Abstract
Dark energy accounts for most of the matter-energy content of our universe, yet current theories of its origin rely on radical physical assumptions such as the holographic principle or controversial anthropic arguments. We give a better motivated explanation for dark energy, claiming that it arises from a small negative scalar-curvature present even in empty spacetime. The vacuum has this curvature because spacetime is fundamentally discrete and there are more ways for a discrete geometry to have negative curvature than positive. We explicitly compute this effect using a variant of the well known dynamical-triangulations (DT) model for quantum gravity. Our model predicts a time-varying non-zero cosmological constant with a current value, 
in natural units, in agreement with observation. This calculation is made possible by a novel characterization of the possible DT action values combined with numerical evidence concerning their degeneracies.
Citation: Trout AD (2013) Dark Energy from Discrete Spacetime. PLoS ONE 8(12):
e80826.
https://doi.org/10.1371/journal.pone.0080826
Editor: Garret Cotter, University of Oxford, United Kingdom
Received: July 19, 2013; Accepted: October 15, 2013; Published: December 3, 2013
Copyright: © 2013 Aaron D. Trout. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The author has no support or funding to report beyond his university salary.
Competing interests: The author has declared that no competing interests exist.
Introduction
Multiple independent sets of empirical data [1]–[4] indicate that about 70% of the matter and energy in our universe comes from a mysterious repulsive gravitational effect known as “dark energy”. Understanding the origin of this energy is one of the most important problems in physics. Our only current theories involve speculative physical assumptions or finely tuned parameters. One popular assumption is the holographic principle: the idea that the degrees of freedom in a region of space are encoded on the region’s boundary [5]–[14]. Other explanations assume the existence of exotic matter fields or modify the Lagrangian defining general relativity. One recent theory [15] even hypothesizes a connection between dark matter and dark energy. See [16], [17] for reviews of various explanations of dark energy.
Our work provides a simpler, better motivated model for dark energy set within the well-known dynamical triangulations (DT) approach to quantum gravity. This model assumes no holographic principle, uses no additional matter fields or finely tuned parameters, and does not modify general relativity beyond the geometric discretization inherent in dynamical-triangulations spacetimes. In our model, a positive vacuum energy of the correct observed magnitude spontaneously arises from the entropic bias toward negative curvature states inherent in DT geometries. Note that treating gravity as an emergent mean-field phenomenon driven by entropic forces is a popular research perspective at the moment [18]–[24].
A reasonable prediction for dark-energy within a quantum-gravity theory is only significant if the theory approximates general relativity well at large distances. Why should we believe this about a theory that uses DT spacetime states? The progenitor of the DT theory, called the Regge calculus, has been used successfully in numerical general-relativity and quantum gravity for nearly five decades [25]–[31]. The DT model itself [32]–[36] and its descendent, causal dynamical triangulations (CDT) [32], [34], [35], [37]–[40] have been studied for nearly two decades. The numerous successes achieved by these theories give confidence that our model can describe general relativity at length-scales much larger than Planck’s length.
The model presented in this paper uses the same discretization of geometry and the same action as the DT theory. However, it is not identical to DT because it puts restrictions on the set of triangulations which contribute to the partition function. These kind of restrictions are also what distinguish DT from CDT although our restrictions are distinct from those in CDT. Note that it is not our purpose to advocate “triangulations” as the ultimate structure of spacetime. Indeed, in our calculation the discrete nature of geometry may be removed at the end without altering the predicted vacuum energy. We suspect that the effect described in this paper is actually a generic feature of any quantum-gravity theory which predicts a discrete spacetime geometry and which has general relativity as its large-distance limit.
Background Material
General relativity can be written in the Lagrangian formalism using the Einstein-Hilbert action, which in natural units is
(1)
Here 
is a closed 
-manifold, 
a Lorentzian metric, 
scalar-curvature, 
the cosmological constant, 
the Lagrangian for matter and 
the standard volume element. See Table 1 for a list of commonly used symbols. Note, both 
and 
depend on 
while 
does not. Also note that 
is the only term in this action with a physically distinguished zero value. In quantum field theory on a fixed background geometry, an arbitrary constant can be added to 
without changing the observed physics, allowing one to simply set 
to zero. Thus, it is reasonable to argue, as we do in this paper, that the observed non-zero value of 
arises from quantum effects related to the scalar-curvature field 
.
