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1、arXiv:hep-th/0010232v4 26 Jan 2001Trace anomaly driven inflationS.W. Hawking, T. Hertog,DAMTP, Centre for Mathematical Sciences, University of CambridgeWilberforce Road, Cambridge CB3 0WA, United KingdomandH.S. ReallPhysics Department, Queen Mary and Westfield College,Mile End Road, London E1 4NS, U

2、nited KingdomPreprint DAMTP-2000-92, QMW-PH/00-1025 October 2000AbstractThis paper investigates Starobinskys model of inflation driven by the trace anomalyof conformally coupled matter fields. This model does not suffer from the problem ofcontrived initial conditions that occurs in most models of in

3、flation driven by a scalar field.The universe can be nucleated semi-classically by a cosmological instanton that is muchlarger than the Planck scale provided there are sufficiently many matter fields. There aretwo cosmological instantons: the four sphere and a new “double bubble” solution. Thispaper

4、 considers a universe nucleated by the four sphere. The AdS/CFT correspondence isused to calculate the correlation function for scalar and tensor metric perturbations duringthe ensuing de Sitter phase. The analytic structure of the scalar and tensor propagators isdiscussed in detail. Observational c

5、onstraints on the model are discussed. Quantum loopsof matter fields are shown to strongly suppress short scale metric perturbations, whichimplies that short distance modifications of gravity would probably not be observable in thecosmic microwave background. This is probably true for any model of i

6、nflation providedthere are sufficiently many matter fields.This point is illustrated by a comparison ofanomaly driven inflation in four dimensions and in a Randall-Sundrum brane-world model.email: S.W.Hawkingdamtp.cam.ac.ukemail: T.Hertogdamtp.cam.ac.ukemail: H.S.Reallqmw.ac.uk1Contents1Introduction

7、32O(4) Instantons62.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.2The trace anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.3Energy conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.4Shape of t

8、he instanton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.5Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113Metric perturbations113.1Scalars, vectors and tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . .113.2Matter effectiv

9、e action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123.3Solving the Einstein equations: scalars and vectors. . . . . . . . . . . . . . . .153.4Tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173.5The gravitational action. . . . . . . . . . .

10、 . . . . . . . . . . . . . . . . . . . .183.6Metric correlation functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .214Analytic structure of propagators224.1Flat space limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224.2Scalar propagator on the sphere . .

11、. . . . . . . . . . . . . . . . . . . . . . . . .234.3Tensor propagator on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . .244.4Complex poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265Lorentzian two-point correlators275.1Scalar propagator . . . . . .

12、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275.2Tensor propagator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .286Observational constraints296.1Duration of inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .296.2Amplitude of perturbations. .

13、 . . . . . . . . . . . . . . . . . . . . . . . . . . .317Short distance physics327.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327.2Randall-Sundrum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337.3Brane-world perturbations . . . . .

14、 . . . . . . . . . . . . . . . . . . . . . . . . .348Conclusions3721IntroductionInflation 1 in the very early universe seems the only natural explanation of many observedfeatures of our universe, in particular the recent measurements of a Doppler peak in the cosmicmicrowave background fluctuations 2

15、. However, while it provides an appealing explanation forseveral cosmological problems, it provokes the natural question of why the conditions were suchas to start inflation in the first place.The new inflationary scenario 3, 4 was proposed primarily to overcome the problem ofobtaining a natural exi

16、t from the inflationary era. In this model, the value of the scalar issupposed to be initially confined to zero by thermal effects. As the universe expands and coolsthese effects disappear, leaving the scalar field miraculously exposed on a mountain peak of thepotential. If the low temperature poten

17、tial is sufficiently flat near = 0 then slow roll inflationwill occur, ending when the field reaches its true minimum c. This scenario seems implausiblebecause a high temperature would confine only the average or expectation value of the scalar tozero. Rather than be supercooled to a state with 0 lo

18、cally, the field fluctuates and rapidlyforms domains with near cThe dynamics of the phase transition is governed by the growthand coalescence of these domains and not by a classical roll down of the spatially averaged field 5. Because this and other problems, new inflation was largely abandoned in f

