CALT 68-2797

Unitary Evolution and Cosmological Fine-Tuning

Sean M. Carroll and Heywood Tam

California Institute of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### 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 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.

###### Contents

  1. 1 Introduction
  2. 2 The Evolution of our Comoving Patch
    1. 2.1 Autonomy
    2. 2.2 Unitarity
  3. 3 The Canonical Measure
  4. 4 Minisuperspace
    1. 4.1 Canonical scalar field
    2. 4.2 Scalar perfect fluid
    3. 4.3 The flatness problem
    4. 4.4 Likelihood of inflation
  5. 5 Perturbations
    1. 5.1 Description of perturbations
    2. 5.2 Computation of the measure
    3. 5.3 Likelihood of inflation
  6. 6 Discussion
  7. 7 Appendix: Eternal Inflation

##  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 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 beginning. 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 trajectories. 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 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 reversibility: 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 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.111The 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. In the
Hamiltonian formulation of classical mechanics, a state is an element of phase
space, specified by coordinates $q^{i}(t)$ and momenta $p_{i}(t)$. Time
evolution is governed by Hamilton’s equations,

$\dot{q}^{i}=\frac{\partial{\mathcal{H}}}{\partial
p_{i}}\,,\quad\dot{p}_{i}=-\frac{\partial{\mathcal{H}}}{\partial q^{i}}\,,$
(1)

where ${\mathcal{H}}$ is the Hamiltonian. In quantum mechanics, a state is
given by a wave function $|\psi(t)\rangle$ which defines a ray in Hilbert
space. Time evolution is governed by the Schrödinger equation,

$\hat{{\mathcal{H}}}|\Psi\rangle=i\partial_{t}|\Psi\rangle,$ (2)

where $\hat{{\mathcal{H}}}$ is the Hamiltonian operator, or equivalently by
the von Neumann equation,

$\partial_{t}\hat{\rho}=-i[\hat{{\mathcal{H}}},\hat{\rho}]\,,$ (3)

where $\hat{\rho}(t)=|\psi(t)\rangle\langle\psi(t)|$ 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 justification 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 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
elements: 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.

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.

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,
$\Sigma$, 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 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 quantum 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 compatible 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 $\lambda_{\rm IR}\sim aH_{0}^{-1}$, where $a$ is the
scale factor (normalized to unity today) and $H_{0}$ is the current Hubble
parameter. But there is also a fixed ultraviolet cutoff at the Planck length,
$\lambda_{\rm UV}\sim L_{\rm pl}=\sqrt{8\pi G}$. 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
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 inhomogeneities 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 $S_{\Sigma}(t_{0})\sim 10^{104}$
[17]. If all the matter were part of a single black hole it would be as large
as $S_{\Sigma}({\rm BH})\sim 10^{122}$. At early times, when inhomogeneities
were small and local gravitational effects were negligible, the entropy was of
order $S_{\Sigma}({\rm RD})\sim 10^{88}$. 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 $\exp[10^{104}]\gg\exp[10^{88}]$).

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 constraints. The
state of a classical system is described by a point $\gamma$ in a phase space
$\Gamma$, with canonical coordinates $q^{i}$ and momenta $p_{i}$. The index
$i$ goes 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

$V=\frac{\partial{\mathcal{H}}}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial{\mathcal{H}}}{\partial q^{i}}\frac{\partial}{\partial
p_{i}}.$ (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 $\Gamma$:

$\omega=\sum_{i=1}^{n}{\rm d}p_{i}\wedge{\rm d}q^{i}\,,\qquad{\rm
d}\omega=0\,.$ (5)

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

$\Omega=\frac{(-1)^{n(n-1)/2}}{n!}\omega^{n}\,.$ (6)

This is the Liouville measure, a $2n$-form on $\Gamma$. It corresponds to the
usual way of integrating distributions over regions of phase space,

$\int f(\gamma)\Omega=\int f(q^{i},p_{i})d^{n}qd^{n}p\,.$ (7)

The Liouville measure is conserved under Hamiltonian evolution. If we begin
with a region $A\subset\Gamma$, and it evolves into a region $A^{\prime}$,
Liouville’s theorem states that

$\int_{A}\Omega=\int_{A^{\prime}}\Omega\,.$ (8)

The infinitesimal version of this result is that the Lie derivative of
$\Omega$ with respect to the vector field $V$ vanishes,

${\cal L}_{V}\Omega=0\,.$ (9)

These results can be traced back to the fact that the original symplectic form
$\omega$ is also invariant under the flow:

${\cal L}_{V}\omega=0\,,$ (10)

so any form constructed from powers of $\omega$ will 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.

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,

$C=\Gamma/\\{{\mathcal{H}}={\mathcal{H}}_{*}\\}\,.$ (11)

For Robertson-Walker cosmology, the Hamiltonian precisely vanishes for either
open or closed universes, so we can take ${\mathcal{H}}_{*}=0$. Then we
consider the space of classical trajectories within this constraint
hypersurface:

$M=C/V\,,$ (12)

where the quotient by the evolution vector field $V$ means 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 $C$ always stay within $C$, because
the Hamiltonian is conserved.

