ArticlePDF Available

# Unitary Evolution and Cosmological Fine-Tuning

Authors:

## Abstract and Figures

Inflationary cosmology attempts to provide a natural explanation for the flatness and homogeneity of the observable universe. In the context of reversible (unitary) evolution, this goal is difficult to satisfy, as Liouville's theorem implies that no dynamical process can evolve a large number of initial states into a small number of final states. We use the invariant measure on solutions to Einstein's equation to quantify the problems of cosmological fine-tuning. The most natural interpretation of the measure is the flatness problem does not exist; almost all Robertson-Walker cosmologies are spatially flat. The homogeneity of the early universe, however, does represent a substantial fine-tuning; the horizon problem is real. When perturbations are taken into account, inflation only occurs in a negligibly small fraction of cosmological histories, less than $10^{-6.6\times 10^7}$. We argue that while inflation does not affect the number of initial conditions that evolve into a late universe like our own, it nevertheless provides an appealing target for true theories of initial conditions, by allowing for small patches of space with sub-Planckian curvature to grow into reasonable universes.
Content may be subject to copyright.
CALT 68-2797
Unitary Evolution and Cosmological Fine-Tuning
Sean M. Carroll and Heywood Tam
California Institute of Technology
seancarroll@gmail.com, heywood.tam@gmail.com
Abstract
Inﬂationary cosmology attempts to provide a natural explanation for the ﬂatness
and homogeneity of the observable universe. In the context of reversible (unitary)
evolution, this goal is diﬃcult to satisfy, as Liouville’s theorem implies that no dynam-
ical process can evolve a large number of initial states into a small number of ﬁnal
states. We use the invariant measure on solutions to Einstein’s equation to quantify
the problems of cosmological ﬁne-tuning. The most natural interpretation of the mea-
sure is the ﬂatness problem does not exist; almost all Robertson-Walker cosmologies
are spatially ﬂat. The homogeneity of the early universe, however, does represent a
substantial ﬁne-tuning; the horizon problem is real. When perturbations are taken into
account, inﬂation only occurs in a negligibly small fraction of cosmological histories,
less than 106.6×107. We argue that while inﬂation does not aﬀect the number of initial
conditions that evolve into a late universe like our own, it nevertheless provides an
appealing target for true theories of initial conditions, by allowing for small patches of
space with sub-Planckian curvature to grow into reasonable universes.
1
arXiv:1007.1417v1 [hep-th] 8 Jul 2010
Contents
1 Introduction 3
2 The Evolution of our Comoving Patch 5
2.1 Autonomy..................................... 6
2.2 Unitarity...................................... 8
3 The Canonical Measure 9
4 Minisuperspace 12
4.1 Canonicalscalareld............................... 13
4.2 Scalarperfectuid ................................ 15
4.3 Theatnessproblem ............................... 17
4.4 Likelihoodofination .............................. 20
5 Perturbations 22
5.1 Description of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Computation of the measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Likelihoodofination .............................. 25
6 Discussion 28
7 Appendix: Eternal Inﬂation 31
2
1 Introduction
Inﬂationary cosmology [1, 2, 3] has come to play a central role in our modern understanding
of the universe. Long understood as a solution to the horizon and ﬂatness problems, the
success of inﬂation-like perturbations (adiabatic, Gaussian, approximately scale-invariant)
at explaining a multitude of observations has led most cosmologists to believe that some
implementation of inﬂation is likely to be responsible for determining the initial conditions
of our observable universe.
Nevertheless, our understanding of the fundamental workings of inﬂation lags behind our
progress in observational cosmology. Although there are many models, we do not have a
single standout candidate for a speciﬁc particle-physics realization of the inﬂaton and its
dynamics. The fact that the scale of inﬂation is likely to be near the Planck scale opens the
door to a number of unanticipated physical phenomena. Less often emphasized is our tenuous
grip on the deep question of whether inﬂation actually delivers on its promise: providing a
dynamical mechanism that turns a wide variety of plausible initial states into the apparently
ﬁnely-tuned conditions characteristic of our observable universe.
The point of inﬂation is to make the evolution of our observable universe seem natural.
One can take the attitude that initial conditions are simply to be accepted, rather than
explained – we only have one universe, and should learn to deal with it, rather than seek
explanations for the particular state in which we ﬁnd it. In that case, there would never be
any reason to contemplate inﬂation. The reason why inﬂation seems compelling is because
we are more ambitious: we would like to understand why the universe seems to be one
way, rather than some other way. By its own standards, the inﬂationary paradigm bears
the burden of establishing that inﬂation is itself natural (or at least more natural than the
alternatives).
It has been recognized for some time that there is tension between this goal and the
underlying structure of classical mechanics (or quantum mechanics, for that matter). A key
feature of classical mechanics is conservation of information: the time-evolution map from
states at one time to states at some later time is invertible and volume-preserving, so that
the earlier states can be unambiguously recovered from the later states. This property is
encapsulated by Liouville’s theorem, which states that a distribution function in the space of
states remains constant along trajectories; roughly speaking, a certain number of states at one
time always evolves into precisely the same number of states at any other time. In quantum
mechanics, an analogous property is guaranteed by unitarity of the time-evolution operator;
most of our analysis here will be purely classical, but we will refer to the conservation of the
number of states as “unitarity” for convenience.
The conﬂict with the philosophy of inﬂation is clear. Inﬂation attempts to account for
the apparent ﬁne-tuning of our early universe by oﬀering a mechanism by which a relatively
natural early condition will robustly evolve into an apparently ﬁnely-tuned later condition.
But if that evolution is unitary, it is impossible for any mechanism to evolve a large number
of states into a small number, so the number of initial conditions corresponding to inﬂation
must be correspondingly small, calling into question their status as “relatively natural.” This
point has been emphasized by Penrose [4], and has been subsequently discussed elsewhere
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. As long as it operates within the framework of unitary
3
evolution, the best inﬂation can do is to move the set of initial conditions that creates a
smooth, ﬂat universe at late times from one part of phase space to another part; it cannot
increase the size of that set.
As a logical possibility, the true evolution of the universe may be non-unitary. Indeed,
discussions of cosmology often proceed as if this were the case, as we discuss below. The
justiﬁcation for this perspective is that a comoving patch of space is smaller at earlier times,
and therefore can accommodate fewer modes of quantum ﬁelds. But there is nothing in
quantum ﬁeld theory, or anything we know about gravity, to indicate that evolution is
fundamentally non-unitary. The simplest resolution is to imagine that there are a large
number of states that are not described by quantum ﬁelds in a smooth background (e.g.,
with Planckian spacetime curvature or the quantum-mechanical version thereof). Even if we
don’t have a straightforward description of the complete set of such states, the underlying
principle of unitarity is suﬃcient to imply that they must exist.
If unitary evolution is respected, there is nothing special about “initial” states; the state
at any one moment of time speciﬁes the evolution just as well as the state at any other time.
In that light, the issue of cosmological ﬁne-tuning is a question about histories, not simply
about initial conditions. Our goal should not be to show that generic initial conditions give
rise to the early universe we observe; Liouville’s theorem forbids it. Given the degrees of
freedom constituting our observable universe, and the macroscopic features of their current
state, the vast majority of possible evolutions do not arise from a smooth Big Bang begin-
ning. Therefore, a legitimate explanation for cosmological ﬁne-tuning would show that not
all histories are equally likely – that the history we observe is very natural within the actual
evolution of the universe, even though it belongs to a tiny fraction of all conceivable trajec-
tories. In particular, a convincing scenario would possess the property that when the degrees
of freedom associated with our observable universe are in the kind of state we currently ﬁnd
them in, it is most often in the aftermath of a smooth Big Bang.
We can imagine two routes to this goal: either our present condition only occurs once,
and the particular history of our universe is simply highly non-generic (perhaps due to
an underlying principle that determines the wave function of the universe); or conditions
like those of our observable universe occur many times within a much larger multiverse,
and the dynamics has the property that most appearances of our local conditions (in some
appropriate measure) are associated with smooth Big-Bang-like beginnings. In either case,
inﬂation might very well play a crucial role in the evolution of the universe, but it does not
by itself constitute an answer to the puzzle of cosmological ﬁne-tuning.
In this paper we try to quantify the issues of cosmological ﬁne-tuning in the context
of unitary evolution, using the canonical measure on the space of solutions to Einstein’s
equations developed by Gibbons, Hawking, and Stewart [5]. Considering ﬁrst the measure
on purely Robertson-Walker cosmologies (without perturbations) as a function of spatial
curvature, there is a divergence at zero curvature. In other words, curved RW cosmologies
are a set of measure zero – the ﬂatness problem, as conventionally understood, does not exist.
This divergence has no immediate physical relevance, as the real world is not described by
a perfectly Robertson-Walker metric. Nevertheless, it serves as a cautionary example for
the importance of considering the space of initial conditions in a mathematically rigorous
way, rather than relying on our intuition. We therefore perform a similar analysis for the
4
case of perturbed universes, to verify that there is not any hidden divergence at perfect
homogeneity. We ﬁnd that there is not; any individual perturbation can be written as an
oscillator with a time-dependent mass, and the measure is ﬂat in the usual space of coordinate
and momentum. The homogeneity of the universe represents a true ﬁne tuning; there is no
reason for the universe to be smooth.
We also use the canonical measure to investigate the likelihood of inﬂation. In the
minisuperspace approximation, we ﬁnd that inﬂation can be very probable, depending on
the inﬂaton potential considered. However, this approximation is wildly inappropriate for
this problem; it is essential to consider perturbations. If we restrict ourselves to universes
that look realistic at the epoch of matter-radiation equality, we ﬁnd that only a negligible
fraction were suﬃciently smooth at early times to allow for inﬂation. This simply reﬂects
the aforementioned fact that there are many more inhomogeneous states at early times than
smooth ones.
We are not suggesting that inﬂation plays no role in cosmological dynamics; only that
it is not suﬃcient to explain how our observed early universe arose from generic initial
conditions. Inﬂation requires very speciﬁc conditions to occur – a patch of space dominated
by potential energy over a region larger than the corresponding Hubble length [15] – and these
conditions are an extremely small fraction of all possible states. However, while inﬂationary
conditions are very few, there is something simple and compelling about them. Without
inﬂation, when the Hubble parameter was of order the Planck scale our universe needed
to be smooth over a length scale many orders of magnitude larger than the Planck length.
With inﬂation, by contrast, a smooth volume of order the Planck length can evolve into our
entire observable universe. There are fewer such states than those required by conventional
Big Bang cosmology, but it is not hard to imagine that they are somehow easier to create.
In other words, given that the history of our observable universe seems non-generic by
any conceivable measure, it seems very plausible that some hypothetical theory of initial
conditions (or multiverse dynamics) creates the necessary initial conditions through the
mechanism of inﬂation, rather than by creating a radiation-dominated Big Bang universe
directly. We argue that this is the best way to understand the role of inﬂation, rather than
as a solution to the horizon and ﬂatness problems.
The lesson of our investigation is that the state of the universe does appear unnatural
from the point of view of the canonical measure on the space of trajectories, and that no
choice of unitary evolution can alleviate that ﬁne-tuning, whether it be inﬂation or any other
mechanism. Inﬂation can alter the set of initial conditions that leads to a universe like ours,
but it cannot make it any larger. Inﬂation does not remove the need for a theory of initial
conditions; it brings that need into sharper focus.
2 The Evolution of our Comoving Patch
For many years, the paradigm for fundamental physics has been information-conserving
dynamical laws applied to initial data. A consequence of information conservation is re-
versibility: the state of the system at any one time is suﬃcient to recover its initial state, or
indeed any state in the past or future. The goal of this section is to lay out the motivations
5
for treating the degrees of freedom of our observable universe as a system obeying reversible
dynamics, and to establish the limitations of that approach.
