Can multi-field inflation produce a testably large tensor tilt for LiteBIRD: realistic model building and robust forecasts

In single-field slow-roll inflation with a canonical kinetic term, the tensor-to-scalar ratio $r$ and the tensor spectral index $n_T$ obey the consistency relation $r=-8n_T$. Jinno, Kohri, Moroi, Takahashi and Hazumi (JCAP 03 (2024) 011) showed that LiteBIRD measurements of $r$ and $n_T$, combined with Planck's constraint on the scalar spectral index, can test multi-field (spectator/curvaton) inflation, but that testing the consistency relation itself requires a large tilt $|n_T|\sim\mathcal{O}(0.1)$, which in their analysis only an engineered three-field construction achieves. They left open: (i) whether realistic multi-field microphysical models generically produce such large tilts while satisfying all other observables and a viable reheating history; and (ii) whether the forecast conclusions survive end-to-end component separation and delensing, non-separable potentials, non-quadratic spectators, and spectator dynamics away from the stochastic equilibrium. No resolution of these questions is known in the literature to date; the problem is open.

Background

Inflation — a phase of accelerated expansion in the very early Universe — is the leading explanation of the observed cosmic microwave background (CMB) anisotropies. Quantum fluctuations of the inflaton, the scalar field driving inflation, seed the scalar (density) perturbations whose spectral index $n_s$ Planck measures to be about 0.965; quantum fluctuations of the metric itself generate a stochastic background of primordial gravitational waves (tensor perturbations), observable as CMB $B$-mode (curl-type) polarization. The tensor amplitude relative to the scalar perturbations is the tensor-to-scalar ratio $r$; its scale dependence is the tensor spectral index $n_T$. LiteBIRD is a JAXA-led satellite mission (planned launch in the late 2020s) that will map the full-sky CMB polarization in 15 frequency bands between 34 and 448 GHz with the goal of measuring or bounding $r$ at the $\delta r\sim10^{-3}$ level; achieving this requires separating the Galactic synchrotron and dust foregrounds (component separation) and removing the lensing $B$-mode signal (delensing), since both contaminate the primordial tensor signal.

For single-field slow-roll inflation with a canonical kinetic term, tensor amplitude and tilt are both controlled by the first slow-roll parameter $\epsilon\equiv-\dot H/H^{2}$ (with $H$ the Hubble rate): $r=16\epsilon$ and $n_T=-2\epsilon$, giving the consistency relation $r=-8n_T$. Testing this relation is a classic target of $B$-mode cosmology because many non-standard mechanisms violate it.

In multi-field (spectator) scenarios such as the curvaton model, a second light scalar field $\sigma$ contributes to or dominates the primordial density perturbation while the inflaton sets the expansion rate. Writing $R_\sigma$ for the ratio of the spectator's to the inflaton's contribution to the curvature power spectrum, the tensor amplitude is diluted, $r=16\epsilon/(1+R_\sigma)$, while the tilt $n_T=-2\epsilon$ is unchanged, so $r=-8n_T/(1+R_\sigma)$: the consistency relation is violated (Ichikawa, Suyama, Takahashi and Yamaguchi 2008). Whether the violation is observable is another matter: LiteBIRD constrains $n_T$ only at the $\mathcal{O}(0.1)$ level for currently allowed $r$ (Paoletti et al. 2022), so a decisive test needs $|n_T|\sim\mathcal{O}(0.1)$, i.e. $\epsilon\simeq0.05$; since $\epsilon\simeq1/N$ with $N$ the number of e-folds before the CMB pivot scale exits the horizon, this forces $N\simeq20$, in tension with a standard post-inflationary history unless extra dynamics (for example a second inflationary phase) is introduced. The initial amplitude of a light spectator is often estimated with the stochastic formalism, which treats the coarse-grained spectator dynamics as a diffusion process in quasi-de Sitter space with an equilibrium distribution, giving a typical initial value $\sigma_\ast\sim H_\ast^{2}/m_\sigma$ for a quadratic spectator potential of mass $m_\sigma$.

