1. Introduction
The production of cosmic defects through the Kibble mechanism is a generic prediction of grand unified scenarios [
1]. These defect networks can leave behind a variety of observable cosmological signatures, provided that they survive long enough and their characteristic energy scale is sufficiently high (see, for example, Ref. [
2] and references therein). An accurate description of the cosmological evolution of cosmic defects is crucial to performing accurate predictions of their observational consequences. This is usually achieved either by resorting to Nambu-Goto [
3,
4,
5,
6] or field theory numerical simulations [
7,
8,
9,
10] or by using semi-analytical models describing the evolution of a few thermodynamic variables that characterize the large-scale dynamics of the networks [
11,
12,
13,
14]. While the first approach might be preferable for specific models inspired by particle physics, the latter is more versatile since it can easily accommodate a variety of well-motivated scenarios or even the phenomenological modeling of a priori unknown physics. These semi-analytical models have been used with great success to constrain cosmological scenarios with cosmic strings [
15,
16,
17,
18,
19,
20,
21,
22] and domain walls [
23,
24]. Although the energy scale of domain-wall-forming phase transitions is limited to be smaller than 0.92 Mev [
24], the existence of domain walls has not been ruled out, and they may still play a relevant role in cosmology—for instance, as a subdominant contributor to dark energy [
25]. This further motivates the study of their cosmological consequences and the development of more accurate and practical models to study the cosmological evolution of domain wall networks.
A velocity-dependent one-scale (VOS) model for the evolution of the characteristic length
L and of the root-mean-squared velocity
of domain wall networks in flat expanding or collapsing homogeneous and isotropic universes was developed in [
26,
27], generalizing previous work on cosmic strings [
11,
12] (see also for [
13,
14,
28] for a unified description of
p-brane dynamics). This model has been shown to provide an accurate description of domain wall dynamics not only in cosmology but also in the context of non-relativistic systems in condensed matter [
26] and biology [
29], wherein the dynamics are dominated by the curvature of domain interfaces. Unfortunately, this approach relies on numerical simulations for the (cosmology-dependent) calibration of its two phenomenological parameters, usually referred to as energy-loss and momentum parameters (see also [
10,
30] for a six-parameter extension of the standard parametric domain wall VOS model).
Recently, the evolution of cosmological domain walls in an expanding universe characterized by a power law evolution of the scale factor
a with the cosmic time
t (
) has been investigated using a parameter-free VOS model [
31,
32]. The model predictions have been compared with the results of field theory numerical simulations, with a notable agreement obtained for
. Although for slower expansion rates, the values of the characteristic length
L and root-mean-squared velocity
predicted by the parameter-free VOS model can be larger by up to
than the reported numerical results, a number of potential problems with the determination of
L and
in field theory numerical simulations have been identified. These problems are expected to manifest themselves, especially in the relativistic regime [
9,
31], and need to be addressed before a more meaningful comparison with the model predictions can be performed.
In the present paper, we use the parameter-free VOS model to derive an analytical approximation for the evolution of the characteristic length
L and root-mean-squared velocity
of standard frictionless domain wall networks in homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) universes undergoing a power law evolution of the scale factor
a with cosmic time
t (
, with constant
). In
Section 2, we briefly review the parameter-free VOS model for the evolution of cosmological domain walls. In
Section 3, we derive several analytical results for the dynamics of cylindrical and spherical domain walls and use them to determine the linear scaling parameters of the parameter-free VOS model for
and
. We also briefly discuss the curvature-dominated evolution of individual domains in the
limit. In
Section 4, we shall use the approximation found for
close to unity, in combination with the exact results found for
, to obtain a fit to the model predictions valid for
with a maximum error of the order of
. We also confront the predictions of the model with the results of numerical simulations. In
Section 5, we provide an alternative demonstration of the dependence of the linear scaling parameters on
, in the
limit, using the standard VOS model for domain walls and determine the value of the phenomenological energy-loss parameter in this limit. We also discuss the implications of this result for more general multi-parameter domain wall VOS models. Finally, we conclude in
Section 6.
We shall use fundamental units with , where c is the speed of light in a vacuum.
2. Parameter-Free Domain Wall VOS Model
In this section, we briefly describe the parameter-free domain wall VOS model for the evolution of standard frictionless domain wall networks proposed in [
31]. We shall consider a flat
-dimensional homogeneous and isotropic FLRW universe with line element
where
a is the cosmological scale factor,
is the conformal time,
t is the physical time and
are comoving spatial coordinates. In this paper, we focus on expanding cosmologies that have a power law evolution of the scale factor with the physical time:
with
, so that
Our parameter-free VOS model [
31] is motivated by field theory simulations of standard domain wall network evolution, which show that domain wall intersections are rare [
10] and by the fact that thin domain walls are not expected to produce significant amounts of scalar radiation, except in the final stages of collapse [
33]. In this model, the network is assumed to be composed of infinitely thin domain walls that are taken to be either all spherical (
) or all cylindrical (
)—for
, all the domain walls are assumed to be oriented along parallel axes. The corresponding energy or energy per unit length (for spherical or cylindrical walls, respectively) is equal to
where
q represents the comoving radius of each domain wall,
,
, and
is the proper domain wall energy per unit area. The consideration of both spherical and cylindrical configurations makes it possible to evaluate the impact of geometry on the model predictions.
