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\begin{document}

%\twocolumn[\hsize\textwidth\columnwidth\hsize\csname
%@twocolumnfalse\endcsname
%%
% OU-TAP
%%
%\tighten
%\draft
%%
\title{Quantitative evolution of global strings from the Lagrangian view point}
%%
%%
%%
%
\author{Masahide Yamaguchi}
\affiliation{Physics Department, Brown University, Providence, RI
  02912, USA \\ and \\
Research Center for the Early Universe, University of
  Tokyo, Tokyo 113-0033, Japan}
%
\author{Jun'ichi Yokoyama}
\affiliation{Department of Earth and Space Science, Graduate School of
Science, Osaka University, Toyonaka 560-0043, Japan}
%%%

\date{\today}
%%

%\maketitle

\begin{abstract}
We completely elucidate quantitative nature of cosmological evolution
of global string network, that is, energy density, peculiar velocity,
velocity squared, Lorentz factor, formation rate of loops, and emission
rate of Nambu-Goldstone (NG) bosons,
based on a new type of numerical simulations of scalar fields in
 Eulerian meshes.  We give a detailed explanation of a method to extract
 above-mentioned quantities to characterize string evolution by
 analyzing scalar fields in Eulerian meshes from a Lagrangian
 view point.  We confirm our previous claim that the
number of long strings per horizon volume in global string network is
smaller than the case of local string network by a factor of $\sim$ 10
even under the cosmological situations and its reason is clarified.
\end{abstract}

\pacs{98.80.Cq \hspace{6cm} BROWN-HET-1332, OU-TAP-188}

\maketitle

%]

\section{Introduction}

The idea of spontaneous symmetry breaking in high energy physics has a
profound implication on vacuum structure of the universe and our
Universe has presumably experienced various phase transitions, which
are thermal \cite{Kibble} or nonthermal \cite{KVY}. Their consequences
may be traced by topological defects that may have been produced with
them.  Indeed, a lot of implications of topological defects on
cosmology have been investigated \cite{VS}. Furthermore, recently,
defect formations have also been discussed in the context of phase
transitions which occur in the laboratory. For example, defect
formations in $^{3}$He \cite{He3} and $^{4}$He \cite{He4} are studied
in detail. Thus cosmological scenario of defect formation can be
tested by experiments in the laboratory \cite{Zurek}.
  
Among several types of topological defects, strings hold a unique
position in cosmology, because they do not overclose the universe
unlike magnetic monopoles or domain walls, settling down to a scaling
solution in which the typical scale of the network grows in proportion
to the horizon scale \cite{Kibble,Kibble2}.  The key mechanism to
achieve a scaling behavior is intercommutation of infinite strings to
dissipate their energy by producing closed loops which subsequently
decay to radiation of relativistic particles or gravitational waves
depending on their property.  While one may understand the qualitative
nature of the scaling solution analytically, more quantitative
features such as number of long strings per horizon volume or size
spectrum of loops produced cannot be obtained unless full numerical
analyses are performed.

Although there exist two types of strings, namely local and global
strings depending on the nature of the symmetry breaking, only the
former have been investigated extensively as for the numerical
analysis of their evolution for a long time.  By numerically solving
the equation of motion of string segments derived from the Nambu-Goto
action \cite{NG}, several groups have confirmed the scaling behavior
\cite{AT,AT2,BB,AS}, and estimated the scaling parameter $\xi$ as
$\xi\simeq 10$ in the radiation dominated universe \cite{AT2,BB,AS}.
Here, $\xi$ is defined as
%
\beq
  \xi = \rho_{\infty} t^{2} / \mu
  \label{eq:xi}
\eeq
%
where $\rho_{\infty}$ is the energy density of long strings and $\mu$
is the string tension per unit length. Thanks to this feature, local
strings generate density fluctuations with a scale-invariant spectrum
and their cosmological consequences were investigated extensively some
time ago.  Recent observations of the cosmic microwave background
(CMB) anisotropy \cite{cmb}, however, disfavor the cosmic-string
scenario of structure formation and the motivations to investigate
local strings as a source of primordial density fluctuations have
somewhat diminished, although a hybrid model of structure formation
may still be viable, where primordial fluctuations are comprised of
adiabatic fluctuations induced by inflation and isocurvature
perturbations by topological defects \cite{TDCMB}\footnote{Note that
  topological defects can be compatible with cosmic inflation
  \cite{KVY}. Note also that local strings may be still important in
  that they may emit massive particles as sources of ultra high energy
  cosmic rays \cite{VHS}.}.

On the other hand, global strings are much better motivated in the
context of axion cosmology. They are formed as a consequence of
breaking of the Peccei-Quinn U(1) symmetry \cite{PQ,VE}, which is
introduced to solve the strong CP problem in quantum chromodynamics.
These global strings radiate axions as associated Nambu-Goldstone (NG)
bosons \cite{Davis}, which are one of the most promising candidates of
cold dark matter. Despite their importance, cosmological evolution of
global strings has been less studied and the results of local strings
have often been borrowed even though there is a decisive difference
between them.  For local strings, gradient energy of scalar fields are
cancelled out by gauge fields far from the core. The string core is
well localized and false vacuum energy of the core dominates the
system.  Hence Nambu-Goto action is suitable as an effective action in
order to study cosmological evolution of local strings except at
crossing \cite{NG}. On the other hand, for global strings, there are
no gauge fields to cancel gradient energy of the NG scalar field,
which dominates over the false vacuum energy of the core.  The
effective action appropriate for global strings is not Nambu-Goto
action but Kalb-Ramond action, which incorporates NG bosons and their
couplings with the core \cite{KR}. Also, due to gradient energy of the
NG scalar field, long-range force works between global strings. Thus,
the behavior of both types of strings is expected to be different from
each other and it is nontrivial whether global strings relax to a
scaling regime.  In fact, for example, the behavior of global
monopoles is quite different from that of local monopoles due to
long-range forces. While the former may be useful in cosmology
\cite{monopole,Yamaguchi}, the latter causes a disaster \cite{Preskill}
unless diluted
by inflation.  Furthermore, it has been shown in the literature that
local and global strings behave differently in two dimensional space
\cite{YB,2dim}.

There have been several attempts to investigate cosmological evolution
of the global string network by use of the Kalb-Ramond action
\cite{BS,sharp}.  However, Kalb-Ramond action is too complicated to be
dealt with numerically. It has difficulty of logarithmic divergence
due to self-energy of the string.  In such a situation the authors and
Kawasaki made first numerical investigations of cosmological evolution
of global strings without resort to Kalb-Ramond action.  We instead
solved equations of motion for scalar fields forming strings in three
dimensional Eulerian meshes \cite{YKY}.  We found that the global
string network would also go into a scaling regime but the scaling
parameter $\xi$ was found to be of order of unity \cite{YKY,YYK},
which is significantly smaller than the case of local strings ($\xi
\simeq 10$).

Recently, however, this quantitative difference was questioned and it
was claimed that smallness of the scaling parameter of global strings
might be apparent due to small dynamic range of numerical simulations
\cite{MS,MSM}. The authors of \cite{MS,MSM} claimed that in our
simulations global strings lost their energy by excessive direct
emission of NG bosons.  Based on the speculation that such direct
emission of NG bosons from long strings would be negligible in
cosmological scales, they reached a conjecture that both types of
strings behave quantitatively in the same way in cosmological context,
namely $\xi \simeq 10$, with the only difference between them being
energy loss mechanism of closed loops.