Both quantum mechanics and classical mechanics feature this kind of unitary evolution.1
In the Hamiltonian formulation of classical mechanics, a state is an element of phase space,
speciﬁed by coordinates qi(t) and momenta pi(t). Time evolution is governed by Hamilton’s
equations,
˙qi=H
∂pi
,˙pi=H
∂qi,(1)
where His the Hamiltonian. In quantum mechanics, a state is given by a wave function
|ψ(t)iwhich deﬁnes a ray in Hilbert space. Time evolution is governed by the Schr¨odinger
equation, ˆ
H|Ψi=i∂t|Ψi,(2)
where ˆ
His the Hamiltonian operator, or equivalently by the von Neumann equation,
tˆρ=i[ˆ
H,ˆρ],(3)
where ˆρ(t) = |ψ(t)ihψ(t)|is the density operator. In either formalism, knowledge of the state
at any one moment of time is suﬃcient (given the Hamiltonian) to determine the state at all
other times. While we don’t yet know the complete laws of fundamental physics, the most
conservative assumption we could make would be to preserve the concept of unitarity. Even
without knowing the Hamiltonian or the space of states, we will see that the principle of
unitarity alone oﬀers important insights into cosmological ﬁne-tuning problems.
Although the assumption of unitary evolution seems like a mild one, there are challenges
to applying the idea directly to an expanding universe. We can only observe a ﬁnite part of
the universe, and the physical size of that part changes with time. The former feature implies
that the region we observe is not a truly closed system, and the latter implies that the set of
ﬁeld modes within this region is not ﬁxed. Both aspects could be taken to imply that, even
if the underlying laws of fundamental physics are perfectly unitary, it would nevertheless
be inappropriate to apply the principle of unitarity to the the part of the universe we can
observe.
We will take the stance that it is nevertheless sensible to proceed under the assumption
that the degrees of freedom describing our observable universe evolve according to unitary
dynamical laws, even if that assumption is an approximation. In this section we oﬀer the jus-
tiﬁcation for this assumption. In particular we discuss two separate parts to this claim: that
the observable universe evolves autonomously (as a closed system), and that this autonomous
evolution is governed by unitary laws.
2.1 Autonomy
We live in an expanding universe that is approximately homogeneous and isotropic on large
scales. We can therefore consider our universe as a perturbation of an exactly homogenous
1The collapse of the wave function in quantum mechanics is an apparent exception. We will not address
this phenomenon, implicitly assuming something like the many-worlds interpretation, in which wave function
collapse is only apparent and the true evolution is perfectly unitary.
6
time
observability
cuto surface
our comoving patch Σ
(”the observable universe”)
past
light
cone
expansion of space
us
comoving
worldlines
Figure 1: The physical system corresponding to our observable universe. Our comoving patch
is deﬁned by the interior of the intersection of our past light cone with a cutoﬀ surface, for
example the surface of last scattering. This illustration is not geometrically faithful, as the
expansion is not linear in time. Despite the change in physical size, we assume that the
space of states is of equal size at every moment.
and isotropic (Robertson-Walker) background spacetime. Deﬁning a particular map from the
background to our physical spacetime involves a choice of gauge. Nothing that we are going
to do depends on how that gauge is chosen, as long as it is deﬁned consistently throughout
the history of the universe. Henceforth we assume that we’ve chosen a gauge.
The map from the RW background spacetime to our universe provides two crucial ele-
ments: a foliation into time slices, and a congruence of comoving geodesic worldlines. The
time slicing allows us to think of the universe as a ﬁxed set of degrees of freedom evolving
through time, obeying Hamilton’s equations. At each moment in time there exists an exact
value of the (background) Hubble parameter and all other cosmological parameters.
The notion of comoving worldlines, orthogonal to spacelike hypersurfaces of constant
Hubble parameter, allows us to deﬁne what we mean by our comoving patch. If there is a
Big Bang singularity in our past, there is a corresponding particle horizon, deﬁned by the
intersection of our past light cone with the singularity. However, independent of the precise
nature of the Big Bang, there is an eﬀective limit to our ability to observe the past; in
practice this is provided by the surface of last scattering, although in principle observations
of gravitational waves or other particles could extend the surface backwards. The precise
details of where we draw the surface aren’t important to our arguments. What matters is
that there exists a well-deﬁned region of three-space interior to the intersection of our past
light cone with the observability surface past which we can’t see. Our comoving patch, Σ,
is simply the physical system deﬁned by the extension of that region forward in time via
comoving worldlines, as shown in Figure 1.
Our assumption is that this comoving patch can be considered as a set of degrees of
7
freedom evolving autonomously through time, free of inﬂuence from the rest of the universe.
This is clearly an approximation, as an observer stationed close to the boundary of our patch
would see particles pass both into and out of that region; our comoving patch isn’t truly a
closed system. However, the fact that the observable universe is homogenous implies that
the net eﬀect of that exchange of particles is very small. In particular, we generally don’t
believe that what happens inside our observable universe depends in any signiﬁcant way on
what happens outside.
Note that we are not necessarily assuming that our observable universe is in a pure quan-
tum state, free of entanglement with external degrees of freedom; such entanglements don’t
aﬀect the local dynamics of the internal degrees of freedom, and therefore are complete com-
patible with the von Neumann equation (3). We are, however, assuming that the appropriate
Hamiltonian is local in space. Holography implies that this is not likely to be strictly true,
but it seems like an eﬀective approximation for the universe we observe.
2.2 Unitarity
Autonomy implies that we can consider our comoving patch as a ﬁxed set of degrees of
freedom, evolving through time. Our other crucial assumption is that this evolution is
unitary (reversible). Even if the underlying fundamental laws of physics are unitary, it is
not completely obvious that the eﬀective evolution of our comoving patch evolves this way.
Indeed, this issue is at the heart of the disagreement between those who have emphasized
the amount of ﬁne-tuning required by inﬂationary initial conditions [4, 8, 9, 11, 12] and those
who have argued that they are natural [10, 14].
The issue revolves around the time-dependent nature of the cutoﬀ on modes of a quantum
ﬁeld in an expanding universe. Since we are working in a comoving patch, there is a natural
infrared cutoﬀ given by the size of the patch, a length scale of order λIR aH1
0, where ais
the scale factor (normalized to unity today) and H0is the current Hubble parameter. But
there is also a ﬁxed ultraviolet cutoﬀ at the Planck length, λUV Lpl =8πG. Clearly
the total number of modes that ﬁt in between these two cutoﬀs increases with time as the
universe expands. It is therefore tempting to conclude that the space of states is getting
larger.
We can’t deﬁnitively address this question in the absence of a theory of quantum gravity,
but for purposes of this paper we will assume that the space of states is not getting larger –
which would violate the assumption of unitarity – but the nature of the states is changing.
In particular, the subset of states that can usefully be described in terms of quantum ﬁelds
on a smooth spacetime background is changing, but those are only a (very small) minority
of all possible states.
The justiﬁcation for this view comes from the assumed reversibility of the underlying
laws. Consider the macrostate of our universe today – the set of all microstates compatible
with the macroscopic conﬁguration we observe. For any given amount of energy density,
there are two solutions to the Friedmann equation, one with positive expansion rate and
one with negative expansion rate (unless the expansion rate is precisely zero, when the
solution is unique). So there are an equal number of microstates that are similar to our
8
current conﬁguration, except that the universe is contracting rather than expanding. As the
universe contracts, each of those states must evolve into some unique state a ﬁxed time later;
therefore, the number of states accessible to the universe for diﬀerent values of the Hubble
parameter (or diﬀerent moments in time) is constant.
Most of the states available when the universe is smaller, however, are not described
by quantum ﬁelds on a smooth background. This is reﬂected in the fact that spatial in-
homogeneities would be generically expected to grow, rather than shrink, as the universe
contracted. The eﬀect of gravity on the state counting becomes signiﬁcant, and in particular
we would expect copious production of black holes. These would appear as white holes in
the time-reversed expanding description. Therefore, the overwhelming majority of states at
early times that could evolve into something like our current observable universe are not
relatively smooth spacetimes with gently ﬂuctuating quantum ﬁelds; they are expected to
be wildly inhomogeneous, ﬁlled with white holes or at least Planck-scale curvatures.
We do not know enough about quantum gravity to explicitly enumerate these states,
although some attempts to describe them have been made (see e.g. [16]). But we don’t
need to know how to describe them; the underlying assumption of unitarity implies that
they are there, whether we can describe them or not. (Similarly, the Bekenstein-Hawking
entropy formula is conventionally taken to imply a large number of states for macroscopic
black holes, even if there is no general description for what those individual states are.)
This argument is not new, and it is often stated in terms of the entropy of our comoving
patch [4, 12]. In the current universe, this entropy is dominated by black holes, and has
a value of order SΣ(t0)10104 [17]. If all the matter were part of a single black hole it
would be as large as SΣ(BH) 10122. At early times, when inhomogeneities were small and
local gravitational eﬀects were negligible, the entropy was of order SΣ(RD) 1088. If we
assume that the entropy is the logarithm of the number of macroscopically indistinguishable
microstates, and that every microstate within the current macrostate corresponds to a unique
predecessor at earlier times, it is clear that the vast majority of states from which our
present universe might have evolved don’t look anything like the smooth radiation-dominated
conﬁguration we actually believe existed (since exp[10104]exp[1088]).
This distinction between the number of states implied by the assumption of unitarity
and the number of states that could reasonably be described by quantum ﬁelds on a smooth
background is absolutely crucial for the question of how ﬁnely-tuned are the conditions
necessary to begin inﬂation. If we were to start with a conﬁguration of small size and very
high density, and consider only those states described by ﬁeld theory, we would dramatically
undercount the total number of states. Unitarity could possibly be violated in an ultimate
theory, but we will accept it for the remainder of this paper.
3 The Canonical Measure
In order to quantify the issue of ﬁne-tuning in the context of unitary evolution, we review
the canonical measure on the space of trajectories, as examined by Gibbons, Hawking, and
Stewart [5]. Despite subtleties associated with coordinate invariance, GR can be cast as
a conventional Hamiltonian system, with an inﬁnite-dimensional phase space and a set of
9
constraints. The state of a classical system is described by a point γin a phase space Γ,
with canonical coordinates qiand momenta pi. The index igoes from 1 to n, so that phase
space is 2n-dimensional. The classical equations of motion are Hamilton’s equations (1).
Equivalently, evolution is generated by a Hamiltonian phase ﬂow with tangent vector
V=H
∂pi
∂qiH
∂qi
∂pi
.(4)
Phase space is a symplectic manifold, which means that it naturally comes equipped with
a symplectic form, which is a closed 2-form on Γ:
ω=
n
X
i=1
dpidqi,dω= 0 .(5)
The existence of the symplectic form provides us with a naturally-deﬁned measure on phase
space,
Ω = (1)n(n1)/2
n!ωn.(6)
This is the Liouville measure, a 2n-form on Γ. It corresponds to the usual way of integrating
distributions over regions of phase space,
Zf(γ)Ω = Zf(qi, pi)dnqdnp . (7)
The Liouville measure is conserved under Hamiltonian evolution. If we begin with a
region AΓ, and it evolves into a region A0, Liouville’s theorem states that
ZA
Ω = ZA0
.(8)
The inﬁnitesimal version of this result is that the Lie derivative of Ω with respect to the
vector ﬁeld Vvanishes,
LV = 0 .(9)
These results can be traced back to the fact that the original symplectic form ωis also
invariant under the ﬂow:
LVω= 0 ,(10)
so any form constructed from powers of ωwill be invariant.
In classical statistical mechanics, the Liouville measure can be used to assign weights
to diﬀerent distributions on phase space. That is not equivalent to assigning probabilities
to diﬀerent sets of states, which requires some additional assumption. However, since the
Liouville measure is the only naturally-deﬁned measure on phase space, we often assume that
it is proportional to the probability in the absence of further information; this is essentially
Laplace’s “Principle of Indiﬀerence.” Indeed, in statistical mechanics we typically assume
that microstates are distributed with equal probability with respect to the Liouville measure,
consistent with known macroscopic constraints.
