Detectability of Nearby Binary Neutron Stars with Future sub-mHz Gravitational Wave Missions

This paper estimates that future sub-mHz gravitational wave missions, such as LISAmax, Folkner, and eASTROD, will significantly outperform current detectors by identifying hundreds of Galactic binary neutron stars and several in the Large Magellanic Cloud over 5–10 years, thereby offering unprecedented insights into binary evolution and nuclear physics.

The Gist

Imagine the universe is a giant, cosmic orchestra. For years, we've only been able to hear the loudest, highest-pitched instruments (like the final, screaming notes of colliding black holes) using ground-based detectors like LIGO. But there's a whole section of the orchestra playing a deep, slow, rumbling bass line that we've been missing. This "bass line" is made of gravitational waves from binary neutron stars (two dead stars dancing around each other) that are still in the early stages of their dance.

This paper is about building a new, super-sensitive "ear" to hear that deep bass line, specifically focusing on a frequency range that is too low for our current space telescopes but too high for our pulsar timing arrays.

Here is the breakdown of the paper in simple terms:

1. The Problem: The "Frequency Gap"

Think of gravitational wave detectors like radio tuners.

• Ground detectors (LIGO/Virgo): Tune into high frequencies (fast spins). They hear the end of the dance when stars crash.
• Pulsar Timing Arrays: Tune into ultra-low frequencies (very slow spins). They hear the background hum of the whole galaxy.
• Space detectors (LISA, Taiji, Tianqin): Tune into the middle (millihertz). They are great, but they miss the very slow, early stages of the dance.

There is a "gap" in the middle-low frequencies (sub-millihertz). The authors are proposing new, next-generation space missions (named LISAmax, Folkner, and eASTROD) designed specifically to fill this gap. These new detectors are like upgrading from a standard radio to a high-fidelity studio microphone that can hear the faintest whispers of the universe.

2. The Target: The "Eccentric Dancers"

The stars they are looking for are Binary Neutron Stars (BNSs).

• The "Circular" Dancers: Most stars eventually get pulled into a perfect circle by gravity before they crash. By the time they reach the high-frequency range, they are spinning in perfect circles.
• The "Eccentric" Dancers: Some stars are born with a wild, elliptical (oval-shaped) orbit because of a violent "kick" from a supernova explosion. They spin in a lopsided, stretched-out path.

Why does this matter?
Current detectors can't see these "eccentric" dancers well because they spin too slowly (low frequency) and their orbits are too stretched out. But these new sub-mHz detectors are designed to catch them while they are still spinning in those wild, oval paths. This is like catching a dancer mid-twirl before they smooth out their steps. It tells us how they were born and what kind of "kick" they got at birth.

3. The Simulation: A Cosmic Crystal Ball

Since we can't wait 10 years to see if these detectors work, the authors used a computer simulation (a "crystal ball") to predict what would happen.

• They created a fake universe with millions of binary stars, simulating how they are born, how they move through the galaxy, and how their orbits change over billions of years.
• They then "ran" the simulation through the noise profiles of the three new detectors (LISAmax, Folkner, eASTROD) to see how many stars they could hear.

4. The Results: A Symphony of Discovery

The results are exciting! Here is what they found:

• The Numbers: Over a 5-to-10-year mission, these new detectors could hear hundreds to over a thousand of these binary neutron stars in our own galaxy (the Milky Way).
  • LISAmax is the specialist for the "wild" dancers (highly eccentric orbits). It might find about 500–900 stars.
  • Folkner and eASTROD are the all-rounders. They might find even more (780–1,370) because they are super-sensitive to the slow, low-frequency rumbles.
• The "Verification" Stars: The authors identified seven real stars that we already know about (found by radio telescopes). They calculated that the new detectors would hear these stars loud and clear.
  • The Star of the Show: One system, J0737-3039, is so close and loud that it would be the "VIP" of the concert. It would have a signal-to-noise ratio of about 100. This means it would be the perfect test case to prove the new detectors actually work.
• The Magellanic Clouds: They also looked at the Large and Small Magellanic Clouds (small galaxies next to ours). They expect to hear a few stars in the Large Cloud, but the Small Cloud is too far away and too quiet to hear much.

