Hiding the cosmological constant

Perhaps standard effective field theory arguments are right, and vacuum fluctuations really do generate a huge cosmological constant. I show that if one does not assume homogeneity and an arrow of time at the Planck scale, a very large class of general relativistic initial data exhibit expansions, shears, and curvatures that are enormous at small scales, but quickly average to zero macroscopically. Subsequent evolution is more complex, but I argue that quantum fluctuations may preserve these properties. The resulting picture is a version of Wheeler's `spacetime foam,' in which the cosmological constant produces high curvature at the Planck scale but is nearly invisible at observable scales.


The cosmological constant problem
Quantum fluctuations of the vacuum are believed to generate very high energy densities, which should manifest themselves as an enormous cosmological constant. We do not know how to calculate this quantity exactly, and it remains possible that it is suppressed, exponentially [1] or otherwise [2], but standard effective field theory arguments predict a value Λ ∼ ±1/ℓ 2 , where the cut-off length is usually taken to be the Planck length ℓ P [3,4]. The sign of Λ depends on the exact particle content of the universe-bosons and fermions contribute with opposite signs-but unless a remarkable cancellation occurs, the predicted value is huge.
We do, in fact, observe an accelerated expansion of the universe that could be explained by a cosmological constant. But for this, the Planck-scale estimate is some 120 orders of magnitude too large, making it what has been called "the worst theoretical prediction in the history of physics" [5]. It is generally assumed that Λ must either be canceled by incredibly precise fine tuning or eliminated by some other form of special pleading: anthropic selection [6], nonlocal modifications of the gravitational action [7], or the like. The problem is especially intractable because of the mixture of scales: Λ is generated near the Planck scale, but observed at cosmological scales.
Here I propose a simple but radical alternative. Perhaps our universe really does have a cosmological constant of order 1/ℓ 2 P . If one assumes homogeneity, this is, of course, immediately ruled out by observation. But if Λ is generated by Planck scale quantum fluctuations, there is no reason to expect homogeneity at that scale. This notion was anticipated by Wheeler [8], who called the resulting picture "spacetime foam." For a Riemannian space-one with a positive definite metric-it is easy to imagine high curvature at small scales that averages to zero macroscopically. For a spacetime, though, one might worry that a cosmological constant entails exponential expansion (at least is the anisotropy is not too big [9]), and it is not obvious how such behavior can be averaged away. But general relativity is time reversal invariant; for every expanding solution there is a corresponding contracting solution, and these may indeed compensate.
In what follows, I will make this idea more concrete. Starting with the initial value formulation of general relativity with a large cosmological constant, I demonstrate that a very large class of initial data has a local Hubble constant that is huge at the Planck scale but tiny macroscopically. For an infinite subset of data, the macroscopic spatial curvature is also very small, and has a vanishing first time derivative. It is worth emphasizing that a "macroscopic" average need not require a very large volume: a cubic centimeter contains some 10 100 Planck-size regions.
An initial value formulation is not enough, of course. One must also show that these features are preserved dynamically. Higher order time derivatives depend on finer details, and while they are also strongly suppressed, it is not clear whether this is enough. If the initial inhomogeneity is generated by quantum fluctuations, though, I argue that these fluctuations should also preserve the crucial properties that camouflage the cosmological constant.
These arguments do not provide a complete answer to the cosmological constant problem. They do not, for example, explain the apparent existence of a very small Λ at macroscopic scales. More generally, one would have to show that long wavelength excitations on this foamlike background obey a macroscopic version of the Einstein field equations, a form of the muchstudied but unresolved "averaging problem" [10]. But the results here suggest, at least, that we may have been looking for answers at the wrong scales.

The initial value formulation
Let Σ be a compact three-dimensional manifold, interpreted as a Cauchy surface of a spacetime. The initial data for general relativity on Σ consist of a spatial metric g ij and an extrinsic curvature K i j . These are not arbitrary, but must satisfy the constraints where R is the scalar curvature of the metric g ij , D i is the covariant derivative compatible with the metric, K = K i i , and I have assumed that the matter stress-energy tensor is negligible compared to the cosmological constant. This is the classical formalism that translates most natural into a canonical quantum theory; the constraint (2.1a) becomes the Wheeler-DeWitt equation, while (2.1b) implies spatial diffeomorphism invariance.
