Commutative and non-commutative signatures of anisotropic inflation: constraints from planck PR4 data

Abstract We present a comparative study of anisotropic inflationary signatures in both commutative and non-commutative spacetime frameworks, using three anisotropic metrics (Models I–III) as inflationary backgrounds. For each model, we consider the primordial scalar power spectrum computed with the in–in formalism, obtaining the leading real (quadrupolar) modulation from geometric shear and the leading imaginary (dipolar) modulation from spacetime non-commutativity. These spectra are propagated to CMB anisotropies and tested against Planck PR4 temperature data ( Commander NPIPE) via a unified, signal-matched likelihood framework based on complex Bipolar Spherical Harmonic (BipoSH) coefficients, specifically targeting the dipole sector L = 1 coupling adjacent multipoles ( ℓ , ℓ + 1). This unified framework ensures methodological consistency in model comparison by defining a common data vector D ˆ that encompasses both real-valued (commutative) and imaginary-valued (non-commutative) anisotropic signatures, ensuring mathematically valid Bayesian evidence calculations. We implement a Bayesian pipeline that jointly samples standard ΛCDM parameters and model parameters (initial shear amplitude, orientation, and non-commutative scale), estimating posterior distributions and model evidences on an equal statistical footing. Our analysis yields stringent upper limits on residual inflationary shear ( σ / H ) < 1.8 × 1 0 − 5 and non-commutative effects [ α < 0.0069at95% C.L.], with Bayesian evidence favoring non-commutative models over geometric shear by Δ ln B ≈ 3 – 5 , though both remain disfavored relative to isotropic ΛCDM. We discuss implications for the cosmic no-hair conjecture and prospects for improved tests with future CMB polarization and higher-resolution datasets.