- Email: [email protected]
Relaxing the cosmological constant
Physics LettersB North-Holland
301 (1993)
351-357
Relaxing the cosmological N.C. Tsamis
PHYSICS
LETTERS
B
constant
’
Department of Physics, University of Crete, GR- 71409 Iraklion, Crete, Greece
and R.P. Woodard 2 Department of Physics, University of Florida, Gainesville, FL 32611, USA Received
22 December
1992
A non-zero cosmological constant allows massless gravitons to self-interact via a coupling of dimension three. If the cosmological constant is positive then the background geometry will subject gravitons to an enormous redshift. These two facts are together responsible for the severe infrared divergences which detailed, explicit calculations [ N.C. Tsamis and R.P. Woodard, Strong infrared effects in quantum gravity, preprint CRETE-92-17, UFIFT-92-241 reveal in loop corrections to the gravitational force law. We argue that, as a result, the cosmological interaction has a finite lifetime. Furthermore, this lifetime is consistent wi h inflation.
The most general lagrangian based upon the metric and restricted to two derivatives of this field is given by #’
It possesses an apparently free parameter, A, whose value is observed to be less then 10-120G - ’ [ 11. Opinion is nearly unanimous that this can only be explained through quantum effects [ 21. With a few exceptions, however, the search for a solution has focused on making a clever choice for the matter theory which is coupled to gravity. There is something to be said for this approach but it has an obvious difficulty in that any mechanism for canceling /1 must operate in the far infrared where the particle spectrum is believed to be well known. It is preferable to consider the possibility that the job is done by an already ob’ E-mail address: TSAMIS @ IESL.FORTH.GR. ’ E-mail address: WOODARD @ UFHEPA.PHYS.UFL.EDU. #’ Our metric has spacelike signature and R’,=T“,,,+.... Greek indices run from 0 to 3, while Latin indices run from to 3. We are also using units where c= fi = 1.
0370-2693/93/$06.00
0 1993 Elsevier Science Publishers
I
served massless particle rather than by a new one which has somehow escaped detection. And the resolution of greatest aesthetic appeal would be for gravity to cure its own problem using nothing more than the graviton. Very little attention has been paid to the last possibility for two reasons. First, there is a widespread prejudice against any appeal to quantum gravity because there is no consistent theory for it. Second, simple arguments purport to show that quantum gravitational effects should be weak in the infrared. The oldest such argument is that the theory’s natural coupling constant, ~~ = 16nG, goes like inverse mass squared. This means that perturbation theory must generate a series in positive powers of K2p2, where p2 is some combination of the momenta or particle masses appearing in whatever process is under investigation. Therefore, loop corrections are suppressed in the infrared. A more sophisticated study by Weinberg [ 31 concluded long ago that the leading infrared corrections of quantum gravity are similar to those of QED. That is, the most significant infrared effects are soft graviton emission from, and exchange between, the external legs of an exclusive amplitude. Soft
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graviton corrections to soft graviton lines are suppressed because each vertex contains two derivatives and these vanish in the infrared. The same effect also dilutes contributions from other massless particle loops on soft graviton lines. The first objection to considering quantum gravity is certainly invalid: the non-renormalizability of Einstein’s theory in no way prevents us from using it reliably in the far infrared. The fact that we perceive a finite gravitational force implies that there must be a consistent quantum theory of it. The fact that we perceive only a weak gravitational force, which is so well described by general relativity, implies that there can be no significant coupling between large distance phenomena and the details of whatever Planck scale mechanism through which this fundamental theory contrives to free itself of ultraviolet divergences. On cosmological distance scales it suffices to use the appropriate local counterterms to subtract off the ultraviolet divergences of ( 1) as they occur, order by order in perturbation theory. The dominant infrared effects at any fixed order reside in ultravioletfinite and nonlocal terms that are not affected by how we make these subtractions and which dominate the result at large distances once ultraviolet finiteness is achieved. This is why Bloch and Nordsieck [4] were able to resolve the infrared problem of QED more than a decade before renormalization was understood. The same principle underlies Weinberg’s [ 31 analysis of infrared quantum gravity for A = 0. The second objection is also invalid: when A#0 quantum gravity should become very strong in the far infrared. How particles influence the physics of large distances is determined by their masses and by the dimensionalities of their self-interactions. The dominant effect comes from the lightest particles which self-interact via couplings of the lowest canonical field dimension. Gravitons are massless for any value of the cosmological constant, but for A = 0 their lowest self-interaction has dimension five - there are two derivatives distributed between three graviton fields. This is why conventional quantum gravity is so much weaker in the infrared than QCD, which allows massless gluons to self-interact via a coupling of dimension four. However, when /1# 0 gravitons can self-interact through a coupling of dimension three since the cosmological interaction provides a threepoint vertex with no derivatives. This version of 352
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quantum gravity is so different from the conventional theory that we distinguish it with the name “quantum cosmological gravity”, or QCG for short. Based on the preceding considerations QCG ought to be stronger in the far infrared than QCD. The infrared sector of QCD is responsible for confinement and it might seem difficult to imagine an even stronger effect. The nearest analog in the familiar physics of flat space is the theory of a massless scalar with a cubic self-interaction:
(2) Of course this model differs from QCG in that the background @= 0 is not even classically stable whereas de Sitter space and anti-de Sitter space do seem to be classically stable backgrounds for QCG [ 5 1. However, @= 0 is at least a classical solution and the quantum perturbation theory based on this background is a good analog for the infrared behavior of QCG. Of special significance is the fact that the one loop self-energy of the scalar becomes infinitely negative on shell #*: C(P2) = 73;;
In (p’) + constant
.
