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. This conclusion can be evaded by exploiting \emph{field-space geometry}. For a minimal complex PQ scalar with a $U(1)$-symmetric potential and a nonlinear sigma-model kinetic term $d\sigma^{2}=dR^{2}+f^{2}(R)\,d\theta^{2}$, the observable axion fluctuation is $\delta\theta\sim H_{\rm inf}/f(R)$, so an enhanced effective decay constant $f(R)$ suppresses isocurvature without explicit PQ breaking, extreme radial displacements, or additional couplings. We specialize to a hyperbolic metric $f(R)\propto \sinh(R/L)$ with curvature scale $L$. The same geometry also induces a time-dependent $\mathcal{O}(H_{\rm inf})$ effective mass for the canonical axial mode during radial slow-roll, and fixing the tilt and running of isocurvature. Thus, CMB-scale isocurvature is suppressed while a characteristic blue-tilted spectrum is generated. 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 `observable' 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.