10
In cosmology, we don’t typically imagine choosing a random state of the universe, subject
to some constraints. When we consider questions of ﬁne-tuning, however, we are comparing
the real world to what we think a randomly-chosen history of the universe would be like. The
assumption of some sort of measure is absolutely necessary for making sense of cosmological
ﬁne-tuning arguments; otherwise all we can say is that we live in the universe we see, and
no further explanation is needed. (Note that this measure on the space of solutions to
Einstein’s equation is conceptually distinct from a measure on observers in a multiverse,
which is sometimes used to calculate expectation values for cosmological parameters based
on the anthropic principle.)
GHS [5] showed how the Liouville measure on phase space could be used to deﬁne a unique
measure on the space of solutions (see also [6, 7, 13]). In general relativity we impose the
Hamiltonian constraint, so we can consider the (2n1)-dimensional constraint hypersurface
of ﬁxed Hamiltonian,
C= Γ/{H =H}.(11)
For Robertson-Walker cosmology, the Hamiltonian precisely vanishes for either open or closed
universes, so we can take H= 0. Then we consider the space of classical trajectories within
this constraint hypersurface:
M=C/V , (12)
where the quotient by the evolution vector ﬁeld Vmeans that two points are equivalent if
they are connected by a classical trajectory. Note that this is well-deﬁned, in the sense that
points in Calways stay within C, because the Hamiltonian is conserved.
As Mis a submanifold of Γ, the measure is constructed by pulling back the symplectic
form from Γ to Mand raising it to the (n1)th power. GHS constructed a useful explicit
form by choosing the nth coordinate on phase space to be the time, qn=t, so that the
conjugate momentum becomes the Hamiltonian itself, pn=H. The symplectic form is then
ω=eω+ dH ∧ dt , (13)
where
eω=
n1
X
i=1
dpidqi.(14)
The pullback of ωonto Cthen has precisely the same coordinate expression as (14), and
we will simply refer to this pullback as eωfrom now on. It is automatically transverse to
the Hamiltonian ﬂow (eω(V) = 0), and therefore deﬁnes a symplectic form on the space of
trajectories M. The associated measure is a (2n2)-form,
Θ = (1)(n1)(n2)/2
(n1)! eωn1.(15)
We will refer to this as the GHS measure; it is the unique measure on the space of trajectories
that is positive, independent of arbitrary choices, and respects the appropriate symmetries
[5].
11
To evaluate the measure we need to deﬁne coordinates on the space of trajectories. We can
choose a hypersurface Σ in phase space that is transverse to the evolution trajectories, and
use the coordinates on phase space restricted to that hypersurface. An important property
of the GHS measure is that the integral over a region within a hypersurface is independent
of which hypersurface we chose, so long as it intersects the same set of trajectories; if S1
and S2are subsets, respectively, of two transverse hypersurfaces Σ1and Σ2in C, with the
property that the set of trajectories passing through S1is the same as that passing through
S2, then ZS1
Θ = ZS2
Θ.(16)
The property that the measure on trajectories is local in phase space has a crucial im-
plication for studies of cosmological ﬁne-tuning. Imagine that we specify a certain set of
trajectories by their macroscopic properties today – cosmological solutions that are approx-
imately homogeneous, isotropic, and spatially ﬂat, suitably speciﬁed in terms of canonical
coordinates and momenta. It is immediately clear that the measure on this set is indepen-
dent of the behavior in very diﬀerent regions of phase space, e.g. for high-density states
corresponding to early times. Therefore, no choice of early-universe Hamiltonian can make
the current universe more or less ﬁnely tuned. No new early-universe phenomena can change
the measure on a set of universes speciﬁed at late times, because we can always evaluate the
measure on a late-time hypersurface without reference to the behavior of the universe at any
earlier time.2At heart, this is a direct consequence of Liouville’s theorem.
4 Minisuperspace
In this section, we evaluate the measure on the space of solutions to Einstein’s equation in
minisuperspace (Robertson-Walker) cosmology with a scalar ﬁeld, applying the results to
the ﬂatness problem and the likelihood of inﬂation. We will look at two speciﬁc models:
a scalar with a canonical kinetic term and a potential, and a scalar with a non-canonical
kinetic term chosen to mimic a perfect-ﬂuid equation of state.
A scalar ﬁeld coupled to general relativity is governed by an action
S=Zd4xg1
2R+P(X, φ),(17)
where Ris the curvature scalar and Pis the Lagrange density of the scalar ﬁeld φ. We have
set m2
pl = 8πG = 1 for convenience. The scalar Lagrangian is taken to be a function of the
ﬁeld value and and the kinetic scalar X, deﬁned by
X≡ −1
2gµν µφνφ. (18)
2On the other hand, if the eﬀective Hamiltonian is time-dependent, what looks like a generic state at
early times can evolve into a non-generic state at later times, as energy can be injected into the system. This
is related to the recent proposal of weak gravity in the early universe [18].
12
We will consider homogeneous scalar ﬁelds φ(t) deﬁned in a Robertson-Walker metric,
ds2=N2dt2+a2(t)dr2
1kr2+r2d2,(19)
where the spatial curvature parameter kcan be normalized to 1, 0, or +1 (so that a(t0) is
not normalized to unity). Nis the lapse function, which acts as a Lagrange multiplier. We
then have
X=1
2N2˙
φ2.(20)
4.1 Canonical scalar ﬁeld
P(X, φ) = XV(φ).(21)
The Lagrangian for the combined gravity-scalar system in minisuperspace is
L=3N1a˙a2+ 3Nak +1
2N1a3˙
φ2Na3V(φ).(22)
The canonical coordinates can be taken to be the lapse function N, the scale factor a, and
the scalar ﬁeld φ. The conjugate momenta are given by pi=∂L/∂qi, implying
pN= 0 , pa=6N1a˙a , pφ=N1a3˙
φ . (23)
The vanishing of pNreﬂects the fact that the lapse function is a non-dynamical Lagrange
multiplier. We can do a Legendre transformation to calculate the Hamiltonian, obtaining
H=Xpi˙qiL(pi, qi) (24)
=Np2
a
12a+p2
φ
2a3+a3V(φ)3ak.(25)
Varying with respect to Ngives the Hamiltonian constraint, H= 0, which is just the
Friedmann equation,
H2=1
31
2˙
φ2+V(φ)3k
a2.(26)
Henceforth we will set N= 1 (consistent with the equations of motion), leaving us with
a four-dimensional phase space,
Γ = {φ, pφ, a, pa}.(27)
The GHS measure on the space of trajectories is just the the Liouville measure subject to
the constraint that H= 0,
13
Note that the measure in this example is a two-form; the full phase space is four-dimensional,
the Hamiltonian constraint surface is three dimensional, so the space of trajectories is two-
dimensional.
To express the measure in a convenient form, we use the Friedmann equation to eliminate
one of the phase-space variables. Solving for pφgives us
pφ=1
6a2p2
a2a6V(φ)+6a4k1/2
.(29)
We can change variables from pato Husing pa=6a2H, so that
pφ=6a6H22a6V+ 6a4k1/2.(30)
Our coordinates on the constraint hypersurface Care therefore {φ, a, H}. The basis one-
forms appearing in (28) are
dpa=12aHda 6a2dH (31)
and
dpφ=6a4HdH a4V0+ 6a(3a2H2a2V+ 2k)da
(6a2H22a2V+ 6k)1/2,(32)
where V0(φ) = dV/dφ. Plug into the expression (28) for the measure, whose components
become
ΘφH =6a4
(6a2H22a2V+ 6k)1/2
ΘHa =6a2
Θ= 6 3a3H2a3V+ 2ak
(6a2H22a2V+ 6k)1/2.(33)
The measure is calculated by choosing some transverse surface Σ in phase space, and
integrating Θ over a subset of that surface. If we choose coordinates such that one coordinate
is constant over Σ, we simply integrate the orthogonal component of Θ with respect to the
other coordinates. One possible choice of the surface Σ is to ﬁx the Hubble parameter,
Σ : {H=H}.(34)
Any consistent deﬁnition is equally legitimate; however, this choice corresponds to our infor-
mal idea that initial conditions are set in the early universe when the Hubble parameter is
near the Planck scale. The measure evaluated on a surface of constant His then the integral
of Θ,
µ=6ZH=H
3a3H2
a3V+ 2ak
(6a2H2
where the minus sign indicates that we have chosen an orientation that will give us a positive
ﬁnal answer. We can make this expression look more physically transparent by introducing
variables
˙
φ˙
φ2
6H2
,VV(φ)
3H2
,k≡ − k
a2H2
,(36)
14
so that the Friedmann equation is equivalent to
˙
φ+ ΩV+ Ωk= 1.(37)
The scale factor is strictly positive, so that integrating over all values of Ωkis equivalent to
integrating over all values of a. Note that k/k= 1/|k|. We therefore have
da =1
2H|k|3/2dk,(38)
and the measure becomes
µ= 3r3
2H2
ZH=H
1V2
3k
|k|5/2(1 Vk)1/2dk(39)
= 3r3
2H2
ZH=H
˙
φ1
3k
|k|5/21/2
˙
φ
dkdφ, (40)
where ˙
φ(φ, k) is deﬁned by (37).
This integral is divergent. One divergence clearly occurs for small values of the curvature
parameter, Ωk0, as the denominator includes a factor of |k|5/2. The integrand also
blows up at Ω ˙
φ= 0 (or equivalently at ΩV+ Ωk= 1), but the integral in that region remains
ﬁnite. The integral would also diverge if Ωkor ˙
φwere allowed to become arbitrarily large,
but that could be controlled by only integrating over a ﬁnite range for those quantities, e.g.
under the theory that Planckian energy densities or curvatures should not be included in
this classical description.
The important divergence, therefore, is the one at Ωk0, i.e. for ﬂat universes. We
discuss the implications of this divergence in Section 4.3.
4.2 Scalar perfect ﬂuid
In conventional Big Bang cosmology, we generally consider perfect-ﬂuid sources of energy
such as matter or radiation, rather than using a single scalar ﬁeld. This situation is slightly
more diﬃcult to analyze as a problem in phase space, as homogeneity and isotropy are only
recovered after averaging over many individual particles. However, we can model a perfect
ﬂuid with an (almost) arbitrary equation of state by a scalar ﬁeld with a non-canonical
kinetic term [19].
Consider the action (17), where the scalar Lagrangian takes the form P(X, φ), where
X=(µφ)2/2. In a Robertson-Walker background, the energy-momentum tensor takes
the form of a perfect ﬂuid,
Tµν = (ρ+P)UµUνP gµν ,(41)
where the pressure is equal to the scalar Lagrange density itself (thereby accounting for the
choice of notation). The ﬂuid has four-velocity
Uµ= (2X)1/2µφ(42)
15
and energy density
ρ= 2XXPP. (43)
We will be interested in a vanishing potential but a non-canonical kinetic term,
P(X, φ) = 2n1
nXn=1
2nN2n˙
φ2n.(44)
This gives a ﬂuid with a density
ρ=2n1
2nN2n˙
φ2n,(45)
corresponding to a constant equation-of-state parameter
w=P=1
2n1,(46)
as can easily be checked. Therefore we can model the behavior of radiation (w= 1/3) by
choosing n= 2, and approximate matter (w= 0) by choosing nvery large.
