General properties and kinetics of spontaneous baryogenesis

General features of spontaneous baryogenesis are studied. The relation between the time derivative of the (pseudo) Goldstone field and the baryonic chemical potential is revisited. It is shown that this relation essentially depends upon the representation chosen for the fermionic fields with nonzero baryonic number (quarks). The calculations of the cosmological baryon asymmetry are based on the kinetic equation generalized to the case of nonstationary background. The effects of the finite interval of the integration over time are also taken into consideration. All these effects combined lead to a noticeable deviation of the magnitude of the baryon asymmetry from the canonical results.

Figure 1

Left: Evolution of ${\xi }_B(\eta )$ according to Eq. (5.8) where $C_B{T}_{\mathrm{in}}^2/(5f^2)=0.1$ for $\mathrm{K}=\mathrm{N}=20$ (thick), 5 (dashed), 1 (dotted), and 0.3 (dot dashed). Right: Ratio of ${\xi }_B(\eta )$ to its equilibrium value, ${\xi }_B^{\mathrm{eq}}(\eta )$, determined by (5.9) for the same values of the parameters.

Figure 2

Left: Evolution of $\dot{\theta }(\eta )$, normalized to its initial value, ${\dot{\theta }}_{\mathrm{in}}$, for $C_B{T}_{\mathrm{in}}^2/(5f^2)=0.1$ and $\mathrm{K}=\mathrm{N}=20$ (thick), 5 (dashed), 1 (dotted), and 0.3 (dot dashed). Right: The same with $C_B{T}_{\mathrm{in}}^2/(5f^2)=1$ and $\mathrm{K}=\mathrm{N}=50$ (thick), 30 (dashed), 10 (dotted), and 3 (dot dashed).

Figure 3

Left: The rhs of Eq. (7.13), denoted by $\mathrm{\Xi }$, as a function of $\dot{\theta }/T$ for the cutoff of the time integration in Eq. (8.1): $\tau \equiv t_{\max }T=30$ (dashed), 10 (dotted), 3 (dot dashed), and $\dot{\theta }/T$ (thick). Right: The relative difference: $\zeta =\mathrm{\Xi }/(\dot{\theta }/T)-1$, as a function of $\dot{\theta }/T$ for $\tau =30$ (thick), 10 (dashed), 5 (dotted), and 3 (dot dashed).

Figure 4

Left: The relative difference, $\zeta =\mathrm{\Xi }/(\dot{\theta }/T)-1$, as a function of $\dot{\theta }/T$ for the cutoff of the time integration $\tau =30$ (thick), 10 (dashed), 5 (dotted), and 3 (dot dashed) for fixed $\ddot{\theta }/T^2=0.1$. Right: The same difference as a function of $\ddot{\theta }/T^2$ for fixed time of integration $\tau =10$ and different $\dot{\theta }/T=0.1$ (thick), 0.2 (dashed), 0.3 (dotted), and 0.4 (dot dashed).

Figure 5

Exact (thick) and approximate (dashed) expressions for the amplitude $A$ (7.3) with $\theta (t)$ determined by Eq. (8.6) as functions of $mt$ for ${\theta }_0=1$ (left) and ${\theta }_0=0.3$ (right).

Figure 6

Left: Thick curve, the rhs of Eq. (7.13), $\mathrm{\Xi }$, for $m/T=0.1$ and the maximal time of interation $\tau =30$; dashed curve, $\dot{\theta }/T$ as functions of ${\theta }_0$; see Eq. (8.5). Right: The same as the maximal time of integration $\tau =3$.