Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets

Abstract

A large class of the dynamical laws for causal sets described by a classical process of sequential growth yield a cyclic universe, whose cycles of expansion and contraction are punctuated by single `origin elements' of the causal set. We present evidence that the effective dynamics of the immediate future of one of these origin elements, within the context of the sequential growth dynamics, yields an initial period of de Sitter like exponential expansion, and argue that the resulting picture has many attractive features as a model of the early universe, with the potential to solve some of the standard model puzzles without any fine tuning. Comment: 16 pages, 10 figures; minor changes

Full text

Page 1

arXiv:0909.4771v2 [gr-qc] 25 Nov 2009
Indications of deSitter Spacetime from Classical Sequential Growth Dynamics of
Causal Sets
Maqbool Ahmed
Centre for Advanced Mathematics and Physics,
Campus of College of E&ME
National University of Sciences and Technology,
Peshawar Road, Rawalpindi, 46000, Pakistan
David Rideout
Perimeter Institute for Theoretical Physics
31 Caroline Street North, Waterloo,
Ontario N2L 2Y5, Canada
(Dated: 6 November 2009)
A large class of the dynamical laws for causal sets described by a classical process of sequential
growth yield a cyclic universe, whose cycles of expansion and contraction are punctuated by single
‘origin elements’ of the causal set. We present evidence that the effective dynamics of the immediate
future of one of these origin elements, within the context of the sequential growth dynamics, yields
an initial period of de Sitter-like exponential expansion, and argue that the resulting picture has
many attractive features as a model of the early universe, with the potential to solve some of the
standard model puzzles without any fine tuning.
Contents
I. Introduction
1
II. Causal Sets
A. Kinematics
B. Dynamics
C. Cosmic Renormalization
2
3
3
4
III. Originary Percolation and Random Trees
A. Random Tree Era
B. Originary Percolation
4
4
6
IV. Volume of an Alexandrov Neighbourhood in deSitter Spacetime
7
V. Simulation details
9
VI. Results
10
VII. Summary and Conclusions
12
References
16
I. INTRODUCTION
Cosmology is the natural playground for theories of quantum gravity for many reasons. The most obvious is the
fact that all quantum gravity theories are formulated at such high energies (or small distances) that it is practically
impossible for an earth based laboratory or accelerator to test them. However, the early universe can provide such
a laboratory. In addition cosmology is one of the most natural applications of general relativity, which is exactly
what a theory of quantum gravity hopes to “unify” with quantum theory. On the other hand the standard model of
cosmology [1, 2] also suffers from some problems/puzzles [3] that warrant extensions to it. Most of these have their
origins in our lack of understanding of the physics of very high energies and thus a successful theory of quantum
gravity should be able to address these problems. Quantum gravity has recently began to shed some light on the
puzzles of cosmology, such as resolution of the cosmic singularity [4, 5, 6], providing some hints as to alternatives to

