Inflationary assessment of $F(\mathcal{R},\tilde{\mathcal{R}})$ Einstein-Cartan models

TL;DR

This work examines inflation in Einstein-Cartan gravity with an action depending on the Ricci scalar and the Holst invariant, allowing cubic Holst terms. By integrating out torsion and moving to the Einstein frame, the dynamics reduce to a single pseudoscalar field with a potential that is modified by cubic curvature terms. Two models are studied: a cubic-Holst model and a Weyl-invariant model, with detailed slow-roll analyses showing that cubic corrections can either improve or degrade agreement with Planck and ACT data depending on parameter choices. The results highlight that cubic terms can enable acceptable inflation where quadratic forms fail, but can also shift viable regions toward BAO-preferred domains, emphasizing a delicate balance between linear, quadratic, and cubic contributions. Overall, cubic curvature corrections enrich the landscape of inflationary solutions in Einstein-Cartan gravity, with specific viable parameter ranges identified for both models.

Abstract

In the framework of $F(\mathcal{R},\tilde{\mathcal{R}})$ Einstein-Cartan gravity with an action depending both of the Ricci scalar and the so-called Holst-invariant curvature we consider models that include cubic terms of the latter in the action and study their inflationary behavior. These terms can have a considerable effect either positive or negative in relation to the agreement with present observational data, depending on parameters. In parameter regions where the quadratic models fail to produce results consistent with observational data, the presence of these additional cubic terms can lead to compatible predictions.

Figures (6)

• Figure 1: In \ref{['CUBIC-MODEL']} we have plotted the potential for the cubic model, given by eq. \ref{['UEXACT']}. We have used the value $\beta=70$ for the parameter of the linear Holst term, while $\alpha$ is the parameter of the corresponding cubic term. The scale $M^2$ is fixed by the scalar power spectrum amplitude $A_s=2.1\cdot 10^{-9}$. In \ref{['WEYL-MODEL']} we have plotted the potential for the Weyl-invariant model, given by eq. \ref{['WEYL-POTENTIAL']}. We have used the parameter values $q=100$ and $q_1=150$, while parameter $\bar{\epsilon}$ scales the cubic correction. The parameter $V_0$ is fixed by the amplitude of the scalar power spectrum $A_s=2.1\cdot10^{-9}$.
• Figure 2: The plots (a) and (b) depict $r$ vs $n_s$ for the cubic-Holst model for two different values of the parameter $\beta$ that parametrizes the linear Holst term, while $\alpha$ corresponds to $\tilde{\mathcal{R}}^3$. The scale $M^2$ is fixed by the observational value of the scalar power spectrum amplitude $A_s=2.1\cdot10^{-9}$. With dark gray and light gray we denote the $1\sigma$ and $2\sigma$ regions respectively for Planck/BK18/BAO data set Planck:2018jriPlanck:2018vyg, while the dark purple and light purple correspond to the Planck/ACT/LB/BK18 data set ACT:2025fjuACT:2025tim.
• Figure 3: The plots (a) and (b) depict $r$ vs $n_s$ for the cubic-Holst model for two different values of the parameter $\beta$ that parametrizes the linear Holst term, while $\alpha$ corresponds to $\tilde{\mathcal{R}}^3$. The scale $M^2$ is fixed by the observational value of the scalar power spectrum amplitude $A_s=2.1\cdot10^{-9}$. With dark gray and light gray we denote the $1\sigma$ and $2\sigma$ regions respectively for Planck/BK18/BAO data set Planck:2018jriPlanck:2018vyg, while the dark purple and light purple correspond to the Planck/ACT/LB/BK18 data set ACT:2025fjuACT:2025tim.
• Figure 4: The plots (a) and (b) depict $r$ vs $n_s$ for the Weyl-cubic model for different values of the parameters $q$ and $q_1$, given by eq. \ref{['parametrization']}. The parameter $V_0$ is fixed by the observed value of the scalar power spectrum amplitude $A_s=2.1\cdot 10^{-9}$, while the parameter $\bar{\epsilon}$ scales the cubic term. With dark gray and light gray we have denoted the $1\sigma$ and $2\sigma$ regions respectively for Planck/BK18/BAO data set Planck:2018jriPlanck:2018vyg, while the dark purple and light purple correspond to the Planck/ACT/LB/BK18 data set ACT:2025fjuACT:2025tim.
• Figure 5: The plots (a) and (b) depict $r$ vs $n_s$ for the Weyl-cubic model for different values of the parameters $q$ and $q_1$, given by eq. \ref{['parametrization']}. The parameter $V_0$ is fixed by the observed value of the scalar power spectrum amplitude $A_s=2.1\cdot 10^{-9}$, while the parameter $\bar{\epsilon}$ scales the cubic term. With dark gray and light gray we have denoted the $1\sigma$ and $2\sigma$ regions respectively for Planck/BK18/BAO data set Planck:2018jriPlanck:2018vyg, while the dark purple and light purple correspond to the Planck/ACT/LB/BK18 data set ACT:2025fjuACT:2025tim.