19、avor of chaoticinflation 6 in which it is just assumed that that the scalar field was initially displaced fromthe minimum of the potential. One attempt to explain these initial conditions for inflation interms of quantum fluctuations of the scalar field seems to lead to eternal inflation at the Plan

20、ckscale 7, at which the theory breaks down. Another attempt, using the Hartle-Hawking “noboundary” proposal 8, found that the most probable universes did not have enough inflation9. No satisfactory answer to the question of why the scalar field was initially displaced fromthe minimum of its potentia

21、l has been found.In this paper we will reconsider an earlier model, in which inflation is driven by the traceanomaly of a large number of matter fields. The Standard Model of particle physics containsnearly a hundred fields.This is at least doubled if the Standard Model is embedded in asupersymmetri

22、c theory. Therefore there were certainly a large number of matter fields presentin the early universe, so the large N approximation should hold in cosmology, even at thebeginning of the universe. In the large N approximation, one performs the path integral overthe matter fields in a given background

23、 to obtain an effective action that is a functional of thebackground metric:exp(Wg) =Zdexp(S;g).(1.1)One then argues that the effect of gravitational fluctuations is small in comparison to the largenumber of matter fluctuations. Thus one can neglect graviton loops, and look for a stationarypoint of

24、the combined gravitational action and the effective action for the matter fields. This isequivalent to solving the Einstein equations with the source being the expectation value of thematter energy momentum tensor:Rij12Rgij= 8GhTiji,(1.2)wherehTiji = 2gWgij.(1.3)3Finally, one can calculate linearize

25、d metric fluctuations about this stationary point metric andcheck they are small. This is confirmed observationally by measurements of the cosmic mi-crowave background, which indicate that the primordial metric fluctuations were of the orderof 10510.Matter fields might be expected to become effectiv

26、ely conformally invariant if their massesare negligible compared to the spacetime curvature. Classical conformal invariance is brokenat the quantum level 11 (see 12, 13 for reviews), leading to an anomalous trace for theenergy-momentum tensor:gijhTiji 6= 0.(1.4)This trace is entirely geometrical in

27、origin and therefore independent of the quantum state. Ina maximally symmetric spacetime, the symmetry of the vacuum implies that the expectationvalue of the energy momentum tensor can be expressed in terms of its traceh0|Tij|0i =14gijgklh0|Tkl|0i.(1.5)Thus the trace anomaly acts just like a cosmolo

28、gical constant for these spacetimes. Hence apositive trace anomaly permits a de Sitter solution to the Einstein equations 14.This is very interesting from the point of view of cosmology, as pointed out by Starobinsky15. Starobinsky showed that the de Sitter solution is unstable, but could be long-li

29、ved, anddecays into a matter dominated Friedman-Robertson-Walker (FRW) universe. The purpose ofStarobinskys work was to demonstrate that quantum effects of matter fields might resolve theBig Bang singularity1. From a modern perspective, it is more interesting that the conformalanomaly might have bee

30、n the source of a finite but significant period of inflation in the earlyuniverse. This inflation would be followed by particle production and (p)reheating during thesubsequent matter dominated phase. Starobinskys work is reviewed and extended by Vilenkinin 17. For a more recent discussion of the St

31、arobinsky model, see 18.Starobinsky showed that the de Sitter phase is unstable both to the future and to the past, soit was not clear how the universe could have entered the de Sitter phase. However, this problemcan be overcome by an appeal to quantum cosmology, which predicts that the de Sitter ph

32、aseof the universe is created by semi-classical tunneling from nothing. This process is mediatedby a four sphere cosmological instanton 17. One of the results of this paper is that the foursphere is not the only cosmological instanton in this model.In order to test the Starobinsky model, it is neces

33、sary to compare its predictions for thefluctuations in the cosmic microwave background (CMB) with observation. This was partlyaddressed by Vilenkin 17. Using an equation derived by Starobinsky 19, Vilenkin showedthat the amplitude of long wavelength gravitational waves could be brought within observ

