Narrowband searches for continuous gravitational waves from known pulsars in the first two parts of the fourth LIGO--Virgo--KAGRA observing run

This paper presents the largest narrowband continuous gravitational wave search to date in the advanced detector era, analyzing data from the first two parts of the fourth LIGO–Virgo–KAGRA observing run for 34 known pulsars (including binary systems) using the 5n-vector pipeline with expanded frequency derivative parameters, which yielded no detections but established stringent upper limits, including a tight constraint on the Crab pulsar at less than 0.04% of its spin-down power.

The Gist

Imagine the universe as a vast, dark ocean. For decades, we've been listening for the loud, crashing waves caused by massive collisions—like two black holes smashing together. These are the "transient" signals that made headlines a few years ago.

But this new paper is about listening for something much quieter: the hum of a spinning top.

The Big Idea: Listening to the Cosmic Hum

Neutron stars are the densest objects in the universe, essentially giant atomic nuclei the size of a city. When they spin, they are usually perfectly round, like a smooth beach ball. If they spin perfectly, they are silent in the gravitational wave sense.

However, if a neutron star has a tiny "bump" or "mountain" on its surface (even if that mountain is only a few millimeters high), or if it's slightly squashed, it's like a beach ball with a pebble stuck to it. As it spins, that wobble creates a continuous, rhythmic ripple in spacetime. This is a Continuous Gravitational Wave (CW).

The problem? These ripples are incredibly faint. They are like trying to hear a whisper from a thousand miles away while standing next to a roaring waterfall (the detector's own noise).

The Detective Work: Narrowband Searches

The scientists in this paper (a massive team from the LIGO, Virgo, and KAGRA collaborations) decided to hunt for these whispers from 34 specific known pulsars.

Think of a known pulsar like a lighthouse. We know exactly where it is, how fast it spins, and when its light flashes.

• The Old Way (Targeted Search): We assumed the gravitational "hum" was perfectly synchronized with the light flashes. We listened only at that exact frequency.
• The New Way (Narrowband Search): The scientists realized that sometimes, the gravitational hum might be slightly out of tune with the light. Maybe the "bump" on the star is wobbling differently than the light beam.

So, instead of listening to just one single note, they listened to a tiny range of notes around the expected frequency. It's like if you were trying to find a friend in a crowded room who is humming a specific song. Instead of listening for exactly "C-Sharp," you listen for "C-Sharp plus or minus a tiny bit." This makes the search much more robust if the friend is slightly off-key.

The New Tools: What Changed?

This paper introduces two major upgrades to their listening strategy:

1. The "Second-Order" Spin: Neutron stars slow down over time. Usually, scientists account for a steady slowdown. But sometimes, the slowdown itself changes speed (like a car braking harder or softer). This paper is the first to look for these subtle changes in the rate of the slowdown. It's like listening not just for the pitch of the hum, but for how the pitch is changing over time.
2. The Binary Dance: Some of these pulsars have a partner star they orbit. This adds a complex Doppler effect (the pitch goes up and down as they dance around each other). This is the first time the team has successfully applied this "narrowband" search to pulsars in binary systems. It's like trying to hear a hummingbird while it's flying in a circle around a tree, rather than just hovering in one spot.

The Results: Silence is Golden

After analyzing data from the first two parts of their fourth major observing run (O4a and O4b), the team found no gravitational waves.

Does this mean they failed? Absolutely not. In science, a "null result" is often a victory.

• The "Spin-Down" Limit: Imagine a spinning top losing energy. If it loses energy only by making gravitational waves, there is a theoretical maximum limit to how loud it can be.
• The Victory: For 20 of the pulsars they studied, the team proved that the gravitational waves are quieter than this theoretical maximum.
• The Champion: The most famous result is for the Crab Pulsar. The team proved that the gravitational waves it emits are less than 2% of the maximum possible energy it could be losing. In other words, less than 0.04% of the Crab Pulsar's energy is being turned into gravitational waves. The rest is likely going into light, wind, and heat.

Why Does This Matter?

