Non-Canonical Inflation in Supergravity

TL;DR

We address how non-canonical kinetic terms in $N=1$ supergravity affect inflation. By adding the leading higher-derivative correction with coefficient $T$, we show that the auxiliary field $F$ obeys a cubic equation and induces large corrections to the inflaton potential that can dominate over naive kinetic corrections. The analysis, including explicit examples and the $T\, ext{ge} ext{0}$ and $T<0$ regimes, reveals that the potential shaping inflation can be dramatically altered and that a three-branch structure arises for $T<0$ with a singularity in the speed of sound on the physically connected branch. The results underscore that EFTs of inflation in string-inspired supergravity must incorporate full supergravity dynamics of higher-derivative terms, as they can substantially modify inflationary dynamics and observational predictions.

Abstract

We investigate the effect of non-canonical kinetic terms on inflation in supergravity. We find that the biggest impact of such higher-derivative kinetic terms is due to the corrections to the potential that they induce via their effect on the auxiliary fields, which now have a cubic equation of motion. This is in contrast to the usual (non-supersymmetric) effective field theory expansion which assumes that mass-suppressed higher-derivative terms do not affect the lower-derivative terms already present. We demonstrate with several examples that such changes in the potential can significantly modify the inflationary dynamics. Our results have immediate implications for effective descriptions of inflation derived from string theory, where higher-derivative kinetic terms are generally present. In addition we elucidate the structure of the theory in the parameter range where there are three real solutions to the auxiliary field's equation of motion, studying the resulting three branches of the theory, and finding that one of them suffers from a singularity in the speed of propagation of fluctuations.

Figures (16)

• Figure 1: The auxiliary field $F$ as a function of $X$ and $T$, for $T>0$ and with $C_1 = C_2 = 0.1$. We see that $F$ is a negative function, which, starting from its standard $T=0$ solution (here arbitrarily set at the value $F=-1$), approaches zero asymptotically as $X$ and $T$ increase.
• Figure 2: The blue line indicates the inflationary attractor (\ref{['infsolution']}) for $P_{Fl}(X, \phi)$ with $V_0=10, \, T=10^{-3}$ (left). The line is solid until $\eta_X$ and $\eta_\Pi$ become greater than $0.5,$ and dashed from then on. The figure on the right shows the behavior of the generalized slow-roll parameters, evaluated along the blue solid/dashed line (\ref{['infsolution']}).
• Figure 3: Trajectories in $(\Pi, \phi)$ phase space for $c=3$ and $T = 0.01$ with $P_{CanK}(X, \phi)$ (left) and $P_{QuadK}(X, \phi)$ (right). The blue line is the slow-roll approximation, which is a solid line until one of the generalized slow-roll trajectories reaches $1$.
• Figure 4: Trajectories in $(\Pi, \phi)$ phase space for $P_{Fl}(X, \phi)$ with $V_0=10$ and $T = 0.0001$.
• Figure 5: Trajectories in $(\Pi, \phi)$ (left) and $(\dot \phi, \phi)$ (right) phase space for $P_{QuadK}(X, \phi)$ with $c = 3$ and $T = 0.01$. The same trajectories for $P_{QuadK}(X, \phi)$ with non-canonical kinetic terms switched off are in blue.