Cosmological $\mathcal{C}\mathcal{P}$-odd axion field as the coherent Berry’s phase of the Universe

We consider a dark matter model offering a very natural explanation of the observed relation, ${\mathrm{\Omega }}_{\mathrm{dark}}\sim {\mathrm{\Omega }}_{\text{visible}}$. This generic consequence of the model is a result of the common origin of both types of matter (dark and visible) which are formed during the same QCD transition. The masses of both types of matter in this framework are proportional to one and the same dimensional parameter of the system, ${\mathrm{\Lambda }}_{\mathrm{QCD}}$. The focus of the present work is the detailed study of the dynamics of the $\mathcal{C}\mathcal{P}$-odd coherent axion field $a(x)$ just before the QCD transition. We argue that the baryon charge separation effect on the largest possible scales inevitably occurs as a result of merely the existence of the coherent axion field in the early Universe. It leads to preferential formation of one species of nuggets on the scales of the visible Universe where the axion field $a(x)$ is coherent. A natural outcome of this preferential evolution is that only one type of the visible baryons remains in the system after the nuggets complete their formation. This represents a specific mechanism of how the baryon charge separation mechanism (when the Universe is neutral, but the visible part of matter consists of the baryons only) replaces the conventional “baryogenesis” scenario.

Figure 1

The conjectured phase diagram. The plot is taken from Ref. [4]. Possible cooling paths are denoted as path 1, 2, or 3. The phase diagram is in fact much more complicated as the dependence on the third essential parameter; the $\theta$ is not shown as it is largely unknown. It is assumed that the final destination of the nuggets is the CS region with $T_{\mathrm{form}}\approx 41\mathrm{MeV}$, $\mu >{\mu }_c$, and $\theta \approx 0$.

Figure 2

Numerical solutions of (anti)nuggets evolving in the background of axion field before QCD transition. The blue and orange lines represent the evolutions of $R^{-}(s)$ and $R^{+}(s)$, respectively. We plot the upper four subfigures in (a) with $s_c={10}^{-2}$ and the lower four subfigures in (b) with $s_c={10}^{-5}$. The numerical values of parameters $m_a$ and $a_c$ that we use in calculating each subfigure can be seen in the upper edge and right edge of the graph.

Figure 3

Dependence on viscosity $\eta$. Amplitudes of $R^{-}$ (blue) and $R^{+}$ (orange) are plotted. The solid lines correspond to $\eta =8.4m_{\pi }^3(\times {10}^9)$, and the dashed lines correspond to $\eta =1.0m_{\pi }^3(\times {10}^9)$. Here, parameters $m_a={10}^{-4}\mathrm{eV}$ and $s_c={10}^{-5}$ are chosen.

Figure 4

This plot shows that ${\omega }_R/{\omega }_{\theta }$ is always much larger than unity. This behavior justifies our adiabatic approximation in numerical analysis when the axion mass $m_a(T)$ and the axion field $\theta (T)$ are kept constant. This plot essentially shows that the nuggets make a very large number of oscillations, while the axion field $\theta (T)$ slowly varies.

Figure 5

The first few oscillations of $R^{+}$ in the lower left subfigure of Fig. 2. We choose this as an example to show that there is no cuspy problem.