Even though they didn't find the waves, they set the strictest rules ever on how "bumpy" these stars can be.

• The Mountain Analogy: If a neutron star has a "mountain" on it, how big can it be? The new limits tell us that for the Crab Pulsar, any mountain must be smaller than a few millimeters.
• Physics of the Impossible: This helps physicists understand the "equation of state" of neutron stars—basically, what happens to matter when it is crushed so hard that atoms break down. If the stars were bumpier, we would have heard them. Since we didn't, we know the "crust" of these stars is incredibly strong and smooth.

Summary

Think of this paper as the world's most sensitive microphone being pointed at 34 specific cosmic lighthouses. The team didn't hear the hum, but by proving the hum is quieter than ever before, they have taught us that the universe's most extreme objects are smoother and more stable than we thought. They have tightened the net, making it harder for these "mountains" to hide, bringing us one step closer to finally hearing the cosmic hum.

Technical Summary

1. Problem Statement

Continuous Gravitational Waves (CWs) are long-duration, nearly monochromatic signals expected from rapidly rotating, non-axisymmetric neutron stars (pulsars). Detecting these signals is crucial for understanding the internal structure, equation of state, and deformation mechanisms (e.g., "mountains" or magnetic fields) of neutron stars.

However, CW signals are extremely weak, often buried in detector noise. A major challenge in searching for CWs from known pulsars is the potential mismatch between the electromagnetic (EM) timing solutions (derived from radio, X-ray, or gamma-ray observations) and the actual gravitational wave emission. This mismatch can arise from:

• Timing noise and glitches (sudden changes in rotation frequency).
• Decoupling between the superfluid interior and the solid crust.
• Differences in emission mechanisms (e.g., free precession).

Fully coherent searches assume the GW phase is perfectly locked to the EM phase; if this assumption fails, sensitivity is lost. Narrowband searches address this by scanning a small frequency and spin-down range around the EM-predicted values, offering robustness against phase mismatches while maintaining high sensitivity.

2. Methodology

Data Sources:

• GW Data: The analysis utilized data from the first two parts of the fourth observing run (O4a and O4b) of the LIGO–Virgo–KAGRA (LVK) collaboration. Specifically, data from the LIGO Livingston (L1) and LIGO Hanford (H1) detectors were used. Virgo and KAGRA were excluded as they joined the run in later parts.
  • Observing Time: Approximately 360 days for L1 and 300 days for H1.
  • Calibration: Standard calibrated strain data were used, with specific handling for a period of poor 60 Hz line subtraction for the Crab pulsar search.
• EM Data: Timing solutions were derived from a multi-messenger approach using data from Fermi-LAT (gamma-ray), Nancay, Jodrell Bank, IAR, CHIME, MeerKAT (radio), Chandra, and NICER (X-ray).
  • Glitch Handling: For pulsars that experienced glitches during the run, the timing models included glitch parameters (step changes in frequency and derivatives, and exponential recovery terms). Data segments surrounding glitches were excluded to prevent contamination.

Search Algorithm:

• Pipeline: The search employed the 5n-vector narrowband pipeline, which performs frequency-domain matched filtering based on the 5-vector formalism.
• Parameter Space:
  • Frequency ($f$): Scanned a narrow band around $2f_{rot}$ (twice the rotational frequency).
  • Spin-down ($\dot{f}$): Scanned a narrow range around the EM-predicted $\dot{f}$.
  • Second Derivative ($\ddot{f}$): A novel feature of this work was the inclusion of a search range in the second time derivative of frequency ($\ddot{f}$), allowing for better tracking of complex spin evolution.
  • Binary Systems: For the first time in a narrowband search, the pipeline was adapted to target pulsars in binary systems, accounting for orbital modulation using Keplerian parameters.
• Template Bank: The search grid covered the $f - \dot{f} - \ddot{f}$ space. The resolution was determined by the observation time ($T_{obs}$), with grid spacing $\delta f \approx 1/T_{obs}$, $\delta \dot{f} \approx 1/T_{obs}^2$, and $\delta \ddot{f} \approx 2/T_{obs}^3$.
• Statistical Analysis:
  • Data were Doppler-corrected and resampled.
  • Matched filtering was applied to estimate GW polarizations.
  • The detection statistic ($S$) was calculated by coherently combining results from L1 and H1.
  • A threshold was set to achieve a false-alarm probability ($p_{fa}$) of 1%.
  • Upper Limits (ULs) at 95% confidence level were calculated by injecting simulated signals into real data.