The scalar-Einstein Lagrangian in a Robertson-Walker background takes the form
L=3N1a˙a2+ 3Nak +1
2nN(2n1)a3˙
φ2n,(47)
and the Friedmann equation is
H2˙a
a2
=2n1
6n˙
φ2nk
a2,(48)
where we have set N= 1. We can duplicate the steps taken in the previous section, to
evaluate the GHS measure in terms of coordinates {φ, a, H}. We end up with
ΘφH =2n1
2n1/2n6a6n/(2n1)
[a6n/(2n1)3H2+ 3a2(n+1)/(2n1)k]1/2n
ΘHa =6a2
Θ= 6(2n1)(12n)/2nna(4n+1)/(2n1)3H2+ (n+ 1)a3/(2n1)k
[2na6n/(2n1)3H2+ 6na2(n+1)/(2n1)k]1/2n.(49)
To calculate the measure of a set of trajectories over a surface of constant H=H, we
integrate Θover aand φ. This yields
µ=32n1
6n(12n)/2nZH=H
a2H2
+(n+1)
3na2k
(H2
Note that the integrand has no dependence on φ, since there was no potential in the original
action. We therefore deﬁne
x32n1
6n(12n)/2nZdφ, (51)
16
which contributes an overall multiplicative constant to the measure. As before, it is conve-
nient to change variables from ato Ωk=k/a2H2
. This leaves us with
µ=x
2H(n+1)/n
Z1(n+1)
3nk
|k|5/2(1 k)1/2ndk.(52)
This will diverge for small Ωkfor any value of n; all of the measure is at spatially ﬂat
universes. This of course includes the case of radiation, n= 2. Therefore, the divergence we
found in the previous subsection for ﬂat universes does not seem to depend on the details of
the matter action.
4.3 The ﬂatness problem
Let’s return to the expression for the measure (40) we derived for Robertson-Walker universes
with a scalar ﬁeld featuring a canonical kinetic term and a potential,
µZH=H
1V2
3k
|k|5/2(1 Vk)1/2dkdφ. (53)
We have left out the numerical constants in front, as the overall normalization is irrelevant.
It is clear that this is non-normalizable as it stands; the integral diverges near Ωk= 0,
which is certainly a physically allowed region of parameter space. This non-normalizability
is problematic if we would like to interpret the measure as determining the relative fraction
of universes with diﬀerent physical properties.
We propose that the proper way of handling such a divergence is to regularize it. That
is, we deﬁne a series of integrals that are individually ﬁnite, and which approach the original
expression as the regulator parameter is taken to zero. We can then isolate an appropriate
power of by which we can divide the regulated expression, so that we isolate the ﬁnite part
of the result as goes to zero.
The divergence in (53) can be regulated by “smoothing” the factor |k|5/2in an -
neighborhood around Ωk= 0 to get a ﬁnite integral. Consider the function
f(x) = (|x|5/2if |x| ≥ ,
5/2if |x|< .(54)
Clearly lim0f(x) = |x|5/2, our original function. The integral of f(x) over all values of
xis 10
33/2. So we obtain a normalized integral by introducing the function
F(x) =
33/2
10|x|5/2if |x| ≥ ,
3
10if |x|< ,
(55)
17
which satisﬁes RF(x)dx = 1. We can therefore regularize the integral in (53) by replacing
|k|5/2by F(Ωk), and take the limit as 0:
µlim
03/2ZH=H
F(Ωk)1V2
3k
(1 Vk)1/2dkdφ. (56)
The multiplicative factor of 3/2goes to inﬁnity in the limit, but only the ﬁnite integral
is physically relevant. We interpret this integral as deﬁning the normalized measure on the
space of cosmological spacetimes.
However, it is clear that the limit of F(x) is simply a delta function,
lim
0F(x) = δ(x),(57)
in the sense that the integral over a test function ψ(x) gives
lim
0Z
−∞
F(x)ψ(x)dx =ψ(0).(58)
Consequently, the measure is entirely concentrated on exactly ﬂat universes; universes with
nonvanishing spatial curvature are a set of measure zero. The integrated measure (56) is
equivalent to
µZH=Hp1Vdφ, (59)
with Ωkﬁxed to be 0.
Therefore, our interpretation is clear: almost all universes are spatially ﬂat. In terms of
the measure deﬁned by the classical theory itself, a “randomly chosen” cosmology will be ﬂat
with probability one. The ﬂatness problem, as conventionally understood, does not exist;
it is an artifact of informally assuming a ﬂat measure on the space of initial cosmological
parameters. Of course, any particular speciﬁc theory of initial conditions might actually
have a ﬂatness problem, if it predicts spatially-curved universes with high probability; but
that problem is not intrinsic to the standard Big Bang model by itself.
Classical general relativity is not a complete theory of gravity, and our notions of what
constitutes a “natural” set of initial conditions are inevitably informed by our guesses as to
how it will ultimately be completed by quantum gravity. At the level of the classical equations
of motion, initial data for a solution may be speciﬁed at any time; Hamilton’s equations then
deﬁne a unique solution for the complete past and future. However, we generally impose a
cutoﬀ on the validity of a classical solution when some quantity – the energy density, Hubble
parameter, or spatial curvature – reaches the Planck scale. It therefore makes sense to us
to imagine that some unknown physical process sets the initial conditions near the Planck
regime. In Robertson-Walker cosmology, we might imagine that the space of allowed initial
conditions consists of all values of the phase-space variables such that the energy density and
curvatures are all sub-Planckian; in terms of the density parameters Ωi, this corresponds to
|i,pl|<1, where the subscript “pl” denotes that the quantity is evaluated when Hmpl.
18
-1.0
-0.5
0.0
0.5
1.0
Wk
1
2
3
4
Figure 2: Two measures as a function of the curvature parameter Ωk. The GHS measure
is highly peaked near the origin, indicating a divergence at spatially ﬂat universes. (We’ve
drawn the unnormalized measure; a normalized version would simply be a δ-function.) This
is in stark contrast with the ﬂat distribution generally assumed in the discussion of the
ﬂatness problem, which we’ve plotted for Ωkbetween ±1.
What this means in practice is that we tend to assign equal probability – a ﬂat prior –
to all the allowed Ωi,pl’s when contemplating cosmological initial conditions. As a matter of
principle, it is necessary to invoke some kind of prior in order to sensibly discuss ﬁne-tuning
problems; a quantity is ﬁnely-tuned if it is drawn from a small (as deﬁned by some measure)
region of parameter space. The lesson of the GHS measure is that a ﬂat prior on Ωi,pl ignores
the structure of the classical theory itself, which comes equipped with a unique well-deﬁned
measure. In Figure 2 we plot two diﬀerent measures on the value of Ωkat the Planck scale;
the informal ﬂat prior assumed in typical discussions of the ﬂatness problem, and the GHS
measure (evaluated at ΩV= 0 for convenience). We see that using the measure deﬁned by
the classical equations of motion leads to a dramatic diﬀerence in the probability density.
Note that the model of a canonical scalar ﬁeld with a potential will allow for the possibility
of inﬂation if the potential is chosen appropriately; however, the divergence at ﬂat universes
is not because inﬂation is secretly occurring. For one thing, the divergence appears for any
choice of potential, and also in the perfect-ﬂuid model where there is no potential at all. For
another, we could always choose to evaluate the measure at late times – i.e., we could pick
the ﬁxed Hubble parameter Hto be very small. The measure on trajectories is independent
of this choice, so the divergence for ﬂat universes cannot depend on whether inﬂation occurs.
This divergence was noted in the original GHS paper [5], where it was attributed to
19
“universes with very large scale factors” due to a diﬀerent choice of variables. This is
not the most physically transparent characterization, as any open universe will eventually
have a large scale factor. It is also discussed by Gibbons and Turok [13], who correctly
attribute it to nearly-ﬂat universes. However, they advocate discarding all such universes
as physically indistinguishable, and concentrating on the non-ﬂat universes. To us, this
seems to be throwing away almost all the solutions, and keeping a set of measure zero. It is
true that universes with almost identical values of the curvature parameter will be physically
indistinguishable, but that doesn’t aﬀect the fact that almost all universes have this property.
In Hawking and Page [6] and Coule [7] the divergence is directly attributed to ﬂat universes,
but they do not seem to argue that the ﬂatness problem is therefore an illusion.
The real world is not precisely Robertson-Walker, so in some sense the ﬂatness problem
is not rigorously deﬁned; a super-Hubble-radius perturbation could lead to a deviation from
Ω = 1 in our observed universe, even if the background cosmology were spatially ﬂat. Nev-
ertheless, the unanticipated structure of the canonical measure in minisuperspace serves as
a cautionary example for the importance of considering the space of initial conditions in a
rigorous way. More directly, it raises an obvious question: if the canonical measure is concen-
trated on spatially ﬂat universes, might it also be concentrated on smooth universes, thereby
calling into question the status of the horizon problem as well as the ﬂatness problem? (We
will ﬁnd that it is not.)
4.4 Likelihood of inﬂation
A common use of the canonical measure has been to calculate the likelihood of inﬂation
[5, 6, 7]. Most recently, Gibbons and Turok [13] have argued that the fraction of universes
that inﬂate is extremely small. However, they threw away all but a set of measure zero of
trajectories, on the grounds that they all had negligibly small spatial curvature and therefore
physically indistinguishable. Inﬂation, of course, tends to make the universe spatially ﬂat,
so this procedure is potentially unfair to the likelihood of inﬂation. We therefore re-examine
this question, following the philosophy suggested by the above analysis, which implies that
almost all universes are spatially ﬂat. We choose to look only at ﬂat universes, and calculate
the fraction that experience more than sixty e-folds of inﬂation. We will look at two choices
of potential: a massive scalar, and a pseudo-Goldstone boson. (We will argue in the next
section that these results are physically irrelevant, as perturbations play a crucial role.)
We start with a massive scalar ﬁeld with mφ= 3 ×103mpl , which yields an amplitude
of perturbations that agrees with observations. We choose to evaluate the measure on the
hypersurface H= 1/3, so that the Friedmann equation becomes 1 = 1
2˙
φ2+1
2m2φ2. After
replacing the divergence at zero curvature in (52) by a delta function, the normalized measure
becomes
µ=2m
πZH=1/3r11
2m2φ2dφ. (60)
(Recall that we have set mpl = 1/8πG = 1.) The range of integration is |φ| ≤ q2/m2
φ(or
ρ˙
φ1), corresponding to V1.
20
We used the Euler method with a time step ∆t= 103to numerically follow the evolution
of the scale factor and the scalar ﬁeld. We ﬁnd that the universe undergoes more than sixty e-
folds of inﬂation for all initial values of φexcept for the range 24 to 6 if ˙
φ > 0 (for ˙
φ < 0, the
range would be 6 to 24 due to the symmetry of the potential). For simplicity, we disregard
in our calculation the expansion that occurs after the ﬁrst period of slow-roll inﬂation. (We
veriﬁed numerically that subsequent periods of slow-roll expansion are relatively brief and
lead to very little further expansion.) Excluding the region 24 φ6, the measure (60)
integrates to 0.99996. It seems highly likely to have more than sixty e-folds of inﬂation by
this standard.
As another example we consider inﬂation driven by a pseudo-Goldstone boson [20] with
potential
V(φ) = Λ4(1 + cos(φ/f)).(61)
In our calculation, we use f=8πand Λ = 103, so that the model is consistent with
WMAP3 data [21]. We evaluate the measure on the hypersurface H=H=p4/2, so
that 3H2
= 2Vmax = 4Λ4. In this case, the normalized measure becomes
µ=1
8πE[1] ZH=4/2s11
41 + cos φ
8πdφ, (62)
where E[m] is the elliptic integral R2π
0p1msin2tdt. Numerically we ﬁnd that the universe
expands by more than 60 efolds for 4.0< φ < 2.4 if ˙
φ > 0 at H=H=p4/2. (The
evenness of the potential allows us to consider only this branch of solutions.) Evaluating the
measure gives a probability of 0.171. Notice that this is not too diﬀerent from the calculation
in [20], which gives 0.2 by assuming that φis randomly distributed between 0 and 8π. We
also note that the probability is rather sensitive to the value of f; numerical evidence suggest
that it increases with f(a ﬂatter potential).