In order to clarify evolution of global strings on cosmological scales, 
we should first study
which the dominant energy loss mechanism of long strings in numerical
simulations is, loop production or direct emission of NG bosons.
We note that the master equation for the energy density of long
strings, $\rho_{\infty}$ can be expressed as
\beq
  \frac{d\rho_{\infty}}{dt} = - 2 H (1 + \la v^{2} \ra) \rho_{\infty}
    - \Gamma_{\rm loop}\rho_{\infty}
      - \Gamma_{\rm NG}\rho_{\infty},
  \label{eq:energyloss}
\eeq
where the second and the third terms on the right-hand-side represent
energy loss due to loop formation and direct emission of NG bosons or
axions, respectively, and $\la v^{2} \ra$ denotes average square
velocity of string segments. 
For our purpose, we need to calculate both loop production rate 
$\Gamma_{\rm loop}$ 
and emission rate of NG bosons $\Gamma_{\rm NG}$, and compare their
magnitude in simulations. 

If the system relaxes to the scaling regime, which will be confirmed
to be the case shortly, the string network is described by the
so-called one-scale model\footnote{Although improvements of the
  simplest one-scale model have been proposed by taking into account
  effects of small-scale structures \cite{ACK} and time evolution of
  velocity \cite{MS,MS2}, the original one scale model is sufficient
  here because small-scale structures are smeared in the case of
  global strings and velocity remains constant in the scaling regime
  as will be shown later. } with the characteristic scale $L \equiv
\sqrt{\mu / \rho_{\infty}}$, which grows with the horizon scale $L
\propto t$.  If we introduce the loop production coefficient $c$ and
the emission coefficient $\kappa$ of NG bosons by
%
\bea
\Gamma_{\rm loop}\rho_{\infty}= c \frac{\rho_{\infty}}{L},~~~~~
  \Gamma_{\rm NG}\rho_{\infty} 
     = \kappa \frac{\rho_{\infty}}{L}, 
  \label{eq:ckappa}
\eea
%
these parameters remain constant in the one-scale picture and are
related to
the scaling parameter $\xi$ as
%
\beq
  \xi = \lmk \frac{1 - \la v^{2} \ra}{c + \kappa} \rmk^{2}.
  \label{eq:relation}
\eeq
Indeed if $\kappa$ incorrectly turned out to be much larger than $c$, we
would find a smaller value of $\xi$ than it should really be.
Hence it is essential to evaluate 
the parameters $c$ and $\kappa$ in the scaling regime.

In our previous simulations \cite{YKY}, however, it was impossible to
calculate these quantities for two reasons.  First in the previous work
a lattice was identified as a part of string based on the value of
potential energy there, which had a microscopic problem that
we occasionally found disconnected string pieces, although  overall
feature was traced reasonably well. 
Second it was impossible to monitor intercommutation of strings 
for lack of dynamical informations, namely, velocity of each string segment.
These problems are overcome by our new identification scheme and
Lagrangian analysis of evolution of strings \cite{YY}.

The rest of the paper is organized as follows.  In \S II we present
the method of our new procedure 
to follow Lagrangian evolution of global strings in Eulerian
simulations, that is, new methods of identification of strings,
measurement of string velocity, and estimation of intercommutation rate.
In \S III the results are described and applied to cosmological
situations.  Finally \S IV is devoted to discussion and conclusion.

\section{numerical simulations}

\subsection{Formulation}
\label{sub:formulation}

We
consider the following Lagrangian density for two-component real scalar
fields $\phi_{a}(x)$ ($a = 1, 2$) which can accommodate global strings,
%
\beq
  \CL[\phi_{a}] = \frac12 g_{\mu\nu}
                   (\del^{\mu}\phi_{a})(\del^{\nu}\phi_{a})
                    - V[\phi_{a},T],
  \label{eq:lagrangian}
\eeq 
%
in the spatially flat Robertson-Walker spacetime,
%
\beq
 ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}=dt^2-R^2(t)d\vect{x}^2,
\eeq
%
with $R(t)$ being the scale factor.  We adopt the following potential
at finite temperature $T$,
%
\bea
  V[\phi_{a},T] &=& \frac{\lambda}{4}(\phi^{2} - \sigma^2)^2 
                 + \frac{\lambda}{6}T^2\phi^{2} \\
                &=& \frac{\lambda}{4}(\phi^{2} - \eta^2)^2
                 +  \frac{\lambda}{4}(\sigma^{4} - \eta^4).
  \label{eq:potential}
\eea
%
Here $\phi^{2} \equiv \phi_{1}^{2} + \phi_{2}^{2}$ and $\eta^{2}
\equiv \sigma^{2} - T^{2}/3 = \sigma^{2}(1 - T^{2}/T_{c}^{2})$ with
$T_{c} \equiv \sqrt{3}\sigma$ being the critical temperature. Below
the critical temperature, global U(1) symmetry is broken to form
global strings.


