Neutrino Isocurvature: Theoretical Frameworks & Constraints

• Neutrino isocurvature is defined as primordial entropy perturbations in the neutrino density or velocity that deviate from standard adiabatic conditions.
• It is characterized by unique transfer functions affecting CMB temperature, polarization, and bispectra, serving as probes of early universe physics.
• Its study provides actionable insights into multi-field inflation, curvaton decay, dark radiation sectors, and gravitational wave induction across cosmological scales.

Neutrino isocurvature denotes primordial entropy perturbations in the neutrino sector, i.e. perturbations in neutrino density or velocity relative to the photon–baryon plasma at fixed total curvature. In the standard taxonomy, the two canonical neutrino modes are neutrino density isocurvature, usually called NID or NDI, and neutrino velocity isocurvature, usually called NIV or NVI. In contemporary treatments, these modes are studied not only as alternatives to purely adiabatic initial conditions, but also as diagnostics of multi-field inflation, curvaton decay, lepton-asymmetry generation, dark-radiation sectors, and dark-matter production. Their phenomenology spans CMB temperature and polarization power spectra, CMB bispectra, spectral distortions, large-scale structure, ultraviolet luminosity functions, and even pulsar-timing-array probes of induced gravitational waves (Langlois et al., 2012, Gerlach et al., 23 Apr 2025).

1. Definitions, conventions, and mode taxonomy

On super-Hubble scales in the radiation era, adiabatic initial conditions require equal fractional number-density perturbations among species, which can be written as

$$\delta_c=\delta_b=\frac{3}{4}\delta_\nu=\frac{3}{4}\delta_\gamma.$$

A standard gauge-invariant entropy definition is

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$

where $\zeta_i$ is the curvature perturbation on uniform-density hypersurfaces for species $i$. For neutrinos, a convenient density entropy variable is

$S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$

while on superhorizon scales one may equivalently use

$$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$

In this language, neutrino density isocurvature is a primordial neutrino–photon entropy perturbation, whereas neutrino velocity isocurvature is a primordial relative-velocity perturbation $V_{\nu\gamma}\neq 0$ or, in the notation of some CMB initial-condition analyses, $q_{\nu r}\neq 0$ (Co et al., 11 May 2026, Langlois et al., 2012, Savelainen et al., 2013).

For NID, the leading super-Hubble relation may be written

$$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$

For NIV, the defining feature is a compensated initial velocity pattern with adiabatic energy densities,

$$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$

together with

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$0

In CAMB-based phenomenological analyses, equivalent total-radiation conventions are often used, such as $S_{ij}\equiv 3(\zeta_i-\zeta_j),$1 and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$2, where $S_{ij}\equiv 3(\zeta_i-\zeta_j),$3 (Langlois et al., 2012, Savelainen et al., 2013).

A further extension replaces “neutrino” by a free-streaming dark-radiation sector. In that case the dark-radiation entropy mode $S_{ij}\equiv 3(\zeta_i-\zeta_j),$4 is cosmologically equivalent to a neutrino-density-type mode, and in the limit of vanishing extra radiation $S_{ij}\equiv 3(\zeta_i-\zeta_j),$5 one has $S_{ij}\equiv 3(\zeta_i-\zeta_j),$6. This equivalence underlies many analyses that map free-streaming dark-radiation isocurvature onto NID-like phenomenology (Kawasaki et al., 2011).

A more recent generalization observes that realistic neutrino-isocurvature generation often produces both neutrino and matter isocurvature. That admixture can be parameterized by a mixing angle

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$7

with $S_{ij}\equiv 3(\zeta_i-\zeta_j),$8 corresponding to pure matter isocurvature and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$9 to pure neutrino isocurvature. In this framework, the commonly used compensated neutrino isocurvature mode corresponds to a specific angle,

$\zeta_i$0

This makes “pure neutrino isocurvature” a special limit rather than the generic outcome of all neutrino-sector scenarios (Gerlach et al., 23 Apr 2025).

2. Linear evolution and characteristic signatures

Neutrino isocurvature modes are distinguished observationally by their transfer functions. In the CMB bispectrum and power-spectrum literature, the adiabatic transfer $\zeta_i$1 exhibits the standard acoustic-peak pattern and Sachs–Wolfe behavior, whereas the NID transfer $\zeta_i$2 has a different acoustic phase relative to both the adiabatic and CDM-isocurvature cases. Its large-scale TT and EE spectra fall off less steeply than CDM isocurvature and remain appreciable to high multipole. The NIV transfer $\zeta_i$3 is driven by an initial velocity pattern and has a particularly strong polarization response, especially in squeezed configurations, because the relevant low-$\zeta_i$4 kernel grows steeply when moving from temperature to $\zeta_i$5-polarization (Langlois et al., 2012).

