Cosmic Strings and Domain Walls in Models with Goldstone and Pseudo-Goldstone Bosons
带有戈尔登和赝戈尔登玻色子的模型中的宇宙弦和畴壁

PACS numbers: 98.80.Bp, 12.10.-g, 11.30.Qc
PACS 编号：98.80.Bp，12.10.-g，11.30.Qc

Gauge theories of elementary particles with spontaneous symmetry breaking predict that the early universe passes through a sequence of phase transitions as it cools after the big bang. These phase transitions can give rise to vacuum structures-vacuum domain walls, strings, or monopoles, depending on the topology of the gauge theory. ${}^1$${}^1$^(1){ }^{1} Domain walls are produced if $M$$M$MM, the manifold of degenerate vacuums after symmetry breaking, consists of two or more disconnected components. Strings are formed if $M$$M$MM contains incontractible loops (that is, if $M$$M$MM is not simply connected), and monopoles are formed if $M$$M$MM contains incontractible two-dimensional surfaces. The cosmological evolution of these structures has been discussed previously. ${}^{1-6}$${}^{1-6}$^(1-6){ }^{1-6}
量子场论中的基本粒子规范理论预测，大爆炸后早期宇宙在冷却过程中会经历一系列相变。这些相变可能会产生真空结构——真空畴壁、弦或磁单极子，这取决于规范理论的拓扑结构。 ${}^1$${}^1$^(1){ }^{1} 如果对称性破缺后的简并真空流形 $M$$M$MM 包含两个或更多个不连通的分量，则会产生畴壁。如果 $M$$M$MM 包含不可收缩的环路（即 $M$$M$MM 不是单连通的），则会形成弦，如果 $M$$M$MM 包含不可收缩的二维曲面，则会形成磁单极子。这些结构的宇宙学演化已在之前讨论过。 ${}^{1-6}$${}^{1-6}$^(1-6){ }^{1-6}
Besides gauge symmetries, some particle models have exact or approximate global symmetries which can also be spontaneously broken. Well-known examples are the $\mathrm{U}(1)B-L$$\mathrm{U}(1)B-L$U(1)B-L\mathrm{U}(1) B-L symmetry of $\mathrm{SU}(5),{}^7$$\mathrm{SU}(5),{}^7$SU(5),^(7)\operatorname{SU}(5),{ }^{7} and the approximate $U(1)$$U(1)$U(1)U(1) chiral symmetry in models with axions. ${}^{8,9}$${}^{8,9}$^(8,9){ }^{8,9} In the present paper, we shall discuss the vacuum structures arising in such models and their cosmological evolution. We shall concentrate on the most interesting case of a global $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) symmetry.
除了规范对称性之外，一些粒子模型还具有精确或近似的全局对称性，这些对称性也可以自发破缺。著名的例子包括 $\mathrm{U}(1)B-L$$\mathrm{U}(1)B-L$U(1)B-L\mathrm{U}(1) B-L 对称性以及 $\mathrm{SU}(5),{}^7$$\mathrm{SU}(5),{}^7$SU(5),^(7)\operatorname{SU}(5),{ }^{7} 的 $U(1)$$U(1)$U(1)U(1) 近似手征对称性。 ${}^{8,9}$${}^{8,9}$^(8,9){ }^{8,9} 在本文中，我们将讨论此类模型中产生的真空结构及其宇宙学演化。我们将重点讨论最有趣的全局 $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) 对称性的情况。
Let us first consider the case of an exact $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) symmetry. The character of vacuum structures is insensitive to the details of the model, and all relevant physics is present in a simple Goldstone model described by the Lagrangian
让我们首先考虑一个精确的 $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) 对称性。真空结构的特征对模型的细节不敏感，所有相关的物理都包含在一个由拉格朗日量描述的简单戈尔登模型中

