Golden and Silver Dark Sirens for precise H0 measurement with HETDEX

This paper demonstrates that using the HETDEX spectroscopic survey to obtain precise redshifts for both rare "golden" and common "silver" dark sirens from the upgraded LIGO network could enable a few-percent measurement of the Hubble constant, offering a critical independent path to resolving the Hubble tension.

The Gist

The Big Problem: The Universe is Expanding, But We Can't Agree on How Fast

Imagine the universe is a giant balloon being blown up. Scientists have been trying to measure exactly how fast the air is being pumped in (the expansion rate, known as the Hubble Constant or $H_0$).

For decades, we've had two different ways to measure this, and they don't match.

1. The "Baby Picture" method: Looking at the oldest light in the universe (the Cosmic Microwave Background) suggests the balloon is expanding at one speed.
2. The "Local Neighborhood" method: Looking at nearby exploding stars (Supernovae) suggests it's expanding faster.

This disagreement is called the "Hubble Tension." It's like two people measuring the same room with different tape measures and getting different results. One of them might be wrong, or maybe the rules of physics are different than we thought.

The New Tool: "Standard Sirens"

To solve this, scientists are using Gravitational Waves. When two heavy objects like black holes crash into each other, they create ripples in space-time. These ripples are like sound waves, so scientists call them "sirens."

• The Bright Siren: If we can see the crash with a telescope (light) and hear it with gravitational wave detectors, we know exactly where it is and how far away it is. This is like seeing a car crash and hearing the horn; you know exactly where it happened.
• The Dark Siren: Most of the time, we only hear the crash (gravitational waves) but don't see it (no light). We know how far away it is based on how loud the "sound" is, but we don't know exactly where in the sky it happened. It's like hearing a car crash in the distance but not knowing which street it's on.

The Paper's Solution: "Golden" and "Silver" Sirens

This paper focuses on the "Dark Sirens." Since we don't know the exact location, we have to guess which galaxy the crash happened in. The authors propose a strategy to make these guesses much smarter by categorizing the events:

1. Golden Dark Sirens: These are the lucky ones where the gravitational wave detectors are so precise that the "search area" is tiny (less than 0.1 square degrees). It's like narrowing the search for a lost key down to a single room. There might only be one or two galaxies in that tiny spot.
2. Silver Dark Sirens: These are more common but less precise. The search area is a bit bigger (up to 1 square degree). It's like narrowing the search down to a whole neighborhood. There are more houses (galaxies) to check, but it's still manageable.

The Detective Work: HETDEX and VIRUS

To solve the mystery of the Dark Sirens, we need a list of all the "suspects" (galaxies) in the search area.

The paper suggests using a specific telescope setup called HETDEX (Hobby-Eberly Telescope Dark Energy Experiment) and its instrument, VIRUS.

• The Analogy: Imagine you are looking for a specific person in a crowded stadium. You need a camera that can take a picture of everyone in the stadium instantly and tell you their name and address.
• How it works: The VIRUS instrument is like a massive, super-fast camera that can take a "spectrum" (a chemical fingerprint) of every galaxy in a specific patch of sky. This tells us exactly how fast those galaxies are moving away from us (their redshift).
• The Claim: The authors tested this using data from the "COSMOS" and "SHELA" fields (patches of sky already mapped by HETDEX). They found that VIRUS is incredibly good at finding almost every galaxy in these areas, even the faint ones, up to a certain distance.

The Results: Cracking the Case

The team ran a simulation (a "mock data challenge") to see what would happen if we used this method with future, more powerful gravitational wave detectors (called LIGO-A#).

• The Setup: They simulated 1 year of observations.
• The Findings:
  • With the new, super-sensitive detectors, they expect to find a few "Golden" events and many "Silver" events.
  • By combining the gravitational wave data (distance) with the HETDEX galaxy data (speed), they can calculate the expansion rate of the universe.
  • The Result: They predict that after just one year of this combined observation, they could measure the Hubble Constant with an accuracy of about 1% to 2%.

Why This Matters

This paper argues that we don't need to wait for a miracle "Bright Siren" (a crash we can see and hear) to solve the Hubble Tension. Instead, by using our "Dark Sirens" and a powerful galaxy catalog like HETDEX, we can statistically solve the puzzle.

• Golden Sirens are the "smoking gun" (very precise, few suspects).
• Silver Sirens are the "strong evidence" (many suspects, but enough data to win the case).

The authors conclude that this method is robust. Even though the search areas are fuzzy, having a complete list of galaxies in those areas allows us to pinpoint the expansion rate of the universe with high precision, potentially settling the debate between the "Baby Picture" and "Local Neighborhood" measurements.

In short: We are learning to listen to the universe's "dark" crashes and cross-reference them with a super-detailed map of galaxies to finally measure how fast our cosmic balloon is inflating.

Technical Summary

Technical Summary: Golden and Silver Dark Sirens for Precise $H_0$ Measurement with HETDEX

Problem Statement
The paper addresses the "Hubble tension," the persistent 4–6$\sigma$ discrepancy between the Hubble constant ($H_0$) derived from the Cosmic Microwave Background (Planck) and local distance ladder measurements (SH0ES). While Gravitational Waves (GWs) from compact binary coalescences (CBCs) offer a direct, standard siren measurement of luminosity distance independent of the cosmic distance ladder, the majority of these events (dominated by black hole mergers) lack identified electromagnetic (EM) counterparts. These "dark sirens" require statistical association with galaxies in existing catalogs to infer redshift. The primary challenge is achieving sufficient sky localization precision and obtaining deep, complete spectroscopic redshift catalogs for potential host galaxies to constrain $H_0$ to the percent level.

