aroaches. It should also be noted that, from a quantum gravity ersective, there are really no adjustable arameters in Eq. (1.1), excet for the new nonerturbative curvature scale : both c and are in rincile finite and calculable numbers. The mass scale m 1 in Eq. (1.1) is suosed to determine the magnitude of quantum deviations from the classical theory, and searates the short-distance, ultraviolet regime with characteristic momentum scale m, where nonerturbative quantum corrections are negligible, from the long-distance regime where quantum corrections become significant. The magnitude of itself involves, in a rather nontrivial way, the dimensionless bare couling G, the fixed oint value G c, and the ultraviolet cutoff, 1 / ex Z G dg 0 G 0 G!Gc jg G c j 1=0 G c : (1.) Ultimately to make rogress and determine the actual hysical value for the nonerturbative scale some hysical inut is needed, as the underlying theory cannot fix it 1 Already in ordinary Einstein gravity one finds for very short distances r l P corrections to the static otential, which can be comuted erturbatively [9]. In general for such short distances string corrections and/or higher derivative terms should be considered as well. (the ratio of the hysical Newton s constant to can be as small as one desires, rovided the bare couling G is very close to its fixed oint value G c ). It seems natural to identify 1= with either some very large average satial curvature scale, or erhas more aroriately with the Hubble constant (as measured today) determining the macroscoic exansion rate of the Universe via the corresondence 1=H; (1.3) in a system of units for which the seed of light equals one. Let us briefly digress here, and recall that in non-abelian SUN gauge theories a similar set of results is known to hold for the renormalization-grou induced running of the gauge couling g, so it will be instructive to draw further on the analogy with QCD, and non-abelian gauge theories in general. Of course one crucial difference between gravity and ordinary gauge theories lies in the fact that, in the latter case, the evolution of the couling constant can be systematically comuted in erturbation theory due to asymtotic freedom, a statement which is known to reflect the fact that such theories become noninteracting at short distances, u to logarithmic corrections, making erturbation theory consistently alicable. It is well known that for weak enough gauge couling in SUN gauge theories one has 1 g 1 g MS 0 log (1.4) MS with 0 the coefficient of the lowest order term in the beta function, 1=r an arbitrary momentum scale, MS 0 MeV a nonerturbative scale arameter, and the dots denoting higher loo effects. Instead of the MS arameter one could just as well use some other hysical scale, such as the inverse of the gauge correlation length, m 0 1, where the 0 denotes the lowest glueball state (the Slavnov-Taylor identities revent of course the gluon from acquiring a mass to any order in erturbation theory). For the urose of comaring to gravity, one should erhas emhasize that confining non-abelian gauge theories such as QCD do not, and cannot, directly determine the scale MS, which needs to be ultimately fixed by exeriment from say a direct measurement of the size of scaling violations. Its magnitude involves in a nontrivial way the bare gauge couling g and the ultraviolet cutoff, MS / ex Z g dg 0 g 0 (1.5) which is very much analogous to Eq. (1.). The correson- A ossible scenario is one in which 1 H 1 lim t!1 Ht H 0 with H1 8G 3 3, where is the observed cosmological constant, and for which the horizon radius is R 1 H

3 NONLOCAL EFFECTIVE GRAVITATIONAL FIELD... PHYSICAL REVIEW D 7, (005) dence with QCD and non-abelian gauge theories would therefore suggest 1 $ MS, with the gravitational a new nonerturbative scale, ultimately also to be determined from exeriment. Although not always necessarily advantageous (most erturbative calculations, being based on Feynman diagrams, are eventually done in momentum sace and do not seem to benefit significantly from this aroach), the running of the gauge couling g can be reformulated in terms of an effective action, involving the d Alembertian acting on functions of the field strength. One sets 1 g 1 g MS 0 log MS (1.6) with 0 11N n f =4 for non-abelian SUN gauge theories with n f massless fermion flavors, and with the log of the d Alembertian suitably defined, for examle, via log Z 1 dm 1 0 m 1 m (1.7) leading to a one-loo corrected effective action of the form I eff 1 Z 1 dxf x 4 g 0 log 0 F x (1.8) with an aroriately chosen mass scale [18]. In the gravitational case the corrections described by Eq. (1.1) have a more comlicated structure, and, in articular, are no longer logarithmic. But they can be viewed, for examle, as arising from a resummation of an infinite number of loo logarithms, as in the exansion X nlog 1 1=: n (1.9) n! n0 In the next section we shall describe how the renormalization-grou induced running of the gravitational constant can be imlemented in a simle way via a nonlocal set of manifestly covariant correction terms arising in the effective, long-distance gravitational field equations. These effective equations can then be used as a basis for a systematic discussion of various quantum corrections to the standard solutions of the classical field equations. II. EFFECTIVE GRAVITATIONAL ACTION AND EFFECTIVE FIELD EQUATIONS In general terms, a quantum-mechanical running of the gravitational couling imlies the relacement G! Gr (.1) in classical hysical observables. This is easier said than done, as in gravity the r in the running couling Gr is coordinate deendent, and as such can lead to considerable ambiguities regarding the interretation of exactly which distance r is involved. A more satisfactory aroach would relace Gr in the gravitational action I 1 16G Z dx g R (.) with a manifestly covariant object, intended to correctly reresent an invariant distance, and incororating the running of G as exressed in Eq. (1.1),! 