where $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi is a complex scalar field. The manifold $M$$M$MM of degenerate vacuum states in this model is a circle. It is not simply connected, and thus we must have strings. The strings can be classified according to their winding number, $n=\mathrm{\Delta }\theta /2\pi$$n=\mathrm{\Delta }\theta /2\pi$n=Delta theta//2pin=\Delta \theta / 2 \pi, where $\mathrm{\Delta }\theta$$\mathrm{\Delta }\theta$Delta theta\Delta \theta is the change of phase of the vacuum expectation value (VEV) $⟨\mathrm{\Phi }⟩$$⟨\mathrm{\Phi }⟩$(:Phi:)\langle\Phi\rangle as we go around the string. The lowest energy strings have $n=1$$n=1$n=1n=1.
其中 $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi 是一个复标量场。该模型中简并真空态的流形 $M$$M$MM 是一个圆。它不是单连通的，因此我们必须有弦。弦可以根据它们的缠绕数 $n=\mathrm{\Delta }\theta /2\pi$$n=\mathrm{\Delta }\theta /2\pi$n=Delta theta//2pin=\Delta \theta / 2 \pi 进行分类，其中 $\mathrm{\Delta }\theta$$\mathrm{\Delta }\theta$Delta theta\Delta \theta 是当我们绕弦时真空期望值（VEV） $⟨\mathrm{\Phi }⟩$$⟨\mathrm{\Phi }⟩$(:Phi:)\langle\Phi\rangle 的相位变化。最低能量的弦有 $n=1$$n=1$n=1n=1 。

The magnitude of the VEV $|⟨\mathrm{\Phi }⟩|$$|⟨\mathrm{\Phi }⟩|$|(:Phi:)||\langle\Phi\rangle| is substantially different from $\eta$$\eta$eta\eta only inside the string core of width $\lambda \sim (h\eta {)}^{-1}$$\lambda \sim (h\eta {)}^{-1}$lambda∼(h eta)^(-1)\lambda \sim(h \eta)^{-1} and is equal to zero on the string axis. The energy per unit length of the core is ${\mu }_{\text{core}}\sim h^2{\eta }^4{\lambda }^2\sim {\eta }^2$${\mu }_{\text{core }}\sim h^2{\eta }^4{\lambda }^2\sim {\eta }^2$mu_("core ")∼h^(2)eta^(4)lambda^(2)∼eta^(2)\mu_{\text {core }} \sim h^{2} \eta^{4} \lambda^{2} \sim \eta^{2}. All this applies equally to the cases of local and global $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) symmetries. The basic differences are to be found outside the core. Consider an infinite straight string parallel to the $z$$z$zz axis. In cylindrical coordinates $r$$r$rr, $\phi ,z$$\phi ,z$varphi,z\varphi, z, and for $r\gg \lambda$$r\gg \lambda$r≫lambdar \gg \lambda,
VEV $|⟨\mathrm{\Phi }⟩|$$|⟨\mathrm{\Phi }⟩|$|(:Phi:)||\langle\Phi\rangle| 的幅度仅在宽度为 $\lambda \sim (h\eta {)}^{-1}$$\lambda \sim (h\eta {)}^{-1}$lambda∼(h eta)^(-1)\lambda \sim(h \eta)^{-1} 的弦核内与 $\eta$$\eta$eta\eta 有显著差异，并且在弦轴上等于零。核的单位长度的能量是 ${\mu }_{\text{core}}\sim h^2{\eta }^4{\lambda }^2\sim {\eta }^2$${\mu }_{\text{core }}\sim h^2{\eta }^4{\lambda }^2\sim {\eta }^2$mu_("core ")∼h^(2)eta^(4)lambda^(2)∼eta^(2)\mu_{\text {core }} \sim h^{2} \eta^{4} \lambda^{2} \sim \eta^{2} 。这对局部和全局 $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) 对称性的情况都适用。基本差异存在于核外。考虑一个与 $z$$z$zz 轴平行的无限长直弦。在圆柱坐标 $r$$r$rr ， $\phi ,z$$\phi ,z$varphi,z\varphi, z 和对于 $r\gg \lambda$$r\gg \lambda$r≫lambdar \gg \lambda ，