Equations of motion for the scalar fields are given by
%
\beq
  \ddot{\phi_{a}}(x) + 3H(t)\dot{\phi_{a}}(x) 
    - \frac{1}{R(t)^2}\nabla^2\phi_{a}(x)
      + \frac{\del V}{\del \phi_{a}} = 0,
  \label{eq:EOM}
\eeq
%
where a dot represents the time derivative.  Discretization of these
differential equations is given in Appendix \ref{app:1}. Our numerical
calculations are based on the staggered leapfrog method with second
order accuracy both in time and in space. In the radiation dominated
universe, the Hubble parameter $H(t) = \dot R(t)/R(t)$ and cosmic time
$t$ are given by
%
\bea
  H(t)^2 = \frac{8\pi}{3 \mpl^2} \frac{\pi^2}{30} g_{*} T^4,
   ~~~~~
  t = \frac{1}{2H} \equiv \frac{\epsilon}{T^2},
  \label{eq:hubble}
\eea
%
where $\mpl = 1.2 \times 10^{19}$GeV is the Plank mass and $g_{*}$ is
the total number of degrees of freedom for the relativistic particles.
We define a dimensionless parameter $\zeta$ as,
%
\beq
  \zeta \equiv \frac{\epsilon}{\sigma}  = \lmk \frac{45\mpl^2}
      {16\pi^3g_{*} \sigma^2}
  \rmk^{1/2},
  \label{eqn:zeta}
\eeq
%
and take $\zeta=10$ and the self coupling $\lambda = 0.08$ for
definiteness, but these particular choices do not affect on the
results.  We start numerical simulation at the temperature $T_{i} = 2
T_{c}$ corresponding to $t_{i} = t_{c}/4$ and adopt as an initial
condition the thermal equilibrium state with a mass equal to the
inverse curvature of the potential at that time.  In this state
$\phi_a$ and $\dot{\phi}_a$ are Gaussian distributed with the
correlation functions,
%
\bea
  \la \beta|\phi_a(\vect x)\phi_b(\vect y)|\beta
                             \ra_{\rm equal\hbox{-}time} &=&
   \int \frac{d^3 k}{(2\pi)^3} \frac1{2\sqrt{\vect k^2 + m^2}}
           \coth{\frac{\beta\sqrt{\vect k^2 + m^2}}{2}}
             e^{i\vect k \cdot (\vect x-\vect y)}\delta_{ab} \:,   \\
             && \non \\
  \langle \beta|\dot\phi_a(\vect x)\dot\phi_b(\vect y)|\beta
                             \rangle_{\rm equal\hbox{-}time} &=&
   \int \frac{d^3 k}{(2\pi)^3} \frac{\sqrt{\vect k^2 + m^2}}{2}
           \coth{\frac{\beta\sqrt{\vect k^2 + m^2}}{2}}
             e^{i\vect k \cdot (\vect x-\vect y)}\delta_{ab} \:, 
\eea
%
where $m^2=V''[\phi_a,T_i]$ and $\beta=T^{-1}_i$.  $\phi_a(\vect x)$
and $\dot\phi_a(\vect y)$ are uncorrelated for $\vect x \ne \vect y$.
We generate initial configuration in the momentum space, where the
scalar fields are uncorrelated.  Then they are transformed into the
position spaces by the Fast Fourier Transformation.

We perform simulations in five different sets of lattice sizes and
spacings as shown in TABLE \ref{tab:set1} to investigate their effects
on the results. In all cases, the time step is taken as $\Delta t =
0.01 t_{i}$, and the periodic boundary condition  is adopted. In
the case (a), the box size is nearly equal to the horizon length
$H^{-1}$ and the lattice spacing to typical core width $\delta \sim
1.0/(\sqrt{2\lambda}\sigma)$ of a string at the final time $t_{f} =
200 t_i$.  Other cases have equal or larger simulation volume.

In this type of numerical calculations, it is a
nontrivial task to identify string cores from data of scalar fields
because a point with $\phi_{a} = 0$ corresponding to a string core is
not necessarily situated at a lattice point. So, in the next subsection,
we give our new method of identification of string cores.

\subsection{Identification method of strings}
\label{sub:identification}

In the previous works \cite{YKY,YYK}, whether a lattice point was a
part of string core was judged based on potential energy density
there.  That is, a lattice point was identified to be in the core of a
string if the potential energy density there was found to be larger
than that of a static cylindrically-symmetric solution of a global
string at $r=\Delta x_{phys}/\sqrt{2}$ off center, where $\Delta
x_{phys}$ is the physical length of lattice spacing at each time.

Although this method worked fairly well to evaluate the scaling
parameter, it was inadequate to fully identify strings particularly
for small loops which do not resemble the static
cylindrically-symmetric solution due to the curvature.  As a result we
occasionally found disconnected string segments that should not exist
in reality.  Furthermore, it was impossible to find more correct
position of string core in a box beyond the lattice spacing, which is
essentially important to evaluate the length and the velocity of a
string correctly.

Here we develop a new method of string identification based on the fact
that strings lie on the intersection of two surfaces
$\phi_1(\vect{x},t)=0$ and $\phi_2(\vect{x},t)=0$, so that if a strings
penetrates a (sufficiently small) plaquette, $\phi_a$ has a different
sign at one or two corners of the plaquette from the rest for each $a$.
First we classify relative phase of scalar
fields into three groups as shown in Fig.\,\ref{fig:phase}(Left), that is,
%
\bea
   \theta \equiv \arccos\frac{\phi_{2}}{\sqrt{\phi_{1}^2+\phi_2^2}}
 +2\pi \lkk 1-\Theta (\phi_2) \rkk, 
\eea \bea
  {\rm (i)}   &&\quad 0 \le \theta < \frac{\pi}{2}, \non \\
  {\rm (ii)}  &&\quad \frac{\pi}{2} \le \theta < \frac{3\pi}{2},  \\
  {\rm (iii)} &&\quad \frac{3\pi}{2} \le \theta < 2\pi , \non
\eea
%
where arccosine should take the principal value and $\Theta (\phi_2)$
is the step function.

Then, we judge whether a string penetrates a plaquette by monitoring
phase rotation around it just as Vachaspati-Vilenkin  algorithm
\cite{VV}.  If we assigned equal range $2\pi/3$ of relative phase to
all regions [(i)-(iii)] as in the original Vachaspati-Vilenkin
 algorithm and judged the
presence of a string from phase rotation, we would occasionally
identify a plaquette as containing a string even if $\phi_1$ or
$\phi_2$ takes the same sign at its four corners.  This is the main
reason why all regions [(i)-(iii)] do not have equal range of relative
phase $\theta$ in our scheme, in which such a case is avoided and
$\phi_a$ takes different sign for each $a$ at one or two corners of
each plaquette along which a circular phase rotation is observed.
Then using the values of $\phi_a$ at its four corners we can draw two
lines corresponding to $\phi_1=0$ and $\phi_2=0$ and their
intersection is identified with the point a string penetrates the
plaquette as shown in Fig.\,\ref{fig:cross1}.

In fact, the intersecting point could be found outside of the
plaquette. In such a case, the nearest point on the edge of the
plaquette from the intersection is identified as the position of a
string as shown in Fig.\,\ref{fig:cross2}.  In our simulations we
encountered such a case very rarely and it did not cause any serious
problems.