As $M$ is a submanifold of $\Gamma$, the measure is constructed by pulling
back the symplectic form from $\Gamma$ to $M$ and raising it to the $(n-1)$th
power. GHS constructed a useful explicit form by choosing the $n$th coordinate
on phase space to be the time, $q^{n}=t$, so that the conjugate momentum
becomes the Hamiltonian itself, $p_{n}={\mathcal{H}}$. The symplectic form is
then

$\omega={\widetilde{\omega}}+{\rm d}{\mathcal{H}}\wedge{\rm d}t\,,$ (13)

where

${\widetilde{\omega}}=\sum_{i=1}^{n-1}{\rm d}p_{i}\wedge{\rm d}q^{i}\,.$ (14)

The pullback of $\omega$ onto $C$ then has precisely the same coordinate
expression as (14), and we will simply refer to this pullback as
${\widetilde{\omega}}$ from now on. It is automatically transverse to the
Hamiltonian flow (${\widetilde{\omega}}(V)=0$), and therefore defines a
symplectic form on the space of trajectories $M$. The associated measure is a
$(2n-2)$-form,

${\Theta}=\frac{(-1)^{(n-1)(n-2)/2}}{(n-1)!}{\widetilde{\omega}}^{n-1}\,.$
(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].

To evaluate the measure we need to define coordinates on the space of
trajectories. We can choose a hypersurface $\Sigma$ 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 $S_{1}$ and $S_{2}$ are subsets, respectively, of two
transverse hypersurfaces $\Sigma_{1}$ and $\Sigma_{2}$ in $C$, with the
property that the set of trajectories passing through $S_{1}$ is the same as
that passing through $S_{2}$, then

$\int_{S_{1}}{\Theta}=\int_{S_{2}}{\Theta}\,.$ (16)

The property that the measure on trajectories is local in phase space has a
crucial implication for studies of cosmological fine-tuning. Imagine that we
specify a certain set of trajectories by their macroscopic properties today –
cosmological solutions that are approximately 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 independent
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.222On 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]. At 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

$S=\int d^{4}x\,\sqrt{-g}\left[\frac{1}{2}R+P(X,\phi)\right],$ (17)

where $R$ is the curvature scalar and $P$ is the Lagrange density of the
scalar field $\phi$. We have set $m_{\rm pl}^{-2}=8\pi G=1$ for convenience.
The scalar Lagrangian is taken to be a function of the field value and and the
kinetic scalar $X$, defined by

$X\equiv-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi.$ (18)

We will consider homogeneous scalar fields $\phi(t)$ defined in a Robertson-
Walker metric,

$ds^{2}=-N^{2}{\rm d}t^{2}+a^{2}(t)\left[\frac{{\rm
d}r^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right]\,,$ (19)

where the spatial curvature parameter $k$ can be normalized to $-1$, $0$, or
$+1$ (so that $a(t_{0})$ is not normalized to unity). $N$ is the lapse
function, which acts as a Lagrange multiplier. We then have

$X=\frac{1}{2}N^{-2}\dot{\phi}^{2}.$ (20)

###  4.1 Canonical scalar field

We start with the canonical case,

$P(X,\phi)=X-V(\phi).$ (21)

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

$L=-3N^{-1}a\dot{a}^{2}+3Nak+\frac{1}{2}N^{-1}a^{3}\dot{\phi}^{2}-Na^{3}V(\phi)\,.$
(22)

The canonical coordinates can be taken to be the lapse function $N$, the scale
factor $a$, and the scalar field $\phi$. The conjugate momenta are given by
$p_{i}=\partial L/\partial q^{i}$, implying

$p_{N}=0\,,\quad p_{a}=-6N^{-1}a\dot{a}\,,\quad
p_{\phi}=N^{-1}a^{3}\dot{\phi}\,.$ (23)

The vanishing of $p_{N}$ reflects the fact that the lapse function is a non-
dynamical Lagrange multiplier. We can do a Legendre transformation to
calculate the Hamiltonian, obtaining

$\displaystyle{\mathcal{H}}$ $\displaystyle=$ $\displaystyle\sum
p_{i}\dot{q}^{i}-L(p_{i},q^{i})$ (24) $\displaystyle=$ $\displaystyle
N\left(-\frac{p_{a}^{2}}{12a}+\frac{p_{\phi}^{2}}{2a^{3}}+a^{3}V(\phi)-3ak\right).$
(25)

Varying with respect to $N$ gives the Hamiltonian constraint,
${\mathcal{H}}=0$, which is just the Friedmann equation,

$H^{2}=\frac{1}{3}\left(\frac{1}{2}{\dot{\phi}}^{2}+V(\phi)-\frac{3k}{a^{2}}\right).$
(26)

Henceforth we will set $N=1$ (consistent with the equations of motion),
leaving us with a four-dimensional phase space,