where we have assumed that $n=1$$n=1$n=1n=1. In the case of a local $U(1)$$U(1)$U(1)U(1) symmetry, the variation of $⟨\mathrm{\Phi }⟩$$⟨\mathrm{\Phi }⟩$(:Phi:)\langle\Phi\rangle is balanced by that of the gauge field, so that the field outside the core is pure gauge. All the energy of the string is in its core, and the energy per unit length is $\mu \sim {\eta }^2$$\mu \sim {\eta }^2$mu∼eta^(2)\mu \sim \eta^{2}. For a global symmetry, there is no gauge field, and the energy per unit length of the string diverges logarithmically,
在这里我们假设了 $n=1$$n=1$n=1n=1 。在局部 $U(1)$$U(1)$U(1)U(1) 对称性的情况下， $⟨\mathrm{\Phi }⟩$$⟨\mathrm{\Phi }⟩$(:Phi:)\langle\Phi\rangle 的变化被规范场的变化所平衡，因此核心外的场是纯规范场。弦的能量全部在其核心中，单位长度的能量是 $\mu \sim {\eta }^2$$\mu \sim {\eta }^2$mu∼eta^(2)\mu \sim \eta^{2} 。对于全局对称性，没有规范场，弦的单位长度能量对数发散，

For this reason, such strings are usually thought to be illegitimate. ${}^{10}$${}^{10}$^(10){ }^{10}
因此，人们通常认为这样的弦是不合法的。 ${}^{10}$${}^{10}$^(10){ }^{10}
We note, however, that the divergence of the integral (3) does not introduce any difficulties in a cosmological context. If the scale of the system of strings is $\sim \xi$$\sim \xi$∼xi\sim \xi (that is, the typical curvature radius of the strings and the typical distance between adjacent strings are $\sim \xi )$$\sim \xi )$∼xi)\sim \xi), then $\xi$$\xi$xi\xi provides a natural cutoff in the integral (3) and we obtain
然而，我们注意到积分（3）的发散在宇宙学背景下不会引起任何困难。如果弦系统的尺度是 $\sim \xi$$\sim \xi$∼xi\sim \xi （即弦的典型曲率半径和相邻弦之间的典型距离是 $\sim \xi )$$\sim \xi )$∼xi)\sim \xi) ，那么 $\xi$$\xi$xi\xi 在积分（3）中提供了一个自然的截止，我们得到

[Note that the energy of a closed loop is always finite, $E\sim {\eta }^{2}R\ln (R/\lambda )$$E\sim {\eta }^{2}R\ln (R/\lambda )$E∼eta^(2)R ln(R//lambda)E \sim \eta^{2} R \ln (R / \lambda), where $R$$R$RR is the size of the loop and we have assumed that the loop is momentarily at rest. We note also that global-symmetry strings are well known to condensed matter physicists in the form of quantized vortex lines in liquid helium.] Two approximately parallel strings with opposite signs of $\mathrm{\Delta }\theta$$\mathrm{\Delta }\theta$Delta theta\Delta \theta attract one
[注意，闭合环的能量总是有限的， $E\sim {\eta }^{2}R\ln (R/\lambda )$$E\sim {\eta }^{2}R\ln (R/\lambda )$E∼eta^(2)R ln(R//lambda)E \sim \eta^{2} R \ln (R / \lambda) ，其中 $R$$R$RR 是环的大小，我们假设环暂时静止。我们还注意到，全局对称性弦在凝聚态物理学家中作为液氦中的量化涡线而众所周知。] 两条近似平行的弦具有相反的 $\mathrm{\Delta }\theta$$\mathrm{\Delta }\theta$Delta theta\Delta \theta 符号相互吸引
another, the force per unit length being $F_{\text{int}}\sim \mu /$$F_{\text{int }}\sim \mu /$F_("int ")∼mu//F_{\text {int }} \sim \mu / $\xi$$\xi$xi\xi. The force of tension in curved strings is of the same order of magnitude. It appears that the presence of interaction between the strings does not substantially alter the scenarios of Refs. 1, 3 , and 4 ; thus the scale of the system of strings at cosmic time $t$$t$tt is $\sim t$$\sim t$∼t\sim t. The energy density due to the strings is
另一个，单位长度的力是 $F_{\text{int}}\sim \mu /$$F_{\text{int }}\sim \mu /$F_("int ")∼mu//F_{\text {int }} \sim \mu / $\xi$$\xi$xi\xi 。弯曲弦的张力具有相同的数量级。看起来弦之间的相互作用并没有显著改变参考文献 1、3 和 4 中的情况；因此，宇宙时间 $t$$t$tt 时弦系统的尺度是 $\sim t$$\sim t$∼t\sim t 。弦产生的能量密度是