Page 2

2
inflation [7], and providing potential explanations of density perturbations [8]. Additionally the prediction of a non-
zero cosmological constant arises naturally from the discreteness expected from quantum gravity [9]. Thus quantum
gravity is beginning to show some promise in resolving some of the paradoxes of the standard model. However, they
are far from any consensus in this regard, so cosmologists generally look for alternative explanations.
The most common path taken is an extension via particle physics that introduces a scalar field with very special
properties in the early universe. If the potential of the field satisfies certain conditions it can be arranged that the
universe undergoes rapid expansion in a “de Sitter-like” phase. This rapid expansion, called the inflationary era or
inflation, gets rid of many of the standard model puzzles/problems such as the so-called “horizon problem”, “the
flatness puzzle”, “the monopole problem”, etc. For a more complete discussion see references [10, 11]. Despite the
fact that this scenario has “problems” of its own, as has been pointed out by many authors [7, 12], the resulting
features are very attractive and hard to ignore. On the other hand the lack of any competing model leaves a scientific
void that needs to be filled if the case for or against inflation has to be decided.
Fortunately Causal Set theory has reached a stage in its evolution where some predictions about cosmology have
come out [9, 13]. Based on a mixture of “classical dynamics” (referred to as the classical sequential growth [14]
or CSG models) and some expectations about “quantum dynamics”, Causal Set theory predicts fluctuations in the
cosmological term around a mean value. If this mean value is taken to be zero, the fluctuations are of the right
magnitude to explain the present observations. Computer simulations of the behavior of the universe with such a
fluctuating cosmological term have shown that the energy density in Λ follows the total energy density of the universe
and is roughly of the same order. This is the first time that such testable predictions have come out of a fundamental
theory of quantum gravity.
Causal Set theory is also different from most other quantum gravity theories in the sense that it assumes fundamental
discreteness. There have been many arguments for discreteness at the most basic level, but all of them have either
been merely philosophical, or regarded only as methods of regulating the infinities and singularities in particle physics
and general relativity. In the absence of real predictions it has always been difficult to see if these considerations have
more than a philosophical value. Prediction about fluctuations in Λ, however, draws life directly from fundamental
discreteness, and it is but natural to wonder if Causal Set theory has more to say about cosmology. In this paper
we will present evidence that many of the CSG models produce a de Sitter-like early universe, and thus may prove
helpful towards solving another puzzle — why is the universe so large when it is not so old?
A general question which naturally arises in Causal Set theory is whether causal sets which are well approximated
by continua arise dynamically. It has been shown that the sequential growth models possess continuum limits, as
N → ∞ and p → 0, however the resulting continua look nothing like spacetime manifolds of dimension > 1 [15].
However it may still be the case that something resembling a spacetime arises at finite p. We consider this latter
question here.
This paper is organized as follows. In section II we briefly describe that portion of Causal Set theory which is
relevant to the current work. In section III we describe the behavior of the “originary percolation” dynamics, which
arises as an effective dynamics of the “early universe” of CSG models. Then in section IV we compute the spacetime
volume of ‘Alexandrov neighborhoods’ (‘causal diamonds’) in de Sitter space of arbitrary (integer) dimension. In
section V we describe the particular simulation we perform, with results in section VI, and wrap up with some
concluding remarks in section VII.
II. CAUSAL SETS
A causal set, or ‘causet’ for short, is a locally finite partially ordered set, whose elements can be thought of as
irreducible1‘atoms of spacetime’. A partially ordered set C consists of a ‘ground set’, which one generally labels
with integers from 0 to N − 1 (N can be infinite), along with a binary relation ≺ which is irreflexive (x ⊀ x) and
transitive (x ≺ y ≺ z ⇒ x ≺ z). Local finiteness is the condition that every order interval (or simply interval)
[x,y] = {y|x ≺ y ≺ z} ∀x,z ∈ C has finite cardinality.
1It may be necessary to add matter degrees of freedom to the causet elements, and at some stage it may be important to ‘coarse grain’
the causet so that a single element may stand in for many, but for the moment we can think of the elements as not containing any
internal information.

