Distinguishing Neutron Star vs. Low-Mass Black Hole Binaries with Late Inspiral & Postmerger Gravitational Waves $-$ Sensitivity to Transmuted Black Holes and Non-Annihilating Dark Matter

This paper demonstrates that next-generation gravitational wave detectors with enhanced high-frequency sensitivity can distinguish between binary neutron star and low-mass black hole mergers during the late inspiral and postmerger phases, thereby enabling the disentanglement of their respective merger rates and placing constraints on heavy, non-annihilating dark matter interactions that could cause neutron stars to collapse into transmuted black holes.

The Gist

Imagine the universe as a giant, dark ocean. For a long time, we've been listening to the ripples in this ocean—gravitational waves—to understand what's happening beneath the surface. Recently, we've heard a lot of splashes from two objects crashing together. The big mystery? Are we hearing two giant balls of neutron star dough collide, or are we hearing two tiny, invisible black holes smash into each other?

To the naked ear (and current detectors), these two events sound almost identical in the beginning. It's like trying to tell the difference between a piano and a synthesizer playing the same note at a low volume; they sound the same. But as they get closer to the crash, the sounds should diverge.

This paper is about building better "ears" (future gravitational wave detectors) to finally tell the difference, and in doing so, solving a cosmic mystery about Dark Matter.

Here is the breakdown in simple terms:

1. The Great Cosmic Imposter

For years, astronomers have been confused. They see two heavy objects merging. If they weigh less than about 3 suns, they usually assume they are Neutron Stars (super-dense balls of neutrons). But what if they are actually Low-Mass Black Holes?

• The Problem: In the early stages of the crash (the "inspiral"), a neutron star and a black hole of the same mass sing the exact same song.
• The Difference: Neutron stars are like soft, squishy pillows. Black holes are like hard, pointy needles. When they get very close, the "pillow" gets squished by gravity, changing the sound. The "needle" doesn't change.
• The Catch: Current detectors (like LIGO) are great at hearing the low notes but miss the high-pitched squeals where the "squishing" happens.

2. The New "Super-Ears"

The authors of this paper looked at future detectors: NEMO, Cosmic Explorer, and the Einstein Telescope.

• NEMO is like a specialized microphone tuned to hear the high-pitched "squeal" of the crash. It's perfect for hearing the difference between the pillow and the needle.
• Cosmic Explorer and Einstein Telescope are like massive concert halls that can hear the whole song from the first note to the last, with incredible clarity.

The Result: With these new tools, we won't just guess anymore. We will be able to say with 99% certainty: "That was a neutron star collision!" or "That was a black hole collision!"

3. The Dark Matter Detective Story

Here is where it gets really cool. Why do we care if we can tell them apart? Because of Dark Matter.

• The Theory: Dark Matter is the invisible stuff holding galaxies together. Some scientists think heavy dark matter particles might get trapped inside neutron stars over billions of years.
• The Transformation: Imagine a neutron star as a giant sponge. If you keep soaking it with dark matter, it eventually gets so heavy it collapses into a black hole. The authors call this a "Transmuted Black Hole" (TBH).
• The Connection: If these TBHs exist, they would look exactly like low-mass black holes. If we start seeing too many low-mass black holes, it might mean our neutron stars are secretly turning into black holes because of dark matter.

4. The "Misclassification" Game

The paper does a clever statistical trick.

1. They assume every crash we see is actually a Neutron Star crash (the "True" scenario).
2. They ask: "How often would our new detectors mistake a neutron star crash for a black hole crash?"
3. They found that with current tech, we'd make a lot of mistakes. But with the new "Super-Ears," we can tell them apart almost perfectly.

Why does this matter?
If we can perfectly distinguish them, we can count how many black holes there really are. If we find more black holes than standard physics predicts, it's a smoking gun for Dark Matter turning neutron stars into black holes.