The energy density of matter is $\rho \sim 1/G{t}^2$$\rho \sim 1/G{t}^2$rho∼1//Gt^(2)\rho \sim 1 / G t^{2}, and thus
物质的能量密度是 $\rho \sim 1/G{t}^2$$\rho \sim 1/G{t}^2$rho∼1//Gt^(2)\rho \sim 1 / G t^{2} ，因此

It has been shown in Refs. 3 and 4 that strings can produce cosmological density fluctuations sufficient to explain galaxy formation ${}^{11}$${}^{11}$^(11){ }^{11} if ${}^{12}G\mu$${}^{12}G\mu$^(12)G mu{ }^{12} G \mu $\sim {10}^{-5}$$\sim {10}^{-5}$∼10^(-5)\sim 10^{-5}. Then Eq. (6) gives $\eta \sim {10}^{15}-{10}^{16}\mathrm{GeV}$$\eta \sim {10}^{15}-{10}^{16}\mathrm{GeV}$eta∼10^(15)-10^(16)GeV\eta \sim 10^{15}-10^{16} \mathrm{GeV}. We conclude that a model in which a global $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) symmetry is spontaneously broken at the grand unification mass scale can provide a solution to the problem of cosmological density fluctuations. It should be noted that grand unified models with stable strings resulting from a gauge-symmetry breaking are difficult to construct ${}^{13}$${}^{13}$^(13){ }^{13}; no realistic model of this kind has yet been suggested.
参考文献中 3 和 4 表明，弦可以产生足够解释星系形成的宇宙学密度涨落 ${}^{11}$${}^{11}$^(11){ }^{11} ，如果 ${}^{12}G\mu$${}^{12}G\mu$^(12)G mu{ }^{12} G \mu $\sim {10}^{-5}$$\sim {10}^{-5}$∼10^(-5)\sim 10^{-5} 。那么公式（6）给出 $\eta \sim {10}^{15}-{10}^{16}\mathrm{GeV}$$\eta \sim {10}^{15}-{10}^{16}\mathrm{GeV}$eta∼10^(15)-10^(16)GeV\eta \sim 10^{15}-10^{16} \mathrm{GeV} 。我们得出结论，一个在 grand unification 质量尺度上自发破缺全局 $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) 对称性的模型可以为宇宙学密度涨落问题提供一个解决方案。应该注意的是，由于规范对称性破缺而产生的稳定弦的 grand unified 模型很难构建 ${}^{13}$${}^{13}$^(13){ }^{13} ；目前还没有提出任何现实的模型。
One possible candidate for the role of $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) global is the $B-L$$B-L$B-LB-L symmetry of $\mathrm{SU}(5)$$\mathrm{SU}(5)$SU(5)\mathrm{SU}(5). Models with spontaneous breaking of this symmetry can be readily constructed. ${}^7$${}^7$^(7){ }^{7} It can also be shown ${}^7$${}^7$^(7){ }^{7} that the resulting Goldstone boson is virtually unobservable and does not give rise to a $1/r^2$$1/r^2$1//r^(2)1 / r^{2} force competing with gravity.
$\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) 全局对称的一个可能候选者是 $B-L$$B-L$B-LB-L 对称性 $\mathrm{SU}(5)$$\mathrm{SU}(5)$SU(5)\mathrm{SU}(5) 。可以轻易构建具有这种对称性自发破缺的模型。 ${}^7$${}^7$^(7){ }^{7} 还可以证明 ${}^7$${}^7$^(7){ }^{7} ，由此产生的戈尔登玻色子几乎无法观测，并且不会产生与引力竞争的 $1/r^2$$1/r^2$1//r^(2)1 / r^{2} 力。
We now turn to the case of an approximate global $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) symmetry. Again we shall consider a simple model with one complex scalar field,
现在我们转向近似全局 $\mathrm{U}(1)$$\mathrm{U}(1)$U(1)\mathrm{U}(1) 对称性的情况。我们仍然将考虑一个具有一个复标量场的简单模型，

