# Remarks on the Axion Domain Wall Problem

Michael Dine

###### Abstract

Theories in which the Peccei-Quinn phase transition occurs after inflation
tend to suffer from problematic domain walls. One possible solution involves a
small, explicit breaking ot the symmetry. But this raises other potential
issues. We review some aspects of axion domain walls, focussing especially on
this proposed solution. We argue, in disagreement with some recent literature,
that there is little axion radiation from the system until the domains
actually collapse. The same applies to gravitational waves and electromagnetic
radiation. The final stages of the collapse yields small numbers of extremely
energetic axions, which interact only rarely with ordinary matter, and are
thus relatively harmless. We then note that, if one accepts a remarkable
coincidence, this solution can be acceptable. We consider a possible
explanation of the required coincidence.

##  1 To Do

Consider domain walls bounded by strings, along the lines of the old paper by
Sikivie et al. Is there any change in the picture, e.g. due to attractive and
repulsive forces between string elements? Might imagine always have order one
domain per horizon before considering bias. Suppose, e.g., ${\cal N}$. In
light travel time to cross the horizon, system develops a large $\gamma$.

Ask what fraction of domain wall energy might be in hadrons and interpret in
terms of final collapse.

Think about final collapse in terms of particle collisions. If principally
axions, what are the collision products?

##  2 Introduction

A Peccei-Quinn symmetry[1, 2, 3, 4] has the potential to solve the strong CP
problem and account for the dark matter of the universe [5, 6, 7]. Before
considering cosmology, the axion decay constant, a priori, can take a broad
range of values. Stellar astrophysics places a lower bound in the range of
$10^{9}-10^{10}$ GeV[8]. Big bang cosmology, with the assumption that the
universe, in the past, was hotter than a GeV or so, places an upper limit of
about $10^{12}$ GeV. Attaining a symmetry of sufficient quality[9, 10] to
solve the strong CP problem, however, is quite a challenge. String theory, and
more general considerations of quantum gravity, rule out exact, continuous
global symmetries. So one expects that in the effective field theory at low
energies, at the very least there will be Planck-suppressed operators which
violate the symmetry. Even for the low range of $f_{a}$, operators of very
high dimension must be suppressed to account for the smallness of $\theta$[9].
One might try to account for this suppression by discrete symmetries[11], but
the symmetries must be quite large. String theory appears capable of avoiding
this problem[12], in the sense that PQ symmetries may be violated only by non-
perturbative effects, which can be extremely small if the theory generates a
small coupling constant. But in this case, a value of $f_{a}$ much smaller
than, say, typical scales associated with coupling constant unification, would
be surprising. Larger scales are admissible if the universe was never much
hotter than nucleosynthesis temperatures in the past. This might occur if
there was a period where the universe was dominated by moduli; see, for
example[13].

These observations arguably cast doubt on the PQ solution, and in any case,
would seem to favor large values of $f_{a}$ and a modified cosmology. In this
paper, however, we will adopt the conventional picture that the universe was
quite hot in the past and we will assume that there was a PQ transition after
inflation, and focus on the problem of domain walls.

In the scenario in which there is a PQ transition after inflation, domain
walls are potentially problematic[14].If the PQ symmetry is an exact,
continuous global symmetry (up to anomalies), the theory has stable domain
walls provided the coefficient of the QCD anomaly is (suitably normalized) an
integer different than one. The domain wall energy density falls off as $1/R$,
where $R$ is the scale factor, as opposed to the radiation energy density
($1/R^{4}$) or the matter dominated energy density $1/R^{3}$. Typically, the
domain walls dominate well before the present era, spoiling the successes of
the Standard Cosmology.

Two plausible solutions to this problem were put forward in[14]:

  1. 1.

The coefficient of the anomaly is unity.

  2. 2.

There is a small explicit breaking of the PQ symmetry, large enough to lead to
collapse of the domain wall system.

