Observationally discriminating the chiral gravitational-wave background of spectator axion-SU(2) inflation from the vacuum background: completing the CMB and laser-interferometer program

Inflation can source gravitational waves either from quantum vacuum fluctuations of the metric or from matter fields amplified during inflation, and these origins are difficult to tell apart with two-point CMB statistics alone. In the spectator axion-SU(2) model, an axion-like pseudoscalar rolling for a few e-folds amplifies one helicity of an SU(2) gauge-field fluctuation, producing a fully chiral gravitational-wave background (GWB) with a log-Gaussian spectrum peaking at an arbitrary scale. Thorne, Fujita, Hazumi, Katayama, Komatsu and Shiraishi (Phys. Rev. D 97, 043506 (2018)) forecast detection of this chiral signal through CMB TB/EB correlations with LiteBIRD and through the circular-polarization Stokes parameter V with advanced two-constellation space interferometers, and concluded that CMB two-point statistics alone give at most signal-to-noise of order 2-3 in the then-allowed parameter space. They explicitly deferred the analyses needed for a decisive discrimination between the sourced and vacuum GWBs. The open problem is to extend that program along the analyses the authors identified as needed but left open, which the extension is to cover in full (the paper presents two of them — the consistency-relation test and the higher-order-statistics route — as alternative routes to a highly significant discrimination): tensor non-Gaussianity forecasts, above all of the B-mode bispectrum; joint CMB-plus-interferometer tests of the single-field consistency relation $n_T=-r/8$ via the spectral lever arm; the phenomenology of the regime in which the spectator axion later acts as a curvaton; the degradation of circular-polarization sensitivity by realistic polarized astrophysical foregrounds and non-ideal mission designs; validated analytic templates outside the slow-roll regime; and a full model-selection comparison of vacuum versus sourced backgrounds. Later literature has resolved the tensor bispectrum and trispectrum theory and tightened constraints with Planck and LIGO-Virgo-KAGRA data, but the curvaton regime, realistic-foreground V forecasts, and the comprehensive multi-band model selection remain open, and the observational execution awaits LiteBIRD- and LISA-era data.

Background

A stochastic gravitational-wave background (GWB) is expected from inflation: quantum vacuum fluctuations of the tensor modes of the metric generate a nearly scale-invariant spectrum whose amplitude relative to the scalar curvature perturbations is measured by the tensor-to-scalar ratio $r$. In single-field slow-roll inflation the tensor spectral index obeys the consistency relation $n_T=-r/8$, so detecting a violation of this relation would point to a non-standard production mechanism. An alternative origin for primordial tensors is matter sources: fields present during inflation can amplify tensor fluctuations far beyond the vacuum level. The spectator axion-SU(2) model of Dimastrogiovanni, Fasiello and Fujita (JCAP 01 (2017) 019) is a leading example. A spectator pseudoscalar field $\chi$ (an axion, i.e. a field whose shift symmetry is weakly broken by a cosine potential $\mu^4[1+\cos(\chi/f)]$) is coupled to an SU(2) gauge field through a Chern-Simons-type term $\frac{\lambda}{4f}\chi F\tilde{F}$, where $F$ is the SU(2) field-strength tensor and $\tilde{F}$ its dual. The gauge field admits an isotropic classical background $A_i^a=aQ\delta_i^a$, and its transverse-traceless fluctuation $t_{ij}$ develops a transient tachyonic instability around horizon crossing that amplifies one circular polarization (say the left-handed one) while the scalar sector remains stable for effective gauge mass $m_Q\equiv gQ/H\ge\sqrt{2}$. The result is a fully chiral sourced tensor spectrum well approximated by a log-Gaussian template, $\mathcal{P}_h^{\rm Sourced}(k)=r_*\mathcal{P}_\zeta\exp[-\frac{1}{2\sigma^2}\ln^2(k/k_p)]$, peaking at wavenumber $k_p$ with peak amplitude $r_*$ (the tensor-to-scalar ratio at the peak scale) and width $\sigma$; the opposite helicity is essentially unsourced. (Throughout, an $e$-fold is the time over which the cosmic scale factor grows by a factor of $e$.) Although Thorne et al. treat $r_*$, $k_p$ and $\sigma$ as free phenomenological parameters, the template derivation in their Appendix A expresses them through microscopic parameters of the model: the maximal effective gauge mass $m_*$ (the peak value reached by $m_Q$ while the axion rolls), the gauge-field energy fraction at that moment, $\epsilon_{B*}\equiv H^2m_*^4/(g^2M_{\rm Pl}^2)$, with $H$ the Hubble scale during inflation and $M_{\rm Pl}$ the Planck mass, and $\Delta N$, which measures the duration of the axion roll in $e$-folds, with spectrum width $\sigma\simeq\sqrt{0.15}\,\Delta N$; this mapping relies on a slow-roll approximation that degrades when $\Delta N$ is small, i.e. when the axion moves rapidly. Chirality is observable because it breaks parity. In the CMB (cosmic microwave background), the polarization decomposes into a gradient-like E mode and a curl-like B mode; a chiral tensor background generates the parity-odd temperature-B-mode (TB) and E-mode-B-mode (EB) cross-correlation spectra, which vanish in a parity-symmetric universe. For direct detection, a stochastic background is characterized by Stokes parameters: intensity $I$ and circular polarization $V$, with $V$ the difference between right- and left-handed amplitudes; $V$ is a clean observable for a chiral GWB, but planar single-constellation interferometers such as baseline LISA (the Laser Interferometer Space Antenna) cannot measure it, so Thorne et al. adopted the optimized two-constellation space-interferometer designs of Smith and Caldwell (Phys. Rev. D 95, 044036 (2017)), which achieve roughly equal sensitivity to $I$ and $V$. Thorne et al. found that LiteBIRD (a next-generation CMB polarization satellite) could detect the chirality (via TB/EB) only for $r_*\gtrsim0.03$ while $r_*<0.07$ was required by contemporary CMB data, and that even cosmic-variance-limited observations would cap the two-point chirality signal-to-noise near 3; interferometers could detect large-$k_p$, large-$\sigma$ spectra invisible to the CMB. A B-mode bispectrum (BBB, the three-point function of the CMB B-mode map) had been shown for the related axion-U(1) model to reach greater-than-2-sigma significance (Shiraishi et al., Phys. Rev. D 94, 043506 (2016); Namba et al., JCAP 01 (2016) 041), motivating the same analysis for the SU(2) case. A curvaton is a field that is subdominant during inflation but whose energy fraction $\Omega_\chi$ grows afterwards so that its decay contributes substantially to the observed curvature perturbation; in this model large $\sigma$ with a suppressed axion decay rate could realize this regime, invalidating the simplifying assumption that scalar observables are unaffected.

