Dynamical fractional chaotic inflation

Chaotic inflation based on a simple monomial scalar potential, $V(\varphi )\propto {\varphi }^p$, is an attractive large-field model of inflation capable of generating a sizable tensor-to-scalar ratio $r$. Therefore, assuming that future cosmic microwave background observations will confirm the large $r$ value reported by BICEP2, it is important to determine what kind of dynamical mechanism could possibly endow the inflaton field with such a simple effective potential. In this paper, we answer this question in the context of field theory, i.e. in the framework of dynamical chaotic inflation, where strongly interacting supersymmetric gauge dynamics around the scale of grand unification dynamically generate a fractional power-law potential via the quantum effect of dimensional transmutation. In constructing explicit models, we significantly extend our previous work, as we now consider a large variety of possible underlying gauge dynamics and relax our conditions on the field content of the model. This allows us to realize almost arbitrary rational values for the power $p$ in the inflaton potential. The present paper may hence be regarded as a first step toward a more complete theory of dynamical chaotic inflation.

Figure 1

Power $p$ in the inflaton potential as a function of $N_f$ and $N_d$ for $N_c=3$. Here, the gray-shaded region is excluded by construction, as $N_d=N_f-N_f^{\text{eff}}$ must satisfy $0<N_d\le N_f$, i.e. as $N_f^{\text{eff}}$ must satisfy $0\le N_f^{\text{eff}}<N_f$.

Figure 2

Hierarchy among all the relevant energy scales involved in dynamical chaotic inflation ($H_0$: Hubble rate; ${\mu }_0$: Wilsonian cutoff scale; $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_{\text{eff}}$: fundamental and effective dynamical scale, respectively; $\lambda \tau$: inflaton-dependent heavy quark mass; $M_{\mathrm{Pl}}$: Planck scale; $\tau$: real inflaton field). Here, we have chosen the following exemplary parameter values: $p=2$, $\mathrm{\Lambda }=1.0\times 10^{15}\mathrm{GeV}$, $\lambda =1.2\times 10^{-2}$, $N_e=50$, which also result in the correct value for the scalar spectral amplitude $A_s$; cf. the black dot in Fig. 4. The red dots indicate the inflaton field value $\tau$ as well as the inflaton-dependent heavy quark mass $\lambda \tau$ at the time 15 $e$-folds before the end of inflation.

Figure 3

Schematic shape of the scalar potential for the canonically normalized inflaton field $\tau$. At large field values, $\lambda \tau \gg \mathrm{\Lambda }$, the inflaton slowly rolls in a power-law potential [cf. Eq. (115)], thereby giving rise to a stage of chaotic inflation. As the inflaton field decreases to very small values, $\lambda \tau \ll \mathrm{\Lambda }$, the strongly interacting sector reaches the phase of $s$ confinement and the inflaton begins to oscillate in a quadratic potential [cf. Eq. (127)].

Figure 4

Viable region in the $(\lambda ,\mathrm{\Lambda })$ plane. The green lines indicate where the amplitude of the scalar power spectrum can be successfully reproduced for different values of the power $p$, while the gray contours correspond to different values of the inflaton mass $m_{\varphi }\sim \lambda \mathrm{\Lambda }$. The regions of parameter space where one or more of our theoretical requirements listed in Eqs. (116), (118) and (119) are violated are also correspondingly marked. The black dot marks the position of the parameter point corresponding to the parameter values used in Fig. 2.

Figure 5

Viable ranges for the dynamical scale $\mathrm{\Lambda }$ (left panel) and the inflaton Yukawa coupling $\lambda$ (right panel) according to the bounds in Eqs. (116) and (119) and after imposing the condition that the scalar power spectrum be correctly normalized, $A_s=A_s^{\mathrm{obs}}\simeq 2.21\times 10^{-9}$ [56], for fixed $N_e$ and as functions of the power $p$ [cf. Eq. (140)]. The black dots mark again the position of the parameter point corresponding to the parameter values used in Fig. 2.

Figure 6

Comparison between our predictions for the scalar spectral tilt $n_s$ and the tensor-to-scalar ratio $r$ [cf. Eq. (138)] and the constraints on these two observables according to the PLANCK Collaboration [56] (left panel) and the authors of Ref. [63] (right panel), respectively. Note the linear relation given in Eq. (141).

Figure 7

Predictions of all DCI models constructed in this paper for the power $p$ and the tensor-to-scalar ratio $r$, respectively. For definiteness, we have set $N_e$ to 50. In the case of the $SP(N_c)$ theories, the four different columns respectively refer to (from right to left) $N_c=2$ (black lines), $N_c=3$ (orange lines), $N_c=4$ (green lines), $N_c=5$ (purple lines). As for the $SP(1)$ case, the predictions for $p$ and $r$ are identical to those obtained in the general $SU(N_c)$ case [cf. Eq. (a20)]. The small numbers next to each vertical bar denote the respective numbers of additional massive quark flavors $N_m$ coupling to the inflaton field. Hence, the bars labeled with a red 0 correspond to the models without any additional mass input scales. For these models, we state the predicted power explicitly in each column.