🔭 astrophysics

Search for Planetary-mass Black Holes with an Improved Viterbi Algorithm

This paper presents and validates a novel search pipeline utilizing an improved Viterbi algorithm and enhanced time-frequency analysis to effectively detect and characterize gravitational waves from planetary-mass primordial black hole binaries in LIGO O3 data, achieving high recovery rates and accurate chirp mass estimation across Galactic scales.

Original authors: Raul Rodriguez, George Alestas, Sachiko Kuroyanagi, Juan Garcia-Bellido

Published 2026-07-22

📖 9 min read🧠 Deep dive

CC BY 4.0

Original authors: Raul Rodriguez, George Alestas, Sachiko Kuroyanagi, Juan Garcia-Bellido

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, cosmic ocean, and hidden within its depths are invisible monsters called black holes. For a long time, scientists thought these monsters were only the size of stars or even bigger. But recently, a new theory suggests there might be a whole hidden population of tiny black holes, some as light as a planet or even an asteroid. These are called "primordial black holes," and they were born in the very first moments of the universe, long before stars existed. If we could find them, they might hold the key to solving one of the biggest mysteries in physics: what is "dark matter"? Dark matter is the invisible stuff that holds galaxies together, but we can't see it. If these tiny black holes are the dark matter, finding them would change everything we know about the cosmos.

To find these invisible ghosts, scientists listen for "gravitational waves"—ripples in the fabric of space-time caused when two black holes dance toward each other and eventually crash. Usually, when big black holes merge, they make a quick, loud "chirp" that lasts only a fraction of a second. But if the black holes are tiny, like the planetary ones we are looking for, their dance is incredibly slow. They could be spiraling toward each other for months or even years, making a very faint, long, and slow sound that is almost impossible to hear over the static noise of the universe. It's like trying to hear a single drop of water falling in a hurricane.

This is where a team of researchers steps in with a clever new trick. They have built a digital detective tool designed specifically to hunt down these slow, planetary-mass black holes. Instead of looking for a quick explosion, their tool listens for a long, drawn-out whisper that slowly changes pitch over time. They tested this tool on real data from the LIGO observatory (a giant ear listening to the universe) and found that it works. It can spot these faint signals even when they are very far away, and it can even guess how heavy the black holes are just by listening to their song. While they haven't found a real planetary black hole yet, they have proven that their new method is ready to find one if it's out there.

The Great Cosmic Chirp Hunt

Imagine you are trying to find a specific song playing on a radio, but the station is full of static, and the song is playing so slowly that it takes hours to finish just one verse. That is the challenge scientists face when looking for primordial black holes (PBHs) that are as light as planets. These tiny black holes, if they exist, would be circling each other for months or years before merging. Their signal is a "chirp"—a sound that slowly rises in pitch—but because they are so light, the chirp is incredibly long and faint, getting lost in the noise of the universe.

For a long time, scientists had two main ways to listen for black holes. One method, called "matched filtering," is like having a library of every possible song and checking them one by one. But for these tiny, slow black holes, the songs are so long that the library would need billions of entries, making the search impossible with current computers. The other method, "continuous wave" searches, is great for steady, unchanging sounds, but these black holes do change pitch, just very slowly. They were stuck in the middle, unable to catch these "intermediate" signals.

The Viterbi Detective

The authors of this paper, led by Raúl Rodríguez and his team, decided to build a new kind of detective: the Viterbi algorithm. Think of this algorithm as a super-smart hiker trying to find the best path up a foggy mountain. The mountain is covered in fog (noise), and there are many possible paths (tracks) the hiker could take. The hiker doesn't know the exact path the "signal" (the black hole) took, but they know the general rules of how the path should look.

The Viterbi algorithm looks at all the possible paths the signal could have taken through the noisy data and picks the one that is most likely to be real. It's like a detective who looks at a messy crime scene and figures out the most logical sequence of events, ignoring the random distractions. The team used a software package called SOAP to run this algorithm, which is already known for finding steady signals, but they tweaked it to handle these long, slow chirps.

A New Way to See the Sound

The biggest problem with the old way of looking at this data was the map they used. Usually, scientists look at a "spectrogram," which is like a musical score where time is on the bottom and pitch is on the side. For a slow black hole, the signal looks like a curved line that gets steeper and steeper. It's hard to follow a curve that changes shape so much.

