Primordial density perturbations may contain, besides the dominant adiabatic curvature perturbation, isocurvature (entropy) fluctuations of the cold-dark-matter density, neutrino density, or neutrino velocity, possibly correlated with the adiabatic mode. Any such mode would prove that more than one primordial degree of freedom was dynamically relevant during inflation, and concrete early-Universe scenarios (curvaton, axion-like dark matter, Affleck-Dine-type baryogenesis) generically make these perturbations non-Gaussian, with an observable signature in the cosmic microwave background (CMB) bispectrum. The Wilkinson Microwave Anisotropy Probe (WMAP)-era analysis of Hikage, Kawasaki, Sekiguchi and Takahashi (2013) constrained only the two leading nonlinearity-parameter combinations of the correlated case and looked forward to more accurate CMB measurements from Planck; that analysis assumed tree-level dominance (omitting loop contributions to the primordial bispectrum), neglected second-order (secondary) CMB contributions, excluded the neutrino-velocity mode because no concrete super-horizon generation models existed, and flagged survey-specific systematics for higher-sensitivity analyses to control. Planck subsequently constrained all six local-type nonlinearity parameters per isocurvature mode without a detection, and second-order intrinsic bispectra from isocurvature initial conditions have been computed. The same determination nonetheless remains open at detection level: no isocurvature non-Gaussianity has been detected and the mixed nonlinearity parameters are still only broadly bounded, while related theoretical questions - loop corrections to the mixed primordial bispectrum, concrete neutrino-velocity generation models, and the non-local bispectrum shapes that realistic models predict - remain unresolved and bear on the sensitivity and interpretation of any future determination.

Background

The cosmic microwave background (CMB, the relic radiation released when the primordial plasma became transparent) has anisotropies that trace perturbations laid down during inflation, an early phase of accelerated expansion. In the simplest single-field inflation models there is only one primordial degree of freedom, so all particle species share the same fractional density perturbation; this is the adiabatic mode, described by the curvature perturbation ζ\zeta. If a second light scalar field fluctuated during inflation - for example the curvaton, a massive field that later decays and contributes to the curvature perturbation, or an axion-like field making up part of the dark matter - the entropy between species can also fluctuate. These isocurvature perturbations are defined as Si=3(ζi−ζtot)S_i=3(\zeta_i-\zeta_{\rm tot}), the difference between the curvature perturbation of species ii and that of total matter. At the epoch of recombination there are four physically distinct isocurvature modes: cold-dark-matter density (CI), baryon density (nearly indistinguishable from CI in the CMB), neutrino density (NID), and neutrino velocity (NIV, in which the initial velocity perturbations of neutrinos and photons differ). A detection of any isocurvature mode would rule out all single-field inflation models. No isocurvature mode has been detected; CMB power-spectrum data bound their amplitude, conventionally quoted as the ratio α=PSS(k∗)/Pζζ(k∗)\alpha=P_{SS}(k_*)/P_{\zeta\zeta}(k_*) of isocurvature to adiabatic power at a reference scale, to α≲0.01\alpha\lesssim 0.01 for correlated CI.

Because the nonlinear transfer of a second field's fluctuations can be substantial, isocurvature perturbations can be markedly non-Gaussian, and their non-Gaussianity (NG) can constrain early-Universe models in parameter regions where power-spectrum limits are weak. The lowest-order NG statistic is the bispectrum, the three-point function of the CMB harmonic coefficients aℓma_{\ell m}. For the 'local' ansatz ζ=ζG+(3/5)fNL(ζG2−⟨ζG2⟩)\zeta=\zeta_{\rm G}+(3/5)f_{\rm NL}(\zeta_{\rm G}^2-\langle\zeta_{\rm G}^2\rangle), built from a Gaussian field ζG\zeta_{\rm G}, the primordial bispectrum is largest for squeezed triangle configurations (one momentum much smaller than the other two) and is fully characterized by the nonlinearity parameter fNLf_{\rm NL}. When one isocurvature mode SS with arbitrary correlation with ζ\zeta is present, Langlois and van Tent showed that the local-type bispectrum decomposes into six elementary pieces with amplitudes fNLI,JKf_{\rm NL}^{I,JK} (I,J,K∈{ζ,S}I,J,K\in\{\zeta,S\}, symmetric in J,KJ,K): one purely adiabatic, one purely isocurvature, and four mixed ones such as the ⟨ζζS⟩\langle\zeta\zeta S\rangle bispectrum. CMB E-mode polarization (the curl-free component of the CMB polarization field) is disproportionately powerful for these parameters, because the isocurvature polarization signal survives to smaller angular scales than the temperature signal, improving the constraints by roughly a factor of six.