(3)
This is very interesting because the one loop self-energy contributes at two loops to the vacuum energy, which in QCG is an important constituent of the cosmological constant. If it were correct to useflat space propagators in QCG, and to take the resulting infrared divergences seriously, this scalar result would imply that quantum corrections from the far infrared have an enormous impact on the observed value of A. To proceed further we must answer two questions. First, how is the scalar result modified in QCG by the necessarily non-flat background? Second, how should we interpret infrared divergences in QCG? For the case of positive cosmological constant the natural background is the open submanifold of de x2 We note in passing that the ultraviolet
divergences in this quantity are removed by a mass counterterm which affects only the constant. The fact that the leading infrared effect is nonlocal, ultraviolet finite and completely untouched by this subtraction exemplifies the general rule enunciated in the prepenultimate paragraph.
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Sitter space [ 61, a convenient which #3 ds*= -dt*+exp(2Ht)
H2GiA.
coordinatization
(d_x*+dy*+dz*)
for
, (4)
We have been able to completely solve linearized QCG on this background [ 7 1, and to work out in full the structure of perturbative QCG [ 61. The result is a spectacular enhancement of the infrared divergences which appear in massless e3 [ 8 1. The reason is simple: since spatial coordinates are exponentially
expanded with increasing time, their Fourier conjugates, the spatial momenta, are redshifted to zero. Infrared effects derive from the low end of the momentum spectrum so they are strengthened when this sector is more densely populated. In fact we believe this temporally increasing effect to be the only physical infrared problem for n > 0 since it is very doubtful that a realistic cosmological evolution could have resulted in the establishment of an initial de Sitter background over distances much larger than the Hubble radius. With the redshift one does not need an initially infinite space in order to see infrared divergences because arbitrarily large spatial separations appear in the course of time evolution. Note that a particle whose coupling to gravity is conformally invariant cannot experience the de Sitter redshift because the de Sitter geometry is conformally flat. All observed particles other than the graviton are either massive - in which case they decouple from the infrared - or else they are conformally invariant in four dimensions. Since conformally invariant particles also lack the dimension three self-coupling of QCG we see that gravitons must completely
dominate the far infrared when the cosmological constant is positive. We do not yet have a good understanding of perturbative QCG on the anti-de Sitter background that would be appropriate for a negative cosmological constant. We believe that the infrared divergences of M Though these coordinates
cover only a portion of the full de Sitter manifold they provide a completely valid arena for physics because no causal evolution off the submanifold is possible. Indeed, if one wishes to imagine inflation as a phase in the development of a previously open Friedmann-Robertson-Walker universe there is no alternative to using such an open coordinate system.
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the scalar analog survive provided the full anti-de Sitter manifold is used [ 8 1, but the establishment of an initially infinite anti-de Sitter background does not seem any more plausible than does the de Sitter analog. The crucial question, to which we do not know the answer, is what happens on an anti-de Sitter bubble of large but finite size. We turn now to the physical interpretation of infrared divergences in QCG. To prove that these signal physical effects we have constructed a QCG analog of the Wilson loop [ 8 1. Our observable measures the force needed to hold a small mass m at a fixed invariant distance from a large mass M which is freely falling in the background geometry. Like the Wilson loop, our force observable is gauge invariant and of finite spatial and temporal extent. These features protect the Wilson loop from contamination by the infrared divergences that plague the on-shell S-matrix of QCD but they do not save our QCG force observable. For the case of n > 0, where we understand how to compute perturbatively, the force observable can be shown to suffer from infrared divergences at one loop [ 8 1. As expected from the nature of gravitation, the infrared singular corrections to the force are attractive. The origin of the infrared problem for (1> 0 is diagrams where an interaction vertex strays into the infinite future while all the vertices to which it is connected by a only single propagator remain at finite times. The generic effect is simple to estimate if we cut the divergence off by restricting interactions to the period between t=O and t= T in the coordinate system (4). In this case the leading infrared divergence in the N-point Green function behaves like GA exp (HT) to a positive power which depends simply upon N and the number of loops [ 8 1. Because the infrared divergences of QCG affect even gauge invariant, quasi-localized and apparently physical quantities such as our force observable, they cannot be avoided by averaging over degenerate ensembles as one does in QED and QCD. According to the theorem of Lee and Nauenberg [9] this only works when the on-shell self-energy is infrared finite. But the one loop self-energy of QCG contains infrared divergences, just as do all the other Green functions #4. The interpretation of these divergences lt4 For footnote see next page. 3.53
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is the same as that of the analogous problem in massless e3 theory: they signal a physical instability of the theory that it cures by adjusting the effective value of its lowest dimension coupling. In massless g3 the vacuum decays and the resulting mass cuts off the infrared problem. Although the QCG background can change, this cannot induce a graviton mass because such a term would increase the number of dynamical degrees of freedom. What happens instead is that the effective value of the cosmological constant - which is the lowest dimension coupling in quantum gravity - relaxes to the only value consistent with an infrared finiteness, namely zero. The relaxation mechanism can be seen in a completely conventional fashion using zero temperature asymptotic quantum field theory [ 81, although this is not the natural setting for it. The procedure is to assume a finite lifetime T for the cosmological interaction: a finite, positive ,4 appears at t = 0 and disappears at t = T. The turn-on can be justified as describing what happens in any realistic cosmology when the high initial temperature falls rapidly below (,JI/G)‘/~; the point of the calculation is to show that if one also assumes a cutoff for t > Tthen the effective value of the cosmological constant passes through zero at a certain time. The assumption becomes self-consistent if the decay is sudden and if the cutoff T can be chosen to coincide with the decay time. If the decay is not sudden then it is not consistent to use a sharp cutoff, but there is still a decay. In this case there is a complicated problem of back-reaction whereby the decay affects the process which drives it. The focus of the calculation is on the amputated one-point function:
= &4 + (loop corrections)
,
(5)
where g,,, is the de Sitter background with cosmological constant ,4 and L7~yp0is the QCG kinetic operator. By virtue of manifest spatial translation invariance we need only consider the dependence on time. Expression (5) must vanish if the expectation value #’ Actually
the Lee-Nauenberg technique would produce infrared finite QCG transition amplitudes at one loop. The essential breakdown occurs one loop beyond the point where the self-energy becomes infrared divergent [ 9 1.
