Inflationary Axion Isocurvature in the CMB across All Ultralight Masses

Daniel Grin; Rayne Liu; Wayne Hu

REVIEW 3 major objections 4 minor 67 references

Inflationary Axion Isocurvature in the CMB across All Ultralight Masses

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Planck's cold dark matter isocurvature bound, translated through the axion abundance–initial-field mapping, implies $r f_{\rm dm} < 0.076\,(m_a/10^{-27}\,{\rm eV})^{-1/2}$ for dark-matter-like ultralight axions, a constraint stronger than…

desk verdict Careful, honest extension of AxiECAMB to isocurvature modes with a strong headline bound that is conditionally correct pending resolution of a factor-of-4 normalization discrepancy with prior work. read the letter →

arxiv 2608.07803 v1 pith:DHG57IVN submitted 2026-08-07 astro-ph.CO hep-ph

Rayne Liu , Wayne Hu , Daniel Grin This is my paper · ORCID

classification astro-ph.COhep-ph

keywords axionisocurvatureultralightaxionsCMBanisotropyinflationtensor-to-scalarratiomisalignmentdarkmatterenergy

topics Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to turn axion isocurvature fluctuations into a quantitative probe of inflation across the entire ultralight axion mass range, from dark-energy-like ($m_a \lesssim H_0$) to dark-matter-like ($m_a \gg 10^{-28}\,{\rm eV}$) axions. It claims that, once the axion abundance is accurately calibrated to the initial field value, Planck's bound on cold dark matter isocurvature implies $r f_{\rm dm} < 0.076\,(m_a/10^{-27}\,{\rm eV})^{-1/2}$ in the dark-matter regime, which is stronger than the current BICEP tensor bound for $m_a f_{\rm dm}^2 \gtrsim 10^{-26.4}\,{\rm eV}$. In the intermediate and dark-energy regimes the isocurvature power is strongly suppressed by Jeans and horizon effects, so the paper concludes that any primary-CMB detection there would signal physics beyond the standard frozen-field misalignment scenario. The payoff is a small region near $m_a \sim 10^{-25}\,{\rm eV}$, $f_{\rm dm} \gtrsim 0.1$, where both axion isocurvature and tensor modes could be discovered below current bounds while easing the $S_8$ tension.

What carries the argument

The load-bearing mechanism is the effective time average (ETA) implemented in the Boltzmann code AxiECAMB, which replaces the exact Klein-Gordon evolution of a light axion with a fluid description after a switch epoch $m_a/H_*$, chosen here as 20 with AccuracyBoost=2 for isocurvature modes. The ETA decomposition and its validity criterion (Eq. A5) determine when the averaging is trustworthy; for isocurvature modes with $k \gg k_J$ the criterion fails and the code falls back to instantaneous values, a limitation the paper quantifies. The other load-bearing element is the abundance fitting formula (Eqs. 8–10), which connects the initial field value $\phi_{\rm ini}$ to the present abundance $\Omega_a h^2$ across all masses, and through it converts the isocurvature amplitude $A_{\rm iso}$ into the tensor-to-scalar ratio $r_a$ via $A_{\rm iso}/A_s = (r_a/2)(M_{\rm pl}/\phi_{\rm ini})^2$.

What would settle it

Compute the isocurvature CMB power spectrum for $m_a = 10^{-28}\,{\rm eV}$, $f_{\rm dm}=1$ with a switch at $m_a/H_* = 50$ and AccuracyBoost=3; if the low-$\ell$ plateau or quadrupole moves by more than the quoted percent-level accuracy, the ETA-based numerical claim is not converged. Alternatively, detect isocurvature at $m_a \sim 10^{-30}\,{\rm eV}$ at the level predicted by $r=0.036$; the paper predicts that signal is unobservable, so such a detection refutes its conclusion.

