Non parametric constraints of gravitational-electromagnetic luminosity distance ratio

This paper introduces a new non-parametric method to constrain the ratio between gravitational-wave and electromagnetic-wave luminosity distances, which, when applied to GWTC-3 binary black hole merger data, yields results consistent with General Relativity and previous parametric analyses.

The Gist

Imagine the universe as a giant, expanding ocean. For a long time, we've been trying to measure how fast this ocean is stretching (a rate called the Hubble Constant). To do this, scientists usually use "lighthouses" in space.

There are two types of lighthouses:

1. Electromagnetic Lighthouses (EM): These are stars or galaxies we can see with telescopes. We know how bright they should be. If they look dim, we know they are far away.
2. Gravitational Wave Lighthouses (GW): These are ripples in the fabric of space-time caused by massive objects crashing together (like black holes). We can't "see" them, but we can "hear" them. By analyzing the sound of the crash, we can calculate how far away they are.

The Big Question: Do the Rules Change?

In our current understanding of physics (General Relativity), both types of lighthouses should tell us the exact same distance. If a black hole merger is 1 billion light-years away, the "sound" distance and the "light" distance should match perfectly.

However, some scientists suspect that gravity might behave differently as the universe gets older and bigger. Maybe gravity gets slightly "weaker" or "stronger" over time, or maybe it leaks into other dimensions. If that's true, the gravitational wave distance would be different from the light distance.

The Problem with Previous Methods

Previously, to test this, scientists had to guess a specific mathematical formula for how gravity might change. It was like trying to find a lost key by guessing, "It's either in the kitchen, the bedroom, or the garage." If the key was actually in the basement, you'd never find it because you didn't guess that option.

The New Solution: A "Shape-Shifting" Ruler

This paper introduces a clever new method that doesn't guess the formula. Instead, it builds a flexible, non-parametric ruler.

Think of it like a connect-the-dots game.

• Instead of drawing a straight line or a perfect curve based on a guess, the scientists place a few "knots" (dots) at different distances in the universe (redshifts).
• They then use a special mathematical tool (called PCHIP) to draw a smooth, flexible line connecting these dots.
• This line can bend, stretch, or shrink however the data tells it to. It doesn't assume a specific shape. It just asks the data: "Where do you want to go?"

This allows them to map the relationship between the "sound distance" and the "light distance" without forcing it into a pre-made box.

The Experiment: Listening to 42 Cosmic Crashes

The authors took data from the GWTC-3 catalogue, which contains records of 42 black hole collisions detected by LIGO and Virgo.

Since most of these black holes didn't have a visible "light" counterpart (they were "Dark Sirens"), the scientists had to be detectives. They looked at the sky location of the crash and checked galaxy maps to guess which galaxies might have hosted the event. By combining the "sound" distance with the statistical "light" distance of those potential host galaxies, they could test their flexible ruler.

The Results: Einstein Was Right (Again)

After running the numbers, the flexible ruler showed something very reassuring:

• The line connecting the dots stayed almost perfectly flat at 1.0.
• This means the distance measured by gravity and the distance measured by light are identical.
• The results are consistent with General Relativity. There is no evidence that gravity is changing or leaking into other dimensions—at least not within the limits of our current data.

Why This Matters

Even though the result confirms Einstein, the method is the real breakthrough.

• Before: We were like people trying to describe a shape by only looking at a circle, a square, or a triangle.
• Now: We have a piece of clay that can be molded into any shape. If gravity does start acting weird in the future, this new method will be able to spot the weirdness immediately, without us having to guess what it looks like first.

In short: The scientists built a super-flexible, guess-free tool to measure the universe. They used it to check if gravity is playing by the rules, and for now, gravity is playing fair. But if it ever decides to break the rules, this tool will be the first to know.

Technical Summary

1. Problem Statement

The ratio between the gravitational wave (GW) luminosity distance ($d_L^{GW}$) and the electromagnetic wave (EMW) luminosity distance ($d_L^{EM}$), denoted as $r(z)$, is a critical observable for testing the nature of gravity.

• General Relativity (GR) Prediction: In GR, GWs and EMWs propagate identically, implying $r(z) = 1$ for all redshifts $z$.
• Modified Gravity: Many alternative theories predict a deviation where $r(z) \neq 1$, often due to a time-varying effective Planck mass or a speed of GWs different from the speed of light.
• Current Limitation: Previous analyses of GW data (specifically from the GWTC-3 catalogue) have relied on parametric models (e.g., $r(z) = \Xi_0 + (1-\Xi_0)(1+z)^{-n}$) to constrain this ratio. These models assume a specific functional form for the deviation, which may bias results or miss features not captured by the chosen parametrization.
• Goal: The authors aim to develop a model-independent (non-parametric) method to reconstruct $r(z)$ directly from observational data without assuming a specific theoretical functional form.

2. Methodology

The authors developed a new non-parametric framework implemented within the existing Bayesian inference package gwcosmo.

A. Non-Parametric Reconstruction (PCHIP)

Instead of fitting a global equation, the authors reconstruct $r(z)$ using Piecewise Cubic Hermite Interpolating Polynomials (PCHIP).