Both of these examples lead to the conclusion that inﬂation has a very reasonable chance
of occurring. Indeed, it is sometimes claimed that inﬂation is an “attractor” (see e.g. [24]),
but that is a misleading abuse of nomenclature. It is a basic feature of dynamical systems
theory that there are no attractors in true Hamiltonian mechanics; Liouville’s theorem im-
plies that the total volume of a region of phase space remains constant under time evolution.
Attractors, in the rigorous sense of the word, only occur for systems with dissipation. In-
ﬂation appears to be an attractor only because it is often convenient to portray “phase
portraits” in terms of the inﬂaton φand its time derivative, ˙
φ. But ˙
φis not the momentum
conjugate to φ; as seen in (23), with the lapse function set to N= 1, it is pφ=a3˙
φ. Tra-
jectories drawn on a (φ, ˙
φ) plot tend to approach a ﬁxed point, but only because the scale
factor ais dramatically increasing, not because of any true attractor behavior.
These calculations of the likelihood of inﬂation are of dubious physical relevance. Exam-
ining a single scalar ﬁeld in minisuperspace is an extremely unrealistic scenario. At a very
simple level, if there are other massless ﬁelds in the problem, any of them may share some
of the energy density, reducing the probability that the inﬂaton potential dominates. More
importantly, the role of perturbations is crucial. The real reason why inﬂation is unlikely
21
from the point of view of the canonical measure is not because it is unlikely in minisuper-
space, but because perturbations can easily be suﬃciently large to prevent inﬂation from
ever occurring. We examine this issue in detail in the next section.
5 Perturbations
The horizon problem is usually formulated in terms of the absence of causal contact between
widely-separated points in the early universe. Operationally, however, it comes down to the
fact that the universe is smooth over large scales. We can investigate the measure associated
with such universes by looking at perturbed Robertson-Walker cosmologies. While the set of
all perturbations deﬁnes a large-dimensional phase space, in linear perturbation theory we
can keep things simple by looking at a single mode at a time. We will ﬁnd that, in contrast
with the surprising result of the last section, the measure on perturbations is just what we
would expect – there is no divergence at nearly-smooth universes. However, this implies that
only an imperceptibly small fraction of spacetimes were suﬃciently smooth at early times to
allow for inﬂation to occur.
To calculate the measure for scalar perturbations, we need to ﬁrst compute the corre-
sponding action. We are interested in universes dominated by hydrodynamical matter such
as dust or radiation. For linear scalar perturbations, the coupled gravity-matter system can
be described by a single independent degree of freedom, as discussed by Mukhanov, Feld-
man and Brandenberger [22]; we will follow closely the discussion in [23]. After obtaining
the action, we can isolate the dynamical variables and construct the symplectic two-form
on phase space, which can then be used to compute the measure on the set of solutions
to Einstein’s equations. A slight subtlety arises because the corresponding Hamiltonian is
time-dependent, but this is easily dealt with.
5.1 Description of perturbations
In this section it will be convenient to switch to conformal time,
η=Za1dt. (63)
Derivatives with respect to ηare denoted by the superscript 0, and e
Ha0/a is related to
the Hubble parameter H= ˙a/a by e
H=aH. The Friedmann equations become
e
H2=8πG
3a2¯ρk, (64)
e
H0=4πG
3a2(¯ρ+ 3¯p),(65)
where ¯ρand ¯pare the background density and pressure. In a ﬂat universe with only matter
η(RD) = a
H0aeq
,e
H(RD) = η1,(66)
22
where aeq is the scale factor at matter-radiation equality, and now we set the current scale
factor to unity, a0= 1. Numerically, the conformal time in the radiation-dominated era is
approximately
η(T)5×1030
T(eV) eV1.(67)
The metric for a ﬂat RW universe in conformal time with scalar perturbations is
ds2=a2(η)(1 + 2Φ)2+ 2B,idηdx2+ ((1 2Ψ)δij + 2E,ij )dxidxj,(68)
where Φ, Ψ, E, and Bare scalar functions characterizing metric perturbations, and commas
denote partial derivatives. It is useful to deﬁne the gauge-invariant Newtonian potential,
Φ = φ1
a[a(BE)0]0,(69)
and the gauge-invariant energy-density perturbation,
e
δρ =δρ ¯ρ0(BE0).(70)
For scalar perturbations in the absence of anisotropic stress, these are related by
e
δρ =1
4πGa2h2Φ3e
H0+e
HΦ)i.(71)
For adiabatic perturbations (δS = 0), the potential obeys an autonomous equation,
Φ00 + 3(1 + c2
s)e
HΦ0c2
s2Φ + [2 e
H0+ (1 + 3c2
s)e
H2]Φ = 0,(72)
where c2
s=∂p/∂ρ is the speed of sound squared in the ﬂuid. This equation simpliﬁes if we
introduce the rescaled perturbation variable
uΦ
¯ρ+ ¯p,(73)
and the time-dependent parameter
θ= exp 3
2Z(1 + c2
s)e
HΦ = 1
a"2
3 1e
H0
e
H2!#1/2
.(74)
In terms of these (72) becomes
u00 c2
s2uθ00
θu= 0.(75)
The variable uis a single degree of freedom that encodes both the gravitational potential
[through (73)] and the density perturbation [through (71)]. The equation of motion (75)
corresponds to an action
Su=1
2Zd4xu02c2
su,iu,i +θ00
θu2.(76)
23
Deﬁning the conjugate momentum pu=L/∂u0=u0, we can describe the dynamics in terms
of a Hamiltonian density for an individual mode with wavenumber k,
H=1
2p2
u+1
2c2
sk2θ00
θu2.(77)
This is simply the Hamiltonian for a single degree of freedom with a time-dependent eﬀective
mass m2=c2
sk2θ00.
5.2 Computation of the measure
Given the Hamiltonian (77), we can straightforwardly compute the invariant measure on
phase space. One caveat is that now the Hamiltonian is time-dependent, because the eﬀective
mass evolves. The carrier manifold of the Hamiltonian therefore has an odd number of
dimensions. We can retain the symplecticity of a time-dependent Hamiltonian system (which
requires an even number of dimensions) by promoting time to be an addition canonical
coordinate, qn+1 =t. The conjugate momentum is minus the Hamiltonian, pn+1 =−H. We
can then deﬁne an extended Hamiltonian by
H+=H(p, q, t) + pn+1.(78)
This is formally time-independent, and recovers the original Hamiltonian equations via
˙qi=H+
∂pi,˙pi=H+
∂qi,(79)
along with two additional trivial equations ˙
t= 1 and ˙
H=H/∂t.
With tpromoted to a coordinate, the time-dependent Hamiltonian system also comes
equipped naturally with a closed symplectic two-form, now with an additional term:
ω=
n
X
i=1
dpidqidH ∧ dt. (80)
The invariance of the form of Hamilton’s equations ensures that the Lie derivative of ωwith
respect to the vector ﬁeld generated by H+vanishes. The top exterior power of ωis then
guaranteed to be conserved under the extended Hamiltonian ﬂow, and can thus play the role
of the Liouville measure for the augmented system. The GHS measure can then be obtained
by pulling back the Liouville measure onto a hypersurface intersecting the trajectories and
satisfying the constraint H+= 0.
In our case, the original system, with coordinate uand conjugate momentum pu, is
augmented to one with two coordinates uand ηand their conjugate momenta puand −H.
The extended Hamiltonian,
H+=1
2p2
u+1
2c2
sk2θ00
θu2− H,(81)
24
is time-independent and set to zero by the equations of motion. Its conservation is analogous
to the Friedmann equation constraint in the analysis of the ﬂatness problem. Using (80),
the GHS measure Θ for the perturbations is the two-form
Θ = dpudu (dH ∧ )|H+=0
=dpudu 1
2dp2
u+c2
sk2θ00
θu2
=dpudu pu(dpu)uc2
sk2θ00
θdu dη . (82)
One convenient hypersurface in which we can evaluate the ﬂux of trajectories is η=
η= constant. (This is equivalent to a surface of H= constant or a= constant, although
those are not coordinates in the phase space of the perturbation.) As ηis always positive in
a matter- and radiation-dominated universe, this surface intersects all trajectories exactly
once. We then have
µ=Zη=η
Θpuududpu
=Zη=η
dudpu.(83)
The ﬂux of trajectories crossing this surface is unity, implying that all values for uand puare
equally likely. There is nothing in the measure that would explain the small observed values
of perturbations at early times. Hence, the observed homogeneity of our universe does imply
considerable ﬁne-tuning; unlike the ﬂatness problem, the horizon problem is real.
5.3 Likelihood of inﬂation
We can use the canonical measure on perturbations to estimate the likelihood of inﬂation.
Our strategy will be to consider universes dominated by matter and radiation – i.e., the
supposed post-inﬂationary era in the universe’s history – and ask what fraction of them
could have begun with inﬂation. This is somewhat contrary to the conventional approach,
begin; but it is fully consistent with the philosophy of unitary and autonomous evolution,
and takes advantage of the feature of the canonical measure that it can be evaluated at any
time.
If inﬂation does occur, perturbations will be very small when it ends. Indeed, pertur-
bations must be sub-dominant if inﬂation is to begin in the ﬁrst place [15], and by the end
of inﬂation only small quantum ﬂuctuations in the energy density remain. It is therefore
a necessary (although not suﬃcient) condition for inﬂation to occur that perturbations be
small at early times. For convenience, we will take inﬂation to end near the GUT scale,
TG= 1016 GeV (ηG6×105eV1), although this choice is not crucial.
We therefore want to calculate what fraction of perturbed Robertson-Walker universes
are relatively smooth near the GUT scale. We take “smooth” to mean that both the density
25
contrast δ=e
δρ/ ¯ρand the Newtonian potential Φ are less than one. Because the phase space
is unbounded and the measure (83) is ﬂat, it is necessary to cut oﬀ the space of perturbations
in some way. We might deﬁne “realistic” cosmologies as those that match the homogeneity
of our observed universe at the redshift of recombination z1200, when CMB temperature
anisotropies are observed. Our expressions will be much less cumbersome, however, if we
demand smoothness at matter-radiation equality, zeq 3000 (ηeq 1031 eV1), within an
order of magnitude of recombination. Since the observed temperature anisotropies are of
order one part in 105, we therefore deﬁne a realistic universe as one with δeq 105and
Φeq 105.
There is a long-distance cutoﬀ on the modes we consider given by the size of our comoving
observable universe, extrapolated back to matter-radiation equality. The size L0of our
observable universe today is a few times H1
0= 1033 eV1, and the size of our comoving
patch at equality is aeq = 1/3000 times that, so
Leq 1030 eV1.(84)
We will also impose a short-distance cutoﬀ at the Hubble radius at equality,
H1
eq mpl a2
eq
T2
01028 eV1.(85)
The total number of modes we consider is therefore
n=Leq
H1
eq 3
106.(86)
Our short-distance cutoﬀ is chosen primarily for convenience; there is a natural ultraviolet
cutoﬀ set by the scale below which the hydrodynamical approximation becomes invalid, but
that is much shorter than H1
eq . It is clear that we are neglecting a large number of modes
that could plausibly have large amplitudes at early times; our result will therefore represent a
generous overestimate of the fraction of inﬂationary spacetimes. The ﬁnal numerical answer
will be small enough that this shortcut won’t matter.