Watch

Extended reading notes

Core claim

The central discovery is that axion isocurvature is not merely a CDM analogue but a mass-dependent CMB observable with three regimes. For $m_a \gg H_{\rm eq}$ the axion isocurvature transfer function is indistinguishable from CDM isocurvature, so Planck's $\beta_{\rm iso} < 0.038$ bound applies directly and yields the $r f_{\rm dm}$ inequality of Eq. (22). For $H_0 \ll m_a \lesssim H_{\rm eq}$, Jeans suppression cuts off the spectrum with a $\sim(\ell_J/\ell)^6$ falloff, breaking the degeneracy with CDM and leaving unique scale-dependent signatures, but also making the inflationary signal undetectable even under optimistic cosmic-variance-limited forecasts. For $m_a \lesssim H_0$, the axion behaves as dark energy and the isocurvature signal peaks at the quadrupole, scaling as $(m_a/H_0)^2$; a detection there would contradict the standard inflationary production mechanism. The paper's quantitative claim is that the crossover where isocurvature constraints beat tensor constraints occurs at $m_a f_{\rm dm}^2 \gtrsim 10^{-26.4}\,{\rm eV}$, with an open coexistence window near $10^{-25}\,{\rm eV}$.

Load-bearing premise

The whole mapping rests on the axion's symmetry being broken before inflation ends with the field frozen in a nearly quadratic potential, so that its present-day density and the size of its quantum fluctuations both follow from one initial field value; change that setup and the derived bounds on the inflationary energy scale no longer follow.

Editorial extensions

If this is right

In the dark-matter-like regime, any measurement of axion mass and abundance plus a null isocurvature search strengthens the inflation probe: Planck data already require $r f_{\rm dm} < 0.076\,(m_a/10^{-27}\,{\rm eV})^{-1/2}$ (95% CL).
For $m_a f_{\rm dm}^2 \gtrsim 10^{-26.4}\,{\rm eV}$, the isocurvature bound is the tighter of the two independent inflationary probes, superseding the BICEP tensor limit.
In the intermediate mass range $10^{-32} \lesssim m_a/{\rm eV} \lesssim 10^{-28}$, the Jeans-scale cutoff makes the isocurvature spectrum distinguishable from CDM isocurvature, but even a cosmic-variance-limited CMB experiment cannot detect the inflationary signal; a detection would require non-inflationary generation.
In the dark-energy regime $m_a \lesssim H_0$, only the CMB quadrupole carries the isocurvature signal, and the BICEP tensor bound forces it far below detectability, so any observed signal would indicate a breakdown of the frozen-field misalignment scenario.
The coexistence window near $m_a \sim 10^{-25}\,{\rm eV}$, $f_{\rm dm} \gtrsim 0.1$ is a near-term discovery target: both axion isocurvature and tensor modes could appear just below current limits while simultaneously lowering $S_8$ to the weak-lensing preferred range.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not stated in the paper, but a direct corollary: the same $r f_{\rm dm}$ calibration means that a deep null search in the coexistence window would push the tensor-to-scalar bound below the current BICEP value using CMB temperature and polarization data already being analyzed.
A testable extension the paper leaves implicit: run an exact (non-time-averaged) field integration over $m_a \sim 10^{-30}{-}10^{-28}\,{\rm eV}$, where its own validity criterion (Eq. A5) fails for $k \gg k_J$; the result would either confirm the order-of-magnitude accuracy claimed there or sharpen the light-mass bounds.
The abundance fit (Eqs. 8–10) is also a tool for QCD axion relic-density computations, since the same field-to-fluid switching errors that the paper documents for isocurvature modes have historically biased axion abundance estimates by factors of order unity.
If a future CMB experiment detects isocurvature in the light regimes, the paper's logic implies that the amplitude, not just the presence, of the signal could be used to infer how much the axion's effective potential or initial field velocity departs from the frozen quadratic picture; that inversion is not performed here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper is worth engaging. It does something genuinely new: extends the ETA approach to axion isocurvature perturbations, provides analytic abundance fits across a huge mass range, and derives a constraint mapping that shifts the relevant mass window by orders of magnitude. The technical core is plausible. The initial conditions look right, the switch criterion in Eq. (A5) is a sensible guard against the ETA failing for fast Jeans oscillations, and the accuracy tests in Appendix A2 are thorough, including comparisons against exact KG solutions and a documented order-of-magnitude regime. The code is public, and the paper is honest about its own limits. Footnote 5 flags the normalization discrepancy with Ref. [12], and Appendix A1 states where accuracy degrades.