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of the metric operator is really gfiy. Tadpole diagrams that would spoil the relation (g,,(t)) =g,,(t) are supposed to be canceled by the cosmological counterterm, -K-’ iM~~$?“hpy. Since 84 is a constant, this cancellation can only be enforced at one instant. We must choose that instant to be t = 0 in accordance with the statement that the initial cosmological constant has strength /1. If the cosmological interaction were allowed to persist for all time then the expectation value of hpv( t) could only be some multiple of g,,(t) and the vanishing of ( 5) at t= 0 would imply its vanishing for all time. However, since the cosmological interaction must be cut off in order to regulate infrared divergences, one has to allow for the possibility that the amputated one-point function becomes non-zero for t > 0. Although loop corrections to (5) contain both ultraviolet and infrared divergences we emphasize that there is no coupling between them. This follows from the cancellation of overlapping divergences, even in BPHZ renormalization. Accordingly, &l is the sum of an irrelevant ultraviolet counterterm, 6/lUv, and a piece &ii,, which at t = 0 cancels the reliable infrared contributions. Since it is only by virtue of the infrared cutoff that ( 5 ) can be non-zero, we need only consider &I,, and the associated infrared tadpoles. Explicit calculations at two loops reveal [ 81 that 6A ,R _ -#/i [ GA exp (HT) ] 2. The number in front is not too important but the minus sign is, as is the fact that an odd power of A appears. We can use the analogous scalar model (2) to understand why both of these properties should persist to all orders. Suppose that we suppress the cubic interaction before t = 0 and after t = T, and that we cancel the amputated scalar one-point function at t = 0 using a counterterm of the form -k@ Since quantum corrections would otherwise cause the scalar to begin rolling down its potential, we see that a positive cubic coupling constant il leads to infrared singular tadpoles which make a negative contribution to the expectation value of @ Topological considerations imply that these diagrams always contain an odd number of cubic vertices. It follows that k must be -A times a positive, even function of il. Since the most infrared singular diagrams of QCG have the same topologies as those of the scalar model, and since the signs of the propagators and the interaction vertices are the same in the
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infrared, we conclude that &I,, should be -A times a positive, even function of /1. As t approaches the cutoff time T, the soft gravitons responsible for the infrared effect cannot propagate very long in the presence of the cosmological interaction. Consequently, the infrared singular contributions to ( 5) dissipate and an ever smaller fraction of the constant counterterm &Iin is needed to make the amputated one-point function vanish. If we call the remainder &4,,(t) then the effective cosmological constant is the decreasing function /i,& t) =A + 6A,,, (t ). At this stage we should properly redo the calculation with a background geometry appropriate to n,,(t), but then things become intractable - the more so as perturbation theory necessarily breaks down when the infrared loop corrections become strong. The simplifying assumption which we have already been making implicitly - is that the extinction of n is sharp enough that backreactions can be ignored. If so, it follows that &I,, can be made arbitrarily negative for fixed A by merely increasing T. We can obviously then adjust T to enforce &( T) = 0, and if we could compute &( t ) reliably it would be possible to check the consistency of assuming a sharp cutoff. If the assumption is not selfconsistent the preceding considerations still indicate a tendency for n,,(t) to diminish, although we can no longer be certain that zero is reached. Computing the relaxation time with any precision is a very difficult problem that involves non-perturbative effects and non-linear back-reactions. We cannot do this sort of calculation analytically in QCD and it would unreasonable to expect QCG to differ in this respect. However, just as one can estimate the radius of light mesons from the scale at which QCD becomes strong so one can roughly gauge the QCG relaxation time from the fact that infrared effects become strong when the cutoff time T is such that GA exp (HT) x 1 [ 8 1. Equating Twith the relaxation time gives the following estimate for the number of e-foldings in the open coordinate system (4):
This is probably a considerable underestimate since it neglects the most important back-reaction whereby the infrared effect weakens as/i relaxes. Although we do not know how to carry out the
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above analysis for the case of an initially negative cosmological constant it is perhaps worth noting that we can make the argument on the level of the analogous scalar model (2 ). The result follows trivially by changing the sign of 1 and exploiting the previously noted oddness of the constant k. This suggests that for an initially negative cosmological constant the counterterm &tin would be positive. If so, the rest of the analysis would proceed as for the case of positive A. Note that if we adhere to the usual conventions then the induced infrared vacuum energy is negative in both cases. This is the paradox of signs: because
vacuum energy can couple to the graviton one-point function through a vertex which carries a factor of A, the generation of negative vacuum energy seems to relax a negative cosmological constant as weII as a positive one. The cosmological history we envisage commences, as usual, with a very large initial temperature. We shall assume that the renormalized cosmological constant is positive and smaller than the Planck scale - but nowhere near as small as the 120 orders of magnitude of the current observational bound [ 11. For instance, a typical GUT scale cosmological constant would be an acceptable initial condition. Thermal effects will dominate the stress-energy as long as the temperature is very much greater than (/i/G)‘/“; in this case thermal fluctuations will also suppress the long-lived correlations that drive the infrared effect of zero temperature QCG. In this period it is as though there is no cosmological constant at all. Eventually the universe cools sufficiently that its temperature drops below (n/G)‘14. We assume that this happens nearly simultaneously over a large but not infinite volume of space. At this stage it is as if the cosmological interaction was turned on within the cool region: ng,, begins to dominate the classical stress-energy and inflation begins. Long-lived correlations also become sustainable so the infrared effect described previously begins to develop a negative vacuum energy. The latter process lags the onset of inflation because it derives from the propagation of virtual gravitons for long periods in an inflating universe. There is no effect at all when inflation begins; it builds up only after a number of e-foldings. In time enough negative vacuum energy is produced to cancel the initial cosmological constant and inflation ceases. 355
