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

Technical Summary: Decaying Vector Dark Matter with Low Reheating Temperature for KM3NeT Signal and its Impact on Gravitational Waves

Problem Statement
The paper addresses the tension between the recent detection of ultra-high-energy neutrinos by the KM3NeT collaboration (specifically the KM3-230213A event at $\mathcal{O}(100)$ PeV) and the non-observation of similar signals by the IceCube detector. While IceCube has detected neutrinos up to $\mathcal{O}(1)$ PeV, it has not observed events in the 100 PeV range despite its significantly larger effective area. The authors propose that this discrepancy, along with the need to explain the KM3NeT signal, can be resolved by a decaying Super-Heavy Dark Matter (SHDM) scenario. However, standard thermal freeze-out mechanisms struggle to accommodate SHDM with masses in the PeV range ($M_{DM} \gtrsim 10^5$ GeV) without violating unitarity bounds or overproducing dark matter density. Furthermore, standard cosmological scenarios often fail to produce the correct relic density for such heavy candidates without invoking extremely fine-tuned parameters.

Methodology
The authors propose a new model extending the Standard Model (SM) gauge sector with an abelian dark gauge symmetry $U(1)_D$ and a SM singlet scalar $\phi_D$.

1. Model Construction: The $U(1)_D$ gauge boson ($W_D$) serves as the vector Dark Matter candidate, acquiring mass via the spontaneous symmetry breaking of $\phi_D$. The model incorporates kinetic mixing between $U(1)_D$ and the SM hypercharge $U(1)_Y$, which allows the DM to decay into SM particles (quarks, leptons, and neutrinos).
2. Low Reheating Scenario: To overcome the overabundance problem of heavy DM, the authors employ a low reheating temperature scenario. In this framework, the inflaton (with a quadratic potential) decays continuously into the SM bath, generating entropy. This continuous entropy injection dilutes the DM abundance produced via thermal freeze-out or freeze-in mechanisms.
3. Production Mechanisms: The study analyzes both the freeze-out (WIMP) and freeze-in (FIMP) mechanisms. The dilution effect allows for the correct relic density to be achieved for PeV-scale masses with reasonable gauge couplings, bypassing the unitarity limits that restrict standard thermal freeze-out.
4. Phenomenological Analysis:
   • Neutrino Flux: The authors calculate the neutrino flux from both Galactic and extragalactic DM decay, utilizing the NFW profile for the Galactic distribution and accounting for redshift effects for the extragalactic component. They use the HDMSpectra package to generate decay spectra.
   • Gravitational Waves (GWs): The spontaneous breaking of $U(1)_D$ at a high scale generates cosmic strings. The authors calculate the resulting GW spectrum using the Velocity-Dependent One-Scale (VOS) model. They specifically investigate how the low reheating temperature (where the scale factor $a \propto T^{-3/8}$ during the inflaton-dominated era) modifies the GW spectrum compared to standard radiation-dominated evolution.
5. Tools: Numerical analysis is performed using micrOMEGAs (v6.2.3) for relic density calculations and HDMSpectra for decay spectra.

Key Contributions and Results

• Resolution of KM3NeT Tension: The model successfully explains the KM3NeT signal while remaining consistent with IceCube limits. By adjusting the DM mass ($M_{WD} \sim 4.4 \times 10^8$ GeV), lifetime ($\tau_{WD} \sim 10^{29}$ s), and the fraction of DM density ($f_{WD}$), the predicted flux peaks at the KM3NeT energy scale but remains below the IceCube upper bounds. The authors note that for very small DM fractions, the signal is dominated by the extragalactic component, as the Galactic DM would have decayed by the present epoch.
• Relic Density in Low Reheating: The study demonstrates that the low reheating scenario effectively solves the overproduction problem for SHDM. It allows for the correct relic density to be achieved via both freeze-out (with gauge couplings up to the perturbative limit) and freeze-in mechanisms, provided the reheating temperature is sufficiently low to induce significant entropy dilution.
• Decay Channels: The dominant decay modes are identified as $q\bar{q}$, followed by $l\bar{l}$ and $\nu_l\bar{\nu}_l$. The decay to $W^+W^-$ is suppressed due to the mutual cancellation of mixing angles. This branching ratio structure is crucial for producing the observed neutrino flux without exceeding photon flux limits from experiments like LHAASO-KM2A.
• Gravitational Wave Signatures: The model predicts a stochastic GW background from cosmic strings. Due to the large vacuum expectation value (vev) required for SHDM, the string tension ($G\mu$) is high enough to be potentially detectable by future experiments (SKA, LISA, DECIGO, BBO, ET, CE).
• Spectral Suppression: A distinct feature of the model is the modification of the GW spectrum at high frequencies. Because the universe evolves as a matter-dominated era ($a \propto T^{-3/8}$) during the reheating phase rather than a radiation-dominated era ($a \propto T^{-1}$), the GW spectrum is suppressed at higher frequencies. This suppression depends on the reheating temperature.

Significance
The paper claims that its work provides a unified explanation for the KM3NeT neutrino event and the potential detection of gravitational waves from cosmic strings within a consistent theoretical framework.

• Multimessenger Connection: It links a specific high-energy neutrino anomaly to a specific early-universe cosmological history (low reheating) and a specific BSM particle physics model (vector DM with kinetic mixing).
• Cosmological Evolution: The authors highlight that the detection of the specific suppression pattern in the GW spectrum at high frequencies would serve as indirect evidence for a non-standard reheating period in the early universe where the inflaton behaves as a matter-like field.
• Viability of SHDM: The work demonstrates that super-heavy dark matter, often considered problematic in standard cosmology due to overproduction, can be a viable candidate if produced in a low reheating scenario with entropy dilution.

The authors conclude that their model not only resolves the tension between KM3NeT and IceCube data but also offers testable predictions for future gravitational wave observatories, thereby providing a pathway to indirectly confirm non-standard evolution in the early universe.