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On Infrared Effects in de~Sitter Background

We have estimated higher order quantum gravity corrections to de~Sitter spacetime. Our results suggest that, while the classical spacetime metric may be distorted by the graviton self-interactions, the corrections are relatively weaker than previously thought, possibly growing like a power rather than exponentially in time.

As is very well known Einstein field equations permit an addition of an extra term proportional to the metric tensor,ḡ µν : Here T µν is the energy-momentum tensor of matter and Λ is the so called cosmological constant. (We put an bar over all quantities here to distinguish them from the rescaled ones, see Eqs. (4,5)). The last term in Eq. (1) may be interpreted as the vacuum energymomentum tensor. Any natural estimate gives Λ or ρ vac = Λ/8πG much larger (by 50-100 orders of magnitude) than the observed upper bound in the present day Universe. This mysterious discrepancy is known as the cosmological constant problem and presents one of the most interesting challenges in the modern physics (for a review see Refs. [1,2]).
de Sitter spacetime is a solution of the Einstein equations with a dominant cosmological term. Written in the Robertson-Walker form for the special case of the spatially flat section the metric takes the form where a(t) = exp(Ht) with H = Λ/3. The assumption of spatial flatness is not essential and the results obtained below are true also for open and closed geometries. It is convenient to rewrite the metric in terms of conformal coordinates where, up to an overall scale factor, it has the Minkowski form: The conformal time τ is related to the physical one as dτ = − exp(−Ht)dt and a(τ ) = −1/Hτ . Note that when t tends to future infinity, τ tends to 0−. The de Sitter metric possesses the same degree of high symmetry as Minkowski one and for this reason, it is the simplest non-trivial curved background for a quantum field theory. Despite its simplicity, it has several very interesting properties and in particular generates an infrared instability of a massless scalar field φ minimally coupled to gravity. It was shown [3,4] that the vacuum expectation value (VEV) φ 2 is singular at the zero mass limit, φ 2 ∼ H 4 /m 2 . With the advent of the inflationary scenario, this phenomenon was rediscovered in a number of papers [5,6,7] where it was argued that φ 2 rises linearly with time in the zero mass limit. This quantity however does not have a direct physical meaning and, in particular, the contribution of the field φ to the energy density does not rise with time. Nevertheless, this infrared instability afflicts the scalar field propagator and needs to be resolved to make a sensible quantum theory. Note in passing that this instability is a result of broken conformal invariance. To ensure the latter one needs the coupling to the curvature scalar, Rφ 2 /6, which gives rise to an infrared cut-off. In fact, any nonzero mass m φ or coupling ξRφ 2 with a nonnegative coefficient is sufficient for infrared stability. In any case, this infrared divergence is a sign that the true vacuum state is not de Sitter invariant.
Conformal invariance is also broken for gravitons [8] and, for this reason, there should be infrared instability of the de Sitter vacuum due to quantum gravity effects. If this is indeed the case, the solution of the long standing cosmological constant problem may be found in this direction. One-loop graviton quantum corrections to the de Sitter metric were considered in this connection in Ref. [9] where it was found that they are time independent.
However, it was suggested that higher loops may show evidence of this infrared instability.
Indeed, it was claimed recently in several papers [10,11,12] that higher loop effects are much stronger, giving corrections that rise exponentially with physical time t or as a power of 1/τ with conformal time. This is a very exciting result and, if confirmed, would mean that de Sitter space cannot exist indefinitely, opening a beautiful way for the solution of the cosmological constant problem in the framework of the normal quantum gravity without any drastic assumptions.
Here we have reconsidered results of papers [10,11,12] using a different formalism and have found, unfortunately, that such a strong instability does not set in. The corrections at most behave as a power 2 of log |τ | ∼ t. 2 We understand that these authors now also do not get power law singularity but only powers of log τ . (R.
