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arXiv:hep-th/0111030v2 10 Oct 2002
A Cyclic Model of the Universe
Paul J. Steinhardt1∗and Neil Turok2
1Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USAA
2DAMTP, CMS, Wilberforce Road, Cambridge, CB3 0WA, UK
∗To whom correspondence should be addressed; E-mail: steinh@princeton.edu.
We propose a cosmological model in which the universe undergoes
an endless sequence of cosmic epochs each beginning with a ‘bang’
and ending in a ‘crunch.’ The temperature and density are finite at
each transition from crunch to bang. Instead of having an inflation-
ary epoch, each cycle includes a period of slow accelerated expansion
(as recently observed) followed by slow contraction. The combina-
tion produces the homogeneity, flatness, density fluctuations and
energy needed to begin the next cycle.
Introduction
The current standard model of cosmology combines the original big bang model and the
inflationary scenario. (1, 2, 3, 4, 5, 6, 7) Inflation, a brief period (10−30 s) of very rapid
cosmic acceleration occurring shortly after the big bang, can explain the homogeneity
and isotropy of the universe on large scales (>100 Mpc), its spatial flatness, and also
the distribution of galaxies and the fluctuations in the cosmic microwave background.
However, the standard model has some cracks and gaps. The recent discoveries of cosmic
1
acceleration indicating self-repulsive dark energy (8,9,10,11) were not predicted and have
no clear role in the standard model. (1, 2, 3) Furthermore, no explanation is offered for
the ‘beginning of time’, the initial conditions of the universe, or the long-term future.
In this paper, we present a new cosmology consisting of an endless sequence of cycles
of expansion and contraction. By definition, there is neither a beginning nor an end of
time, nor a need to specify initial conditions. We explain the role of dark energy, and
generate the homogeneity, flatness, and density fluctuations without invoking inflation.
The model we present is reminiscent of oscillatory models introduced in the 1930’s
based on a closed universe that undergoes a sequence of expansions, contractions and
bounces. The oscillatory models had the difficulty of having to pass through a singularity
in which the energy and temperature diverge. Furthermore, as pointed out by Tolman
(12, 13), entropy produced during one cycle would add to the entropy produced in the
next, causing each cycle to be longer than the one before it. Extrapolating backward
in time, the universe would have to have originated at some finite time in the past so
that the problem of explaining the ‘beginning of time’ remains. Furthermore, recent
measurements of the cosmic microwave background anisotropy and large scale structure
favor a flat universe over a closed one.
In our cyclic model, the universe is infinite and flat, rather than finite and closed. We
introduce a negative potential energy rather than spatial curvature to cause the rever-
sal from expansion to contraction. Before reversal, though, the universe undergoes the
usual period of radiation and matter domination, followed by a long period of accelerated
expansion (presumably the acceleration that has been recently detected (8, 11)). The
accelerated expansion, caused by dark energy, is an essential feature of our model, needed
to dilute the entropy, black holes and other debris produced in the previous cycle so that
the universe is returned to its original pristine vacuum state before it begins to contract,
2
bounce, and begin a cycle anew.
Essential Ingredients
As in inflationary cosmology, the cyclic scenario can be described in terms of the evolu-
tion of a scalar field φin a potential V(φ) according to a conventional four-dimensional
quantum field theory. The essential differences is in the form of the potential and the
couplings between the scalar field, matter and radiation.
The analysis of the cyclic model follows from the action Sdescribing gravity, the scalar
field φ, and the matter and radiation fluids:
S=Zd4x√−g1
16πG R − 1
2(∂φ)2−V(φ) + β4(φ)(ρM+ρR),(1)
where gis the determinant of the metric gµν,Gis Newton’s constant and Ris the Ricci
scalar. The coupling β(φ) between φand the matter (ρM) and radiation (ρR) densities
is crucial because it allows the densities to remain finite at the big crunch/big bang
transition.
The line element for a flat, homogeneous universe is −dt2+a2dx2, where ais the
Robertson-Walker scale factor. The equations of motion following from Eq. (1) are,
H2=8πG
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2˙
φ2+V+β4ρR+β4ρM,(2)
¨a
a=−8πG
3˙
φ2−V+β4ρR+1
2β4ρM,(3)
where a dot denotes a derivative with respect to tand H≡˙a/a is the Hubble parameter.