This qualitative difference is visible already in WMAP-era phenomenology. For scale-invariant spectra of equal amplitude, CDM and neutrino density isocurvature place the first TT acoustic peak to the right of the adiabatic first peak, whereas neutrino velocity isocurvature places it slightly to the left. In the WMAP-9 analysis, the nonadiabatic contribution to the CMB temperature variance could be as large as $\zeta_i$6 for NDI and $\zeta_i$7 for NVI, reflecting the different phase structure and preferred sign of correlation in the fits (Savelainen et al., 2013).

In more recent analyses that combine high-resolution CMB data with late-time probes, NDI is found to produce little Sachs–Wolfe plateau, with power rising from $\zeta_i$8, peaking around $\zeta_i$9, and then declining, while leaving the high-$i$0 damping tail comparatively less altered than a neutrino-mass signal. This separation in scale dependence is precisely why current $i$1 constraints weaken only marginally when an uncorrelated NDI component is added: NDI primarily alters large-scale CMB and intermediate-scale matter clustering, while massive neutrinos suppress the damping tail and broad-band small-scale growth (Hou et al., 16 Jun 2026).

The generalized neutrino–matter mixing-angle picture makes the large-scale structure of the signal explicit. In that notation, the large-scale Sachs–Wolfe contribution can be written as

$i$2

The observable therefore depends not only on the isocurvature amplitude but also on whether the perturbation points more strongly in the matter or neutrino direction. This is the basis for the statement that Planck constraints are weakest near the pure-neutrino limit $i$3 (Gerlach et al., 23 Apr 2025).

3. CMB bispectrum and primordial non-Gaussianity

With one adiabatic mode $i$4 and one isocurvature mode $i$5, the local-type reduced CMB bispectrum admits a six-component decomposition,

$i$6

with the six amplitudes

$i$7

In the squeezed limit, the bispectrum factorizes into a product of large-$i$8 angular power and a low-$i$9 kernel, which makes mixed adiabatic–isocurvature components especially important: they inherit large short-scale adiabatic power while remaining sensitive to the low-$S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$0 isocurvature kernel. For NID, adding $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$1-polarization improves all six amplitudes rather uniformly; for NIV, polarization is decisive for mixed and pure isocurvature pieces, particularly $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$2, $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$3, and $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$4 (Langlois et al., 2012).

Under Planck-like assumptions with $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$5, the marginalized $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$6 uncertainties for NID with $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$7 are

$S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$8

to be compared with $S_{\nu\gamma}\equiv 3(\zeta_\nu-\zeta_\gamma),$9-only

$$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$0

For NIV, the corresponding $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$1 uncertainties are

$$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$2

versus $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$3-only

$$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$4

These numbers encapsulate a central qualitative result: polarization improves NID constraints broadly, but it improves NIV constraints dramatically because NIV has unusually strong EE sensitivity (Langlois et al., 2012).

An earlier WMAP-7 temperature-only bispectrum analysis of NID found no significant deviation from Gaussianity. In the uncorrelated case the constrained combination was $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$5, with representative non-marginalized values $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$6 or $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$7 depending on tilt and foreground marginalization. In the totally correlated case the leading constrained combination was $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$8, with representative non-marginalized values $$S_{\nu d}\equiv \frac{3}{4}(\delta_\nu-\delta_\gamma).$$9 or $V_{\nu\gamma}\neq 0$0. The results were consistent with Gaussian primordial perturbations at around the $V_{\nu\gamma}\neq 0$1 level (Hikage et al., 2012).

A subsequent joint power-spectrum–bispectrum treatment in a two-field inflation framework reached a more nuanced conclusion. For Planck, because neither isocurvature nor primordial non-Gaussianity was detected, the joint likelihood does not improve constraints in the general five-parameter model. For LiteBIRD+CMB-S4, however, the region of parameter space where a joint analysis helps is much larger for neutrino isocurvature than for CDM isocurvature; for NIV it can cover about half of the Planck-allowed region, with isocurvature error bars reduced by up to $V_{\nu\gamma}\neq 0$2, and standard-parameter errors improved by up to $V_{\nu\gamma}\neq 0$3 for $V_{\nu\gamma}\neq 0$4 (Montandon et al., 2020).