1 16G Z 1 dx g 1 c 1= O 1= R (.3) with the covariant d Alembertian oerator defined through an aroriate combination of covariant derivatives g r r : (.4) Multilication by the coordinate r gets therefore relaced by the action of, whose Green s function in D sacetime dimensions is known to behave as <xj 1 1 jy>rdx;yjg : (.5) D r Here d would be the distance along a minimal ath z connecting the oints x and y in a fixed background geometry characterized by the metric g, and given by dx;yjg Z s y d g z dz dz : (.6) x d d As a result 1= can be envisioned as a coordinate indeendent way of defining consistently what is meant by r in the running of Gr, Gr!G: (.7) The above rescrition has in fact been used successfully for some time to systematically incororate the effects of radiative corrections in an effective action formalism [19 1]. It should be noted that the coefficient c in Eq. (1.1) is exected to be a calculable number of order one, but not necessarily the same as the coefficient c,asr and 1= are clearly rather different entities to begin with. 3 One should recall here that in general the form of the covariant d Alembertian oerator deends on the secific tensor nature of the object it is acting on, T g r r T : (.8) 3 In the lattice theory c was originally estimated from the invariant curvature correlations at around c 0:01, while more recently it was estimated at c 0:06 from the correlation of Wilson lines [17]. It is imortant to note that while the exonent is universal, c in general deends on the secific choice of regularization scheme (i.e. lattice regularization versus dimensional regularization or momentum subtraction scheme)

4 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) Thus on scalar functions one obtains the fairly simle result Sx g g Sx (.9) whereas on second rank tensors one has the significantly more comlicated exression T g r r T. Furthermore one should recognize that the form for the effective gravitational action of Eq. (.3) is ossibly not unique. A more integration-by-arts symmetric exression would be, for examle, I 1 Z 1 1= dx g R 1 c... 16G R : (.10) In general the covariant oerator aearing in the above exression, namely 1 1= A c (.11) has to be suitably defined by analytic continuation from ositive integer owers. The latter can be done either by comuting n for ositive integer n and then analytically continuing to n!1=, or alternatively by making use of the identity Z 1 1 1n ds s n1 exis (.1) n n 0 and subsequent use of the Schwinger-DeWitt reresentation for the kernel exis of the massless oerator. Within the limited scoe of this aer, we will be satisfied with comuting the effects of ositive integer owers n of the covariant d Alembertian, and then analytically continue the answer to fractional n 1=. In the following the above analytic continuation from ositive integer n will always be understood. 4 It should be stressed here that the action in Eq. (.10) should be treated as a classical effective action, with dominant radiative corrections at short distances (r ) already automatically built in, and for which a restriction to generally smooth field configurations does make some sense. In articular one would exect that in most instances it should be ossible, as well as meaningful, to neglect terms involving large numbers of derivatives of the metric 4 We notice in assing that in this aroach it is not obvious how to formulate a running cosmological constant, as the d Alembertian in r R dx R g! dx g 1 c1= has no function of the metric left to act on []. This situation is not entirely surrising as, lacking derivatives, the effect of the term is just to control the overall scale. In ure lattice gravity the bare is trivially scaled out and does not run [8,17]. In this scenario the hysical long-distance cosmological constant 1=, being related to an average curvature, is considered a hysical quantity to be ket fixed as the gravitational couling Gr slowly evolves with scale. in order to comute the effects of the new contributions aearing in the effective action. 5 A number of useful results can already be obtained from the form of the effective action in Eq. (.10). In articular, once a secific metric is chosen, the running of G can be readily exressed in terms of the coordinates aroriate for that metric. Later in this work we will illustrate extensively this statement for the secific, and hysically relevant, case of the Robertson-Walker (RW) metric. The next major ste involves a derivation of the effective field equations, incororating the running of G. As will be shown below this is not entirely straightforward, as the variation of the nonlocal effective action is comlicated by the resence of a differential oerator raised to a fractional ower, acting on what are rather comlicated functions of the metric to begin with. We shall therefore ostone a discussion of this asect to the Aendix, which focuses on this secific toic. Had one not considered the action of Eq. (.10) as a starting oint for constructing the effective theory, one would naturally be led (following Eq. (.7)) to consider the following effective field equations R 1 g R g 8G1 AT ; (.13) the argument again being the relacement Gr!G1 A involving the invariant object. Here, following