Problem Statement

For the spectator axion-SU(2) model, in which an axion rolling for a few e-folds during inflation tachyonically amplifies one helicity of the SU(2) gauge-field tensor fluctuation and sources a fully chiral gravitational-wave background with a log-Gaussian spectrum parameterized by $(r_*, k_p, \sigma)$ on top of the unpolarized vacuum contribution, determine whether this sourced chiral GWB can be observationally distinguished from the conventional vacuum GWB, extending the forecast program of Thorne et al. (Phys. Rev. D 97, 043506 (2018)) to the analyses they identified as needed but explicitly left open. The source frames its two routes to a highly significant discrimination disjunctively — "it will be necessary to test deviations from the self-consistency relation, or use higher order statistics to leverage the non-Gaussianity of the model" — so that either route can in principle deliver the discrimination, and the deferred analyses that the extension is required to cover are all of the following: forecasts of the CMB detectability of the model's higher-order tensor statistics, above all the B-mode bispectrum (BBB), addressing whether non-Gaussianity can separate the sourced from the vacuum background where two-point statistics cannot; joint CMB-plus-interferometer constraints on the tensor tilt $n_T$ with a lever arm sufficient to test deviations from the single-field slow-roll consistency relation $n_T=-r/8$; the phenomenology of the regime in which the spectator axion later acts as a curvaton (its energy fraction $\Omega_\chi$ growing after inflation so that it contributes substantially to the scalar curvature perturbation), and the corresponding changes in the allowed parameter space and observables; the quantitative degradation of circular-polarization ($V$) sensitivity at space interferometers — both the two-constellation benchmark designs adopted in the source and network configurations such as LISA-Taiji — once realistic polarized astrophysical foregrounds and practical, non-ideal mission designs are included; delineation, against full numerical solutions, of the parameter region where the slow-roll analytic templates connecting the phenomenological parameters $(r_*, k_p, \sigma)$ to the microscopic parameters $(m_*, \epsilon_{B*}, \Delta N)$ (the maximal effective gauge mass, the peak gauge-field energy fraction, and the duration of the axion roll in $e$-folds) fail, i.e. at small $\Delta N$ with rapid axion motion; and a full model-selection comparison between the vacuum and sourced GWBs, which the source explicitly did not carry out. Because the relevant LiteBIRD- and LISA-era data do not yet exist, the discrimination is to be assessed at the forecast level, through projected sensitivities rather than through actual measurements; the observational execution of the program itself lies beyond the present assessment and awaits those data. The deliverable is the forecast-level determination of whether the sourced chiral GWB can be so distinguished, and a solution is complete only when the extension has covered all six of these analyses at the forecast level; addressing only a subset — or the parent question without them — does not resolve the task.

References

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