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Note that scalar fields are unnecessary in our model, although they may still be present. The onset of inflation and its cessation are mediated by gravity, without the need for any sort of fine tuning. The same may also be true for reheating if enough of the vacuum energy locked up in the initial cosmological constant can be thermalized late in the relaxation process. The fact that QCG induces a negative vacuum energy only means that a state of lower energy is available, not that it is reached. To the extent that total energy is conserved, the universe can only pass to a background geometry of lower energy by producing radiation. (The associated massive entropy increase explains why such transitions are favored). Abbott and Deser [ 51 have shown how to construct a de Sitter energy functional which is conserved on the classical level; since there is nowhere for energy to go we shall assume that a quantum extension is possible. Nothing much happens in the early stages of relaxation. Little radiation is produced initially, and this is rapidly redshifted to insignificance by the inflating geometry. Our perturbative analysis indicates that the infrared effect eventually becomes strong but we cannot follow the evolution during this period. In the absence of reliable calculations many possibilities can be entertained. For example, the onset of large infrared effects might be sudden or slow; and the approach of/i,& t) to zero might be monotonic or there might be an “overshoot” and then damped oscillations between de Sitter and anti-de Sitter phases. This last possibility bears a striking analogy to the coherent scalar oscillations of the usual reheating scenario. For a sudden onset with subsequent oscillations one would expect a reheating temperature comparable with the initial scale of inflation, TRH z (.4/G) “4. For the case of a slow, monotonic approach one would expect a rather low value of TRH. An important constraint on any mechanism proposed for canceling /1 is that it should permit enough inflation to explain the homogeneity and isotropy of the present observed universe. The resulting lower limit on the number of e-foldings is
(7) 356
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where M= (n/G) ‘I4 is the scale of inflation [ lo]. Substituting our sharp cessation estimate of T,, =A4 and comparing with expression (6) shows that to sustain inflation over such a period implies a low but not unreasonable scale, MS 10 I2 GeV. A value of TRH as small as 10 MeV would add only an order of magnitude. The larger effect derives from our probable underestimate of the relaxation time; a factor of two increase in NmT raises the bound on A4 by a factor of about 300. A non-zero cosmological constant is liable to be generated whenever the universe experiences a phase transition much faster than the natural relaxation time. When such a transition induces a positive cosmological constant there will be a period of inflation and relaxation; we believe that relaxation occurs as well when a negative cosmological constant is induced. Of the many episodes of inflation and deflation which may have preceded the current epoch the final ones obviously have the greatest phenomenological significance. Two candidates for the last fast phase transition are the breaking of electroweak symmetry at 250 GeV and the breaking of chiral symmetry in QCD at 250 MeV. If either of these was fast and induced a positive cosmological constant, the number of inflationary e-foldings would be adequate - formula (6 ) gives NToT= 154 and NT0T.z 18 1 respectively - but the standard model would require extension to permit sufficient baryogenesis at low reheating temperatures. Although we pretend to no expertise in this field, the thing does not seem impossible [ Ill. To sum up, it seems beyond doubt that perturbative quantum gravity manifests enormously strong infrared effects when the cosmological constant is positive. Chief among these effects is the generation of a negative graviton self-energy that diverges when the infrared cutoff is removed. Our proposal is that the associated build-up of negative vacuum energy gradually extinguishes the universe’s (assumed) initially positive cosmological constant. The manner in which this occurs may have profound implications for low energy phenomenology. We wish to emphasize that the relaxation mechanism described in this work is a proposal, not a proven fact. It has been rigorously shown to operate only for so long as perturbation theory remains valid. The reader should bear in mind that nonperturbative ef-
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fects may supervene to produce an entirely different outcome; it cannot even be ruled out with the technology currently at our disposal that /i&t) eventually increases. The situation is remarkably similar to the challenge of perceiving confinement in QCD. In both cases there is strong experimental evidence from different regimes - deep inelastic scattering for the strong interaction, astrophysics for gravity - that the theory in question ought to be correct in the far infrared. In both cases there is a qualitative argument - the attraction between QCD lines of force, the generation of negative vacuum energy for QCG - that physics in the far infrared ought to manifest the phenomenon that is actually observed. In both cases explicit perturbative calculations support these arguments, and in both cases perturbation theory breaks down as soon as the observed phenomenon begins to occur. We thank R. Brandenberger, S. Deser, poulos, E.G. Floratos, G.T. Horowitz, C. H.B. Nielsen, P. Ramond, P. Sikivie, M. and T.N. Tomaras. Support comes from
S. DimoKounnas, Srednicki DOE-DE-
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FG05-86-ER40272, SC 1-0430-c.
11 March
NATO-CRC-9
1993
10627 and EEC-
References [ 1] A. Sandage, Observatory 88 ( 1968) 91. [ 21 S. Weinberg, Rev. Mod. Phys. 6 I ( 1989) 1. [3] S. Weinberg, Phys. Rev. B 138 (1965) 988. [4] F. Bloch and H. Nordsieck, Phys. Rev. 52 ( 1937) 54. [ 51 S. Deser and L.F. Abbott, Nucl. Phys. B 195 (1982) 76. N.C. Tsamis and R.P. Woodard, The structure of perturbative quantum gravity on a de Sitter background, preprint CRETE-92-I 1, UFIFT-92-14, Commun. Math. Phys., to appear. N.C. Tsamis and R.P. Woodard, Phys. Lett. B 292 (1992) 269. N.C. Tsamis and R.P. Woodard, Strong infrared effects in quantum gravity, preprint CRETE-92-17, UFIFT-92-24. [ 91 T.D. Lee and M. Nauenberg, Phys. Rev. B 133 ( 1964) 1549. [ IO] E.W. Kolb and M.S. Turner, The early Universe (AddisonWesley, Reading, MA, 1990). [ 111 S. Dimopoulos and L.J. Hall, Phys. Lett. B 196 (1987) 135; L. McLerran, M. Shaposhinikov, N. Turok and M. Voloshin, Phys. Lett. B 256 (1991) 451; A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Lett. B 263 (1991) 86.