common notation, is the scaled cosmological constant, not to be confused with the ultraviolet cutoff. Being manifestly covariant, these exressions at least satisfy some of the requirements for a set of consistent field equations incororating the running of G. The above effective field equation can then be easily recast in a form similar to the classical field equations R 1 g R g 8G ~T (.14) with ~T 1AT defined as an effective, or gravitationally dressed, energy-momentum tensor. Just like the ordinary Einstein gravity case, in general ~T might not be covariantly conserved a riori, r ~T 0, but ultimately the consistency of the effective field equations demands that it be exactly conserved in consideration of the Bianchi identity satisfied by the Riemann tensor. The ensuing new covariant conservation law r ~T r 1 AT 0 (.15) can be then be viewed as a constraint on ~T (or T ) 5 Dominant contributions to the original Feynman ath integral for the underlying quantum gravity theory are, on the other hand, resumably nowhere differentiable, the smooth configurations having ultimately zero measure in the gravitational functional integral [3]. Furthermore, issues related to causality, unitarity, and ositivity are better referred to the original, local microscoic action, which resumably shares all of these roerties

5 NONLOCAL EFFECTIVE GRAVITATIONAL FIELD... PHYSICAL REVIEW D 7, (005) which, for examle, in the secific case of a erfect fluid, will imly again a definite relationshi between the density t, the ressure t, and the RW scale factor Rt, just as it does in the standard case. This oint is sufficiently imortant that we wish to elaborate on it further. In ordinary Einstein gravity the energy-momentum tensor is defined via the variation of the matter action I M 1 Z dx g g T : (.16) But when the above arbitrary variation g is taken to be a gauge variation, g g g (.17) integration-by-arts in Eq. (.16) immediately yields the covariant conservation law r T 0, as a direct consequence of the gauge invariance of the matter action. On the other hand, in the modified field equations of Eq. (.13), the object which will be required to be conserved by the consistency of the field equations is the gravitationally dressed energy-momentum tensor, namely 1 AT, and not the original bare T itself. Referring therefore to the original T as the energymomentum tensor would aear to be imroer, since, for the consistency of the effective field equations of Eq. (.13), the latter is no longer required to be covariantly conserved. 6 In a sense, the effective field equations of Eq. (.13) can be seen simly as a consequence of having changed the exression in Eq. (.16) to I 0 M 1 Z dx g g 1 AT : (.18) Let us make a few more comments regarding the above effective field equations, in which we will set the cosmological constant 0 from now on. One simle observation is that the trace equation only involves the (simler) scalar d Alembertian, acting on the trace of the energymomentum tensor R 8G1 AT : (.19) Furthermore, to the order one is working here, the above 6 This can be illustrated further by the secific case of the erfect fluid, for which the energy-momentum tensor is usually written as T ttu u g t. In general its covariant divergence is not zero, but consistency of the Einstein field equations demands r T 0, which for the RW metric forces a definite relationshi between Rt, t, and t, namely _t3tt Rt=Rt _ 0, irresective of the equation-of-state relating to. In the effective field equations of Eq. (.13) the erfect fluid form for T can still be used (as it still satisfies all the original symmetry requirements), but the covariant conservation law has the new form dislayed in Eq. (.15), which imoses a new constraint on the scale factor Rt, as well as on the underlying t and t. effective field equations should be equivalent to 1 AOA R 1 g R 8GT (.0) where the running of G has been moved over to the gravitational side. Indeed it has recently been claimed [] that equations similar to the above effective field equations (at least for ositive integer ower n, including the classical case n 0) can be derived from a nonlocal extension of the Einstein-Hilbert action. In the classical case (A 0) one writes a new nonlocal action I 1 16G Z dx g R 1 g R 1 R (.1) whose variation, it is argued, gives the correct field equations u to curvature squared terms g R 1 g R OR 0: (.) For nonvanishing A the above construction can then be generalized to I 1 Z dx g R 1 16G g R 1 A OA 1 R (.3) whose variation can now be shown to give g 1 AOA R 1 g R OR 0 (.4) and which would coincide with the reviously roosed effective field equations, again u to higher order curvature terms. III. COVARIANT D ALEMBERTIAN ON SCALAR FUNCTIONS As a first ste in solving the new set of effective field equations, consider first the trace of the field equation in Eq. (.19), written as 1 AOA R 8GT (3.1) where R is the scalar curvature. Here we have made the choice to move the oerator A over on the gravitational side, so that it now acts on functions of the metric only, using the binomial exansion of 1=1 A. A discussion of the full tensor equations and their added comlexity will be ostoned to the next section. To roceed further, one needs to comute the effect of A on the scalar curvature. The d Alembertian oerator acting on scalar functions Sx is given by