The soft spots are real but localized. The unresolved factor-of-4 discrepancy with Ref. [12] is the main one. The authors attribute it to an improperly normalized analytic expression in that earlier work, but they do not rerun a Planck MCMC to confirm that their new mapping holds up against actual data. The old MCMC may have captured degeneracies or transfer-function effects that the analytic route misses. Since the headline bound in Eq. (22) comes from the analytic abundance relation plus the Planck beta_iso constraint, this unresolved factor directly shifts the mass threshold where isocurvature beats the tensor bound, by up to a factor of ~16 in m_a. That is not a cosmetic issue; it moves the claimed coexistence window. The paper should either run the MCMC or explain quantitatively why the old chain was wrong. The abstract's "sub-percent" accuracy claim also overstates the body, which reports 1-10% accuracy at the lighter end. That is minor, but worth fixing. The k >> k_J failure is a genuine limitation, though it affects the suppressed regimes rather than the main dark-matter bound.

Who is this for? Anyone working on axion cosmology, CMB constraints on inflation, or ultralight dark matter. It deserves a serious referee. My recommendation: send it to peer review, with a requirement that the authors resolve the Ref. [12] normalization discrepancy before publication. The paper does the field a service by providing tools and scaling relations, but the headline number should rest on a verified foundation.

Referee Report

3 major / 4 minor

Summary. This manuscript extends the effective time average (ETA) treatment in AxiECAMB v1.1 to inflationary axion isocurvature perturbations, gives a fitting formula for the present axion abundance as a function of the initial field value φ_ini across masses from the dark-energy regime to the dark-matter regime, and uses that mapping to translate the Planck bound on CDM isocurvature, β_iso < 0.038, into the constraint r_a f_dm < 0.076 (m_a/10^-27 eV)^-1/2 (Eq. 22). The paper also derives analytic scalings for Jeans-suppressed intermediate masses and for the quadrupole-dominated dark-energy regime, presents Fisher forecasts, and identifies a coexistence window near m_a ~ 10^-25 eV, f_dm ≳ 0.1 where axion isocurvature and tensor modes could both be detected. The appendices document the code implementation, the criterion (Eq. A5) for when ETA is valid, and accuracy tests against lower-accuracy codes.

Significance. If the central constraint in Eq. (22) is correct, the paper establishes the CMB isocurvature channel as a competitive probe of the inflationary energy scale for ultralight axions, with a bound that is stronger than BICEP/Keck in a well-defined mass range and with falsifiable predictions in three distinct mass regimes. The submission has notable strengths: the code is public, the analytic fitting forms are explicitly calibrated and accompanied by accuracy tests in Appendix A, and the limitations of the ETA approximation are stated honestly, including order-of-magnitude accuracy in the k >> k_J regime. These features make the results reproducible. The significance is conditional on resolving the normalization discrepancy with an existing full Planck MCMC analysis and on qualifying the low-accuracy regime, both discussed below.

major comments (3)

[Footnote 5; Eq. (22)] The headline bound is not yet settled. The paper reports r_a < 0.0024 f_dm^-1 at m_a = 10^-24 eV while the full Planck MCMC analysis of Ref. [12] with overlapping authorship found r_a ≲ 10^-2 at f_dm = 1, a factor of about four. The footnote attributes this to an improper normalization in Ref. [12] and states that the issue is not explored further. Because Eq. (22) is derived from the analytic abundance mapping and not from a new MCMC, this discrepancy directly affects the claimed threshold where isocurvature beats the tensor bound (Eq. 23) and the coexistence window in Fig. 5; a factor of four in r_a shifts the mass threshold by about a factor of sixteen. I request either a Planck MCMC rerun with AxiECAMB v1.1 or a detailed quantitative reconciliation showing that the old normalization error accounts for the full difference and that no degeneracy or transfer-function effect is missing.
[Appendix A1; Sec. III B] The abstract's claim that AxiECAMB v1.1 'accurately evolve[s] these perturbations across the full axion mass range' is stronger than the validation supports. Equation (A5) fails for k >> k_J, and the code then reverts to instantaneous field values with only order-of-magnitude accuracy; this is the Jeans-suppressed regime that produces the unique intermediate-mass and lightest dark-energy signatures in Figs. 6 and 7. The authors should either improve the treatment of this regime or explicitly restrict the accuracy claims to the masses and multipoles for which the predictions are better than order of magnitude, and adjust the abstract and Section III accordingly.
[Eqs. (17), (30), Appendix A2] The calibrations C_P = 0.32 and C_Q = 0.0097 are fit to AxiECAMB at ℓ = 2 and then used as analytic predictions, while the abundance formula in Eqs. (8)-(10) is calibrated to and validated against the same code. This means the quoted sub-percent accuracy and the plateau/quadrupole estimates are internally consistent but not independent validations of the isocurvature transfer functions. I ask for at least one independent cross-check: e.g., the CDM isocurvature spectrum from CAMB for the m_a ≫ H_eq plateau, or a direct likelihood comparison with Ref. [12], together with a statement of how much the derived r_a bounds change under the calibration uncertainty.