By joining these points, strings are completely connected and more
accurate total length of a global string  can be obtained.  One
may wonder if our biased classification of relative phase may cause
some artificial effects on the result of numerical simulations. So
we have tried another classification of relative phase as shown in
Fig.\,\ref{fig:phase}(Right), that is,
%
\bea
  {\rm (i)}   &&\quad 0 \le \theta < \frac{\pi}{2}, \non \\
  {\rm (ii)}  &&\quad \frac{\pi}{2} \le \theta \le \pi, \quad  \\
  {\rm (iii)} &&\quad \pi \le \theta < 2\pi , \non
\eea
%
and confirmed that the results do not depend on the classification
scheme of relative phase.

At each time step, we can evaluate total length of the global string
by the method shown above.  Since strings typically move at a speed
close to unity, we should multiply the length of each string segment
by $\mu = \gamma \mu_{s}$ to calculate the total energy density of
strings.  Here $\gamma$ is the Lorentz factor and $\mu_{s}\simeq
2\pi\eta^2\ln\lmk t/(\delta\xi^{1/2})\rmk$ is the line density of a
static string \cite{MSM}.  So it is important to establish a method to
calculate velocity and Lorentz factor of string segments, which will
be presented shortly.  We can then evaluate the scaling parameter
$\xi$ from Eq.\,(\ref{eq:xi}).

\subsection{Velocity of strings}
\label{sub:velocity}

Here we describe our method to evaluate velocity of strings, which is
a nontrivial task in Eulerian calculations of scalar field
configurations.  First we expand scalar fields $\phi_{a}(\vect
x,t_0+\delta t)$ around $\phi_{a}(\vect x_{0},t_{0})$ up to the first
order,
%
\bea
  \phi_{a}(\vect x,t_0+\delta t) \cong \phi_{a}(\vect x_{0},t_{0}) 
      + \nabla\phi_{a}(\vect x_{0},t_{0}) \cdot (\vect x-\vect x_{0}) 
     %  \non \\
       + \dot\phi_{a}(\vect x_{0},t_{0})\delta t,
      \qquad ( a = 1, 2).
  \label{eq:expansion}
\eea
%
Suppose that a string core exists at a point $\vect x_0$ at time $t_0$
and moves to a point $\vect x$ at $t=t_0 + \delta t$, that is,
$\phi_a(\vex_0,t_0)=0$ and $\phi_a(\vex,t_0+\delta t)=0$ for each $a$.
Then, from (\ref{eq:expansion}) we find the loci of string core at
$t=t_0+\delta t$ lie on the intersection of two planes,
%
\beq
 \nabla\phi_{a}(\vect x_{0},t_{0}) \cdot (\vect x-\vect x_{0})
 +\dot\phi_{a}(\vect x_{0},t_{0})\delta t=0,  \label{plane}
\eeq
%
with $a=1,2$. Since motion tangential to a string is a gauge mode, we
should evaluate velocity normal to it. Suppose that the line normal to
the string segment at $(\vect x_0,t_0)$ reaches across the above
intersection line at $\vect x =\vect x_{l}(\vect x_0,t_0,\delta t)$.
Then we can easily obtain the velocity of this string segment as
%
\bea
  \vect{v}(\vect x_0,t_0) &=& \lim_{\delta t \longrightarrow 0}
\frac{\vect x_{l}(\vect x_0,t_0,\delta t) - \vect x_{0}}{\delta t}
\nonumber \\
    &=& \left.\frac{\dot\phi_{1}\nabla\phi_{2} - \dot\phi_{2}\nabla\phi_{1}}
           {|\nabla\phi_{1} \times \nabla\phi_{2}|}\right|_{\vect{x_0},t_0}.
  \label{eq:velocity}
\eea    
%
We calculate velocity at each point where strings cross plaquettes.
For this purpose we need to evaluate $\dot\phi_{a}$ and
$\nabla\phi_{a}$ at arbitrary points on a plaquette, as shown
explicitly in the Appendix \ref{app:2} where $\dot\phi_{a}$ and
$\nabla\phi_{a}$ are given by the quantities on lattice points up to
the second order.  Collecting the value of (\ref{eq:velocity}) at each
intersection of strings with plaquettes, we can obtain the average of
velocity, velocity squared, and Lorentz factor.

\subsection{Intercommutation of strings}
\label{sub:intercommutation}

Now we discuss energy loss rate from infinite strings to loops by
calculating the intercommutation rate $c$.  In simulations of cosmic
strings based on Nambu-Goto action, one has to assign the
intercommutation probability of intersecting strings by hand due to
the lack of microscopic informations.  Since strings are identified
from scalar field configuration in our simulation, we can
unambiguously calculate how strings intersect with each other, namely
whether they simply pass through or intercommute upon collision.  We
can therefore identify new loops by monitoring length of all long
strings and loops at each time step and comparing the data with those
at the previous time step.


\subsection{NG boson emission}
\label{sub:emission}

It is very difficult to evaluate directly the NG boson emission rate
because separation of emitted axions and NG phase associated with
string core is nontrivial. Since loops also emit these particles, it
is formidable to identify axions radiated from long strings only.
However, we have already described how we can obtain $\rho_{\infty}$,
$\langle v^2 \rangle$, and $\Gamma_{\rm loop}$ from simulation data at
each time step, and $d\rho_{\infty}/dt$ can also be calculated from
$\rho_{\infty}$ at two adjacent time steps.  So all the quantities in
the master equation (\ref{eq:energyloss}) can be calculated from our
simulation data except for the last term, which can be found from the
equation itself.  In particular, in the scaling regime, when the
one-scale description suffices, we can find $\kappa$ from \beq \kappa
= \frac{1 - \la v^{2} \ra}{\sqrt{\xi}} - c .
  \label{eq:relationk}
\eeq

\section{Results}

\subsection{Scaling parameter}

In Fig.\,\ref{fig:xi}, time evolution of $\xi$ is depicted for all
cases [(a)-(e)]. Filled squares represent time development of $\xi$
for new identification method.  For comparison, the results of our
previous identification method based on the potential energy density
\cite{YKY,YYK} are shown by blank circles, which slightly
overestimates $\xi$ by a factor of $1.2$. The results of the original
Vachaspati-Vilenkin identification scheme are indicated by blank
squares which are obtained by counting the number of boxes through
which a string passes as was done in Ref. \cite{MSM}. This method
overestimates $\xi$ by a factor of $1.4$.

We find that the difference in the simulation settings except the box
size normalized by the final horizon scale does not affect the overall
behavior significantly.  $\xi$ becomes smaller in the smallest box
size with $= H^{-1}$ at the final time. This is mainly because under
the periodic boundary condition, a string feels an attractive force
from the boundary and tend to disappear.  Indeed, in the case (a) this
artificial effect starts to operate before the system really relaxes
to the scaling solution, so $\xi$ continues decreasing without a
plateau.  On the other hand, in the case (b), the boundary effect
becomes operative after $t\simeq 130$ when $\xi$ starts to decrease
from the scaling value.  The other three cases, whose box size is