The dynamical evolution of this data is described by the equations where L n is the Lie derivative along the unit normal to Σ, essentially a covariant time derivative, and N is the lapse function. It can be useful to split off the trace of the extrinsic curvature, writing Then K is the expansion; by (2.2a), it is the local Hubble constant, the logarithmic derivative of the volume element. The quantity σ i j is the shear tensor; the shear scalar is σ 2 = 1 2 σ i j σ j i . We will need two properties of the initial value formulation: 1. The equations are time reversal invariant: if (g, K) is allowed initial data for a manifold Σ, so is (g, −K).
2. Two manifolds Σ 1 and Σ 2 with initial data (g 1 , K 1 ) and (g 2 , K 2 ) can be "glued" to form a manifold Σ 1 #Σ 2 for which the initial data is unchanged outside arbitrarily small neighborhoods of the points where the gluing is performed [11,12].
More precisely, topologically Σ 1 #Σ 2 is the connected sum of Σ 1 and Σ 2 , the manifold formed by cutting balls out of Σ 1 and Σ 2 and identifying the boundaries. Geometrically, pick open sets U 1 ⊂ Σ 1 and U 2 ⊂ Σ 2 , restricted only by the (generic) condition that the initial data in each set is "not too symmetric," in the technical sense that the domains of dependence of U 1 and U 2 contain no Killing vectors. Pick points p 1 ∈ U 1 and p 2 ∈ U 2 , cut geodesic balls B 1 and B 2 of arbitrarily small radius ǫ around each, and join the boundaries. Then Σ 1 #Σ 2 admits initial data, still satisfying the constraints, that exactly coincides with the original data outside U 1 ∪ U 2 and is close to the original data, in a suitable norm, inside We start with a preliminary construction. Pick a three-manifold Σ with a fixed open set U and a point p ∈ U , and specify initial data (g, K). LetΣ be an identical copy of Σ, but with initial data (g, −K). Glue the two manifolds symmetrically at p to form a connected sum Σ = Σ#Σ. By symmetry, Σ will have an isometry (g, K) → (g, −K). While the definition of an averaged tensor has ambiguities [10], it is clear that any average that respects this symmetry will give K i j = 0. Now consider the more general case of a large collection of manifolds Σ 1 , Σ 2 , . . . , Σ N , each with its own initial data (g α , K α ). Form the glued manifold In general, the extrinsic curvatures of the components will have no particular relationship, but for any particular Σ α , the data (g, K) and (g, −K) are equally likely, so, again, any sensible average over a large enough number of components should give K i j = 0. Exactly how fast the average will go to zero depends on the number and distribution of manifolds and initial data sets, but, as noted above, a cubic centimeter already contains some 10 100 Planck-size regions.
These results imply that L n √ g = 0 and σ i j = 0. It is also easy to check that L n R = 0. To first order, the averaged spatial geometry is thus stationary and homogeneous. To match our universe, we would also like the average spatial curvature to be small. The only restriction comes from averaging the constraint (2.1a): Evidently if K 2 is large (for positive Λ) or σ 2 is large (for negative Λ), the cosmological constant can be "absorbed" in fluctuations of extrinsic curvature. Let me stress that I am not starting with a spacetime and searching for a special hypersurface on which K i j = 0. That would be an artificial procedure, and there would be no reason to expect such a hypersurface to be physically interesting. Rather, I am taking an arbitrary hypersurface and giving it initial data chosen randomly from a large collection. Further requirements may be added to make this data "nice," but as long as these allow Planck-scale inhomogeneity and do not pick out a microscopic arrow of time, the outcome will not change.