$\Gamma=\\{\phi,p_{\phi},a,p_{a}\\}\,.$ (27)

The GHS measure on the space of trajectories is just the the Liouville measure
subject to the constraint that $\mathcal{H}=0$,

${\Theta}=(dp_{a}\wedge da+dp_{\phi}\wedge d\phi)|_{\mathcal{H}=0}.$ (28)

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 $p_{\phi}$ gives us

$p_{\phi}=\left[\frac{1}{6}a^{2}p_{a}^{2}-2a^{6}V(\phi)+6a^{4}k\right]^{1/2}.$
(29)

We can change variables from $p_{a}$ to $H$ using $p_{a}=-6a^{2}H$, so that

$p_{\phi}=\left(6a^{6}H^{2}-2a^{6}V+6a^{4}k\right)^{1/2}.$ (30)

Our coordinates on the constraint hypersurface $C$ are therefore
$\\{\phi,a,H\\}$. The basis one-forms appearing in (28) are

$dp_{a}=-12aHda-6a^{2}dH$ (31)

and

$dp_{\phi}=\frac{6a^{4}HdH-a^{4}V^{\prime}d\phi+6a(3a^{2}H^{2}-a^{2}V+2k)da}{(6a^{2}H^{2}-2a^{2}V+6k)^{1/2}},$
(32)

where $V^{\prime}(\phi)=dV/d\phi$. Plug into the expression (28) for the
measure, whose components become

$\displaystyle{\Theta}_{\phi H}$ $\displaystyle=$
$\displaystyle-\frac{6a^{4}}{\left(6a^{2}H^{2}-2a^{2}V+6k\right)^{1/2}}$ (33)
$\displaystyle{\Theta}_{Ha}$ $\displaystyle=$ $\displaystyle-6a^{2}$ (34)
$\displaystyle{\Theta}_{a\phi}$ $\displaystyle=$ $\displaystyle
6\frac{3a^{3}H^{2}-a^{3}V+2ak}{\left(6a^{2}H^{2}-2a^{2}V+6k\right)^{1/2}}.$
(35)

The measure is calculated by choosing some transverse surface $\Sigma$ in
phase space, and integrating ${\Theta}$ over a subset of that surface. If we
choose coordinates such that one coordinate is constant over $\Sigma$, we
simply integrate the orthogonal component of ${\Theta}$ with respect to the
other coordinates. One possible choice of the surface $\Sigma$ is to fix the
Hubble parameter,

$\Sigma:\\{H=H_{*}\\}\,.$ (36)

Any consistent definition is equally legitimate; however, this choice
corresponds to our informal 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 $H$ is then the integral of
${\Theta}_{a\phi}$,

$\mu=-6\int_{H=H_{*}}\frac{3a^{3}H_{*}^{2}-a^{3}V+2ak}{\left(6a^{2}H_{*}^{2}-2a^{2}V+6k\right)^{1/2}}dad\phi,$
(37)

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

$\Omega_{\dot{\phi}}\equiv\frac{\dot{\phi}^{2}}{6H_{*}^{2}},\quad\Omega_{V}\equiv\frac{V(\phi)}{3H_{*}^{2}},\quad\Omega_{k}\equiv-\frac{k}{a^{2}H_{*}^{2}},$
(38)

so that the Friedmann equation is equivalent to

$\Omega_{\dot{\phi}}+\Omega_{V}+\Omega_{k}=1.$ (39)

The scale factor is strictly positive, so that integrating over all values of
$\Omega_{k}$ is equivalent to integrating over all values of $a$. Note that
$-k/\Omega_{k}=1/|\Omega_{k}|$. We therefore have

$da=-\frac{1}{2H_{*}|\Omega_{k}|^{3/2}}d\Omega_{k},$ (40)

and the measure becomes

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
3\sqrt{\frac{3}{2}}H_{*}^{-2}\int_{H=H_{*}}\frac{1-\Omega_{V}-\frac{2}{3}\Omega_{k}}{|\Omega_{k}|^{5/2}\left(1-\Omega_{V}-\Omega_{k}\right)^{1/2}}\,d\Omega_{k}d\phi$
(41) $\displaystyle=$ $\displaystyle
3\sqrt{\frac{3}{2}}H_{*}^{-2}\int_{H=H_{*}}\frac{\Omega_{\dot{\phi}}-\frac{1}{3}\Omega_{k}}{|\Omega_{k}|^{5/2}\Omega_{\dot{\phi}}^{1/2}}\,d\Omega_{k}d\phi,$
(42)

where $\Omega_{\dot{\phi}}(\phi,\Omega_{k})$ is defined by (39).