6 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) g g Sx (3.) and for the RW metric, acting on functions of t only, one obtains a fairly simle result in terms of the scale factor Rt R 3 R 3 Ft: As a next ste one comutes the action of on the scalar curvature R, which gives 6k Rt5 R_ t RtRt R t3rt RtR _ 3 t R tr 4 t=r 3 t (3.4) and then on R which gives 66k R_ t Rt15 R_ 4 t Rt6kRt R t45rt R_ t R t1r t R 3 t6krt RtR _ 3 t 15Rt R_ 3 tr 3 t41r t Rt _ RtR 3 t5r 3 tr 3 tkr tr 4 t9r t R_ tr 4 t 7R 3 t RtR 4 t4r 3 t RtR _ 5 trt 4 R 6 t=rt 5 ; (3.5) etc. It should already become clear at this oint that the comuted exressions are raidly becoming quite comlicated. Nevertheless some of the higher order terms can, for examle, be interreted as higher derivative curvature contributions, since for Riemann squared, Ricci squared, and scalar curvature squared, one has resectively R R 1k k R_ t R_ 4 t R t R t=rt 4 ; (3.6) R R 1k k R_ tkrt Rt R_ 4 t R t R trt R_ t Rt=Rt 4 ; (3.7) with R 36k _ R trt Rt =Rt 4 ; (3.8) R R 1 6 R R 1 R 0 (3.9) for arbitrary scale factor Rt. But in the following we will just simly set Rt R 0 t, in which case R R 1 1 t 4 ; (3.10) R R t 4 ; (3.11) R 36 1 t 4 ; (3.1) and for the scalar curvature (here allowing for k 0, see Eq. (A9) in Aendix A) k 1 6 R 0t t : (3.13) Acting with n on the above scalar curvature now gives for k 0 61 t ; (3.14) t 4 ; (3.15) t 6 ; (3.16) t 8 ; (3.17) for n 0, 1,, and 3, resectively, and therefore for arbitrary ower n with the coefficient c n given by c n 61 t n (3.18) n 131 n c n 4 31 : (3.19) n Here use has been made of the relationshi d z c dz 1 1 z c (3.0) to analytically continue the above exressions to negative fractional n [4]. For n 1= the correction on the scalar curvature term R is therefore of the form with 1 c t= 1= 61 t (3.1) 1 1 c 1= : (3.) In articular for =3 (the classical value for a ressureless erfect fluid) and 1=3 one has c (3.3) 4 whereas, for examle, for 1= and 1=3 one obtains c 5 4 =7 4. Putting everything together, one then obtains for the trace art of the effective field equations

7 NONLOCAL EFFECTIVE GRAVITATIONAL FIELD... PHYSICAL REVIEW D 7, (005) t 1= c c Ot= = 8Gt: n t! 4 n 1 n1 t (3.4) The new term can now be moved back over to the matter side (since the correction is assumed to be small), in accordance with the structure of the original effective field equations Eqs. (.13) and (.19), and thus avoids the roblem of having to deal with the binomial exansion of 1=1 A. One then has 6 1 t t 1= 8G 1 c c Ot= = t (3.5) which is the RW metric form of Eq. (.19). If one assumes for the matter density t 0 t, then matching owers when the new term starts to take over at larger distances gives the first result 1=: (3.6) Thus the density decreases faster in time than the classical value would indicate. The exansion aears therefore to be accelerating, but before reaching such a conclusion one needs to determine the time deendence of the scale factor Rt (or ) as well. One might be troubled by the fact that some of the Gamma functions aearing in the exression for c can diverge for secific choices of, e.g. when 1=n 1 as in Eq. (3.) for n integer. But further thought reveals that this is not necessarily a concern here, as the coefficient c actually has to be divided out and then multilied by c (which, as discussed in the introduction and in [17], is exected to be a number of order one) to get the correct magnitude for the correction. One has therefore c c c (3.7) so that the correction eventually ends u as 1 c t= 1=, as it should, in accordance with Eq. (1.1) for Gr (the t here is like r there). Having comleted the calculation of the quantum correction term acting on the scalar curvature, as in Eq. (3.1), one can alternatively ursue the following exercise in order to check the overall consistency of the aroach. Consider n acting on T t instead, as in the trace of the effective field equation Eq. (.19) R 8G1 AT (3.8) for 0 and t 0. Fort 0 t and Rt R 0 t one finds in this case n31 n 0t n (3.9) which again imlies 1= as in Eq. (3.6) for large(r) times, when the quantum correction starts to become imortant (since the left-hand side of Einstein s equation always goes like 1=t, no matter what the value for is, at least for k 0). IV. COVARIANT D ALEMBERTIAN ON TENSOR FUNCTIONS Next we will examine the full effective field equations (as oosed to just their trace art) of Eq. (.13) with 0, R 1 g R 8G1 AT : (4.1) Here the d Alembertian oerator g r r (4.) acts on a second rank tensor, r T T T I ; r r I I I I (4.3) and would thus seem to require the calculation of 190 terms, of which fortunately many vanish by symmetry. Next assume that T has the erfect fluid form, for which one obtains _ T tt 6tt Rt Rt 3_t _ Rt Rt t; T rr 1 1 kr ftt Rt _ 3 _trt Rt _ trt g; T r 1 kr T rr ; T r 1 kr sin T rr (4.4) with the remaining comonents equal to zero. Note that a nonvanishing ressure contribution is generated in the effective field equations, even if one assumes initially a ressureless fluid, t 0. As before, reeated alications of the d Alembertian to the above exressions leads to raidly escalating comlexity (for examle, eighteen distinct terms are generated by for each of the above contributions), which can only be tamed by introducing some further simlifying assumtions. In the following we will therefore assume that T has the erfect fluid form aroriate for nonrelativistic matter, with a ower-law behavior for the density, t 0 t, and