The equation of motion for φis
¨
φ+ 3H˙
φ=−V,φ −β,φβ3ρM(4)
and the fluid equation of motion for matter (M) or radiation (R) is
ˆadρi
dˆa=a∂ρi
∂a +β
β′
∂ρi
∂φi
=−3(ρi+pi), i = M,R,(5)
3
V(φ)
φ
1
2
3
4
5
67
}ρQ
0
Figure 1: Schematic plot of the potential V(φ) versus field φ. In M theory, φdetermines
the distance between branes, and φ→ −∞ as the branes collide. We define φto be zero
where V(φ) crosses zero and, therefore, φis positive when the branes are at their maximal
separation. Far to the right, the potential asymptotes to ρ0
Q, the current value of the
quintessence (dark energy) density. The solid circles represent the dark energy dominated
stage; the grey represent the contracting phase during which density fluctuations are
generated; the open circles represent the phase when the scalar kinetic energy dominates;
and the broken circle represents the stage when the universe is radiation dominated.
Further details of the sequence of stages are described in the article.
where ˆa≡aβ(φ) and pis the pressure of the fluid component with energy density ρ. The
implicit assumption is that matter and radiation couple to β2(φ)gµν (with scale factor ˆa)
rather than the Einstein metric gµν alone (or the scale factor a). Note that the radiation
term in Eq. (1) is actually independent of φ(since ρR∝ˆa−4) so only ρMenters the φ
equation of motion.
We assume the potential V(φ) has the following features, illustrated in Fig. 1: (i)
V(φ) must approach zero rapidly as φ→ −∞; (ii) the potential must be negative for
intermediate φ; and, (iii) as φincreases, the potential must rise to a shallow plateau with
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a positive value V0. An example of a potential with these properties is
V(φ) = V0(1 −e−cφ)F(φ),(6)
where from this point onwards we adopt units in which 8πG = 1. F(φ) is a function we
introduce to ensure that V(φ)→0 as φ→ −∞. Without loss of generality, we take F(φ)
to be nearly unity for φto the right of potential minimum. The detailed manner in which
it tends to zero is not crucial for the main predictions of the cyclic model. A quantitative
analysis of this potential (Ref. 14) shows that a realistic cosmology can be obtained by
choosing c≥10 and V0equal to today’s dark energy density (about 6 ×10−30 g/cm3) in
Eq. (6).
We have already mentioned that the coupling β(φ) is chosen so that ˆaand, thus, the
matter and radiation density are finite at a= 0. This requires β(φ)∼e−φ√6as φ→ −∞,
but this is precisely the behavior expected in M-theory (see below). The presence of β(φ)
and the consequent coupling of φto nonrelativistic matter represent a modification of
Einstein’s theory of general relativity. Because φevolves by an exponentially small amount
between nucleosynthesis (t∼1 s) and today (t∼1017 s), deviations from standard general
relativity are small and easily satisfy current cosmological constraints. (14) However, the
coupling of matter to φproduces other potentially measurable effects including a fifth
force which violates the equivalence principle. Provided (lnβ),φ ≪10−3, for today’s value
of φ, these violations are too small to be detected. (14, 15, 16) We shall assume this to be
the case. Hence, the deviations from general relativity are negligible today.
The final crucial ingredient in the cyclic model is a matching rule which determines
how to pass from the big crunch to the big bang. The transition occurs as φ→ −∞
and then rebounds towards positive φ. Motivated again by string theory (see below), we
propose that some small fraction of the φ-field kinetic energy is converted to matter and
5
radiation. The matching rule amounts to
˙
φe√3/2φ→ −(1 + χ)˙
φe√3/2φ(7)
where χis a parameter measuring the efficiency of production of radiation at the bounce.
Both sides of this relation are finite at the bounce.
Stringy motivation
From the perspective of four-dimensional quantum field theory, the introduction of a scalar
field, a potential and the couplings to matter in the cyclic model is no more arbitrary or
tuned than the requirements for the inflationary models. However, the ingredients for the
cyclic model are also strongly motivated by string theory and M-theory. This connection
ties our scenario into the leading approach to fundamental physics and quantum gravity.
The connection should not be overemphasized. String theory remains far from proven
and quantum gravity effects may be unimportant for describing cosmology at wavelengths
much longer than a Planck length (10−33 cm). If the reader prefers, the connection to
string theory can be ignored. On the other hand, we find the connection useful because
it provides a natural geometric interpretation for the scenario. Hence, we briefly describe
the relationship.