4. Nonlinear and small-scale probes beyond CMB anisotropy power

On scales well below those directly accessible to CMB anisotropy spectra, neutrino isocurvature can be probed through dissipation and second-order mode coupling. In the spectral-distortion formalism, NID and NIV source $V_{\nu\gamma}\neq 0$5- and $V_{\nu\gamma}\neq 0$6-distortions through Silk damping with mode-dependent heating efficiencies. For the fiducial $V_{\nu\gamma}\neq 0$7, the paper gives

$V_{\nu\gamma}\neq 0$8

Thus, for the same small-scale power, NID dissipates about $V_{\nu\gamma}\neq 0$9 times less efficiently than the adiabatic mode, while NIV is lower than adiabatic by about a factor $q_{\nu r}\neq 0$0. Using $q_{\nu r}\neq 0$1-distortion forecasts, PIXIE could exclude scale-invariant NID power with amplitude $q_{\nu r}\neq 0$2 at $q_{\nu r}\neq 0$3 at $q_{\nu r}\neq 0$4 if $q_{\nu r}\neq 0$5 is not detected, while COBE/FIRAS implies $q_{\nu r}\neq 0$6; the corresponding NIV constraints are about $q_{\nu r}\neq 0$7 times tighter (Chluba et al., 2013).

A different nonlinear channel is anisotropic $q_{\nu r}\neq 0$8-distortion sourced by a non-Gaussian NID field of Gaussian-squared form,

$q_{\nu r}\neq 0$9

In that model, the CMB power spectrum constrains $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$0 at $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$1, but the induced bispectrum can still be large. For uncorrelated, scale-invariant NID with $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$2, the mean distortion and temperature cross-correlation can reach

$$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$3

and a blue tilt $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$4 enhances $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$5 by about an order of magnitude (Ota et al., 2014).

At much smaller scales, PTA-frequency modes enter the horizon before neutrino decoupling. In that regime the standard free-streaming neutrino initial conditions are not appropriate. A dedicated pre-decoupling treatment shows that neutrino density isocurvature is phenomenologically similar to baryon or CDM isocurvature up to an overall coefficient

$$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$6

so that

$$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$7

for induced gravitational waves. For a representative $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$8, the paper recasts the CDI peak constraint near $$\frac{3}{4}\delta_\nu=S_{\nu d}+\frac{3}{4}\delta_\gamma,\qquad \delta_b=\delta_c=\frac{3}{4}\delta_\gamma.$$9 into

$$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$0

This extends neutrino-isocurvature phenomenology into the PTA band (Cai et al., 9 Dec 2025).

Second-order Boltzmann initial conditions after neutrino decoupling introduce an additional technical layer. Mixed modes such as AD$$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$1NDI, NDI$$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$2CDI, NDI$$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$3BDI, and CIP$$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$4NDI produce nontrivial super-horizon evolution even when one constituent mode is trivial at linear order. By contrast, neutrino velocity isocurvature generates non-regular decaying modes at second order and therefore requires special handling in nonlinear Boltzmann solvers (Carrilho et al., 2018).

5. Empirical constraints across scales

The WMAP-9 mixed-mode analysis remains a useful benchmark for phenomenological amplitudes. In the uncorrelated case, the $$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$5 upper limits on the primordial NDI fraction were $$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$6 at $$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$7 and $$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$8 at $$\mathcal V_\nu=S_{\nu v},\qquad \mathcal V_{\gamma b}=-\frac{7}{8}N_\nu\left(\frac{4}{11}\right)^{4/3}S_{\nu v},$$9; for NVI the corresponding limits were $S_{ij}\equiv 3(\zeta_i-\zeta_j),$00 and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$01. For maximally correlated and maximally anticorrelated NDI, the bound tightened to $S_{ij}\equiv 3(\zeta_i-\zeta_j),$02 and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$03, respectively. Bayesian model comparison favored pure adiabatic initial conditions over mixed neutrino-isocurvature models, with the least-disfavored mixed case being uncorrelated NDI at odds $S_{ij}\equiv 3(\zeta_i-\zeta_j),$04 relative to pure adiabaticity (Savelainen et al., 2013).

Planck sharpened the density-mode constraints considerably. In a joint $S_{ij}\equiv 3(\zeta_i-\zeta_j),$05–NID analysis with Planck 2013 temperature data plus WMAP low-$S_{ij}\equiv 3(\zeta_i-\zeta_j),$06 polarization, the posterior showed no evidence for a neutrino isocurvature density component: $S_{ij}\equiv 3(\zeta_i-\zeta_j),$07 for Planck+WP, and

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$08

for Planck+WP+HST. The anti-correlated branch was especially constrained once the HST prior was imposed, because it worsened the $S_{ij}\equiv 3(\zeta_i-\zeta_j),$09 tension rather than alleviating it (Valentino et al., 2014).

