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

TL;DR

The work addresses the axion isocurvature constraint that typically limits high-scale inflation with large $f_a$ in pre-inflationary PQ scenarios. It introduces a minimal two-field PQ model with a curved target-space metric, specifically a hyperbolic geometry with $d\sigma^{2}=dR^{2}+L^{2}\sinh^{2}(R/L)\,d\theta^{2}$, which endows the angular fluctuation with a time-dependent geometric mass during inflation. This geometric mass suppresses CMB-scale isocurvature while producing a blue-tilted spectrum on smaller scales, thereby reopening parameter space with $H_{\rm inf}\sim 10^{13}$ GeV and $f_a\sim 10^{14}$–$10^{16}$ GeV without explicit PQ breaking. A compact phenomenological template links the isocurvature scale dependence to the field-space curvature $R/L$, enabling direct tests of PQ geometry in future observations and motivating UV completions and extensions to more general target spaces.

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.

Figures (4)

• Figure 1: Illustrative constraints in the $(H_{\rm inf},f_a)$ plane for pre-inflationary QCD axion DM from misalignment. For the standard flat-metric case $f_{\rm eff}=f_a$, the isocurvature bound restricts QCD axion to the blue shaded region. Gray bands indicate additional excluded regions. In this work, we show that the white region can be reopened using geometric suppression. (Redrawn with inspiration from Graham:2025iwx.)
• Figure 2: Time-dependent mass-squared of the angular fluctuation ($m_\psi^2$) and background radial mass-scale ($m_R^2$) for B1.
• Figure 3: Total matter isocurvature spectra for the benchmarks, assuming QCD axions constitute all DM via misalignment ($r_a=1$). We take $H_{\rm inf}=10^{13}\,{\rm GeV}$. Adiabatic spectrum also shown for $A_s=2.1\times10^{-9}$ and $n_s=0.967$Planck:2018jri.
• Figure 4: Isocurvature spectral index $n_{\rm iso}(k)$ inferred from the numerical spectra. The red dashed curve shows the semi-analytic estimate in Eq. (\ref{['eq:niso']}) for B2. The gray curve indicates the corresponding flat-metric expectation.