where  其中

As a prototype for this model we used models with “invisible” axions ${}^9$${}^9$^(9){ }^{9} designed to solve the strong $CP$$CP$CPC P problem. In such models $\eta \sim {10}^{15}\mathrm{GeV}$$\eta \sim {10}^{15}\mathrm{GeV}$eta∼10^(15)GeV\eta \sim 10^{15} \mathrm{GeV} and $m\sim 0.1\mathrm{GeV}$$m\sim 0.1\mathrm{GeV}$m∼0.1GeVm \sim 0.1 \mathrm{GeV}. (We shall use these values for estimations below.) The approximate $U(1)$$U(1)$U(1)U(1) symmetry of (7) corresponds to the anomalous chiral Peccei-Quinn symmetry ${}^{8,9}$${}^{8,9}$^(8,9){ }^{8,9} which is explicitly broken by instantons at the strong interaction mass scale.
作为这个模型的原型，我们使用了旨在解决强 $CP$$CP$CPC P 问题的“隐形”轴子 ${}^9$${}^9$^(9){ }^{9} 模型。在这样的模型中 $\eta \sim {10}^{15}\mathrm{GeV}$$\eta \sim {10}^{15}\mathrm{GeV}$eta∼10^(15)GeV\eta \sim 10^{15} \mathrm{GeV} 和 $m\sim 0.1\mathrm{GeV}$$m\sim 0.1\mathrm{GeV}$m∼0.1GeVm \sim 0.1 \mathrm{GeV} 。（我们将在下面使用这些值进行估算。）(7) 的近似 $U(1)$$U(1)$U(1)U(1) 对称性对应于异常手征佩西-昆恩对称性 ${}^{8,9}$${}^{8,9}$^(8,9){ }^{8,9} ，该对称性在强相互作用质量尺度上被瞬子显式地破缺。
The field $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi acquires a nonzero VEV when the universe cools to a temperature $T\sim \eta$$T\sim \eta$T∼etaT \sim \eta. At such temperatures the last term in (7) is negligible and the behavior of the model is the same as that
当宇宙冷却到温度 $T\sim \eta$$T\sim \eta$T∼etaT \sim \eta 时，场 $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi 获得非零的真空期望值。在这种温度下，公式 (7) 中的最后一项可以忽略不计，模型的行为与金戈斯坦模型 (1) 相同。
of the Goldstone model (1). The phase transition gives rise to strings which evolve as discussed above. At sufficiently low temperatures the $\theta$$\theta$theta\theta dependent term in (7) becomes important. The minimum of energy corresponds to $\theta \equiv 0$$\theta \equiv 0$theta-=0\theta \equiv 0 (mod $2\pi$$2\pi$2pi2 \pi ). However, the phase $\theta$$\theta$theta\theta cannot become equal to zero everywhere, since it changes by $\mathrm{\Delta }\theta =2\pi$$\mathrm{\Delta }\theta =2\pi$Delta theta=2pi\Delta \theta=2 \pi around the strings. It is natural to assume that $\theta$$\theta$theta\theta settles down to zero at all points around a string, except within a relatively thin wall, so that $\theta$$\theta$theta\theta changes by $2\pi$$2\pi$2pi2 \pi across the wall. ${}^{14}$${}^{14}$^(14){ }^{14} The possibility that strings can get connected by domain walls has been first suggested, in a different context, by Kibble, Lazarides, and Shafi. ${}^{15}$${}^{15}$^(15){ }^{15}
相变产生了如上所述演化的弦。在足够低的温度下，公式 (7) 中的 $\theta$$\theta$theta\theta 相关项变得重要。能量的最小值对应于 $\theta \equiv 0$$\theta \equiv 0$theta-=0\theta \equiv 0 (模 $2\pi$$2\pi$2pi2 \pi )。然而，相 $\theta$$\theta$theta\theta 不能处处为零，因为它在弦周围变化为 $\mathrm{\Delta }\theta =2\pi$$\mathrm{\Delta }\theta =2\pi$Delta theta=2pi\Delta \theta=2 \pi 。自然地，我们假设 $\theta$$\theta$theta\theta 在弦周围的各点都趋于零，除了在相对较薄的壁内，因此 $\theta$$\theta$theta\theta 在壁上变化为 $2\pi$$2\pi$2pi2 \pi 。 ${}^{14}$${}^{14}$^(14){ }^{14} 弦可能被畴壁连接的可能性首先由 Kibble、Lazarides 和 Shafi 在不同的背景下提出。 ${}^{15}$${}^{15}$^(15){ }^{15}
To see that the model (7) indeed has domainwall solutions, we note that, away from the string cores, the VEV of $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi is $⟨\mathrm{\Phi }⟩=\eta e^{i\theta }$$⟨\mathrm{\Phi }⟩=\eta e^{i\theta }$(:Phi:)=etae^(i theta)\langle\Phi\rangle=\eta e^{i \theta}, and the effective Lagrangian for $\theta$$\theta$theta\theta is
为了证明模型 (7) 确实有畴壁解，我们注意到，在弦核之外， $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi 的真空期望值是 $⟨\mathrm{\Phi }⟩=\eta e^{i\theta }$$⟨\mathrm{\Phi }⟩=\eta e^{i\theta }$(:Phi:)=etae^(i theta)\langle\Phi\rangle=\eta e^{i \theta} ，并且 $\theta$$\theta$theta\theta 的有效拉格朗日量是