minor comments (4)

[Sec. II, around Eq. (12)] The double use of τ for conformal time and reionization optical depth is acknowledged in the text, but replacing one of the symbols would reduce confusion.
[Appendix A1, Eq. (A5)] The threshold 10^2 in Eq. (A5) is set without a derivation; a brief derivation or a convergence test justifying this value would strengthen the appendix.
[Fig. 1 caption] The caption refers to 'points, blue solid curve' for the AxiECAMB result, but the figure appears to show only a solid curve; please clarify the legend.
[Sec. IV] The statement that the analytic estimates of Ref. [12] shift the relevant mass range by 'orders of magnitude' is hard to reconcile with the factor-of-four normalization discrepancy quoted in footnote 5; please state the resulting mass shift precisely.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: Eq. (22) is anchored by Planck's external beta_iso bound and the analytic F_1/3 normalization; only the internally calibrated abundance and power-spectrum fits are self-referential and non-load-bearing.

fitted input called prediction [Sec. II, Eqs. (8)-(10) and Fig. 1; Sec. III.A, Eq. (17); Sec. III.C, Eq. (30)]
"We use the numerical solutions from AxiECAMB to interpolate between these scaling behaviors across the full mass range. ... In Fig. 1, we show the comparison between the numerical result and the analytic fitting function F. ... Numerical evaluation with AxiECAMB gives C_P = 0.32 instead of the analytic estimate given in Eq. (16)."

The analytic fitting form F(x) and the constants x0, d, C in Eqs. (8)-(10), plus the normalization constants C_P = 0.32 and C_Q = 0.0097, are calibrated to AxiECAMB output. Equations (11)-(12) then use this same fitted phi_ini to obtain the isocurvature amplitude Aiso, and Eq. (17) uses the AxiECAMB-calibrated C_P to predict the low-ell plateau. The sub-percent agreement in Fig. 1 therefore only demonstrates that the fit reproduces the numerical data it was fitted against, not an independent check. This is a genuine self-referential calibration loop, but it is not load-bearing for the headline dark-matter bound (Eq. 22), which instead uses the analytic radiation-dominated normalization F_1/3 (Eq.

full rationale

The central quantitative claim, Eq. (22), is not circular. It is obtained by combining the externally measured Planck bound beta_iso < 0.038 (Eq. 19) with the analytic radiation-dominated normalization F_1/3 (Eq. 4) and the definition of beta_iso (Eq. 20); none of the constants entering Eq. (22) are fitted to the data being constrained. The mapping between axion and CDM isocurvature in the dark-matter regime is checked against the independent CAMB CDM-isocurvature transfer functions (Figs. 3-4). The ETA method is imported from Refs. [32,34], which share authorship, but the paper adds its own convergence tests at ma/H* = 20 with AccuracyBoost = 2, and openly documents where the ETA fails (Eq. A5 and Appendix A1); this is a methodological lineage rather than a logical circle. The abundance fitting formula (Eqs. 8-10) and the C_P/C_Q calibrations are indeed fit to AxiECAMB, so their quoted sub-percent accuracy is self-referential, but these fits are not used to derive Eq. (22). The footnote-5 factor-of-four discrepancy with Ref. [12] is an unresolved correctness risk, since the authors do not rerun the Planck MCMC with AxiECAMB to confirm the new normalization; however, rejecting an earlier self-result is not the same as deriving a result from its own inputs. On balance, the load-bearing bound is anchored by external data and analytic asymptotics, with only a minor self-calibration loop, so the circularity score is low.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The paper's central mapping (Eqs 8-12) rests on three fitted constants (x0, d, C) and two calibrated observables (C_P, C_Q), all obtained from the authors' own AxiECAMB code; the DM-regime normalization itself is analytic (F_1/3, Eq 4). The physical assumptions are the standard frozen-field misalignment scenario and spectator-field dynamics during inflation, both stated and both explicitly conditional in Sec IV. No new particles, forces, or fields are introduced: the ultralight axion is a pre-existing BSM candidate, and the ETA fluid is a numerical representation.