larger than $2 H^{-1}$, are free from the boundary effects and the
scaling parameter $\xi$ remains constant.  Hence we conclude that
after the relaxation period the global string network goes into the
scaling regime.  From TABLE \ref{tab:set2}, which shows average values
of $\xi$ after some relaxation period ($t > 80$), $\xi$ is found to be
$\xi \simeq 0.80$ in the scaling regime.

\subsection{String velocity}

In Figs. \ref{fig:velocity} -- \ref{fig:gamma}, time evolution of
average velocity $\la v\ra$, average square velocity $\la v^2\ra$, and
Lorentz factor $\la \gamma \ra$ are shown. After some relaxation
period, they become fairly constant in all cases.  The average values
are given in TABLE \ref{tab:set2}. We find that the average values
become larger in the smallest box size. This is mainly because under
the periodic boundary condition, a string feels an attractive force
from the boundary and tend to be accelerated. However, such an effect
becomes negligible for the box size larger than $2 H^{-1}$ as observed
in TABLE \ref{tab:set2}.  Thus we find the average velocity $\la v \ra
\simeq 0.60$ and the average square velocity $\la v^{2} \ra \simeq
0.50 \gg \la v \ra^{2}$, in the scaling regime.  On the other hand,
the average Lorentz factor has large scatter in time, although
long-time average is fairly constant with $\langle\bar{\gamma}\rangle
\simeq 1.8$.  Since string segments moving with a speed close to light
velocity have extremely large Lorentz factor and push up the average
dramatically, fluctuation in the number of such string segments
results in such large scatter.  This is the reason the average Lorentz
factor, $\langle\gamma\rangle =1.8\pm 0.2$ is much larger than
$1/\sqrt{1 - \la v^{2} \ra}$ and $ 1/\sqrt{1 - \la v \ra^{2}}$.  Thus
the energy per unit length of a string is enhanced by a factor
$\langle\bar{\gamma}\rangle \simeq 1.8$ compared with a static string.

\subsection{Energy dissipation coefficients}

As explained in \ref{sub:intercommutation} we can obtain the
intercommutation rate $c$ without assigning its probability by hand.
The result is given in TABLE \ref{tab:set2}, where we find that $c$ is
larger in the smallest box size. This can be understood in the same
way, namely, under the periodic boundary condition, a string feels an
attractive force from the boundary and tend to intercommute more
often. Since such an effect is unimportant for the box size larger
than $2 H^{-1}$ as discussed above, $c$ is found to be $c = 0.40 \pm
0.04$.

Taking account of the relation (\ref{eq:relationk}), the emission
parameter $\kappa$ is given in TABLE \ref{tab:set2}. In simulations
with the smallest box, strings intercommute more often due to the
boundary effect, which suppresses NG boson emission. But again such an
effect is inoperative for the cases with their box size larger than
$2H^{-1}$.  So we conclude the NG boson emission rate to be $\kappa =
0.16 \pm 0.04$.

\section{Discussion and conclusion}

In this paper we have investigated cosmological evolution of the
global string network in detail. In our numerical simulations,
equations of motion of the two-component real scalar field are solved
in Eulerian meshes. In order to follow time evolution of global
strings, we have developed a new identification method which enables
us to find more correct position of string core in a box beyond the
lattice spacing. Furthermore, we have given a detailed explanation for
the method to extract Lagrangian quantities, such as velocity and
intercommutation, characterizing evolution of global strings. The NG
boson emission rate is obtained from the master equation
(\ref{eq:energyloss}). Thus quantitative nature of cosmological
evolution of global string network is elucidated without setting the
intercommutation probability of two intersecting strings by hand.
Specifically, we find the scaling parameter characterizing energy
density $\xi \simeq 0.80$, peculiar velocity $\la v \ra \simeq 0.60$,
velocity squared $\la v^2 \ra \simeq 0.50$, Lorentz factor
$\langle\gamma\rangle = 1.8 \pm 0.2$, formation rate of loops $c =
0.40 \pm 0.04$, and emission rate of Nambu-Goldstone (NG) bosons
$\kappa = 0.16 \pm 0.04$ from our results.

In our simulations, due to the limit of the dynamic range of the
lattices, the logarithm of the ratio of the Hubble radius to the string
width, which appears in the expression of the effective line density of
global strings, took $\ln(t/\delta)\sim 5$, while in cosmological
setting it can be as large as $\CO (100)$.  The authors of
\cite{MS,MSM} argue that $\kappa$ is inversely proportional to 
 $\ln(t/\delta)$, although $c$ and $\la v^2 \ra$ do not have such
dependence.  Based on this speculation they claimed that we obtained the
smaller value of the scaling parameter $\xi$ in our previous work
 \cite{YKY} than it should really be, because $\kappa$ was incorrectly
large there. They also argue that in cosmological situation loop
production is the dominant mechanism of energy dissipation of long
strings and that $\xi$ would take the same value as local strings.

Now that we have all the values of relevant quantities from our
simulation data, we can disprove their claim.  Indeed, we find
$\kappa$ is smaller than $c$ and hence direct emission of NG bosons
does not dominate energy loss mechanism even in our setting of
numerical simulations with relatively small dynamic range,
$\ln(t/\delta)$.  If we extrapolate the value of $\kappa$ with the
scaling $\kappa \propto 1/\ln(t/\delta)$, we find $\kappa \lesssim
0.01$ in cosmological situation.  Even then the scaling parameter
(\ref{eq:relation}) does not increase much, yielding $\xi = 1.6 \pm
0.3$, which is still much smaller than that of local strings.

Thus we conclude that there is a quantitative difference between
cosmological evolution of global strings and that of local strings.
It is based on the qualitative difference that global strings have a
long-range force and intercommute more often with larger $\la v^2 \ra$
than local strings.  Note that it is by no means surprising that
global defects behave differently than local defects in cosmology.  In
the case of monopoles, as discussed in the Introduction, global
monopoles evolve in a strikingly different manner than magnetic
monopoles. In the case of strings, both local and global defects
relax to a scaling solution and their difference is more subtle.  So
it is not until our numerical analysis of evolution of the latter from
Lagrangian point of view is performed that their difference is fully
elucidated.

Finally we use our results to obtain a constraint on the symmetry
breaking scale $\eta\equiv f_a$ of the Peccei-Quinn U(1) symmetry.