Page 3

3
A.Kinematics
The connection to macroscopic spacetime arises via the notion of a “sprinkling”, in which one selects events of
a spacetime at random by a Poisson process, identifies them with causal set elements, and then deduces a partial
ordering among the elements from the causal structure of the spacetime. One regards a continuum spacetime as being
a good approximation to an underlying causal set if that causal set is likely to have arisen from a sprinkling into that
spacetime. For an extensive review of the causal set program, see [16, 17, 18, 19].
The connection between familiar concepts from continuum geometry and their discrete counterparts on the causal
set is the domain of causal set kinematics. We will make extensive use of two results in this regard.
The first is stated as a definition at this stage of the theory’s development, and is implicit in the description in the
previous paragraph of the correspondence of the causal set with the continuum. In order for a causal set to be likely
to arise from a sprinkling, it must be the case that the number of elements sprinkled into any region of spacetime
with volume V is Poisson distributed, with a mean of V . This Poisson fluctuation in the correspondence between
spacetime volume and number of elements plays a crucial role in the prediction of a fluctuating cosmological constant
[9].
The second result relates to the correspondence between the length of chains and proper time. A chain is a subset
of a causal set for which each pair of elements is related. The length L of a chain is the number of elements in the
chain minus 1. In Minkowski space of any dimension, it has been proven that the length of the longest chain between
any pair of elements is proportional to the proper time between the events at which they are sprinkled, in the limit
of infinite sprinkling density [20, 21]. In [22] the proportionality is claimed to hold for any spacetime. Following [21],
we define m to be the constant of proportionality, so that
τ = mL .(2.1)
B.Dynamics
There are a number of approaches to constructing a dynamical law for causal sets. Perhaps the most developed
to date is the classical sequential growth model [14, 23, 24] mentioned in the Introduction. It describes the causal
set as growing via a sort of “cosmological accretion” process, in which elements of the causet arise one at a time,
each selecting some subset of the causal set to be its past. The process of growth in the model is stochastic; each
newborn element selects a “precursor set” at random, with probabilities which satisfy a discrete analog of general
covariance and a causality condition akin to that used to derive the Bell inequalities. This randomness is regarded
as fundamental, and yet purely classical in nature, because it does not allow for any quantum interference among
alternative outcomes. Given the classical nature of the probability distribution, the dynamics is incomplete, but can be
seen as a stepping stone toward formulating a fully quantum process, which could then be regarded as a generalization
of classical probability theory. Although the dynamically generated causal sets do not lead to orders which are readily
approximated by smooth spacetime manifolds, they do have a number of striking cosmological features, which we
explore further in this paper.
The sequential growth dynamics is described to take place in “stages”, though it is important to emphasize that the
discrete general covariance condition enforces that this ordering in which the causet elements arise is “pure gauge”
— it has no effect on the probability of forming a particular (order equivalence class of) causal set. At stage n the
causal set has n elements “so far”, and the task is to select a precursor set for the new element which arises in this
stage. The probabilities of the CSG model derive from a sequence of nonnegative “coupling constants” (tn),n ≥ 0.
With these weights, the probability for selecting a precursor set S is proportional to t|S|.2Thus the probability to
choose a particular set S is
Pr(S) =
t|S|
?n
i=0
?n
i
?ti
.
Once a precursor is chosen, to be to the past of the new element, all the relations implied by transitivity are included
as well. Thus it is the “past closure” of S which forms the past of the newly generated element.
2The definition of precursor set used here differs from that in [14, 23]. There only precursor sets which contained their own past were
included. This definition allows for a much simpler expression of the transition probabilities (e.g. as given in [15]), which better reveals
the physical meaning of the tn.

Page 4

4
The particular sequence tn= tn, for a single non-negative real number t, gives rise to a dynamics called transitive
percolation [14]. This sequence plays an important role, as we will see in a moment. The rule for deciding which
elements to select for the past of a new element is particularly simple for transitive percolation. The newborn
element simply considers each already existing element in turn, and selects it to be to its past with a fixed probability
p = t/(1+t). It then adds to its past every element which precedes any of the originally selected elements, to maintain
transitivity.
C.Cosmic Renormalization
Consider an element of a causal set, called in the combinatorics literature a ‘post’, which is related to every other
element of the causal set. This would resemble an initial or final singularity of a universe, in that the entirety of
the universe is causally related to it. Now for any finite p, it has been proven that a(n infinite) causet generated by
transitive percolation almost surely contains an infinite number of posts [25]. It is the large scale behavior of the
universe subsequent to one of these posts which is the subject of this paper. We will present evidence that the period
immediately following a post is one of rapid expansion of spacetime volume with respect to proper time. Thus at
the largest scales the causal sets generated by transitive percolation resemble a bouncing universe, which periodically
undergoes collapse down to a final singularity and an ensuing re-expansion.
It has further been shown that a large class of CSG models, which includes the sequence tn= (α/ln(n))nfor n > 0,
α > π2/3 also lead to causets which almost surely contain an infinite number of posts [26]. Now the presence of
posts in the dynamical model suggests an interesting possibility, as described in [5], that the dynamics following a
post can be regarded as growing an entirely new universe, except with coupling constants which are ‘renormalized’
with respect to those of the previous era. Much is now known about the flow of the coupling constants (tn) under
this ‘cosmic renormalization’ [27, 28], in particular that the transitive percolation dynamics family tn= tnforms a
unique attractive fixed point. The sequence tn= tn/n! has also been studied in some detail [5, 29]. There it is shown
that the region immediately subsequent to a post behaves like transitive percolation, with a parameter t which gets
driven toward zero under the cosmic renormalization (t →?t/N for N elements to the past of the current era’s post
(or “origin element”)).
III. ORIGINARY PERCOLATION AND RANDOM TREES
Note that the effective dynamics following a post comes with a caveat: each element is required to be related to
the post, by definition. Therefore we find an originary dynamics, for which the possibility of being born unrelated to
any other element is excluded, and all remaining probabilities are normalized correspondingly. Thus the probabilities
of an originary dynamics are equal to those of an ordinary CSG model, conditioned on the event that the newborn
element connects to at least one other element. (The originary dynamics is in fact one of the general class of solutions
to the covariance and causality conditions on sequential growth, described in [30].)
Originary percolation is the originary version of transitive percolation. As mentioned in section IIB, at each stage
of the growth process, the newborn element considers each existing element x in turn, and selects x to be in its past
with probability p. In order to maintain transitivity of the order, if it chooses x for part of its past, it includes all
ancestors of x as well. In the event that no element x is selected in this process, it simply ‘tries again’, so as to
maintain the condition of originarity. At stage n (meaning that there are n elements currently in the causet), the
probability to select a particular subset S of existing elements is
Pr(S) =p|S|qn−|S|
1 − qn
,
where q = 1 − p, and the factor 1/(1 − qn) accounts for the originary condition, which excludes the possibility of not
connecting to anything (which occurs with probability qn). Once a set S is chosen, the past closure of S becomes the
past of the newborn element x.
A.Random Tree Era
For small values of p, the early universe of originary percolation, by which we mean the structure of that portion
of the causal set which is born shortly after the origin element, forms a random tree, with high probability. To see