357
301 (1993)
351-357
Relaxing the cosmological N.C. Tsamis
PHYSICS
LETTERS
B
constant
’
Department of Physics, University of Crete, GR- 71409 Iraklion, Crete, Greece
and R.P. Woodard 2 Department of Physics, University of Florida, Gainesville, FL 32611, USA Received
22 December
1992
A non-zero cosmological constant allows massless gravitons to self-interact via a coupling of dimension three. If the cosmological constant is positive then the background geometry will subject gravitons to an enormous redshift. These two facts are together responsible for the severe infrared divergences which detailed, explicit calculations [ N.C. Tsamis and R.P. Woodard, Strong infrared effects in quantum gravity, preprint CRETE-92-17, UFIFT-92-241 reveal in loop corrections to the gravitational force law. We argue that, as a result, the cosmological interaction has a finite lifetime. Furthermore, this lifetime is consistent wi h inflation.
The most general lagrangian based upon the metric and restricted to two derivatives of this field is given by #’
It possesses an apparently free parameter, A, whose value is observed to be less then 10-120G - ’ [ 11. Opinion is nearly unanimous that this can only be explained through quantum effects [ 21. With a few exceptions, however, the search for a solution has focused on making a clever choice for the matter theory which is coupled to gravity. There is something to be said for this approach but it has an obvious difficulty in that any mechanism for canceling /1 must operate in the far infrared where the particle spectrum is believed to be well known. It is preferable to consider the possibility that the job is done by an already ob’ E-mail address: TSAMIS @ IESL.FORTH.GR. ’ E-mail address: WOODARD @ UFHEPA.PHYS.UFL.EDU. #’ Our metric has spacelike signature and R’,=T“,,,+.... Greek indices run from 0 to 3, while Latin indices run from to 3. We are also using units where c= fi = 1.
0370-2693/93/$06.00
0 1993 Elsevier Science Publishers
I
served massless particle rather than by a new one which has somehow escaped detection. And the resolution of greatest aesthetic appeal would be for gravity to cure its own problem using nothing more than the graviton. Very little attention has been paid to the last possibility for two reasons. First, there is a widespread prejudice against any appeal to quantum gravity because there is no consistent theory for it. Second, simple arguments purport to show that quantum gravitational effects should be weak in the infrared. The oldest such argument is that the theory’s natural coupling constant, ~~ = 16nG, goes like inverse mass squared. This means that perturbation theory must generate a series in positive powers of K2p2, where p2 is some combination of the momenta or particle masses appearing in whatever process is under investigation. Therefore, loop corrections are suppressed in the infrared. A more sophisticated study by Weinberg [ 31 concluded long ago that the leading infrared corrections of quantum gravity are similar to those of QED. That is, the most significant infrared effects are soft graviton emission from, and exchange between, the external legs of an exclusive amplitude. Soft
B.V. All rights reserved.
351
Volume 301, number
4
PHYSICS
graviton corrections to soft graviton lines are suppressed because each vertex contains two derivatives and these vanish in the infrared. The same effect also dilutes contributions from other massless particle loops on soft graviton lines. The first objection to considering quantum gravity is certainly invalid: the non-renormalizability of Einstein’s theory in no way prevents us from using it reliably in the far infrared. The fact that we perceive a finite gravitational force implies that there must be a consistent quantum theory of it. The fact that we perceive only a weak gravitational force, which is so well described by general relativity, implies that there can be no significant coupling between large distance phenomena and the details of whatever Planck scale mechanism through which this fundamental theory contrives to free itself of ultraviolet divergences. On cosmological distance scales it suffices to use the appropriate local counterterms to subtract off the ultraviolet divergences of ( 1) as they occur, order by order in perturbation theory. The dominant infrared effects at any fixed order reside in ultravioletfinite and nonlocal terms that are not affected by how we make these subtractions and which dominate the result at large distances once ultraviolet finiteness is achieved. This is why Bloch and Nordsieck [4] were able to resolve the infrared problem of QED more than a decade before renormalization was understood. The same principle underlies Weinberg’s [ 31 analysis of infrared quantum gravity for A = 0. The second objection is also invalid: when A#0 quantum gravity should become very strong in the far infrared. How particles influence the physics of large distances is determined by their masses and by the dimensionalities of their self-interactions. The dominant effect comes from the lightest particles which self-interact via couplings of the lowest canonical field dimension. Gravitons are massless for any value of the cosmological constant, but for A = 0 their lowest self-interaction has dimension five - there are two derivatives distributed between three graviton fields. This is why conventional quantum gravity is so much weaker in the infrared than QCD, which allows massless gluons to self-interact via a coupling of dimension four. However, when /1# 0 gravitons can self-interact through a coupling of dimension three since the cosmological interaction provides a threepoint vertex with no derivatives. This version of 352
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quantum gravity is so different from the conventional theory that we distinguish it with the name “quantum cosmological gravity”, or QCG for short. Based on the preceding considerations QCG ought to be stronger in the far infrared than QCD. The infrared sector of QCD is responsible for confinement and it might seem difficult to imagine an even stronger effect. The nearest analog in the familiar physics of flat space is the theory of a massless scalar with a cubic self-interaction:
(2) Of course this model differs from QCG in that the background @= 0 is not even classically stable whereas de Sitter space and anti-de Sitter space do seem to be classically stable backgrounds for QCG [ 5 1. However, @= 0 is at least a classical solution and the quantum perturbation theory based on this background is a good analog for the infrared behavior of QCG. Of special significance is the fact that the one loop self-energy of the scalar becomes infinitely negative on shell #*: C(P2) = 73;;
In (p’) + constant
.