According to M-theory, the universe consists of a four dimensional ‘bulk’ space bounded
by two three-dimensional domain walls, known as ‘branes’ (short for membranes), one with
positive and the other with negative tension. (19,20, 21) The branes are free to move along
the extra spatial dimension, so that they may approach and collide. The fundamental
theory is formulated in ten spatial dimensions, but six dimensions are compactified on
a Calabi-Yau manifold, which for our purposes can be treated as fixed, and therefore
ignored. Gravity acts throughout the five dimensional spacetime, but particles of our
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visible universe are constrained to move along one of the branes, sometimes called the
visible brane. Particles on the other brane interact only through gravity with matter on
the visible brane and hence behave like dark matter.
The scalar field φwe want is naturally identified with the field that determines the
distance between branes. The potential V(φ) is the inter-brane potential caused by non-
perturbative virtual exchange of membranes between the boundaries. The interbrane
force is what causes the branes to repeatedly collide and bounce. At large separation
(corresponding to large φ), the force between the branes should become small, consistent
with the flat plateau shown in Fig. 1. Collision corresponds to φ→ −∞. But the string
coupling gs∝eγφ, with γ > 0, so gsvanishes in this limit (22). Non-perturbative effects
vanish faster than any power of gs, for example as e−1/g2
sor e−1/gs, accounting for the
prefactor F(φ) in Eq. (6).
The coupling β(φ) also has a natural interpretation in the brane picture. Particles
reside on the branes, which are embedded in an extra dimension whose size and warp are
determined by β. The effective scale factor on the branes is ˆa=a β(φ), not a, and ˆais
finite at the big crunch or big bang. The function β(φ) is in general different for the two
branes (due to the warp factor) and for different reductions of M-theory. However, the
standard Kaluza-Klein behavior β(φ)∼e−φ/√6as φ→ −∞ is universal, since the warp
factor becomes irrelevant as the branes approach one another. (14, 22)
Most importantly, the brane-world provides a natural resolution of the cosmic singu-
larity. (14, 22) One might say that the big crunch is an illusion, because the scale factors
on the branes (→ˆa) are perfectly finite there. That is why the matter and radiation
densities, and the Riemannian curvature on the branes, are finite. The only respect in
which the big crunch is singular is that the one extra dimension separating the two branes
momentarily disappears. Our scenario is built on the hypothesis (23) that the branes sep-
7
arate after collision, so the extra dimension immediately reappears. This process cannot
be completely smooth, because the disappearance of the extra dimension is non-adiabatic
and leads to particle production. That is, the brane collision is partially inelastic. Prelim-
inary calculations of this effect are encouraging, because they indicate a finite density of
particles is produced (17, 18). The matching condition, Eq. (7), parameterizes this effect.
Ultimately, a well-controlled string-theoretic calculation (14, 17, 18, 22) should determine
the value of χ.
Dark energy and the cyclic model
The role of dark energy in the cyclic scenario is novel. In the standard big bang and
inflationary models, the recently discovered dark energy and cosmic acceleration (8, 11)
are an unexpected surprise with no clear explanation. In the cyclic scenario, however, not
only is the source of dark energy explained, but the dark energy and its associated cosmic
acceleration are actually crucial to the consistency of the model. Namely, the associated
exponential expansion suppresses density perturbations and dilutes entropy, matter and
black holes to negligible levels. By periodically restoring the universe to an empty, smooth
state, the acceleration causes the cyclic solution to be a stable attractor.
Right after a big bang, the scalar field φis increasing rapidly. However, its motion is
damped by the expansion of the universe and φessentially comes to rest in the radiation
dominated phase [stage (1) in Figure 1]. Thereafter it remains nearly fixed until the dark
energy begins to dominate and cosmic acceleration commences. The positive potential
energy density at the current value of φacts as a form of quintessence, (24) a time-
varying energy component with negative pressure that causes the present-day accelerated
expansion. This choice entails tuning V0, but it is the same degree of tuning required in
any cosmological model (including inflation) to explain the recent observations of cosmic
8
acceleration (8, 11). In this case, because the dark energy serves several purposes, the
single tuning resolves several problems at once.
The cosmic acceleration is nearly 100 orders of magnitude smaller than considered
in inflationary cosmology. Nevertheless, if sustained for hundreds of e-folds (trillions of
years) or more, the cosmic acceleration can flatten the universe and dilute the entropy,
black holes, and other debris (neutron stars, neutrinos, etc.) created over the preceding
cycle, overcoming the obstacle that has blocked previous attempts at a cyclic universe.
