Axion-Induced Isocurvature Perturbations

• Axion-induced isocurvature perturbations are primordial fluctuations in the axion field that modulate the local dark matter abundance without changing the total energy density.
• Different production regimes—including pre- and post-inflationary scenarios and hilltop enhancements—lead to varied spectral shapes and correlations with adiabatic modes.
• Suppression strategies like increasing the axion mass during inflation or using low-scale inflation are crucial to reconcile these models with CMB and large-scale structure observations.

Searching arXiv for recent and foundational papers on axion isocurvature perturbations to ground the article. Axion-induced isocurvature perturbations are primordial fluctuations in the axion sector that modulate the composition of the cosmic fluid without initially perturbing the total energy density in the same way as inflaton-generated adiabatic modes. They arise whenever an axion or axion-like field is dynamically relevant during or after inflation and later contributes to cold dark matter through misalignment or related mechanisms. The subject spans several distinct cosmological regimes: pre-inflationary Peccei–Quinn breaking with spectator-field quantum fluctuations, correlated and cross-correlated isocurvature in multi-axion inflation, low-scale inflation scenarios in which a rolling axion sources isocurvature classically, and post-inflationary axion-like-particle realignment in which causality enforces a white-noise spectrum on large scales (Kobayashi et al., 2013, Kadota et al., 2015, Caputo et al., 2023, Feix et al., 2020).

1. Definition and basic mechanism

Axion-induced isocurvature perturbations are usually discussed in the setting where the Peccei–Quinn symmetry is broken before or during inflation, so that the axion is present as a light field during inflation and acquires nearly scale-invariant quantum fluctuations. Because the axion later behaves as cold dark matter, these primordial field fluctuations become fluctuations in the local axion abundance rather than fluctuations in the total energy density, and therefore correspond to cold-dark-matter isocurvature perturbations (Kobayashi et al., 2013, Harigaya et al., 2015).

For the QCD axion, the inflationary fluctuation amplitude is conventionally written as

$$\delta a_*(k) \simeq \frac{H_{\rm inf}}{2\pi}, \qquad P_{\delta a_*}(k)=\frac{H_*^2}{2k^3},$$

or in angular variables

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$

These fluctuations perturb the initial misalignment angle and hence the axion abundance generated by coherent oscillations (Kobayashi et al., 2013, Schmitz et al., 2018).

The entropy perturbation is commonly expressed as

$$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$

or, in the axion-specific notation of the general analytical treatment,

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$

In the pre-inflationary misalignment scenario, the linear isocurvature amplitude is controlled by the sensitivity of the relic abundance to the initial misalignment angle,

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$

with an additional factor $r=\Omega_\theta/\Omega_c$ if the axion is only a fraction of the dark matter (Allali et al., 8 Oct 2025).

A complementary formulation emphasizes that the axion density perturbation follows from the mapping between the local field value and the late-time abundance. In the small-angle limit one has

$$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$

and

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$

which makes explicit the parametric dependence on $H_{\rm inf}$, $f_a$, and the axion dark-matter fraction (Schmitz et al., 2018).

2. Pre-inflationary axions: quantum fluctuations, misalignment, and observational bounds

In the standard pre-inflationary scenario, the axion is effectively massless during inflation and its de Sitter fluctuations source an approximately scale-invariant, usually uncorrelated, cold-dark-matter isocurvature mode. This is the setting in which CMB data impose the familiar upper bound on the inflationary Hubble scale. A representative bound quoted for the QCD axion is

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$0

with

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$1

and the resulting upper limit

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$2

for $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$3 near the smaller end of the axion dark matter window in hilltop initial conditions (Kobayashi et al., 2013).

A more general Planck-era bound frequently used in low-scale hybrid inflation analyses is

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$4

which translates into

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$5

and then into

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$6

The same analysis gives

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$7

showing how stringent the isocurvature bound becomes for high-scale inflation if axions make up an appreciable fraction of dark matter (Schmitz et al., 2018).

The general analytical treatment of axion perturbations recasts the same physics in a background-independent language. On super-horizon scales, the isocurvature part of the axion perturbation is simply

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$8

and more generally

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$9

This identifies axion isocurvature as the variation of the homogeneous background trajectory with respect to the initial misalignment angle, independent of whether the cosmological background is radiation dominated, matter dominated, or more general (Allali et al., 8 Oct 2025).

A common misconception is that the axion perturbation is purely isocurvature at all times. The general treatment shows that an adiabatic axion mode is also present and is guaranteed by Weinberg’s theorem. For the axion,

$$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$0

so an initially homogeneous axion field can evolve into one with nontrivial adiabatic perturbations once the field begins to roll or oscillate (Allali et al., 8 Oct 2025).

