Starobinsky inflation and the latest CMB data: a subtle tension?

Abstract We analyze the Starobinsky inflation model and the impact of curvature corrections, particularly a cubic $$R^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> 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 $$n_s=0.9752 \pm 0.0030$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.9752</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.0030</mml:mn> </mml:mrow> </mml:math> , when combined with BAO data from DESI DR2, which appears to exclude the pure Starobinsky model at approximately the $$2\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> 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 $$N_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> 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 $$R^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> term can shift its predictions to better align with the Planck and ACT measurements.