Reheating in geometric Weyl-invariant Einstein-Cartan gravity

TL;DR

The paper investigates a Weyl-invariant gravity model in the Einstein–Cartan framework that reduces to Einstein gravity with a single axion-like inflaton. The parity-violating Holst term, encoded in the parameter $\theta = \epsilon/(2\gamma)$, shapes the inflaton potential and yields an inflationary plateau that, for large $\theta$, reproduces Starobinsky-like predictions; in the absence of the parity-odd term, the potential is exponential and incompatible with observations. Reheating is treated in a model-independent way using $N_{\rm reh}$, the equation-of-state parameter $w$, and the reheating temperature $T_{\rm reh}$, and is shown to significantly modify the mapping from horizon exit to observables $n_s$ and $r$. In the Starobinsky-like regime, data prefer $w>1/3$ with $T_{\rm reh}$ ranging from near BBN scales up to about $10^{13}$ GeV, while smaller values of $\theta$ allow a broader set of reheating histories. The work emphasizes that reheating must be incorporated when confronting gravity-based inflationary models with data and highlights the role of future CMB polarization experiments in distinguishing between scenarios.

Abstract

We study Weyl-invariant purely gravitational theories formulated within the Einstein-Cartan framework. In the Einstein-frame description, these models are dynamically equivalent to standard general relativity coupled to an axion-like pseudoscalar degree of freedom, which naturally drives a period of cosmic inflation. Without committing to a specific microscopic mechanism for reheating, we demonstrate that the post-inflationary reheating dynamics play a crucial role in shaping the inflationary predictions. In particular, we show that assumptions about the reheating temperature and the equation-of-state parameter can significantly affect the predicted values of inflationary observables, highlighting the necessity of consistently incorporating reheating effects in the phenomenological analysis of inflationary models.

Figures (4)

• Figure 1: Tensor-to-scalar ratio $r$ versus scalar spectral index $n_s$ (upper panel) and running of the scalar spectral index ${\rm d}n_s/{\rm d}\ln k$ versus $n_s$ (lower panel), evaluated at the pivot scale $k_\star = 0.05\,{\rm Mpc}^{-1}$. The $1\sigma$ and $2\sigma$ confidence regions obtained from the latest combination of Planck, BICEP/Keck, and BAO data Planck:2018jriBICEP:2021xfz and from Planck, ACT, and DESI data AtacamaCosmologyTelescope:2025nti are shown in gray and purple, respectively. The gray solid curve corresponds to the predictions of the model assuming instantaneous reheating, with the parameter $\theta$ varying in the range $\theta \in [15,3000]$. The dark blue dashed and light blue dashed curves show the predictions for a non-instantaneous reheating scenario with equation-of-state parameters $w=1$ and $w=-1/3$, respectively, for representative values of the parameter $\theta = 15, 25, 3000$. Colored dots indicate representative reheating temperatures associated with BBN, electroweak, and leptogenesis scales, as well as the maximum temperature obtained in the instantaneous reheating limit. For comparison, the black solid curve shows the predictions of the Starobinsky model Starobinsky:1980te.
• Figure 2: The normalized product $\theta^2 V_0$ as a function of $\theta$. In the large-$\theta$ regime, $\theta^2 V_0$ asymptotes to a constant, signalling the recovery of the Starobinsky limit (see Eq. \ref{['eq:pot_staro']}).
• Figure 3: Reheating temperature $T_{\rm reh}$ as a function of the scalar spectral index $n_s$ for $\theta = 3000$ (left), $\theta = 25$ (middle), and $\theta = 15$ (right). Dashed curves correspond to non-instantaneous reheating scenarios with equation-of-state parameters $w = -1/3, 0, 1/4$, and $1$ (color coded as shown in the figure), while the instantaneous reheating case ($w = 1/3$) is represented by the solid vertical line. All curves intersect at the point where reheating occurs instantaneously. Gray horizontal lines indicate the characteristic energy scales discussed in Fig. \ref{['Fig:fig1']}. The observational confidence regions are the same as those shown in Fig. \ref{['Fig:fig1']}.
• Figure 4: Reheating temperature $T_{\rm reh}$ as a function of the number of $e$-folds $N_\star$ (left) and $N_{\rm reh}$ (right), for $\theta = 25$ and equation-of-state parameters $w = -1/3, 0, 1/4, 1/3$, and $1$, as in Fig. \ref{['Fig:fig2']}.