For the conformally flat background metric we introduce the quantum graviton field h µν in the usual wayḡ The issue is whether, taking into account loop corrections, the VEV h µν is non-zero and, in particular, whether it is time-dependent. This would suggest that the background de Sitter metric is not self-consistent, although it must then be shown that this is a physical effect by calculating, for example, the curvature for the modified metric. Even then, it is important to ascertain whether the inconsistency involves strong coupling or whether it is an essentially negligible effect. The Einstein action (with nonzero cosmological constant) in terms of the new metric g µν can be rewritten as with κ 2 ≡ 16π/m 2 P l . This implies the following equations of motion: where tensor indices are raised and lowered with the new metric g µν . To zeroth order in h µν , we get the usual equation for the scale factor of the classical background metric, with solution a(τ ) = −1/Hτ (see notation after Eq. (3).) It is of course well-known that this is a non-renormalizable quantum field theory, but we regard the Einstein action as the first two terms in an infinite series of local operators of increasing dimension. This is an effective field theory, presumed valid on scales below m P l , and is renormalizable in the sense that all divergences involving these vertices may be absorbed in a renormalization of one of the infinite number of coefficients of these local operators. While we have not written such terms explicitly, they should be understood to be present. In the following, we will deal exclusively with renormalized couplings and operators, with the tacit understanding that the counterterms are included in our interaction. To Woodard, private communication) consistently quantize the theory, we must add a gauge-fixing term L gf and the corresponding Faddeev-Popov ghosts L F P . We choose L gf = − 1 2 F µ F ν η µν as in Refs. [10,11], with The ghost Lagrangian L F P may be found in Ref. [12]. Here and subsequently when we consider perturbation theory in h µν , the indices are raised with the Minkowski tensor η µν .
With these gauge conditions, the linear part of the equation of motion for h µν has very simple form: One can easily verify that time components h 0µ (µ = 0, 1, 2, 3) are conformally invariant in the sense that the rescaled functions χ 0µ = a(τ )h 0µ satisfy the free field equations of motion, More interesting are the space-space components of metric h ij . The equations of motion are not diagonal for them but after a simple linear redefinition f ij = h ij − δ ij h 00 they are diagonalized and have the form: This is the same equation which is satisfied by a massless minimally coupled scalar field.
The solutions with definite momenta can be written as These nonconformal modes differ from the conformal modes by the presence of the i/k term.
The conventionally normalized Heisenberg field operator for quantum fluctuations of the gravitational field is defined as ψ µν ≡ h µν /κ. This may be expressed in the standard way in terms of creation-annihilation operators a µν (k) by the decomposition: 3 with a and a † satisfying the commutation relations All other commutators are zero. For the nonconformal modes, the wave function ψ k (τ, x) is given by Eq. (10), while for the conformal modes, they are plane waves Hτ exp(i k x − ikτ ).
Respectively, the propagator for a conformal field (for example for the field h 0α , see above) is proportional to that in flat space. It means that the evolution of h 0α is trivial and in particular the corresponding Green's functions are obtained from the ones for the flat spacetime by the rescaling: Now, propagators (two-point functions) for nonconformal fields exhibit new features since the solutions (10) blow up at small k as compared to the solution for conformal fields. In particular, this implies that the anticommutator correlator G 1 is in fact not defined in the infrared. To be explicit, it is given by where ∆τ ≡ τ − τ ′ . If we calculate the Fourier transform of (14) then it is logarithmically divergent at small k and undefined. However, derivatives of it are well defined.
It might also worth mentioning that in a de Sitter background, the Green function for nonconformal field can be also given in a closed form in four dimensional notations. Namely we have where, as usual, depending on the iǫ prescription we get either Feynman or retarded, or advanced propagator. (The derivation of (15) follows the lines presented in the appendix to Ref [17] where the first terms in the WKB expansion are considered for arbitrary background field. In a de Sitter background, a similar argument fixes the exact propagator.) In particular the Feynman propagator takes the form [11] Note that the argument of the log is not defined, as a manifestation of the same infrared instability. What is specific about the de Sitter background (and ξ = 1/6 or 0) is that the expansion in k −2 in Eq. (15) terminates on the second term.
In fact, this infrared blowup of the Feynman propagator is the basic observation which gave rise to the hopes that quantum corrections to the de Sitter metric grow with time.
Indeed, the quantum propagators are not vanishing in causally forbidden regions and may bring information on the overall expansion in the future, characteristic of the de Sitter solution, revealing in this way an instability of pure classical solution. We are going to scrutinize this suggestion.
The best way to approach the problem is to keep as close to classical consideration as possible. Indeed, the classical solutions are known to be stable against perturbations [18]. The crucial difference between classical and quantum problems is that development of classical fluctuations is governed by the retarded Green function G R which is free of the infrared divergencies mentioned above. The Green function is obtained in the standard way: where we have suppressed the various tensor indices. Although commonly expressed in terms of a VEV, it is important to recognize that G R for the linearized theory is a purely classical construct that may be obtained directly from the classical equations of motion. Accordingly, it is completely independent of the definition of the vacuum state.
A simple calculation gives [12] G R (x, where r = | x − x ′ |, ∆τ = (τ − τ ′ ) and is the retarded Green's function in the flat spacetime. We suppressed here the evident tensor indices. The second term in Eq. (18) is connected with the broken conformal invariance. For what follows it is essential that it vanishes for ∆τ = 0. In the mixed (τ, k) representation we find and in the limit if k → 0 the retarded Green function for a nonconformal field is no more singular than for a conformal (or in flat space).