Jinno, Kohri, Moroi, Takahashi and Hazumi (2024) performed likelihood forecasts for $(r,n_T)$ with LiteBIRD, modeling residual foregrounds through a single residual fraction $\sigma_{BB,\rm FG}=0.2$ tuned to reproduce the $\Delta r=5\times10^{-4}$ benchmark of the official LiteBIRD forecast at $r=0$, and assuming a separable potential $V(\phi,\sigma)=W(\phi)+U(\sigma)$ with a quadratic spectator $U=\frac12 m_\sigma^{2}\sigma^{2}$. They found that multi-field models with a light spectator are identified or stringently constrained once $r$ is detected, and that the consistency relation becomes testable only for a negatively large tilt $n_T\sim-\mathcal{O}(0.1)$; their Appendix B gives one explicit three-field construction (a quartic inflaton, a quadratic curvaton with $R_\sigma\gg1$, and a second-phase field with $\chi_\ast\simeq10M_{\rm pl}$ (with $M_{\rm pl}$ the reduced Planck mass)) achieving $n_T\simeq-0.1$ with acceptable $n_s$ and scalar running $\alpha_s\sim10^{-3}$. Higher-order slow-roll corrections shift the single-field relation itself at the percent level (Bianchi and Gamonal 2024), so any test must be defined against the relation's precision.

Problem Statement

The problem, as left open by Jinno et al. (2024), is to determine whether — and under what microphysical conditions — LiteBIRD tensor observations can genuinely test multi-field (spectator/curvaton) inflation. It has two linked parts serving this single objective:

1. Model building: construct, or rule out, realistic multi-field inflationary models that produce a tensor tilt of magnitude $|n_T|\sim\mathcal{O}(0.1)$ (or other testably large deviations from $r=-8n_T$), consistent with the observed $n_s$, the current upper bound on $r$, the bounds on primordial non-Gaussianity and on the scalar running $\alpha_s$, and a viable reheating and e-fold history; and establish whether such large tilts are generic in motivated multi-field constructions or require fine-tuned arrangements such as the paper's engineered second inflationary phase.

2. Forecast robustness: determine how the expected LiteBIRD constraints on the multi-field $(r,n_T)$ parameter space change when the simplified analysis assumptions are dropped, addressing at least one of the following relaxations: end-to-end map-level component separation and delensing with realistic foreground models instead of the single tuned residual-foreground parameter; non-separable potentials; non-quadratic spectator potentials; or spectator dynamics away from the stochastic-equilibrium distribution.

An answer addressing any part must state how it advances the shared objective of assessing the observational testability of multi-field inflation; a complete answer characterizes the class of multi-field models that LiteBIRD can distinguish from single-field inflation and the conditions under which the single-field consistency relation can be tested or excluded.

References

1. Ryusuke Jinno, Kazunori Kohri, Takeo Moroi, Tomo Takahashi, Masashi Hazumi, 'Testing multi-field inflation with LiteBIRD', JCAP 03 (2024) 011, DOI: 10.1088/1475-7516/2024/03/011, arXiv:2310.08158, https://arxiv.org/abs/2310.08158
2. Masazumi Honda, Ryusuke Jinno, Koki Tokeshi, 'Exactly solvable stochastic spectator', JCAP 12 (2024) 044, DOI: 10.1088/1475-7516/2024/12/044, arXiv:2409.16272, https://arxiv.org/abs/2409.16272
3. Daniela Paoletti, Fabio Finelli, Jussi Valiviita, Masashi Hazumi, 'Planck and BICEP/Keck Array 2018 constraints on primordial gravitational waves and perspectives for future B-mode polarization measurements', Phys. Rev. D 106 (2022) 083528, DOI: 10.1103/PhysRevD.106.083528, arXiv:2208.10482, https://arxiv.org/abs/2208.10482
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