In this picture, we are presently about 14 billion years into the current cycle, and have
just begun the trillions years of cosmic acceleration. After this amount of accelerated
expansion, the number of particles in the universe may be suppressed to less than one per
Hubble volume before the cosmic acceleration ends. Ultimately, the scalar field begins to
roll back towards −∞, driving the potential to zero. The scalar field φis thus the source of
the currently observed acceleration, the reason why the universe is homogeneous, isotropic
and flat before the big crunch, and the root cause for the universe reversing from expansion
to contraction.
A brief tour of the cyclic universe
Putting together the various concepts that have been introduced, we can now present the
sequence of events in each cycle beginning from the present epoch, stage (1) in Figure 1.
The universe has completed radiation and matter dominated epochs during which φis
nearly fixed. We are presently at the time when its potential energy begins to dominate,
ushering in a period of slow cosmic acceleration lasting trillions of years or more, in which
the matter, radiation and black holes are diluted away and a smooth, empty, flat universe
results. Very slowly the slope in the potential causes φto roll in the negative direction,
as indicated in stage (2). Cosmic acceleration continues until the field nears the point of
9
zero potential energy, stage (3). The universe is dominated by the kinetic energy of φ, but
expansion causes this to be damped. Eventually, the total energy (kinetic plus negative
potential) reaches zero. From Eq. (2), the Hubble parameter is zero and the universe
is momentarily static. From Eq. (3), ¨a < 0, so that abegins to contract. While ais
nearly static, the universe satisfies the ekpyrotic conditions for creating a scale-invariant
spectrum of density perturbations. (23, 25) As the field continues to roll towards −∞,a
contracts and the kinetic energy of the scalar field grows. That is, gravitational energy
is converted to scalar field kinetic energy during this part of the cycle. Hence, the field
races past the minimum of the potential and off to −∞, with kinetic energy becoming
increasingly dominant as the bounce nears, stage (5). The scalar field diverges as atends
to zero. After the bounce, radiation is generated and the universe is expanding. At first,
scalar kinetic energy density (∝1/a6) dominates over the radiation (∝1/a4), stage (6).
Soon after, however, the universe becomes radiation dominated, stage (7). The motion of
φis rapidly damped away, so that it remains close to its maximal value for the rest of the
standard big bang evolution (the next 15 billion years). Then, the scalar field potential
energy begins to dominate, and the field rolls towards −∞, where the next big crunch
occurs and the cycle begins anew.
Obtaining scale-invariant perturbations
One of the most compelling successes of inflationary theory was to obtain a nearly scale-
invariant spectrum of density fluctuations that can seed large-scale structure. (4) Here,
the same feat is achieved using different physics during an ultra-slow contraction phase
[stage (2) in Fig. 1]. (23, 25) In inflation, the density fluctuations are created by very
rapid expansion, causing fluctuations on microscopic scales to be stretched to macro-
scopic scales. (4) In the cyclic model, the fluctuations are generated during a quasistatic,
10
contracting universe where gravity plays no significant role. (23) Simply because the
potential V(φ) is decreasing more and more rapidly, quantum fluctuations in φare ampli-
fied as the field evolves downhill. (23,26,27) Instabilities in long-wavelength modes occur
sooner than those in short wavelength modes, thereby amplifying long wavelength power
and, curiously, nearly exactly mimicking the inflationary effect. The nearly scale-invariant
spectrum of fluctuations in φcreated during the contracting phase transform into a nearly
scale-invariant spectrum of density fluctuations in the expanding phase. (25) Current ob-
servations of large-scale structure and fluctuations of the cosmic microwave background
cannot distinguish between inflation and the cyclic model because both predict a nearly
scale-invariant spectrum of adiabatic, gaussian density perturbations.
Future measurements of gravitational waves may be able to distinguish the two pic-
tures. (23) In inflation, where gravity is paramount, quantum fluctuations in all light
degrees of freedom are subject to the same gravitational effect described above. Hence,
not only is there a nearly scale-invariant spectrum of energy density perturbations, but
also there is a scale-invariant spectrum of gravitational waves. In the cyclic and ekpy-
rotic models, where the potential, rather than gravity, is the cause of the fluctuations,
the only field which obtains a nearly scale-invariant spectrum is the one rolling down the
potential, namely φ, which only produces energy density fluctuations. The direct search
for gravitational waves or the search for their indirect effect on the polarization of the
cosmic microwave background (28) are the crucial tests for distinguishing inflation from
the cyclic model.
