Constraining the gravitational-wave emission of core-collapse supernovae with ground-based detectors

Using cross-correlation data from Advanced LIGO and Advanced Virgo during their third observing run, this study establishes a $95\%$credibility upper limit of$0.01~M_\odot c^2$ on the gravitational-wave energy emitted by core-collapse supernovae, representing a two-order-of-magnitude improvement over previous constraints and suggesting that third-generation detectors will likely detect individual supernova events before the gravitational-wave background.

The Gist

Imagine the universe is a giant, noisy party. For the last decade, we've been able to hear the loudest guests: the "crashes" of black holes and neutron stars smashing into each other. These are the Compact Binary Coalescences (CBCs). We've heard them clearly, like distinct thunderclaps.

But there's another kind of noise at this party. Imagine thousands of people whispering at the same time. Individually, you can't hear a single whisper. But if you stand in the middle of the room, you hear a constant, low-level hum. In physics, this is called the Gravitational Wave Background (GWB).

This paper is about trying to figure out how loud the "whispers" of Core-Collapse Supernovae (CCSNe) are. These are massive stars that run out of fuel and collapse in on themselves, exploding into supernovas. We know they happen, but we've never heard their specific "whisper" in the gravitational wave background.

Here is a breakdown of what the scientists did, using simple analogies:

1. The Problem: The "Whisper" vs. The "Thunder"

The scientists wanted to know: How much energy do these exploding stars release as gravitational waves?

The problem is that the "thunder" (black hole collisions) is so loud it drowns out the "whispers" (supernovas). It's like trying to hear a single person whispering in a stadium while a rock band is playing. To solve this, the researchers had to be very clever about how they listened.

2. The Method: The "Super-Listener"

Instead of looking for one specific explosion, they looked at the entire hum of the universe using data from the LIGO and Virgo detectors (which are like giant, ultra-sensitive ears).

• The Strategy: They used a technique called cross-correlation. Imagine two people standing in different parts of the stadium, both wearing headphones. They record the noise. If they both hear the exact same pattern of static at the exact same time, they know it's a real signal coming from the universe, not just random noise in their headphones.
• The Result: They didn't find a specific "whisper" from a supernova. But, by listening carefully, they could say: "Okay, we didn't hear it, but we know it can't be louder than X."

3. The Big Discovery: Tightening the Rules

In the past, scientists thought a supernova might release a huge amount of energy (like a massive explosion). This paper says: "No, if they are making gravitational waves, they are much quieter than we thought."

• The Old Limit: Previous guesses allowed for a supernova to release energy equivalent to half a sun turning into pure gravitational waves.
• The New Limit: This paper says the energy is likely less than 1% of a sun's mass (specifically, about 0.01 solar masses).
• The Analogy: It's like we used to think a whisper might be as loud as a jet engine. Now, we've proven that even the loudest possible whisper is actually just a normal human voice. We haven't heard it yet, but we know exactly how quiet it must be.

4. The Future: Better Ears for the Future

The paper also looked ahead to the next generation of detectors, like the Einstein Telescope and Cosmic Explorer. These are like upgrading from a cheap radio to a high-end studio microphone.

• The Prediction: The scientists ran simulations and found something interesting:
  • Individual Events: We will likely hear a single supernova explosion (a clear whisper) before we can hear the background hum of all of them combined.
  • Why? The new detectors are so sensitive that they can pick up a single loud event from far away. But to hear the "hum" of the whole universe, the collective noise needs to be even louder.

5. Why Does This Matter?

You might ask, "If we didn't find the signal, why is this a win?"

• Ruling Out Bad Theories: By proving the signal is quieter than we thought, they have ruled out some wild theories about how stars explode. It's like a detective narrowing down a suspect list by proving the criminal is too short to have been at the scene.
• Setting the Goal: Now, the scientists building the next generation of detectors know exactly how sensitive they need to be. They know they need to be able to hear a "whisper" that is 100 times quieter than what we could hear before.

Summary

Think of this paper as a noise pollution report for the universe.

1. We listened to the cosmic background noise.
2. We didn't find the specific sound of exploding stars, but we proved that if they are making sound, it's very quiet.
3. We told the future engineers: "Your new microphones need to be this sensitive to finally hear the stars explode."

It's a step forward not because we found the treasure, but because we finally drew a very accurate map of where the treasure isn't, making it much easier to find it next time.

Technical Summary

Here is a detailed technical summary of the paper "Constraining the gravitational-wave emission of core-collapse supernovae with ground-based detectors" by Jingwang Diao and Xingjiang Zhu.

1. Problem Statement

Despite the detection of numerous gravitational wave (GW) events from compact binary coalescences (CBCs), the Gravitational-Wave Background (GWB) arising from unresolved astrophysical sources remains undetected. One potential contributor to this background is Core-Collapse Supernovae (CCSNe).

• The Challenge: Direct detection of GWs from individual CCSNe is extremely difficult due to the intrinsically weak emission and the low rate of Galactic events (estimated at 1–2 per century).
• The Gap: Previous constraints on the average GW energy emitted per CCSN event were derived from initial LIGO data and were relatively loose ($\sim 0.5 - 2 M_\odot c^2$). Furthermore, previous analyses often neglected the significant contribution of the CBC background, which could mask or mimic the CCSN signal in cross-correlation searches.
• Objective: To place tighter upper limits on the average GW energy emitted by CCSNe using data from the third observing run (O3) of Advanced LIGO and Advanced Virgo, while explicitly accounting for the CBC background contribution. Additionally, the authors aim to forecast the detection prospects for future third-generation detectors.

