Bayesian analysis of $α$-Starobinsky model with Planck, ACT and DESI data

Abstract

We present a joint Bayesian analysis to impose constraints on the generalized $α$-Starobinsky inflationary model using the high-precision cosmological datasets: Planck, CMB lensing from ACT DR6, and Baryon Acoustic Oscillations (BAO) from DESI DR2. For the parameter inference, we introduce an alternative sampling approach. Rather than imposing priors on the cosmological parameters of the inflationary potential $(V_0, \, α, \, N_*)$, we place priors directly on the primordial physical observables $(A_s,\, n_s,\, r)$ through using analytical slow-roll consistency relations, our pipeline internally maps these sampled observables to the corresponding $α$-Starobinsky parameters. These values are then passed to a modified version of $\tt{CLASS}$, which solves the exact inflationary dynamics fully numerically. This pipeline ensures that the final reported posteriors for the observables are computed exactly, completely free from the slow-roll approximation. Applying this methodology, we explore the viability of the $α$-Starobinsky model. We show that, when the full combined dataset is considered, the pure Starobinsky model (i.e., the canonical limit $α= 1$) faces an apparent discrepancy: it requires a large number of $e$-folds of inflation after horizon crossing ($N_* > 60$) due to the shift in the scalar spectral index, $n_s$. In contrast, allowing the deformation parameter $α$ as a free parameter yields a clear $1σ$ preference for $\log_{10} α> 0$ present across all datasets. By favoring a broader inflationary plateau, the $α$-Starobinsky model elegantly reconciles theoretically sound expansion histories with empirical data. Notably, we also show that the addition of ACT DR6 lensing data introduces no significant impact on these primordial constraints, confirming that our robust posteriors are primarily driven by Planck and DESI measurements.

Figures (8)

• Figure 1: Dimensionless inflationary potential, $V(\phi)/V_0$, for the $\alpha$-Starobinsky model in Eq. \ref{['eq:alpha_starobinsky']}. Colored curves indicate different values of the deformation parameter $\alpha$, interpolating between a sharper rise at small field values ($\alpha=0.25$) and an extended, flatter plateau ($\alpha=100$). The dashed curve marks the Starobinsky limit $\alpha=1$. Increasing $\alpha$ flattens the potential and delays its approach to the asymptotic value $V \rightarrow V_0$, effectively widening the inflationary plateau.
• Figure 2: CMB temperature anisotropy power spectrum $\mathcal{D}_\ell^{TT} \equiv \ell(\ell+1)C_\ell^{TT}/2\pi$ for the $\alpha$-Starobinsky model. Numerical predictions obtained from our modified Boltzmann solver are compared against Planck 2018 data (teal points). The panels illustrate the impact of systematically varying the model's core parameters: (a) the deformation parameter $\alpha$ (plotted as $\log_{10}\alpha \in [-1, 1]$), (b) the potential scale $V_0$ ($\log_{10}V_0 \in [-14, -11]$), and (c) the number of $e$-folds $N_*$ ($N_* \in [40, 80]$). In each panel, while one parameter is swept across the copper colormap, the other two are held fixed at their fidual values. The prominent vertical dispersion in (a) and (b) illustrates the strong dependence of the overall scalar amplitude ($A_s$) on both $\alpha$ and $V_0$. Despite these large amplitude shifts, the stable horizontal placement of the acoustic peaks confirms that the varying parameters preserve the fundamental acoustic structure without inducing gross scale-dependent deformations.
• Figure 3: Posterior constraints for the Starobinsky model ($\alpha=1$). One- and two-dimensional marginalized distributions compare Planck-only (dashed orange) with the joint Planck+ACT+DESI analysis (solid gray). The geometrical rigidity of the model enforces a tight correlation between $n_s$ and $r$. Incorporating DESI and ACT shifts the preferred tilt toward higher mean value ($n_s = 0.969$) and, within the $\alpha=1$ track, this is accommodated by a larger mean value of $e$-folds ($N_* = 63.4$), pushing against the known theoretical canonical $50 \leq N_* \leq 60$ range.
• Figure 4: Marginalized 1D and 2D posteriors for $\{\log_{10} V_0, \log_{10}\alpha, N_*\}$ in the $\alpha$-Starobinsky model. Contours (68% and 95% credible regions) are shown for Planck (gray), Planck+DESI (blue), Planck+ACT (purple), and the full combination (red). Dotted lines indicate the Starobinsky limit $\log_{10}\alpha=0$. While $\alpha=1$ remains compatible at $2\sigma$, the joint dataset disfavors it at $1\sigma$ and prefers $\log_{10}\alpha>0$, indicating a mild but data-driven tilt toward a broader plateau. The near overlap of Planck and Planck+ACT contours shows that ACT adds little constraining power for these specific inflationary parameters beyond Planck.
• Figure 5: Joint 68% and 95% CL constraints in the $(n_s, r)$ plane at $k=0.05\,\mathrm{Mpc}^{-1}$ for Planck (light blue) and Planck+DESI (dark blue). The red contour shows the Planck constraint restricted to the $\alpha=1$ model. Theoretical $\alpha$-Starobinsky tracks (black) are overlaid for selected $\alpha \in \{0.1, 1, 5, 10, 20\}$; filled circles mark $N=\{50,60\}$, and the zoom highlights the $\alpha=1$ locus with open circles for $N=\{50,60,70,80\}$. DESI shifts the preferred $n_s$ higher; within $\alpha=1$ this implies larger $N$, whereas allowing $\alpha > 1$ raises $r$ at fixed $N$, easing to be inconsistent. The gray band ($r<10^{-3}$) indicates the target sensitivity of future CMB-S4 2019arXiv190704473A2022ApJ...926...54A. Large-$N$ slow-roll estimates are used for the theory tracks: $n_s \simeq 1-2/N$ and $r \simeq 12\alpha/N^2$ (and $r \simeq 8/N$ for quadratic chaotic inflation).