The Heisenberg picture of quantum field theory is most close to the classical one and we expect that the infrared problem is most easily treated within this approach. With the initial condition ψ µν (τ 0 , x) = ψ in µν (τ 0 , x), the operator equation (8) can be rewritten in the integral form: where V αβ (τ, x) is the interaction term. More specifically, where Γ α µν = (ψ α µ,ν + ψ α ν,µ − ψ ,α µν )/2 is the Christoffel symbol associated with the metric g µν = η µν + κψ µν to first order in ψ µν . 4 Note that the last two terms that are explicitly singular in τ are multiplied by combinations of fields whose wave functions, in lowest order, vanish when τ → 0. This result persists in higher order, as we shall argue below.
We will be looking into the perturbative solution of (21). The use of (21) insures that at least one propagator attached to each vertex is a retarded one and is smooth in infrared 4 We remind the reader that counterterms are implicitly included in V µν .
limit. Another important advantage of the Heisenberg representation is that the state vectors in Hilbert space remain constant and the evolution of the system is contained in the time dependence of the field operators. Therefore one may study the time evolution of the physical quantities such as, e.g., energy density as the expectation values of the corresponding field operators over the initial quantum state |in which we take to be a no-particle, de Sitter invariant state. The time evolution of this state is quite a complicated problem by itself since the notion of particle production is ambiguous in general relatively (for discussion see, e.g., [16]). In the Heisenberg picture we do not confront this problem.
Although analytically quite complicated, it is straightforward in principle to develop the perturbative solution of the integral equation (21) (See Fig. 1a.) Each three-point vertex carries a factor of 1/m P l , so, for consistency, we should also include in V µν higher-point vertices obtained from the expansion of Eq. (6) as well as vertices of the same order from higher dimensional operators implicitly included in the action Eq. (5). We will discuss that in more detail in a subsequent paper where we will also consider in greater detail the renormalization of the theory. Here we concentrate only on a possible existence of singularities at τ = 0. The contribution to ψ from the conformal Green's function reads where | r| = τ −τ ′ and the angular integration is made over directions of r. The nonconformal part coming from (18) contains two-terms, the first of the form of Eq. (23) and the second given by Our interest is in the VEV of this expression. Using translation invariance, one may simply set the spatial arguments to 0 to obtain The generic form of the loop expansion is illustrated in Fig. 1b. In calculating the VEV, one encounters quantum correlation functions of the type this is a much weaker singularity at τ = 0 than previously suggested, [10,11,12], this may nevertheless lead to a breakdown a perturbation theory and leave open the question of the ultimate future of the de Sitter metric.
It might worth emphasizing that it is not only the use of G R that softens the infrared behaviour but the form of V µν as well. With one exception, the nonconformal fields ψ ij enter the expression V µν in the form ψ ij,α , and the derivative renders their contributions infrared finite. The one exception is the second-to-last term in Eq. (22), which includes ψ ij Γ 0 ij /τ. The connection Γ 0 ij involves space derivatives of conformal modes ψ 0i as well as the time derivative of the nonconformal modes ψ ij,0 . Despite what one might think, ψ ij,0 can be shown to satisfy a conformal field equation, so that all these terms in Γ 0 ij involve an explicit factor of τ and satisfy the conformal integral equation. Thus, this term in V µν is a kind of cross product of a nonconformal field with a conformal field. It is only the expectation value of squares of conformal fields that manifest infrared problems, so one would not expect this to give trouble. In conclusion, there are no infrared divergences in quantities of interest to us here, and we may ignore for our purposes the fact that the true vacuum state is not de Sitter invariant.
Let us consider higher loop corrections to V (τ, 0) . The two-loop corrections take the generic form Here we have gone over to a mixed time-momentum representation, which is more convenient, since momentum is conserved, and power-counting is more familiar in momentum space. G R is, as before, the retarded Green function, while G 1 is the anticommutator function 5 given in Eq. (26). Each partial derivative in Eq. (28) is to be interpreted either as a derivative with respect to time or multiplication by one power of momentum. As mentioned previously, these remove the potential infrared problem normally associated with the presence of G 1 . With regard to the momentum integration, there will be various terms. In a mass-independent renormalization prescription, such as dimensional regularization with minimal subtraction, all divergences proportional to a power of the cutoff are completely removed by the counterterms. Thus, we need only concern ourselves with terms that are logarithmically divergent.