5. The Challenge: The "Crowded Room"

There is one big hurdle. The galaxy is full of other gravitational wave sources, mostly Double White Dwarfs (two dead, smaller stars). Imagine trying to hear a single violin in a room where 10,000 other violins are playing at the same time. This creates "confusion noise."

• The authors are optimistic. They believe that as we get better at modeling the data, we can filter out this background noise (like noise-canceling headphones) and isolate the binary neutron stars. If we can't, the number of detectable stars drops significantly.

The Big Picture

This paper is a blueprint for the future. It says: "If we build these specific types of space telescopes, we will unlock a new chapter in astronomy."

Instead of just hearing the "crash" of stars, we will be able to listen to their entire "relationship history"—how they were born, how they danced in wild, oval orbits, and how they slowly tightened their grip before merging. It's like going from watching a movie only at the very end to watching the whole story from the beginning.

Technical Summary

1. Problem Statement

Gravitational wave (GW) astronomy currently operates in three distinct frequency bands:

• Nanohertz (nHz): Probed by Pulsar Timing Arrays (PTAs), detecting supermassive black hole binaries.
• Millihertz (mHz): Probed by space-based missions like LISA, Taiji, and Tianqin, targeting massive black holes and extreme mass-ratio inspirals.
• Hertz (Hz): Probed by ground-based detectors (LIGO/Virgo/KAGRA), detecting stellar-mass compact binary coalescences.

There exists a significant frequency gap between the nHz and mHz bands (specifically the sub-mHz to microhertz range). While Binary Neutron Stars (BNSs) are crucial sources for understanding stellar evolution and nuclear physics, their early inspiral stages often possess significant orbital eccentricities. Ground-based detectors observe BNSs only after gravitational radiation has circularized their orbits ($e \lesssim 10^{-5}$), erasing information about their formation channels (e.g., supernova natal kicks). While LISA can detect some residual eccentricity, its sensitivity drops significantly in the sub-mHz range.

The paper addresses the potential of next-generation sub-mHz space-based detectors (specifically LISAmax, Folkner, and eASTROD) to detect nearby inspiraling BNSs, particularly those with high eccentricities, thereby bridging the gap in multi-band GW astronomy.

2. Methodology

The authors employed a multi-step simulation and analysis framework:

• Population Synthesis (CBPS):

  • Utilized an updated version of the Binary Stellar Evolution (BSE) code to generate mock BNS populations in the Milky Way (MW), Large Magellanic Cloud (LMC), and Small Magellanic Cloud (SMC).
  • Initial Conditions: Assumed a Kroupa initial mass function, uniform mass ratios, and specific distributions for binary separation.
  • Physics: Incorporated common envelope (CE) ejection, supernova explosions (with natal kicks), and gravitational wave-driven orbital decay.
  • Equation of State (EOS): Adopted the SLy EOS ($M_{TOV} \approx 2.06 M_\odot$) but verified that stiffer EOSs (e.g., DD2) do not significantly alter results.
  • Dynamics: Simulated the dynamical evolution of BNSs within the Galactic potential (using the galpy package) from formation to the present day, accounting for natal kicks that displace systems from the thin disk.

• Signal-to-Noise Ratio (SNR) Calculation:

  • Calculated SNR ($\rho$) for observation periods of 5 and 10 years.
  • Eccentricity Handling: Unlike circular binaries, eccentric BNSs emit GWs at harmonics of the orbital frequency ($f_{GW} = n f_{orb}$). The authors summed the SNR contributions from all harmonics using the characteristic strain formula derived by Peters & Mathews.
  • Noise Modeling: Compared the sensitivity curves of sub-mHz detectors (LISAmax, Folkner, eASTROD) against mHz detectors (LISA, Taiji, Tianqin).
  • Confusion Noise: Optimistically assumed that the confusion noise from unresolved Galactic Double White Dwarfs (DWDs) can be modeled and subtracted, leaving only a residual noise floor. A detection threshold of $\rho_{th} = 8$ was applied.