The corresponding wave equation is
相应的波动方程是

where $m_a=m^2/\eta \sim {10}^{-8}\mathrm{eV}$$m_a=m^2/\eta \sim {10}^{-8}\mathrm{eV}$m_(a)=m^(2)//eta∼10^(-8)eVm_{a}=m^{2} / \eta \sim 10^{-8} \mathrm{eV} is the axion mass. ${}^9$${}^9$^(9){ }^{9} Equation (10) is the so-called sine-Gordon equation which is known to have domain-wall solutions (solitons) ${}^{10}$${}^{10}$^(10){ }^{10} :
其中 $m_a=m^2/\eta \sim {10}^{-8}\mathrm{eV}$$m_a=m^2/\eta \sim {10}^{-8}\mathrm{eV}$m_(a)=m^(2)//eta∼10^(-8)eVm_{a}=m^{2} / \eta \sim 10^{-8} \mathrm{eV} 是轴子质量。 ${}^9$${}^9$^(9){ }^{9} 方程（10）是所谓的正弦-戈登方程，该方程已知具有畴壁解（孤立子） ${}^{10}$${}^{10}$^(10){ }^{10} ：

where the $x$$x$xx axis is perpendicular to the wall. The thickness of the wall is $\delta \sim m_a{}^{-1}\sim {10}^4\mathrm{cm}$$\delta \sim m_a{}^{-1}\sim {10}^4\mathrm{cm}$delta∼m_(a)^(-1)∼10^(4)cm\delta \sim m_{a}{ }^{-1} \sim 10^{4} \mathrm{~cm} and the energy per unit area is
其中 $x$$x$xx 轴垂直于畴壁。畴壁的厚度为 $\delta \sim m_a{}^{-1}\sim {10}^4\mathrm{cm}$$\delta \sim m_a{}^{-1}\sim {10}^4\mathrm{cm}$delta∼m_(a)^(-1)∼10^(4)cm\delta \sim m_{a}{ }^{-1} \sim 10^{4} \mathrm{~cm} ，单位面积的能量为