Such contributions cannot change the power behavior of the time dependence, so that we simply have the result that one would guess on the basis of dimensional analysis. Thus, at worst, F 2 ∼ (log |τ ′ |) p /τ ′2 for some power p. When inserted into the integral equation for 5 There are also two-loop contributions from the commutator function.
Thus it is safe to say that quantum correction may bring only logarithmic dependence on τ . Terms proportional to powers of ln |τ | are not inherently problems for perturbation theory. Consider, for example, the effect of a finite renormalization of the curvature, so that indicates in fact two distinct steps in the calculation. First, we calculate the one-loop effective action which involves acausal propagators. Then we iterate this action with the help of the retarded (i.e., classical) propagators. Our procedure so far was to look for log dependence of the generic term (28). We did not check that the result adds up to a general covariant function. In fact, as we will demonstrate now, this constraint may eliminate some infrared logs.
To this end let us consider the effective action generated by a scalar field of mass m and parameter ξ (see, e.g., chapter 6 of book [16]): where γ is Euler's constant, and Here R αβγδ , R αβ , R are the Riemann tensor, Ricci tensor and scalar curvature, respectively.
Since quantum part of the gravitational field, h µν , is a collection of fields with ξ = 0 or ξ = 1/6 (see above for a discussion,) Eq. (31) can be directly used to evaluate the effective action generated by quantum gravity. Note also that to regularize the expression in ultraviolet and in infrared one introduces both dimensional regularization, n = 4, and a nonvanishing mass, Eq (30) is valid in arbitrary background field. For our purposes we need to calculate R αβγδ for the metric specified by (4) and expand the result in ψ µν . What is specific about de Sitter background is that This means that in the limit n → 4 only anomalous terms survive where, by the anomalous terms, we now understand terms proportional to (n − 4) −1 . Thus the correct result for the variation is not zero but local terms.
This kind of technique is widely used to fix the trace anomaly (see, e.g., [16]) in arbitrary background. What is specific about the de Sitter background is that there are no "normal" terms in one-loop expectation value of the energy-momentum tensor T µν ; the whole of T µν is anomalous both for ξ = 1/6 and for ξ = 0. While considering logs piecewise, without collecting terms into R 2 or other invariants, we lose the property (32) which in fact extends to second derivative as well, as we argue next.
Indeed, what is special about the terms in the r.h.s. of eq (31) is that, upon multiplication by √ −g they do not depend on the Hubble constant H. Therefore δ n (δH) n ( √ gR 2 ) = 0 (33) for any n. The same is true for all the invariants quadratic in R αβγδ entering a 2 (x). On the other hand, in conformal coordinates, the variation of H is equivalent to a variation of metric proportional to the metric of de Sitter space itself: Thus there exists a direction along which even finite variation of g µν leaves √ −ga 2 (x) invariant. High symmetry of the de Sitter solution implies then that variations of √ −ga 2 (x) are strongly constrained for arbitrary change of g µν as well. In particular, the first variation is always proportional to g µν itself: where r(x) is function of coordinates. Since we know that the variation vanishes for the particular choice (34) of δg µν we conclude that in fact r(x) = 0.
For our purposes here we need the second variation of the effective action above. This vanishes again for n = 4. Indeed the second derivative generically takes the form: where N αβγδ is a tensor which in case of de Sitter space is constructed on g µν alone. Moreover it vanishes upon multiplying by the variation of the special form (34). It means, N αβγδ ∼ g γα g δβ − g γβ g δα (37) and the second derivative (36) vanishes for any variation δg µν symmetric in µν. Thus we have shown that the second variation of the action can produce only anomalous local terms.
Such terms are to be included in renormalization of the bare action and are not relevant to infrared logs. So far we relied on infrared regularization by finite mass. If we put the mass equal to zero from the very beginning, then one may argue [19] that one has ln(R) instead of ln(m). This form of the infrared cutoff does not modify our conclusions. Indeed, if one calculates variations of such an effective action, the result is again local terms.
To summarize, we have demonstrated that quantum corrections to the classical de Sitter solution generated by higher loops in quantum gravity can be at worst powers of logs of the conformal time τ . Moreover calculation of these logs is no longer a pure infrared problem.
We presented then further arguments based on consideration of the one-loop effective action according to which one-loop effects in de Sitter background reduce to renormalization of the action for ψ µν , with no logs involved. One needs also to include the effects of the FP ghosts as well as higher-point vertices of the same order in 1/m P l .
We would like to thank R. Woodard for interesting discussions and for communicating his results prior to publication. We are also grateful to   http://arxiv.org/ps/gr-qc/9403056v1