Even for the first solution, one has to consider the effects of cosmic
strings[15]. The second solution has troubling features. The strength of the
leading symmetry breaking operator is restricted to a narrow range. First, it
must be small enough that the resulting $\theta$ satisfies the current
experimental bounds. Typically this requires that the leading operator which
breaks the continuous PQ symmetry is of very high dimension[9]. Second, the
operator must be large enough that, if there are domain walls, these disappear
before they come to dominate the energy density of the universe. Naively, for
interesting values of $f_{a}$, the axion decay constant, these two conditions
limit the symmetry breaking operator to a narrow range (we will take
$f_{a}=10^{12}$ GeV as our benchmark). If the suppression of high dimension
operators is a consequence of discrete symmetries, these symmetries must be
very large, but not too large[11].

Recently, the authors of [16] have revisited the domain wall problem. They put
forward and then rule out a third possible solution: a bias in the domain wall
ensemble favoring one of the ground states.They then argue that the second
solution above is unlikely to work, except for relatively small values of the
axion decay constant, small enough as to be problematic for stellar processes.
They argue, in particular, that the collapsing domain wall system produces too
much dark matter.

Motivated by the appearance of high quality PQ symmetries in string theory,
the present author has long been an advocate for high scale breaking of the PQ
symmetry, which requires a breakdown of the Standard Big Bang Cosmology at
temperatures not much higher than nucleosynthesis temperature. But in this
note, we consider the possibility of a post-inflationary PQ transition and
smaller axion decay constants, with a conventional thermal history for the
university at least up to temperatures of a few GeV. We will focus on the
issues associated with the second solution, small breaking of the Peccei-Quinn
symmetry. We will consider the dark matter issue, demonstrating that, until
the domains collapse, most of the excess energy is converted into kinetic
energy of the domain walls. Once the domains shrink to sizes of order
$m_{a}^{-1}$, this energy is converted to ultrarelativistic axions and
hadrons. Provided that the the domain walls never dominate, these objects are
relatively harmless.

We will review the problem of accounting for small symmetry breaking,
focussing on models where the PQ symmetry is an accident of a large discrete
symmetry[11]. Such symmetries, at least at first sight, are not particularly
plausible. The requirements, indeed, yield extremely large symmetries, yet the
symmetry also cannot be too large if the breaking is to be sufficient to avoid
a domain wall dominated universe. Given the seeming absurdity of the
requirements on the symmetry breaking, we ask whether there might be some
anthropic explanation. Taking a very conservative approach to the anthropic
principle, where one asks whether the change of one particular parameter can
rule out the existence of observers, anthropic constraints have the potential
to restrict the strength of the symmetry breaking to the required range.

This note is organized as follows. In the next section, we review some aspects
of domain walls and the cosmic strings which bound them[14]. In particular, we
discuss the sense in which one can systematically construct the domain wall
from the chiral lagrangian, and also provide a simple, analytic domain wall
solution in a particular limit. In section 4.1, we study the system in the
presence of explicit breaking. We discuss constraints on the size of the
breaking and the axion decay constant. We focus, particularly, on models where
the PQ symmetry arises from a large discrete symmetry, noting that the
symmetry must be quite large to accommodate the current limits on $\theta$ but
(anticipating our cosmology discussion) can’t be appreciably larger than this
if it is to avoid domain wall catastrophes. In section 4 we turn to cosmology.
After reviewing aspects of the cosmological domain wall problem, we
demonstrate that the wall collisions principally produce gravitational waves
and note that their energy density can readily be in a suitable range. We then
turn to the coincidence problem in section 5, arguing that it places
requirements on a theory which call out, if a PQ symmetry is realized in
nature in this fashion, for an anthropic solution. As noted above, we will see
that such a solution is plausible. It is hard to see how otherwise there would
be any solution at all. Our conclusions are presented in section 6.

##  3 Domain Wall Generalities

Suppose we have an exact Peccei-Quinn symmetry up to the anomaly. Under the PQ
symmetry, the various fields transform by phases $e^{iq_{pq}\alpha}$. By
convention, we take the PQ charges, $q_{pq}$ to be integers. $\theta$ changes,
in general, by $2\pi{\cal N}$ for some integer ${\cal N}$ under this
transformation. Given $2\pi$ periodicity of $\theta$, the symmetry is in fact
$Z_{\cal N}$. For ${\cal N}\neq 1$, there are domain walls.