(3)
This is very interesting because the one loop self-energy contributes at two loops to the vacuum energy, which in QCG is an important constituent of the cosmological constant. If it were correct to useflat space propagators in QCG, and to take the resulting infrared divergences seriously, this scalar result would imply that quantum corrections from the far infrared have an enormous impact on the observed value of A. To proceed further we must answer two questions. First, how is the scalar result modified in QCG by the necessarily non-flat background? Second, how should we interpret infrared divergences in QCG? For the case of positive cosmological constant the natural background is the open submanifold of de x2 We note in passing that the ultraviolet
divergences in this quantity are removed by a mass counterterm which affects only the constant. The fact that the leading infrared effect is nonlocal, ultraviolet finite and completely untouched by this subtraction exemplifies the general rule enunciated in the prepenultimate paragraph.
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Sitter space [ 61, a convenient which #3 ds*= -dt*+exp(2Ht)
H2GiA.
coordinatization
(d_x*+dy*+dz*)
for
, (4)
We have been able to completely solve linearized QCG on this background [ 7 1, and to work out in full the structure of perturbative QCG [ 61. The result is a spectacular enhancement of the infrared divergences which appear in massless e3 [ 8 1. The reason is simple: since spatial coordinates are exponentially
expanded with increasing time, their Fourier conjugates, the spatial momenta, are redshifted to zero. Infrared effects derive from the low end of the momentum spectrum so they are strengthened when this sector is more densely populated. In fact we believe this temporally increasing effect to be the only physical infrared problem for n > 0 since it is very doubtful that a realistic cosmological evolution could have resulted in the establishment of an initial de Sitter background over distances much larger than the Hubble radius. With the redshift one does not need an initially infinite space in order to see infrared divergences because arbitrarily large spatial separations appear in the course of time evolution. Note that a particle whose coupling to gravity is conformally invariant cannot experience the de Sitter redshift because the de Sitter geometry is conformally flat. All observed particles other than the graviton are either massive - in which case they decouple from the infrared - or else they are conformally invariant in four dimensions. Since conformally invariant particles also lack the dimension three self-coupling of QCG we see that gravitons must completely
dominate the far infrared when the cosmological constant is positive. We do not yet have a good understanding of perturbative QCG on the anti-de Sitter background that would be appropriate for a negative cosmological constant. We believe that the infrared divergences of M Though these coordinates
cover only a portion of the full de Sitter manifold they provide a completely valid arena for physics because no causal evolution off the submanifold is possible. Indeed, if one wishes to imagine inflation as a phase in the development of a previously open Friedmann-Robertson-Walker universe there is no alternative to using such an open coordinate system.
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the scalar analog survive provided the full anti-de Sitter manifold is used [ 8 1, but the establishment of an initially infinite anti-de Sitter background does not seem any more plausible than does the de Sitter analog. The crucial question, to which we do not know the answer, is what happens on an anti-de Sitter bubble of large but finite size. We turn now to the physical interpretation of infrared divergences in QCG. To prove that these signal physical effects we have constructed a QCG analog of the Wilson loop [ 8 1. Our observable measures the force needed to hold a small mass m at a fixed invariant distance from a large mass M which is freely falling in the background geometry. Like the Wilson loop, our force observable is gauge invariant and of finite spatial and temporal extent. These features protect the Wilson loop from contamination by the infrared divergences that plague the on-shell S-matrix of QCD but they do not save our QCG force observable. For the case of n > 0, where we understand how to compute perturbatively, the force observable can be shown to suffer from infrared divergences at one loop [ 8 1. As expected from the nature of gravitation, the infrared singular corrections to the force are attractive. The origin of the infrared problem for (1> 0 is diagrams where an interaction vertex strays into the infinite future while all the vertices to which it is connected by a only single propagator remain at finite times. The generic effect is simple to estimate if we cut the divergence off by restricting interactions to the period between t=O and t= T in the coordinate system (4). In this case the leading infrared divergence in the N-point Green function behaves like GA exp (HT) to a positive power which depends simply upon N and the number of loops [ 8 1. Because the infrared divergences of QCG affect even gauge invariant, quasi-localized and apparently physical quantities such as our force observable, they cannot be avoided by averaging over degenerate ensembles as one does in QED and QCD. According to the theorem of Lee and Nauenberg [9] this only works when the on-shell self-energy is infrared finite. But the one loop self-energy of QCG contains infrared divergences, just as do all the other Green functions #4. The interpretation of these divergences lt4 For footnote see next page. 3.53
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is the same as that of the analogous problem in massless e3 theory: they signal a physical instability of the theory that it cures by adjusting the effective value of its lowest dimension coupling. In massless g3 the vacuum decays and the resulting mass cuts off the infrared problem. Although the QCG background can change, this cannot induce a graviton mass because such a term would increase the number of dynamical degrees of freedom. What happens instead is that the effective value of the cosmological constant - which is the lowest dimension coupling in quantum gravity - relaxes to the only value consistent with an infrared finiteness, namely zero. The relaxation mechanism can be seen in a completely conventional fashion using zero temperature asymptotic quantum field theory [ 81, although this is not the natural setting for it. The procedure is to assume a finite lifetime T for the cosmological interaction: a finite, positive ,4 appears at t = 0 and disappears at t = T. The turn-on can be justified as describing what happens in any realistic cosmology when the high initial temperature falls rapidly below (,JI/G)‘/~; the point of the calculation is to show that if one also assumes a cutoff for t > Tthen the effective value of the cosmological constant passes through zero at a certain time. The assumption becomes self-consistent if the decay is sudden and if the cutoff T can be chosen to coincide with the decay time. If the decay is not sudden then it is not consistent to use a sharp cutoff, but there is still a decay. In this case there is a complicated problem of back-reaction whereby the decay affects the process which drives it. The focus of the calculation is on the amputated one-point function:
= &4 + (loop corrections)
,
(5)
where g,,, is the de Sitter background with cosmological constant ,4 and L7~yp0is the QCG kinetic operator. By virtue of manifest spatial translation invariance we need only consider the dependence on time. Expression (5) must vanish if the expectation value #’ Actually
the Lee-Nauenberg technique would produce infrared finite QCG transition amplitudes at one loop. The essential breakdown occurs one loop beyond the point where the self-energy becomes infrared divergent [ 9 1.