Hikage, Kawasaki, Sekiguchi and Takahashi (2013) applied an optimal (minimum-variance, likelihood-based) CMB bispectrum estimator to the 7-year temperature maps of the Wilkinson Microwave Anisotropy Probe (WMAP) and constrained the nonlinearity parameters for uncorrelated and totally correlated CI and NID modes. In the correlated case they constrained only the two leading observable combinations, fNL=f~NLζ,ζζf_{\rm NL}=\tilde f_{\rm NL}^{\zeta,\zeta\zeta} and αfNLISO=f~NLS,ζζ\alpha f_{\rm NL}^{\rm ISO}=\tilde f_{\rm NL}^{S,\zeta\zeta}, because the remaining four parameters are suppressed by powers of the small ratio α\alpha; they verified that including the O(α)\mathcal{O}(\sqrt\alpha) and O(α)\mathcal{O}(\alpha) terms shifts their results by at most ∼0.1σ\sim 0.1\sigma at α=0.003\alpha=0.003. Their analysis explicitly omitted loop contributions to the primordial bispectrum (a tree-level dominance assumption), secondary NG generated by second- and higher-order cosmological perturbation theory, and the NIV mode, for which they stated no concrete super-horizon generation models existed; they also flagged survey-specific systematics - error inflation from inhomogeneous noise and sky cuts for NID, and central-value biases up to 0.5σ0.5\sigma from foreground template marginalization - as items for higher-sensitivity analyses to control.

The observational situation has since advanced. The Planck satellite's dedicated NG analyses (2015 and final 2018) constrained all six fNLI,JKf_{\rm NL}^{I,JK} per isocurvature mode - CDM density, neutrino density, and neutrino velocity - from temperature and E-mode polarization, both independently and jointly, with the gravitational-lensing bispectrum bias subtracted, and found no signal. Montandon, Patanchon and van Tent (2021) then built the joint CMB power-spectrum-plus-bispectrum likelihood for Planck 2018 data, including the NIV mode, and forecast the gains achievable with the planned LiteBIRD satellite and CMB-S4 ground-based experiment. On the theory side, Carrilho and Malik developed second-order Boltzmann transfer with isocurvature initial conditions and computed the intrinsic (secondary) CMB bispectrum those initial conditions generate, while Geller, Kumar and Wang (2024) showed that concrete multi-field models predict mixed ⟨ζζS⟩\langle\zeta\zeta S\rangle bispectra whose shapes differ substantially from the local template, so that local-template searches can lose factors of 22-1010 in sensitivity. What has not been achieved: any detection, or a constraint or forecast improving on the Planck 2018 joint power-spectrum-plus-bispectrum analysis and the LiteBIRD/CMB-S4 forecasts of Montandon et al. (2021); a quantification of loop corrections to the mixed primordial bispectrum; concrete NIV generation models; and shape-optimized searches applied to real data.

Problem Statement

Are primordial isocurvature perturbations - CDM-density, neutrino-density, or neutrino-velocity entropy fluctuations, uncorrelated or correlated with the dominant adiabatic mode - non-Gaussian, and at what amplitude? The question is posed observationally on the full set of local-type nonlinearity parameters fNLI,JKf_{\rm NL}^{I,JK} (six per isocurvature mode, including the four mixed ones and the purely isocurvature one) from CMB temperature and polarization, where the Planck 2018 limits found no signal and left the mixed parameters only broadly bounded. Any data or analysis route that materially advances this determination is relevant - next-generation CMB experiments (e.g., LiteBIRD, CMB-S4) and/or large-scale-structure surveys, joint power-spectrum-bispectrum likelihood analyses, or searches for the non-local mixed bispectrum shapes that concrete early-Universe models (e.g., an axion-like dark-matter field kinetically mixed with a curvaton) predict, for which local-template analyses can lose factors of 22-1010 in sensitivity. A route materially advances the determination when it delivers a constraint or forecast that goes beyond the joint Planck 2018 power-spectrum-plus-bispectrum analysis and the LiteBIRD/CMB-S4 forecasts of Montandon et al. (2021) - either on one or more of the six fNLI,JKf_{\rm NL}^{I,JK} per isocurvature mode, from a reanalysis of existing data or a forecast for a named experiment, or on the amplitude of the non-local mixed bispectrum shape that a concrete early-Universe model predicts, together with how that amplitude maps onto those local-type parameters; such a constraint or forecast is the required deliverable, and the mapping and the other theoretical items named below condition its interpretation, so that work addressing only those items is a partial contribution and must state what it leaves open. Theoretical progress conditions the interpretation of any answer: the tree-level local ansatz omits loop corrections, whose impact on the mapping from model parameters (curvaton or axion parameters, correlations) to observable bispectrum amplitudes remains unquantified; the secondary and instrumental contributions - the intrinsic second-order bispectrum from isocurvature initial conditions, the lensing bispectrum, foreground template marginalization, inhomogeneous noise, and sky cuts - need to be accounted for at the precision of the determination; and the concrete model classes within which a detection would be interpreted include the curvaton, axion-like dark matter, and Affleck-Dine baryogenesis, a mechanism in which the evolution and decay of a complex scalar field generates the matter-antimatter asymmetry.

References

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