From $\xi=1.6\pm0.3$ and $\la \gamma\ra=1.8\pm0.2$, we find $f_a
\lesssim (0.16 - 1.2)\times 10^{12}$GeV for the normalized Hubble
parameter $h=0.7$ \cite{YKY}.  
We also note that the Lagrangian method developed
in this paper is directly applicable to other species of extended
objects such as topological defects like local strings and
non-topological solitons like Q-balls as well.

\section*{Acknowledgements}

M.Y.\ is grateful to Robert Brandenberger for his hospitality at Brown
University, where the final part of the work was done.  J.Y.\ would
like to thank Toru Tsuribe for useful comments on numerical analysis.
This work was partially supported by the JSPS Grant-in-Aid for
Scientific Research, Nos.\ 12-08555(MY) 5(JY).

\appendix

\section{Discretization of differential equations}
\label{app:1}

The equation of motion of the scalar fields is given by
%
\beq
  \ddot{\phi_{a}}(x) + 3H(t)\dot{\phi_{a}}(x) 
    - \frac{1}{R(t)^2}\nabla^2\phi_{a}(x)
      + \frac{\del V}{\del \phi_{a}} = 0.
\eeq
%
In order to discretize the above equation, it is reduced to first-order
differential equations,
%
\bea
  \dot{\phi}_{a} &\equiv& \pi_{a} \non \\
  \dot{\pi}_{a} &=& -3H(t)\pi_{a}
        - \frac{1}{R(t)^2}\nabla^2\phi_{a}(x) 
        - \frac{\del V}{\del \phi_{a}}. \non \\
\eea
%
Expanding $\phi_{a}(t,\vect x)$ and $\phi_{a}(t+\Delta t,\vect x)$ 
around the intermediate time step
 $t+\frac12 \Delta t$ we find
%
\beq
  \phi_{a}(t+\Delta t,\vect x) - \phi_{a}(t,\vect x) =
      \Delta t\,\dot{\phi}_{a}(t+\frac12 \Delta t,\vect x)
      + \CO\bigl((\Delta t)^3\bigr),               
\eeq
up to the third-order.
%
Thus $\pi_{a}$ at the intermediate time step is given by
%
\beq
  \pi_{a}(t+\frac12 \Delta t,\vect x) = 
     \frac{\phi_{a}(t+\Delta t,\vect x) - \phi_{a}(t,\vect x)}{\Delta t}
      + \CO\bigl((\Delta t)^2\bigr).
\eeq
%
In the same way, $\dot{\pi}_{a}$ and $\pi_{a}$ at the time step $t$ 
are represented
by the quantities at the two adjacent intermediate steps
$t\pm\frac12 \Delta t$, 
%                 
\bea
  \dot{\pi}_{a}(t,\vect x) &=&
     \frac{\pi_{a}(t+\frac12\Delta t,\vect x) 
          -\pi_{a}(t-\frac12\Delta t,\vect x)}
          {\Delta t}
      + \CO\bigl((\Delta t)^2\bigr) \non \\
  \pi_{a}(t,\vect x) &=&
     \frac{\pi_{a}(t+\frac12\Delta t,\vect x) 
          +\pi_{a}(t-\frac12\Delta t,\vect x)}
          {2} 
      + \CO\bigl((\Delta t)^2\bigr). \label{a6}
\eea
%
The second-order derivatives are approximated up to the second order in 
$\Delta$ as,  
%
\bea
  \frac{\del^2 \phi_{a}(t,\vect x)}{\del x^2} &=&
    \frac{\phi_{a}(t,x+\Delta,y,z)
         -2\phi_{a}(t,x,y,z)
         +\phi_{a}(t,x-\Delta,y,z)}
         {\Delta^2}
    + \CO(\Delta^2) \non \\
  \frac{\del^2 \phi_{a}(t,\vect x)}{\del y^2} &=&
    \frac{\phi_{a}(t,x,y+\Delta,z)
         -2\phi_{a}(t,x,y,z)
         +\phi_{a}(t,x,y-\Delta,z)}
         {\Delta^2}
    + \CO(\Delta^2) \non \\
  \frac{\del^2 \phi_{a}(t,\vect x)}{\del z^2} &=&
    \frac{\phi_{a}(t,x,y,z+\Delta)
         -2\phi_{a}(t,x,y,z)
         +\phi_{a}(t,x,y,z-\Delta)}
         {\Delta^2}
    + \CO(\Delta^2).
\eea 

Thus the fundamental equations are discretized up to the second order
both in space and time,   
%
\bea
  && \hspace{-1.0cm} 
  \ddot{\phi_{a}}(x) + 3H(t)\dot{\phi_{a}}(x) 
    - \frac{1}{R(t)^2}\nabla^2\phi_{a}(x)
      + \frac{\del V(\phi_{a}(x))}{\del \phi_{a}}
  \non \\
  &&
  =
  \frac{\dot{\phi}_{a}(t+\frac12\Delta t,\vect x) 
       -\dot{\phi}_{a}(t-\frac12\Delta t,\vect x)}{\Delta t}    
  + 3H(t)\frac{\dot{\phi}_{a}(t+\frac12\Delta t,\vect x) 
               +\dot{\phi}_{a}(t-\frac12\Delta t,\vect x)}
              {2} 
  \non \\
  && \qquad
  - \frac{1}{R(t)^2} 
      \Sigma_{l = x,y,z} \biggl\{ \frac{ 
                                \phi_{a}(t,\vect x+\vect \Delta_{l})
                                -2\phi_{a}(t,\vect x)
                                + \phi_{a}(t,\vect x-\vect \Delta_{l}) 
                                       }
                                       {\Delta^2}   
                         \biggr\}
  + \frac{\del V(\phi_{a}(t,\vect x))}{\del \phi_{a}} 
  + \CO\bigl((\Delta t)^2,\Delta^2 \bigr) \non \\
  && = 0,
\eea
%
where $\Delta_{x}=(\Delta,0,0),\Delta_{y}=(0,\Delta,0)$, and
$\Delta_{z}=(0,0,\Delta)$.

In summary, in our numerical simulations ${\phi}_{a}$ and
%
\bea
  \dot{\phi}_{a}(t+\frac12\Delta t,\vect x) 
    &=& \frac{1}{1+\frac{3H(t)\Delta t}{2}}
      \Biggl[
        \lmk 1-\frac{3H(t)\Delta t}{2} \rmk 
          \dot{\phi}_{a}(t-\frac12\Delta t,\vect x) \non \\
    && \quad    
        + \Delta t
            \lmk 
              \frac{1}{R(t)^2} 
                \Sigma_{l = x,y,z} \biggl\{ \frac{ 
                                \phi_{a}(t,\vect x+\vect \Delta_{l})
                                -2\phi_{a}(t,\vect x)
                                + \phi_{a}(t,\vect x-\vect \Delta_{l}) 
                                       }
                                       {\Delta^2}   
                         \biggr\}
              - \frac{\del V(\phi_{a}(t,\vect x))}{\del \phi_{a}}  
            \rmk
      \Biggr] \non \\
  \phi_{a}(t+\Delta t,\vect x) &=& 
      \phi_{a}(t,\vect x) 
     + \Delta t\,\dot{\phi}_{a}(t+\frac12\Delta t,\vect x).     
\eea
%
The value of $\dot{\phi}_{a}$ at the time step $t$, which is required
to calculate string velocity, is evaluated from (\ref{a6}).
 
\section{Quantities at an arbitrary point on a plaquette}
\label{app:2}

In order to evaluate velocity of a string correctly, we need to evaluate
quantities at an arbitrary point on a plaquette.  That is,
$\dot{\phi}_{a}$ and $\nabla{\phi}_{a}$ within a plaquette should be
expressed by their values at its four corners.  As an example, let us
consider a plaquette parallel to the $z$-plane with four corners
$(x,y,z)$, $(x+\Delta,y,z)$, $(x,y+\Delta,z)$, and
$(x+\Delta,y+\Delta,z)$ and express
$\dot{\phi}_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)$ and
$\nabla\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)$, which we denote
collectively by $\Pi(x+\alpha\Delta,y+\beta\Delta)$ 
below, in terms of their values at these
four points.  Here $0 \le \alpha \le 1$ and $0 \le \beta \le 1$.

We express $\Pi$ at the four corners by an expansion around
$(x+\alpha\Delta,y+\beta\Delta)$ as
%
\bea
  \Pi(x,y) 
           &=& \Pi(x+\alpha\Delta,y+\beta\Delta) 
               - \alpha\Delta\frac{\del\Pi}{\del x}
               - \beta\Delta\frac{\del\Pi}{\del y} \non \\
           &&  + \frac12\alpha^2\Delta^2\frac{\del^2\Pi}{\del x^2}
               + \alpha\beta\Delta^2\frac{\del^2\Pi}{\del x\del y}
               + \frac12\beta^2\Delta^2\frac{\del^2\Pi}{\del y^2}