Hilbert and Einstein showed that the critical points 
of this action satisfy the equations of motion
(2)
These are, of course, the field equations for general relativity. Here, 
is the Ricci curvature tensor and 
the stress-energy tensor for matter. In this work we restrict attention to the Einstein-Hilbert action for the vacuum with zero cosmological constant
(3)
The critical points of 
are metrics which satisfy the vacuum field equations. These metrics are Ricci flat everywhere (
at every point) and therefore also scalar flat everywhere (
at every point.) Thus these metrics have action exactly zero. Finally, in dimensions less than four, the Ricci tensor determines the full curvature tensor 
, so critical points of 
in these dimensions must actually be flat everywhere (
at each point.).
In his influential 1961 paper [25] Regge proposed a discretized version of 
which applies to triangulated piecewise-linear (PL) manifolds. A triangulation 
of a closed 
-manifold 
is a combinatorial 
-manifold homeomorphic to 
given as an abstract simplicial complex. Assigning a length 
to each edge in 
uniquely defines a piecewise-linear metric on 
provided these lengths satisfy some natural compatibility conditions. If we let 
denote the number of 
-simplices in 
, the Regge action is given by
(4)
In this equation, the sum runs over all codimension-2 simplices of 
(called hinges), 
is the total dihedral angle around the hinge 
, and 
is that hinge’s volume. It is easy to insert a cosmological constant into this action, although here we do not. The possibility of incorporating matter fields into 
is a currently active topic of research. See [41]–[43].
Note that 
has a nice geometric interpretation. The summand in this action is the angle defect in a small triangle enclosing and perpendicular to the hinge 
, weighted by the volume of that hinge. Given the close relationship in classical non-euclidean geometry between angle defect and curvature, it is natural to interpret 
as a discrete measure of total curvature. Because of the success of the Regge action in describing general relativity, we will interpret 
as a discrete measure corresponding to the Einstein-Hilbert action, and thus to total scalar-curvature. Interpreting 
as a total curvature is also supported by the fact that, like point-wise curvature bounds in Riemannian geometry, bounds on the angle-defect for all hinges have profound topological consequences for 
. See [44]–[47] for examples.
The Dynamical-Triangulation Action
Suppose we fix the abstract simplicial complex 
and consider 
as a function of the edge-lengths 
only. There is a large body of numerical evidence [26]–[29] that the critical points of this action define PL-metrics which behave like solutions 
to the vacuum field equations, at least at length scales much larger than the maximum edge-length. See [36] for a overview of this work, known as the Regge calculus. In this paper, however, we will require all edges to have a single fixed length 
so that the action is determined only by the structure of 
as an abstract simplicial complex, i.e. only on the way the simplices in 
are attached together. This form of the Regge action has been studied extensively as part of the dynamical triangulations (DT) approach to quantum gravity. We write this action as
(5)where 
is the volume of a 
-simplex with all edges of length 
, 
is the dihedral angle in such a simplex, and 
, called the degree of 
, is the number of 
-simplices in 
with 
as a face. Usually, we will suppress the dependence on 
and write simply 
.
Now for some terminology and preliminary results. Let 
denote the set of all triangulations of a fixed closed 
-manifold 
. We will write 
for the set of all triangulations of 
containing exactly 
-simplices, and 
for those with 
-simplices and DT-action 
. Since there are only finitely many ways to attach together the faces of a finite collection of 
-simplices, 
and 
are finite sets. We define 
to be the spacetime entropy of 
for 
-simplices and action 
. We will also need notation for the average hinge degree of a triangulation T,
(6)
By double-counting arguments we may alternately write this as
(7)
Proof. Suppose we examine each 
-simplex 
in 
and place a mark on each 
-simplex with 
as a face. Clearly we have placed 
marks. On the other hand, each 
-simplex has 
codimension-2 faces, so the number of marks is also 
. Dividing through by 
gives the first equality. Next, suppose we examine each 
-simplex 
in 
and place a mark on each of the two 
-simplices incident at 
. We have obviously placed 
marks. However, each 
-simplex has 
codimension-1 faces, so the number of marks is also 
and we have 
. Plugging into the previous equality and simplifying finishes the proof.
The first part of equation (7) lets us nicely express 
as a function of the number of 
-simplices in 
and its average hinge-degree. We get
(8)where 
is called the flat hinge-degree. Why do we call 
the flat hinge-degree? It is the number of regular 
-simplices needed around a hinge to provide a total dihedral angle of exactly 
, the expected quantity in a flat space. Note that, except in dimension two (where 
) the quantity 
is not an integer.
Proof of Equation (8). We begin with the DT action (5) and distribute the sum into the summand to obtain
(9)
By equation (7) we can replace 
with 
and the summation by 
to get
(10)
Finally, moving a factor of 
to the front finishes the derivation.
