Natural inflation in supergravity and beyond

Supergravity models of natural inflation and its generalizations are presented. These models are special examples of the class of supergravity models proposed in [R. Kallosh and A. Linde, J. Cosmol. Astropart. Phys. 11 (2010) 011; R. Kallosh, A. Linde, and T. Rube, Phys. Rev. D 83, 043507 (2011)], which have a shift symmetric Kähler potential, superpotential linear in goldstino, and stable Minkowski vacua. We present a class of supergravity models with arbitrary potentials modulated by sinusoidal oscillations, similar to the potentials associated with axion monodromy models. We show that one can implement natural inflation in supergravity even in the models of a single axion field with axion parameters $O(1)$. We also discuss the irrational axion landscape in supergravity, which describes a potential with an infinite number of stable Minkowski and metastable dS minima.

Figure 1

Potential of the natural inflation model (7) in supergravity at $S=0$ and $T=\frac{\beta +i\varphi }{\sqrt{2}}$. During inflation $\beta =0$ at its minimum and $\varphi$ is the inflaton field with the potential (1). This plot is made for $a=0.1$. All fields are given in Planck units, and the potential is in units ${\mathrm{\Lambda }}^4$.

Figure 2

Irrational axion potential for $A=B=1$, $a=0.01\sqrt{3}$, $b=0.005\sqrt{7}$. The field is shown in Planck units, from 0 to 10,000. This may create an impression that the potential is very steep, but in fact the potential is very flat and allows chaotic inflation. Just as in the string landscape scenario [13, 14, 15, 16, 17, 18], inflation may end in any of the infinitely many metastable dS vacua with different values of the cosmological constant [12].

Figure 3

Inflationary potential for natural inflation in the theory of two scalar fields, $T=\frac{\beta +i\varphi }{\sqrt{2}}$ and $U=\frac{\gamma +i\chi }{\sqrt{2}}$, as a function of the fields $\varphi$ and $\chi$. The potential shown in units of ${\mathrm{\Lambda }}^4$, with all other parameters of the superpotential $O(1)$. Nevertheless, flat inflationary valleys are formed for $|a-b|\ll 1$ or $|c-d|\ll 1$.

Figure 4

Potential of the modulated chaotic inflation in supergravity (30) at $S=0$ and $T=\frac{\beta +i\varphi }{\sqrt{2}}$, for $a=1$, $b=1.3$. It is similar to the potentials encountered in axion monodromy models [9, 10]. Fields are shown in Planck mass units, the scale of the potential is in units ${\mathrm{\Lambda }}^4$.