More recent CMB-SPA, DESI DR2, and DES Year 5 analyses have incorporated NDI directly into late-time parameter estimation. In $S_{ij}\equiv 3(\zeta_i-\zeta_j),$10CDM, allowing an uncorrelated NDI component changes the $S_{ij}\equiv 3(\zeta_i-\zeta_j),$11 upper bound on the neutrino mass sum only from $S_{ij}\equiv 3(\zeta_i-\zeta_j),$12 eV to $S_{ij}\equiv 3(\zeta_i-\zeta_j),$13 eV; in CPL dark energy the limit shifts from $S_{ij}\equiv 3(\zeta_i-\zeta_j),$14 eV to $S_{ij}\equiv 3(\zeta_i-\zeta_j),$15 eV. The inferred NDI amplitudes at both primordial pivots remain consistent with zero, so current neutrino-mass bounds are robust against this specific extension of initial conditions (Hou et al., 16 Jun 2026).

At intermediate and small scales, the first UVLF-based limits now reach neutrino isocurvature directly. Using Planck PR4/NPIPE, BAO, supernovae, HST UVLF, and JWST UVLF, the analysis constructs $S_{ij}\equiv 3(\zeta_i-\zeta_j),$16 and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$17 envelopes for uncorrelated NDI and NIV power over $S_{ij}\equiv 3(\zeta_i-\zeta_j),$18. For both modes, adding UVLF data improves the small-scale upper bounds by roughly $S_{ij}\equiv 3(\zeta_i-\zeta_j),$19–$S_{ij}\equiv 3(\zeta_i-\zeta_j),$20 at $S_{ij}\equiv 3(\zeta_i-\zeta_j),$21, and the high-$S_{ij}\equiv 3(\zeta_i-\zeta_j),$22 allowed envelope rises approximately like $S_{ij}\equiv 3(\zeta_i-\zeta_j),$23 with $S_{ij}\equiv 3(\zeta_i-\zeta_j),$24. For NDI, the $S_{ij}\equiv 3(\zeta_i-\zeta_j),$25 envelopes from broken and running parameterizations agree very well over $S_{ij}\equiv 3(\zeta_i-\zeta_j),$26, indicating that the two-sigma result is largely insensitive to the assumed spectral shape across the CMB–UVLF overlap (Co et al., 11 May 2026).

A complementary general analysis combines CMB, BAO, Ly-$S_{ij}\equiv 3(\zeta_i-\zeta_j),$27 forest, and CMB spectral distortions to constrain NDI across

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$28

For a broken power-law NDI spectrum, the $S_{ij}\equiv 3(\zeta_i-\zeta_j),$29 bound on $S_{ij}\equiv 3(\zeta_i-\zeta_j),$30 is approximately constant for $S_{ij}\equiv 3(\zeta_i-\zeta_j),$31 and rises as $S_{ij}\equiv 3(\zeta_i-\zeta_j),$32 for $S_{ij}\equiv 3(\zeta_i-\zeta_j),$33 in the CMB-dominated regime, and again as $S_{ij}\equiv 3(\zeta_i-\zeta_j),$34 once the Ly-$S_{ij}\equiv 3(\zeta_i-\zeta_j),$35 window is passed. For delta-function spectra, NDI distortion limits are nearly flat in $S_{ij}\equiv 3(\zeta_i-\zeta_j),$36 inside the $S_{ij}\equiv 3(\zeta_i-\zeta_j),$37- and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$38-windows because the transfer factor satisfies $S_{ij}\equiv 3(\zeta_i-\zeta_j),$39 (Buckley et al., 27 Feb 2025).

6. Model realizations, generalized frameworks, and open issues

One long-standing realization of neutrino density isocurvature is the curvaton scenario with a neutrino chemical potential or lepton asymmetry. In that setting,

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$40

and a nonzero $S_{ij}\equiv 3(\zeta_i-\zeta_j),$41 can leave residual NID after curvaton decay. Planck-era analyses therefore constrained $S_{ij}\equiv 3(\zeta_i-\zeta_j),$42 and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$43 jointly rather than independently (Valentino et al., 2014).

