Generic no-scale inflation inspired from string theory compactifications

We propose the generic no-scale inflation inspired from string theory compactifications. We consider the Kähler potentials with an inflaton field $\phi$, as well as one, two, and three Kähler moduli. Also, we consider the renormalizable superpotential of $\phi$ in general. We study the spectral index and tensor-to-scalar ratio in details, and find the viable parameter spaces which are consistent with the Planck and BICEP/Keck experimental data on the cosmic microwave background (CMB). The spectral index is $n_s\simeq 1-2/N\sim 0.965$ for all models, and the tensor-to-scalar ratio $r$ is $r\simeq 12/N^2$, $83/N^4$ and $4/N^2$ for the one, two and three moduli models, respectively. The particular $r$ for two moduli model comes from the contributions of the non-negligible higher order term in potential. In the three moduli model, the scalar potential is similar to the global supersymmetry, but the Kähler potential is different. The E model with $\alpha =1$ and T model with $\alpha =1/3$ can be realized in the one modulus model and the three moduli model, respectively. Interestingly, the models with quadratic and quartic potentials still satisfy the current tight bound on $r$ after embedding into no-scale supergravity.

Figure 1

Inflationary potential (a) and the $r$ versus $n_s$ predictions (b) for the inflaton potential in Eq. (26) with parameter $d=\pm \delta$.

Figure 2

The evolution of the slow-roll parameters (a) $\epsilon$ and (b) $\eta$ for the inflaton potential in Eq. (26).

Figure 3

Inflationary potential (a) and the $r$ versus $n_s$ predictions (b) for the inflaton potential in Eq. (30) with parameter $d<-1$.

Figure 4

The evolution of the slow-roll parameters (a) $\epsilon$ and (b) $\eta$ for the inflaton potential in Eq. (30).

Figure 5

Inflationary potential (a) and the evolution of the slow-roll parameters (b) for the inflaton potential in Eq. (31). The solid and dashed lines correspond to $\epsilon$ and $\eta$, respectively.

Figure 6

Inflationary potential (a) and the $r$ versus $n_s$ predictions (b)–(c) for the inflaton potential in Eq. (23). Also, (b) has $B_1<0$, and (c) has $A_1<0$, $B_1>0$ and $A_1^2>4B_1$.

Figure 7

The evolution of the slow-roll parameters. Solid and dashed lines are corresponding to $ϵ$ and $\eta$, respectively.

Figure 8

The inflaton potential in Eq. (37). (a) The potential remains the same after both parameter $d$ and field $\chi$ become negative. (b) The potential with $d\ge 0$.

Figure 9

The evolution of the slow-roll parameters (a) $\epsilon$ and (b) $\eta$ for the inflaton potential in Eq. (37).

Figure 10

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (37) with parameter $d=\sqrt{27/32}\delta$.

Figure 11

Inflaton potential in Eq. (41).

Figure 12

Inflaton potential in Eq. (43).

Figure 13

The $r$ versus $n_s$ predictions (b),(c) for the inflaton potential (a) in Eq. (45).

Figure 14

Inflaton potential in Eq. (54).

Figure 15

(a) The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (54) with $d\le 1/2$, (b) $n_s$ versus the parameter $d$, and (c) $r$ versus the parameter $d$. The circles and triangles correspond to $d=0$ (T model for ${\phi }^2$) and $d=1/2$, respectively.

Figure 16

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (54) with (a) $1/2<d\le 1$ and (b) $d>1$.

Figure 17

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (54). The circles and triangles correspond to $d=0$ (T model for ${\phi }^2$) and $d=1/2$, respectively.

Figure 18

Inflaton potential in Eq. (60).

Figure 19

The parameter $d$ with respect to inflaton $\chi /M_{\mathrm{Pl}}$.

Figure 20

The evolutions of the slow-roll parameters $ϵ$, $\eta$ for the inflaton potential in Eq. (60).

Figure 21

The CMB predictions for the model in Eq. (60). The circles and triangles are corresponding to the T and the E model, respectively.

Figure 22

Inflaton potential in Eq. (64) (a) and its possible parameter space for $d$ (b).

Figure 23

The evolution of the slow-roll parameters $ϵ$ and $\eta$.

Figure 24

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (64). The circles and triangles correspond to T and E models, respectively.

Figure 25

Inflaton potential in Eq. (47) with $A>0$ (a) and $A<0$ (b).

Figure 26

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (47) with $A_3\ge 2$ and $B_3<0$. (a) In the region $[{\chi }_M,{\chi }_m]$; the circles are in the limit $B_3\to -\infty$. The dashed and solid lines are corresponding to $N=50$ and $N=60$. (b) In the region $[{\chi }_m,\infty ]$.

Figure 27

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (47) with $A_3>2$, $B_3>0$, and $A_3>2B_3$. (a) In the region $[-\infty ,{\chi }_M]$. (b) In the region $[{\chi }_M,\infty ]$.

Figure 28

CMB observations for the model in Eq. (47) with $0\le A_3<2$, $A_3<2B_3$. Inflation occurs on the right side of ${\chi }_m$.

Figure 29

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (47) with $A_3<0$, $B_3>0$, and $A_3^2>4B_3$. Inflation occurs in the regions (a) $[-\infty ,{\chi }_{m1}]$ and (b) $[{\chi }_{m2},\infty ]$.

Figure 30

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (47) with $A_3<0$, $B_3>0$, and $A_3^2<4B_3$. Inflation occurs in the regions (a) $[-\infty ,{\chi }_m]$ and (b) $[{\chi }_m,\infty ]$.

Figure 31

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (47) with $A_3<0$, $B_3<0$, and $A_3<2B_3$. Inflation occurs in the regions (a) $[-\infty ,{\chi }_m]$ and (b) $[{\chi }_m,{\chi }_M]$.

Figure 32

The $r$ versus $n_s$ predictions for the inflaton potential in Eq. (47) with $A_3<0$, $B_3<0$, and $A_3>2B_3$. Inflation occurs in the regions (a) $[-\infty ,{\chi }_m]$ and (b) $[{\chi }_m,\infty ]$. The circles and triangles correspond to T and E models, respectively.