8 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) t 0. Thus all comonents of T vanish in the fluid s rest frame, excet the tt one, which is simly t. Setting k 0 and Rt R 0 t one then finds 4 T tt! t ; (4.1) T tt t ; T rr R 0 t 0 t (4.5) which again shows that the tt and rr comonents get mixed by the action of the oerator, and that a nonvanishing rr comonent gets generated, even though it was not originally resent. Higher owers of the d Alembertian acting on T can then be comuted as well, but it is easier to introduce the slightly more general auxiliary diagonal tensor V with comonents V tt 0 t, V rr 1 t, V r V rr, and V r sin V rr, with an arbitrary ower. One then finds V tt t 6 R 1t ; 0 t V rr R 0 t 0 t t (4.6) as well as V r V rr ; V r sin V rr ; (4.7) and zero for the remaining comonents. The above exressions can then be used conveniently to generate n acting on T to any desired ower n. But since the roblem at each ste involves a two by two matrix acting on the energy-momentum tensor, it would seem rather comlicated to get a closed form solution for arbitrary n. But a comarison with the left-hand (gravitational) side of the effective field equation, which always behaves like 1=t for k 0, shows that in fact a solution can only be achieved at order n rovided the exonent satisfies n, or since n 1=, 1= (4.8) as was found reviously from the trace equation, Eqs. (.19) and (3.6). As a result one obtains a much simler set of exressions, which now read T tt! 6 0 t ; (4.9) T tt! t ; (4.10) 3 T tt! t ; (4.11) 5 T tt! t ; (4.13) etc., here for owers n 1 to n 5, resectively, and with changing with n in accordance with Eq. (4.8). For general n one can then write n T tt! c tt ; 0 t (4.14) and similarly for the rr comonent n T rr! c rr ; R 0 t 0 t : (4.15) But remarkably (see also Eq. (4.4)) one finds for the two coefficients the simle identity c rr ; 1 3 c tt; (4.16) as well as c r c rr and c r sin c rr. Then for large times, when the quantum correction starts to become imortant, the tt and rr field equations reduce to and 3 t 8Gc tt ; 0 t (4.17) 3 R 0 t 8Gc rr ; R 0 t 0 t ; (4.18) resectively. But the identity c rr 1 3 c tt imlies, simly from the consistency of the tt and rr effective field equations at large times, c rr ; c tt ; (4.19) whose only ossible solution finally gives the second sought-for result, namely 1 : (4.0) For the secific value of 1 one can then show that the coefficients c tt obey the recursion relation c tt n 4n 7n 1c tt n1 (4.1) with initial condition c tt n1 3=. Consequently a closed form exression for c tt and c rr c tt =3 can be written down, either in terms of the Pochhammer symbol x n xx 1...x n 1 x n=x, or more directly in terms of ratios of Gamma functions as

9 NONLOCAL EFFECTIVE GRAVITATIONAL FIELD... PHYSICAL REVIEW D 7, (005) c tt 1 ;n1= of radiation. Thus one can visualize the covariant gravitational vacuum-olarization contribution as behaving to some extent like classical radiation, here in the form of a n1 1 3n 8 n 8 n : dilute gas of virtual gravitons (4.) V. CONCLUSIONS Still, the above exression does not seem to be articularly illuminating at this oint, excet for roviding an exlicit roof that the coefficients c tt and c rr exist and are finite for secific values of n, such as n 1= 3=. One might worry at this oint whether the above solution is consistent with covariant energy conservation. With the assumed form for T it is easy to check that energy conservation yields for the t comonent r n T t! 31=c tt 3c rr 0 t 1=1 0 (4.3) when evaluated for n 1=, and zero for the remaining three satial comonents. But from the solution for the matter density t at large times one has 1=, so the above zero condition gives again c rr =c tt 3 =3, exactly the same relationshi reviously imlied by the consistency of the tt and rr field equations. In conclusions the values for 1= of Eq. (4.0) and 1= of Eq. (4.8), determined from the effective field equations at large times, are found to be consistent with both the field equations and covariant energy conservation. More imortantly, the above solution is also consistent with what was found reviously by looking at the trace of the effective field equations, Eq. (.19), which also imlied the result 1=, Eq. (3.6). Together these results imly that for sufficiently large times the scale factor Rt behaves as and the density t as Rtt t 1= (4.4) tt t 1= Rt 1= : (4.5) Thus the density decreases significantly faster in time than the classical value (tt ), again a signature of an accelerating exansion at later times. It is amusing to note that the vacuum-olarization term we have been discussing so far behaves very much like a ositive ressure term, as should already have been clear from the fact that the covariant d Alembertian g r r causes, in the RW metric case, a mixing of the tt and rr comonents in the field equations. Furthermore, within the classical FRW model, the value 1= corresonds to an equation-of-state arameter! 1=3 in Eq. (A0), with 31! ; (4.6) where t!t, and which is therefore characteristic The main results of this aer are the effective field equations of Eq. (.13), R 1 g R g 8G1 AT ; (5.1) their trace in (Eq. (.19)), and the solution for the trace and full equations for the secific case of the RW metric and 0 outlined in Sections III and IV, resectively. The combined results for the density t 0 t, namely 1= for large times (Eqs. (3.6) and (4.8)), and for the scale factor RtR 0 t, namely 1= (Eq. (4.0)) again for large times, imly that for 0 and for sufficiently large times the density falls off as tt 1= Rt 1= : (5.) Thus the matter density decreases significantly faster in time than redicted by the classical value (tt ), a signature of an accelerating exansion at later times. Within the Friedmann-Robertson-Walker (FRW) framework the gravitational vacuum-olarization term behaves in many ways like a ositive ressure term. The value 1= corresonds to! 1=3 in Eq. (A0), 31! ; (5.3) where we have taken the ressure and density to be related by t!t, and which is therefore characteristic of radiation. One can therefore visualize the gravitational vacuum-olarization contribution as behaving like ordinary radiation, in the form of a dilute virtual graviton gas. It should be emhasized though that the relationshi between density t and scale factor Rt is very different from the classical case. The results of Section IV show that the effective Friedmann equations for a universe filled with nonrelativistic matter ( 0) have the following aearance k _ R t Rt 8Gt t 1 Rt 3 3 8G 3 1 c t= 1= Ot= = t 1 ; (5.4)

