# A Review of Gravitational Waves from Cosmic Domain Walls

## Abstract

**:**

## 1. Introduction

## 2. Domain Walls and Cosmology

#### 2.1. Field Theory

#### 2.2. Cosmological Evolution

#### 2.3. Biased Domain Walls

## 3. Estimation of Gravitational Waves from Domain Walls

## 4. Particle Physics Models

#### 4.1. Standard Model Higgs Field

#### 4.2. Axion Models

#### 4.3. Supersymmetric Models

## 5. Implications for Present and Future Observations

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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1. | We emphasize that the condition is not robust, and we can consider several loopholes depending on the details of the models. For instance, if the $\varphi $ field never thermalizes, domain walls may not be formed even when ${T}_{\mathrm{max}}>{m}_{\varphi}$ is satisfied. We can also consider the case where the effective mass ${m}_{\varphi ,\mathrm{eff}}$ during inflation is different from the bare mass ${m}_{\varphi}$. In such a case, it is possible to avoid the formation of domain walls even when ${H}_{\mathrm{inf}}>{m}_{\varphi}$ is satisfied [34,35]. |

2. | We note that the results of numerical simulations imply ${\rho}_{\mathrm{wall}}\propto {t}^{-\nu}$, where the exponent $\nu $ slightly deviates from $\nu =1$. At this point, it is unclear whether this deviation represents some physical effect or just a numerical artifact which could be removed if we improve the dynamical range of the simulation. In this article, we carry out the analysis by assuming that the evolution of domain walls is described by the exact scaling law [Equation (17)]. |

3. | It is also possible to avoid the domain wall problem by assuming an asymmetric probability distribution for initial field fluctuations [49] instead of introducing the energy bias in the potential. Here, we do not consider such a scenario, since it depends on the models of the evolution of the early universe, which must produce an appropriate initial field distribution. |

4. | The domain wall domination does not directly imply a cosmological disaster. In principle, it can happen in the early universe without causing any trouble with the standard cosmology if such domain walls are annihilated before the epoch of Big Bang nucleosynthesis (BBN). For instance, the possibilities of diluting unwanted relics in the domain wall dominated universe are discussed in Refs. [51,52]. However, little is known about the detailed dynamics of domain walls in the domain wall dominated universe, and their behavior in such a scenario is uncertain. Therefore, in this work, we just focus on the case in which the energy density of domain walls never dominates the critical energy density of the universe, and use the condition of the domain wall domination to indicate the potential uncertainties. |

5. | |

6. | This assumption is not rigorous since the collapse of domain walls is not instantaneous, and they may continue to produce GWs until they completely disappear. This ambiguity can be incorporated into the definition of ${T}_{\mathrm{ann}}$ or the uncertainty of the parameter ${C}_{\mathrm{ann}}$ in Equation (28). |

7. | In some exceptional cases, domain walls may be formed even if strings do not exist. For instance, if the initial value of the axion field is tuned to the location which is very close to the top of the cosine potential (7) and its fluctuations are sufficiently large, domain walls without strings can be formed around the time of QCD phase transition. Domain walls without strings can also be formed due to the level crossing between the axion and an ALP [84,85], if there exists an ALP whose mass is comparable to the axion mass around the epoch of QCD phase transition. |

8. | String-wall systems may eventually collapse into a single string bundle, since the vacuum may not be disconnected along the direction of the unbroken U(1). However, such a collapse is not likely to occur if ${N}_{\mathrm{ax}}$ is large and ${\u03f5}_{i}$ is sufficiently small, since in this case strings obey the scaling solution, i.e., the number of strings per horizon remains $\mathcal{O}\left(1\right)$, before the formation of domain walls and the size of the hybrid object, which evolves toward the bundle and contains exponentially large number of strings, is far outside the horizon [101]. |

9. | From Equation (11), we see that the choice ${V}_{0}={\sigma}^{4/3}$ corresponds to ${\delta}^{-1}\sim {\sigma}^{1/3}$, which is satisfied in the toy model given by Equations (1) and (2) if $\lambda \simeq \mathcal{O}\left(1\right)$. We also note that the condition (26) is always satisfied for domain walls in axion models [Equation (7)], since Equations (9), (11), (26), and (27) imply $m\gtrsim {H}_{\mathrm{ann}}$, which holds after the formation of domain walls. |

**Figure 2.**The spectrum of gravitational waves (GWs) [Equation (35)] for five different conformal times $\tau =20$, 30, 40, 50, and 60 obtained in Ref. [46]. All dimensionful quantities are shown in the unit of $v=1$, where v is the vacuum expectation value (VEV) of the scalar field [see Equation (2)].

**Figure 3.**The schematics of the sensitivities of present/future GW experiments and GW signatures from domain walls. Solid lines represent the present upper limits on the GW background obtained by European Pulsar Timing Array (EPTA) (red) and Advanced LIGO O1 (blue). Dashed lines represent the sensitivities of future experiments including SKA (orange), eLISA (green), DECIGO (cyan), Ultimate DECIGO (gray), Advanced LIGO design (blue), and Einstein Telescope (ET) (purple). The sensitivity curves for DECIGO and Ultimate DECIGO contain both the instrumental noise and the white-dwarf (WD) confusion noise. Light colored regions represent typical spectra of GWs from domain walls for ${\sigma}^{1/3}={10}^{5}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ and ${T}_{\mathrm{ann}}=0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ (light red), ${\sigma}^{1/3}={10}^{9}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ and ${T}_{\mathrm{ann}}={10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ (light green), and ${\sigma}^{1/3}={10}^{11}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ and ${T}_{\mathrm{ann}}={10}^{8}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ (light blue).

**Figure 4.**Sensitivities of present/future GW experiments in the parameter space of ${T}_{\mathrm{ann}}$ and ${\sigma}^{1/3}$. In the colored regions, the peak amplitude of GWs from domain walls estimated based on Equation (41) exceeds the sensitivity curves plotted in Figure 3. Gray regions correspond to the parameter space in which domain walls overclose the universe [not satisfying Equation (30)] or they are not formed since the energy bias is too large [Equation (54) with ${V}_{0}={\sigma}^{4/3}$]. The thin black lines represent the contours for the peak amplitude of GWs from domain walls, ${\mathsf{\Omega}}_{\mathrm{gw}}{h}_{\mathrm{peak}}^{2}={10}^{-8}$ (solid), ${10}^{-14}$ (dashed), and ${10}^{-20}$ (dot-dashed). The dashed magenta lines represent the contours for the peak frequency with ${f}_{\mathrm{peak}}={10}^{-8}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$, ${10}^{-5}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$, ${10}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$, and $10\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$.

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Saikawa, K. A Review of Gravitational Waves from Cosmic Domain Walls. *Universe* **2017**, *3*, 40.
https://doi.org/10.3390/universe3020040

**AMA Style**

Saikawa K. A Review of Gravitational Waves from Cosmic Domain Walls. *Universe*. 2017; 3(2):40.
https://doi.org/10.3390/universe3020040

**Chicago/Turabian Style**

Saikawa, Ken’ichi. 2017. "A Review of Gravitational Waves from Cosmic Domain Walls" *Universe* 3, no. 2: 40.
https://doi.org/10.3390/universe3020040