34、ationallimits at the expense of some fine-tuning of the coefficients parameterizing the trace anomaly.Density perturbations were discussed by Starobinsky in 20.The analysis of Starobinsky and Vilenkin was complicated by the fact that tensor pertur-bations destroy the conformal flatness of a FRW back

35、ground, making the effective action formatter fields hard to calculate. However, we now have a way of calculating the effective actionfor a particular theory, namely N = 4 U(N) super Yang-Mills theory, using the AdS/CFT1Another paper 16 which discussed the effects of the trace anomaly in cosmology f

36、ailed to obtain non-singularsolutions because it included a contribution from a classical fluid.4correspondence 21. In this paper we will calculate the effective action for this theory in aperturbed de Sitter background. This enables us to calculate the correlation function for metricperturbations a

37、round the de Sitter background. We can then compare our results with observa-tions. The fact that we are using the N = 4 Yang-Mills theory is probably not significant, andwe expect our results to be valid for any theory that is approximately massless during the deSitter phase. One might think that o

38、ur results could shed light on the effects of matter interac-tions during inflation since AdS/CFT involves a strongly interacting field theory. However, aswe shall explain, our results are actually independent of the Yang-Mills coupling.Our calculations will be performed in Euclidean signature (on t

39、he four sphere), and thenanalytically continued to Lorentzian de Sitter space. The condition that all perturbations areregular on the four sphere defines the initial quantum state for Lorentzian perturbations. Thefour sphere instanton is much larger than the Planck scale (since we are dealing with a

40、 large Ntheory), so there is a clear cut separation into background metric and fluctuation.We shall include in our action higher derivative counterterms, which arise naturally in therenormalization of the Yang-Mills theory. There are three independent terms that are quadraticin the curvature tensors

41、: the Euler density, the square of the Ricci scalar and the square of theWeyl tensor. The former just contributes a multiple of the Euler number to the action. Metricperturbations do not change the Euler number, so this term has no effect. The square of theRicci scalar has the important effect of ad

42、justing the coefficient of the 2R term in the traceanomaly. It is precisely this term that is responsible for the Starobinsky instability, so by varyingthe coefficient of the R2counter term we can adjust the duration of inflation. The Weyl-squaredcounterterm does not affect the trace anomaly but it

43、can contribute to suppression of tensorperturbations. The effects of this term were neglected by Starobinsky and Vilenkin. They alsoneglected the effects of the non-local part of the matter effective action. We shall take fullaccount of all these effects.Vilenkin showed that the initial de Sitter ph

44、ase is followed by a phase of slow-roll inflationbefore inflation ends and the matter-dominated phase begins.Since the horizon size growssignificantly during this slow-roll phase, it is important to investigate whether modes we observetoday left the horizon during the de Sitter phase or during the s

45、low-roll phase. If the presenthorizon size left during the de Sitter phase, we find that the amplitude of metric fluctuationscan be brought within observational bounds if N, the number of colours, is of order 105. Sucha large value for N is rather worrying, which leads us to the second possibility,

46、that the presenthorizon size left during the slow-roll phase. Our results then suggest that the coefficient ofthe R2term must be at most of order 108, and maybe much lower, but N is unconstrained(except by the requirement that the large N approximation is valid so that AdS/CFT can beused). We also f

47、ind that the tensor perturbations can be suppressed independently of the scalarperturbations by adjusting the coefficient of the Weyl-squared counterterm in the action.Inflation blows up small scale physics to macroscopic scales. This suggests that inflationmay lead to observational consequences of

48、small-scale modifications of Einstein gravity, such asextra dimensions. However, we find that the non-local part of the matter effective action has theeffect of strongly suppressing tensor fluctuations on very small scales, a result first noted in flatspace by Tomboulis 22. This suggests that any sm

49、all-scale modifications to four dimensionalEinstein gravity would be unobservable in the CMB since matter fields would dominate thegraviton propagator at the scales at which such modifications might be expected to becomeimportant. This result is probably not restricted to trace anomaly driven inflat

50、ion since it is5simply a consequence of the presence of a large number of matter fields. As we have mentioned,there really are a large number of matter fields in the universe and these will suppress small-scalegraviton fluctuations in any model of inflation.We illustrate this point by considering a