Spectral distortions in the decaying QCD dark matter scenario

TL;DR

This work analyzes how energy injection from a QCD-like dark sector, featuring confinement at scale $a_c$ and decays with rate $\Gamma_\chi$ and efficiency $\Sigma_\chi$, imprints on the CMB as $\mu$- and $y$-type spectral distortions. A unified fluid framework is developed to model both relativistic and non-relativistic regimes and fast/slow decays, covering exponential, power-law, oscillatory, and cascade decay histories. The authors show that oscillatory and cascade decays can be effectively mapped to exponential forms with renormalized parameters, and that the dominant SDs are controlled by the decay epoch and lifetime, with $v_{\chi c}$ only relevant in the ultra-relativistic limit. Using FIRAS limits and joint $\mu$/$y$ constraints, they localize viable regions in $(a_c, \Gamma_\chi, \Sigma_\chi)$ space, finding favored lifetimes around $\Gamma_\chi \sim \mathcal{O}(10^{-3})\,\mathrm{yr}^{-1}$ and small energy fractions, while very fast decays become observationally negligible. Future missions such as PIXIE or PRISM could substantially tighten these bounds, probing regions of parameter space that are currently inaccessible and offering a powerful test of dark-sector confinement dynamics.

Abstract

We study the QCD--DM scenario by analyzing the imprint of energy injection from decaying dark-sector particles on the spectral distortions (SDs) of the Cosmic Microwave Background (CMB). We adopt a unified framework capable of describing both relativistic and non-relativistic particles, as well as fast and slow decay regimes. Within this approach, we model exponential, power-law, oscillatory, and two-step decays, computing the resulting $μ$- and $y$-type distortions across the parameter space spanned by the confinement scale $a_c$, decay rate $Γ_χ$, energy-transfer efficiency $Σ_χ$, and velocity $v_{χc}$. We find that power-law, oscillatory, and cascade decays can be effectively mapped onto exponential models with appropriate rescaling. The dominant factors controlling SDs are the decay epoch and lifetime, with $v_{χc}$ becoming relevant only in the ultra-relativistic limit. FIRAS observations impose tight constraints on early energy injection, with $μ$-type distortions placing the strongest bounds on $Σ_χ$. Simultaneous matching of both $μ_{\rm firas}$ and $y_{\rm firas}$ breaks the degeneracy between $Γ_χ$ and $Σ_χ$, localizing preferred decay rates around $Γ_χ\lesssim (3.3-4.4)\times10^{-3}~{\rm yr}^{-1}$ and $Σ_χ\lesssim 8.5\times10^{-4}$ for relativistic particles, while fast decays with $Γ_χ\gtrsim 6.5~{\rm yr}^{-1}$ become observationally negligible. Our results show that CMB spectral distortions are a powerful probe of dark-sector dynamics. Future missions such as PIXIE or PRISM could extend current limits by several orders of magnitude and test previously inaccessible regions of parameter space.

Figures (10)

• Figure 1: Schematic diagrams representing the decay of a dark hadron $\chi_h$ into two Standard Model (SM) particles through an effective interaction vertex. The hatched circle denotes the effective operator or mediator responsible for the decay process.
• Figure 2: Evolution of the energy density of dark-sector components in the QCD--DM scenario, normalized to $\rho_{dmc}$ and $a_c$. The red lines show the stable QCD--DM component before and after confinement, while the blue lines depict the decaying species $\chi_1 (v_{\chi 1}=0.9, f_\chi=0.1)$ for different decay rates $\Gamma_{\chi_1}=[1.0,\,0.01,\,0.001]\,4H_c$. Green lines correspond to the daughter particles $\chi_2$ for fixed $\Gamma_{\chi_1}=0.01\,\Gamma_{\chi 1}$ and varying $\Gamma_{\chi_2}$. The dashed black line represents standard CDM, and vertical markers indicate the confinement and non-relativistic transition epochs.
• Figure 3: Energy-injection rate $\dot Q/(H\rho_\gamma)$ normalized to the Hubble expansion as a function of $z/z_c$, where $z_c$ is stablish at the beginning of the $\mu$-era. (a) Exponential energy-injection histories for decay rates $\Gamma=[10,\,1,\,0.5,\,0.1]\Gamma_{\chi}$ and power-law indices $\alpha=\{0.5,6\}$ with fixed $\Gamma_{\chi} = t_\chi =4H_c$. Vertical dashed lines mark epochs when $\Gamma = 4H$, corresponding to the onset of strong decay. (b) Comparison of different decay mechanisms: exponential (reference), oscillatory, and two-step decay, showing that an exponential decay can effectively approximate the latter. For oscillatory decay, Case 1 (green) corresponds to a fast decay ($\omega/\Gamma=100$), and Case 2 (yellow) to a moderate decay ($\omega/\Gamma=1$). For the cascade decay, Case 3 (magenta) corresponds to $\Gamma_{\chi_2}=0.5\,\Gamma_{\chi_1}$, while Case 4 (brown) to $\Gamma_{\chi_2}=0.1\,\Gamma_{\chi_1}$, and both cases have $\Gamma_{\chi1} = \Gamma_\chi$. See Table \ref{['tab:decay_osc_2step']} for the resulting spectral distortions.
• Figure 4: Contour plot of the constraints on $\Sigma_\chi$ in the $a_c$–$v_{\chi c}$ parameter space. The left and right panels show isolines of constant $\mu=\mu_{\mathrm{FIRAS}}$ and $y=y_{\mathrm{FIRAS}}$, respectively. The cyan vertical line marks the upper bound on $a_c$ from CMB data, while the red isoline corresponds to the FIRAS limits for $\mu_{\rm firas}$ and $y_{\rm firas}$. The white isoline indicates the $\Lambda$CDM prediction ($\mu_{\Lambda\mathrm{cdm}}$, $y_{\Lambda\mathrm{cdm}}$). The magenta vertical line denotes the QCD–DM case with $a_c = 3.18\times10^{-7}$, equivalent to a 3 keV warm–DM model, and the brown isoline shows the late-time $y$-distortion reference value ($y=y_{\mathrm{sz}}$). Positive $\Sigma_\chi$ values represent the inverse scaling factor applied to the $\mu$- or $y$-type distortions relative to the FIRAS reference.
• Figure 5: Contour plots illustrating the magnitude of the $\mu$-type SD in the parameter space defined by the decay rate and the fractional abundance parameter ($\Gamma_{\chi}-\Sigma_{\chi}$) for three fixed values of the transition scale factor $a_c$: upper bound for $\mu_{\rm firas}$ constraint (left panel), upper CMB limit with $v_{dmc} = 1/\sqrt{2}$ (middle panel), and upper CMB limit with $v_{dmc} = 0$ (right panel). The solid white line labeled $\Lambda$CDM corresponds to the $\mu$-SD value predicted by the standard cosmological model. The blue isoline labeled FIRAS represents the observational upper limit from FIRAS. Regions above FIRAS isoline are excluded (red-shaded areas). The grey-shaded areas represent parameter regions producing produce negligible distortions (undetectable or at/below $\Lambda$CDM). Vertical dashed lines mark the critical decay rate $\Gamma_{cr}(a_c)$ distinguishing short- from long-lived regimes.