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of the metric operator is really gfiy. Tadpole diagrams that would spoil the relation (g,,(t)) =g,,(t) are supposed to be canceled by the cosmological counterterm, -K-’ iM~~$?“hpy. Since 84 is a constant, this cancellation can only be enforced at one instant. We must choose that instant to be t = 0 in accordance with the statement that the initial cosmological constant has strength /1. If the cosmological interaction were allowed to persist for all time then the expectation value of hpv( t) could only be some multiple of g,,(t) and the vanishing of ( 5) at t= 0 would imply its vanishing for all time. However, since the cosmological interaction must be cut off in order to regulate infrared divergences, one has to allow for the possibility that the amputated one-point function becomes non-zero for t > 0. Although loop corrections to (5) contain both ultraviolet and infrared divergences we emphasize that there is no coupling between them. This follows from the cancellation of overlapping divergences, even in BPHZ renormalization. Accordingly, &l is the sum of an irrelevant ultraviolet counterterm, 6/lUv, and a piece &ii,, which at t = 0 cancels the reliable infrared contributions. Since it is only by virtue of the infrared cutoff that ( 5 ) can be non-zero, we need only consider &I,, and the associated infrared tadpoles. Explicit calculations at two loops reveal [ 81 that 6A ,R _ -#/i [ GA exp (HT) ] 2. The number in front is not too important but the minus sign is, as is the fact that an odd power of A appears. We can use the analogous scalar model (2) to understand why both of these properties should persist to all orders. Suppose that we suppress the cubic interaction before t = 0 and after t = T, and that we cancel the amputated scalar one-point function at t = 0 using a counterterm of the form -k@ Since quantum corrections would otherwise cause the scalar to begin rolling down its potential, we see that a positive cubic coupling constant il leads to infrared singular tadpoles which make a negative contribution to the expectation value of @ Topological considerations imply that these diagrams always contain an odd number of cubic vertices. It follows that k must be -A times a positive, even function of il. Since the most infrared singular diagrams of QCG have the same topologies as those of the scalar model, and since the signs of the propagators and the interaction vertices are the same in the
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infrared, we conclude that &I,, should be -A times a positive, even function of /1. As t approaches the cutoff time T, the soft gravitons responsible for the infrared effect cannot propagate very long in the presence of the cosmological interaction. Consequently, the infrared singular contributions to ( 5) dissipate and an ever smaller fraction of the constant counterterm &Iin is needed to make the amputated one-point function vanish. If we call the remainder &4,,(t) then the effective cosmological constant is the decreasing function /i,& t) =A + 6A,,, (t ). At this stage we should properly redo the calculation with a background geometry appropriate to n,,(t), but then things become intractable - the more so as perturbation theory necessarily breaks down when the infrared loop corrections become strong. The simplifying assumption which we have already been making implicitly - is that the extinction of n is sharp enough that backreactions can be ignored. If so, it follows that &I,, can be made arbitrarily negative for fixed A by merely increasing T. We can obviously then adjust T to enforce &( T) = 0, and if we could compute &( t ) reliably it would be possible to check the consistency of assuming a sharp cutoff. If the assumption is not selfconsistent the preceding considerations still indicate a tendency for n,,(t) to diminish, although we can no longer be certain that zero is reached. Computing the relaxation time with any precision is a very difficult problem that involves non-perturbative effects and non-linear back-reactions. We cannot do this sort of calculation analytically in QCD and it would unreasonable to expect QCG to differ in this respect. However, just as one can estimate the radius of light mesons from the scale at which QCD becomes strong so one can roughly gauge the QCG relaxation time from the fact that infrared effects become strong when the cutoff time T is such that GA exp (HT) x 1 [ 8 1. Equating Twith the relaxation time gives the following estimate for the number of e-foldings in the open coordinate system (4):
This is probably a considerable underestimate since it neglects the most important back-reaction whereby the infrared effect weakens as/i relaxes. Although we do not know how to carry out the
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above analysis for the case of an initially negative cosmological constant it is perhaps worth noting that we can make the argument on the level of the analogous scalar model (2 ). The result follows trivially by changing the sign of 1 and exploiting the previously noted oddness of the constant k. This suggests that for an initially negative cosmological constant the counterterm &tin would be positive. If so, the rest of the analysis would proceed as for the case of positive A. Note that if we adhere to the usual conventions then the induced infrared vacuum energy is negative in both cases. This is the paradox of signs: because
vacuum energy can couple to the graviton one-point function through a vertex which carries a factor of A, the generation of negative vacuum energy seems to relax a negative cosmological constant as weII as a positive one. The cosmological history we envisage commences, as usual, with a very large initial temperature. We shall assume that the renormalized cosmological constant is positive and smaller than the Planck scale - but nowhere near as small as the 120 orders of magnitude of the current observational bound [ 11. For instance, a typical GUT scale cosmological constant would be an acceptable initial condition. Thermal effects will dominate the stress-energy as long as the temperature is very much greater than (/i/G)‘/“; in this case thermal fluctuations will also suppress the long-lived correlations that drive the infrared effect of zero temperature QCG. In this period it is as though there is no cosmological constant at all. Eventually the universe cools sufficiently that its temperature drops below (n/G)‘14. We assume that this happens nearly simultaneously over a large but not infinite volume of space. At this stage it is as if the cosmological interaction was turned on within the cool region: ng,, begins to dominate the classical stress-energy and inflation begins. Long-lived correlations also become sustainable so the infrared effect described previously begins to develop a negative vacuum energy. The latter process lags the onset of inflation because it derives from the propagation of virtual gravitons for long periods in an inflating universe. There is no effect at all when inflation begins; it builds up only after a number of e-foldings. In time enough negative vacuum energy is produced to cancel the initial cosmological constant and inflation ceases. 355
