Content uploaded by Sean M. Carroll

Author content

All content in this area was uploaded by Sean M. Carroll

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 10−6.6×107. 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 Canonicalscalarﬁeld............................... 13

4.2 Scalarperfectﬂuid ................................ 15

4.3 Theﬂatnessproblem ............................... 17

4.4 Likelihoodofinﬂation .............................. 20

5 Perturbations 22

5.1 Description of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Computation of the measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 Likelihoodofinﬂation .............................. 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 ∼aH−1

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

∂

∂qi−∂H

∂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

dpi∧dqi,dω= 0 .(5)

The existence of the symplectic form provides us with a naturally-deﬁned measure on phase

space,

Ω = (−1)n(n−1)/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 A⊂Γ, and it evolves into a region A0, Liouville’s theorem states that

ZA

Ω = ZA0

Ω.(8)

The inﬁnitesimal version of this result is that the Lie derivative of Ω with respect to the

vector ﬁeld Vvanishes,

LVΩ = 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 (2n−1)-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 (n−1)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=

n−1

X

i=1

dpi∧dqi.(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 (2n−2)-form,

Θ = (−1)(n−1)(n−2)/2

(n−1)! eωn−1.(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=Zd4x√−g1

2R+P(X, φ),(17)

where Ris the curvature scalar and Pis the Lagrange density of the scalar ﬁeld φ. We have

set m−2

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 ∇µφ∇νφ. (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

1−kr2+r2dΩ2,(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

2N−2˙

φ2.(20)

4.1 Canonical scalar ﬁeld

We start with the canonical case,

P(X, φ) = X−V(φ).(21)

The Lagrangian for the combined gravity-scalar system in minisuperspace is

L=−3N−1a˙a2+ 3Nak +1

2N−1a3˙

φ2−Na3V(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=−6N−1a˙a , pφ=N−1a3˙

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˙qi−L(pi, qi) (24)

=N−p2

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,

Θ = (dpa∧da +dpφ∧dφ)|H=0.(28)

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

a−2a6V(φ)+6a4k1/2

.(29)

We can change variables from pato Husing pa=−6a2H, so that

pφ=6a6H2−2a6V+ 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 −a4V0dφ + 6a(3a2H2−a2V+ 2k)da

(6a2H2−2a2V+ 6k)1/2,(32)

where V0(φ) = dV/dφ. Plug into the expression (28) for the measure, whose components

become

ΘφH =−6a4

(6a2H2−2a2V+ 6k)1/2

ΘHa =−6a2

Θaφ = 6 3a3H2−a3V+ 2ak

(6a2H2−2a2V+ 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 Θaφ,

µ=−6ZH=H∗

3a3H2

∗−a3V+ 2ak

(6a2H2

∗−2a2V+ 6k)1/2dadφ, (35)

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

Ω˙

φ≡˙

φ2

6H2

∗

,ΩV≡V(φ)

3H2

∗

,Ωk≡ − k

a2H2

∗

,(36)

14

so that the Friedmann equation is equivalent to

Ω˙

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/Ωk= 1/|Ωk|. We therefore have

da =−1

2H∗|Ωk|3/2dΩk,(38)

and the measure becomes

µ= 3r3

2H−2

∗ZH=H∗

1−ΩV−2

3Ωk

|Ωk|5/2(1 −ΩV−Ωk)1/2dΩkdφ (39)

= 3r3

2H−2

∗ZH=H∗

Ω˙

φ−1

3Ωk

|Ωk|5/2Ω1/2

˙

φ

dΩkdφ, (40)

where Ω ˙

φ(φ, Ωk) is deﬁned by (37).

This integral is divergent. One divergence clearly occurs for small values of the curvature

parameter, Ωk→0, as the denominator includes a factor of |Ωk|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 Ω ˙

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 Ωk→0, 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)UµUν−P 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)−1/2∇µφ(42)

15

and energy density

ρ= 2X∂XP−P. (43)

We will be interested in a vanishing potential but a non-canonical kinetic term,

P(X, φ) = 2n−1

nXn=1

2nN−2n˙

φ2n.(44)

This gives a ﬂuid with a density

ρ=2n−1

2nN−2n˙

φ2n,(45)

corresponding to a constant equation-of-state parameter

w=P/ρ =1

2n−1,(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=−3N−1a˙a2+ 3Nak +1

2nN−(2n−1)a3˙

φ2n,(47)

and the Friedmann equation is

H2≡˙a

a2

=2n−1

6n˙

φ2n−k

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 =−2n−1

2n1/2n6a6n/(2n−1)

[a6n/(2n−1)3H2+ 3a2(n+1)/(2n−1)k]1/2n

ΘHa =−6a2

Θaφ = 6(2n−1)(1−2n)/2nna(4n+1)/(2n−1)3H2+ (n+ 1)a3/(2n−1)k

[2na6n/(2n−1)3H2+ 6na2(n+1)/(2n−1)k]1/2n.(49)

To calculate the measure of a set of trajectories over a surface of constant H=H∗, we

integrate Θaφ over aand φ. This yields

µ=−32n−1

6n(1−2n)/2nZH=H∗

a2H2

∗+(n+1)

3na−2k

(H2

∗+a−2k)1/2ndadφ. (50)

Note that the integrand has no dependence on φ, since there was no potential in the original

action. We therefore deﬁne

x≡32n−1

6n(1−2n)/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)

3nΩk

|Ωk|5/2(1 −Ωk)1/2ndΩk.(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,

µ∝ZH=H∗

1−ΩV−2

3Ωk

|Ωk|5/2(1 −ΩV−Ωk)1/2dΩkdφ. (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 |Ωk|−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 lim→0f(x) = |x|−5/2, our original function. The integral of f(x) over all values of

xis 10

3−3/2. 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

|Ωk|−5/2by F(Ωk), and take the limit as →0:

µ∝lim

→0−3/2ZH=H∗

F(Ωk)1−ΩV−2

3Ωk

(1 −ΩV−Ωk)1/2dΩkdφ. (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

µ∝ZH=H∗p1−ΩVdφ, (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

|Ωi,pl|<1, where the subscript “pl” denotes that the quantity is evaluated when H∼mpl.