This integral is divergent. One divergence clearly occurs for small values of
the curvature parameter, $\Omega_{k}\rightarrow 0$, as the denominator
includes a factor of $|\Omega_{k}|^{5/2}$. The integrand also blows up at
$\Omega_{\dot{\phi}}=0$ (or equivalently at $\Omega_{V}+\Omega_{k}=1$), but
the integral in that region remains finite. The integral would also diverge if
$\Omega_{k}$ or $\Omega_{\dot{\phi}}$ were 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 $\Omega_{k}\rightarrow 0$,
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
$P(X,\phi)$, where $X=-(\nabla_{\mu}\phi)^{2}/2$. In a Robertson-Walker
background, the energy-momentum tensor takes the form of a perfect fluid,

$T_{\mu\nu}=(\rho+P)U_{\mu}U_{\nu}-Pg_{\mu\nu},$ (43)

where the pressure is equal to the scalar Lagrange density itself (thereby
accounting for the choice of notation). The fluid has four-velocity

$U_{\mu}=(2X)^{-1/2}\nabla_{\mu}\phi$ (44)

and energy density

$\rho=2X\partial_{X}P-P.$ (45)

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

$P(X,\phi)=\frac{2^{n-1}}{n}X^{n}=\frac{1}{2n}N^{-2n}\dot{\phi}^{2n}.$ (46)

This gives a fluid with a density

$\rho=\frac{2n-1}{2n}N^{-2n}\dot{\phi}^{2n},$ (47)

corresponding to a constant equation-of-state parameter

$w=P/\rho=\frac{1}{2n-1},$ (48)

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 $n$
very large.

The scalar-Einstein Lagrangian in a Robertson-Walker background takes the form

$L=-3N^{-1}a\dot{a}^{2}+3Nak+\frac{1}{2n}N^{-(2n-1)}a^{3}\dot{\phi}^{2n},$
(49)

and the Friedmann equation is

$H^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{2n-1}{6n}\right)\dot{\phi}^{2n}-\frac{k}{a^{2}},$
(50)

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 $\\{\phi,a,H\\}$.
We end up with

$\displaystyle{\Theta}_{\phi H}$ $\displaystyle=$
$\displaystyle-\left(\frac{2n-1}{2n}\right)^{1/2n}\frac{6a^{6n/(2n-1)}}{\left[a^{6n/(2n-1)}3H^{2}+3a^{2(n+1)/(2n-1)}k\right]^{1/2n}}$
(51) $\displaystyle{\Theta}_{Ha}$ $\displaystyle=$ $\displaystyle-6a^{2}$ (52)
$\displaystyle{\Theta}_{a\phi}$ $\displaystyle=$ $\displaystyle
6(2n-1)^{(1-2n)/2n}\frac{na^{(4n+1)/(2n-1)}3H^{2}+(n+1)a^{3/(2n-1)}k}{\left[2na^{6n/(2n-1)}3H^{2}+6na^{2(n+1)/(2n-1)}k\right]^{1/2n}}.$
(53)

To calculate the measure of a set of trajectories over a surface of constant
$H=H_{*}$, we integrate ${\Theta}_{a\phi}$ over $a$ and $\phi$. This yields

$\mu=-3\left(\frac{2n-1}{6n}\right)^{(1-2n)/2n}\int_{H=H_{*}}a^{2}\frac{H_{*}^{2}+\frac{(n+1)}{3n}a^{-2}k}{\left(H_{*}^{2}+a^{-2}k\right)^{1/2n}}dad\phi.$
(54)

Note that the integrand has no dependence on $\phi$, since there was no
potential in the original action. We therefore define

$x\equiv 3\left(\frac{2n-1}{6n}\right)^{(1-2n)/2n}\int d\phi,$ (55)

which contributes an overall multiplicative constant to the measure. As
before, it is convenient to change variables from $a$ to
$\Omega_{k}=-k/a^{2}H_{*}^{2}$. This leaves us with

$\mu=\frac{x}{2H_{*}^{(n+1)/n}}\int\frac{1-\frac{(n+1)}{3n}\Omega_{k}}{|\Omega_{k}|^{5/2}\left(1-\Omega_{k}\right)^{1/2n}}d\Omega_{k}.$
(56)

This will diverge for small $\Omega_{k}$ for 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 (42) we derived for Robertson-
Walker universes with a scalar field featuring a canonical kinetic term and a
potential,

$\mu\propto\int_{H=H_{*}}\frac{1-\Omega_{V}-\frac{2}{3}\Omega_{k}}{|\Omega_{k}|^{5/2}\left(1-\Omega_{V}-\Omega_{k}\right)^{1/2}}\,d\Omega_{k}d\phi.$
(57)

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 $\Omega_{k}=0$, 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 $\epsilon$
is taken to zero. We can then isolate an appropriate power of $\epsilon$ by
which we can divide the regulated expression, so that we isolate the finite
part of the result as $\epsilon$ goes to zero.

