Planck 2013 results. XXII. Constraints on inflation

We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to \$n\_s = 0.9603 \backslash pm 0.0073\$, ruling out exact scale invariance at over 5 \$\backslash sigma\$. Planck establishes an upper bound on the tensor-to-scalar ratio of r < 0.11 (95\% CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V'' < 0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n > 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining \$d n\_s/d ln k = -0.0134 \backslash pm 0.0090\$. Several analyses dropping the slow-roll approximation are carried out, including detailed model comparison and inflationary potential reconstruction. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit \$\backslash chi\^{}2\$ by \~{} 10; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with bounds on \$f\_\backslash mathrm\{NL\}\$ measured by Planck. The fractional primordial contribution of CDM isocurvature modes in the curvaton and axion scenarios has upper bounds of 0.25\% or 3.9\% (95\% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelation can improve \$\backslash chi\^{}2\$ by approximatively 4 as a result of slightly lowering the theoretical prediction for the \$\backslash ell<40\$ multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.