The Luminosity of the Darkness: Schechter function in dark sirens

This paper demonstrates that neglecting the redshift evolution of the Schechter function in galaxy luminosity distributions introduces biases in Hubble constant ($H_0$) measurements from dark sirens, highlighting the necessity of incorporating evolving galaxy population models to accurately constrain both $H_0$ and merger rate parameters.

The Gist

The Big Picture: Measuring the Universe's Speed

Imagine the universe is a giant, expanding balloon. Scientists want to know exactly how fast this balloon is inflating. This speed is called the Hubble Constant ($H_0$).

For a long time, scientists have been arguing about the exact number. Some measurements say it's one speed, and others say it's faster. This disagreement is known as the "Hubble Tension."

To solve this, scientists are using Gravitational Waves (ripples in space-time caused by colliding black holes or neutron stars) as a new tool. These ripples act like "standard sirens"—think of them as cosmic lighthouses. By measuring how loud the ripple is, we know how far away it is. But to know how fast the universe is expanding, we also need to know how fast the source is moving away from us (its redshift).

The Problem: The "Dark" Sirens and the Missing Map

Usually, when a siren goes off, we look for the light (the host galaxy) to get its speed. But most of these events are "Dark Sirens." We hear the crash, but we can't see the light because the event is too far away or too faint.

To find the speed of a Dark Siren, scientists use a Galaxy Catalogue (a massive list of known galaxies) to guess which galaxy the siren came from.

• The Issue: Our current galaxy lists are like a map of a city that only shows the downtown area. They are very complete for nearby galaxies, but as you look further out (higher redshift), the map becomes empty. We are looking into the "darkness" where we don't have a list of galaxies.

The Solution: Guessing the Darkness

Since we can't see the distant galaxies, we have to guess how many of them are out there and how bright they are. Scientists use a mathematical recipe called the Schechter Function.

Think of the Schechter Function as a recipe for a galaxy soup:

• It tells you how many galaxies are in a pot of a certain size.
• It says there are lots of tiny, faint galaxies and very few giant, bright ones.

The Old Way: For years, scientists assumed this recipe never changed. They thought the "soup" looked exactly the same 1 billion years ago as it does today.

The New Way (This Paper): The authors say, "Wait a minute! Galaxies evolve." Just like a forest looks different when it's young versus when it's old, the population of galaxies changes over time.

• In the past, galaxies might have been brighter or more numerous in specific ways.
• The authors updated their recipe to account for this evolution. They made the "soup recipe" change depending on how far back in time (redshift) we are looking.

The Experiment: Testing the Recipe

The team took 46 gravitational wave events (mostly black hole collisions) and ran their calculations twice:

1. Scenario A: Using the old, static recipe (galaxies never change).
2. Scenario B: Using the new, evolving recipe (galaxies change over time).

They also tested two conditions:

• With a Map: Using the existing galaxy list (GLADE+).
• Without a Map: Pretending the list is empty and relying entirely on the recipe.

The Results: What Did They Find?

1. The Map Matters Most: When they had a good galaxy list (the "downtown" part of the map), the difference between the old and new recipes was tiny. The map did the heavy lifting.
2. The Darkness is Tricky: When they looked at events far away (where the map is empty), the recipe mattered more.
   • If you use the old recipe, you might slightly underestimate the speed of the universe ($H_0$) because you aren't accounting for how bright galaxies used to be.
   • If you use the new, evolving recipe, you get a slightly more accurate number.
3. The "Rate" Confusion: The authors found that the "recipe" for how galaxies change is mathematically linked to the "rate" of how often black holes crash. If you don't let the recipe change, you might get the crash rate wrong, which then messes up your speed calculation. However, if you let both the speed and the crash rate float freely in the math, the bias disappears.

The Takeaway: Why This Matters

Imagine you are trying to guess the speed of a car driving away into a fog.

• If you assume the car's headlights have always been the same brightness, you might guess it's further away (or closer) than it really is.
• This paper says: "The headlights actually get brighter or dimmer as the car gets older."

By updating our understanding of how galaxies "age" (evolve), we can better guess where the Dark Sirens are coming from.

Why do we care?
As our telescopes get better and we detect more distant, fainter events, the "fog" will clear, but we will be looking deeper into the past. If we don't update our "galaxy recipes" to account for cosmic evolution, our measurements of the universe's expansion will have hidden errors. This paper is a warning: To measure the universe's speed accurately, we must understand how the stars and galaxies have changed over time.

Technical Summary

1. Problem Statement

Gravitational wave (GW) astronomy offers a unique method to measure the Hubble constant ($H_0$) using "dark sirens"—GW events without electromagnetic counterparts where the host galaxy is identified statistically using galaxy catalogues. However, current galaxy catalogues (specifically GLADE+) are incomplete at high redshifts ($z > 0.1$), while GW detectors observe events up to $z \sim 1.2$.

To account for galaxies missing from these catalogues (the "out-of-catalogue" part), researchers must model the galaxy luminosity distribution. Currently, this is done using a constant-in-redshift Schechter function (SF). The paper addresses the potential systematic bias introduced by ignoring the evolution of the galaxy luminosity function with redshift. Specifically, it investigates how assuming a static galaxy population versus an evolving one affects the redshift prior, the inferred merger rates, and the final measurement of $H_0$.