The divergence in (57) can be regulated by “smoothing” the factor
$|\Omega_{k}|^{-5/2}$ in an $\epsilon$-neighborhood around $\Omega_{k}=0$ to
get a finite integral. Consider the function

$f_{\epsilon}(x)=\begin{cases}|x|^{-5/2}&\text{if $|x|\geq\epsilon$,}\\\
\epsilon^{-5/2}&\text{if $|x|<\epsilon$.}\end{cases}$ (58)

Clearly $\lim_{\epsilon\rightarrow 0}f_{\epsilon}(x)=|x|^{-5/2}$, our original
function. The integral of $f_{\epsilon}(x)$ over all values of $x$ is
$\frac{10}{3}\epsilon^{-3/2}$. So we obtain a normalized integral by
introducing the function

$F_{\epsilon}(x)=\begin{cases}\displaystyle{\frac{3\epsilon^{3/2}}{10|x|^{5/2}}}&\text{if
$|x|\geq\epsilon$,}\\\ \displaystyle{\frac{3}{10\epsilon}}&\text{if
$|x|<\epsilon$,}\end{cases}$ (59)

which satisfies $\int F_{\epsilon}(x)dx=1$. We can therefore regularize the
integral in (57) by replacing $|\Omega_{k}|^{-5/2}$ by
$F_{\epsilon}(\Omega_{k})$, and take the limit as $\epsilon\rightarrow 0$:

$\mu\propto\lim_{\epsilon\rightarrow
0}\epsilon^{-3/2}\int_{H=H_{*}}F_{\epsilon}(\Omega_{k})\frac{1-\Omega_{V}-\frac{2}{3}\Omega_{k}}{\left(1-\Omega_{V}-\Omega_{k}\right)^{1/2}}\,d\Omega_{k}d\phi.$
(60)

The multiplicative factor of $\epsilon^{-3/2}$ goes 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 $F_{\epsilon}(x)$ is simply a delta
function,

$\lim_{\epsilon\rightarrow 0}F_{\epsilon}(x)=\delta(x),$ (61)

in the sense that the integral over a test function $\psi(x)$ gives

$\lim_{\epsilon\rightarrow
0}\int_{-\infty}^{\infty}F_{\epsilon}(x)\psi(x)\,dx=\psi(0).$ (62)

Consequently, the measure is entirely concentrated on exactly flat universes;
universes with nonvanishing spatial curvature are a set of measure zero. The
integrated measure (60) is equivalent to

$\mu\propto\int_{H=H_{*}}\sqrt{1-\Omega_{V}}\,d\phi,$ (63)

with $\Omega_{k}$ fixed 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 $\Omega_{i}$, this corresponds to $|\Omega_{i,{\rm
pl}}|<1$, where the subscript “pl” denotes that the quantity is evaluated when
$H\sim m_{\rm pl}$.

What this means in practice is that we tend to assign equal probability – a
flat prior – to all the allowed $\Omega_{i,{\rm pl}}$’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 $\Omega_{i,{\rm pl}}$ 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 $\Omega_{k}$ at the
Planck scale; the informal flat prior assumed in typical discussions of the
flatness problem, and the GHS measure (evaluated at $\Omega_{V}=0$ for
convenience). We see that using the measure defined by the classical equations
of motion leads to a dramatic difference in the probability density.

Figure 2: Two measures as a function of the curvature parameter $\Omega_{k}$.
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 $\delta$-function.) This is in stark contrast with
the flat distribution generally assumed in the discussion of the flatness
problem, which we’ve plotted for $\Omega_{k}$ between $\pm 1$.

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 “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 $\Omega=1$ in our observed universe, even if
the background cosmology were spatially flat. Nevertheless, 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 concentrated 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 $m_{\phi}=3\times 10^{-3}m_{\rm
pl}$, which yields an amplitude of perturbations that agrees with
observations. We choose to evaluate the measure on the hypersurface
$H=1/\sqrt{3}$, so that the Friedmann equation becomes
$1=\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}m^{2}\phi^{2}$. After replacing the
divergence at zero curvature in (56) by a delta function, the normalized
measure becomes

$\mu=\frac{\sqrt{2}m}{\pi}\int_{H=1/\sqrt{3}}\sqrt{1-\frac{1}{2}m^{2}\phi^{2}}\,d\phi.$
(64)

(Recall that we have set $m_{\rm pl}=1/\sqrt{8\pi G}=1$.) The range of
integration is $|\phi|\leq\sqrt{2/m_{\phi}^{2}}$ (or $\rho_{\dot{\phi}}\leq
1$), corresponding to $V\leq 1$.

We used the Euler method with a time step $\Delta t=10^{-3}$ to 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 $\phi$ except for the range $-24$ to $6$ if $\dot{\phi}>0$ (for
$\dot{\phi}<0$, 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 $-24\leq\phi\leq 6$,
the measure (64) 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

$V(\phi)=\Lambda^{4}(1+\cos(\phi/f)).$ (65)

In our calculation, we use $f=\sqrt{8\pi}$ and $\Lambda=10^{-3}$, so that the
model is consistent with WMAP3 data [21]. We evaluate the measure on the
hypersurface $H=H_{*}=\sqrt{4/3}\Lambda^{2}$, so that
$3H_{*}^{2}=2V_{max}=4\Lambda^{4}$. In this case, the normalized measure
becomes

