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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 $n_s = 0.9603 \pm 0.0073$, ruling out exact scale invariance at over 5$\sigma$. 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 $d n_s/d ln k = -0.0134 \pm 0.0090$. 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 $\chi^2$ 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 $f_\mathrm{NL}$ 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 $\chi^2$ by approximatively 4 as a result of slightly lowering the theoretical prediction for the \ell&lt;40 multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.

We use the angular cross-correlation between a Luminous Red Galaxy (LRG) sample from the DR9 DESI Legacy Survey and the $Planck$ PR4 CMB lensing maps to constrain the local primordial non-Gaussianity parameter $f_{\rm NL}$ using the scale-dependent galaxy bias effect. The galaxy sample covers $\sim$ 40% of the sky and contains galaxies up to $z \sim 1.4$, and is calibrated with the LRG spectra that have been observed for the DESI Survey Validation. We apply a nonlinear imaging systematics treatment based on neural networks to remove observational effects that could potentially bias the $f_{\rm NL}$ measurement. Our measurement is performed without blinding, but the full analysis pipeline is tested with simulations including systematics. Using the two-point angular cross-correlation between LRG and CMB lensing only ($C_\ell^{\kappa G}$) we find $f_{\rm NL} = 39_{-38}^{+40}$ at 68% confidence level, and our result is robust in terms of systematics and cosmology assumptions. If we combine this information with the autocorrelation of LRG ($C_\ell^{GG}$) applying a $\ell_{\rm min}$ scale cut to limit the impact of systematics, we find $f_{\rm NL} = 24_{-21}^{+20}$ at 68% confidence level. Our results motivate the use of CMB lensing cross-correlations for measuring $f_{\rm NL}$ with future datasets given its stability in terms of observational systematics compared to the angular auto-correlation.