The tension of the domain walls is of order

$T\sim m_{a}f_{a}^{2}\sim m_{\pi}f_{\pi}f_{a}.$ (3.1)

We will have in mind $f_{a}\sim 10^{11}-10^{12}~{}{\rm GeV}$ in what follows.
It is of interest to ask whether, within the framework of chiral perturbation
theory and/or large $N$, we can write a strict equality for the domain wall
tension.

###  3.1 Domain Wall Solutions from the Chiral Lagrangian

Just as one can compute the axion mass using the chiral lagrangian, one can
obtain the domain wall solutions in the case that the system supports axionic
domain walls. Suppose that the light quarks have PQ charges $q_{i}$ (we will
mainly write explicit formulas for the case of two light quarks). Suppose, as
well, that the PQ symmetry has anomaly

$\partial_{\mu}j^{\mu}_{PQ}={{\cal N}\over 32\pi^{2}}F\tilde{F}.$ (3.2)

Then we can define an anomaly free current, $\tilde{j}^{mu}_{PQ}$ by
subtracting off a non-conserved current with the same anomaly:

$j^{5\mu}={\cal
N}(\bar{u}\gamma^{\mu}\gamma^{5}u+\bar{d}\gamma^{\mu}\gamma^{5}d)$ (3.3)

So now:

$\partial_{\mu}\tilde{j}_{PQ}^{\mu}={\cal
N}(m_{u}\bar{u}\gamma^{5}u+m_{d}\bar{d}\gamma^{5}d).$ (3.4)

In computing the axion mass, one sometimes makes a different choice[17], so
that the divergence of the current does not have matrix elements between
vacuum and the single pion state, but this is not convenient for the domain
wall problem, where one needs to consider a finite axion field range.

We can explore the effect of finite transformations generated by
$\tilde{Q}_{PQ}$,

$U(\alpha)=e^{i\alpha\tilde{Q}_{PQ}}.$ (3.5)

Under this transformation the quark mass terms are not invariant; these
transform as:

$m_{u}\bar{u}u+m_{d}\bar{d}d\rightarrow(m_{u}\bar{u}u+m_{d}\bar{d}d)\cos({\cal
N}\alpha)+(m_{u}\bar{u}\gamma_{5}u+m_{d}\bar{d}\gamma_{5}d)\sin({\cal
N}\alpha).$ (3.6)

Now consider the chiral lagrangian. Our goal is to integrate out the pion
fields. We do this by replacing $\bar{u}u$ and $\bar{d}d$ by their expectation
values as functions of the pseudogoldstone fields, and solving for the minimum
of the $\vec{\pi}$ potential as a function of $a=\alpha f_{a}$. Switching to a
two component notation, and letting $f,g$ denote flavor indices:

$\bar{\psi}(x)_{f}\psi(x)_{g}=\langle\bar{\psi}(0)\psi(0)\rangle\left(e^{i{\vec{\pi}\cdot\sigma\over
2f_{\pi}}}\right)_{fg}$ (3.7)

We only have to solve for $\pi_{0}$.

A particularly simple case is that of $m_{u}=m_{d}$. Then the potential is
independent of $\pi_{0}$:

$V(a)=-m_{\pi}^{2}f_{\pi}^{2}\cos({\cal N}{a\over
f_{a}})=-{f_{a}^{2}m_{a}^{2}{\cal N}^{-2}}\cos({\cal N}{a\over f_{a}}).$ (3.8)

The presence of domains for ${\cal N}\neq\pm 1$ is manifest; the potential has
${\cal N}$ degenerate minimima with

${a\over f_{a}}={2\pi k\over{\cal N}};k=0,\dots,{\cal N}-1,$ (3.9)

and the existence of domain wall solutions follows. The domain wall solution
can be written down explicitly; it is the static soliton of the Sine-Gordan
theory:

$a(x)=f_{a}\left(4{\arctan(e^{m_{a}x})\over{\cal N}}+{2\pi k\over{\cal
N}}\right).$ (3.10)

The tension of the domain wall satisfies:

$T\propto f_{a}^{2}m_{a}\propto{f_{a}m_{\pi}f_{\pi}}.$ (3.11)

It $m_{u}\neq m_{d}$, there is an additional contribution from the pion fields
proportional to

$\delta T\propto{m_{u}-m_{d}\over m_{u}+m_{d}}{f_{a}m_{\pi}f_{\pi}}.$ (3.12)

So, in general, the pions make an order one contribution to the tension. In
the final collapse of the domains, this will be associated with production of
energetic hadrons.

##  4 Domain Wall Cosmology

Domain walls, if they come to dominate the energy density of the universe, are
problematic[8]. The domain wall energy density decreases as $1/R$, so it can
quickly overwhelm the density of radiation or matter, falling as $1/R^{4}$ or
$1/R^{3}$, respectively. So it is necessary that there either never were
domain walls at all, or that they disappear relatively quickly, typically by
times of order a few seconds after the big bang. Reference [16] considers, in
addition to the two proposed solutions we mentioned earlier, a third
possibility, that of a biased domain wall distribution, but rules it out. They
then argue that the constraints associated with PQ violating operators have
been underestimated. We will address their critique shortly.

Reference [14] analyzed explicit PQ symmetry violation, tilting the axion
potential and causing all but one type of domain to collapse. Suppose the
splitting between states is:

$\Delta V=\epsilon~{}10^{-10}~{}m_{\pi}^{2}f_{\pi}^{2}.$ (4.1)

This corresponds to a potential for the axion roughly of the form:

$\Delta V(a)=\epsilon~{}10^{-10}m_{a}^{2}f_{a}~{}a.$ (4.2)

$\epsilon$ cannot be extremely small if the domain wall system is not to
dominate the energy of the universe before it disappears.

When the temperature is of order $f_{\pi}$, the domain walls form. The
corresponding Hubble parameter is $H_{0}={m_{\pi}f_{\pi}\over M_{p}}$. Calling
the corresponding scale factor $R_{0}$, the domain wall density is
subsequently of order:

$\rho_{DW}=(f_{a}m_{\pi}f_{\pi}){m_{\pi}f_{\pi}\over M_{p}}{R_{0}\over R}.$
(4.3)

So domain walls dominate when

$\left({R_{0}\over R}\right)^{3}\approx{f_{a}\over M_{p}}$ (4.4)

or

$\left({R_{0}\over R}\right)\approx 10^{-2}\left({f_{a}\over
10^{12}}\right)^{1/3}.$ (4.5)

This corresponds to a temperature of order $1$ MeV, or times of order 10
seconds. How quickly the domains collapse is the subject of the next section.

###  4.1 Fate of the Domain Walls

As noted in the literature, when the walls collapse, their energy can be
converted to kinetic energy of the domain walls, to axions, gravitational
waves, electromagnetic radiation, and possibly other types of matter or
radiation. We will shortly argue that, before collapse, the energy is
principally converted to kinetic energy of the walls, followed by highly
relativistic axions. At the final collapse, this energy is converted to
extremely relativstic axions. Gravitational and electormagnetic radiation are
minor components of the energy budget. These axions would still be highly
relativistic today.

When formed, the bubbles have radius, $r_{0}$, of order:

$r_{0}\approx{M_{p}\over m_{\pi}f_{\pi}}.$ (4.6)

Initially their acceleration due to $\Delta V$ is slightly less than that due
to the Hubble expansion, $H$:

$a\sim{\Delta V\over f_{a}f_{\pi}^{2}}\approx\epsilon
10^{-22}f_{\pi}^{2};~{}~{}H\sim 10^{-19}f_{\pi}^{2}.$ (4.7)

They become comparable when the temperature decreases by a factor of order
$10^{2}$. beyond this point we can, to first approximation, neglect the
expansion. The velocity becomes of order one in less than a Hubble time, and
collapse occurs in such a time. We can then ask: if we ignore gravitational
radiation, what is the velocity of the domain wall once the domain shrinks to
a microscopic size (more precisely, how large is the Lorentz $\gamma$ factor
for the wall.