3. Enhancement mechanisms: hilltops, dynamical decay constants, and deformed dynamics

The magnitude of axion-induced isocurvature is highly model dependent because it is controlled not only by $$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$1 and $$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$2, but also by the response of the relic abundance to the initial condition. The most studied enhancement mechanism is the hilltop regime of the periodic axion potential. For the QCD axion,

$$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$3

the onset of oscillations is delayed as the initial misalignment angle approaches $$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$4, and this delay amplifies the dependence of the final abundance on the initial field value (Kobayashi et al., 2013).

In the attractor formalism for the time-dependent QCD axion potential, the onset of oscillations is defined by

$$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$5

with

$$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$6

The quantity

$$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$7

encodes anharmonic effects: $$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$8 for a quadratic potential and $$S(\vec x)\equiv 3\left(\zeta_c(\vec x)-\zeta_r(\vec x)\right),$$9 near the hilltop. The isocurvature power spectrum then receives a large enhancement from the factor

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$0

which becomes strongly enhanced in the hilltop regime (Kobayashi et al., 2013).

The corresponding semi-analytic isocurvature spectrum is

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$1

while the local isocurvature nonlinearity parameter is

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$2

The paper emphasizes that the power spectrum is strongly enhanced toward the hilltop, whereas the non-Gaussianity increases only slowly (Kobayashi et al., 2013).

A different enhancement mechanism appears when the axion decay constant evolves in time. For a QCD axion with a dynamical decay constant $$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$3, the mode equation before the potential turns on is

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$4

Assuming

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$5

the super-horizon dynamical mode becomes significant in the matter-dominated post-inflation case when

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$6

Then a shrinking decay constant suppresses CMB-scale isocurvature if $$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$7, but amplifies smaller-scale perturbations, with a characteristic enhancement scale

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$8

The resulting fluctuation plateau can become so large that

$$S_{\theta\gamma}\equiv 3(\zeta_\theta-\zeta_\gamma).$$9

triggering the formation of stringless axionic domain walls and cosmological overclosure independently of $$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$0 (Kobayashi et al., 2016).

A third suppression mechanism operates through deformed axion dynamics with a field-dependent wavefunction factor,

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$1

For

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$2

in an intermediate regime, the canonical potential develops a tracking phase. In that phase, the entropy perturbation satisfies

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$3

so the system dynamically erases sensitivity to the initial condition. This suppresses axionic dark-matter isocurvature and can also suppress baryonic isocurvature in axion-driven spontaneous baryogenesis models (Bae et al., 2018).

4. Correlated and cross-correlated axion isocurvature

Axion-induced isocurvature is not generically uncorrelated with adiabatic perturbations. In string-inspired multi-axion inflation, one axion may drive inflation while a second lighter axion generates cold-dark-matter isocurvature. Non-perturbative effects then induce a mixed potential of the form

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$4

so the inflaton and spectator axion fluctuations are coupled (Kadota et al., 2015).

Two concrete realizations are studied. In natural inflation with a sinusoidal correction,

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$5

with

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$6

while in axion monodromy inflation with a sinusoidal correction,

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$7

In both cases the second sinusoidal term mixes the inflaton and the light axion (Kadota et al., 2015).

The curvature and isocurvature perturbations are defined as

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$8

The cross-correlation arises because expanding the interaction term

$$S_{\theta\gamma} = \frac{\partial \ln \Omega_\theta}{\partial \theta_{\rm ini}} \frac{H_I}{2\pi f_a},$$9

around the background produces a quadratic interaction of the form

$r=\Omega_\theta/\Omega_c$0

The resulting cross-power spectrum is computed with the in-in formalism (Kadota et al., 2015).

For natural inflation the cross-power is

$r=\Omega_\theta/\Omega_c$1

numerically approximated as

$r=\Omega_\theta/\Omega_c$2

For monodromy inflation,

$r=\Omega_\theta/\Omega_c$3

The correlation coefficient is

$r=\Omega_\theta/\Omega_c$4

so the sign of the correlation is controlled by $r=\Omega_\theta/\Omega_c$5 (Kadota et al., 2015).

The same theme reappears in low-scale inflation, but through a different mechanism. If the axion is rolling classically during inflation, then different local durations of inflation generate different end-of-inflation axion values. In the separate-universe picture,

$r=\Omega_\theta/\Omega_c$6

leading to

$r=\Omega_\theta/\Omega_c$7

The resulting axion isocurvature modes are fully correlated with the adiabatic ones and have nearly the same spectral index (Caputo et al., 2023).

In that low-scale mechanism the spectra are

$r=\Omega_\theta/\Omega_c$8

and

$r=\Omega_\theta/\Omega_c$9

The paper states that the isocurvature perturbations are fully correlated with the adiabatic perturbations and that their spectral index satisfies

$$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$0

up to corrections of order

$$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$1

(Caputo et al., 2023).