With this setup in place, we would like to compare the measure on trajectories that are
smooth near the GUT scale to the measure on those that are realistic near matter-radiation
equality. We therefore only need to consider a single kind of evolution – long-wavelength
modes (super-Hubble-radius, cskη < 1) in a radiation-dominated universe. In that case the
general solution to our evolution equation (75) is
u=c1θ+c2θZη0
θ2dη, (87)
where c1and c2are constants. During radiation domination we have
θ=3
2aeqH0
η1.(88)
26
The solution for uis therefore
u=αη1+βη2,(89)
where αand βare constants. The conjugate momentum is
pu=αη2+ 2βη. (90)
The potential is related to uby (73). In the radiation era we have
(¯ρ+ ¯p)1/2=γη2,(91)
where we have deﬁned
γ=2mpl
aeqH0
.(92)
Our general solution is therefore
Φ = γ(αη3+β).(93)
Finally we turn to the density perturbation, which is given by (71). The 2Φ = k2Φ term
is negligible for long wavelengths, so we’re left with
e
δρ =12m3
pl
a3/2
eq H3
0
(2αη7βη4),(94)
which in turn implies
δe
δρ
¯ρ= 2γ(2αη3β).(95)
To calculate the measure, it is convenient to use αand βas the independent variables
that specify a mode. The measure is simply
µ=Zdudpu= 3 Zdαdβ. (96)
This comes from taking the derivative of (89) and (90), treating αand βas the independent
variables, and computing dudpu. No η-dependent factors appear when we write the measure
in terms of αand β. We can also express it in terms of the density contrast and Newtonian
potential,
dΦ= 6γ2η3dαdβ. (97)
Therefore, a region in the Φ-δplane at time ηGhas a measure that is larger than the same
coordinate region at time ηeq by a factor of
ηeq
ηG3
1076.(98)
27
The coordinate area of our initial region at the GUT scale is ∆ΦGδG1, while the
coordinate area of our region at equality is ∆Φeqδeq (105)2= 1010 . For each mode,
we therefore have
µ(inﬂationary)
µ(realistic) =ηG
ηeq 3∆ΦGδG
∆Φeqδeq 1066.(99)
This is saying that, for a given wave vector, only 1066 of the allowed amplitudes that are
realistic at matter-radiation equality are small at the GUT scale. To allow for inﬂation, we
require that modes of every ﬁxed comoving wavelength and direction be less than unity at
the GUT scale; the fraction of realistic cosmologies that are eligible for inﬂation is therefore
P(inﬂation) (1066)n106.6×107.(100)
This is a small number, indicating that a negligible fraction of universes that are realistic
at late times experienced a period of inﬂation at very early times. We derived this particular
value by assuming the universe was realistic at matter-radiation equality, but similarly tiny
fractions would apply had we started with any other time in the late universe. We also looked
at only a fraction of possible modes, so a more careful estimate would yield a much smaller
number. Indeed, using entropy as a proxy for the number of states yields estimates of order
1010122 [4]. Clearly, the precise numerical answer is not of the essence; the conclusion is that
inﬂationary trajectories are a negligible fraction of all possible evolutions of the universe.
A crucial feature of this analysis is that we allowed for the possibility of decaying cos-
mological perturbations; if all we know about the perturbations is that they are small at
matter-radiation equality, the generic case is that many have been decaying since earlier
times. Such decaying modes are often neglected in cosmology, but for our purposes that
would be begging the question. A successful theory of cosmological initial conditions will
account for the absence of such modes, not presume it.
6 Discussion
We have investigated the issue of cosmological ﬁne-tuning under the assumption that our
observable universe evolves unitarily through time. Using the invariant measure on cosmo-
logical solutions to Einstein’s equation, we ﬁnd that the ﬂatness problem is an illusion; in
the context of purely Robertson-Walker cosmologies, the measure diverges on ﬂat universes.
In the case of deviations from homogeneity, however, we recover something closer to the
conventional result; in appropriate variables, the measure on the phase space of any partic-
ular mode of perturbation is ﬂat, so that a generic universe would be expected to be highly
inhomogeneous.
Inﬂation by itself cannot solve the horizon problem, in the sense of making the smooth
early universe a natural outcome of a wide variety of initial conditions. The assumptions
of unitarity and autonomy applied to our comoving patch imply that any set of states at
late times necessarily corresponds to an equal number of states at early times. Diﬀerent
choices for the Hamiltonian relevant in the early universe cannot serve to focus or spread the
trajectories, which would violate Liouville’s theorem; they can only deﬂect the trajectories
28
in some overall way. Therefore, whether or not a theory allows for inﬂation has no impact
on the total fraction of initial conditions that lead to a universe that looks like ours at late
times.
This basic argument has been appreciated for some time; indeed, its essential features
were outlined by Penrose [4] even before inﬂation was invented. Nevertheless, it has failed
to make an important impact on most discussions of inﬂationary cosmology. Attitudes
toward this line of inquiry fall roughly into three camps: a small camp who believe that the
implications of Liouville’s theorem represent a signiﬁcant challenge to inﬂation’s purported
ability to address ﬁne-tuning problems [8, 9, 11, 12, 13]; an even smaller camp who explicitly
argue that the allowed space of initial conditions is much smaller than the space of later
conditions, in apparent conﬂict with the principles of unitary evolution [10, 14]; and a very
large camp who choose to ignore the issue or keep their opinions to themselves.
But this issue is crucial to understanding the role of inﬂation (or any alternative mech-
anism) in accounting for the apparent ﬁne-tuning of our universe. The part of the universe
we observe consists of a certain set of degrees of freedom, arranged in a certain way – a few
hundred billion galaxies, distributed approximately uniformly through an expanding space
– and apparently evolving from a very ﬁnely-tuned smooth Big-Bang-like beginning. Un-
derstanding why things are this way could have crucial consequences for our view of other
features of the universe, much as the inﬂationary scenario revolutionized our ideas about the
origin of cosmological perturbations.
There seem to be two possible ways we might hope to account for the apparent ﬁne-tuning
of the history of the observable universe:
1. The present conﬁguration of the universe only occurs once. In this case, the evolution
from the Big Bang to today is highly non-generic, and the question becomes why this
evolution, rather than some other one. The answer might be found in properties of the
wave function of the universe (e.g. [25]).
2. Degrees of freedom arrange themselves in conﬁgurations like the observable universe
many times in the history of a much larger multiverse. In this case, there is still
hope that the overall evolution may be generic, if it can be shown that conﬁgurations
like ours most often occur in the aftermath of a smooth Big Bang. The apparent
restrictions of Liouville’s theorem may be circumvented by imagining that the degrees
of freedom of our current universe do not describe a closed system for all time, but
interact strongly with other degrees of freedom at some times (e.g. by arising as baby
universes [12]).
In either case, inﬂation could play an important role as part of a more comprehensive
picture. While inﬂation does not make universes like ours more numerous in the space of
all possible universes, it might provide a more reasonable target for a true theory of initial
conditions, from quantum cosmology or elsewhere. (This is a possible reading of [10, 14],
although those authors seem to exclude non-smooth initial conditions a priori, rather than
relying on some well-deﬁned theory of initial conditions.)
As we have shown in this paper, most universes that are smooth at matter-radiation
equality were wildly inhomogeneous at very early times. But the converse is not true; most
29
universes that were wildly inhomogeneous at early times simply stay that way. The process
of smoothing out represents a violation of the Second Law of Thermodynamics, as entropy
decreases along the way. Even though the vast majority of trajectories that are smooth at
matter-radiation equality were inhomogeneous at early times, it seems intuitively unlikely
that the real universe behaves this way; much more plausible is the conventional supposition
that the universe was smoother (and entropy was lower) all the way back to the Big Bang.
One way of expressing why this seems more natural to us is that the corresponding initial
states are very simple to characterize: they are smooth within an appropriate comoving
volume. In contrast, the much more numerous histories that begin inhomogeneously and
proceed to smooth out are impossible to characterize in terms of macroscopically observable
quantities at early times; the fact that they will ultimately smooth out is hidden in extremely
subtle correlations between a multitude of degrees of freedom.3It seems much easier to
imagine that an ultimate theory of initial conditions will produce states that are simple to
describe rather than ones that feature an enormous number of mysterious and inaccessible
correlations. It may be true that a randomly-chosen universe like ours would have begun in
a wildly inhomogeneous state; but it’s clear that the history of our observable universe is
not a randomly-chosen evolution of the corresponding degrees of freedom.
Given that we need some theory of initial conditions to explain why our universe was
not chosen at random, the question becomes whether inﬂation provides any help to this
unknown theory. There are two ways in which it does. First, inﬂation allows the initial
patch of spacetime with a Planck-scale Hubble parameter to be physically small, while
conventional cosmology does not. If we extrapoloate a matter- and radiation-dominated
universe from today backwards in time, a comoving patch of size H1
0today corresponds to
a physical size 1026H01034Lpl 1 cm when H=mpl . In contrast, with inﬂation, the
same patch needs to be no larger than Lpl when H=mpl, as emphasized by Kofman, Linde,
and Mukhanov [10, 14]. If our purported theory of initial conditions, whether quantum
cosmology or baby-universe nucleation or some other scheme, has an easier time making
small patches of space than large ones, inﬂation would be an enormous help.
The other advantage is in the degree of smoothness required. Without inﬂation, a perfect-
ﬂuid universe with Planckian Hubble parameter would have to be extremely homogeneous
to be compatible with the current universe, while an analogous inﬂationary patch could
accommodate any amount of sub-Planckian perturbations. While the actual number of
trajectories may be smaller in the case of inﬂation, there is a sense in which the requirements
seem more natural. Within the set of initial conditions that experience suﬃcient inﬂation,
all such states give us reasonable universes at late times; in a more conventional Big Bang
cosmology, the perturbations require an additional substantial ﬁne tuning. Again, we have
a relatively plausible target for a future comprehensive theory of initial conditions: as long
as inﬂation occurs, and the perturbations are not initially super-Planckian, we will get a
reasonable universe.
3The situation resembles the time-reversal of a glass of water with an ice cube that melts over the course
of an hour. At the end of the melting process, if we reverse the momentum of every molecule in the glass, we
will describe an initial condition that evolves into an ice cube. But there’s no way of knowing that, just from
the macroscopically available information; the surprising future evolution is hidden in subtle correlations
between diﬀerent molecules.
30
These features of inﬂation are certainly not novel; it is well-known that inﬂation allows
for the creation of a universe such as our own out of a small and relatively small bubble
of false vacuum energy. We are nevertheless presenting the point in such detail because we
believe that the usual sales pitch for inﬂation is misleading; inﬂation does oﬀer important
advantages over conventional Friedmann cosmologies, but not necessarily the ones that are
often advertised. In particular, inﬂation does not by itself make our current universe more
likely; the number of trajectories that end up looking like our present universe is unaﬀected
by the possibility of inﬂation, and even when it is allowed only a tiny minority of solutions
feature it. Rather, inﬂation provides a speciﬁc kind of set-up for a true theory of initial
conditions – one that is yet to be deﬁnitively developed.
Acknowledgments
This work was supported in part by the U.S. Dept. of Energy and the Gordon and Betty
Moore Foundation. We thank Andy Albrecht, Adrienne Erickcek, Don Page, and Paul
7 Appendix: Eternal Inﬂation
Eternal inﬂation [26, 27, 28, 29] is sometimes held up as a solution to the puzzle of the
unlikeliness of inﬂation occurring. In many models of inﬂation, the process is eternal – while
some regions reheat and become radiation-dominated, other regions (increasing in physical
volume) continue to inﬂate. This may be driven by the back-reaction of large quantum
ﬂuctuations in the inﬂaton during slow-roll inﬂation, or simply by the failure of percolation
in a false vacuum with a suﬃciently small decay rate.
Through eternal inﬂation, a small initially inﬂating volume grows without bound, creating
an ever-increasing number of pocket universes that expand and cool in accordance with
conventional cosmology. Therefore, the reasoning goes, it doesn’t matter how unlikely it is
that inﬂation ever begins; as long as there is some nonzero chance that it starts, it creates
an inﬁnite number of universes within the larger multiverse, and questions of probability
become moot.
If unitary evolution is truly respected, this reasoning fails. Consider the state of the
universe at some late time t(long after inﬂation has begun), in some particular slicing. Let
us imagine that the basic idea of eternal inﬂation is correct, and the multiverse consists of
more and more localized universes of ever-increasing volume as time passes. According to
the reasoning developed in this paper, the macroscopic state of the multiverse (that is, the
set of microstates with macroscopic features identical to the multiverse at time t) will be
compatible with a very large number of past histories, only a very small fraction of which
will begin in a single inﬂating patch. The more the volume grows and the more universes
that are created, the less likely it is that this particular conﬁguration began with such a
patch. It requires more and more ﬁne-tuning to take all of the degrees of freedom and
evolve them all backward into their vacuum states in a Planck-sized region. Therefore, while
31
eternal inﬂation can create an ever-larger volume, it does so at the expense of starting in an
ever-smaller fraction of the relevant phase space.
