Cosmology and modified GW propagation from the BNS mass function at third-generation detector networks

This paper presents conservative forecasts demonstrating that third-generation gravitational-wave detector networks, particularly when combining the Einstein Telescope with Cosmic Explorer, can achieve percent-level precision in constraining the Hubble constant and modified gravitational-wave propagation parameters by leveraging the binary neutron star mass function through joint cosmology-population Bayesian inference.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have been trying to measure exactly how fast this balloon is inflating (a rate called the Hubble Constant, or $H_0$) and whether the fabric of space itself is behaving exactly as Einstein predicted, or if there are some "secret rules" of gravity we haven't discovered yet.

This paper is a forecast—a crystal ball gazing into the future of how we will measure these things using the next generation of gravitational wave detectors.

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Messengers: "Standard Sirens"

In the past, astronomers measured the universe's expansion using "Standard Candles" (like supernovas), which are objects with a known brightness. If they look dim, they are far away.

Gravitational waves (ripples in space-time caused by colliding objects) act as "Standard Sirens."

• The Analogy: Imagine you hear a siren from a fire truck. If you know how loud the siren should be, you can tell how far away the truck is just by how quiet it sounds.
• The Problem: In the universe, we don't know the "loudness" of the siren perfectly because of a trick called the Mass-Redshift Degeneracy. It's like hearing a siren but not knowing if it's a loud siren far away or a quiet siren close by. Usually, we need to see the light (electromagnetic signal) from the crash to know which galaxy it came from, but most of these crashes are "dark" (no light).

2. The New Trick: The "Weight" of the Objects

This paper proposes a clever new way to solve the "loudness vs. distance" puzzle without needing to see the light.

• The Analogy: Imagine you are trying to guess how far away a person is shouting, but you can't see them. However, you know that people in your town generally have a specific range of voices (some deep, some high). If you hear a very deep voice, you know it's likely a large person. If you hear a high pitch, it's likely a smaller person.
• The Science: The authors use the mass function of Binary Neutron Stars (BNS). They know the statistical "distribution of weights" for these colliding stars. By analyzing the pattern of masses they detect across the universe, they can mathematically untangle the distance from the mass. It's like using the average size of people in a crowd to figure out how far away the crowd is, even if you can't see them clearly.

3. The Future Detectors: The "Big Ears"

The paper looks at two future super-detectives:

1. Einstein Telescope (ET): A massive underground detector in Europe. It can be built in a triangle shape (3 arms) or as two separate "L" shapes (2L).
2. Cosmic Explorer (CE): A massive detector in the US (40km long arms).

The authors simulate what happens when these detectors listen to thousands of neutron star collisions. They are very picky, though: they only listen to the loudest signals (Signal-to-Noise Ratio > 50).

• Why be picky? It's like trying to hear a whisper in a noisy room. If you only listen to the people shouting, you get a clear picture. If you try to listen to the whispers too, the background noise confuses your brain. By focusing on the loudest events, they get a very clean, though smaller, sample of data.

4. The Two Big Questions They Answered

Question A: How fast is the universe expanding? ($H_0$)

• The Result: Using just the Einstein Telescope, they can measure the expansion rate with about 10-12% precision.
• The Upgrade: If they team up with the Cosmic Explorer (US), the precision jumps to ~9%.
• The Metaphor: It's like going from guessing the speed of a car by looking at it through a foggy window, to using a high-speed camera that can measure the speed almost exactly.

Question B: Is Gravity different at large distances? ($\Xi_0$)

Einstein's General Relativity says gravity travels through space exactly as predicted. But some theories say gravity might "leak" or change as it travels across the vast universe.

• The Result: With just the Einstein Telescope, they can test this with 18% precision.
• The Upgrade: With the ET + CE team, they can test this with 6% precision.
• The Metaphor: Imagine throwing a ball across a field. If the air is thick (modified gravity), the ball slows down or changes shape. If the air is normal (Einstein's gravity), it flies straight. These detectors are so sensitive they can tell if the "air" of space is slightly thicker than we thought.

5. The "Conservative" Warning

The authors are very humble about their results. They admit they are being very conservative.