               + \CO(\Delta^3) \non \\
  \Pi(x+\Delta,y) 
           &=& \Pi(x+\alpha\Delta,y+\beta\Delta) 
               + (1-\alpha)\Delta\frac{\del\Pi}{\del x}
               - \beta\Delta\frac{\del\Pi}{\del y} \non \\
           &&  + \frac12(1-\alpha)^2\Delta^2\frac{\del^2\Pi}{\del x^2}
               - (1-\alpha)\beta\Delta^2\frac{\del^2\Pi}{\del x\del y}
               + \frac12\beta^2\Delta^2\frac{\del^2\Pi}{\del y^2}
               + \CO(\Delta^3) \non \\
  \Pi(x,y+\Delta) 
           &=& \Pi(x+\alpha\Delta,y+\beta\Delta) 
               - \alpha\Delta\frac{\del\Pi}{\del x}
               + (1-\beta)\Delta\frac{\del\Pi}{\del y} \non \\
           &&  + \frac12\alpha^2\Delta^2\frac{\del^2\Pi}{\del x^2}
               - \alpha(1-\beta)\Delta^2\frac{\del^2\Pi}{\del x\del y}
               + \frac12(1-\beta)^2\Delta^2\frac{\del^2\Pi}{\del y^2}
               + \CO(\Delta^3) \non \\
  \Pi(x+\Delta,y+\Delta) 
           &=& \Pi(x+\alpha\Delta,y+\beta\Delta) 
               + (1-\alpha)\Delta\frac{\del\Pi}{\del x}
               + (1-\beta)\Delta\frac{\del\Pi}{\del y} \non \\
           &&  + \frac12(1-\alpha)^2\Delta^2\frac{\del^2\Pi}{\del x^2}
               + (1-\alpha)(1-\beta)\Delta^2\frac{\del^2\Pi}{\del x\del y}
               + \frac12(1-\beta)^2\Delta^2\frac{\del^2\Pi}{\del y^2}
               + \CO(\Delta^3).
\eea

Making an appropriate combination, $\Pi(x+\alpha\Delta,y+\beta\Delta)$
can be expressed by its values at the four vertices in the plaquette,
%
\bea
  && \hspace{-1.5cm} 
  (1-\alpha)(1-\beta)\,\Pi(x,y) + \alpha(1-\beta)\,\Pi(x+\Delta,y)
    + (1-\alpha)\beta\,\Pi(x,y+\Delta)+\alpha\beta\,\Pi(x+\Delta,y+\Delta)
           \non \\
  && = \Pi(x+\alpha\Delta,y+\beta\Delta)
    +\frac12\alpha(1-\alpha)\Delta^2\frac{\del^2\Pi}{\del x^2}
    +\frac12\beta(1-\beta)\Delta^2\frac{\del^2\Pi}{\del y^2}
    +\CO(\Delta^3) \label{eq:third} \\
  && = \Pi(x+\alpha\Delta,y+\beta\Delta) + \CO(\Delta^2).
\eea
%
In particular, replacing $\Pi(x+\alpha\Delta,y+\beta\Delta)$ by
$\dot{\phi}_{a}(x+\alpha\Delta,y+\beta\Delta)$, it can be expressed by
the quantities on the lattice points up to the second order,
%
\bea
  \dot{\phi}_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)
   &=& (1-\alpha)(1-\beta)\,\dot{\phi}_{a}(t,x,y,z)
      + \alpha(1-\beta)\,\dot{\phi}_{a}(t,x+\Delta,y,z) \non \\
   && \quad
      + (1-\alpha)\beta\,\dot{\phi}_{a}(t,x,y+\Delta,z)
      + \alpha\beta\,\dot{\phi}_{a}(t,x+\Delta,y+\Delta,z)
      + \CO(\Delta^2) .  
\eea
%
In the same way, inserting $\del\phi_{a}/\del x$ into $\Pi$, we find
%
\bea
  \frac{\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)}{\del x}
   &=& (1-\alpha)(1-\beta)\,\frac{\partial\phi_{a}(t,x,y,z)}{\del x}
      + \alpha(1-\beta)\,\frac{\partial\phi_{a}(t,x+\Delta,y,z)}{\del x} 
          \non \\
   && \quad
      + (1-\alpha)\beta\,\frac{\partial\phi_{a}(t,x,y+\Delta,z)}{\del x}
      + \alpha\beta\,\frac{\partial\phi_{a}(t,x+\Delta,y+\Delta,z)}{\del x}
      + \CO(\Delta^2) \\  
   &=& \frac{1}{2\Delta} \biggl[
     (1-\alpha)(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y,z)-\phi_{a}(t,x-\Delta,y,z) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x+2\Delta,y,z)-\phi_{a}(t,x,y,z) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + (1-\alpha)\beta\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y+\Delta,z)-\phi_{a}(t,x-\Delta,y+\Delta,z) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha\beta\,
       \Bigl\{
         \phi_{a}(t,x+2\Delta,y+\Delta,z)
        -\phi_{a}(t,x,y+\Delta,z) 
       \Bigr\}
       \biggr]
      + \CO(\Delta^2).
\eea
%
Here we have used the relations,
%
\bea
  \frac{\phi_{a}(x+\Delta,y,z)-\phi_{a}(x-\Delta,y,z)}{2\Delta}
   &=& \frac{\del\phi_{a}(x,y,z)}{\del x} + \CO(\Delta^2), \non \\
  \frac{\phi_{a}(x+2\Delta,y,z)-\phi_{a}(x,y,z)}{2\Delta}
   &=& \frac{\del\phi_{a}(x+\Delta,y,z)}{\del x} + \CO(\Delta^2).
\eea

Interchanging $x$ for $y$ and $\alpha$ for $\beta$, $\del\phi_{a}/\del
y$ is also given by
%
\bea
  \frac{\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)}{\del y}
   &=& (1-\alpha)(1-\beta)\,\frac{\partial\phi_{a}(t,x,y,z)}{\del y}
      + \alpha(1-\beta)\,\frac{\partial\phi_{a}(t,x+\Delta,y,z)}{\del y} 
          \non \\
   && \quad
      + (1-\alpha)\beta\,\frac{\partial\phi_{a}(t,x,y+\Delta,z)}{\del y}
      + \alpha\beta\,\frac{\partial\phi_{a}(t,x+\Delta,y+\Delta,z)}{\del y}
      + \CO(\Delta^2) \\  
   &=& \frac{1}{2\Delta} \biggl[
     (1-\alpha)(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x,y+\Delta,z)-\phi_{a}(t,x,y-\Delta,z) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y+\Delta,z)-\phi_{a}(t,x+\Delta,y-\Delta,z) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + (1-\alpha)\beta\,
       \Bigl\{
         \phi_{a}(t,x,y+2\Delta,z)-\phi_{a}(t,x,y,z) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha\beta\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y+2\Delta,z)
        -\phi_{a}(t,x+\Delta,y,z) 
       \Bigr\}
       \biggr]
%          \non \\
%   && \quad
      + \CO(\Delta^2).
\eea

Using the relations obtained above and Eq.\,(\ref{eq:third}),
$\del\phi_{a}/\del z$ is calculated as
%
\bea
%  && \hspace{-1.0cm} 
  \frac{\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)}{\del z}
%          \non \\ 
   &=& \frac{\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z+\Delta)
              -\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z-\Delta)}
             {2\Delta} + \CO(\Delta^2) 
          \non \\
   && = \frac{1}{2\Delta} \biggl[
     (1-\alpha)(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x,y,z+\Delta)-\phi_{a}(t,x,y,z-\Delta) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y,z+\Delta)-\phi_{a}(t,x+\Delta,y,z-\Delta) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + (1-\alpha)\beta\,
       \Bigl\{
         \phi_{a}(t,x,y+\Delta,z+\Delta)-\phi_{a}(t,x,y+\Delta,z-\Delta) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha\beta\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y+\Delta,z+\Delta)
        -\phi_{a}(t,x+\Delta,y+\Delta,z-\Delta) 
       \Bigr\}
       \biggr]
          \non \\
   && \quad
      - \frac14\alpha(1-\alpha) \Delta
       \lhk   
         \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z+\Delta)}
              {\del x^2} -                 
         \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z-\Delta)}
              {\del x^2}           
       \rhk  
          \non \\
   && \quad
      - \frac14\beta(1-\beta) \Delta
       \lhk   
         \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z+\Delta)}
              {\del y^2} -                 
         \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z-\Delta)}