5. Spectral shapes beyond the scale-invariant case

Although the best-known axion isocurvature signal is nearly scale invariant, several well-motivated mechanisms predict nontrivial spectral shapes. One such class arises when the Peccei–Quinn symmetry-breaking field is displaced during inflation, so that the axion acquires a temporary mass of order $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$2. In this regime, the axion isocurvature spectrum is strongly blue tilted on scales exiting the horizon while the field is displaced, then transitions to a flat spectrum once the axion becomes effectively massless again (Chung et al., 2017, Chung et al., 2016).

This “axionic blue isocurvature” spectrum is characterized by a blue tilt at $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$3, a break scale $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$4, and a flat plateau at $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$5. The numerical study of the supersymmetric Kasuya–Kawasaki model showed that the transition region contains a bump rather than a monotonic interpolation. The fitted spectrum is described by three phenomenological parameters: the break scale $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$6, the blue spectral index $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$7, and an amplitude parameter $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$8 (Chung et al., 2016).

For $$S_{\rm iso} \simeq F_{\rm DM}^a \frac{2\sigma_{\bar\theta}}{\bar{\theta}_{\rm ini}},$$9, the dimensionless spectrum behaves as

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$0

whereas for $$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$1 it approaches a constant. In the explicit SUSY construction, the blue tilt is determined by the Hubble-induced mass parameter $$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$2,

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$3

The numerical analysis found that the bump near the transition can enhance the signal by nearly a factor of two for a steep blue tilt (Chung et al., 2016).

A subsequent cosmological fit using Planck and BOSS DR11 data found a mild hint for such a blue-tilted component, with best-fit parameters approximately

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$4

but $$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$5 remained allowed at about the $$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$6 level, so the result was explicitly characterized as a mild, non-significant hint rather than a detection (Chung et al., 2017).

A qualitatively different spectral shape appears in the post-inflationary axion-like-particle scenario. If Peccei–Quinn breaking occurs after inflation, the field takes random values in causally disconnected Hubble patches, and the resulting primordial isocurvature fluctuations are fixed by causality rather than by inflationary vacuum fluctuations. On large scales the density power spectrum is white noise,

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$7

so

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$8

The turnover scale is

$$\mathcal{P}_{\rm iso}\simeq \left(F_{\rm DM}^a\frac{H_{\rm inf}}{\pi f_a\bar{\theta}_{\rm ini}}\right)^2,$$9

and for the ALP masses studied it lies around $H_{\rm inf}$0–$H_{\rm inf}$1 (Feix et al., 2020).

This post-inflationary case differs conceptually from pre-inflationary spectator-axion isocurvature. The fluctuations arise from the spatial randomness of post-inflationary field values and the horizon-sized coherence domains set by the Kibble mechanism, not from super-horizon quantum fluctuations during inflation (Feix et al., 2020).

6. Suppression mechanisms and model-building strategies

Because axion-induced isocurvature so often constrains inflationary model building, a substantial literature is devoted to suppression mechanisms. One strategy is to make the axion heavy during inflation. If the axion mass satisfies $H_{\rm inf}$2, the standard super-horizon fluctuation amplitude is strongly suppressed and the usual isocurvature bound on $H_{\rm inf}$3 is relaxed (Higaki et al., 2014, Jeong et al., 2013).

Two distinct heavy-axion constructions are described in the literature summarized here. In one, the Peccei–Quinn symmetry is explicitly broken down to a discrete subgroup $H_{\rm inf}$4 by an operator such as

$H_{\rm inf}$5

with the breaking enhanced during inflation so that

$H_{\rm inf}$6

and

$H_{\rm inf}$7

The late-time explicit breaking must remain tiny,

$H_{\rm inf}$8

so if the operator persists to the present vacuum one typically needs $H_{\rm inf}$9 for $f_a$0 and $f_a$1 TeV (Higaki et al., 2014).

In another heavy-axion mechanism, QCD itself becomes stronger during inflation because the Higgs field has a very large expectation value, raising the effective QCD scale. The resulting inflationary axion potential is

$f_a$2

and the axion mass during inflation is

$f_a$3

The condition $f_a$4 becomes

$f_a$5

providing a direct route to suppressing isocurvature (Jeong et al., 2013).

Another broad class of suppression mechanisms lowers the inflationary Hubble scale rather than raising the axion mass. Low-scale supersymmetric hybrid inflation is an example. In F-term hybrid inflation and D-term hybrid inflation with hidden-sector SUSY breaking, one can match $f_a$6 and $f_a$7 while keeping $f_a$8 low enough to evade the axion isocurvature bound. The resulting viable parameter space requires at least one small coupling,

$f_a$9

depending on the model and regime. The corresponding gravitino-mass scales are

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$00

for large $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$01 (Schmitz et al., 2018).