10 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) k _ R t Rt Rt Rt Rt 8G 3 c Equivalently, substituting t Rt= Rt, _ one can, as an t= 1= examle, rewrite the first Friedman equation as Ot= = t k _ (5.5) R t Rt 8G _ 1 c Rt 3 = 1= Rt 1= Rt with the running G aroriate for the RW metric aearing exlicitly in the first equation, 7 Gt G1 c t= 1= Ot= = (5.6) and used, in the second equation, the result c tt 3c rr of Eq. (4.16). We have also restored the cosmological constant term, with a scaled cosmological constant 1=. One can therefore sensibly talk about an effective density eff t Gt G t 1 c t= 1= t (5.7) and an effective ressure eff t 1 3 Gt G 1 t 1 3 c t= 1= t (5.8) with eff t= eff t 1 3 GtG=Gt. Strictly seaking, the above results can only be roven if one assumes that the ressure s time deendence is given by a ower-law (as discussed in Section IV). 7 Corrections to the above formulae are exected to be fixed by higher order terms in the renormalization-grou -function. In the vicinity of the fixed oint at G c one writes G 0 G G c 1 G G c and obtains then by integration 1 C m j ex Z G dg 0 G 0 j G!Gc C m jg G c j 1=0 G c with an exonent given by 0 G c 0 1=, C m a numeric constant and the ultraviolet cutoff. After relacing! 1=r and G!Gr one finds for the scale deendence of G 1 r 1= Gr G c 1 1 G c 1 1 G c C 1= m 1 1 G c C = m r = : Note that 0 1= < 0, and that for 1 < 0 the second correction term is ositive as well. If one restricts oneself to the lowest order term, valid in the vicinity of the ultraviolet fixed oint, then for a given static source of mass M one has for the gravitational otential the additional contribution Vr MG= 3 r for 1=3, as discussed in [17]. t 1 : (5.9) 3 The effective Friedman equations of Eqs. (5.4) and (5.5) also bear a suerficial degree of resemblance to what might be obtained in scalar-tensor theories of gravity [5 7] (for recent reviews and further references see [8,9]), S Z dx 1 g 16G fr V S matter ; (5.10) where the gravitational Lagrangian is some arbitrary function of the scalar curvature [30]. It is also well known that often these theories can be reformulated in terms of ordinary Einstein gravity couled aroriately to a scalar field [31]. In the FRW case one has for the scalar curvature in terms of the scale factor R 6k _ R trt Rt=R t (5.11) and therefore for k 0 and Rt R 0 t, R 6 1 t : (5.1) The quantum correction in Eq. (5.4) is therefore, at this level, indistinguishable from an inverse curvature term of the tye R 1=. But the resemblance is seen here merely as an artifact due to the articularly simle form of the RW metric, with the coincidence of several curvature invariants (see for examle, Eqs. (3.8) and (3.1)) not exected to be true in general. Finally let us note that the effective field equations incororating a vacuum-olarization-driven running of G, Eq. (.13)), could otentially run into serious difficulties with exerimental constraints on the time variability of G. These have recently been summarized in [3 36], where it is argued on the basis of detailed studies of the cosmic background anisotroy that the variation of G at the recombination eoch is constrained as jgz z rec G 0 j=g 0 < 0:05. Solar system measurements also severely restrict the time variation of Newton s constant to j G=Gj _ < 10 1 yr 1 [3]. It would seem though that these constraints can still be accommodated rovided the scale entering the effective field equations of Eq. (.13) is chosen to be sufficiently large, at least of the order of >3H 1, given that in the resent model one has j G=Gj _ 1 c t 1=1 = 1= and therefore j G=Gj3c _ t = 3 for 1=

11 NONLOCAL EFFECTIVE GRAVITATIONAL FIELD... PHYSICAL REVIEW D 7, (005) ACKNOWLEDGMENTS The authors are grateful to Gabriele Veneziano for his close involvement in the early stages of this work, and for bringing to our attention the ioneering work of Vilkovisky and collaborators on nonlocal effective actions for gauge theories. Both authors also wish to thank the Theory Division at CERN for warm hositality and generous financial suort during the Summer of 004. One of the authors (H. W. H.) wishes to thank James Bjorken for useful discussions on issues related to the subject of this aer. The work of Ruth Williams was suorted in art by the UK Particle Physics and Astronomy Research Council. APPENDIX A: CLASSICAL FIELD EQUATIONS AND CONVENTIONS This aendix is mostly about notation, but also collects a few simle results used extensively in the rest of the aer. We will write the Robertson-Walker (RW) metric as dr ds dt R t 1 kr r d sin d (A1) and note that with this choice of signature (i.e. a minus sign for the dt term), is a ositive oerator (on functions of t). Also g detg r sinr 3 t= 1 kr. The energy-momentum tensor for a erfect fluid is T ttu u g t (A) giving in the fluid s rest frame T diag; R =1 kr ;r R ;r sin R, with trace T 3tt: (A3) The field equations are then written as R 1 g R 8GT : The tt comonent of the Einstein tensor reads 3k R_ t=r t; while the rr comonent is (A4) (A5) 1 1 kr k R_ trt Rt; (A6) and the comonent r k R_ trt Rt; (A7) and finally the comonent r sin k R_ trt Rt: (A8) The