The team came up with a brilliant idea: remap the music. They changed the way they looked at the pitch. Instead of plotting the pitch directly, they plotted a special mathematical version of the pitch (specifically, the pitch to the power of -8/3). When they did this, the messy, curving line of the black hole's song turned into a perfectly straight line!

Imagine trying to follow a winding river on a map; it's confusing. But if you could magically straighten the river out, it would be easy to see where it goes. This new map made the signal look like a straight line, which is much easier for the Viterbi algorithm to track. It also meant that the steepness of the line told them exactly how heavy the black holes were (a property called "chirp mass").

The Two-Step Filter

Once the algorithm found a potential straight line, the team didn't just trust it immediately. They knew that random noise could sometimes look like a line by accident. So, they built a two-step filter to separate the real signals from the fake ones:

The Power Check ( $n\sigma$ ): First, they asked, "Is this line loud enough to be real?" They measured how much "power" (energy) was in the line compared to the background noise. If the line was just a little bit louder than the noise, it was probably just a fluke. If it was significantly louder, it was a candidate.
The Shape Check (NMSE): Second, they asked, "Does this line look like a black hole?" Even if a line was loud, it might be a weird shape caused by a glitch in the machine. They checked if the line was straight and followed the exact mathematical rule that a black hole should follow. If the line wobbled or curved the wrong way, they threw it out.

By using both checks, they could be sure that if they found something, it was likely a real black hole and not just a random noise blip.

The Big Test

To see if their new detective tool actually worked, the team didn't just guess; they ran a massive simulation. They took nearly 1,000 hours of real data from the LIGO Hanford detector (from the O3b observing run) and secretly "injected" hundreds of fake planetary-mass black hole signals into it. These fake signals were hidden at different distances and with different masses, just like real ones might be.

They ran their pipeline over this data and asked: "Can we find the fake signals we hid?"

The results were impressive. The pipeline successfully found most of the fake signals, even when they were very faint. Specifically:

They set a rule that they would only accept a signal if there was less than a 3% chance it was a false alarm (a false positive).
Under this rule, they could detect signals from distances up to 135,000 parsecs (about 135 kpc) in the best cases. This is huge because it means they could see signals from anywhere within our entire galaxy, the Milky Way, and even a bit beyond.
For the most common sizes of these tiny black holes (around $10^{-2}$ solar masses), they could find them with 95% efficiency out to distances of about 50 kpc.

The team also checked if their tool could guess the weight of the black holes. They found that for the signals they successfully found, their estimate of the mass was very accurate, usually within a few percent of the true value. This is important because if they find a real signal, they will immediately know how heavy the black holes are.

What They Didn't Find (Yet)

It is important to note that this paper is about building the tool, not about finding the treasure yet. The team did not claim to have discovered a real planetary-mass black hole. They only proved that their tool works by finding the fake signals they put in themselves.

They also found a small limitation: the tool works best for black holes that are "slow" enough to be tracked as a long line. If the black holes are too heavy (closer to the size of a small star), they merge so fast that the signal becomes a short burst again, and the tool has a harder time finding them. However, even in this difficult range, the tool still performed reasonably well.

The Future

This paper is a major step forward because it provides a fully working "end-to-end" pipeline. It takes raw data, processes it, finds the best tracks, and gives scientists a list of candidates to investigate. The code is even available for other scientists to use.

The authors suggest that in the future, they could improve the tool by looking at data from multiple detectors at once (which would help filter out local noise) and by adjusting the math to account for black holes that might be spinning or have weird orbits. But for now, they have shown that the universe's quietest, longest whispers are finally within reach of our listening ears. If these tiny black holes exist and are hiding in our galaxy, this new method gives us a very good chance of hearing them.

Technical Summary: Search for Planetary-mass Black Holes with an Improved Viterbi Algorithm

Problem Statement
Primordial black holes (PBHs) in the planetary-mass range ( $10^{-4} M_\odot \lesssim M_{\text{PBH}} \lesssim 10^{-1} M_\odot$ ) represent a compelling candidate for dark matter, supported by recent observational hints from microlensing surveys and pulsar timing arrays. However, detecting gravitational waves (GWs) from binary PBHs in this mass range presents a unique challenge. These systems undergo inspirals that last significantly longer than typical binary black hole mergers (spanning months or years) but evolve too rapidly for standard continuous-wave (CW) search methods, which assume negligible frequency drift. Conversely, matched-filter searches for compact binary coalescences (CBCs) are computationally prohibitive due to the enormous number of templates required for such long-duration signals. Consequently, an intermediate detection method is required to bridge the gap between standard CBC and CW techniques.