Action Spectrum in Dimension Three
From this point forward, we will restrict attention to dimension three. What can we say about the possible values of 
on 
when 
? This is a formidable problem, since even for a small number of tetrahedra 
the set 
is quite large and complicated. We begin with an elementary result: for any triangulation 
of a closed 3-manifold 
we have
(15)where 
and 
. This means that for a fixed number of 3-simplices, the effect of increasing 
(or equivalently, decreasing 
) is to decrease both the number of vertices 
and the number of edges 
in the triangulation.
Proof of Equation (15). We begin with a well-known topological fact: every closed 3-manifold has Euler characteristic zero. That is, for any triangulation 
of a closed 3-manifold 
we have 
. Now, we use equation (7) to replace 
by 
to get
(16)
Using equation (7) again to replace 
by 
and then rearranging gives
(17)as desired. Finally, we plug this 
back into equation (16) and simplify to obtain
(18)completing the proof.
Equation (15) tells us that to understand the possible values for 
we must understand the possible combinations of 
and 
that can occur in a triangulation of a given closed 3-manifold. A 1970 paper [48] by Walkup tells us all we need to know.
Theorem (Walkup). For every closed 3-manifold 
there is a smallest integer 
so that any two positive integers 
and 
which satisfy
(19)are given by 
and 
for some 
. The quantity 
is a topological invariant which satisfies 
for all closed 3-manifolds 
.
Note that 
is known for many manifolds 
, see [49], although we will not need this information.
Walkup’s Theorem, together with equation (15) and some algebra suffice to prove the central mathematical result in this paper:
Theorem. Let 
be a closed 3-manifold and 
a fixed number of tetrahedra. Then, there are mean actions
(20)and
(21)so that if 
is an integer for which 
lies in the interval 
then 
for some triangulation 
of 
with 
tetrahedra and 
edges. These 
are regularly spaced over the entire interval 
, each separated from the next by
(22)where 
. This is the smallest possible separation given fixed 
, so these 
are all possible mean-actions on 
.
Note that in most applications, the number of tetrahedra 
will be large and the energy densities given in the theorem will be approximately
(23)
Also note that when edges are Planck’s length (
in our units) the magnitude of these energy densities is enormous, about 
Joules per cubic meter.
Proof of Main Theorem. Let 
be a closed 3-manifold. We start by showing that if two given integers 
and 
satisfy
(24)then there is some triangulation 
of 
with 
and 
. We define 
. Note that 
by the first inequality in (24). A bit of algebra applied to the second inequality in (24) implies
(25)Now, consider the upward opening parabola 
which has largest root 
. The first inequality in (24) implies 
which is just 
. Since 
is the largest root of an upward opening parabola, we conclude 
. By our definition of 
and 
, this tells us
(26)
By Walkup’s theorem, inequalities (25) and (26) imply that some triangulation 
has 
and 
. Finally, by equation (15), we know 
as desired.
Next, we divide the inequality (24) by 
and take reciprocals to get
(27)where 
. Thus, if 
is fixed and 
is an integer for which 
lies in this interval, then 
for some triangulation 
with 
tetrahedra. By equation (12) the change in mean-action for each increment of 
is as claimed in equation (22), completing the proof.
The N-Action Model
The model used in this paper is designed to be dominated by states near a particular chosen target value 
for the mean-action. For a fixed number of tetrahedra 
let 
be the closest attainable mean-action to 
. For each 
, our model admits triangulations with mean-action 
along with those having one of the 
mean-action values on either side of 
. In this paper our target mean-action will be 
since the Einstein-Hilbert action for the vacuum in classical general-relativity is zero. Recall that, unlike actions in quantum field theory, the values of the Einstein-Hilbert and Regge actions are well-defined physical observables. This makes such a targeting strategy physically reasonable.
Why not simply start with a model containing only those triangulation 
for which 
? It turns out that there are no such triangulations. That is, for any triangulation 
of a closed 3-manifold 
we have 
, or equivalently 
. This follows from the irrationality of 
and equation (12). We know 
is irrational due to work [50] by Conway, Radin and Sadun on what are called called geodetic angles. Note that these angles are actually interesting mathematical objects on their own and are central to the solution to Hilbert’s third problem on the scissor-congruence of polyhedra.
So, let 
be the mean-actions in the model and 
the corresponding total actions. Our main theorem implies that for any 
and spacetime volume 
there is an 
small enough so that all of the 
mean-action values 
lie within the range 
where attainable action-values are regularly spaced. For such 
our model has partition function
(28)where 
is spacetime entropy at action 
. The expected action for this model is then
(29)A Euclidean version 
of this expected value can be found by applying the standard Wick rotation 
to the expression above.