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Note that scalar fields are unnecessary in our model, although they may still be present. The onset of inflation and its cessation are mediated by gravity, without the need for any sort of fine tuning. The same may also be true for reheating if enough of the vacuum energy locked up in the initial cosmological constant can be thermalized late in the relaxation process. The fact that QCG induces a negative vacuum energy only means that a state of lower energy is available, not that it is reached. To the extent that total energy is conserved, the universe can only pass to a background geometry of lower energy by producing radiation. (The associated massive entropy increase explains why such transitions are favored). Abbott and Deser [ 51 have shown how to construct a de Sitter energy functional which is conserved on the classical level; since there is nowhere for energy to go we shall assume that a quantum extension is possible. Nothing much happens in the early stages of relaxation. Little radiation is produced initially, and this is rapidly redshifted to insignificance by the inflating geometry. Our perturbative analysis indicates that the infrared effect eventually becomes strong but we cannot follow the evolution during this period. In the absence of reliable calculations many possibilities can be entertained. For example, the onset of large infrared effects might be sudden or slow; and the approach of/i,& t) to zero might be monotonic or there might be an “overshoot” and then damped oscillations between de Sitter and anti-de Sitter phases. This last possibility bears a striking analogy to the coherent scalar oscillations of the usual reheating scenario. For a sudden onset with subsequent oscillations one would expect a reheating temperature comparable with the initial scale of inflation, TRH z (.4/G) “4. For the case of a slow, monotonic approach one would expect a rather low value of TRH. An important constraint on any mechanism proposed for canceling /1 is that it should permit enough inflation to explain the homogeneity and isotropy of the present observed universe. The resulting lower limit on the number of e-foldings is
(7) 356
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where M= (n/G) ‘I4 is the scale of inflation [ lo]. Substituting our sharp cessation estimate of T,, =A4 and comparing with expression (6) shows that to sustain inflation over such a period implies a low but not unreasonable scale, MS 10 I2 GeV. A value of TRH as small as 10 MeV would add only an order of magnitude. The larger effect derives from our probable underestimate of the relaxation time; a factor of two increase in NmT raises the bound on A4 by a factor of about 300. A non-zero cosmological constant is liable to be generated whenever the universe experiences a phase transition much faster than the natural relaxation time. When such a transition induces a positive cosmological constant there will be a period of inflation and relaxation; we believe that relaxation occurs as well when a negative cosmological constant is induced. Of the many episodes of inflation and deflation which may have preceded the current epoch the final ones obviously have the greatest phenomenological significance. Two candidates for the last fast phase transition are the breaking of electroweak symmetry at 250 GeV and the breaking of chiral symmetry in QCD at 250 MeV. If either of these was fast and induced a positive cosmological constant, the number of inflationary e-foldings would be adequate - formula (6 ) gives NToT= 154 and NT0T.z 18 1 respectively - but the standard model would require extension to permit sufficient baryogenesis at low reheating temperatures. Although we pretend to no expertise in this field, the thing does not seem impossible [ Ill. To sum up, it seems beyond doubt that perturbative quantum gravity manifests enormously strong infrared effects when the cosmological constant is positive. Chief among these effects is the generation of a negative graviton self-energy that diverges when the infrared cutoff is removed. Our proposal is that the associated build-up of negative vacuum energy gradually extinguishes the universe’s (assumed) initially positive cosmological constant. The manner in which this occurs may have profound implications for low energy phenomenology. We wish to emphasize that the relaxation mechanism described in this work is a proposal, not a proven fact. It has been rigorously shown to operate only for so long as perturbation theory remains valid. The reader should bear in mind that nonperturbative ef-
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fects may supervene to produce an entirely different outcome; it cannot even be ruled out with the technology currently at our disposal that /i&t) eventually increases. The situation is remarkably similar to the challenge of perceiving confinement in QCD. In both cases there is strong experimental evidence from different regimes - deep inelastic scattering for the strong interaction, astrophysics for gravity - that the theory in question ought to be correct in the far infrared. In both cases there is a qualitative argument - the attraction between QCD lines of force, the generation of negative vacuum energy for QCG - that physics in the far infrared ought to manifest the phenomenon that is actually observed. In both cases explicit perturbative calculations support these arguments, and in both cases perturbation theory breaks down as soon as the observed phenomenon begins to occur. We thank R. Brandenberger, S. Deser, poulos, E.G. Floratos, G.T. Horowitz, C. H.B. Nielsen, P. Ramond, P. Sikivie, M. and T.N. Tomaras. Support comes from
S. DimoKounnas, Srednicki DOE-DE-
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References [ 1] A. Sandage, Observatory 88 ( 1968) 91. [ 21 S. Weinberg, Rev. Mod. Phys. 6 I ( 1989) 1. [3] S. Weinberg, Phys. Rev. B 138 (1965) 988. [4] F. Bloch and H. Nordsieck, Phys. Rev. 52 ( 1937) 54. [ 51 S. Deser and L.F. Abbott, Nucl. Phys. B 195 (1982) 76. N.C. Tsamis and R.P. Woodard, The structure of perturbative quantum gravity on a de Sitter background, preprint CRETE-92-I 1, UFIFT-92-14, Commun. Math. Phys., to appear. N.C. Tsamis and R.P. Woodard, Phys. Lett. B 292 (1992) 269. N.C. Tsamis and R.P. Woodard, Strong infrared effects in quantum gravity, preprint CRETE-92-17, UFIFT-92-24. [ 91 T.D. Lee and M. Nauenberg, Phys. Rev. B 133 ( 1964) 1549. [ IO] E.W. Kolb and M.S. Turner, The early Universe (AddisonWesley, Reading, MA, 1990). [ 111 S. Dimopoulos and L.J. Hall, Phys. Lett. B 196 (1987) 135; L. McLerran, M. Shaposhinikov, N. Turok and M. Voloshin, Phys. Lett. B 256 (1991) 451; A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Lett. B 263 (1991) 86.
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