This construction allows Σ to have an arbitrarily complicated topology. Indeed, any orientable compact three-manifold has a unique decomposition as a connected sum of "prime" manifolds [13]. But Σ may also be topologically trivia: if each Σ α is a three-sphere, the connected sum is also a three-sphere. We can thus reach a large set of initial data by starting with any initial data, cutting out a collection of Planck-size balls, changing the data on the balls, and gluing them back. The geometry of the "necks" connecting components in this construction is rather special, though, and it is an open question how much of the space of initial data can be reached this way.
For the special case of local spherical symmetry, one can perform a similar construction much more explicitly [14]. This case is "too symmetric" to meet the genericity condition for the gluing theorem. Nevertheless, it is possible to explicitly construct a "ring" of three-spheres with arbitrary signs of K, though more general gluings typically require breaking the symmetry or making more nonlocal changes in initial data.

Evolution
We have established that on an initial hypersurface, a large class of initial data can hide the macroscopic effects of a cosmological constant. But is this feature preserved in time?
This is inherently a quantum mechanical question. Strictly classical evolution of an initial data set (2.4) will typically lead to singularities or Cauchy horizons, and there is strong independent evidence that Planck-scale fluctuations can disrupt the causal structure of spacetime [15], requiring a quantum treatment. But more than that, the putative Planck-scale structure in the initial data presumably originates from the same vacuum fluctuations that generate the cosmological constant. These fluctuations know nothing about about our particular choice of an initial surface, and should thus reproduce the same structure on arbitrary time slices. Hence a proper, fully quantum mechanical treatment of evolution should, to a good approximation, preserve the condition K i j = 0. We do not, of course, know how to do a full quantum gravitational calculation. As a first step, we can ask to what extent the classical evolution (2.2a)-(2.2b) preserves the averaged structure. This is not unlike Buchert's question, in a somewhat different context [16], of whether a non-equilibrium "cosmic equation of state" can lead to a stationary averaged configuration.
Unfortunately, any attempt answer this question must confront deep ambiguities in defining time derivatives of averages. First, the derivative of any average • over a region U will depend on how U changes in time. If U is fixed in terms of some set of coordinates, the result will not be invariant; if it is defined in terms of geometric quantities, it will typically be timedependent. Second, averages are often (although not always [17]) defined in term of integrals with a dynamical integration measure, which provides another source of time dependence [18]. Together, these features imply that differentiation and averaging do not commute. To make things worse, in the absence of a preferred coordinate, it is not even clear what "time" should be used to define a derivative. The derivative with respect to proper time, for instance, is 1 N ∂ t , where the lapse function N depends on position and cannot simply be passed into an integral.
A preliminary question might be whether any classical evolution-any choice of a (positive) lapse function N -preserves the condition K = 0. Let us follow [18] and define an average Then if the region U is somehow defined in a time-independent way, we have As long as the integrand doesn't have a definite sign-that is, as long as the shear (for positive Λ) or expansion (for negative Λ) is large in some regions-there will be an infinite number of choices of lapse functions for which the right-hand side of (3.2) vanishes. Furthermore, the quantity multiplying N in the integrand is invariant under (g, K) → (g, −K). By choosing N to have the same symmetry, we can ensure that any integrals containing odd powers of the extrinsic curvature will continue to vanish, even if they also contain factors of N .
One might worry that the condition N (R − 3Λ) = 0 could force average curvature R to be large, in conflict with observation. If N were required to be constant, this would be true. But the lapse function is a part of the spacetime metric, and should itself have Planck-scale structure, so N R can be very different from R .
The second derivative is also simple: The last term contains an odd power of K, and vanishes for U large. The first term is of exactly the same form as (3.2), andṄ can be specified independently, so if the first derivative can be made to vanish, the second derivative can as well.
Higher derivatives are more complicated. L 3 n K, for instance, contains derivative terms like K∆K and higher order correlations like K 4 − K 2 2 , which probe shorter distances. But each new derivative also comes with a new time derivative of N , which can be specified independently. There is thus no obvious obstruction to choosing a time-slicing for which K = 0 on every slice.