5. The Bottom Line

• Old Way: We hear a crash, guess what it is, and hope we're right.
• New Way (with this paper): We use high-tech detectors to listen to the "crunch" at the very end of the crash.
• The Payoff: This allows us to:
  1. Accurately count how many neutron stars and black holes exist.
  2. Set strict limits on how heavy Dark Matter particles can be and how they interact with normal matter.

In a nutshell: This paper is a blueprint for upgrading our cosmic hearing aids. By listening to the final, high-pitched moments of a collision, we can stop confusing "pillows" with "needles" and potentially discover the invisible ghost particles (Dark Matter) that might be turning our stars into black holes.

Technical Summary

1. Problem Statement

The classification of compact binary coalescences (CBCs) with component masses in the "mass gap" ($2.5 - 5 M_\odot$) or the lower mass range ($1 - 2.5 M_\odot$) remains ambiguous. Current gravitational wave (GW) detectors (like LIGO-Virgo-KAGRA) primarily observe the inspiral phase, where waveforms for Binary Neutron Star (BNS) mergers and Binary Low-Mass Black Hole (BLMBH) mergers are nearly identical.

• The Challenge: Without electromagnetic (EM) counterparts (e.g., kilonovae), distinguishing between a true BNS merger and a BLMBH merger is difficult.
• The Consequence: Misclassification leads to biased estimates of the BNS merger rate and obscures the potential discovery of exotic physics, such as Transmuted Black Holes (TBHs). TBHs are hypothetical low-mass black holes formed when neutron stars collapse due to the capture of heavy, non-annihilating Dark Matter (DM) particles. If TBHs exist, they would contribute to the BLMBH merger rate, mimicking BNS signals.

2. Methodology

The authors employ a multi-step statistical and waveform analysis approach to quantify the distinguishability between BNS and BLMBH systems.

A. Waveform Generation & Benchmarking

• BNS Waveforms: Extracted from the CoRe (Computational Relativity) database, utilizing 8 distinct Equation of State (EoS) models ranging from stiff (e.g., 2H, MS1b) to soft (e.g., 2B, SLy). These include both zero-temperature (BAM code) and finite-temperature (THC code) simulations.
• BLMBH Waveforms: Generated using the IMRPhenomD phenomenological model in PyCBC, assuming non-spinning, equal-mass binaries in the $1 - 2.5 M_\odot$ range.
• Parameters: Benchmark systems assume equal masses ($m_1=m_2=1.35 M_\odot$) and non-eccentric orbits.

B. Statistical Metrics

To quantify distinguishability, the authors calculate:

1. Fitting Factor (FF): The maximum overlap (match) between a true BNS signal and a bank of BLMBH templates, optimized over intrinsic parameters.
   • $FF = \max_{\text{templates}} \frac{\langle h_{BNS} | h_{BLMBH} \rangle}{\sqrt{\langle h_{BNS} | h_{BNS} \rangle \langle h_{BLMBH} | h_{BLMBH} \rangle}}$
   • A significant drop in FF ($<0.97$) indicates a loss of signal-to-noise ratio (SNR) and potential distinguishability.
2. Bayes Factor (BF): The ratio of marginal likelihoods favoring the BNS hypothesis ($H_1$) over the BLMBH hypothesis ($H_2$).
   • Approximated as: $B \approx \exp\left(\frac{(1-FF^2)\rho_{opt}^2}{2}\right)$, where $\rho_{opt}$ is the optimal SNR.
   • The analysis is performed separately for the late inspiral ($500 \text{ Hz} - f_{merge}$) and postmerger ($f_{merge} - 4000 \text{ Hz}$) phases.

C. Detector Sensitivity Analysis

The study evaluates performance across four detectors:

• LIGO A+: Current generation (upgraded).
• NEMO: Proposed 2.5-gen detector optimized for high frequencies (postmerger).
• Cosmic Explorer (CE) & Einstein Telescope (ET): Proposed 3rd-generation detectors with superior low-frequency sensitivity.