(the exact expression is ${}^{10}\sigma =16m^2\eta$${}^{10}\sigma =16m^2\eta$^(10)sigma=16m^(2)eta{ }^{10} \sigma=16 m^{2} \eta ).
（精确表达式为 ${}^{10}\sigma =16m^2\eta$${}^{10}\sigma =16m^2\eta$^(10)sigma=16m^(2)eta{ }^{10} \sigma=16 m^{2} \eta ）。
The domain walls can be considered as “formed” when their thickness becomes smaller than the horizon, i.e., at ${}^{16}t>m_a{}^{-1}\sim {10}^{-6}\mathrm{s},T<1\mathrm{GeV}$${}^{16}t>m_a{}^{-1}\sim {10}^{-6}\mathrm{s},T<1\mathrm{GeV}$^(16)t > m_(a)^(-1)∼10^(-6)s,T < 1GeV{ }^{16} t>m_{a}{ }^{-1} \sim 10^{-6} \mathrm{~s}, T<1 \mathrm{GeV}. At later times the system consists of strings connected by domain walls. The linear mass density of the strings is
当畴壁的厚度小于视界时，畴壁可以被认为是“形成”的，即 ${}^{16}t>m_a{}^{-1}\sim {10}^{-6}\mathrm{s},T<1\mathrm{GeV}$${}^{16}t>m_a{}^{-1}\sim {10}^{-6}\mathrm{s},T<1\mathrm{GeV}$^(16)t > m_(a)^(-1)∼10^(-6)s,T < 1GeV{ }^{16} t>m_{a}{ }^{-1} \sim 10^{-6} \mathrm{~s}, T<1 \mathrm{GeV} 。在后期，系统由通过畴壁连接的弦组成。弦的线质量密度为

Strings form the boundaries of the walls and of the holes in the walls. One wall surface can be stretched between a number of strings and can have a complicated topological structure. In addition, there can be closed or infinite wall surface without strings.
弦线构成了墙壁和墙壁上孔洞的边界。一个墙面可以在多条弦线之间拉伸，并且可以有复杂的拓扑结构。此外，还可以存在没有弦线的封闭或无限墙面。

The frictional force acting on the walls as a result of their interaction with particles is negligible, since the width of the walls is much greater than the thermal wavelength of the particles. ${}^{17}$${}^{17}$^(17){ }^{17} The friction of strings is also negligible. ${}^{1,5}$${}^{1,5}$^(1,5){ }^{1,5} The
墙壁由于与粒子的相互作用而产生的摩擦力可以忽略不计，因为墙壁的宽度远大于粒子的热波长。 ${}^{17}$${}^{17}$^(17){ }^{17} 弦线的摩擦也可以忽略不计。 ${}^{1,5}$${}^{1,5}$^(1,5){ }^{1,5}
force of tension in a string of curvature radius $R,F\sim \mu /R$$R,F\sim \mu /R$R,F∼mu//RR, F \sim \mu / R, is greater than the wall tension, $\sigma$$\sigma$sigma\sigma, for $R<\mu /\sigma$$R<\mu /\sigma$R < mu//sigmaR<\mu / \sigma. Therefore, at $t<\mu /\sigma \sim {10}^{-4}\mathrm{s}$$t<\mu /\sigma \sim {10}^{-4}\mathrm{s}$t < mu//sigma∼10^(-4)st<\mu / \sigma \sim 10^{-4} \mathrm{~s} the evolution of the strings will not be qualitatively different from that in model (1). At $t>\mu /\sigma$$t>\mu /\sigma$t > mu//sigmat>\mu / \sigma the typical curvature radius of the strings becomes greater than $\mu /\sigma$$\mu /\sigma$mu//sigma\mu / \sigma, and the dynamics of the system is dominated by the wall tension. Domain walls will tend to shrink, pulling the strings toward one another, and the holes in the walls will increase in size. As the strip of wall connecting two strings shrinks, its energy is transferred to the kinetic and potential (rest) energy of the strings; energetic strings intersect and pass through one another, and a wall surface starts stretching between them again. The whole system thus violently oscillates. The strings frequently intersect and occasionally intercommute; as a result, the strip of domain wall connecting the strings is cut into pieces. If the probability of intercommuting for intersecting strings is $p$$p$pp $\sim 1$$\sim 1$∼1\sim 1, then the whole system breaks into pieces of size $\sim \mu /\sigma$$\sim \mu /\sigma$∼mu//sigma\sim \mu / \sigma at $t\sim \mu /\sigma$$t\sim \mu /\sigma$t∼mu//sigmat \sim \mu / \sigma. (For $p\ll 1$$p\ll 1$p≪1p \ll 1 the size of the pieces is larger.)
弦的张力在曲率半径 $R,F\sim \mu /R$$R,F\sim \mu /R$R,F∼mu//RR, F \sim \mu / R 时，大于壁张力 $\sigma$$\sigma$sigma\sigma ，对于 $R<\mu /\sigma$$R<\mu /\sigma$R < mu//sigmaR<\mu / \sigma 。因此，在 $t<\mu /\sigma \sim {10}^{-4}\mathrm{s}$$t<\mu /\sigma \sim {10}^{-4}\mathrm{s}$t < mu//sigma∼10^(-4)st<\mu / \sigma \sim 10^{-4} \mathrm{~s} 时，弦的演化的定性差异将不同于模型 (1)。在 $t>\mu /\sigma$$t>\mu /\sigma$t > mu//sigmat>\mu / \sigma 时，弦的典型曲率半径大于 $\mu /\sigma$$\mu /\sigma$mu//sigma\mu / \sigma ，系统的动力学由壁张力主导。畴壁将倾向于收缩，将弦拉近彼此，并且畴壁中的孔洞会增大。当连接两条弦的畴壁条带收缩时，其能量转移到弦的动能和势能（静止）上；高能弦相互交叉并通过彼此，并且它们之间开始再次拉伸畴壁表面。因此，整个系统剧烈振荡。弦频繁交叉，偶尔交换；结果，连接弦的畴壁条带被切成碎片。如果交叉弦交换的概率是 $p$$p$pp $\sim 1$$\sim 1$∼1\sim 1 ，那么在 $t\sim \mu /\sigma$$t\sim \mu /\sigma$t∼mu//sigmat \sim \mu / \sigma 时，整个系统会分裂成大小为 $\sim \mu /\sigma$$\sim \mu /\sigma$∼mu//sigma\sim \mu / \sigma 的碎片。（对于 $p\ll 1$$p\ll 1$p≪1p \ll 1 ，碎片的大小更大。）