Dark-radiation constructions generalize this picture. In the mixed inflaton–curvaton formalism, the dark-radiation isocurvature mode $S_{ij}\equiv 3(\zeta_i-\zeta_j),$44 is the neutrino-density-type entropy perturbation of the combined $S_{ij}\equiv 3(\zeta_i-\zeta_j),$45 fluid, and it reduces to $S_{ij}\equiv 3(\zeta_i-\zeta_j),$46 when $S_{ij}\equiv 3(\zeta_i-\zeta_j),$47 is absent. With WMAP7+ACT+BAO+$S_{ij}\equiv 3(\zeta_i-\zeta_j),$48, the quoted $S_{ij}\equiv 3(\zeta_i-\zeta_j),$49 bounds were $S_{ij}\equiv 3(\zeta_i-\zeta_j),$50 for uncorrelated DR isocurvature and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$51 or $S_{ij}\equiv 3(\zeta_i-\zeta_j),$52 for totally correlated or anti-correlated cases. The same framework was applied to SUSY KSVZ and DFSZ axion models and to curvaton-generated extra-radiation scenarios (Kawasaki et al., 2011).

A non-Gaussian extension of that dark-radiation formalism derives six local-type amplitudes $S_{ij}\equiv 3(\zeta_i-\zeta_j),$53 for mixed curvature–DR bispectra and identifies concrete realizations in KSVZ-like, DFSZ-like, and Affleck–Dine lepton-asymmetry models. In the limit where extra species are negligible, the DR mode becomes ordinary neutrino density isocurvature, so these constructions directly generate NID non-Gaussianity as well (Kawakami et al., 2012).

The generalized mixing-angle framework sharpens a conceptual point: realistic cosmological scenarios need not produce pure neutrino isocurvature. Instead, they commonly yield a correlated pair $S_{ij}\equiv 3(\zeta_i-\zeta_j),$54, summarized by $S_{ij}\equiv 3(\zeta_i-\zeta_j),$55. Planck+BAO constraints are weakest near $S_{ij}\equiv 3(\zeta_i-\zeta_j),$56, the pure-neutrino limit, and stronger away from that axis. The paper also reports a mild preference for nonzero isocurvature in that weakly constrained region. This suggests that future analyses should quote limits not only on “neutrino isocurvature” in a fixed convention, but also on its admixture with matter isocurvature (Gerlach et al., 23 Apr 2025).

A physically explicit example is the superhorizon dark-radiation framework in which a decoupled radiation sector carries primordial entropy perturbations. The observable effective-neutrino isocurvature at recombination is then

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$57

while dark-matter production acquires matter isocurvature through Hubble-rate modulation,

$S_{ij}\equiv 3(\zeta_i-\zeta_j),$58

The coefficient $S_{ij}\equiv 3(\zeta_i-\zeta_j),$59 is approximately $S_{ij}\equiv 3(\zeta_i-\zeta_j),$60 for freeze-in and $S_{ij}\equiv 3(\zeta_i-\zeta_j),$61–$S_{ij}\equiv 3(\zeta_i-\zeta_j),$62 for freeze-out, so the sign of the mixed neutrino–matter isocurvature angle becomes a discriminator between the two mechanisms (Gerlach et al., 24 Oct 2025).

An adjacent extension replaces free-streaming neutrino-like radiation by coupled dark radiation. In that setting, the free-streaming case reproduces NID-like initial conditions with variable $S_{ij}\equiv 3(\zeta_i-\zeta_j),$63, while the coupled case yields qualitatively new behavior. One result is that for isocurvature initial conditions, free-streaming dark radiation produces larger CMB anisotropies than coupled dark radiation at fixed amplitude, opposite to the adiabatic case. The same analysis finds that a blue isocurvature tilt is generally preferred and that, once isocurvature is allowed, $S_{ij}\equiv 3(\zeta_i-\zeta_j),$64 can be larger than in adiabatic $S_{ij}\equiv 3(\zeta_i-\zeta_j),$65CDM+$S_{ij}\equiv 3(\zeta_i-\zeta_j),$66 fits (Ghosh et al., 2021).

Across these frameworks, several simplifying assumptions recur: one isocurvature carrier at a time, no adiabatic–isocurvature cross-correlation, equal adiabatic and isocurvature tilts, or local-type non-Gaussianity. These assumptions are often adopted for tractability and because current data do not require more elaborate parameterizations. A plausible implication is that future progress on neutrino isocurvature will depend less on any single dataset than on joint analyses that combine CMB temperature and polarization, higher-order statistics, late-time structure formation, spectral distortions, and small-scale probes under more general initial-condition priors.