free parameters (6)

x0 (tanh transition center) = 0.23 (in log10(ma/Heq))
Center of the tanh transition in the abundance fit (Eq 9), fitted to AxiECAMB numerical abundance across mass.
d (tanh transition width) = 1.52
Width of the tanh transition in Eq 9, fitted to numerical abundance results.
C (matching constant) = 0.2
Constant in Eq 10 adjusted so the oscillating and frozen regimes join at the present epoch; stated to suffice for LCDM cosmologies.
C_P (low-ell plateau calibration) = 0.32
Calibration of the isocurvature low-ell plateau (Eq 17) at ell = 2 from AxiECAMB output, replacing the analytic estimate 0.34.
C_Q (dark energy quadrupole calibration) = 0.0097
Calibration of the DE-regime quadrupole coefficient (Eq 29) from AxiECAMB in the fde = 1, ma << H0 limit.
a_osc^3 = 4 at m_de = 4 (defines m_de ~ 1.58 H0)
Hand-chosen value in Eq 10 so that all lighter masses return the frozen-field abundance of Eq (7).

assumptions (7)

domain assumption PQ symmetry broken before inflation ends, with f_a > H_inf and phi_ini << f_a, giving a nearly quadratic potential (frozen-field misalignment).
Sec I states 'we further assume the axion field acquires a value phi_ini << f_a, then its subsequent evolution is in an effectively quadratic potential'. Load-bearing for Eqs (3)-(12) and for all derived r_a bounds.
domain assumption The axion is a light spectator during inflation, with quantum fluctuations of order H_inf/2pi (Eq 1).
Sec II, Eq (1): standard for a massless spectator; requires m_a << H_inf and forms the link between isocurvature amplitude and the tensor-to-scalar ratio.
domain assumption The ETA decomposition is valid: coefficients vary slowly compared to the mass timescale (Eq A2), with validity check Eq (A5).
Appendix A1. Fails for isocurvature modes with k >> k_J, where the code reverts to instantaneous values and accuracy degrades to order of magnitude.
domain assumption The isocurvature spectrum is uncorrelated with curvature and taken scale invariant (n_a = 0) for the constraints.
Sec II: 'we take a scale-invariant isocurvature spectrum n_a = 0, consistent with tensor upper bounds'. Used in the Planck beta_iso translation.
domain assumption Planck 2018 LCDM fiducial parameters and standard CMB machinery (RecFast recombination, Takahashi halofit for lensing).
Sec I and Appendix A2; inputs from prior literature, not derived in the paper.
standard math Standard synchronous-gauge linear perturbation theory with isocurvature initial conditions in which only the axion field fluctuates (Eq 13).
Sec II, Eq (13) and transfer function definitions (Eqs 14-15); the CAMB family convention.
standard math Separate-universe / compensated-isocurvature picture for superhorizon modes, giving T_iso(k -> 0) ~ (Omega_b + Omega_c)/(Omega_b + Omega_dm).
Sec III.A, Eqs (16)-(18), citing Hu and Sugiyama 1995 and Hu et al. 2016.

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Pith. "Pith review of Inflationary Axion Isocurvature in the CMB across All Ultralight Masses." pith.science (2026). https://pith.science/paper/DHG57IVN