Large Peccei–Quinn field values during inflation can also suppress axion angle fluctuations by increasing the effective decay constant during inflation. In the Linde-type setup,

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$02

This can alleviate isocurvature, but post-inflationary parametric resonance may restore the PQ symmetry and create strings and domain walls, so the isocurvature problem becomes inseparable from the defect problem (Kawasaki et al., 2013, Harigaya et al., 2015).

A further proposal is late entropy production through thermal inflation. For string axions, the late entropy release

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$03

dilutes the axion abundance and therefore suppresses the isocurvature fraction. This mechanism is especially effective for string axions because their shift symmetry forbids the large Hubble-induced mass that would otherwise generate problematic secondary coherent oscillations at the end of thermal inflation (Kawasaki et al., 2014).

7. Domain walls, reheating effects, non-Gaussianity, and observational outlook

Several papers stress that axion-induced isocurvature cannot be treated independently of post-inflationary nonlinear dynamics. Large Peccei–Quinn field values during inflation may suppress isocurvature, but the subsequent oscillation of the PQ field can trigger parametric resonance, restore the symmetry nonthermally, and create strings and domain walls. In a quartic-like PQ potential this resonance is difficult to avoid, whereas a sextet-plus-quartic potential,

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$04

can make the resonance ineffective. In this construction the transition to quartic domination occurs at

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$05

and the model avoids symmetry restoration provided

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$06

This suggests that the viability of isocurvature suppression by large $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$07 depends sensitively on the detailed post-inflationary PQ dynamics (Harigaya et al., 2015).

Lattice studies of models unifying the axion with the inflaton show an even sharper version of the same issue. In two explicit models, nonperturbative reheating effects amplify axion perturbations so strongly that PQ symmetry is restored for

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$08

with a more detailed estimate of roughly

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$09

needed to avoid restoration in one model. Even when restoration does not occur, the low-$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$10 spectrum is amplified so strongly that a naive extrapolation to CMB scales suggests a severe violation of isocurvature bounds (Ballesteros et al., 2021). A plausible implication is that suppressing the horizon-exit spectrum is not sufficient if reheating later reprocesses the super-horizon axion fluctuations.

Non-Gaussianity is another characteristic feature of axion isocurvature. Because the axion abundance is a nonlinear function of the initial misalignment angle, local-type isocurvature non-Gaussianity arises even for Gaussian primordial field fluctuations. In the general analytical treatment,

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$11

while in the hilltop analysis the corresponding $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$12 rises only slowly even when the power spectrum becomes strongly enhanced (Allali et al., 8 Oct 2025, Kobayashi et al., 2013). This suggests that the power-spectrum constraint generally dominates over the non-Gaussianity constraint in dominant-dark-matter hilltop scenarios.

Beyond the power spectrum and bispectrum of density perturbations, axion isocurvature can source other observables. A nearly massless cosmological axion with a parity-violating photon coupling produces anisotropic cosmic birefringence, rotating CMB $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$13-modes into $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$14-modes. The same paper distinguishes this classic isocurvature-induced birefringence from a newer mechanism in which ultralight axion dark matter adiabatic density perturbations source the birefringence, but it explicitly situates the latter as an extension of the standard isocurvature picture (Liu et al., 2016). Another extension is the “axion isocurvature collider,” in which the squeezed-limit bispectrum of axion isocurvature carries a clock signal from a heavy intermediate particle exchanged during inflation (Lu, 2021).

Current observational conclusions depend strongly on the production scenario. For post-inflationary ALP dark matter, the combined current limit

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$15

implies roughly

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$16

for weak temperature dependence and up to

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$17

for stronger temperature dependence. Future cosmic shear, galaxy clustering, and CMB lensing could improve the sensitivity to

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$18

for a conservative scale cut and even

$$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$19

in an optimistic case (Feix et al., 2020).

Taken together, these results show that “axion-induced isocurvature perturbations” is not a single phenomenological template but a family of signatures determined by when the axion is present, whether it is light or massive during inflation, how its abundance depends on initial conditions, whether it is correlated with the adiabatic mode, and how nonlinear post-inflationary dynamics reshape the primordial spectrum. The recurring structural quantity across formulations is the sensitivity of the late-time axion abundance to the initial field value, whether written as $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$20, as $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$21, or through explicit response coefficients in $$\sigma_{\bar\theta} \simeq \frac{H_{\rm inf}}{2\pi f_a}.$$22 language (Allali et al., 8 Oct 2025, Kobayashi et al., 2013, Bae et al., 2018). This suggests that the central theoretical problem is not only the generation of axion fluctuations, but also the nonlinear transfer from axion field space to the dark-matter sector.