$\mu=\frac{1}{8\sqrt{\pi}E[-1]}\int_{H_{*}=\sqrt{4/3}\Lambda^{2}}\sqrt{1-\frac{1}{4}\left(1+\cos{\frac{\phi}{\sqrt{8\pi}}}\right)}d\phi,$
(66)

where $E[m]$ is the elliptic integral
$\int_{0}^{2\pi}\sqrt{1-m\sin^{2}{t}}dt$. Numerically we find that the
universe expands by more than $60$ efolds for $-4.0<\phi<2.4$ if
$\dot{\phi}>0$ at $H=H_{*}=\sqrt{4/3}\Lambda^{2}$. (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 $\phi$ is
randomly distributed between $0$ and $\sqrt{8\pi}$. 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 implies 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. Inflation appears to be an attractor only because it
is often convenient to portray “phase portraits” in terms of the inflaton
$\phi$ and its time derivative, $\dot{\phi}$. But $\dot{\phi}$ is not the
momentum conjugate to $\phi$; as seen in (23), with the lapse function set to
$N=1$, it is $p_{\phi}=a^{3}\dot{\phi}$. Trajectories drawn on a
$(\phi,\dot{\phi})$ plot tend to approach a fixed point, but only because the
scale factor $a$ is dramatically increasing, not because of any true attractor
behavior.

These calculations of the likelihood of inflation are of dubious physical
relevance. Examining 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 from the point of view of the canonical measure is not
because it is unlikely in minisuperspace, 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 corresponding 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, Feldman 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,

$\eta=\int a^{-1}dt.$ (67)

Derivatives with respect to $\eta$ are denoted by the superscript ′, and
$\widetilde{H}\equiv a^{\prime}/a$ is related to the Hubble parameter
$H=\dot{a}/a$ by $\widetilde{H}=aH$. The Friedmann equations become

$\displaystyle\widetilde{H}^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}a^{2}\bar{\rho}-k,$ (68) $\displaystyle\widetilde{H}^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{4\pi G}{3}a^{2}(\bar{\rho}+3\bar{p}),$
(69)

where $\bar{\rho}$ and $\bar{p}$ are the background density and pressure. In a
flat universe with only matter and radiation, in the radiation-dominated era
we have

$\eta({\rm RD})=\frac{a}{H_{0}\sqrt{a_{\rm eq}}},\qquad\widetilde{H}({\rm
RD})=\eta^{-1},$ (70)

where $a_{\rm eq}$ is the scale factor at matter-radiation equality, and now
we set the current scale factor to unity, $a_{0}=1$. Numerically, the
conformal time in the radiation-dominated era is approximately

$\eta(T)\approx\frac{5\times 10^{30}}{T({\rm eV})}\,{\rm eV}^{-1}.$ (71)

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

$ds^{2}=a^{2}(\eta)\left[-(1+2\Phi)d\eta^{2}+2B_{,i}d\eta
dx^{2}+((1-2\Psi)\delta_{ij}+2E_{,ij})dx^{i}dx^{j}\right],$ (72)

where $\Phi$, $\Psi$, $E$, and $B$ are scalar functions characterizing metric
perturbations, and commas denote partial derivatives. It is useful to define
the gauge-invariant Newtonian potential,

$\Phi=\phi-\frac{1}{a}\left[a(B-E)^{\prime}\right]^{\prime},$ (73)

and the gauge-invariant energy-density perturbation,

$\widetilde{\delta\rho}=\delta\rho-\bar{\rho}^{\prime}(B-E^{\prime}).$ (74)

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

$\widetilde{\delta\rho}=\frac{1}{4\pi
Ga^{2}}\left[\nabla^{2}\Phi-3\widetilde{H}(\Phi^{\prime}+\widetilde{H}\Phi)\right].$
(75)

For adiabatic perturbations ($\delta S=0$), the potential obeys an autonomous
equation,

$\Phi^{\prime\prime}+3(1+c_{s}^{2})\widetilde{H}\Phi^{\prime}-c_{s}^{2}\nabla^{2}\Phi+[2\widetilde{H}^{\prime}+(1+3c_{s}^{2})\widetilde{H}^{2}]\Phi=0,$
(76)

where $c_{s}^{2}=\partial p/\partial\rho$ is the speed of sound squared in the
fluid. This equation simplifies if we introduce the rescaled perturbation
variable

$u\equiv\frac{\Phi}{\sqrt{\bar{\rho}+\bar{p}}},$ (77)

and the time-dependent parameter

$\theta=\exp\left[\frac{3}{2}\int(1+c_{s}^{2})\widetilde{H}d\eta\right]\Phi=\frac{1}{a}\left[\frac{2}{3}\left(1-\frac{\widetilde{H}^{\prime}}{\widetilde{H}^{2}}\right)\right]^{-1/2}.$
(78)

In terms of these (76) becomes

$u^{\prime\prime}-c_{s}^{2}\nabla^{2}u-\frac{\theta^{\prime\prime}}{\theta}u=0.$
(79)