scalar curvature is simly 6k R_ trt Rt=R t: (A9) Thus the tt comonent of the Einstein equation becomes 3k R_ t=r t 8Gt (A10) while the rr comonent reads 1 1 kr k R_ trt Rt 8G 1 1 kr tr t: (A11) The trace equation is 6k R_ trt Rt=R t 8Gt3t: (A1) Covariant conservation of the energy-momentum tensor, r T 0 imlies a definite relationshi between Rt, t, and t, which reads _t3tt Rt=Rt _ 0 (A13) (and which the tensor of Eq. (A) in its most general form does not satisfy). Next consider the case k 0 (satially flat) and 0 (nonrelativistic matter). If Rt R 0 t and t 0 t, then the tt field equation 3 8G t 0 t (A14) imlies and 8G 0 =3, while the rr field equation 3 R 0 t 0 imlies =3. Also both of these together imly tt t =3 3 1=Rt 3 : The trace equation now reads (A15) (A16) 6 1 8G t 0 t (A17) and imlies again and 6 1 8G 0. The latter combined with the tt equation gives 3 6 1, or again =3. Finally covariant energy conservation imlies 3 0 t 0 (A18) or 3 0, which does not add to what already comes out of the tt and rr (or, equivalently, tt and trace) equations, but is consistent with it. In conclusion the tt and rr (or tt and trace) equations are sufficient to determine both and. The case of nonvanishing ressure can be dealt with in the same way. In most instances one is interested in a fairly simle equation-of-state t!t, with! a constant. For nonrelativistic matter! 0, for radiation! 1=3, while the cosmological term can be modeled by! 1. The consistency of the tt and rr equations now requires

12 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) 3! 3 (A19) Z Z Y dg detg Y 1= x dg which gives 31! (A0) for 1 <! 1=3. Furthermore from the covariant energy conservation law one has 31! 0 which imlies again. Therefore (A1) Rtt =31! ; trt 31! : (B.) These results are well known and have been collected here for convenient reference. APPENDIX B: SCALE TRANSFORMATIONS AND GRAVITATIONAL FUNCTIONAL INTEGRAL Consider the (Euclidean) Einstein-Hilbert action with a cosmological term I E 0 Z 4 dx g 0 Z dx g R: (B1) Here 0 is the bare cosmological constant, 0 1=16G 0 with G 0 the bare Newton s constant. Also, and in this section only, we follow customary notation used in cutoff field theories and denote by an ultraviolet cutoff, not to be confused with the scaled cosmological constant. The natural exectation is for the bare microscoic, dimensionless coulings to have magnitudes of order one in units of the cutoff, 0 0 O1. Next one can rescale the metric g 0 0g ; g 0 1 g (B) to obtain I E Z q 4 dx g Z q dx g 0 R 0 : (B3) Next consider the fact that the (Euclidean) Feynman ath integral Z Z dg ex Z dx g 0 4 R (B4) 16G 0 includes a functional integration over all metrics, with functional measure [37,38] x Z Y gx Y D4D1=8 x dg! D4 x Z Y Y dg x x with the suermetric over metric deformations given by (B5) G ; gx 1 gx1= g xg xg xg x g xg x: (B6) For our uroses it will be sufficient to note that under a rescaling of the metric the functional measure only icks u an irrelevant multilicative constant. Such a constant automatically dros out when comuting averages. Equivalently one can view a rescaling of the metric as simly a redefinition of the ultraviolet cutoff,! 1=4 0. As a consequence, the nontrivial art of the functional integral over metrics only deends on 0 and 0 through the dimensionless combination 0 = 0 1=16G 0 0. The existence of an ultraviolet fixed oint is then entirely controlled by this dimensionless arameter only, both on the lattice [8,17] and in the continuum [11,15]. It is the only relevant (as oosed to marginal or irrelevant, in statistical mechanics arlance) scaling variable in the ure gravity case, in the sense of having only one ositive (growing) eigenvalue of the linearized renormalization-grou transformation in the vicinity of the fixed oint. The arameter 0 controls the overall scale of the roblem (the volume of sace-time), while the 0 term rovides the necessary derivative or couling term. Since the total volume of sace-time can hardly be considered a hysical observable, quantum averages are comuted by dividing out by the total sace-time volume. For examle, for the quantum exectation value of the Ricci scalar one writes < R dx gxrx> < R dx gx> : (B7) Without any loss of generality one can therefore fix the overall scale in terms of the ultraviolet cutoff, and set the bare cosmological constant 0 equal to one in units of the ultraviolet cutoff. 8 The addition of matter field romts one to do some further rescalings. Thus for a scalar field with action 8 These considerations are not dissimilar from the case of a self-interacting scalar field where one might want to introduce three coulings for the kinetic term, the mass term, and the quartic couling term, resectively. A simle rescaling of the field would then reveal that only two couling ratios are in fact relevant

