Curvaton and QCD axion in supersymmetric theories

A pseudo-Nambu-Goldstone boson as curvaton avoids the $\eta$ problem of inflation which plagues most curvaton candidates. We point out that a concrete realization of the curvaton mechanism with a pseudo-Nambu-Goldstone boson can be found in the supersymmetric Peccei-Quinn mechanism resolving the strong CP problem. In the flaton models of Peccei-Quinn symmetry breaking, the angular degree of freedom associated with the QCD axion can naturally be a flat direction during inflation and provides successful curvature perturbations. In this scheme, the preferred values of the axion scale and the Hubble parameter during inflation turn out to be about ${10}^{10}$ and ${10}^{12}\mathrm{G}\mathrm{e}\mathrm{V}$, respectively. Moreover, it is found that a significant isocurvature component, (anti)correlated to the overall curvature perturbation, can be generated, which is a smoking gun for the curvaton scenario. Finally, non-Gaussianity in the perturbation spectrum at a potentially observable level is also possible.

Figure 1

Log-log plots of $H_{*}/M_P$ with respect to $r$ for all cases: $n=1,2,3$. The dashed line corresponds to the bound $H_{*}/M_P<{10}^{-5}$. The upper graph corresponds to ${\theta }_0\sim 1$, whereas the lower graph corresponds to ${\theta }_0\sim 0.1$. We see that, in the former case, only the case of $n=1$ manages to avoid the bound. On the other hand, in the latter case, all three possibilities are, in principle, allowed. The curves meet when $H_{*}\sim M_P$ as expected.

Figure 2

Log-log plot of ${\theta }_0$ with respect to $r$ for all cases $n=1,2,3$. The solid lines correspond to $H_{\overline{m}}\sim 1\mathrm{G}\mathrm{e}\mathrm{V}$, whereas the dotted lines correspond to $H_{\overline{m}}\sim m_{3/2}$. The horizontal dashed line corresponds to ${\theta }_0\sim 0.1$. We see that, for $H_{\overline{m}}>1\mathrm{G}\mathrm{e}\mathrm{V}$, the value of $r$ is slightly larger, for a given ${\theta }_0$ (it can increase up to an order of magnitude at most). Obviously, ${\theta }_0$ is bounded from above as $\log {\theta }_0<\log \pi \approx 0.5$.

Figure 3

Log-log plots of $H_{*}/M_P$ with respect to ${\theta }_0$ for all cases: $n=1,2,3$ with $H_{\overline{m}}\sim 1\mathrm{G}\mathrm{e}\mathrm{V}$. The dashed line corresponds to the bound $H_{*}/M_P<{10}^{-5}$ (which ensures the compatibility with the observations), while the dotted line corresponds to the bound $H_{*}/M_P<{10}^{-6}$ (which ensures that the inflaton does not contribute significantly to the curvature perturbations). We see that, when ${\theta }_0\sim 1$, only the case of $n=1$ is possible with $H_{*}/M_P\gtrsim {10}^{-6}$, which means that the inflaton generated curvature perturbations are not negligible. For the curvaton to be the sole source of the curvature perturbations, one has to limit oneself to the $n=2,3$ cases, where ${\theta }_0\ll 1$.