Dark radiation from particle decay: cosmological constraints and opportunities

TL;DR

This paper addresses the origin of dark radiation by particle decay and develops model-independent bounds on decays into dark components using $N_\text{eff}$ as the restoring observable. It introduces a parameterization in terms of decay time $\tau$ and mass hierarchy $\delta$, derives kinematic relations, energy-density evolution, and the impact on structure formation, BBN, and CMB observables. The key contributions include analytic bounds on branching ratios from BBN and CMB data, and a comprehensive exploration of one- and two-dark-decay-mode scenarios, showing how decay timing constrains or enables hot dark matter, dark radiation, and potential solutions to the missing satellites problem. The results offer practical guidelines for particle-physics model-building and suggest cosmological observables, including future CMB polarimeters, that could identify or constrain decay-origin dark radiation.

Abstract

We study particle decay as the origin of dark radiation. After elaborating general properties and useful parametrisations we provide model-independent and easy-to-use constraints from nucleosynthesis, the cosmic microwave background and structure formation. Bounds on branching ratios and mass hierarchies depend in a unique way on the time of decay. We demonstrate their power to exclude well-motivated scenarios taking the example of the lightest ordinary sparticle decaying into the gravitino. We point out signatures and opportunities in cosmological observations and structure formation. For example, if there are two dark decay modes, dark radiation and the observed dark matter with adjustable free-streaming can originate from the same decaying particle, solving small-scale problems of structure formation. Hot dark matter mimicking a neutrino mass scale as deduced from cosmological observations can arise and possibly be distinguished after a discovery. Our results can be used as a guideline for model building.

Figures (14)

• Figure 1: Behaviour of comoving energy densities $\rho a^3$ in an expanding universe with dark radiation from particle decay. The full-logarithmic figure is illustrative and not exact. Upper right corner: Nomenclature for the considered two-body decay.
• Figure 2: $\delta_\text{min}$-$\tau$-plane exploiting \ref{['implicitlowboundondelta']}. Values above the corresponding line are considered to be allowed. The thick solid curve corresponds to the hot dark matter constraint, cf. Sec. \ref{['sec:hdm']}, with $\Delta N_\text{eff} =1$. Thin dashed curves below ($\Delta N_\text{eff}=0.52$) and above ($\Delta N_\text{eff}=5.265$) show the dependence of this bound on the produced amount of dark radiation. The thin solid curve corresponds to the non-domination constraint. The analytic approximations \ref{['lowboundondelta']} and \ref{['lowboundondeltaafter']} are overplotted as very thin grey curve with a jump at ${t_\text{eq}}$. The dotted curve (at the lower edge of the thick solid one) gives $\delta$ such that the heavier daughter becomes non-relativistic at photon decoupling $t_\gamma^\text{dpl}$ and the dash-dotted one such that this happens today. These three curves are for $\Delta N_\text{eff}=1$. At the upper edge of the thick solid curve mean values of Benson:2011ut originate from the decay for massless neutrinos. Various important and suggestive times are highlighted by vertical dashed lines: onset $t_\text{bbn}$ and end of BBN $t_\text{bbn}^\text{end}$, the earliest possible time for the heavier daughter to become non-relativistic $(t_2^\text{nr})_\text{min}$ from \ref{['tnrmin']}, re-entry of first observable modes in the CMB $t_\text{cmb}$ and matter-radiation equality ${t_\text{eq}}$. For decays during BBN the increase in $N_\text{eff}$ determined from BBN is smaller than the corresponding increase measured in the CMB. The relative difference depends on the time of decay as quantified in Menestrina:2011mz. In all figures we take into account the evolution of $g_\ast$ and $g_{\ast s}$. Nevertheless, the curves are smooth around $e^+e^-$ annihilation at $t_{e^+e^-}$, which shows that dependencies have cancelled. Within the horizontal dashed lines the mass hierarchy or degeneracy is within an order of magnitude.
• Figure 3: Minimal free-streaming scale of the heavier daughter $(\lambda_2^\text{fs})_\text{min}$ depending on the time of decay for $\Delta N_\text{eff}=1$. Times are highlighted as in Fig. \ref{['fig:delta-tau-dr']}. Horizontal dashed lines indicate the galaxy cluster scale $\lambda_\text{gc} \sim 10 \text{ Mpc}$ and $1 \text{ Mpc}$ as suggestive free-streaming scale of warm dark matter, respectively.
• Figure 4: Upper bound from BBN on the direct branching ratio of the decaying particle into electromagnetically interacting particles as function of its lifetime $\tau$. The thick solid curve represents the weakest bound obtainable. We consider larger branching ratios as excluded. The thin solid curve represents the strongest bound obtainable. It shows how strong the actual bound might become depending on the actual value of $\Delta N_\text{eff}$ and the energy density of the decaying particle after ${t_\text{eq}}$. Times are highlighted as in Fig. \ref{['fig:delta-tau-dr']}.
• Figure 5: Upper bound from BBN on the direct branching ratio of the decaying particle into hadronically interacting particles as function of its lifetime $\tau$. Solid/dashed curves represent weakest/strongest bounds obtainable as in Fig. \ref{['fig:BemmaxBBNdirect']}. Black curves apply to a mass of the decaying particle $m=1 \text{ TeV}$ and grey (green) curves to $m=100\text{ GeV}$. The thin dotted curves correspond in each case to a less conservative bound for the ${}^6$Li$/{}^7$Li ratio. Within the enclosed area the cosmic lithium problems could be solved by the decay. Times are highlighted as in Fig. \ref{['fig:delta-tau-dr']}.