3. Key Contributions

1. Largest Narrowband Target Set: This is the most extensive narrowband search to date in the advanced detector era, targeting 34 known pulsars.
2. Binary System Inclusion: The first narrowband search to explicitly target pulsars in binary systems, expanding the scope of CW searches beyond isolated pulsars.
3. Higher-Order Spin-Down: The implementation of a search range in the $\ddot{f}$ (second derivative of frequency) dimension, improving sensitivity to sources with complex spin evolution.
4. Glitch Robustness: The analysis successfully handled pulsars that glitched during the observation period by splitting data segments and modeling glitch parameters, ensuring no loss of sensitivity due to timing discontinuities.
5. Re-analysis of Previous Data: The authors identified and fixed a bug in the spin-down correction of the pipeline used in a previous O4a-only analysis, confirming that the new results supersede the previous ones with no additional candidates found.

4. Results

• No Detection: No statistically significant evidence for continuous gravitational waves was found.
• Outliers: Two candidates (outliers) were identified for PSR J0117+5914 and PSR J1826−1334. However, these were rejected as astrophysical signals because their frequency tracks crossed a known transient noise feature in the L1 detector (a ~3.4 Hz line interacting with a calibration line) that occurred during the observation period.
• Upper Limits (ULs):
  • Upper limits on the strain amplitude ($h_0^{95\%}$) were established for all 34 targets.
  • Spin-Down Limit: For 20 analyses (counting pre- and post-glitch segments separately), the upper limits were set below the theoretical spin-down limit. This implies that less than 100% of the pulsar's rotational energy loss is being converted into gravitational waves.
  • Tightest Constraint: The most stringent limit was achieved for the Crab pulsar (PSR J0534+2200). The upper limit on the strain amplitude is approximately 2% of its spin-down limit. This corresponds to less than 0.04% of the Crab pulsar's spin-down power being radiated as gravitational waves.
• Sensitivity Depth: The median sensitivity depth ($D$) reached was approximately 230 Hz$^{1/2}$, consistent with previous searches but improved due to the longer observation time ($T_{obs}$).
• Ellipticity Constraints:
  • For young pulsars ($f \lesssim 100$ Hz), ellipticity limits ($\epsilon$) were found in the range $10^{-3}$ to $10^{-5}$.
  • For millisecond pulsars ($f \gtrsim 100$ Hz), limits were in the range $10^{-8}$ to $10^{-7}$.
  • The results constrain the maximum deformation of neutron stars, ruling out "mountains" supported by exotic matter phases (like solid quark phases) for the Crab pulsar at the levels predicted by some theories.

5. Significance

• Astrophysical Constraints: The non-detection places the tightest constraints to date on the ellipticity of the Crab pulsar and other young pulsars, significantly limiting the size of internal deformations.
• Methodological Advancement: The successful integration of binary system searches and $\ddot{f}$ exploration into the narrowband pipeline demonstrates the maturity of CW search techniques. This paves the way for more complex searches in future observing runs (O5 and beyond).
• Robustness: The ability to maintain sensitivity despite glitches and timing mismatches validates the narrowband approach as a critical tool for monitoring known pulsars, complementing fully coherent targeted searches.
• Benchmark for Future Runs: The results provide a baseline for sensitivity improvements expected with the next generation of detectors and longer observation times, where the sensitivity is expected to improve by a factor of $\sqrt{2}$ (due to doubled $T_{obs}$) compared to O4a.

In conclusion, this paper represents a significant step forward in the search for continuous gravitational waves, combining the largest target list to date with advanced pipeline features to set the most stringent constraints on neutron star deformations in the O4 observing era.