It is currently impossible to write 
or 
as exact closed form expressions since the entropies 
are beyond our ability to compute. However, if we replace 
with its first order approximation 
for 
a constant, then a closed-form expression can be found. We used the computer-algebra package Mathematica to show
(30)
A closed form expression for 
can obtained as before by replacing 
with 
in the equation above.
Choosing 
How are we to choose 
? In an ideal world, we would have in hand a fully formed DT-style theory of quantum gravity coupled to matter, which provably reduced to general-relativity at large distances. From this theory we could derive an appropriate 
by computing how far a typical spacetime was from the classical action. We believe such a theory will eventually emerge, but it is not yet available. However, we have set up enough machinery to reasonably guess what such a theory would tell us about 
.
Suppose we fix a total spacetime volume 
and consider the 
-action theory targeting mean-action zero. What happens as we let the edge-length 
approach zero? Because the separation between actions 
goes to zero and 
, if 
is left fixed as 
then even the most extreme action values in the theory, 
, would converge to zero. Since we wish to investigate quantum gravity, this is unacceptable and we are forced to choose an 
which diverges as 
. Now, suppose we make the affine entropy approximation 
. Equation (30) implies that if 
then for large enough 
and small enough 
the expected action is dominated by the final hyperbolic cotangent term and we have 
. This tells us that under these conditions, the model is completely dominated by entropy. The oscillating complex phase 
which suppresses the contribution of states far from 
is swamped by the entropy term involving 
.
Thus, since 
is proportional to 
, it is natural to choose the dimensionless 
to be proportional to 
. For such a choice we can take the 
limit and the theory gives a finite non-zero value for the expected action. Therefore, we choose to use
(31)mean-action values on either side of 
. Notice that by the approximations (23) even though 
diverges as 
, all actions in the model eventually lie within the “regularly spaced” range 
for small enough 
. Also note that as 
all the mean edge-degrees corresponding to these 
converge to the flat mean edge-degree 
.
Finally, for any fixed 
we can use equation (30) to compute the 
limit, obtaining
(32)
For 
we get a purely imaginary standard expectation 
and a Euclidean expectation given by 
Evidence for the Entropies 
The calculation of the expected action as 
given by equation (32) depends on two assumptions about the entropies 
. First, for the states contributing to the model, spacetime entropy must be an approximately linear function of mean-action, i.e. 
, at least for large enough 
. Second, this 
must not approach zero as 
. In this section, we present evidence from Monte-Carlo simulations and small-
enumerations that strongly supports these assumptions.
Monte-Carlo Sampling Results
To measure the dependence of entropy on mean-action we use a Metropolis-Hastings algorithm to take samples 
from 
near a given number of tetrahedra and mean-action. The algorithm wanders among the elements of 
by using the well-known Pachner moves to change from one triangulation to another, repeatedly choosing a random move and executing it with probability 
where 
is some non-negative objective function. Metropolis proved that if we wait long enough between samples, then each sample 
occurs with probability 
. Here, we use a quadratic objective function
(33)with 
and 
fixed constants. This form for 
keeps the sampled triangulations near a target mean-action 
and number of tetrahedra 
.
If there were equally many triangulations at each 
and 
then our sampled pairs 
would form a Gaussian distribution centered at the target point 
. If our samples have a Gaussian distribution but with mean 
significantly displaced from the target, this indicates a linear dependence of spacetime entropy on 
and 
with the magnitude of the dependence proportional to the size of this displacement. Since it is obvious that spacetime entropy is strongly dependent on 
and because the relative deviation from the mean for 
is at most 
in our data, we focus solely on deviation in mean-action 
. From this we can estimate the change entropy (per mean-action step) 
, in nats, using
(34)
The sampling trials conducted for this paper use the 3-sphere 
with target mean-action zero (
) and various targets for the number of tetrahedra 
. In all cases, we take 
and 
. In order to ensure independent samples, the algorithm attempts Pachner moves until 
accepted moves per tetrahedron have occurred. We checked that this wait time was sufficient using standard correlation tests. For these parameters, each sample was uncorrelated from the next. We also checked that the sampled 
and 
were independent. As desired, samples are approximately normally distributed with sample mean 
somewhat displaced from the target 
. This indicates that entropy is approximately a linear function of mean-action near 
, as was assumed in the previous section. See Figure 1 for a histogram of mean-actions for 
samples at 
. For each such distribution, we use equation (34) to infer the approximate change in entropy 
between mean-actions. These 
are comfortably negative and do not appear to approach zero as 
gets larger, validating our second assumption. See Figure 2. Copies of the code used for triangulation sampling are available on request.