We take as the initial time the time when the cosmic acceleration is equal to
the acceleration of the wall:

$H_{0}={\Delta V\over f_{\pi}^{2}f_{a}}\approx\epsilon\times 10^{-24}~{}{\rm
GeV}.$ (4.8)

The energy stored in a horizon sized region of one of the excited states is:

$E\approx\Delta VH_{0}^{-3}\sim\epsilon 10^{58}~{}{\rm GeV}.$ (4.9)

If there is no emission of axions, gravitational or other radiation during the
collapse, then once the region size is of order $m_{a}^{-1}$, the $\gamma$
factor is enormous. The effective mass of the system is of order

$m_{eff}=f_{a}f_{\pi}^{2}m_{a}^{-2}\sim 10^{38}~{}{\rm GeV},$ (4.10)

so

$\gamma\sim\epsilon 10^{20}.$ (4.11)

We now argue that in fact most of the energy gained is transferred to kinetic
energy of the domain wall. Initially the domain is large and the curvature of
the domain wall is negligible on macroscopic scales. To consider axion
radiation, it is helpful to work in the instantaneous rest frame of the domain
wall (more precisely, of a macroscopic segment of the wall). We can define
what we mean as axion radiation by considering a set of domain walls,
instantaneously at rest, described by a classical field configuration,

$\phi(\vec{x})=\phi_{cl}(\vec{x}-\vec{x}_{0}).$ (4.12)

Were it not for the symmetry-breaking potential, $\Delta V$, these
configurations would be solutions of the equations of motion if
$\vec{x}_{0}=\vec{v}t$. But due to the potential, they are not. Axion
radiation corresponds roughly to the difference, $\delta\phi$, of the actual
axion field and the would-be domain wall configuration at a given time. This
is proportional to

$\delta\phi(\vec{x},t)={\rm
C}\ddot{x}_{0}(t)_{i}\cdot\partial_{i}\phi_{cl}(\vec{x},t).$ (4.13)

It would be challenging to compute the energy carried by $\delta\phi$, but
it’s form, for non-relativistic motion, can be determined by simple
considerations. The energy should be rotationally invariant, translationally
invariant, and time reversal invariant (up to very small effects within the
Standard Model coupled to an axion). So the energy per unit time transferred
to $\delta\phi$ behaves as:

${\cal E}={\rm B}f_{a}^{2}|\ddot{x}_{0}^{2}|A$ (4.14)

where $A$ is the area of the domain wall, and ${\rm B}$ is an order one
constant. We can write $\ddot{x}_{0}$ in terms of $\Delta V$ and the tension
of the domain wall,

$\ddot{x}_{0}={\Delta V\over f_{\pi}m_{\pi}f_{a}}.$ (4.15)

Correspondingly, in a frame boosted with Lorentz factor $\gamma$, the energy
per unit time is increased by a factor of $\gamma$ for the energy, but
decreased by a factor of order $\gamma$ from the time dilation, so the
transformation between the two frames behaves as $(\gamma)^{0}$.

A little more precisely, we might consider radiation in a time interval
$\Delta t$ in the instantaneous rest frame of the wall. $\Delta t$ should be
such that the wall is non-relativistic in that frame. For the element of area
$A$, there will be movement of order $\Delta t\times v$ in this time period.
The corresponding time elapsed in our observers frame is:

$\Delta t^{\prime}=\gamma(\Delta t+v\Delta z)$ (4.16)

but the second term is, by assumption, much smaller than the first and can be
neglected.