Domain walls are assumed to start at rest at some early conformal time
with an initial comoving size
(which varies from wall to wall) and, therefore, assumed to never intersect each other. When a domain wall reaches
, it is assumed to decay instantaneously, ceasing to be part of the network. The initial comoving radius of the domain walls that disappear at a conformal time
shall be denoted by
. The probability density function of the initial comoving radii
of the domain walls is given by [
31]
where
is the Heaviside step function. The form of this probability density function was chosen in such a way that the initial energy density of domain walls with
larger than
satisfies
It was also required that it satisfies the normalization condition
.
The total energy density of domain walls at a conformal time
can be computed as
Here,
is the minimum initial comoving radius required for a domain wall to survive until the conformal time
,
, where
is the initial domain wall number density defined as the number of walls per unit volume (spherical case) or per unit area (cylindrical case), and the scale factor is normalized to unity at the initial time (
).
The world history of an infinitely thin featureless domain wall in a flat expanding FRLW universe can be represented by a three-dimensional world-volume obeying the usual Nambu–Goto action. For a cylindrical or a spherical domain wall (
or
, respectively) and a power law expansion described by Equation (
2), the corresponding equations of motion may be written as
where
,
, and
.
The characteristic length of the network, defined by
, satisfies [
31]
where
and
, while the mean-squared velocity of the domain walls is given by [
31]
4. Fitting the Scaling Predictions of the Parameter-Free VOS Model
In
Figure 1, we show the evolution of the reduced radius
(top panel) and velocity
v (bottom panel) of cylindrical domain walls in universes with different expansion rates, parameterized by
. The solid line represents the numerical solution obtained using the Nambu–Goto equations of motion, while the dash-dotted line represents the approximation given in Equations (
10)–(
12).
Figure 1 shows that, independently of the value of
, the agreement between this approximation and the numerical solution is always excellent for sufficiently low values of
in a regime where the domain walls are still non-relativistic. Furthermore, for values of
close to unity, this excellent agreement spans almost the full conformal lifetime of the domain walls since, as shown in the previous section, in this limit, they only become relativistic extremely close to
. We have verified that qualitatively similar results are obtained in the case of spherical domain walls.
In
Figure 2, we compare the value of the domain wall lifetime, parameterized by
, calculated by solving the Nambu–Goto equations of motion (solid lines) numerically with the prediction of the analytical approximation given in (
12) (dashed lines). The blue and orange lines display the results obtained for cylindrical and spherical domain walls, respectively, while the blue and orange circles represent the corresponding exact solutions for
.
Figure 2 shows that there is an excellent agreement between the analytical approximation and the numerical solution for
. On the other hand, for smaller values of
λ, the approximation slightly underestimates the value of
.
Figure 3 summarizes the different results obtained for
(top panel) and
(bottom panel). The solid lines show the (numerically evaluated) prediction of the parameter-free VOS model for cylindrical and spherical domain walls (in blue and orange, respectively), which has an exact analytic solution for
(represented by the blue and orange circles). The dashed and dash-dotted lines represent, respectively, the approximations A1 and A2 discussed in the previous section. Here, we have taken
, thus requiring the model to reproduce the value of
obtained using field theory simulations for
[
30].
Figure 3 shows that both approximations do an excellent job for
but deviate more significantly from the model prediction for smaller values of
λ. Furthermore, notice that while approximation A1 predicts the same values of
ζ and
for both cylindrical and spherical domain wall configurations, the degeneracy between these two different geometries is broken by approximation A2. This is indeed a property shared by the exact solutions, thus showing that the geometry of the domain walls plays a crucial role for small values of
λ in which the relativistic regime of the domain wall evolution lasts for longer.
The unfilled circles in
Figure 3 represent the values of
ζ and
measured in field-theory numerical simulations [
10]. Notice the excellent agreement for
between the predictions of the parameter-free VOS model and the simulation results (see also
Table 1). For smaller values of
λ, there is still good qualitative agreement, but significant discrepancies occur. These discrepancies may be associated with the limitations of numerical field theory simulations in the relativistic limit [
9] and of the methods used to extract
L and
or may stem from the fact that the parameter-free VOS model only considers simple geometries. These are discussed in detail in [
31,
32] and will need to be addressed in future work. An important feature that a comparison between the prediction of the parameter-free VOS model and the results of numerical simulations will have to explain is the fact that both the results for
ζ and
fall below the model predictions.
The analytical results obtained in the present paper can be used to construct an accurate fit to the scaling predictions of the parameter-free VOS model for
and
over the whole range of
λ (λ∈[0, 1]) for which a linear scaling solution with cosmic time is possible in an expanding homogeneous and isotropic FLRW universe. We shall consider a fit of the form
where
represents either
or
,
is the exact solution for
in the
case,
is the first analytical approximation
for
as given in Equations (
14) and (
18),
, and
are the fit parameters given in
Table 2. We have verified that the maximum relative error of the fits for both
and
is of the order of
, allowing us to provide a closed formula to the scaling values of
and
predicted by the parameter-free VOS model.