The variable $u$ is a single degree of freedom that encodes both the
gravitational potential [through (77)] and the density perturbation [through
(75)]. The equation of motion (79) corresponds to an action

$S_{u}=\frac{1}{2}\int d^{4}x\left(u^{\prime
2}-c_{s}^{2}u_{,i}u_{,i}+\frac{\theta^{\prime\prime}}{\theta}u^{2}\right).$
(80)

Defining the conjugate momentum $p_{u}=\partial{\mathcal{L}}/\partial
u^{\prime}=u^{\prime}$, we can describe the dynamics in terms of a Hamiltonian
density for an individual mode with wavenumber $k$,

$\mathcal{H}=\frac{1}{2}p_{u}^{2}+\frac{1}{2}\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)u^{2}.$
(81)

This is simply the Hamiltonian for a single degree of freedom with a time-
dependent effective mass $m^{2}=c_{s}^{2}k^{2}-\theta^{\prime\prime}/\theta$.

###  5.2 Computation of the measure

Given the Hamiltonian (81), 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, $q^{n+1}=t$. The conjugate momentum is minus the Hamiltonian,
$p_{n+1}=-\mathcal{H}$. We can then define an extended Hamiltonian by

$\mathcal{H}_{+}=\mathcal{H}(p,q,t)+p_{n+1}.$ (82)

This is formally time-independent, and recovers the original Hamiltonian
equations via

$\dot{q}_{i}=\frac{\partial\mathcal{H}_{+}}{\partial
p^{i}}\,,\quad\dot{p}^{i}=-\frac{\partial\mathcal{H}_{+}}{\partial q^{i}},$
(83)

along with two additional trivial equations $\dot{t}=1$ and
$\dot{\mathcal{H}}=\partial\mathcal{H}/\partial t$.

With $t$ promoted to a coordinate, the time-dependent Hamiltonian system also
comes equipped naturally with a closed symplectic two-form, now with an
additional term:

$\omega=\sum_{i=1}^{n}dp_{i}\wedge dq^{i}-d\mathcal{H}\wedge dt.$ (84)

The invariance of the form of Hamilton’s equations ensures that the Lie
derivative of $\omega$ with respect to the vector field generated by
$\mathcal{H}_{+}$ vanishes. The top exterior power of $\omega$ is 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
$\mathcal{H}_{+}=0$.

In our case, the original system, with coordinate $u$ and conjugate momentum
$p_{u}$, is augmented to one with two coordinates $u$ and $\eta$ and their
conjugate momenta $p_{u}$ and $-\mathcal{H}$. The extended Hamiltonian,

$\mathcal{H}_{+}=\frac{1}{2}p_{u}^{2}+\frac{1}{2}\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)u^{2}-\mathcal{H},$
(85)

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 (84), the GHS measure $\Theta$ for the
perturbations is the two-form

$\displaystyle\Theta$ $\displaystyle=$ $\displaystyle dp_{u}\wedge
du-(d\mathcal{H}\wedge d\eta)|_{\mathcal{H}_{+}=0}$ (86) $\displaystyle=$
$\displaystyle dp_{u}\wedge
du-\frac{1}{2}d\left[p_{u}^{2}+\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)u^{2}\right]\wedge
d\eta$ $\displaystyle=$ $\displaystyle dp_{u}\wedge du-p_{u}(dp_{u}\wedge
d\eta)-u\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)du\wedge
d\eta\,.$

One convenient hypersurface in which we can evaluate the flux of trajectories
is $\eta=\eta_{*}=\mbox{constant}$. (This is equivalent to a surface of
$H=\mbox{constant}$ or $a=\mbox{constant}$, although those are not coordinates
in the phase space of the perturbation.) As $\eta$ is always positive in a
matter- and radiation-dominated universe, this surface intersects all
trajectories exactly once. We then have

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\int_{\eta=\eta_{*}}\Theta_{p_{u}u}dudp_{u}$ (87)
$\displaystyle=$ $\displaystyle\int_{\eta=\eta_{*}}dudp_{u}.$

The flux of trajectories crossing this surface is unity, implying that all
values for $u$ and $p_{u}$ are 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, perturbations 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, $T_{\rm G}=10^{16}$ GeV ($\eta_{\rm G}\approx 6\times 10^{5}\,{\rm
eV}^{-1}$), 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 contrast $\delta=\widetilde{\delta\rho}/\bar{\rho}$ and
the Newtonian potential $\Phi$ are less than one. Because the phase space is
unbounded and the measure (87) 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 $z\sim 1200$, when CMB temperature anisotropies are observed.
Our expressions will be much less cumbersome, however, if we demand smoothness
at matter-radiation equality, $z_{\rm eq}\sim 3000$ ($\eta_{{\rm eq}}\approx
10^{31}\,{\rm eV}^{-1}$), within an order of magnitude of recombination. Since
the observed temperature anisotropies are of order one part in $10^{5}$, we
therefore define a realistic universe as one with $\delta_{\rm eq}\leq
10^{-5}$ and $\Phi_{\rm eq}\leq 10^{-5}$.