13 NONLOCAL EFFECTIVE GRAVITATIONAL FIELD... PHYSICAL REVIEW D 7, (005) I S 1 Z dx m 0 R g (B8) m 1 ex Z G dg 0 and functional measure G 0 G!Gc jg G c j 1=0 G c Z Z Y d gx 1= dx; (B9) (B13) x with an exonent related to the derivative of the beta the metric rescaling is to be followed by a field rescaling function at the fixed oint 0 x x= 1=4 0 (B10) 0 G c 1=: (B14) with the only surviving change being a rescaling of the bare The overall size of this new scale is controlled by the mass m 0! m 0 = 1=4 distance from the fixed oint G G 0. The scalar functional measure acquires an irrelevant multilicative factor which does not c, which can be made arbitrarily small (in the Regge lattice theory one finds for the critical couling, in units of the ultraviolet cutoff, G c affect quantum averages. The bare mass rescaling is of 0:66, and for the exonent 0:33). course ineffectual if the fields are massless to begin with. Thus a result such as The same set of considerations aly as well to the Euclidean lattice [39,40] regularized version of Eq. (B1), < R dx gxrx> which now reads [41,4] X X < R dx gx> G G c 1 (B15) I L 0 V h l 0 h l A h l (B11) and h Z L Z X X dl exf 0 V h l 0 h l A h l g; h h h (B1) where, as is customary, the lattice ultraviolet cutoff is set equal to one (i.e. all lengths and masses are measured in units of the cutoff). It is known that convergence of the Euclidean lattice functional integral requires a ositive bare cosmological constant 0 > 0 [41 43]. The couling should really not be allowed to run, as the overall sace-time volume is intended to be fixed, not to be itself rescaled under a renormalization-grou transformation. Indeed, in the sirit of Wilson [4], a renormalization-grou transformation allows a descrition of the original hysical system in terms of a new course grained Hamiltonian, whose new oerators are interreted as describing averages of the original system on the finest scale, but within the same hysical volume. This new effective Hamiltonian is still suosed to describe the original hysical system, but does so more economically in terms of a reduced set of degrees of freedom. The ure gravity theory deends only on one couling (the dimensionless G), and only that couling is allowed to run. This is also, to some extent, imlicit in the correct definition of gravitational averages, for examle in Eq. (B7). Physical observable averages such as the one in Eq. (B7) in general have some rather nontrivial deendence on the bare couling G 0, more so in the resence of an ultraviolet fixed oint. Renormalization in the vicinity of the ultraviolet fixed oint invariably leads to the introduction of a new dynamically generated, nonerturbative scale for G>G c referring here to an average curvature on the largest observable scales (with and some ositive exonents) does not resumably allow one to state whether the average curvature is large or small at large distances (that would clearly deend on the choice of G G c and the cutoff ). 9 But it does establish a definite relationshi between the fundamental scale in Eq. (B13) and say the scale of the curvature at the largest scales, Eq. (B15), as well as with any other observable involving G G c or. It is the latter curvature that most likely should be identified with a hysical, astrohysically measurable, macroscoic cosmological constant (and not in any way with 0 ). While it is natural to assume for the curvature measured on the largest distance scales (for examle via the arallel transort of vectors along very large loos) that R 1=, and therefore, it has roven difficult so far to establish such a result in the lattice theory, due to the great technical difficulties involved in measuring small invariant correlations at large geodesic distances [44]. APPENDIX C: EFFECTIVE ACTION VARIATION In this section we will consider the effective gravitational action of Eq. (.10), I 1 16G Z dx g R 1 A R (C1) and comute its variation. One needs the following elementary variations 9 Pursuing the analogy with quantum chromodynamics, we note that there the nonerturbative gluon condensate deends in a nontrivial way on the corresonding confinement scale arameter, S <F F > 50 MeV 4 4 with 1 QCD MS

14 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 7, (005) g R 1A R g R 1A g r r g r ; (C10) R g R 1 A R g R 1A one needs the variation of given by Eq. (C6), which R : (C) then gives Using the identity 1 g g g g (C3) g r r r g r 1 r g g g r : (C11) as well as r g 0 one then has Here (or at the end) one also needs to roerly symmetrize 1 the result for the variation of, g g g R 1 A R g R 1 A R 1 n g R A n! Xn k1 nk : (C1) R k1 g R 1 A R : (C4) Next several integrations by arts, involving both the oerator n (with integer n) as well as the oerator Next use is made of the definition of the Ricci scalar, g r r, have to be erformed in order to isolate R g R R g : (C5) the g term. This follows from R R g r V g 1= g V 0 which allows us to reeatedly For the variation of the affine connection one has integrate by arts and move some covariant derivatives 1 around. In general one has to be careful about the ordering g r g r g r g (C6) of covariant derivatives, whose commutator is in general or, equivalently, nonzero in accord with the Ricci identity 1 r g g r g g r g g g g ; and therefore for the variation of the Ricci tensor R r r from which it follows that g R r r g g g g g g r r g ; (C7) (C8) (C9) which is one of the required variations in Eq. (C4). The second term on the right hand side of the last equation is a total derivative in the ordinary Einstein case, but it needs to be ket here. Note also that in general r r, and that g 0 but g 0. For the variation of the covariant d Alembertian r ; r T X i R it X R j T j (C13) with the index in T in the ith osition in the first term, and in the jth osition in the second term. The term involving the variation of the covariant d Alembertian then gives A A nr r R R nr R r R 1 ng r R g A r R (C14) which again needs to be symmetrized with resect to A R $ R, in the way described above. After adding the remaining terms, the effective field equations become R 1 g R A A R g r r A R nr r R R R R A nr R r R 1 ng r R g A r R 8GT ; (C15) where again the last three terms need to be roerly symmetrized in A R $ R, as described above. Taking the covariant divergence of the left-hand side (l.h.s.) gives zero for some of the terms, while the remaining terms give