Whether such a choice is "natural" is a different question. In classical general relativity, quantities like the expansion are coordinate-dependent. K is not completely arbitrary-many spacetimes admit no time slice with K = 0, for instance-but it is far from unique. But as I have argued, if Planck-scale fluctuations of K are caused by quantum fluctuations of the vacuum, slices with K = 0 are "typical," and in that sense, this choice of time-slicing is natural. *

What this proposal does, and does not, do
As early as 1957, Wheeler argued that . . . it is essential to allow for fluctuations in the metric and gravitational interactions in any proper treatment of the compensation problem-the problem of compensation of "infinite" energies that is so central to the physics of fields and particles [19].
What I have proposed here is a concrete realization of this vision. Several previous attempts have been made to model spacetime foam-see, for instance, [20][21][22]-but only a few have addressed the cosmological constant problem [23][24][25][26]. The new ingredients here are the ability to construct explicit initial data and the crucial realization that time reversal invariance allows, and perhaps even requires, the expansion and shear to average to zero.
This proposal addresses the "old" cosmological constant problem, the problem of extremely large vacuum energy. It does not answer the question whether the observed accelerated expansion of the universe is caused by a small residual cosmological constant. It has been proposed elsewhere that higher order correlations of vacuum fluctuations may produce a small cosmological constant [27,28]; these would presumably show up here in higher correlations of the metric and extrinsic curvature, which appear in higher derivatives of averaged expansion and curvature.
While the proposal offers a natural explanation for small macroscopic expansion and shear, the requirement of small spatial curvature seems more arbitrary. It is certainly possible to choose data for which R is small, and there are hints that this may be preferred by the gravitational partition function, but a better understanding is needed. Of course, the answer may be dynamical. In a standard closed FLRW cosmology, after all, the spatial curvature is initially very high and decreases in time. There is some evidence that the same is true here: the second time derivative of the averaged curvature R can be calculated in the manner of the preceeding section, and while the result depends on the lapse function, most of the terms are negative definite.
The proposal also does not attempt to explain the emergence of a macroscopic arrow of time, an important and complicated question but one that is probably logically independent. Nor have I shown that long wavelength disturbances sitting on top of Planck-scale spacetime foam will be described by classical general relativity. This is the notorious "averaging problem" [10,17,18], the problem of how the nonlinearities of general relativity interact with the process of taking averages. Here, though, effective field theory arguments may help [3]. Nothing in this construction has broken spatial diffeomorphism invariance, so at a minimum the effective action should involve only terms invariant under that symmetry. This implies a Hořava-Lifshitz action [29], of which the action of general relativity is a special case. If, as I have argued, there is also nothing "preferred" about the initial time slice, then time reparametrization invariance should also be a symmetry, and the large scale effective action should take the usual Einstein-Hilbert form.
So far, I have treated a quantum gravitational problem semiclassically, appealing to quantum mechanics to generate Planck-scale structure but otherwise relying on classical general relativity to describe constraints and evolution. We might next consider coherent states centered on the configurations described here, and construct more general wave functions as superpositions. But this would force us to confront some of the standard problems of quantum gravity: the metric and extrinsic curvature are not true observables, and to average we would have to figure out what "at the same point" means in different components of the wave function.
As noted earlier, there is another serious ambiguity, even at the classical level, in averaging time derivatives. While I have taken a step toward addressing this problem, ultimately it will be necessary to directly connect the geometric variables used here to large scale physical observables, a task that may require embedding a foam-like model in a macroscopic cosmology.
An interesting set of technical questions remain as well. The gluing construction I have used provides a large set of initial data, but it would be valuable to understand just how much of the total space of initial data is covered. More generally, gluing is certainly not the only way to produce data with no arrow of time at the Planck scale, and a full understanding of the measure such data is still lacking. It would also be useful to further investigate higher order correlations, or, conversely, to see to what extent further restrictions (e.g., L 3 n K = 0) limit the possible initial data.
For all these limitations, though, this proposal suggests a simple and radical solution to a deep and long-standing problem. If a large cosmological constant is generated by Planck scale fluctuations of the vacuum, then maybe the Planck scale is the place to look for answers. I have shown that at least in principle, hiding a Planck scale cosmological constant in Planck scale curvature fluctuations is not only possible, but can be quite natural. Perhaps we have simply been looking in the wrong place.