Axion dark matter from topological defects

The cosmological scenario where the Peccei-Quinn symmetry is broken after inflation is investigated. In this scenario, topological defects such as strings and domain walls produce a large number of axions, which contribute to the cold dark matter of the Universe. The previous estimations of the cold dark matter abundance are updated and refined based on the field-theoretic simulations with improved grid sizes. The possible uncertainties originated in the numerical calculations are also discussed. It is found that axions can be responsible for the cold dark matter in the mass range $m_a=(0.9–1.4)\times 10^{-4}\mathrm{eV}$ for the models with the domain wall number $N_{\mathrm{DW}}=1$, and $m_a\approx \mathcal{O}(10^{-4}–10^{-2})\mathrm{eV}$ with a mild tuning of parameters for the models with $N_{\mathrm{DW}}>1$. Such higher mass ranges can be probed in future experimental studies.

Figure 1

The distribution of the potential energy of the scalar field on the simulation box. These figures are generated from the results of 2D simulations with $N=8192$ and $N_{\mathrm{DW}}=6$ [cases (a-15) and (a-16) in Table 2]. Each subfigure corresponds to a different choice of the bias parameter and the time step: (a) $\mathrm{\Xi }=0,\tau =42$, (b) $\mathrm{\Xi }=0.00006,\tau =42$, (c) $\mathrm{\Xi }=0,\tau =62$, (d) $\mathrm{\Xi }=0.00006,\tau =62$, (e) $\mathrm{\Xi }=0,\tau =82$, and (f) $\mathrm{\Xi }=0.00006,\tau =82$. The size of these figures is set to be a quarter ($4096^2$) of the size of the simulation box ($8192^2$). The green region corresponds to the core of domain walls $V(\mathrm{\Phi })=2m_a^2{\eta }^2/N_{\mathrm{DW}}^2$, and the white region corresponds to the vacuum $V(\mathrm{\Phi })=0$.

Figure 2

Time evolution of the comoving area density $A/V$ of domain walls for various values of $\mathrm{\Xi }$. Each panel shows the results for (a) $N_{\mathrm{DW}}=2$, (b) $N_{\mathrm{DW}}=3$, (c) $N_{\mathrm{DW}}=4$, (d) $N_{\mathrm{DW}}=5$, and (e) $N_{\mathrm{DW}}=6$. In these figures, we plot the mean and standard deviation among ten realizations of numerical simulations.

Figure 3

Time evolution of the area parameter $\mathcal{A}$ for various values of $N_{\mathrm{DW}}$ without the bias ($\mathrm{\Xi }=0$). For the simulations with $N=8192$ and $N=16384$, we plot the mean and standard deviation among ten realizations. For those with $N=32768$, we just plot the result of 1 realization.

Figure 4

Estimated values of $C_d$ for (a) the assumption of exact scaling with 10% criterion, (b) the assumption of exact scaling with 1% criterion, (c) the assumption of deviation from scaling with 10% criterion, and (d) the assumption of deviation from scaling with 1% criterion. In the panels (c) and (d), we only show the results with $N=16384$, since the value of $p$ in Table 5 is obtained from the results of the simulations with $N=16384$. In the panel (b), the result of the simulations with $N=8192$ and $N_{\mathrm{DW}}=6$ is not shown, since in this case the value of $A/V$ does not reach 1% of that with $\mathrm{\Xi }=0$ before the final time of the simulations.

Figure 5

Estimated values of the parameters related to the mean momentum of radiated axions for various values of $n_{\text{bin}}$. Each panel shows the results for (a) $\epsilon$ (global strings), (b) ${\epsilon }_w$ (short-lived string-wall systems with $N_{\mathrm{DW}}=1$), and (c) ${\epsilon }_a$ (long-lived string-wall systems with $N_{\mathrm{DW}}=6$). The points indicated as “old” correspond to the results obtained by the method used in the previous studies [20, 22, 23], while those indicated as “new” correspond to the results obtained by the new estimators shown in Eq. (b5). The results shown in the panel (b) are obtained from the simulations with $\kappa =0.3$.

Figure 6

The values of ${\epsilon }_w$ for various choices of $\kappa$ obtained from the calculation with $n_{\text{bin}}=100$, where the average and error are estimated by using Eq. (b5).

