Planck 2015 results. XX. Constraints on inflation

TL;DR

Planck 2015 full-mission data constrain inflation by precisely measuring the scalar spectral index and its running while placing stringent upper limits on the tensor-to-scalar ratio. Using three complementary methods—P_R(k) parameterization, Hubble-flow function slow-roll, and numerical mode reconstruction—the analysis shows n_s ≈ 0.965 with small running and r ≲ 0.1–0.11, disfavoring V(φ) ∝ φ^2 and natural inflation in favor of models like R^2 inflation. The data remain consistent with adiabatic, Gaussian primordial perturbations, with polarization data significantly reducing degeneracies and refining parameter bounds; BKP cross-analysis further supports the low tensor amplitude. Overall, Planck 2015 tightens inflationary constraints, aligns with prior conclusions, and narrows the viable space of single-field slow-roll models while reinforcing the robustness of the ΛCDM + inflation framework.

Abstract

We present the implications for cosmic inflation of the Planck measurements of the cosmic microwave background (CMB) anisotropies in both temperature and polarization based on the full Planck survey. The Planck full mission temperature data and a first release of polarization data on large angular scales measure the spectral index of curvature perturbations to be $n_\mathrm{s} = 0.968 \pm 0.006$ and tightly constrain its scale dependence to $d n_s/d \ln k =-0.003 \pm 0.007$ when combined with the Planck lensing likelihood. When the high-$\ell$ polarization data is included, the results are consistent and uncertainties are reduced. The upper bound on the tensor-to-scalar ratio is $r_{0.002} < 0.11$ (95% CL), consistent with the B-mode polarization constraint $r< 0.12$ (95% CL) obtained from a joint BICEP2/Keck Array and Planck analysis. These results imply that $V(φ) \propto φ^2$ and natural inflation are now disfavoured compared to models predicting a smaller tensor-to-scalar ratio, such as $R^2$ inflation. Three independent methods reconstructing the primordial power spectrum are investigated. The Planck data are consistent with adiabatic primordial perturbations. We investigate inflationary models producing an anisotropic modulation of the primordial curvature power spectrum as well as generalized models of inflation not governed by a scalar field with a canonical kinetic term. The 2015 results are consistent with the 2013 analysis based on the nominal mission data.

Figures (1)

• Figure 1: Comparison of transfer functions for the scalar and tensor modes. The CMB transfer functions $\Delta _{\ell ,\mathcal{A}}^\mathrm{s}(k)$ and $\Delta _{\ell ,\mathcal{A}}^\mathrm{t}(k)$, where $\mathcal{A}=T,E,B$, define the linear transformations mapping the primordial scalar and tensor cosmological perturbations to the CMB anisotropies as seen by us on the sky today. These functions are plotted for two representative values of the multipole number: $\ell =2$ (in black) and $\ell =65$ (in red).