The Origin of Dark Energy
Taking 
and dividing through by 
in equation (32) gives
(35)
Let us briefly discuss the physical meaning of 
. Our goal was to construct a theory dominated by states close to the classical value of the mean-action, 
. We did this by “slicing” the partition function according to action-value, retaining only states whose actions lie within a certain distance of zero. If the volume of spacetime is large compared to Planck’s volume then we come very close to accomplishing our goal. That is, for 
we do indeed obtain 
in the 
limit. However, there is a small perturbation away from zero because of the relative entropy of action values. Notice that since action values are global observables, this effect is independent of the local details of the “metric”, i.e. the local structure of the triangulation. This leads us to expect that, for a typical triangulation at a given 
, the average action will appear very uniform at length-scales much larger than 
. Finally, recall that everything in the Einstein-Hilbert action except the cosmological constant 
depends on the metric 
. Thus, the basic structure of 
almost demands we interpret our non-zero 
as an emergent cosmological constant given by
(36)
We now turn to the question of applying this result to our own universe. This is a somewhat speculative endeavor since our world appears to be both 
-dimensional and infinite in extent. However, as an entropic effect connected with the pattern of attachment between simplices, we expect the perturbation away from 
identified in this paper to occur quite generally. So, what 
is appropriate for assessing the magnitude of this effect in our particular universe? Considerations of causality give us a reasonable answer: take the volume of space which has had time to causally communicate with our point of observation. That is, we ought to use something like the current Hubble volume 
where 
is the Hubble constant. Plugging in 
in Planck units gives
(37)
which is in general agreement with observation.
At this point, we feel obliged to briefly discuss the term “numerology”. It has long been known that the observed cosmological constant was approximately 
. This and many other unexplained approximate numerical relationships between cosmological parameters are often called large number coincidences. Thinking of them as having explanatory power on their own is surely deserving of the label “numerology”. However, this epithet should not be applied to a physically well-motivated theory which predicts ab-initio such a numerical relationship, as our model does.
Discussion
Our derivation of 
has some interesting features. Using the Hubble parameter to define our characteristic volume 
means that the model actually predicts a time-varying cosmological constant
(38)
where 
is the Hubble parameter at proper time 
. That is, we predict that 
scales like the area of the cosmic horizon. Amazingly, although we made no holographic assumption, this is the same behavior that emerges from holographic dark energy (HDE) theories [5], [6], [10], [12], [13]. In fact, our model shares several other key features with these approaches, including the presence of two “cut-offs” in the theory which are removed in a coordinated fashion. HDE models typically contain both a UV and IR field cut-off which are removed in a way that saturates entropy in the Bekenstein bound. In our theory, the cut-offs 
and 
are chosen to keep the entropic perturbation on 
bounded as 
. While HDE theories are very different in detail from our model, the broad similarities are quite striking. Perhaps both approaches are pointing to the same underlying physical issues. We hope that the relative simplicity of our model can help elucidate these issues.
We should also mention another explanation for 
which shares some features with our approach. In [53] it is argued that the true ground-state vacuum has 
but that we observe 
because the universe has not yet had time to decay into this ground state. The author considers a model in which the true ground state is given by the superposition of two degenerate 
states, one of which describes the universe’s present-day vacuum. Since the decay probability in a given volume and time period is related to the energy density 
, the requirement that no such decay has yet happened in the Hubble volume provides an estimate for 
which agrees with observation. This argument leads, as does our model, to a connection between the Hubble parameter and 
. Also note that both models contain states at or near 
which are suppressed compared to the 
states.
Finally, we note that in the very early universe our model predicts large 
and hence rapid expansion. This raises the tantalizing possibility that big-bang inflation and dark-energy are manifestations of a common effect, though it is likely that a more sophisticated choice for the characteristic volume 
would be needed. See [54] for consideration of this idea in the HDE context.
Acknowledgments
The author would like to thank Cheryl Koester, C. Scott Wylie, Vadas Gintautas, Joe MacNeil, and Larry Viehland for their support and valuable advice. Also, many thanks to Henry Segerman for providing the triangulation census data used in this work and to Leah Langer for assisting in the collection and verification of triangulation sampling data.
Author Contributions
Conceived and designed the experiments: ADT. Performed the experiments: ADT. Analyzed the data: ADT. Wrote the paper: ADT.
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