2. Methodology

Data Sources

• GW Data: The analysis utilizes 47 GW events from the LIGO-Virgo-KAGRA (LVK) O3 run (42 Binary Black Holes, 3 NSBH, 2 BNS) with Signal-to-Noise Ratio (SNR) > 11. GW170817 is excluded as it is a "bright siren."
• EM Data: The GLADE+ galaxy catalogue is used, which combines six different surveys. The analysis focuses on the $K$-band (Vega system) magnitude limits.
• Cosmology: A flat $\Lambda$CDM cosmology is assumed with a fiducial $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$.

Modeling the Schechter Function Evolution

The authors move beyond the standard constant SF model by incorporating redshift evolution for the characteristic magnitude ($M^*$) and the normalization factor ($\Phi^*$), while keeping the faint-end slope ($\alpha$) fixed based on local universe data.

• Evolution Parameters:
  • $M^*(z) = M^*_0 - Qz$
  • $\Phi^*(z) = \Phi^*_0 10^{0.4Pz}$
  • $\alpha(z) = \alpha(0)$ (Fixed at $-1.1$)
• Parameter Derivation: The evolution parameters ($Q$ and $P$) were fitted using data from Arnouts et al. (2007) covering the $K$-band up to $z \approx 1.5$.
  • Fitted values: $M^*_0 = -22.68$, $Q = 0.42$, $\Phi^*_0 = 0.006$, $P = -0.86$.
• Comparison: The evolving model is compared against a constant model derived from Kochanek et al. (2001) ($M^* = -23.39$, $\alpha = -1.09$).

Statistical Framework

The authors use the gwcosmo pipeline to compute the Line-of-Sight (LOS) redshift prior, $p(z)$. This prior is split into two components:

1. In-catalogue: Directly uses galaxies present in GLADE+.
2. Out-of-catalogue: Models the missing galaxies using the Schechter function integrated over the magnitude limit of the survey.
   The probability of a galaxy hosting a GW event is weighted by its luminosity (assuming proportionality to stellar mass). The analysis performs a joint estimation of $H_0$ and the merger rate parameters (specifically the low-redshift slope $\gamma$).

3. Key Contributions

• First Application of Evolving SF: This is the first study to apply a redshift-evolving Schechter function to the dark siren method for $H_0$ estimation.
• Quantification of Systematic Bias: The paper quantifies how neglecting luminosity evolution biases the redshift prior, particularly for events with limited catalogue support (high redshift or poor localization).
• Degeneracy Analysis: It investigates the degeneracy between the evolving luminosity function and the cosmic merger rate evolution.

4. Key Results

Impact on Redshift Priors

• Low Redshift ($z < 0.5$): The evolving SF predicts a higher number density of galaxies compared to the constant SF because it accounts for the fact that galaxies were less luminous in the past (or the characteristic luminosity increases with time). This makes the GLADE+ catalogue appear less complete when interpreted with an evolving model.
• High Redshift ($z > 1$): The evolving SF predicts fewer galaxies than the constant SF.
• Visual Difference: The difference in the LOS prior $p(z)$ is relatively small within the redshift range of currently detected events ($z < 1$), but becomes significant at higher redshifts.

Impact on $H_0$ Measurement

• Empty Catalogue Case: When no galaxy catalogue is used (relying solely on the population model), the evolving SF yields a slightly higher $H_0$ ($64.85^{+13.33}_{-12.12}$) compared to the constant SF ($62.42^{+12.12}_{-12.12}$). The constant SF biases the result toward lower $H_0$ values because it assumes galaxies are intrinsically brighter at high $z$ than they actually are.
• With GLADE+ Catalogue: The difference in $H_0$ posterior distributions is negligible ($O(10^{-5})$ for specific events like GW190814) because the measurement is dominated by the well-localized, low-redshift events with strong catalogue support.
• Current Significance: The bias introduced by the SF model is currently smaller than the uncertainty from the unknown GW population distribution (which has been marginalized over in recent LVK analyses) but is comparable to uncertainties from photometric redshift models.

Impact on Rate Parameters

• Joint Estimation: When $H_0$ and the merger rate parameters are estimated jointly, allowing the rate parameters to vary fixes the bias in $H_0$.
• Degeneracy: However, this creates a degeneracy where the rate parameter $\gamma$ (the slope of the merger rate at low $z$) becomes dependent on the choice of the Schechter function model. If the SF evolution is ignored, the inferred merger rate evolution is biased.

5. Significance and Future Outlook

• Current vs. Future: While the impact of the evolving SF is currently sub-dominant to statistical uncertainties and other systematics (like population models), its importance will grow as:
  1. Detectors (e.g., Einstein Telescope, Cosmic Explorer) become more sensitive, detecting events at much higher redshifts ($z > 2$).
  2. Galaxy catalogues become deeper and more complete (e.g., UpGLADE, Rubin Observatory), shifting the weight of the measurement toward the "in-catalogue" part where redshift evolution modeling becomes critical.
• Recommendation: Future analyses must incorporate evolving Schechter functions, especially when jointly estimating $H_0$ and merger rates, to avoid biases in the inferred cosmic merger rate history.
• Limitations: The faint-end slope $\alpha$ is currently fixed due to a lack of data at high redshifts. Future deep surveys will be required to constrain the evolution of $\alpha$.

Conclusion: The paper concludes that while the "luminosity of the darkness" (the evolving galaxy population) does not currently drastically alter $H_0$ measurements due to the dominance of low-redshift events, it is a crucial systematic to model for the next generation of GW cosmology to ensure unbiased rate parameter estimation and accurate $H_0$ constraints at high redshifts.