We want to compare the energy radiated by the wall per unit time with the
energy the wall acquires per unit time from $\Delta V$. This is

$\Delta VA,$ (4.17)

so the condition that radiation is comparable to the energy increase of the
wall per unit time due to $\Delta V$ is

${\Delta V\over f_{\pi}^{2}m_{\pi}^{2}}>1$ (4.18)

which is never satisfied. In other words, the energy radiated in axions as the
walls collapse is negligible. Similar considerations lead to suppression of
electromagnetic and gravitational radiation. So the walls are extremely
relativistic when they finally shrink to microscopic size. At this point, one
expects that the collapse results in production of extremely relativistic
axions, with $\gamma$ factors comparable to those we discussed above, and very
relativistic hadrons. The hadrons quickly thermalize with the background
hadrons. The number of axions produced would be small, consistent with the
fact that the final domains are microscopic in size. As we explain below, very
little of the axion energy would be degraded in collisions with hadrons. In
any case, provided that the domain walls don’t dominate the energy density of
the universe at the time of their collapse, their cosmological effects would
be minor.

###  4.2 The Final Stage of Domain Wall Collapse

Because of the enormous $\gamma$ factors of the axion produced in the decay of
a domain, most of these axions stream through the universe. Only rarely does
one interact with quarks, gluons. or other axions. For $s$-channel processes,
the mean time between collisions is long. Even assuming an order one coupling
of these high energy axions to nucleons,

${\delta{\cal L}}=a\bar{N}\gamma_{5}N,$ (4.19)

the mean free time for axion collisions with nucleons is

$\tau\sim 10^{10}\gamma m_{a}T^{-3}m_{N}\sim 10^{33}\left({\gamma\over
10^{20}}\right)\left({{\rm MeV}\over T}\right)^{3}\left({m_{a}\over
10^{-14}}\right)$ (4.20)

in ${\rm GeV}$ units, and we have taken ${n_{B}\over n_{\gamma}}\approx
10^{-10}$. This is an enormous time, comparable to even the current age of the
universe. As we have noted, the $\gamma$ factor is actually likely to be
serval orders of magnitude larger than $10^{25}$, giving further suppression.

For $t$-channel exchange, due to the enormous $\gamma$ factor, scattering is
appreciable only at extremely small angles. As a result, the mean scattering
time is, again, extremely large. So most of these axions are still around
today, and are highly energetic. Their numbers, however, are not large, and
their contribution to the energy budget of the present universe is smaller
than the photon contribution (assuming that the domain wall contribution was
always a small fraction of the total energy density at the time of collapse).
Interactions with ordinary matter are very rare.

##  5 Anthropic Considerations

As we have stressed, there is one troubling feature of the Peccei-Quinn
symmetry for these relatively low $f_{a}$ axions in the suppression required
for a Peccei-Quinn symmetry of sufficient quality to solve the strong CP
problem, and another in the requirement of sufficient tilt solution to solve
the domain wall problem. One might, indeed, argue that the requirements to
obtain a PQ symmetry of sufficient quality to solve the strong CP problem are
implausible, For example, if the result of a discrete symmetry, the symmetry
must be very large[9, 11]. On the other hand, we have seen that to avoid a
domain-wall dominated universe, the tilt must be just barely smaller than that
required for the PQ symmetry. One might be inclined to discard axions with
these relatively low decay constants, but it is also tempting to ask whether
such bizarre constraints might be satisfied as a result of anthropic
considerations. In this section, we will examine this possibility. We do not
attempt to understand precisely how, within some sort of landscape, this might
be realized in detail. Rather we ask the simpler question: might the existence
of observers be ruled out if we made a change in a single parameter, holding
others fixed.

If the tilt is controlled by a discrete symmetry, we might imagine that the
symmetry violating terms in the axion potential have the form
$AM_{p}^{-n}\Phi^{n+4}$, where $\Phi=f_{a}e^{i{a\over f_{a}}}$, $A$ is now a
complex constant (not the area) and $\Phi\rightarrow e^{{2\pi i\over N}}\Phi$
under the $Z_{N}$ symmetry, so

$\delta V=|A|f_{a}^{4}\left({f_{a}\over M_{p}}\right)^{N-4}\cos((N{a\over
f_{a}}+\alpha).$ (5.1)

In order that $\theta$ be small enough, assuming $A$ is of order one, $N$ must
be quite large. If $f_{a}=10^{12}$ GeV, we require $N\geq 13$, for example. At
the same time, our discussion of domain wall evolution implies that $N$ can’t
be larger than $13$.

