Starobinsky Inflation and the Latest CMB Data: A Subtle Tension?

TL;DR

The paper tests Starobinsky inflation and a minimal cubic curvature correction against Planck and ACT DR6 data by deriving a reheating-based range for the e-folds $N_k$ and implementing the inflationary potentials directly in the CLASS Boltzmann code. The analysis shows that, when $N_k$ is treated consistently, the pure Starobinsky model remains broadly compatible with observations, with only a mild tension when ACT data are included in a simplified framework. Introducing the $R^3$ term adds a new degree of freedom $oldsymbol{ extalpha}_0$ that shifts $n_s$ and $N_k$ into the reheating window, marginally favoring a negative $oldsymbol{ extalpha}_0$ and improving concordance with both Planck and ACT. Overall, both models remain viable descriptions of inflation under current data, though future surveys (e.g., SO, CMB-S4) will substantially sharpen these tests for curvature corrections.

Abstract

We analyze the Starobinsky inflation model and the impact of curvature corrections, particularly a cubic $R^3$ term, to assess their behavior in light of the latest observational results from the Atacama Cosmology Telescope (ACT). With the recent sixth data release (DR6), the scalar spectral index was measured to be $n_s=0.9743 \pm 0.0034$, which appears to exclude the pure Starobinsky model at approximately the $2σ$ level. In this paper, we implement the Starobinsky inflationary potential directly into the CLASS code, without relying on the slow-roll approximation, and we constrain the number of e-folds of inflation $N_k$ using a theoretically motivated range derived from reheating considerations and standard couplings between matter fields and gravity. We show that it is still possible to identify a significant region of parameter space where the Starobinsky model remains highly consistent with the latest observational data. While the pure Starobinsky model remains a compelling candidate for cosmic inflation, we explore how including a cubic $R^3$ term can shift its predictions to better align with the Planck and ACT measurements.

Figures (3)

• Figure 1: Constraints on the scalar and tensor primordial power spectra for the Starobinsky and $R^3$ model. The constraints on $r$ are driven by the BK18 data, while the constraints on $n_s$ are driven by Planck (blue) or P-ACT (red). The combined dataset also includes CMB lensing and BAO in all cases. For more details about the dataset, see Ref. ACTDR6b. The light green region shows the theoretical evolution of the $R^3$ model. The black circles represent the Starobinsky model for $N_k=53$ (the smaller one) and $N_k=59$ (the bigger one). The darker contours represent the $68\%$ C.L., while the lighter ones correspond to the $95\%$ C.L..
• Figure 2: Confidence regions for the Starobinsky model using P-LB (blue contours) and P-ACT-LB (red contours) datasets. The darker contours represent the $68\%$ confidence level ($1\sigma$), while the lighter ones correspond to the $95\%$ confidence level ($2\sigma$). The vertical dashed lines indicate the theoretical range of $N_k$ derived from the reheating analysis in Eq. \ref{['eq:range']}.
• Figure 3: Confidence regions for the $R^3$ model using P-LB (blue contours) and P-ACT-LB (red contours) datasets. The darker contours represent the $68\%$ confidence level ($1\sigma$), while the lighter ones correspond to the $95\%$ confidence level ($2\sigma$). The vertical dashed lines indicate the theoretical range of $N_k$ derived from the reheating analysis in Eq. \ref{['eq:range']} and the green line indicates $\alpha=0$.