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

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 $d{\sigma }^2=d{R}^2+f^2(R)d{\theta }^2$, the observable axion fluctuation is $\delta \theta \sim H_{\mathrm{i}\mathrm{n}\mathrm{f}}\mathrm{/}f(R)$, 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 $f(R)\propto \sinh (R\mathrm{/}L)$ with curvature scale $L$. The same geometry also induces a time-dependent $\mathcal{O}(H_{\mathrm{i}\mathrm{n}\mathrm{f}})$ 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 $H_{\mathrm{i}\mathrm{n}\mathrm{f}}\sim {10}^{13}\mathrm{G}\mathrm{e}\mathrm{V}$ can be compatible with $f_a\sim {10}^{14}$–${10}^{16}\mathrm{G}\mathrm{e}\mathrm{V}$, 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 $R\mathrm{/}L$, providing a direct phenomenological probe on PQ field-space geometry.