Reheating in geometric Weyl-invariant Einstein-Cartan gravity

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.

• The paper explores inflation and reheating in a Weyl-invariant Einstein–Cartan gravity model, finding that the unique Starobinsky-like potential supports viability with CMB constraints.

Overview

This paper examines the inflationary phenomenology of a purely gravitational, locally Weyl-invariant theory formulated in the Einstein–Cartan framework, with particular emphasis on how post-inflationary reheating reshapes its observational predictions. The action consists of the two independent curvature scalars available in Einstein–Cartan geometry — the Ricci scalar $\mathcal{R}$ and the parity-odd Holst invariant $\tilde{\mathcal{R}}$ — combined into quadratic invariants $\gamma\mathcal{R}^2$, $\delta\tilde{\mathcal{R}}^2$, and $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$ (2602.00317). Weyl invariance forbids dimensionful couplings and linear curvature terms, so this operator content is uniquely selected within the purely gravitational sector.

The central result is that, after integrating out torsion (whose equations of motion are purely algebraic) and fixing the Weyl gauge $\chi = M_P^2/\gamma$, the theory is on-shell equivalent to Einstein gravity minimally coupled to a single canonical axion-like pseudoscalar $\phi$ with potential

$$V(\phi) = V_0 \left(4\theta + \sinh\!\left[\sqrt{\tfrac{2}{3}}\frac{\phi}{M_P} - {\rm arcsinh}(4\theta)\right]\right)^2,$$

where $\theta = \epsilon/(2\gamma)$ and $V_0 = M_P^4/(16(\delta - 4\theta^2))$. The same scalar degree of freedom originates from the Holst term and serves as the inflaton.

Role of parity violation

The parity-violating mixed term $\tilde{\mathcal{R}}$0 is essential for viable inflation. For $\tilde{\mathcal{R}}$1 the potential reduces to a pure exponential, which cannot sustain slow roll consistent with observations. For nonzero $\tilde{\mathcal{R}}$2, an extended plateau develops between the origin and the asymptotic exponential region; for $\tilde{\mathcal{R}}$3 the plateau closely reproduces the Starobinsky form $\tilde{\mathcal{R}}$4, and the model's predictions converge to those of Starobinsky inflation. Notably, the identical potential arises in non-Weyl-invariant parity-odd gravitational actions of the form $\tilde{\mathcal{R}}$5 and in certain limits of $\tilde{\mathcal{R}}$6-attractor-type scenarios, indicating a degree of universality across these constructions.

Inflationary observables are computed numerically from the Hubble-flow parameters rather than the potential slow-roll approximation, since analytic expressions are not tractable for this potential. One methodological point deserves emphasis: the author finds that the condition $\tilde{\mathcal{R}}$7 accurately locates the end of inflation as defined by $\tilde{\mathcal{R}}$8, which contradicts the conclusion reported in the earlier analysis of Karananas et al., where $\tilde{\mathcal{R}}$9 was used. This discrepancy directly affects the estimate of $\gamma\mathcal{R}^2$0 and hence the instantaneous reheating temperature bound.

Reheating analysis

Reheating is treated model-independently through three phenomenological parameters: the number of reheating e-folds $\gamma\mathcal{R}^2$1, an averaged equation-of-state parameter $\gamma\mathcal{R}^2$2, and the reheating temperature $\gamma\mathcal{R}^2$3, bounded above by the instantaneous value $\gamma\mathcal{R}^2$4 and below by the BBN scale of order MeV. These parameters enter the standard relation determining $\gamma\mathcal{R}^2$5, the number of e-folds between pivot-scale horizon exit and the end of inflation.

The key findings are:

Regime	Favored reheating properties
Large $\gamma\mathcal{R}^2$6 ($\gamma\mathcal{R}^2$7, Starobinsky limit)	Stiff equation of state $\gamma\mathcal{R}^2$8; low $\gamma\mathcal{R}^2$9 near BBN bound
Small $\delta\tilde{\mathcal{R}}^2$0 ($\delta\tilde{\mathcal{R}}^2$1)	Soft equation of state $\delta\tilde{\mathcal{R}}^2$2; reduced $\delta\tilde{\mathcal{R}}^2$3
Intermediate $\delta\tilde{\mathcal{R}}^2$4 ($\delta\tilde{\mathcal{R}}^2$5)	Weak sensitivity; broad allowed range of $\delta\tilde{\mathcal{R}}^2$6 and $\delta\tilde{\mathcal{R}}^2$7

Quantitatively, in the Starobinsky limit with $\delta\tilde{\mathcal{R}}^2$8, consistency with Planck/BICEP/Keck/BAO and Planck/ACT/DESI data permits $\delta\tilde{\mathcal{R}}^2$9. For $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$0, agreement requires either canonical reheating ($\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$1) with $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$2 or $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$3 with $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$4. In the large-$\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$5 limit the instantaneous reheating temperature saturates at $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$6, matching the Starobinsky expectation, since $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$7 there so that $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$8 asymptotes to a constant.

A significant implication concerns the ACT results: the original Starobinsky scenario is marginally incompatible with the latest Planck+ACT+DESI constraints, but here compatibility can be restored by stiff reheating with low temperatures — mirroring recent dedicated studies of Starobinsky reheating. Conversely, for small $\epsilon\,\mathcal{R}\tilde{\mathcal{R}}$9, soft equations of state improve agreement by lowering $\chi = M_P^2/\gamma$0. Thus the sign of the reheating-induced shift in $\chi = M_P^2/\gamma$1 reverses across the parameter space, meaning that reheating assumptions do not merely perturb but qualitatively reorganize which regions of $\chi = M_P^2/\gamma$2 are observationally viable.

Limitations and open questions

Several caveats are explicit in the paper. First, the analysis deliberately omits nonminimal couplings such as $\chi = M_P^2/\gamma$3 and torsion-squared terms, restricting attention to a purely gravitational action; whether such operators, which have the same mass dimension and are permitted by Weyl invariance, modify the conclusions is left open. Second, the cosmological constant term $\chi = M_P^2/\gamma$4 present in the Einstein-frame action is assumed to be renormalized by quantum corrections to its observed value, without a mechanism being specified. Third, reheating is treated entirely phenomenologically: no microscopic particle-production mechanism, preheating dynamics, or thermalization history is modeled, so the mapping between the abstract parameters $\chi = M_P^2/\gamma$5 and any concrete completion remains undetermined. Finally, the disagreement over the criterion for the end of inflation ($\chi = M_P^2/\gamma$6 versus $\chi = M_P^2/\gamma$7) relative to prior work is resolved only numerically here and merits analytical clarification.

Conclusion

The paper demonstrates that in Weyl-invariant Einstein–Cartan gravity, the parity-violating $\chi = M_P^2/\gamma$8 term generates a Starobinsky-like plateau in an otherwise exponential potential, yielding a single-field axion-like inflaton fully consistent with current CMB data. Its principal lesson is quantitative: reheating assumptions shift the predicted $\chi = M_P^2/\gamma$9 in opposite directions depending on $\phi$0, so that observational viability of the model cannot be assessed without specifying the post-inflationary equation of state and reheating temperature. As forthcoming tensor-mode experiments tighten bounds on $\phi$1, disentangling the degeneracy between the gravitational-sector parameter $\phi$2 and the reheating parameters will be essential for a definitive test of this class of theories.