Inflation connected to the origin of $CP$ violation

We consider a simple extension of the standard model, which could give a solution for its $CP$ issues such as the origin of both Cabibbo-Kobayashi-Maskawa (CKM) and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) phases and the strong $CP$ problem. The model is extended with singlet scalars which allow the introduction of the Peccei-Quinn symmetry and could cause spontaneous $CP$ violation to result in these phases at low energy regions. The singlet scalars could give a good inflaton candidate if they have a suitable nonminimal coupling with the Ricci scalar. $CP$ issues and inflation could be closely related through these singlet scalars in a natural way. In a case where the inflaton is a mixture of the singlet scalars, we study reheating and leptogenesis as notable phenomena affected by the fields introduced in this extension.

Figure 1

Left: contours of $\mathrm{\Omega }{h}^2=0.12$ are plotted in the $({\lambda }_{+},{\tilde{\lambda }}_3)$ plane by a solid line for $M_{{\eta }_R}=0.9$ (green), 1 (red), and 1.1 (blue) in a TeV unit. Since ${\eta }_R$ should be lighter than the charged components, ${\lambda }_4<0$ should be satisfied, which corresponds to a region above the diagonal black solid line. Direct search bound from Xenon1T [34] is shown by the same color dashed line for each $M_{{\eta }_R}$. Stability conditions (17) restrict the allowed region to the area marked in the upper right quadrant (marked off by the dot-dashed lines), for which ${\tilde{\lambda }}_2=0.1$ is assumed. Right: an example of the running of coupling constants ${\tilde{\lambda }}_i$ for $M=10^{\mu }\mathrm{GeV}$. Initial values of ${\tilde{\lambda }}_3$ and ${\lambda }_4$ are fixed as ${\tilde{\lambda }}_3=0.445$ and ${\lambda }_4=-0.545$, which are included in the allowed region of the left panel for $M_{{\eta }_R}=1\mathrm{TeV}$. ${\tilde{\lambda }}_3+2\sqrt{{\tilde{\lambda }}_1{\tilde{\lambda }}_2}$ and ${\lambda }_{+}+2\sqrt{{\tilde{\lambda }}_1{\tilde{\lambda }}_2}$ are also plotted by red and blue lines, respectively.

Figure 2

Potential of the inflaton $\chi$ for ${\tilde{\xi }}_S=1$ (left panel) and ${\tilde{\xi }}_S=10$ (right panel). In both panels, ${\tilde{\xi }}_S/{\xi }_{\sigma }=20$ and ${\tilde{\kappa }}_S/|{\kappa }_{\sigma S}|=|{\kappa }_{\sigma S}|/{\tilde{\kappa }}_{\sigma }=0.1$ are assumed and ${\tilde{\kappa }}_S$ is fixed by using Eq. (39) for ${\mathcal{N}}_k=55$. As references, we also plot approximated potential $\frac{{\widehat{\kappa }}_S}{6{\tilde{\xi }}_S^2}{M}_{\mathrm{pl}}^2{\chi }^2$ and $\frac{{\widehat{\kappa }}_S}{4}{\chi }^4$ in Eq. (35) as ${\chi }^2$ and ${\chi }^4$. In these plots, a Planck unit $(M_{\mathrm{pl}}=1)$ is used.

Figure 3

Predicted values of the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ in this model. They are read off as the values at intersection points of two lines with a fixed value of ${\tilde{\xi }}_S$ or ${\mathcal{N}}_k$. The coupling constant ${\widehat{\kappa }}_S$ is varied in a range from ${10}^{-7}$ to ${10}^{-10}$.

Figure 4

Left: reheating temperature $T_R$ (GeV) predicted for both the preheating (a) and the perturbative process (b). They are plotted as a function of $g_{\varphi }$. In the case (a), the decay $\varphi \to \overline{q}t$ is assumed. Right: contribution to the reheating temperature due to a component $\sigma$ of the inflaton in a case ${\kappa }_{\varphi S}=0$. The reheating temperature $T_R$ (GeV) is plotted as a function of ${\kappa }_{\varphi \sigma }$. Each line represents $T_R$ for several values of $({\tilde{\kappa }}_S,|{\kappa }_{\sigma S}|/{\tilde{\kappa }}_{\sigma })$, where ${\tilde{\kappa }}_{\sigma }$ is fixed at ${10}^{-4.5}$ and ${10}^{-5.3}$ for ${\tilde{\kappa }}_S={10}^{-8}$ and ${10}^{-9}$, respectively.

Figure 5

Evolution of $Y_{N_1}$ and $Y_L$ obtained as solutions of Boltzmann equations. Results of the case (I) is shown in the left panel for ${\kappa }_{\varphi \sigma }={10}^{-4}$ and ${\kappa }_{\varphi S}={\kappa }_{\eta S}=0$. Results of the case (II) is shown in the right panel for ${\kappa }_{\varphi S}={\kappa }_{\eta S}={10}^{-6}$ and ${\kappa }_{\varphi \sigma }={\kappa }_{\eta \sigma }=0$. Other parameters in each case are given in the text. Initial values for them are fixed as $Y_{N_1}=Y_L=0$ at $z=z_R$. ${\rho }_{N_1}/{\rho }_{\mathrm{R}}$ represents a ratio of the energy density of $N_1$ to the one of radiation.

Figure 6

Left: baryon number asymmetry generated at the expected reheating temperature $T_R$ (GeV) in the case (I). It is plotted by a red solid line for $y_D={10}^{-1.2}$ and by green crosses for $y_D={10}^{-1.5}$. Right: baryon number asymmetry generated at $T_R={10}^9\mathrm{GeV}$ for various values of ${\tilde{\lambda }}_5$ and $y_{N_1}$ in the case (I). In both panels, other parameters are fixed at the ones given in (69).