Cyclic solution as Cosmic Attractor
Not only do cyclic solutions exist for a range of potentials and parameters, but also they
are attractors for a range of initial conditions. The cosmic acceleration caused by the
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positive potential plateau plays the critical role here. For example, suppose the scalar
field is jostled and stops at a slightly different maximal value on the plateau compared
to the exactly cyclic solution. The same sequence of stages ensues. The scalar field is
critically damped during the exponentially expanding phase. So by the time the field
reaches stage (3) where V= 0, it is rolling almost at the same rate as if it had started at
φ= 0, and memory of its initial position has been lost. (14) The argument suggests that
it is natural to expect dark energy and cosmic acceleration following matter domination
in a cyclic universe, in accordance with what has been recently observed.
Comparing cyclic and inflationary model
The cyclic and inflationary models have numerous conceptual differences in addition to
those already described. Inflation requires two periods of cosmic acceleration, a hypothet-
ical period of rapid expansion in the early universe and the observed current acceleration.
The cyclic model only requires one period of acceleration per cycle.
In the inflationary picture, most of the volume of the universe is completely unlike
what we see. Even when inflation ends in one region, such as our own, it continues in
others. Because of the superluminal expansion rate of the remaining inflating regions, they
occupy most of the physical volume of the universe. Regions which have stopped inflating,
such as our region of the universe, represent an infinitesimal fraction. By contrast, the
cyclic model is one in which the local universe is typical of the universe as a whole. All or
almost all regions of the universe are undergoing the same sequence of cosmic events and
most of the time is spent in the radiation, matter, and dark energy dominated phases.
In the production of perturbations, the inflationary mechanism relies on stretching
modes whose wavelength is initially exponentially sub-Planckian, to macroscopic scales.
Quantum gravity effects in the initial state are highly uncertain, and inflationary predic-
12
tions may therefore be highly sensitive to sub-Planckian physics. In contrast, perturba-
tions in the cyclic model are generated when the modes have wavelengths of thousands
of kilometers, using macroscopic physics insensitive to quantum gravity effects.
The cyclic model deals directly with the cosmic singularity, explaining it as a transi-
tion from a contracting to an expanding phase. Although inflation does not address the
cosmic singularity problem directly, it does rely implicitly on the opposite assumption:
that the big bang is the beginning of time and that the universe emerges in a rapidly
expanding state. Inflating regions with high potential energy expand more rapidly and
dominate the universe. If there is a pre-existing contracting phase, then the high potential
energy regions collapse and disappear before the expansion phase begins. String theory
or, more generally, quantum gravity can play an important role in settling the nature of
the singularity and, thereby, distinguishing between the two assumptions.
The cyclic model is a complete model of cosmic history, whereas inflation is only a the-
ory of cosmic history following an assumed initial creation event. Hence, the cyclic model
has more explanatory and predictive power. For example, we have already emphasized
how the cyclic model leads naturally to the prediction of quintessence and cosmic acceler-
ation, explaining them as essential elements of an eternally repeating universe. The cyclic
model is also inflexible with regard to its prediction of no long-wavelength gravitational
waves.
Inflationary cosmology offers no information about the cosmological constant problem.
The cyclic model provides a fascinating new twist. Historically, the problem is assumed
to mean that one must explain why the vacuum energy of the ground state is zero. In
the cyclic model, the vacuum energy of the true ground state is not zero. It is negative
and its magnitude is large, as is obvious from Fig. 1. However, we have shown that the
Universe does not reach the true ground state. Instead, it hovers above the ground state
13
from cycle to cycle, bouncing from one side of the potential well to the other but spending
most time on the positive energy side.
Reviewing the overall scenario and its implications, what is most remarkable is that
the cyclic model can differ so much from the standard picture in terms of the origin
of space and time and the sequence of cosmic events that lead to our current universe.
Yet, the model requires no more assumptions or tunings (and by some measures less) to
match the current observations. It appears that we now have two disparate possibilities:
a universe with a definite beginning and a universe that is made and remade forever. The
ultimate arbiter will be Nature. Measurements of gravitational waves and the properties
of dark energy (14) can provide decisive ways to discriminate between the two pictures
observationally.
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14
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29. We thank M. Bucher, R. Durrer, S. Gratton, J. Khoury, B.A. Ovrut, J. Ostriker,
P.J.E. Peebles, A. Polyakov, M. Rees, N. Seiberg, D. Spergel, A. Tolley, T. Wiseman
and E. Witten for useful conversations. We thank L. Rocher for pointing out historical
references. This work was supported in part by US Department of Energy grant DE-
FG02-91ER40671 (PJS) and by PPARC-UK (NT).
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