Methodology
The authors propose a framework combining next-generation GW detector networks with the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) to create event-specific spectroscopic catalogs for dark sirens.

• GW Detector Networks: The study simulates three network configurations: HLV+ (LIGO Hanford, LIGO Livingston, Virgo at A+ sensitivity), HLI+ (replacing Virgo with LIGO-India at A+), and HLI# (LIGO-India at the more ambitious A# sensitivity).
• Dark Siren Classification: Events are categorized by their 90% credible sky localization area ($\Delta\Omega_{90}$):
  • Golden Dark Sirens: $\Delta\Omega_{90} \le 0.1$ deg$^2$ (potentially containing a single plausible host).
  • Silver Dark Sirens: $0.1 < \Delta\Omega_{90} \le 1$ deg$^2$ (containing multiple potential hosts).
• Spectroscopic Follow-up: The study utilizes the fifth internal data release (HDR5) of HETDEX, employing the Visible Integral-field Replicable Unit Spectrograph (VIRUS). VIRUS is selected for its wide field of view ($>55$ arcmin$^2$) and multiplexing capability, allowing it to survey the entire localization area of a silver siren ($\sim 1$ deg$^2$) in $\sim 60$ pointings. The analysis assumes VIRUS observations provide precise redshifts ($\sim 0.05\text{--}0.15\%$ error) and near-complete detection of galaxies with absolute magnitude $M_g \gtrsim -17.6$ out to $z \approx 0.2$.
• Statistical Framework: A Bayesian inference approach is used to construct the posterior distribution of $H_0$. The likelihood function marginalizes over all potential host galaxies within the GW localization volume, weighted by their probability of hosting the merger. The study assumes a magnitude-complete galaxy catalog and treats spectroscopic redshifts as unbiased. Selection effects (Malmquist bias, detection efficiency) are modeled using Monte Carlo injections with GWBENCH and full Bayesian parameter estimation with bilby for the final $H_0$ inference.
• Mock Data Challenge: The authors simulate GW events and map them onto real galaxy distributions from the COSMOS field (high fill-factor, small area) and the SHELA field (lower fill-factor, large area) to test the robustness of the method against large-scale structure (LSS) fluctuations.

Key Contributions

1. Feasibility of HETDEX for Dark Sirens: The paper demonstrates that VIRUS on the HET is uniquely suited for dark siren follow-up due to its ability to obtain deep spectroscopy over degree-scale areas, ensuring high completeness for potential hosts within $z \le 0.2$.
2. Classification and Yield: The study quantifies the expected yield of golden and silver dark sirens under future detector configurations. It predicts that while current/upcoming A+ networks (HLV+, HLI+) will produce only silver sirens, the A# configuration (HLI#) will yield a significant number of golden dark sirens (4 per year) and a large sample of silver sirens (136 per year).
3. Impact of Sky Background: The analysis highlights the critical role of the sky background in mock studies. It shows that using a small field like COSMOS introduces significant statistical fluctuations in $H_0$ estimates due to local large-scale structure (e.g., galaxy overdensities), whereas larger fields like SHELA provide more stable, unbiased inferences, though with lower fill-factors.

Results

• Event Rates: Under the HLI# (A# sensitivity) configuration, the network is expected to detect $\sim 4$ golden and $\sim 136$ silver dark sirens per year. Over a 10-year period, this scales to 48 golden and 1311 silver events.
• $H_0$ Precision:
  • Combining 25 events (golden + silver) from the HLI# network with COSMOS field data yields $H_0 = 66.35^{+1.10}_{-0.78}$ km s$^{-1}$ Mpc$^{-1}$ (approx. 1.7% precision).
  • Using the SHELA field for the same 25 events yields $H_0 = 69.19^{+0.67}_{-0.61}$ km s$^{-1}$ Mpc$^{-1}$ (approx. 1.0% precision).
  • Golden sirens alone (3 events) provide a less precise constraint ($\sim 12\%$) but demonstrate the potential for single-host identification.
• Systematics: The study finds that galaxy incompleteness in the HETDEX catalog is negligible for the target redshift range ($z \le 0.2$) and luminosity limits. However, peculiar velocities and field-specific large-scale structure clustering in small mock fields can introduce significant biases if not averaged over multiple sky regions.

Significance and Claims
The paper claims that spectroscopic redshift surveys like HETDEX can play a "key role" in realizing high-precision cosmology with dark sirens in the near future. The authors assert that with the upgraded LIGO-A# network and dedicated VIRUS follow-up, a few-percent constraint on $H_0$ is achievable within a single year of observations. The work emphasizes that while "golden" sirens (single host) are rare and dependent on sky position, "silver" sirens are common and, when combined with deep spectroscopic catalogs, provide a robust statistical path to resolving the Hubble tension. The authors caution that their results are based on mock data and that real-world observations will require averaging over diverse sky fields to mitigate large-scale structure effects and will need southern hemisphere spectroscopic capabilities (e.g., 4MOST) to match the sky coverage of the northern HET.