A piece of wall of size $R\gtrsim \mu /\sigma$$R\gtrsim \mu /\sigma$R≳mu//sigmaR \gtrsim \mu / \sigma oscillates at a typical frequency $\omega \sim R^{-1}$$\omega \sim R^{-1}$omega∼R^(-1)\omega \sim R^{-1} and loses its energy by gravitational radiation at a rate
一块尺寸为 $R\gtrsim \mu /\sigma$$R\gtrsim \mu /\sigma$R≳mu//sigmaR \gtrsim \mu / \sigma 的墙壁以典型频率 $\omega \sim R^{-1}$$\omega \sim R^{-1}$omega∼R^(-1)\omega \sim R^{-1} 振荡，并通过引力辐射以速率损失能量

The lifetime of the piece is independent of its size,
该墙壁的寿命与其尺寸无关，

The lifetime can be smaller if the pieces rapidly decay as a result of multiple self-intersections. When the size of the piece becomes smaller than $\mu /\sigma$$\mu /\sigma$mu//sigma\mu / \sigma, its mass is determined mostly by the string, and the decay time is ${}^4\tau \lesssim (G\mu {)}^{-1}R\lesssim (G\sigma {)}^{-1}$${}^4\tau \lesssim (G\mu {)}^{-1}R\lesssim (G\sigma {)}^{-1}$^(4)tau≲(G mu)^(-1)R≲(G sigma)^(-1){ }^{4} \tau \lesssim(G \mu)^{-1} R \lesssim(G \sigma)^{-1}. (It is assumed that sufficiently small pieces decay into elementary particles.)
如果墙壁因多次自相交而迅速衰减，其寿命可以更短。当墙壁的尺寸小于 $\mu /\sigma$$\mu /\sigma$mu//sigma\mu / \sigma 时，其质量主要由弦决定，衰减时间为 ${}^4\tau \lesssim (G\mu {)}^{-1}R\lesssim (G\sigma {)}^{-1}$${}^4\tau \lesssim (G\mu {)}^{-1}R\lesssim (G\sigma {)}^{-1}$^(4)tau≲(G mu)^(-1)R≲(G sigma)^(-1){ }^{4} \tau \lesssim(G \mu)^{-1} R \lesssim(G \sigma)^{-1} 。（假设足够小的墙壁会衰变成基本粒子。）