Bulk viscosity from neutron decays to dark baryons in neutron star matter

This paper investigates how neutron decays into dark baryons affect bulk viscosity in neutron star mergers, finding that while the standard slow decay rate only slightly reduces viscosity, a faster decay rate could significantly enhance it and rapidly dampen merger oscillations.

The Gist

The Big Picture: A Cosmic Mystery and a Sticky Fluid

Imagine a neutron star as a giant, cosmic flywheel made of the densest stuff in the universe. It spins incredibly fast. Sometimes, these stars crash into each other in a spectacular event called a neutron star merger. When they collide, they create a chaotic, super-hot soup of particles that sloshes and vibrates like a giant drum.

Physicists want to know: How fast does this drum stop vibrating?

The answer depends on something called bulk viscosity. Think of bulk viscosity as the "thickness" or "stickiness" of the fluid.

• Low viscosity (Water): If the fluid is like water, the vibrations keep going for a long time. The drum rings out.
• High viscosity (Honey): If the fluid is like honey, the vibrations get swallowed up quickly. The drum stops almost immediately.

This paper asks: Could a secret, invisible ingredient called "Dark Matter" make this cosmic honey even stickier?

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The Mystery: The "Missing" Neutron

To understand the paper, we first need to look at a puzzle on Earth. Scientists have been trying to measure how long a neutron (a particle inside an atom) lives before it decays. They use two different methods:

1. The "Bottle" Method: They trap neutrons in a container and count how many survive.
2. The "Beam" Method: They shoot neutrons through a tube and count how many turn into protons.

The Problem: The "Bottle" says neutrons live about 879 seconds. The "Beam" says they live about 888 seconds. There is a gap. The neutrons seem to be disappearing faster in the bottle than they are turning into protons in the beam.

The Theory: Some physicists think neutrons might be doing something sneaky. Maybe 1% of the time, instead of turning into a proton, a neutron decays into a Dark Baryon (a secret, invisible particle) and a Dark Scalar (a ghostly messenger particle). The "Bottle" counts this as a death, but the "Beam" misses it because it only looks for protons.

The Experiment: What Happens Inside a Star?

The authors of this paper asked: If this sneaky decay happens, does it change how the neutron star merger behaves?

They simulated a neutron star merger with two scenarios:

Scenario A: The "Slow and Steady" Decay (The Standard Theory)

In this version, the neutron decays into the dark particle very slowly (about 1% of the time, matching the Earth mystery).

• The Analogy: Imagine a crowded dance floor (the neutron star). Most dancers are swapping partners normally (turning into protons). A few dancers are secretly slipping out the back door into a dark room (decaying into dark matter).
• The Result: Because the dark room is so far away and the door is so narrow, the dancers leaving don't affect the dance floor much. The "stickiness" (viscosity) of the fluid stays mostly the same. The dark particles just sit there, frozen, not helping to stop the vibrations.
• Conclusion: If the dark decay is slow, it doesn't change the merger much. The "honey" isn't much stickier than before.

Scenario B: The "Fast and Furious" Decay (The "What If" Theory)

The authors then asked: What if the dark decay was actually much faster? Maybe the coupling between normal matter and dark matter is stronger than we thought.

• The Analogy: Now, imagine the back door is wide open. Every time the music speeds up (temperature rises), a huge wave of dancers rushes out the back door and comes back in, constantly changing the crowd's composition.
• The Result: This rapid switching creates a lot of friction. The fluid becomes extremely sticky.
• The Impact: If the decay is fast enough, the bulk viscosity spikes at high temperatures (tens of millions of degrees). This would act like a giant brake, stopping the neutron star's vibrations in just 20 milliseconds.

The "Golden" Insight

The paper makes a fascinating general point about the universe:

• Normal reactions (like neutrons turning into protons) are usually fast enough to keep the star in balance, even when it's hot.
• Dark matter reactions are usually too slow to matter... unless the environment gets incredibly hot (like in a merger).

If dark matter interacts with normal matter just right—not too weak, not too strong—it could create a "sweet spot" where the fluid becomes super-sticky only during the hottest moments of a collision.