Figure 7

Observational constraints on the axion decay constant $F_a$ and the bias parameter $\mathrm{\Xi }$ in the model with $N_{\mathrm{DW}}=3$ based on (a) the assumption of exact scaling and (b) that of deviation from scaling. The red solid line corresponds to the bound obtained from the burst duration of SN 1987A [Eq. (4.10)], and the green area to the left side of this line is excluded. The blue (cyan) solid line corresponds to the constraint of the overclosure of dark matter axions [Eq. (4.1)] with the coefficient $C_d$ estimated based on 10% (1%) criterion. The dotted lines represent uncertainties of ${\mathrm{\Omega }}_{a,\text{tot}}{h}^2$ induced by the numerical parameters $\epsilon$, $\xi$, ${\epsilon }_a$, $\mathcal{A}$ (or ${\mathcal{A}}_{\text{form}}$), and $C_d$. Except for these uncertainties, the red region below the blue (or cyan) line is excluded. The purple solid lines correspond to the NEDM bounds [Eq. (4.9)] for $\delta =1$, $10^{-4}$, and $10^{-8}$. The region above these lines is also excluded. The shaded region corresponds to the parameters satisfying Eq. (4.13), and in this region the axion mass is dominated by the bias term. The exclusion lines shown in these figures are obtained for $g_{*,1}=80$ and ${\mathrm{\Lambda }}_{\mathrm{QCD}}=400\mathrm{MeV}$. Furthermore, we use ${\beta }^{\prime }=58$, $\xi =1.0\pm 0.5$, and $\epsilon =4.02\pm 0.70$ to compute ${\mathrm{\Omega }}_{a,\text{string}}{h}^2$. For parameters required to estimate ${\mathrm{\Omega }}_{a,\text{dec}}{h}^2$, we take ${\epsilon }_a=1.21\pm 0.28$ (the result for $N_{\mathrm{DW}}=3$ in Table 7), $\mathcal{A}=1.10\pm 0.18$ (the result for $N_{\mathrm{DW}}=3$ and $N=16384$ in Table 4), ${\mathcal{A}}_{\text{form}}=0.828\pm 0.032$, and $p=0.926$ (the result for $N_{\mathrm{DW}}=3$ in Table 5). The value for $C_d$ is taken from Table 6, such that $C_d=5.02\pm 0.44$ ($8.15\pm 0.67$) for 10% (1%) criterion with the assumption of exact scaling [panel (a)] and $C_d=7.16\pm 0.53$ ($10.8\pm 0.7$) for 10% (1%) criterion with the assumption of deviation from scaling [panel (b)].

Figure 8

The same figure as Fig. 7 but observational constraints are plotted for the model with $N_{\mathrm{DW}}=6$ based on (a) the assumption of exact scaling and (b) that of deviation from scaling. For parameters required to estimate ${\mathrm{\Omega }}_{a,\text{dec}}{h}^2$, we take ${\epsilon }_a=1.62\pm 0.19$ (the result for $N_{\mathrm{DW}}=6$ in Table 7), $\mathcal{A}=2.23\pm 0.19$ (the result for $N_{\mathrm{DW}}=6$ and $N=16384$ in Table 4), ${\mathcal{A}}_{\text{form}}=1.73\pm 0.06$, and $p=0.932$ (the result for $N_{\mathrm{DW}}=6$ in Table 5). The value for $C_d$ is taken from Table 6, such that $C_d=2.08\pm 0.17$ ($3.14\pm 0.26$) for 10% (1%) criterion with the assumption of exact scaling [panel (a)] and $C_d=2.44\pm 0.20$ ($3.67\pm 0.22$) for 10% (1%) criterion with the assumption of deviation from scaling [panel (b)].

Figure 9

The schematics of the parameter ranges where the axion becomes cold dark matter for the models with $N_{\mathrm{DW}}=1$ (orange interval) and $N_{\mathrm{DW}}=6$ (light blue interval). The width of the orange interval corresponds to the uncertainty of the mass given by Eq. (4.6). The blue intervals represent the allowed region for $\delta =10^{-8}$, $10^{-5}$, and $10^{-3}$, and they also contain the uncertainties appearing in the estimation of ${\mathrm{\Omega }}_{a,\text{tot}}{h}^2$ (see text for details). The vertical dotted line corresponds to the bound from the observation of SN1987A [Eq. (4.10)]. Red and green intervals represent the regions that will be covered by IAXO and ADMX, respectively.