Decaying vector dark matter with low reheating temperature for KM3NeT signal and its impact on gravitational waves

We propose a new model to explain the KM3NeT neutrino event through a low reheating scenario with a suppression in the GW spectrum originating from cosmic string networks. To achieve this, we extend the SM gauge sector by an abelian gauge symmetry and a singlet scalar. Once the abelian gauge symmetry spontaneously breaks, the extra gauge boson acquires mass and becomes a suitable Dark Matter (DM) candidate. Due to the kinetic mixing with the hypercharge gauge group, DM can decay into SM particles. To explain the KM3NeT signal, we need $\mathcal{O}(100)$ PeV DM, which can be produced in the correct order of DM density in a low reheating scenario. In this scenario, the overabundance issue of heavy DM can be tackled by diluting its abundance through the continuous injection of entropy when the matter-like inflaton decays into the SM bath. Using the low reheating scenario, we can obtain the correct value of DM density both for freeze-out and freeze-in mechanisms for super-heavy DM. Moreover, we have studied the Gravitational Waves (GWs) produced from cosmic strings, which fall within the detectable range of future proposed GW experiments. Additionally, the dominance of a quadratic inflaton potential before the reheating temperature changes the temperature-scale factor relation, which suppresses the GW spectrum at higher frequencies. Choosing an arbitrarily low reheating temperature provides only a tiny fraction of the DM density due to dilution from entropy injection. This fraction of the vector DM suggests that only the extragalactic contribution is relevant in the KM3NeT event because DM lifetime is shorter than the age of the Universe.

💡 Research Summary

The paper proposes a unified framework that simultaneously addresses the recent ultra‑high‑energy neutrino event reported by the KM3NeT detector, the relic abundance of super‑heavy dark matter (DM), and a potentially observable stochastic gravitational‑wave (GW) background from cosmic strings. The authors extend the Standard Model (SM) by a dark Abelian gauge group U(1)ₙ and a complex scalar ϕₙ that carries the same charge. When ϕₙ acquires a vacuum expectation value vₙ, the associated gauge boson Wₙ becomes massive (M_Wₙ = gₙ vₙ) and serves as a vector DM candidate. A kinetic‑mixing term ε Fₙ^{μν} F_{Y μν} between the dark gauge field and hypercharge allows Wₙ to decay into SM particles.

Heavy DM production and the over‑abundance problem
In the standard radiation‑dominated cosmology, a freeze‑out calculation shows that for gauge couplings up to the unitarity bound (gₙ ≤ √4π) the relic density exceeds the observed Ω_DM h²≈0.12 once M_Wₙ ≳ 3 × 10⁵ GeV. This is far below the ∼10² PeV mass required to explain the KM3NeT event, which would lead to an enormous over‑production of DM. Freeze‑in can work for very small couplings (gₙ ∼ 10⁻⁴), but still demands a non‑standard cosmological history to avoid over‑closing the Universe.

Low‑reheating scenario
The authors solve the over‑abundance issue by invoking a low reheating temperature (T_R) after inflation. The inflaton ϕ is assumed to have a quadratic potential V(ϕ)=½ m_ϕ² ϕ² and behaves like pressureless matter while it decays perturbatively with rate Γ_ϕ into the SM plasma. The coupled Boltzmann equations for the inflaton energy density ρ_ϕ and the SM entropy density s read

 dρ_ϕ/dt + 3H ρ_ϕ = –Γ_ϕ ρ_ϕ,
 ds/dt + 3H s = Γ_ϕ ρ_ϕ/T.

Continuous entropy injection dilutes the comoving DM number density Y_DM by a factor Δ ≫ 1, where Δ ≈ (s_before / s_after). By choosing T_R as low as a few MeV (well above the BBN bound), the dilution can be strong enough to bring the relic density of a ∼10² PeV vector DM down to the observed value, regardless of whether the DM was produced via freeze‑out or freeze‑in. This mechanism also relaxes the need for extremely tiny gauge couplings.

DM decay and the KM3NeT signal
Through kinetic mixing, Wₙ decays predominantly into SM fermion pairs (q q̄, ℓ ℓ̄) and neutrino pairs (ν ν̄). The decay width scales as Γ ∼ ε² gₙ² M_Wₙ/(12π). By fixing ε ≈ 10⁻⁸–10⁻⁹, the lifetime τ_DM ≈ 10³⁰ s, which is shorter than the age of the Universe but long enough to avoid existing indirect‑detection limits. Consequently, only a tiny fraction f ≈ 10⁻³ of the total DM survives today; the rest has already decayed. Because the surviving component is sub‑dominant, the extragalactic contribution dominates the neutrino flux. This naturally explains why KM3NeT, which observes neutrinos that have traversed ∼147 km of rock and sea, sees an event while IceCube, with a much shorter ice column, does not. The model reproduces the observed event rate without violating IceCube’s upper limits.

Cosmic strings and gravitational waves
Spontaneous breaking of U(1)ₙ at a high scale vₙ generates a network of cosmic strings with tension μ ≈ π vₙ². For vₙ ∼ 10¹⁰ GeV, the resulting stochastic GW background lies within the projected sensitivities of future detectors such as LISA, DECIGO/BBO, the Einstein Telescope, and Cosmic Explorer. The authors compute the GW spectrum using the standard scaling solution for string loops, taking into account the loop size parameter α and the emission efficiency Γ_GW.

A distinctive feature of the scenario is the modification of the temperature–scale‑factor relation during the reheating epoch. Because the inflaton behaves as matter, the scale factor evolves as a ∝ T⁻³⁄⁸ (instead of the usual a ∝ T⁻¹). This alters the red‑shifting of GWs produced before reheating, suppressing the high‑frequency tail of the spectrum (f ≳ 10⁻² Hz) by roughly a factor (T_R/T_eq)^{1/2}. Therefore, a detection of a GW background with a sharp cutoff at high frequencies would provide indirect evidence for a low‑reheating, matter‑dominated phase prior to the standard radiation era.

Conclusions and outlook
The work demonstrates that a simple dark‑U(1) extension can simultaneously (i) generate super‑heavy vector DM, (ii) dilute its abundance via low‑temperature reheating, (iii) give it a lifetime suitable to explain the KM3NeT ∼100 PeV neutrino event, and (iv) predict a testable stochastic GW signal with a characteristic high‑frequency suppression. The model ties together multimessenger observations—high‑energy neutrinos and future GW detections—offering a coherent picture of physics beyond the Standard Model.

Future directions include a detailed study of pre‑heating effects, a full numerical simulation of the cosmic‑string network in the altered cosmological background, and a refined analysis of the directional dependence of the KM3NeT and IceCube event rates. Such investigations will sharpen the predictions and could either confirm or rule out this compelling unified scenario.