Axions on a Hyperbolic Ride: Geometric Suppression of CMB Isocurvature and a Blue-Tilted Spectrum

Abstract: CMB limits on cold-dark-matter isocurvature are often interpreted as excluding the simultaneous realization of high-scale inflation and large QCD axion decay constants in pre-inflationary Peccei--Quinn (PQ) scenarios. We show that this conclusion can be evaded by exploiting \emph{field-space geometry}. For a minimal complex PQ scalar with a $U(1)$-symmetric potential and nonlinear sigma-model kinetic term $dσ^{2}=dR^{2}+f^{2}(R)\,dθ^{2}$, a curved target-space metric endows the axion fluctuation with a time-dependent geometric mass during inflation, suppressing isocurvature without explicit PQ breaking and without extreme radial displacements. Specializing to a hyperbolic metric $f(R)=L\sinh(R/L)$ with curvature scale $L$, we find that for $R\gtrsim L$ the canonically normalized angular mode can be generically $\mathcal{O}(H_{\rm inf})$-heavy during radial slow-roll, dynamically damping CMB-scale fluctuations while producing a characteristic blue-tilted isocurvature spectrum. As a result, inflationary Hubble scales as large as $H_{\rm inf}\sim 10^{13}\,\mathrm{GeV}$ can be compatible with $f_a\sim 10^{14}$--$10^{16}\,\mathrm{GeV}$, reopening parameter space usually regarded as excluded. We present numerical benchmarks and a semi-analytic template that relates the scale-dependence of isocurvature to the geometric lever arm $R/L$, providing a direct phenomenological probe on PQ field-space geometry.