Forecasting Constraints on Non-Thermal Light Massive Relics from Future CMB Experiments (CMB-S4/Simons Observatory)

In this work we present Fisher forecasts on \textit{non-thermal LiMR} models for a CMB Stage IV-like experiment and the Simons Observatory – particularly focusing on a model of inflaton/moduli decay giving rise to non-thermally distributed dark sector particles, and also comparing our results with those for sterile particles following the Dodelson-Widrow distribution. Two independent parameters, $ΔN_\mathrm{eff}$ and $M_\mathrm{sp}^\mathrm{eff}$, influence linear cosmological observables. We find $ΔN_\mathrm{eff}$ to be more tightly constrained (by a factor of $10$) for a less abundant, heavier LiMR which becomes fully non-relativistic around matter-radiation equality than a more abundant, lighter LiMR which becomes fully non-relativistic just after recombination. The uncertainties on $M_\mathrm{sp}^\mathrm{eff}$ differ by a factor of $\sim3$ between the two cases. Our analysis also reveals distinct parameter correlations: the phenomenological parameters ${ΔN_\mathrm{eff},M_\mathrm{sp}^\mathrm{eff}}$ are found to be negatively correlated for the former case and positively correlated for the latter. We obtain similar projected uncertainties on the cosmological parameters (in either case) for both the inflaton/moduli decay and the Dodelson-Widrow models when the first two moments of the LiMR distribution function, related to the phenomenological parameters, are matched. Finally, by constructing a modified distribution that matches the first two moments of the Dodelson-Widrow but deviates maximally in the third moment, we demonstrate that CMB Stage IV data is not expected to be sensitive to higher moments of the distribution.

💡 Research Summary

This paper presents a comprehensive Fisher‑matrix forecast for the detectability of non‑thermal light massive relics (LiMRs) with upcoming Cosmic Microwave Background (CMB) experiments, specifically a CMB‑Stage IV‑like survey (CMB‑S4) and the Simons Observatory Large‑Aperture Telescope (SO‑LAT). The authors focus on two concrete production mechanisms that generate non‑thermal momentum distributions: (i) decay of a heavy scalar field (inflaton or moduli) during an early matter‑dominated era, and (ii) the Dodelson‑Widrow (DW) distribution originally proposed for sterile neutrinos produced via non‑resonant active‑sterile oscillations.

The phenomenology of LiMRs is captured by two independent parameters: ΔN_eff, the extra contribution to the relativistic energy density before photon decoupling, and M_eff^sp, the present‑day energy density of the relic (expressed as an effective mass). Both parameters are directly linked to the first two moments of the underlying momentum distribution function, while the typical free‑streaming velocity ⟨v_fs⟩ is proportional to the product ΔN_eff M_eff^sp.

For the inflaton/moduli decay scenario, the relic momentum distribution is derived from a 1→2 decay process and depends on four microscopic quantities: the parent mass m_ϕ, its lifetime τ, the branching ratio B_sp into the dark sector, and the relic mass m_sp. Analytic expressions (Eqs. 6‑7) relate ΔN_eff and M_eff^sp to these microscopic parameters. In the DW case the distribution is simply a rescaled Fermi‑Dirac form, f(p)=χ/(e^{p/T_ν}+1), where the normalization χ directly equals ΔN_eff and M_eff^sp=χ m_sp.

The authors modify the Boltzmann code CLASS to incorporate these non‑thermal spectra, compute the C_ℓ temperature, E‑mode polarization, and lensing power spectra, and construct the full covariance matrix including instrumental noise (as specified for CMB‑S4 and SO‑LAT). Using the standard Fisher formalism (Eq. 17) they evaluate the information matrix for a parameter set that includes the six ΛCDM parameters plus ΔN_eff and M_eff^sp. Jacobian transformations are employed to propagate uncertainties from the microscopic model parameters to the phenomenological ones.

Key findings:

1. ΔN_eff constraints – For a “heavy, scarce” LiMR that becomes non‑relativistic around matter‑radiation equality, the forecasted 1σ error is σ(ΔN_eff)≈10⁻³, an order of magnitude tighter than for a “light, abundant” LiMR that turns non‑relativistic just after recombination (σ≈10⁻²). The tighter bound arises because the heavier relic’s impact on the high‑ℓ damping tail and phase shift is more pronounced.

2. M_eff^sp constraints – The effective mass is measured roughly three times more precisely for the heavy case than for the light case, reflecting the stronger late‑time ISW and lensing signatures of a more massive relic.

3. Parameter correlations – In the heavy scenario ΔN_eff and M_eff^sp are negatively correlated, whereas in the light scenario they are positively correlated. This reversal stems from the differing ℓ‑dependence of the observables each parameter dominates (low‑ℓ ISW vs. high‑ℓ damping).

4. Model degeneracy – When the first two moments of the inflaton/moduli decay distribution are matched to those of the DW distribution, the Fisher forecasts for both experiments are essentially identical. Thus, CMB‑S4 and SO‑LAT cannot distinguish the underlying production mechanism if only the first two moments are known.

5. Higher‑moment insensitivity – The authors construct an artificial distribution that shares the same ΔN_eff and M_eff^sp as the DW case but maximally differs in the third moment. Forecasts show that even a cosmic‑variance‑limited CMB experiment would not be sensitive to this difference; the 3rd‑moment contribution to the C_ℓ spectra lies well below the expected noise level.

The paper concludes that future CMB experiments will be able to constrain ΔN_eff to the 10⁻³ level and M_eff^sp to ∼10⁻² eV, providing a powerful probe of non‑thermal light relics. However, discriminating between different production mechanisms will require sensitivity to higher moments of the momentum distribution, which likely demands either substantially lower noise, higher angular resolution, or complementary large‑scale‑structure and 21 cm observations. The work thus maps out both the promise and the limitations of next‑generation CMB data for exploring physics beyond the Standard Model in the dark sector.