• The Analogy: They only counted the "shouting" events (SNR > 50). But there are thousands of "whispering" events (lower SNR) that they ignored.
• The Reality: If they included the whispers (which are much more numerous), the precision would likely be much better (perhaps 1% or less). They chose to ignore them to ensure their math didn't get confused by noise. So, the numbers in the paper are actually a "worst-case scenario" for how good these detectors will be.

6. The "Sweet Spot" of the Universe

The paper also found where in the universe these detectors work best.

• The Finding: The Einstein Telescope is best at measuring the universe's expansion at a specific distance (redshift $z \approx 0.25$).
• The Metaphor: It's like a telescope that has a "sweet spot" focus. It sees objects at a medium distance (not too close, not too far) with the sharpest clarity. Adding the Cosmic Explorer shifts this sweet spot slightly further out, allowing us to see deeper into the universe's history.

Summary

This paper is a blueprint for the future. It tells us that when the next generation of gravitational wave detectors comes online, we won't just be listening to black holes; we will be using the weights of neutron stars to measure the speed of the universe and test if Einstein's gravity holds up across cosmic distances.

Even with a very strict filter that ignores most of the data, the results are promising: we are on the verge of measuring the universe's expansion with incredible accuracy and potentially discovering new laws of physics.

Technical Summary

1. Problem Statement

The paper addresses two primary challenges in gravitational-wave (GW) cosmology:

1. The Mass-Redshift Degeneracy: GW signals from compact binary coalescences (CBCs) depend on the combination of source mass and redshift ($m(1+z)$), making it impossible to determine the redshift ($z$) and luminosity distance ($d_L$) independently without an electromagnetic counterpart (a "bright siren"). Since bright sirens are rare, statistical methods using "dark sirens" are required.
2. Modified Gravity Constraints: Many theories of modified gravity predict that GWs propagate differently than in General Relativity (GR), specifically altering the amplitude-distance relation. This is parameterized by a GW luminosity distance $d_L^{gw}$, which differs from the electromagnetic luminosity distance $d_L^{em}$. The deviation is governed by a function $\Xi(z)$, characterized by parameters $\Xi_0$ and $n$.

Goal: The authors aim to forecast the precision with which third-generation (3G) GW detectors can constrain the Hubble constant ($H_0$) and the modified gravity parameter $\Xi_0$ by utilizing the Binary Neutron Star (BNS) mass function as a statistical prior, without relying on galaxy catalogs or electromagnetic counterparts.

2. Methodology

A. Detector Networks

The study forecasts performance for 3G detectors:

• Einstein Telescope (ET): Analyzed in two configurations:
  • ET-$\Delta$: A triangular configuration with 10-km arms.
  • ET-2L: Two L-shaped detectors (15-km arms) located in Sardinia and the Meuse-Rhine region, oriented at 45° relative to each other.
• Cosmic Explorer (CE): A 40-km detector added to the network to form hybrid networks (ET+$\Delta$+CE and ET-2L+CE).

B. Event Selection and Simulation

• Population Model: A synthetic population of BNSs is generated using the Madau-Dickinson star formation rate history and a flat mass distribution between $m_{min}$ and $m_{max}$.
• Selection Criteria: To ensure robust parameter estimation and avoid biases from low-SNR events (where the Fisher matrix approximation breaks down), the authors apply a strict cut: Signal-to-Noise Ratio (SNR) > 50.
  • This results in $\sim 10^2$ events/year for ET alone and $\sim 10^3$ events/year for ET+CE networks.
  • The selected events are restricted to low redshifts ($z < 0.8$).
• Waveform & Analysis: Events are projected using the IMRPhenomD waveform model. Parameter estimation is performed using Fisher matrices, and posterior samples are recovered via direct sampling in the high-SNR limit.

C. Bayesian Inference Framework

The authors employ a hierarchical Bayesian approach using the Icarogw framework to perform a joint inference of:

1. Population Hyper-parameters: Mass distribution limits ($m_{min}, m_{max}$), and star formation rate parameters ($\gamma, \kappa, z_p$).
2. Cosmological Hyper-parameters:
   • $\Lambda$CDM Scenario: Inferring $H_0$ and $\Omega_M$.
   • Modified Gravity Scenario: Inferring $\Xi_0$ and $n$ (with $H_0$ and $\Omega_M$ fixed to Planck 2018 values).