• Knots: The function is defined by values at specific redshift knots: $z_0=0, z_1, \dots, z_n, z_{n+1}=z_{max}$.
• Parameters: The values at these knots, $r_i = r(z_i)$, are the parameters to be constrained.
• Constraints:
  • Low Redshift: $r_0$ is fixed to 1 (consistent with GR and large-scale structure constraints at low $z$).
  • Monotonicity: To ensure the function $d_L^{GW}(z)$ has a well-defined inverse (required for calculating source-frame masses), $r(z)$ must be monotonic. The authors enforce this by defining "increment functions" $\Delta r_i$ that share the same sign.
  • Positivity: A lower bound $r_{min}$ is imposed based on the maximum injection distance and Hubble constant prior to ensure $r(z) > 0$.
• Free Parameters: The reconstruction is driven by free parameters $\rho_i$ that control the increments between knots.

B. Bayesian Framework

The analysis uses a Hierarchical Bayesian approach to jointly infer:

1. Cosmological Parameters: Specifically the Hubble constant ($H_0$).
2. Population Parameters: Describing the merger rate evolution ($\gamma, \kappa, z_p$) and the black hole mass distribution (Power-law + Gaussian peak).
3. Distance Ratio Parameters: The PCHIP parameters ($\rho_1, \rho_2$) governing $r(z)$.

The likelihood function marginalizes over the sky localization and redshift of "dark sirens" (GW events without EM counterparts) using galaxy catalogues (GLADE+ in the K-band) and accounts for selection effects via simulated injections.

C. Data Set

• Catalogue: GWTC-3 (Third Gravitational-Wave Transient Catalog).
• Events: 42 Binary Black Hole (BBH) mergers with Signal-to-Noise Ratio (SNR) > 11.
• Configuration: Two free redshift knots ($n=2$) were chosen at $z_1 = 0.25$ and $z_2 = 0.5$. The maximum redshift was set to $z_{max} = 10$.

3. Key Contributions

1. First Non-Parametric Implementation: This is the first implementation of a non-parametric $r(z)$ reconstruction within the gwcosmo package, moving beyond the limitations of specific parametric forms (like the $\Xi_0-n$ model).
2. Monotonicity Guarantee: The authors designed a specific mathematical formulation using smoothed step functions and increment constraints to guarantee that the reconstructed $r(z)$ is monotonic, a necessary condition for physical consistency in distance calculations.
3. Joint Analysis: The method successfully performs a joint inference of cosmological, population, and distance-ratio parameters, demonstrating that the non-parametric approach is computationally feasible and robust.
4. Consistency Relations: The non-parametric nature of the method makes it directly applicable to consistency relations derived from Effective Field Theory (EFT) of dark energy, which are often expressed in terms of observables rather than specific model parameters.

4. Results

The analysis of 42 BBH mergers yielded the following results:

• Hubble Constant ($H_0$): Estimated at $73.49^{+34.28}_{-28.41}$ km s$^{-1}$ Mpc$^{-1}$. This uncertainty is consistent with previous parametric analyses of the same dataset.
• Distance Ratio Parameters:
  • $\rho_1$ (controlling the first increment): $-0.35^{+1.43}_{-3.71}$.
  • $\rho_2$ (controlling the second increment): $2.46^{+4.15}_{-1.91}$.
• Reconstructed $r(z)$:
  • At $z_1 = 0.25$: $r_1 = 0.97^{+0.14}_{-0.18}$.
  • At $z_2 = 0.5$: $r_2 = 0.9^{+0.51}_{-0.35}$.
• Consistency with GR: The reconstructed $r(z)$ curve, including its 68% and 95% confidence intervals, is fully consistent with the General Relativity prediction of $r(z) = 1$. The results align with previous studies that used parametric models, validating the new non-parametric approach.
• Population Constraints: Most population parameters (e.g., mass distribution indices) were well-constrained, while merger rate evolution parameters ($\kappa, z_p$) remained less constrained, consistent with prior findings.

5. Significance

• Model Independence: By removing the assumption of a specific functional form for modified gravity, this method provides a more robust test of GR. It allows the data to "speak for itself" regarding the shape of $r(z)$, reducing the risk of false negatives (missing deviations because the wrong model was chosen) or false positives (biasing results toward a specific theory).
• Future Prospects: The method is applicable to both "dark sirens" (used here) and "bright sirens" (with EM counterparts). The authors suggest future work could extend this to fully non-parametric population models (mass and merger rate) to detect subtle features in the CBC population that current parametric models might miss.
• Theoretical Bridge: The approach facilitates direct comparison with theoretical consistency relations from dark energy EFT, which do not rely on specific parametrizations, thereby strengthening the link between observational GW data and fundamental theoretical physics.

In conclusion, the paper successfully demonstrates that non-parametric constraints on the GW-EMW luminosity distance ratio are feasible with current data, yielding results consistent with General Relativity while providing a powerful new tool for future, more sensitive gravitational wave observations.