1D Supergravity FLRW Model of Starobinsky

TL;DR

This work embeds Starobinsky $f(R)$ inflation into two one-dimensional supergravity frameworks (N=1 and N=2) using a minimal time-dependent superspace. It derives three equivalent representations of the FLRW Starobinsky model—Ostrogradsky, BF-type scalar-tensor, and standard Weyl-transformed form—revealing the scalaron structure and inflationary dynamics, with $M^2 = \alpha^{-1}$. In the N=1 case, the bosonic sector strictly reproduces $R + \frac{\alpha}{6}R^2$, while the N=2 construction yields an additional massive scalar field, stabilized by a carefully chosen superpotential term that preserves inflation; numerical simulations confirm $R^2$-driven inflation with the extra field in a low-energy state. The paper also develops equivalent tensor-scalar formulations and a canonical Hamiltonian framework, including fermionic contributions, and outlines a path toward quantum cosmology via the Wheeler–DeWitt equation and supersymmetric constraints. Together, these results provide a concrete, supersymmetric, 1D cosmological setting to study inflationary dynamics and pave the way for subsequent quantization and semi-classical analyses.

Abstract

We study two homogeneous supersymmetric extensions for the $f(R)$ modified gravity model of Starobinsky with the FLRW metric. The actions are defined in terms of a superfield $\mathcal{R}$ that contains the FLRW scalar curvature. One model has N=1 local supersymmetry, and its bosonic sector is the Starobinsky action; the other action has N=2, its bosonic sector contains, in additional to Starobinsky, a massive scalar field without self-interaction. As expected, the bosonic sectors of these models are consistent with cosmic inflation, as we show by solving numerically the classical dynamics. Inflation is driven by the $R^2$ term during the large curvature regime. In the N=2 case, the additional scalar field remains in a low energy state during inflation. Further, by means of an additional superfield, we write equivalent tensor-scalar-like actions from which we can give the Hamiltonian formulation.

Figures (2)

• Figure 1: Numerical solution to (\ref{['third']}) with initial values $a=1$, $H=5M$, $\dot{H}=-\frac{1}{6} M^2$, $\ddot{H}=0$ and $M=0.2$. (a) Logarithm of $a(t)$; (b) comoving Hubble length (scale horizon).
• Figure 2: Numerical solutions to Equations (\ref{['scalareq']}). The initial conditions for the scale factor are the same as in Figure \ref{['fig:Mesh2']}. Here, we have an additional scalar field. (a) Scale factor. (b) Comoving Hubble length for pure kinetic initial energy (blue color) and pure potential initial energy (red color) of the field $s$. The dotted line is the same as in the pure Starobinky dynamics of Figure \ref{['fig:Mesh2']}.