• Prior Handling: A specific prior $n \ge 0.3$ is imposed to break the degeneracy between $\Xi_0$ and $n$ (where GR is recovered at $\Xi_0=1$ for any $n$, or $n=0$ for any $\Xi_0$). This prevents unphysical tails in the posterior distribution.

3. Key Contributions

• Mass Function as a Cosmological Probe: The paper demonstrates quantitatively that the intrinsic mass distribution of BNSs can serve as a powerful "spectral siren" to break the mass-redshift degeneracy, enabling cosmological inference without galaxy catalogs.
• Conservative Forecasts: By restricting analysis to high-SNR events ($>50$), the authors provide a conservative lower bound on the capabilities of 3G detectors. They argue that including lower-SNR events would significantly improve constraints but introduces systematic risks.
• Configuration Comparison: A detailed comparison between the ET triangular and 2L configurations, showing they perform comparably for cosmology, while the addition of CE provides the most significant gain.

4. Key Results

A. Precision on Cosmological Parameters

The relative precision (68% confidence level) achieved for different network configurations is summarized below:

Network Configuration	$H_0$ Precision	$\Xi_0$ Precision	Best $H(z)$ Precision (at $z_{best}$)
ET-$\Delta$ (Triangle)	12%	18%	10% at $z=0.23$
ET-2L	11%	18%	6% at $z=0.28$
ET-$\Delta$ + CE	9%	6%	4% at $z=0.37$
ET-2L + CE	9%	6%	3% at $z=0.38$

• Impact of CE: Adding Cosmic Explorer improves constraints on $\Xi_0$ by a factor of 3 (from 18% to 6%) and $H_0$ by ~25%. This is because CE extends the redshift reach, where modified gravity effects (which grow with redshift) are more pronounced.
• ET Configurations: The triangular and 2L configurations of ET yield nearly identical results for cosmological parameters, despite the 2L configuration having a higher event count.

B. Population Parameters

• The mass distribution parameters ($m_{min}, m_{max}$) are well-constrained (precision $\sim 1-2\%$).
• The star formation rate peak ($z_p$) and high-redshift slope ($\kappa$) are poorly constrained due to the lack of high-redshift events in the SNR > 50 sample.

C. Modified Gravity Constraints

• The parameter $\Xi_0$ (representing the deviation from GR at $z \to 0$) is constrained to $\sim 6\%$ precision with ET+CE.
• The parameter $n$ (describing the redshift evolution of the deviation) remains less constrained ($\sim 80-90\%$ precision) due to the limited redshift range of the selected BNS sample.

5. Significance and Future Outlook

• Lower Bound on Capability: The authors emphasize that their results are conservative. If lower-SNR events (down to SNR $\sim 12$) were included, the event count would increase by a factor of 10–50, potentially pushing the precision on $\Xi_0$ to the percent level.
• Complementarity with BBHs: While BNSs are excellent for constraining $H_0$ at low redshifts ($z \sim 0.2-0.4$), Binary Black Holes (BBHs) are detectable to higher redshifts ($z \sim 1.5$). The paper suggests that combining BNS and BBH "dark siren" samples would provide a comprehensive map of the expansion history $H(z)$ and tighter constraints on modified gravity.
• Robustness: The BNS mass function is argued to be more stable and less prone to modeling systematics (e.g., peaks or slope changes) compared to the BBH mass function, making it a reliable anchor for statistical cosmology.
• Synergy: Future work could combine this "spectral siren" method with galaxy catalog methods, leveraging the fact that 3G detectors will probe redshifts ($z < 0.8$) well-covered by upcoming surveys like Euclid and LSST.

In conclusion, the paper establishes that even with a conservative high-SNR cut, 3G detector networks will be capable of measuring the Hubble constant to $\sim 10\%$ and testing modified gravity propagation to $\sim 6\%$ precision solely through the statistical properties of the BNS mass function.