D. Population & Dark Matter Constraints

• Rate Parsing: The authors model the observed CBC rate as a mixture of true BNS and BLMBH events, using the Bayes Factor to determine the probability of misclassification at various luminosity distances ($D_L$).
• DM Constraints: They link the fraction of BLMBHs ($f_{BLMBH}$) to the capture of heavy non-annihilating DM. If DM accumulates in NS cores, it triggers collapse into TBHs. The absence of observed BLMBHs (or the ability to distinguish them from BNSs) sets exclusion limits on the DM mass ($m_\chi$) and DM-nucleon cross-section ($\sigma_{\chi n}$).

3. Key Results

A. Distinguishability via Waveform Features

• Postmerger Signatures: The primary distinction lies in the postmerger phase. BNS mergers produce a characteristic secondary peak in the GW spectrum (due to the remnant's oscillation modes), whereas BLMBH mergers do not.
• Fitting Factors:
  • For LIGO A+, FFs are generally high ($>0.97$), leading to Bayes Factors near unity. Distinguishability is poor even for stiff EoSs.
  • For NEMO, the high-frequency sensitivity allows for significant FF drops in the postmerger phase, yielding strong Bayes Factors ($B > 100$) for stiff EoSs (e.g., 2H, H4) even at $D_L = 300$ Mpc.
  • For CE and ET, the superior low-frequency sensitivity provides high SNR in the inspiral phase, allowing for confident distinction ($B \gg 100$) for all viable EoSs up to $D_L \approx 600$ Mpc.

B. Distance Dependence

• Distinguishability scales as $B \propto \exp(1/D_L^2)$.
• Stiff EoSs (larger radii) are easier to distinguish than Soft EoSs (more compact, mimicking black holes). For the softest EoS (2B), even CE/ET struggle to distinguish BNS from BLMBH at $D_L > 200$ Mpc ($B \approx 11$).

C. Impact on Merger Rates

• Misclassification becomes significant beyond $D_L \approx 120$ Mpc for LIGO A+ and $D_L \approx 1200$ Mpc for ET.
• Without proper distinction, the inferred BNS merger rate could be overestimated by including BLMBH events.
• The study provides 90% exclusion sensitivities for the fraction of BLMBHs ($f_{BLMBH}$) in the CBC population. For ET (1 year), $f_{BLMBH}$ can be constrained to $<1\%$ for stiff EoSs.

D. Dark Matter Constraints

• By translating the exclusion limits on $f_{BLMBH}$ into DM parameters, the authors derive constraints on non-annihilating DM.
• Results:
  • LIGO A+ (10 years): Can probe DM masses $m_\chi \sim 10^7 - 10^{10}$ GeV with cross-sections $\sigma_{\chi n} \sim 10^{-48} - 10^{-45}$ cm$^2$.
  • Einstein Telescope (1 year): Achieves comparable or stronger constraints, effectively probing the parameter space where DM-induced transmutation would occur.
  • The constraints are robust even when accounting for the background of misclassified BNS events, unlike previous studies that assumed zero background.

4. Significance and Conclusion

• Astrophysical Impact: The paper establishes that next-generation detectors (NEMO, CE, ET) will resolve the ambiguity between BNS and BLMBH mergers. This is crucial for accurately determining the BNS merger rate and understanding the neutron star population.
• New Physics Probe: The ability to distinguish these events provides a novel, model-independent method to search for Transmuted Black Holes.
• Dark Matter: The study demonstrates that GW observations can place competitive constraints on heavy, non-annihilating dark matter, specifically ruling out scenarios where DM capture leads to NS collapse within the observable universe.
• Methodological Advance: The work highlights the critical importance of postmerger modeling and high-frequency detector sensitivity. While inspiral data is sufficient for 3rd-gen detectors, 2.5-gen detectors like NEMO rely heavily on the postmerger signal to break degeneracies.

In summary, the paper argues that the "mass gap" ambiguity is solvable with future detectors, turning GW astronomy into a powerful tool for constraining both the Equation of State of nuclear matter and the nature of dark matter.