Page 5

5
0
500
1000
1500
2000
2500
0 5 10 15
t
20 25 30
Nt
FIG. 1: Mean population of levels for an N = 16384 element random tree. The mean and its error are measured from 100
samples of the random tree process. The fitting function proportional to a Poisson distribution is shown. Here A = 16687 ±85
and λ = 9.306 ± .022.
why this occurs, consider the probability that the selected precursor set contains m elements is given by
Pr(|S| = m) =
?n
m
?p|S|qn−|S|
1 − qn
.
For small p this becomes vanishingly small for any m > 0. However the case m = 0 is excluded by the originarity
condition, while it remains true that, as long as p ≪ 1/n, the transitions with m = 1 will be much more likely than
any of the others. In these transitions one element is chosen at random from those already present, with a uniform
distribution. This behaviour yields a simple model of a random tree. It persists until n ∼ 1/p, at which point we
get a percolation phase transition, which heralds the end of the random tree era, and the beginning of a phase of de
Sitter-like expansion, which we describe below. Note that the structure of this earliest random tree era of the universe
is independent of p, save in determining how long it lasts.
We can begin our study of the early universe of originary percolation by studying this simple random tree process.
To get some feel for the initial rate of expansion of the universe, we ask what is the expected number of elements
which arise in ‘level t’, which we define to be those elements whose longest chain to the root element is of length t.
With certainty, the first element appears in level 0, and the second in level 1. At stage n, the probability of joining
level t is proportional to the number of elements in level t − 1. The same exact process has been studied in the
combinatorics literature, under the name “random recursive trees”. There similar questions have been studied, such
as the probability distribution of the level of an element chosen uniformly at random from the tree [31].
Despite the simple recursion obeyed by the ‘joining probability’ above, this problem is not easy to solve, e.g. because
it involves an infinite sequence of distributions. Rather than analyze this problem in detail here, we simply observe
that, after forming a random tree with N elements, the mean cardinality of level t looks very much like a multiple of
a Poisson distribution in t. Thus the mean number Ntof elements in level t is very well fit by the function
Nt=Aλte−λ
t!
,(3.1)
where the normalization factor A>
figure 1, the relation cannot be exact, for example because Ntmust be exactly zero for t > N, which does not occur
in (3.1).
The random tree era will continue until n ∼ 1/p. After this stage a newborn element becomes as likely to choose
more than one parent as not. Thus we expect that the N of the random tree era is ∼ 1/p. As far as the level
∼N and λ ∼ lnN. An example is shown in figure 1. Despite the excellent fit of