The lower limit on the tilt (upper limit on $N$) is a relatively easy one to
explain anthropically. Were domain walls to dominate the energy density of the
universe before their collapse, the universe would not evolve to a situation
with structures of the sort we see in nature (and which support observers).

The upper limit might be understood if we assume that a dark matter density
close to that observed is necessary for (or perhaps optimizes the number of)
observers. Write the tilt contribution to the axion potential as

$\delta V(a)=\epsilon 10^{-10}m_{\pi}^{2}f_{\pi}^{2}\cos({a\over
f_{a}}+\alpha).$ (5.2)

We require $\epsilon<1$. Suppose $\epsilon$ was much larger, corresponding to
$N=11$, for example, and $\epsilon\sim 10^{4}$. Then the axion mass receives a
larger contribution from $\delta V$ then from QCD, and it starts to oscillate
when the temperature is of order $10^{5}$ GeV. As a result, there are far too
few axions to constitute the dark matter. Axions only account for about
$10^{-20}$ of the energy density initially. At temperatures of order $1$ eV,
they are only $10^{-6}$ of the total energy of the universe.

The case of $n=12$ is different. The contribution, at zero temperature, to the
axion mass from $\delta V$ is smaller than that from QCD, but we have to
investigate the behavior of the system as a function of temperature. The axion
mass as a function of temperature behaves as[8]:

$m_{a}(T)\approx m_{a}\left({\Lambda_{QCD}\over T}\right)^{3.7}.$ (5.3)

Without $\delta V$, the axion begins to oscillate when

$3H=m_{a}(T).$ (5.4)

This corresponds to a temperature of order $10$ GeV. But $m_{a}(T)<\delta
m_{a}$ down to a smaller temperature, of order $0.2$ GeV if we take the
formula 5.3 seriously at these temperatures. So there is significant
overproduction of dark matter.

All of this is meant to establish that there is a plausible anthropic
rationale for the size of $\delta V$ being such that axions constitute the
dark matter, and the domain wall system collapses “just in time”. We stress
again that we don’t have a detailed cosmological picture for how this might be
implemented.

##  6 Conclusions

In this paper we have adopted the conventional view of axion cosmology, that
the PQ transition occurred after inflation and that the post inflationary
universe was once at a temperature well above the scales of QCD, and focused
on the resulting question of domain walls. We have recalled that the problem
is generic, and admits only a small number of solutions. Among these, we have
considered the effects of small, explicit breaking of the symmetry. We have
recalled the well-known issue that the constraints of obtaining small enough
$\theta$ and domain wall annihilation before domain wall dominance restrict
the symmetry breaking to a narrow range. We have pointed out that it would be
almost absurd for the size of this breaking to be a consequence of discrete
symmetries. We have noted that, as for other seemingly absurd phenomena
actually observed in nature, one might contemplate an anthropic solution. This
has the virtue that it could explain the remarkable coincidences required.

But most of our attention has been devoted to the fate of the universe in such
a picture. We have studied the question of where the energy in the domain
walls goes. We have noted that gravitational radiation leads to a limiting
Lorentz factor, $\gamma$, which in turn implies that the vast majority of the
domain wall energy is converted to gravitational waves. This is in contrast to
the possibility that much of the energy ends up in non-relativistic, dark
matter axions[16]. The resulting constraints are mild, in the sense that they
are not much stronger than the requirement that the domain wall energy not
dominate the energy density of the universe before annihilation.

## Acknowledgments

We thank Patrick Draper, Guido Festuccia and Pierre Sikivie for conversations
and critical comments. This work was supported in part by U.S. Department of
Energy grant No. DE-FG02-04ER41286.

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