Effects of Radiative Corrections on Starobinsky Inflation

Abstract

We analyze radiative corrections to the Starobinsky model of inflation arising from self-interactions of the inflaton, and from its Yukawa couplings, $y$, to matter fermions, and dimensionful trilinear couplings, $κ$, to scalar fields, which could be responsible for reheating the Universe after inflation. The inflaton self-interactions are found to be of higher order in the Hubble expansion rate during inflation, and hence unimportant for CMB observations. In contrast, matter couplings to the Starobinsky inflaton can have significant effects on the spectral index of scalar CMB perturbations, $n_s$, and on the tensor-to-scalar ratio, $r$. Using a renormalization-group improved analysis of the effective inflationary potential, we find that the Planck measurement of $n_s$ constrains the inflaton coupling to light fermions in the Einstein frame: $y < 4.5 \times 10^{-4}$, corresponding to an upper limit on the reheating temperature $T_{\rm RH} < 2 \times 10^{11}~{\rm GeV}$, whereas the ACT DR6 measurement of $n_s$ corresponds to $3.8 \times 10^{-4} < y < 5.6 \times 10^{-4}$ and $1.7 \times 10^{11} ~{\rm GeV} < T_{\rm RH} < 2.8 \times 10^{11}~{\rm GeV}$, while the upper limits on $r$ provide weaker constraints. Planck data also imply a constraint on a trilinear inflaton coupling to light scalars in the Einstein frame: $κ\leq 4 \times 10^{12}~{\rm GeV}$, corresponding to $T_{\rm RH} \leq 4.2 \times 10^{13}~{\rm GeV}$. We further present constraints on inflaton couplings to massive fermions and scalars, and analyze constraints on couplings in the Jordan frame.

Figures (14)

• Figure 1: The RG-improved effective potential as a function of $\phi$ for the indicated values of $y_0$, compared to the tree-level potential.
• Figure 2: The inflaton mass $m$ as a function of $y_0$, as obtained from the tree-level potential \ref{['treestaro']} (black solid line) and from the one-loop corrected potential \ref{['RGIV']} (blue dashed line).
• Figure 3: The inflaton field values $\phi_*$ and $\phi_{\text{end}}$ as functions of $y_0$, from the tree-level potential \ref{['treestaro']} (solid lines) and from the one-loop corrected potential \ref{['RGIV']} (dashed lines).
• Figure 4: The spectral index $n_s$ from the one-loop corrected potential as a function of the Yukawa coupling $y_0$. The gray shaded region is the constraint on $n_s$ from Planck with $\pm 2\sigma$ uncertainty, while the yellow band is the ACT constraint.
• Figure 5: The number of $e$-folds $N_*$ from the one-loop corrected potential as a function of the Yukawa coupling $y_0$.