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 $L_{0}$ of our observable universe today is a few times
$H_{0}^{-1}=10^{33}~{}{\rm eV}^{-1}$, and the size of our comoving patch at
equality is $a_{\rm eq}=1/3000$ times that, so

$L_{\rm eq}\approx 10^{30}\,{\rm eV}^{-1}.$ (88)

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

$H_{\rm eq}^{-1}\approx m_{\rm pl}\left(\frac{a_{\rm
eq}^{2}}{T_{0}^{2}}\right)\approx 10^{28}\,{\rm eV}^{-1}.$ (89)

The total number of modes we consider is therefore

$n=\left(\frac{L_{\rm eq}}{H_{\rm eq}^{-1}}\right)^{3}\approx 10^{6}.$ (90)

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 $H_{\rm
eq}^{-1}$. 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,
$c_{s}k\eta<1$) in a radiation-dominated universe. In that case the general
solution to our evolution equation (79) is

$u=c_{1}\theta+c_{2}\theta\int_{\eta_{0}}\theta^{-2}\,d\eta,$ (91)

where $c_{1}$ and $c_{2}$ are constants. During radiation domination we have

$\theta=\frac{\sqrt{3}}{2\sqrt{a_{\rm eq}}H_{0}}\eta^{-1}.$ (92)

The solution for $u$ is therefore

$u=\alpha\eta^{-1}+\beta\eta^{2},$ (93)

where $\alpha$ and $\beta$ are constants. The conjugate momentum is

$p_{u}=-\alpha\eta^{-2}+2\beta\eta.$ (94)

The potential is related to $u$ by (77). In the radiation era we have

$(\bar{\rho}+\bar{p})^{1/2}=\gamma\eta^{-2},$ (95)

where we have defined

$\gamma=\frac{2m_{\rm pl}}{\sqrt{a_{\rm eq}}H_{0}}.$ (96)

Our general solution is therefore

$\Phi=\gamma(\alpha\eta^{-3}+\beta).$ (97)

Finally we turn to the density perturbation, which is given by (75). The
$\nabla^{2}\Phi=-k^{2}\Phi$ term is negligible for long wavelengths, so we’re
left with

$\widetilde{\delta\rho}=\frac{12m_{\rm pl}^{3}}{a_{\rm
eq}^{3/2}H_{0}^{3}}(2\alpha\eta^{-7}-\beta\eta^{-4}),$ (98)

which in turn implies

$\delta\equiv\frac{\widetilde{\delta\rho}}{\bar{\rho}}=2\gamma(2\alpha\eta^{-3}-\beta).$
(99)

To calculate the measure, it is convenient to use $\alpha$ and $\beta$ as the
independent variables that specify a mode. The measure is simply

$\mu=\int dudp_{u}=3\int d\alpha d\beta.$ (100)

This comes from taking the derivative of (93) and (94), treating $\alpha$ and
$\beta$ as the independent variables, and computing $du\wedge dp_{u}$. No
$\eta$-dependent factors appear when we write the measure in terms of $\alpha$
and $\beta$. We can also express it in terms of the density contrast and
Newtonian potential,

$d\Phi d\delta=6\gamma^{2}\eta^{-3}d\alpha d\beta.$ (101)

Therefore, a region in the $\Phi$-$\delta$ plane at time $\eta_{\rm G}$ has a
measure that is larger than the same coordinate region at time $\eta_{\rm eq}$
by a factor of

$\left(\frac{\eta_{\rm eq}}{\eta_{\rm G}}\right)^{3}\approx 10^{76}.$ (102)

The coordinate area of our initial region at the GUT scale is $\Delta\Phi_{\rm
G}\Delta\delta_{\rm G}\approx 1$, while the coordinate area of our region at
equality is $\Delta\Phi_{{\rm eq}}\Delta\delta_{{\rm
eq}}\approx(10^{-5})^{2}=10^{-10}$. For each mode, we therefore have

$\frac{\mu(\mathrm{inflationary})}{\mu(\mathrm{realistic})}=\left(\frac{\eta_{\rm
G}}{\eta_{\rm eq}}\right)^{3}\frac{\Delta\Phi_{\rm G}\Delta\delta_{\rm
G}}{\Delta\Phi_{{\rm eq}}\Delta\delta_{{\rm eq}}}\approx 10^{-66}.$ (103)

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

$P({\rm inflation})\approx(10^{-66})^{n}\approx 10^{-6.6\times 10^{7}}.$ (104)

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 $10^{-10^{122}}$ [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_ cosmological 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 cosmological 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 particular 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 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 mechanism) 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.
Understanding 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. 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. 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 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.333The 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. It
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 $H_{0}^{-1}$ today corresponds to
a physical size $\sim 10^{-26}H_{0}\sim 10^{34}L_{\rm pl}\sim 1$ cm when
$H=m_{\rm pl}$. In contrast, with inflation, the same patch needs to be no
larger than $L_{\rm pl}$ when $H=m_{\rm pl}$, 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.

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 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.

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