              {\del y^2}           
       \rhk  
          \non \\
   && \quad
      + \CO(\Delta^2).
\eea
%
Apparently, these approximations are of first-order accuracy. 
However, using the relations,
%
\bea
  && \hspace{-4.0cm} 
  \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z+\Delta)}
       {\del x^2} -                
  \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z-\Delta)}
       {\del x^2}       
     \non \\  
  && \qquad =
  2\Delta  
    \frac{\del^3 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)}
         {\del x^2\del z} + \CO(\Delta^3)                  
    \non \\
  && \hspace{-4.0cm} 
  \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z+\Delta)}
       {\del y^2} -                
  \frac{\del^2 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z-\Delta)}
       {\del y^2}
    \non \\     
  && \qquad =
  2\Delta  
    \frac{\del^3 \phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)}
         {\del y^2\del z} + \CO(\Delta^3),                 
\eea
%
we find they are with second-order accuracy. As a result, 
up to the second order, we find
%
\bea
  \frac{\phi_{a}(t,x+\alpha\Delta,y+\beta\Delta,z)}{\del z}
   &=& \frac{1}{2\Delta} \biggl[
     (1-\alpha)(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x,y,z+\Delta)-\phi_{a}(t,x,y,z-\Delta) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha(1-\beta)\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y,z+\Delta)-\phi_{a}(t,x+\Delta,y,z-\Delta) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + (1-\alpha)\beta\,
       \Bigl\{
         \phi_{a}(t,x,y+\Delta,z+\Delta)-\phi_{a}(t,x,y+\Delta,z-\Delta) 
       \Bigr\}
          \non \\
   && \quad \qquad
      + \alpha\beta\,
       \Bigl\{
         \phi_{a}(t,x+\Delta,y+\Delta,z+\Delta)
        -\phi_{a}(t,x+\Delta,y+\Delta,z-\Delta) 
       \Bigr\}
       \biggr]          \non \\
   && \quad \qquad
      + \CO(\Delta^2).
\eea

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\newpage
\begin{table}
\caption{Five different sets of simulations}
\label{tab:set1}
  \begin{center}
     \begin{tabular}{cccccc}
         Case & Lattice &
         Lattice spacing ($\Delta$) & $\zeta$ & Realization 
         & ${\rm Box~size}/H^{-1}$ \\
         & number & \protect{[}unit = $t_{i}R(t)$\protect{]} &  &  & (at
         final time) \\
        \hline
        (a) & $128^3$ & $\sqrt{3}/8 $ & 10 & 100 & 1 (at 200) \\
        (b) & $256^3$ & $\sqrt{3}/16$ & 10 &  10 & 1 (at 200) \\
        (c) & $128^3$ & $\sqrt{3}/4 $ & 10 &  10 & 2 (at 200) \\
        (d) & $256^3$ & $\sqrt{3}/8 $ & 10 &  10 & 2 (at 200) \\
        (e) & $256^3$ & $\sqrt{3}/4 $ & 10 &  10 & 4 (at 200) \\
     \end{tabular}
  \end{center}
\end{table}

\begin{table}
\caption{Results of numerical simulations}
\label{tab:set2}
  \begin{center}
     \begin{tabular}{ccccccc}
         Case & $\xi$ & $\la v \ra$ & $\la v^2 \ra$ & $\gamma$ 
         & $c$ & $\kappa$ \\
        \hline
        (a) & 0.72 & 0.63 & 0.50 & $2.0\pm0.2$ & $0.52\pm0.05$ 
            & $0.08\pm0.05$ \\
        (b) & 0.79 & 0.65 & 0.50 & $1.9\pm0.3$ & $0.48\pm0.04$
            & $0.08\pm0.04$ \\
        (c) & 0.77 & 0.60 & 0.50 & $1.8\pm0.2$ & $0.40\pm0.02$
            & $0.17\pm0.02$ \\
        (d) & 0.80 & 0.60 & 0.49 & $1.8\pm0.2$ & $0.42\pm0.03$
            & $0.16\pm0.03$ \\
        (e) & 0.80 & 0.60 & 0.50 & $1.8\pm0.2$ & $0.40\pm0.04$
            & $0.16\pm0.04$
     \end{tabular}
  \end{center}
\end{table}

\begin{figure}[htb]
\begin{center}
\begin{minipage}{80mm}
\includegraphics[width=8cm]{phase1.ps}
\begin{center} 
%\caption{Relative phase of the scalar fields is classified into three groups,
% (i) $0 \le \theta < \frac{\pi}{2}$, (ii) $\frac{\pi}{2} \le \theta <
%\frac{3\pi}{2}$, (iii)
%$\frac{3\pi}{2} \le \theta < 2\pi$.}
\label{fig:phase1}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}{80mm}
\includegraphics[width=8cm]{phase2.ps}
\begin{center} 
%\caption{Relative phase of the scalar fields is classified into three groups,
% (i) $0 \le \theta < \frac{\pi}{2}$, (ii) $\frac{\pi}{2} \le \theta <
%\frac{3\pi}{2}$, (iii)
%$\frac{3\pi}{2} \le \theta < 2\pi$.}
\label{fig:phase2}
\end{center}
\end{minipage}
\caption{Left: 
Relative phase of the scalar fields is classified into three groups,
 (i) $0 \le \theta < \frac{\pi}{2}$, (ii) $\frac{\pi}{2} \le \theta <
\frac{3\pi}{2}$, (iii)
$\frac{3\pi}{2} \le \theta < 2\pi$.\\
Right: Another classification of relative phase, (i) $0 \le
\theta < \frac{\pi}{2}$, (ii) $\frac{\pi}{2}\le\theta < \pi$, (iii)
$\pi \le\theta < 2\pi$, was also tried but the numerical results did not
 depend on these choices.}
\label{fig:phase}
\end{center}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=6cm]{cross1.eps} 
\caption{$\vect p_{a}$ and $\vect q_{a}$ are points corresponding to
  $\phi_{a} = 0$ for each $a$.  These points are obtained by linear
  interpolation using the values of $\phi_a$ at two corners of the
  plaquette between which it changes sign.  The line $\phi_{a} = 0$
  for each $a$ is drawn by simply connecting $\vect p_{a}$ and $\vect
  q_{a}$ by a straight line.  The intersection of these two lines is
  identified as a position through which a string penetrates a
  plaquette.}
\label{fig:cross1}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=7.7cm]{cross2.eps} 
\caption{In case the intersection is found outside the plaquette, the
nearest point on the edge of 
the plaquette is identified as the penetration point of a string.}
\label{fig:cross2}
\end{figure}

%\begin{figure}[htb]
%\includegraphics[width=8cm]{phase2.ps} 
%\caption{Another classification of relative phase, (i) $0 \le
%\theta < \frac{\pi}{2}$, (ii) $\frac{\pi}{2}\le\theta < \pi$, (iii)
%$\pi \le\theta < 2\pi$.}
%\label{fig:phase2}
%\end{figure}

\begin{figure}[htb]
\includegraphics[width=18cm]{statistics.ps} 
\caption{Time evolution of the scaling parameter $\xi$. 
  Filled squares represent our new identification method. Blank
  circles correspond to our previous identification method based on
  potential energy density \cite{YKY,YYK}. Blank squares are results
  of the Vachaspati-Vilenkin algorithm.}
\label{fig:xi}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=18cm]{vstatistics.ps} 
\caption{Time evolution of average velocity of global strings is
shown.}
\label{fig:velocity}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=18cm]{svstatistics.ps} 
\caption{Time evolution of average square velocity.}  
\label{fig:svelocity}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=18cm]{fvstatistics.ps} 
\caption{Time evolution of average Lorentz factor.}  
\label{fig:gamma}
\end{figure}

\end{document}