3. Key Contributions

• Quantitative Detection Forecasts: Provided the first comprehensive estimates for the number of detectable Galactic and extragalactic (LMC/SMC) BNSs by future sub-mHz missions.
• Eccentricity Sensitivity: Demonstrated that sub-mHz detectors are uniquely capable of detecting highly eccentric systems ($e > 0.9$) that would be invisible or undetectable by mHz detectors due to frequency limitations.
• Validation Candidates: Identified seven known radio BNSs (including the famous Hulse-Taylor binary and the Double Pulsar J0737-3039) as viable verification sources for these future missions.
• Mission Comparison: Systematically compared the performance of LISAmax, Folkner, and eASTROD, highlighting their distinct sensitivity profiles regarding orbital period and eccentricity.

4. Key Results

A. Detection Rates in the Milky Way

• Total Counts:
  • LISAmax: $\sim 520 - 900$ detectable BNSs (depending on mission duration).
  • Folkner & eASTROD: $\sim 780 - 1,370$ detectable BNSs.
  • Reasoning: Folkner and eASTROD have superior sensitivity at lower frequencies ($< 4 \times 10^{-4}$ Hz), where the majority of the mock population peaks. LISAmax is slightly less sensitive in this low-frequency regime but excels at higher sub-mHz frequencies.
• High Eccentricity Systems:
  • LISAmax is superior at detecting highly eccentric systems ($e > 0.90$) due to its better sensitivity at higher frequencies ($f \gtrsim 10^{-3}$ Hz).
  • LISAmax predicts $\sim 23$ detections with $e > 0.90$, compared to $\sim 17$ (Folkner) and $\sim 20$ (eASTROD).
• Observation Time Dependence: Detection numbers scale non-linearly with time. For a 10-year mission, the SNR scales as $\sqrt{T_{obs}}$, significantly boosting the detectable population compared to shorter missions.

B. Extragalactic Detections (LMC and SMC)

• LMC: Predicted detection of $\sim 4 - 18$ inspiraling BNSs. The detectable population is biased toward shorter orbital periods ($P_{orb} \lesssim 0.14$ days) due to the larger distance ($\sim 49.7$ kpc) reducing the SNR.
• SMC: Detection is highly challenging ($\sim 0.6 - 1$ system), primarily due to lower star formation rates and larger distances ($\sim 62$ kpc).

C. Verification Sources

• Seven Known Systems: The study identified seven radio BNSs detectable by sub-mHz detectors: B1913+16, J1757-1854, J0509+3801, B1534+12, J1906+0746, J1946+2052, and J0737-3039.
• J0737-3039 (The Double Pulsar): This system is the most promising verification source. Due to its proximity ($d_L \sim 1.1$ kpc) and short period, it is predicted to have an extremely high SNR:
  • LISAmax: $\rho \approx 106$
  • Folkner: $\rho \approx 139$
  • eASTROD: $\rho \approx 136$
• Eccentricity Measurement: Only $\sim 30\%$ of detectable mock systems satisfy the criteria (two harmonics with $\rho > 8$) required to measure eccentricity with a fractional precision of $\sim 10\%$.

5. Significance and Implications

• Probing Formation Channels: The ability to detect BNSs with high eccentricities ($e > 0.9$) in the sub-mHz band offers a unique window into supernova natal kicks and binary evolution processes that are otherwise erased by the time the systems reach the LIGO frequency band.
• Multi-Messenger Astronomy: Identifying specific radio BNSs (like J0737-3039) as high-SNR GW sources allows for precise cross-validation between electromagnetic timing and GW observations, testing General Relativity in new regimes.
• Mission Design: The results underscore the scientific necessity of the sub-mHz frequency band. Missions like Folkner and eASTROD, with their extended baselines and lower-frequency sensitivity, are predicted to detect significantly more BNSs than LISAmax, particularly for the bulk of the population.
• Caveats: The authors note that these optimistic estimates rely on the ability to model and subtract the confusion noise from Galactic Double White Dwarfs. If this foreground cannot be removed, the detection rate for MW BNSs could drop drastically to $< 20$.

In conclusion, this study establishes that future sub-mHz GW missions will revolutionize our understanding of BNS formation by detecting a large population of eccentric, inspiraling systems that are currently invisible to existing or planned mHz detectors.