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 H∗to 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 ×10−3mpl , 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/√3r1−1

2m2φ2dφ. (60)

(Recall that we have set mpl = 1/√8πG = 1.) The range of integration is |φ| ≤ q2/m2

φ(or

ρ˙

φ≤1), corresponding to V≤1.

20

We used the Euler method with a time step ∆t= 10−3to 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 ≤φ≤6, 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 = 10−3, so that the model is consistent with

WMAP3 data [21]. We evaluate the measure on the hypersurface H=H∗=p4/3Λ2, so

that 3H2

∗= 2Vmax = 4Λ4. In this case, the normalized measure becomes

µ=1

8√πE[−1] ZH∗=√4/3Λ2s1−1

41 + cos φ

√8πdφ, (62)

where E[m] is the elliptic integral R2π

0p1−msin2tdt. Numerically we ﬁnd that the universe

expands by more than 60 efolds for −4.0< φ < 2.4 if ˙

φ > 0 at H=H∗=p4/3Λ2. (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=Za−1dt. (63)

Derivatives with respect to ηare denoted by the superscript 0, and e

H≡a0/a is related to

the Hubble parameter H= ˙a/a by e

H=aH. The Friedmann equations become

e

H2=8πG

3a2¯ρ−k, (64)

e

H0=−4πG

3a2(¯ρ+ 3¯p),(65)

where ¯ρand ¯pare the background density and pressure. In a ﬂat universe with only matter

and radiation, in the radiation-dominated era we have

η(RD) = a

H0√aeq

,e

H(RD) = η−1,(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) eV−1.(67)

The metric for a ﬂat RW universe in conformal time with scalar perturbations is

ds2=a2(η)−(1 + 2Φ)dη2+ 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 = φ−1

a[a(B−E)0]0,(69)

and the gauge-invariant energy-density perturbation,

e

δρ =δρ −¯ρ0(B−E0).(70)

For scalar perturbations in the absence of anisotropic stress, these are related by

e

δρ =1

4πGa2h∇2Φ−3e

H(Φ0+e

HΦ)i.(71)

For adiabatic perturbations (δS = 0), the potential obeys an autonomous equation,

Φ00 + 3(1 + c2

s)e

HΦ0−c2

s∇2Φ + [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

u≡Φ

√¯ρ+ ¯p,(73)

and the time-dependent parameter

θ= exp 3

2Z(1 + c2

s)e

HdηΦ = 1

a"2

3 1−e

H0

e

H2!#−1/2

.(74)

In terms of these (72) becomes

u00 −c2

s∇2u−θ00

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

2Zd4xu02−c2

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

sk2−θ00

θu2.(77)

This is simply the Hamiltonian for a single degree of freedom with a time-dependent eﬀective

mass m2=c2

sk2−θ00/θ.

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

dpi∧dqi−dH ∧ 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

sk2−θ00

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 = dpu∧du −(dH ∧ dη)|H+=0

=dpu∧du −1

2dp2

u+c2

sk2−θ00

θu2∧dη

=dpu∧du −pu(dpu∧dη)−uc2

sk2−θ00

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,

which might start with an assumed early state of the universe and ask whether inﬂation will

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 (ηG≈6×105eV−1), 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 z∼1200, 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 eV−1), 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 ≤10−5and

Φeq ≤10−5.

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 H−1

0= 1033 eV−1, and the size of our comoving

patch at equality is aeq = 1/3000 times that, so

Leq ≈1030 eV−1.(84)

We will also impose a short-distance cutoﬀ at the Hubble radius at equality,

H−1

eq ≈mpl a2

eq

T2

0≈1028 eV−1.(85)

The total number of modes we consider is therefore

n=Leq

H−1

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 H−1

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

θ−2dη, (87)

where c1and c2are constants. During radiation domination we have

θ=√3

2√aeqH0

η−1.(88)

26

The solution for uis therefore

u=αη−1+βη2,(89)

where αand βare constants. The conjugate momentum is

pu=−αη−2+ 2βη. (90)

The potential is related to uby (73). In the radiation era we have

(¯ρ+ ¯p)1/2=γη−2,(91)

where we have deﬁned

γ=2mpl

√aeqH0

.(92)

Our general solution is therefore

Φ = γ(αη−3+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

(2αη−7−βη−4),(94)

which in turn implies

δ≡e

δρ

¯ρ= 2γ(2αη−3−β).(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 du∧dpu. 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,

dΦdδ = 6γ2η−3dαdβ. (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 ∆ΦG∆δG≈1, while the

coordinate area of our region at equality is ∆Φeq∆δeq ≈(10−5)2= 10−10 . For each mode,

we therefore have

µ(inﬂationary)

µ(realistic) =ηG

ηeq 3∆ΦG∆δG

∆Φeq∆δeq ≈10−66.(99)

This is saying that, for a given wave vector, only 10−66 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) ≈(10−66)n≈10−6.6×107.(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

10−10122 [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 H−1

0today corresponds to

a physical size ∼10−26H0∼1034Lpl ∼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

Steinhardt for helpful conversations.

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