The Takeaway

1. If the dark decay is slow (1%): It's a "ghost" that doesn't really do anything to the neutron star merger. The viscosity stays low, and the vibrations ring out.
2. If the dark decay is faster: It turns the neutron star merger into a "sticky trap." The vibrations would die out almost instantly.

Why does this matter?
If we could listen to the "ringing" of neutron star mergers using gravitational waves (the "sound" of the collision), we might be able to tell if this dark decay is happening.

• If the sound rings out for a while, the dark decay is likely slow.
• If the sound cuts off abruptly, it might be a signature of this secret dark matter interaction.

In short, the paper suggests that neutron star mergers are the ultimate laboratory to test if neutrons are secretly turning into dark matter, and if they are, it would make the universe's most violent collisions much "stickier" than we expected.

Technical Summary

1. Problem Statement

The paper addresses the "neutron decay anomaly," a discrepancy between neutron lifetime measurements obtained via the "beam method" ($\tau \approx 888$ s) and the "bottle method" ($\tau \approx 879$ s). One proposed solution involves a "dark decay" channel where a neutron decays into a dark baryon ($\chi$) and a dark scalar ($\phi$), i.e., $n \to \chi + \phi$, with a branching ratio of $\sim 1\%$.

While this hypothesis has been studied regarding neutron star stability and equation of state (EoS), its impact on transport properties, specifically bulk viscosity, in the hot, dense environments of neutron star mergers remains largely unexplored. Bulk viscosity is crucial for damping density oscillations in merger remnants. The authors investigate how the presence of dark baryons and the specific kinetics of the $n \to \chi + \phi$ decay channel modify the bulk viscosity of neutron star matter, particularly at the high temperatures ($T \sim 10\text{--}100$ MeV) found in merger remnants.

2. Methodology

A. Theoretical Framework

• Model: The authors consider a dark sector with a dark baryon $\chi$ and a massless dark scalar $\phi$. The interaction Lagrangian is $L_{int} = g_\phi (\bar{\chi}n + \bar{n}\chi)\phi$.
• Equation of State (EoS): They utilize the IUF-II relativistic mean-field (RMF) model for standard $npe$ matter, extended to include a population of dark baryons ($npe\chi$).
  • To support $2M_\odot$ neutron stars, the dark baryons are assumed to have a repulsive self-interaction mediated by a dark vector meson ($\omega'$), characterized by a coupling strength $G'$.
  • The system is treated as a single fluid where $npe$ and $\chi$ are thermally equilibrated via unspecified elastic scattering ($n+\chi \to n+\chi$), though the chemical equilibration is driven by the weak decay channels.
• Thermodynamics: The system is analyzed out of beta equilibrium, defined by two independent chemical potential differences:
  1. $\delta\mu_1 = \mu_n - \mu_p - \mu_e$ (Urca channel)
  2. $\delta\mu_2 = \mu_n - \mu_\chi$ (Dark decay channel)

B. Calculation of Bulk Viscosity

The bulk viscosity ($\zeta$) is derived for a fluid element undergoing sinusoidal density oscillations. The general formula for a two-channel system involves the relaxation rates ($\gamma_1, \gamma_2$) and thermodynamic susceptibilities.

• Partial Bulk Viscosities: The total viscosity is a combination of contributions from the standard Urca process ($\zeta_1$) and the dark decay process ($\zeta_2$).
• Reaction Rates:
  • Urca Rate ($\Gamma_{Urca}$): Calculated using standard direct and modified Urca processes.
  • Dark Decay Rate ($\Gamma_{dark}$): This is a key innovation. The authors do not rely on simple phase-space integrals for modified processes (which are computationally difficult at high $T$). Instead, they employ the Nucleon Width Approximation (NWA).
    • The NWA accounts for the collisional broadening of nucleons in dense matter by introducing an imaginary part to the nucleon mass (width $W$).
    • The rate is calculated by integrating the direct decay matrix element over a distribution of effective masses (Breit-Wigner distribution) weighted by the nucleon and dark baryon widths.