@misc{pith2026260807803,
  author       = {Pith},
  title        = {Pith review of: Inflationary Axion Isocurvature in the CMB across All Ultralight Masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHG57IVN}},
  note         = {Machine review of arXiv:2608.07803}
}

abstract

If the Peccei-Quinn symmetry of an ultralight axion is broken before the end of inflation, axion quantum fluctuations seed isocurvature perturbations, linking them to the tensor-to-scalar ratio $r$. We extend the effective time average (ETA) approach of the Boltzmann code ${\rm AxiECAMB}$ to accurately evolve these perturbations across the full axion mass range from dark energy ($m_a \lesssim H_0$) to dark matter ($m_a \gg 10^{-28}$ eV) types. We provide analytic fitting formulae for the axion abundance given the initial field value $\phi_{\rm ini}$, accurate at sub-percent level for $m_a \gg H_0$ and allowed dark matter fraction $f_{\rm dm}$. In the dark matter regime, the Planck bound on CDM isocurvature requires $r\ f_{\rm dm} < 0.08\,(m_a/10^{-27}\,{\rm eV})^{-1/2}$, which becomes stronger than the current BICEP tensor bound for $m_a f_{\rm dm}^2 \gtrsim 10^{-26.4}\,{\rm eV}$. For $10^{-32} \lesssim m_a/{\rm eV} \lesssim 10^{-28}$, Jeans suppression breaks the degeneracy with CDM isocurvature, leaving unique signatures, and in the dark energy regime ($m_a \lesssim H_0$), the isocurvature signal is even more highly suppressed, peaking only at the CMB quadrupole. We provide analytic scalings for both signatures. Given the tensor bound, any primary CMB detection in these two lightest regimes would indicate a non-inflationary origin of the isocurvature modes or a breakdown of the standard frozen-field misalignment scenario. A window of coexistence opens near $m_a \sim 10^{-25}$ eV and $f_{\rm dm} \gtrsim 0.1$ where both axion isocurvature and tensor modes could be discovered just below current bounds while simultaneously alleviating the $S_8$ tension.

Figures

Figures reproduced from arXiv: 2608.07803 by the authors.

**Figure 3.** FIG. 3. CMB temperature power spectrum [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. Axion isocurvature transfer function [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (13 more)

**Figure 5.** Figure 5: FIG. 5. Upper limit (95% CL diagonal lines) on tensor [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: illustrates these effects on the isocurvature TT power spectrum across a wide range of masses ma/eV ∈ [10−34 , 10−24]. Here we adopt fixed parameters at Aiso = 1.6 × 10−7 , Ωah 2 = 0.0024, Ωch 2 = 0.1176 which for the m ≫ H0 cases is equivalent to fdm = 0.02, βiso = 0.…

**Figure 7.** Figure 7: FIG. 7. Axion isocurvature transfer function [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Fisher forecast of the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Fisher forecast of the [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10. Fractional error in the predicted axion abundance [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

**Figure 11.** Figure 11: FIG. 11. Background axion density evolution [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 13.** Figure 13: FIG. 13. Fractional accuracy of the temperature quadrupole [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 12.** Figure 12: FIG. 12. Isocurvature [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 14.** Figure 14: FIG. 14. Axion isocurvature transfer function [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]

**Figure 16.** Figure 16: FIG. 16. Isocurvature [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]

**Figure 17.** Figure 17: FIG. 17. Fractional accuracy of the temperature quadrupole [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]

**Figure 6.** Figure 6: The CQ estimate matches the scaling of the f 2 de dependence of ℓ = 2 observables for all ma ≲ 10−32.5 eV whereas the CP estimate matches the f 2 dm scaling for all ma ≳ 10−31 eV. 3. Isocurvature Parameters and Interface AxiECAMB computes axion isocurvature spectra fro…

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Reviewed August 11, 2026 · model on record in the stance chip above.

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ET A AxiECAMBimplements an effective average over field oscillations on the inverse mass timescale from Ref. [32]. Here the background fieldϕand its pertur- bationδϕare decomposed into ϕ=φ c cos[ma(t−t ∗)] +φ s sin[ma(t−t ∗)], δϕ=δφ c cos[ma(t−t ∗)] +δφ s sin[ma(t−t ∗)],(A1) wheret ∗ is the switch epoch that corresponds to a given H∗/ma. This redundant sy...

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Accuracy T ests a.m a >10H 0 Form a >10H 0, the switch from solving the Klein- Gordon equations and outputting instantaneous values of axion quantities to ETAs occurs before or at the present 13 100 101 102 ma/H 0.01 0.02 0.03 (ρah2/ρcrit) a3 KG AxiECAMB axionCAMB FIG. 11. Background axion density evolutionρaa3 (in units ofρ crit/h2), versusma/Hform a = 1...

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