To say the same thing in a diﬀerent way, if a multiverse mechanism is going to claim to
solve the cosmological ﬁne-tuning problems, it will have to be the case that the mechanism
applies to generic (or at least relatively common) initial data. We should be able to start
from a non-ﬁnely-tuned state, evolve it into the future (and the past), and see universes such
as our own arise. As conventionally presented, models of eternal inﬂation usually presume
a starting point that is a smooth patch with a Planckian energy density – very far from a
generic state. If it could be shown that eternal inﬂation began from generic initial data,
this objection would be largely overcome. Presumably the resulting multiverse would be
time-symmetric on large scales, as in [34, 12, 25].
It is possible that considering the entire multiverse along a single time slice is illegitimate,
and we should follow the philosophy of horizon complemenarity and only consider spacetime
patches that are observable by a single worldline. This approach would run into severe
problems with Boltzmann brains if our current de Sitter vacuum is long-lived [30, 31, 32, 33].
Alternatively, we might argue that the phase space is inﬁnitely big, and there is no sensible
way to talk about probabilities. That may ultimately be true, but represents an abandonment
of any hope of explaining cosmological ﬁne-tuning via inﬂation, rather than a defense of the
strategy.
Analogous concerns apply to cyclic cosmologies [35]. Here, conditions similar to our
observable universe happen multiple times, separated primarily in time rather than in space.
But the burden still remains to show that the conjectured evolution would proceed from
generic initial data. The fact that the multiverse is not time-symmetric (the arrow of time
points in a consistent direction from cycle to cycle) makes this seem unlikely.
References
[1] A. H. Guth, Phys. Rev. D 23, 347 (1981).
[2] A. D. Linde, Phys. Lett. B 108, 389 (1982).
[3] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
[4] R. Penrose, in S. W. Hawking and W. Israel. General Relativity: An Einstein Cente-
nary Survey. Cambridge University Press. pp. 581638. (1979).
[5] G. W. Gibbons, S. W. Hawking and J. M. Stewart, Nucl. Phys. B 281, 736 (1987).
[6] S. W. Hawking and D. N. Page, Nucl. Phys. B 298, 789 (1988).
[7] D. H. Coule, Class. Quant. Grav. 12, 455 (1995) [arXiv:gr-qc/9408026].
[8] W. G. Unruh, In *Princeton 1996, Critical dialogues in cosmology* 249-264.
[9] S. Hollands and R. M. Wald, Gen. Rel. Grav. 34, 2043 (2002) [arXiv:gr-qc/0205058].
32
[10] L. Kofman, A. Linde and V. F. Mukhanov, JHEP 0210, 057 (2002) [arXiv:hep-
th/0206088].
[11] S. Hollands and R. M. Wald, arXiv:hep-th/0210001.
[12] S. M. Carroll and J. Chen, arXiv:hep-th/0410270.
[13] G. W. Gibbons and N. Turok, arXiv:hep-th/0609095.
[14] A. Linde, arXiv:0705.0164 [hep-th].
[15] T. Vachaspati and M. Trodden, Phys. Rev. D 61, 023502 (1999) [arXiv:gr-qc/9811037].
[16] S. D. Mathur, J. Phys. Conf. Ser. 140, 012009 (2008) [arXiv:0803.3727 [hep-th]].
[17] C. A. Egan and C. H. Lineweaver, Astrophys. J. 710, 1825 (2010) [arXiv:0909.3983
[astro-ph.CO]].
[18] B. Greene, K. Hinterbichler, S. Judes and M. K. Parikh, arXiv:0911.0693 [hep-th].
[19] J. Garriga and V. F. Mukhanov, Phys. Lett. B 458, 219 (1999) [arXiv:hep-th/9904176].
[20] K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. 65, 3233 (1990).
[21] K. Freese, C. Savage and W. H. Kinney, Int. J. Mod. Phys. D 16, 2573 (2008)
[arXiv:0802.0227 [hep-ph]].
[22] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203
(1992).
[23] V. F. Mukhanov, Physical Foundations of Cosmology (2005), Cambridge University
Press.
[24] M. Trodden and S. M. Carroll, arXiv:astro-ph/0401547.
[25] J. B. Hartle, S. W. Hawking and T. Hertog, Phys. Rev. D 77, 123537 (2008)
[arXiv:0803.1663 [hep-th]].
[26] A. Vilenkin, Phys. Rev. D 27, 2848 (1983).
[27] A. D. Linde, Mod. Phys. Lett. A 1, 81 (1986).
[28] A. D. Linde, Phys. Lett. B 175, 395 (1986).
[29] A. H. Guth, J. Phys. A 40, 6811 (2007) [arXiv:hep-th/0702178].
[30] L. Dyson, M. Kleban and L. Susskind, JHEP 0210, 011 (2002) [arXiv:hep-th/0208013].
[31] A. Albrecht and L. Sorbo, Phys. Rev. D 70, 063528 (2004) [arXiv:hep-th/0405270].
33
[32] D. N. Page, Phys. Rev. D 78, 063535 (2008) [arXiv:hep-th/0610079].
[33] R. Bousso and B. Freivogel, JHEP 0706, 018 (2007) [arXiv:hep-th/0610132].
[34] A. Aguirre and S. Gratton, Phys. Rev. D 67, 083515 (2003) [arXiv:gr-qc/0301042].
[35] P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 126003 (2002) [arXiv:hep-th/0111098].
34
... To illustrate the extent to which these ambiguities are problematic, consider the concrete results of different authors with different intuitions performing computations of the relative likelihood of cosmic inflation. Advocates for inflation (Kofman, Linde, and Mukhanov 2002;Carroll and Tam 2010) proposed a measure according to which the probability of inflation was found to be infinitesimally close to 1. Inflation skeptics (Gibbons and Turok 2008) proposed an alternative measure where the probability of inflation was found to be 1 part in 10 85 ! This remarkably huge discrepancy reflects the extent to which individual beliefs can affect cosmologist's determinations of the appropriate physical principles used to justify their measure and the difficulties of resolving the tensions between Condition-A1 and Condition-A2. ...
... This dependence on H is significant. As was shown explicitly in Schiffrin and Wald (2012), the different choices of H used by inflation sceptics (Gibbons and Turok 2008) compared with inflations advocates (Kofman, Linde, and Mukhanov 2002;Carroll and Tam 2010) leads to a colossal 85 order of magnitude difference between the estimates of the likelihood of inflation. Because a choice of H corresponds to a choice of initial time, this huge numerical imbalance leads to a significant temporal asymmetry: choosing a more recent value of H gives a dramatically smaller value for the weight of the same region r φ . ...
Article
Many macroscopic physical processes are known to occur in a time-directed way despite the apparent time-symmetry of the known fundamental laws. A popular explanation is to postulate an unimaginably atypical state for the early universe --- a `Past Hypothesis' (PH) --- that seeds the time-asymmetry from which all others follow. I will argue that such a PH faces serious new difficulties. First I strengthen the grounds for existing criticism by providing a systematic analytic framework for assessing the status of the PH. I outline three broad categories of criticism that put into question a list of essential requirements of the proposal. The resulting analysis paints a grim picture for the prospects of providing an adequate formulation for an explicit PH. I then provide a new argument that substantively extends this criticism by showing that any time-independent measure on the space of models of the universe must necessarily break one of its gauge symmetries. The PH then faces a new dilemma: reject a gauge symmetry of the universe and introduce a distinction without difference or reject the time-independence of the measure and lose explanatory power.
... To illustrate the extent to which these ambiguities are problematic, consider the concrete results of different authors with different intuitions performing computations of the relative likelihood of cosmic inflation. Advocates for inflation (Kofman et al., 2002;Carroll and Tam, 2010) proposed a measure according to which the probability of inflation was found to be infinitesimally close to 1. Inflation skeptics (Gibbons and Turok, 2008) proposed an alternative measure where the probability of inflation was found to be 1 part in 10 85 ! This remarkably huge discrepancy reflects the extent to which individual beliefs can affect cosmologist's determinations of the appropriate physical principles used to justify their measure and the difficulties of resolving the tensions between Condition-A1 and Condition-A2. ...
... This dependence on H is significant. As was shown explicitly in Schiffrin and Wald (2012), the different choices of H used by inflation sceptics (Gibbons and Turok, 2008) compared with inflations advocates (Kofman et al., 2002;Carroll and Tam, 2010) leads to a colossal 85 order of magnitude difference between the estimates of the likelihood of inflation. Because a choice of H corresponds to a choice of initial time, this huge numerical imbalance leads to a significant temporal asymmetry: choosing a more recent value of H gives a dramatically smaller value for the weight of the same region r φ . ...
Preprint
Many macroscopic physical processes are known to occur in a time-directed way despite the apparent time-symmetry of the known fundamental laws. A popular explanation is to postulate an unimaginably atypical state for the early universe -- a "Past Hypothesis" (PH) -- that seeds the time-asymmetry from which all others follow. I will argue that such a PH faces serious new difficulties. First I strengthen the grounds for existing criticism by providing a systematic analytic framework for assessing the status of the PH. I outline three broad categories of criticism that put into question a list of essential requirements of the proposal. The resulting analysis paints a grim picture for the prospects of providing an adequate formulation for an explicit PH. I then provide a new argument that substantively extends this criticism by showing that any time-independent measure on the space of models of the universe must necessarily break one of its gauge symmetries. The PH then faces a new dilemma: reject a gauge symmetry of the universe and introduce a distinction without difference or reject the time-independence of the measure and lose explanatory power.
... (C1) Premise 4. The improbability of our Universe cries out for explanation (Dyson 1979, Leslie 1989, Parfit 1998, Rees 2000, Nozick 2001, Rundle 2004, Carroll 2010. ...
Article
Full-text available
This work explores foundational issues related to many-worlds theories in Cosmology. It is argued that the metaphysical picture drawn by these theories arise from metaphysical assumptions made during their formulation—most of which are problematic. I elucidate the nature of these assumptions and examine their legitimacy. I conclude the metaphysical presuppositions responsible for the apparent reliability of many-worlds theories in Cosmology are unmotivated and unwarranted by evidence. On this basis, the questions many-worlds models in Cosmology attempt to solve turn out to be a non-starter because their presupposed metaphysical grounding is ill-founded.
... I will shelve this parameter too. • Dimensionless spatial curvature (κ): this is a special case, as there are reasonable theoretical reasons to give significant probabilistic weight to κ = 0, which is a life-permitting value (Hawking and Page, 1988;Gibbons and Turok, 2008;Carroll and Tam, 2010). This is independent of cosmic inflation, which aims to provide a dynamical mechanism that sets κ ≈ 0. Whether inflation itself is fine-tuned is an interesting question (I have argued in Barnes, 2012 that it probably is), but regardless, we are not justified in assuming the kind of dimensionless, order-unity distribution discussed in regard to Premise [7]. ...
... Their work builds on earlier attempts to construct measures on spaces of trajectories, such as by Gibbons, Hawking, and Stewart (1987). The measure of Gibbons, Hawking, and Stewart (1987), however, diverges for flat FLRW universes [as indeed noted by: Hawking and Page (1988); Coule (1995); Gibbons and Turok (2008); Carroll and Tam (2010); Carroll (2013, 2014); Carroll (2014)], and the work of Carroll (2013, 2014) finds a way around this issue by explicitly constructing measures over trajectories for flat FLRW universes in the context of specific SSF potentials. Yet there remains much work to do in extending such measures to more generic background spacetimes that involve SSF models of inflation (let alone to model-independent EFT-based characterizations of inflation). ...