C. Scenarios Analyzed

1. Benchmark Scenario: Parameters are chosen to satisfy the 1% branching ratio in vacuum (solving the lifetime anomaly). Here, $m_\chi \approx 938$ MeV and $m_\phi = 0$.
2. Rapid Decay Scenario: The dark baryon mass is set to $m_\chi = 940$ MeV (heavier than the neutron), so vacuum decay is forbidden. The coupling $g_\phi$ is increased up to the Raffelt limit (supernova cooling constraints) to test the effect of faster dark decays on merger dynamics.

3. Key Contributions

1. First Calculation of Dark Decay Bulk Viscosity: This is the first work to explicitly calculate the bulk viscosity of neutron star matter including the $n \to \chi + \phi$ channel in the context of merger temperatures.
2. Application of NWA to Dark Decays: The authors extend the Nucleon Width Approximation (previously used for Urca processes) to dark baryon decays. This allows for a rigorous calculation of reaction rates in the high-temperature, non-degenerate regime of neutron star mergers, where traditional Fermi-surface approximations fail.
3. Kinematic Suppression Analysis: They demonstrate that in the benchmark scenario, the dark decay rate is kinematically suppressed in medium due to the mismatch between the neutron and dark baryon Fermi momenta, making the process significantly slower than the Urca process.
4. Two-Fluid vs. One-Fluid Clarification: While assuming a single fluid for simplicity, the paper discusses the implications of dark matter forming a separate fluid and how self-interactions ($G'$) are required to stiffen the EoS.

4. Key Results

A. Benchmark Case (1% Branching Ratio)

• Slow Equilibration: In the medium, the dark baryon equilibration rate ($\gamma_2$) is orders of magnitude slower than the Urca rate ($\gamma_1$), even at temperatures of tens of MeV. The equilibration timescale is $\sim 40$ minutes, far exceeding the millisecond timescales of merger oscillations.
• Effect on Bulk Viscosity: Because the dark baryon population is effectively "frozen" during density oscillations, the bulk viscosity is dominated entirely by the Urca process.
• Magnitude: The presence of dark baryons modifies the EoS (specifically the symmetry energy), leading to a reduction in the Urca bulk viscosity by a factor of 2–3 at most. This change is within current theoretical uncertainties of the nuclear EoS and is unlikely to be distinguishable observationally.

B. Rapid Decay Case (Heavier $\chi$, Stronger Coupling)

• Resonance Potential: If the coupling $g_\phi$ is increased (up to the Raffelt limit) and $m_\chi > m_n$, the dark decay rate increases significantly.
• Enhanced Viscosity: At high temperatures ($T \sim 50$ MeV), the dark equilibration rate approaches the oscillation frequency ($\omega \approx 1$ kHz). This leads to a strong enhancement of the bulk viscosity.
• Damping Timescales: In this scenario, bulk viscosity could damp density oscillations in merger remnants on timescales as short as 20–40 ms at temperatures of 50 MeV, whereas standard Urca viscosity becomes ineffective at these temperatures.

5. Significance and Conclusions

• Signature of Dark Sectors: The paper suggests that while the specific model solving the neutron lifetime anomaly (1% branching ratio) has a negligible impact on merger dynamics, general dark sectors with moderate couplings could leave a distinct signature.
• High-Temperature Transport: A key physical insight is that standard model weak interactions (like Urca) become too fast to generate bulk viscosity at high temperatures ($T > 10$ MeV) because they maintain equilibrium instantly. However, Beyond Standard Model (BSM) interactions with weak couplings might remain slow enough at these high temperatures to be comparable to dynamical timescales, thereby generating significant bulk viscosity.
• Observational Implications: If dark sectors exist with parameters that allow for rapid equilibration in hot matter, they could significantly alter the post-merger gravitational wave signal by damping oscillations more efficiently than standard physics predicts.
• Future Work: The authors note that if dark baryons are truly dark matter, a two-fluid approach (where fluids interact only gravitationally) might be more realistic, which would require a recalculation of the bulk viscosity. Additionally, other decay channels (e.g., to three dark quarks) should be investigated.

In summary, the paper establishes that while the specific "neutron lifetime anomaly" solution does not drastically alter neutron star merger physics, the general mechanism of slow chemical reactions in hot dense matter provided by dark sectors offers a promising avenue for detecting new physics through gravitational wave observations of merger remnants.