2. Methodology

The authors employed a Bayesian inference framework using cross-correlation data from the O3 run.

• Data Source: Coincident strain data from the H1 (LIGO Hanford), L1 (LIGO Livingston), and V1 (Virgo) detectors. The analysis utilized the cross-correlation estimator constructed from Hann-windowed, Fourier-transformed, and inverse-noise-weighted data segments (20–1726 Hz).
• GWB Models:
  1. CCSNe Model: Modeled using a Gaussian energy spectrum in the source frame:
     $$\frac{dE_{GW}}{df_s} = A \exp\left[ -\frac{(f_s - f_{peak})^2}{2\Delta^2} \right]$$
     where $A$ is the amplitude, $f_{peak}$ is the peak frequency, and $\Delta$ is the bandwidth. The total background $\Omega_{CCSNe}(f)$ is integrated over the cosmic star formation rate and the initial mass function (IMF).
  2. CBC Model: Modeled as a power law $\Omega_{CBC}(f) = \Omega_{ref} (f/f_{ref})^\alpha$ with $\alpha = 2/3$, representing the inspiral-dominated background.
  3. Combined Model: $\Omega_{total}(f) = \Omega_{CCSNe}(f) + \Omega_{CBC}(f)$.
• Statistical Analysis:
  • Used nested sampling (via the Dynesty package within Bilby) to explore the parameter space.
  • Defined two prior sets: Prior I (standard ranges for $f_{peak}$ and $\Delta$ up to 1000 Hz) and Prior II (extended ranges up to 2000 Hz to account for potential ultrabroadband signals).
  • The likelihood function assumed a Gaussian distribution for the cross-correlation estimator.
  • The analysis marginalized over the normalization amplitude $A$ to derive constraints on the total GW energy ($E_{GW}$).
• Validation: The pipeline was validated by reproducing the isotropic stochastic background results from the LIGO-Virgo-KAGRA (LVK) O3 analysis (Ref. [42]), showing excellent agreement.

3. Key Contributions

• First Joint Constraint: This work provides the first upper limits on CCSN GW emission derived from O3 data that explicitly include the CBC background contribution, reducing potential biases in the CCSN constraints.
• Significant Improvement in Sensitivity: The study improves upon previous constraints (from initial LIGO) by approximately two orders of magnitude.
• Forecasting Third-Generation Detectors: The paper provides a comprehensive forecast for the Einstein Telescope (ET) and Cosmic Explorer (CE), analyzing both the detectability of the cumulative GWB and individual CCSN events.
• Parameter Space Exploration: The analysis covers a wide range of spectral parameters ($f_{peak}$ and $\Delta$) and tests the robustness of results against different prior assumptions regarding high-frequency emission.

4. Key Results

• Upper Limits on GW Energy ($E_{GW}$):
  • Using Prior I, the 95% credibility upper limit on the average GW energy per CCSN event is $1.1 \times 10^{-2} M_\odot c^2$** (CCSNe only) and **$9.9 \times 10^{-3} M_\odot c^2$ (CCSNe + CBC).
  • Using Prior II (extended frequency range), the limits are slightly more conservative ($\sim 3.2 \times 10^{-2} M_\odot c^2$).
  • These results represent an improvement of $\sim 100\times$ over the previous limit of $\sim 0.5 - 2 M_\odot c^2$.
• Impact of CBC Background: Including the CBC component in the model resulted in a modest ($\approx 10\%$) tightening of the CCSN constraints, confirming that the CBC background is a necessary component for accurate inference but does not dominate the uncertainty in the current non-detection regime.
• Comparison with Theory: The derived limits ($\sim 10^{-2} M_\odot c^2$) sit at the upper end of optimistic phenomenological models (e.g., long-lasting bar-mode instability) which predict emissions in the $10^{-4} - 10^{-2} M_\odot c^2$ range.
• Detection Prospects (Future Detectors):
  • Individual Events vs. GWB: The study finds that individual CCSN events are likely to be detected before the cumulative GWB becomes observable.
  • Thresholds for ET/CE:
    • The GWB becomes detectable (S/N = 3) if $E_{GW} \gtrsim 10^{-4} M_\odot c^2$.
    • Individual events become detectable (1 event/year) if $E_{GW} \gtrsim 10^{-5} M_\odot c^2$.
  • Frequency Dependence: Sensitivity drops significantly for high-frequency signals ($f_{peak} > 1000$ Hz). For signals peaking above 1000 Hz, the required energy for detection increases by a factor of $\sim 10$ compared to signals below 500 Hz, highlighting the need for dedicated kilohertz-band detectors.

5. Significance

• Astrophysical Constraints: By tightening the upper limits on GW energy emission, this work rules out the most extreme theoretical models of CCSN GW emission (those predicting $> 0.1 M_\odot c^2$) and narrows the parameter space for numerical simulations.
• Methodological Benchmark: The successful reproduction of LVK results and the rigorous treatment of the CBC background set a standard for future stochastic background analyses involving multiple source populations.
• Future Observing Strategy: The finding that individual events will be detected before the background suggests that future observing strategies should prioritize targeted searches for specific supernovae (e.g., SN 2023ixf) alongside stochastic searches.
• Detector Design: The strong frequency dependence of the detection efficiency underscores the scientific case for developing dedicated high-frequency (kilohertz) gravitational wave detectors to capture the full spectrum of CCSN emissions.

In conclusion, this paper establishes that while current detectors have not yet detected the GW signature of core-collapse supernovae, they have placed stringent constraints that are now probing the realm of optimistic theoretical predictions. The transition to third-generation detectors promises to either detect these signals or definitively rule out the most energetic emission mechanisms.