Preprint
The leading account of several salient observable features of our universe today is provided by the theory of cosmic inflation. But an important and thus far intractable question is whether inflation is generic, or whether it is finely tuned---requiring very precisely specified initial conditions. In this paper I argue that a recent, model-independent characterization of inflation, known as the 'effective field theory (EFT) of inflation', promises to address this question in a thoroughly modern and significantly more comprehensive way than in the existing literature. To motivate and provide context for this claim, I distill three core problems with the theory of inflation, which I dub the permissiveness problem, the initial conditions problem, and the multiverse problem. I argue that the initial conditions problem lies within the scope of EFTs of inflation as they are currently conceived, whereas the other two problems remain largely intractable: their solution must await a more complete description of the very early universe. I highlight recent work that addresses the initial conditions problem within the context of a dynamical systems analysis of a specific (state-of-the-art) EFT of inflation, and conclude with a roadmap for how such work might be extended to realize the promise claimed above.
... Their work builds on earlier attempts to construct measures on spaces of trajectories, such as by Gibbons, Hawking, and Stewart (1987). The measure of Gibbons, Hawking, and Stewart (1987), however, diverges for flat FLRW universes [as indeed noted by: Hawking and Page (1988); Coule (1995); Gibbons and Turok (2008); Carroll and Tam (2010); Carroll (2013, 2014); Carroll (2014)], and the work of Carroll (2013, 2014) finds a way around this issue by explicitly constructing measures over trajectories for flat FLRW universes in the context of specific SSF potentials. Yet there remains much work to do in extending such measures to more generic background spacetimes that involve SSF models of inflation (let alone to model-independent EFT-based characterizations of inflation). ...
Article
The leading account of several salient observable features of our universe today is provided by the theory of cosmic inflation. But an important and thus far intractable question is whether inflation is generic, or whether it is finely tuned—requiring very precisely specified initial conditions. In this paper I argue that a recent, model-independent characterization of inflation—that treats inflation as an effective field theory (EFT)—promises to address this question in a thoroughly modern and significantly more comprehensive way than in the existing literature. To motivate and provide context for this claim, I distill three core problems with the theory of inflation, which I dub the permissiveness problem, the initial conditions problem, and the multiverse problem. I argue that the initial conditions problem lies within the scope of EFTs of inflation as they are currently conceived, whereas the other two problems remain largely intractable: their solution must await a more complete description of the very early universe. I highlight recent work that addresses the initial conditions problem within the context of a dynamical systems analysis of a specific (state-of-the-art) EFT of inflation, and conclude with a roadmap for how such work might be extended to realize the promise claimed above.
... Stability and FragilityWhile there are a variety of threads that one could pull from these two episodes, leading in many different directions, in this final section I want to keep a hold on the notion of stability and follow it into a more general context, particularly as it appears in dynamical systems theory. The precise notion of stability in dynamical systems theory is usually attributed to Andronov and Pontryagin, in their article "Grubye sistemy" (Coarse systems)(Andronov and Pontryagin, 1937), although this 14 Interestingly, several physicists have made likelihood-based arguments that there is actually no flatness problem(Gibbons et al, 1987;Hawking and Page, 1988;Coule, 1995;Gibbons and Turok, 2008;Carroll and Tam, 2010). Their arguments, however, suffer from problems more or less as serious as those of the flatness problem supporters(Schiffrin and Wald, 2012;McCoy, 2017). ...
Article
I investigate the role of stability in cosmology through two episodes from the recent history of cosmology: (1) Einstein’s static universe and Eddington’s demonstration of its instability, and (2) the flatness problem of the hot big bang model and its claimed solution by inflationary theory. These episodes illustrate differing reactions to instability in cosmological models, both positive ones and negative ones. To provide some context to these reactions, I also situate them in relation to perspectives on stability from dynamical systems theory and its epistemology. This reveals, for example, an insistence on stability as an extreme position in relation to the spectrum of physical systems which exhibit degrees of stability and fragility, one which has a pragmatic rationale, but not any deeper one.
Article
Several authors (including myself) have made claims, none of which has been convincingly rebutted, that the flatness problem, as formulated by Dicke and Peebles, is not really a problem but rather a misunderstanding. In particular, we all agree that no fine-tuning in the early Universe is needed in order to explain the fact that there is no strong departure from flatness, neither in the early Universe nor now. Nevertheless, the flatness problem is still widely perceived to be real, since it is still routinely mentioned as an outstanding (in both senses) problem in cosmology in papers and books. Most of the arguments against the idea of a flatness problem are based on the change with time of the density parameter Ω and normalized cosmological constant λ (often assumed to be zero before there was strong evidence that it has a non-negligible positive value) and, since the Hubble constant H is not considered, are independent of time scale. In addition, taking the time scale into account, it is sometimes claimed that fine-tuning is required in order to produce a Universe which neither collapsed after a short time nor expanded so quickly that no structure formation could take place. None of those claims is correct, whether or not the cosmological constant is assumed to be zero. I briefly review the literature disputing the existence of the flatness problem, which is not as well known as it should be, compare it with some similar persistent misunderstandings, and wonder about the source of confusion.
Chapter
I inquire into the role of stability in cosmology by investigating two episodes from the recent history of cosmology: (1) Einstein’s static universe and Eddington’s demonstration of its instability and (2) the flatness problem of the hot big bang model and its alleged solution by inflationary theory. These episodes illustrate differing reactions to instability in cosmological models, both positive ones and negative ones. To provide some context to these reactions, I situate them in relation to perspectives on stability from dynamical systems theory and its epistemology. This reveals, among other things, that an insistence on stability is an extreme position in light of the broad spectrum of physical systems exhibiting degrees of both stability and fragility, one which has perhaps a pragmatic rationale but not any deeper one.
Chapter
Cosmic Inflation (Guth, Phys Rev D 23:347–356, 1981, [1], Linde, Phys Lett B, 108:389–393, 1982, [2], Albrecht and Steinhardt, Phys Rev Lett 48:1220–1223, 1982, [3], Starobinsky, Phys Lett B 91:99–102, 1980, [4]) is thought to provide a solution to several problems in standard Big Bang theory by dynamically driving a “generic” initial state to a flat, homogeneous and isotropic Universe, while generating a nearly scale-invariant power spectrum of primordial perturbations which is consistent with observations. The question of what constitutes a “generic” initial state is a difficult one, and can only be understood in the context of a quantum theory of gravity. However, regardless of the nature of quantum gravity, a random realisation from the set of all possible initial conditions will not look like an inflationary spacetime, at least initially (Hollands and Wald, Gen Relativ Gravit 34:2043–2055, 2002, [5]), and one should expect the initial conditions from which inflation begins to contain some measure of inhomogeneity.
Article
Full-text available
The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. We review this construction and show how one can visualize the measure in terms of a “magnetic flux” of solutions through phase space. Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat Friedmann-Lemaitre-Robertson-Walker universes. We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. The resulting measure is independent of time and of the choice of coordinates on the space of fields. We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. One such quantity is the probability of inflation in simple scalar field models. We find that, according to our implementation of the canonical measure, the probability for N e-folds of inflation in single-field, slow-roll models is suppressed by of order exp⁡(-3N) and we discuss the implications of this result.
Article
Full-text available
CMB measurements reveal an unnaturally smooth early universe. We propose a mechanism to make this smoothness natural by weakening the strength of gravity at early times, and therefore altering which initial conditions have low entropy.
Article
It is shown that the evolution of the universe in the chaotic inflation scenario has no end and may have no beginning. According to this scenario, the universe consists of an exponentially large number of different mini-universes inside which all possible metastable vacuum states and all possible types of compactifications are realized.
Article
List of contributors; Preface; 1. An introductory survey S. W. Hawking and W. Israel; 2. The confrontation between gravitation theory and experiment C. M. Will; 3. Gravitational-radiation experiments D. H. Douglass and V. B. Braginsky; 4. The initial value problem and the dynamical formulation of general relativity A. E. Fischer and J. E. Marsden; 5. Global structure of spacetimes R. Geroch and G. T. Horowitz; 6. The general theory of the mechanical, electromagnetic and thermodynamic properties of black holes B. Carter; 7. An introduction to the theory of the Kerr metric and its peturbations S. Chandrasekhar; 8. Black hole astrophysics R. D. Blandford and K. S. Thorne; 9. The big bang cosmology - enigmas and nostrums R. H. Dicke and P. J. E. Peebles; 10. Cosmology and the early universe Ya B. Zel'dovitch; 11. Anisotropic and inhomogeneous relativistic cosmologies M. A. H. MacCallum; 12. Singularities and time-asymmetry R. Penrose; 13. Quantum field theory in curved spacetime G. W. Gibbons; 14. Quantum gravity: the new synthesis B. S. DeWitt; 15. The path-integral approach to quantum gravity S. W. Hawking; 16. Ultraviolet divergences in quantum theories of gravitation S. Weinberg; References; Index.
Article
The Henneaux-Gibbons-Hawking-Stewart canonical measure on the set of classical universes is applied to a Friedmann-Robertson-Walker model containing a massive scalar field. Although a uniform probability distribution in this measure would solve the flatness problem, it gives an ambiguous probability for inflation, since both the set of inflationary solutions and the set of noninflationary solutions have infinite measure.
Article
The treatment of first-order phase transitions for standard grand unified theories is shown to break down for models with radiatively induced spontaneous symmetry breaking. It is argued that proper analysis of these transitions which would take place in the early history of the universe can lead to an explanation of the cosmological homogeneity, flatness, and monopole puzzles.
Article
The standard model of hot big-bang cosmology requires initial conditions which are problematic in two ways: (1) The early universe is assumed to be highly homogeneous, in spite of the fact that separated regions were causally disconnected (horizon problem); and (2) the initial value of the Hubble constant must be fine tuned to extraordinary accuracy to produce a universe as flat (i.e., near critical mass density) as the one we see today (flatness problem). These problems would disappear if, in its early history, the universe supercooled to temperatures 28 or more orders of magnitude below the critical temperature for some phase transition. A huge expansion factor would then result from a period of exponential growth, and the entropy of the universe would be multiplied by a huge factor when the latent heat is released. Such a scenario is completely natural in the context of grand unified models of elementary-particle interactions. In such models, the supercooling is also relevant to the problem of monopole suppression. Unfortunately, the scenario seems to lead to some unacceptable consequences, so modifications must be sought.
Article
We present in a manifestly gauge-invariant form the theory of classical linear gravitational perturbations in part I, and a quantum theory of cosmological perturbations in part II. Part I includes applications to several important examples arising in cosmology: a univese dominated by hydrodynamical matter, a universe filled with scalar-field matter, and higher-derivative theories of gravity. The growth rates of perturbations are calculated analytically in most interesting cases. The analysis is applied to study the evolution of fluctuations in inflationary universe models. Part II includes a unified description of the quantum generation and evolution of inhomogeneities about a classial Friedmann background. The method is based on standard canonical quantization of the action for cosmological perturbations which has been reduced to an expression in terms of a single gauge-invariant variable. The spectrum of density perturbations originating in quantum fluctuations is calculated in universe with hydrodynamical matter, in inflationary universe models with scalar-field matter, and in higher-derivative theories of gravity. The gauge-invariant theory of classical and quantized cosmological perturbations developed in parts I and II is applied in part III to several interesting physical problems. It allows a simple derivation of the relation between temperature anistropes in the cosmic microwave background. radiation and the gauge-invariant potential for metric perturbations. The generation and evolution of gravitational waves is studied. As another example, a simple analysis of entropy perturbations and non-scale-invariant spectra in inflationary universe models is presented. The gauge-invariant theory of cosmological perturbations also allows a consistent and gauge-invariant definition of statistical fluctuations.
Article
We define a measure for the set of solutions of hamiltonian systems with an odd number of constraints. We use this to show that almost all Friedmann models containing a massive scalar field undergo a period of inflation.