Frame-Dependence of the Hamilton-Jacobi Formalism for Inflation and Reheating in Non-Minimal Gravity

TL;DR

This work investigates how applying the Hamilton-Jacobi formalism to non-minimally coupled inflation yields frame-dependent results when comparing the Jordan and Einstein frames, across both metric and Palatini formalisms. By deriving two effective potentials, $\hat{V}_J$ and $\hat{V}_E$, corresponding to the two computational paths, the authors show that slow-roll and conformal transformations do not commute, leading to systematic differences in $\hat{n}_s$ and $\hat{r}$ that are more pronounced in Palatini gravity. Numerical analysis demonstrates that inflationary and reheating predictions remain compatible with Planck data in all cases, but the frame choice alters the inferred reheating parameters $\hat{N}_{\mathrm{re}}$ and $\hat{T}_{\mathrm{re}}$, with instantaneous reheating generally allowed. The study quantifies the methodological uncertainty introduced by frame-dependent Hamilton-Jacobi implementations, highlighting the need to consider such frame-path dependencies when making robust predictions in modified gravity scenarios.

Abstract

In this work, we investigate the Hamilton-Jacobi formalism for non-minimally coupled inflation, focusing on the methodological frame-dependence arising from its application in the Jordan and Einstein frames. We systematically compare the physical predictions from two distinct computational schemes: applying the Hamilton-Jacobi approximation before versus after the conformal transformation. This comparison is conducted for both the metric and Palatini formalisms. Our results, consistent with Planck data, reveal significant quantitative differences between the two schemes, highlighting a subtle frame-dependence in the approximation method. These discrepancies, observed in the spectral index, the tensor-to-scalar ratio, and reheating parameters, are more pronounced in the Palatini formalism. Our study emphasizes the sensitivity of cosmological predictions to the computational path chosen, and provides a quantitative analysis of this methodological uncertainty, offering valuable insights into the robustness of predictions in modified gravity.

Figures (6)

• Figure 1: Effective potentials $(1 - \alpha)^2 \hat{V}_i/\lambda^2$ ($i=J,E$) corresponding to the two computational schemes, in the metric (a) and Palatini (b) formalisms. Solid (dashed) lines correspond to the Jordan-frame scheme (Einstein-frame scheme). The red, green and purple lines correspond to $\xi=-0.1, -1$ and $-10$ for $n=1$.
• Figure 2: The predicted values of $\hat{r}$ and $\hat{n}_{\mathrm{s}}$ for the non-minimally coupled inflaton field with various $N_*$ are shown in the metric (a) and Palatini (b) cases. The dashed and solid lines correspond to the results derived from the Hamilton-Jacobi method applied in the Einstein and Jordan frames, respectively. The blue, orange, and red lines correspond to $n=1$ with $\xi=-0.1$, $-1$, and $-10$, respectively. The black dot-dashed line represents $n=1$ with $\xi=0$, and the small and large dots indicate $\hat{N}_*=50$ and $70$, respectively. The green and blue shaded regions depict the constraints on $\hat{n}_{\mathrm{s}}$ and $r$ at the pivot scale $k_*=0.002$$\mathrm{Mpc}^{-1}$ from the Planck $2018$ CMB observations akrami2020planck and the BICEP/Keck survey PhysRevLett.127.151301, respectively. These regions represent the the $68\%$ and $95\%$ CL contours, shown with dark and light shading, respectively.
• Figure 3: Reheating e-folding number $\hat{N}_{\mathrm{re}}$ and reheating temperature $\log_{10}[\hat{T}_{\mathrm{re}}/\mathrm{GeV}]$ as functions of $\hat{n}_s$ for $n=1$ with varying $\xi$ in the metric formulation. Solid and dot-dashed lines represent the Jordan-frame scheme and Einstein-frame scheme results, respectively. The purple, blue, green, orange and red curves correspond to $w_{\mathrm{re}}= -1/3, 0, 1/3, 2/3$ and $1$, respectively. The brown band indicates the $1\sigma$ bound on $\hat{n}_s$ from Planck $2018$ TT, TE, EE+lowE+lensing akrami2020planck. Temperatures below $T_\mathrm{re} < 4$MeV shown as the green region are ruled out by the BBN PhysRevLett.82.4168PhysRevD.62.023506PhysRevD.70.043506Hasegawa_2019.
• Figure 4: Predicted $\hat{N}_*$ versus $\hat{n}_s$ for $n=1$ with varying $\xi$ in the metric formalism. Solid and dashed curves denote Jordan- and Einstein-frame scheme results, respectively. The brown region indicates the $1\sigma$ bound on $\hat{n}_s$ from the Planck $2018$ TT, TE, EE+lowE+lensing akrami2020planck.
• Figure 5: Reheating predictions in the Palatini case for $n=1$: $\hat{N}_{\mathrm{re}}$ and $\log_{10}[\hat{T}_{\mathrm{re}}/\mathrm{GeV}]$ as functions of $\hat{n}_s$ for varying $\xi$. Line styles and shaded regions follow the conventions of Fig. \ref{['fig3']}.