5. Parametric VOS Model
In this section, we will consider the non-relativistic limit of the original (parametric) VOS model for the cosmological evolution of domain wall networks. This model provides a quantitative thermodynamical description of their cosmological evolution by following the evolution of
and
L [
14,
26,
27]:
Here, we introduced the damping length scale and
, which incorporates both the impact of the cosmological expansion (through the Hubble parameter
H) and the effect of the frictional forces caused by the scattering of particles of the domain walls (encoded in the friction length scale,
). In what follows, we shall neglect the effects of friction in the evolution of the network (i.e., we set
).
Notice that this model has two parameters that need to be calibrated against numerical simulations. The first of these,
k, is a dimensionless curvature parameter that, to some extent, describes the average domain wall curvature. The second parameter,
, was introduced to quantify the efficiency of the energy loss associated with the collapse of the domain walls. This energy loss results in a contribution to the evolution of the average domain wall energy density, which, analogously to the cosmic strings’ case [
36], is usually assumed to be of the form
For infinitely-thin and featureless domain walls, Equations (
35) and (
36) may be obtained directly from the generalized Nambu–Goto action assuming an FLRW background and making a couple of approximations, except for the
term that results from this energy loss [
14].
Conservatively, we may expect the rate of variation of energy not associated with the expansion of the cosmological background to be such that
where
is the physical time required for a domain wall to travel across a comoving distance of
. We should then have that
Domain wall networks are known to evolve towards a linear scaling regime of the form
for a power law expansion of the form
, with constant
and
. The parametric VOS model predicts that this regime should be characterized by:
and thus, these equations may be used to calibrate the free parameters
k and
using numerical simulation.
In the non-relativistic regime, with
, this model then predicts that
and
. This necessarily means that the energy loss term in Equation (
37) is, given its dependence on
L and
, irrelevant in this limit. In what follows, we will further show that we necessarily have that
in the non-relativistic limit and that, as a result, the importance of this energy loss term decays significantly faster than Equation (
37) seems to indicate as
. Notice that
in the same limit (here, we have taken into account that
).
The time
a domain wall takes to transverse a comoving distance of
may be estimated as
and hence, it is given approximately by
For
, Equation (
44) yields
and, thus, Equation (
39) implies that we should have
. This shows that, in the relativistic limit, Equations (
39) and (
44) still leave some freedom as to the choice of
. However, in the non-relativistic limit, this is not the case. Taking into account that Equation (
41) implies that
one finds that, in the
limit,
should be infinitely larger than
. It then follows from Equation (
39) that
must necessarily vanish in this limit.
This actually means that, in the non-relativistic limit, the parametric VOS model must reduce to the parameter-free model as it effectively includes no additional source of energy loss (beyond expansion). In fact, in this limit, we should have
which has the same asymptotic behavior as the approximation derived in
Section 3.1 using the parameter-free model. The approximation in
Section 3.1 was derived by taking into account the dominant term in the domain wall acceleration. A simpler approximation can be derived by neglecting the acceleration of the walls in this limit. In this case, domain wall evolution would still be described by Equations (
10) and (
11), but with
. This leads to an approximation similar to Equations (
14) and (
18), but with the substitution
. Notice that
k in Equation (
47) and
in Equation (
14) are equivalent in the
limit.
This coincidence between both models in the
limit is in agreement with the conclusions of [
32], where the parametric and parameter-free models were compared in detail. Therein they found that the parametric model overestimates the strength of the Hubble damping term—as it does not account for the dispersion of
—and that the impact of wall decay on
is not included. In the non-relativistic limit, however, these two effects are expected to be negligible, and thus, the two models should be equivalent in the
. Furthermore, the results of [
32] indicate that
in the
limit, in agreement with our previous discussion. The dependence of
on the rate of expansion seems to suggest that the scaling of the energy loss term with
and
L (which depend on
themselves) should be different from that given in Equation (
37).
The fact that we necessarily have that
in the
limit further implies that if one uses different numerical simulations with a wide range of expansion rates to calibrate the parameters of the original VOS model, one will necessarily find that no unique calibration exists. This is precisely what was found in [
10]: although in the relativistic limit, simulations are well described by a
close to unity, for fast expansion rates, simulations are compatible with
. Given these results, the authors proposed an extension of the VOS model, involving four additional free parameters, which provides a good fit for the results of their simulations. This multi-parameter VOS model drops the assumption that
k is a constant—assuming that
instead—and includes another energy loss term—a “radiation” term that depends on
— while keeping the term in Equation (
37) with a constant
. This multi-parameter model, however, reduces to the original VOS model in the non-relativistic limit (since the additional energy loss term becomes irrelevant in this limit and
k becomes approximately constant) and, as a result, calibration with simulations, unsurprisingly, yielded the
.