Increase of $n_s$ in regularized pole inflation & Einstein-Cartan gravity

TL;DR

This work addresses the tension between ACT hints of a larger scalar spectral index $n_s$ and standard attractor inflation predictions. It introduces regularized pole inflation, where a finite width $\lambda$ regularizes the second-order pole in the inflaton's kinetic term, canonicalizes to a field with a hyperbolic sine relation, and yields an $V(\hat{\varphi})$ that lifts in the large-field regime. The leading observables shift by $\mathcal{O}(\lambda^2)$: $n_s-1\simeq -\frac{2}{N_e}+\frac{2N_e}{3\gamma^4}\frac{V_1^2}{V_0^2}M_{ ext{Pl}}^2\lambda^2$ and $r\simeq \frac{8\gamma^2}{N_e^2}+\frac{8}{3\gamma^2}\frac{V_1^2}{V_0^2}M_{ ext{Pl}}^2\lambda^2$, thereby increasing $n_s$ while maintaining attractor behavior for small $\lambda$. A concrete realization in Einstein–Cartan gravity shows how torsion and associated four-dimensional operators yield a regularized pole and a canonical sinh-type potential; the resulting $n_s$–$r$ predictions align with ACT within $2\sigma$ for $N_e\approx 60$ and smoothly connect to the Starobinsky limit. This mechanism provides a gravity-inspired way to reconcile updated CMB data with the successes of $\alpha$-attractor models.

Abstract

We show that the regularization of the second order pole in the pole inflation can induce the increase of $n_s$, which may be important after the latest data release of cosmic microwave background (CMB) observation by Atacama Cosmology Telescope (ACT). Pole inflation is known to provide a unified description of attractor models that they can generate a flat plateau for inflation given a general potential. Recent ACT observation suggests that the constraint on the scalar spectral index $n_s$ at CMB scale may be shifted to a larger value than the predictions in the Starobinsky model, the Higgs inflation, and the $α$-attractor model, which motivates us to consider the modification of the pole inflation. We find that if we regularize the second order pole in the kinetic term such that the kinetic term becomes regular for all field range, we can generally increase $n_s$ because the potential in the large field regime will be lifted. We have explicitly demonstrated that this type of regularized pole inflation can naturally arise from the Einstein-Cartan formalism, and the inflationary predictions are consistent with the latest ACT data without spoiling the success of the $α$-attractor models.

Figures (4)

• Figure 1: The red and blue dashed lines correspond to ${\rm sign} (\phi) >0$ with $+$ and ${\rm sign} (\phi) <0$ with $-$ in solutions \ref{['eq-separate-branch']}, respectively. The black line corresponds to the solutions \ref{['eq-connect-branch']} with $+$ sign.
• Figure 2: Schematic figure describing the idea of the regularized pole inflation. In the pole inflation, the potential at the pole $\phi_0$ is exponentially flattened by the divergence of the kinetic term, which results in two separated branches. On the other hand, in the regularized pole inflation, the kinetic term is enhanced within a finite field range of $|\phi - \phi_0 | < \lambda M_{\text{Pl}}$ while the point $\phi_0$ is regularized. As a result, the two branches are smoothly connected. Moreover, since a finite field range of the potential is probed by the regularized pole, the attractor behavior is expected only for $\lambda M_{\text{Pl}} \ll \Delta \phi$ with $\Delta \phi$ being the field range where the potential is well approximated by a linear function.
• Figure 3: This is the potential \ref{['eq-potential-example']} with different parameter choices. $\alpha_3 =-2/3$ and $\alpha_R$ is determined by matching the scalar fluctuation $\Delta_s^2$ on CMB. The black dashed line corresponds to the Starobinsky limit $\beta_3 \to -\infty$.
• Figure 4: Predictions of spectral index $n_s$ and tensor-to-scalar ratio $r$ from the model given in Eq. \ref{['eq-model']} as an example of regularized pole inflation. Left: The constraint contours are directly taken from Fig. 5 in Ref. BICEP:2021xfz and Fig. 10 in Ref. ACT:2025tim. The left contours are constraints combining Planck Planck:2018jri with BICEP/Keck BICEP:2021xfz at pivot scale $k/a_0=0.05 \,\mathrm{Mpc}^{-1}$ (and Ref. BICEP:2021xfz has assumed the tensor spectral index $n_t =0$), with $1 \sigma$ and $2\sigma$ regions respectively. The right are those combined with ACT at pivot scale $k/a_0=0.05 \,\mathrm{Mpc}^{-1}$. In this example, $\alpha_3 =-2/3$ coincides with the predictions from the Starobinsky model when $\beta_3 \to -\infty$ (practically we have taken $|\beta_3|=3\times 10^4$ in numerical calculation). The deep blue trajectory is obtained with full numerical solution by fixing $\alpha_3=-2/3$ while changing $\beta_3$ from $-13$ ($n_s$ is too large and lies outside the figure) to $-3\times 10^4$ (corresponding to small $n_s$). Since large-$|\beta_3|$ limit leads to the Starobinsky model, the predictions approach the black line as $|\beta_3|$ increases. The green trajectory is for $\alpha_3 = -1/2$ with the same range of $\beta_3$. Right: This is the zoom-in of the left panel and only the $\alpha_3 =-2/3$ trajectory is kept. We also show the results calculated with the approximated formulae \ref{['eq-approx-ns']} and \ref{['eq-approx-r']} in red dashed lines for $\alpha_3 =-2/3$ and different choices of $\beta_3$. From the left to the right, the red dashed lines correspond to $\beta_3 = -3\times 10^4$, $\beta_3 = -30$, and $\beta_3 = -17$, and the black and blue solid lines are calculated numerically by exact solutions with corresponding $\beta_3$'s. One can see that the approximated formulae work well for the observationally relevant range of parameters, but generally they underestimate the prediction of $n_s$, including that of the Starobinsky inflation limit. As an interesting consequence, in the Starobinsky model case, the predictions by the leading contributions in large $N_e$ expansion is outside the favored regime by ACT, but the results from exact solutions can still reside in the observationally favored contour at the $2 \sigma$ level when $N_e \simeq 60$.