Methodology
The authors present a fully operational search pipeline based on the Viterbi algorithm, a dynamic programming technique adapted from the SOAP package for CW searches. The methodology introduces several key innovations to address the specific characteristics of long-duration inspirals:

Novel Time-Frequency Representation: Instead of standard spectrograms where inspiral signals appear as curved "chirps," the authors apply a coordinate transformation to the frequency axis, mapping $f_{\text{gw}}$ to $f_{\text{gw}}^{-8/3}$ . In this $(t, f^{-8/3})$ space, the frequency evolution of a circular inspiral becomes a straight line, with the slope directly determined by the system's chirp mass. This transformation simplifies the signal model and allows for more efficient tracking.
Optimized Short-Time Fourier Transforms (SFTs): Recognizing that the optimal SFT duration ( $T_{\text{SFT}}$ ) depends on the chirp mass to balance frequency resolution against statistical fluctuations, the pipeline employs multiple coherence times ranging from 2 to 88 seconds. This covers the target mass range while ensuring the signal remains within a single frequency bin for the majority of the observation.
Candidate Isolation and Statistics: The pipeline utilizes a four-stage isolation procedure to extract signal segments from the Viterbi track. Candidates are evaluated using two complementary detection statistics:
- $n\sigma$ : A measure of the accumulated power significance relative to the noise-only background, accounting for the maximization over multiple paths.
- Normalized Mean Square Error (NMSE): A morphological statistic that quantifies the consistency of the recovered track with the expected linear evolution in the transformed frequency map. This is crucial for distinguishing real signals from noise fluctuations that may possess high power but incorrect morphology.

Key Contributions

Pipeline Development: The authors define and validate a complete, end-to-end search pipeline capable of processing real strain data from the LIGO Hanford detector (O3b run) to identify planetary-mass binary candidates.
Algorithmic Enhancement: The work extends previous applications of the Viterbi algorithm by introducing the $f^{-8/3}$ frequency map and a robust candidate isolation procedure tailored to long-duration transients.
Parameter Estimation: The pipeline provides a rapid, albeit rough, estimate of the system's chirp mass by minimizing the NMSE, enabling immediate characterization of detected candidates without full parameter estimation.

Results
The pipeline was validated using 1,000 hours of O3b LIGO Hanford data with approximately 600 injected simulated signals spanning chirp masses from $10^{-4}$ to $10^{-1} M_\odot$ and luminosity distances up to 150 kpc.

Sensitivity: At a fixed false-alarm ratio (FAR) of 3%, the pipeline successfully recovers the majority of injected signals. The detection horizon reaches Galactic scales ( $d_L \gtrsim 50$ kpc) across most of the parameter space. In the most sensitive region (chirp mass $\sim 2 \times 10^{-2} M_\odot$ ), the pipeline achieves 95% recovery efficiency out to distances of $\sim 135$ kpc.
Performance Trends: Sensitivity generally increases with chirp mass due to higher signal-to-noise ratios. However, a slight decrease in sensitivity is observed at the highest masses ( $M \gtrsim 5 \times 10^{-2} M_\odot$ ) where signals evolve too rapidly, spending only minutes in the detector band and challenging the transition matrix constraints of the Viterbi algorithm.
Mass Estimation Accuracy: Within the detectable region, the chirp mass estimates derived from NMSE minimization are highly accurate, with relative errors close to zero. Accuracy degrades only as signals approach the sensitivity limit or at the highest masses where track isolation becomes difficult.
Computational Efficiency: The generation of SFTs is identified as the primary computational bottleneck. Once spectrograms are produced, the candidate evaluation is highly efficient, requiring only seconds for a 10-hour data realization.

Significance
The paper establishes a viable, computationally efficient method for searching for planetary-mass PBH binaries, a regime previously inaccessible to standard GW search techniques. By successfully recovering signals with high efficiency across Galactic distances, the pipeline offers a powerful tool for testing the PBH dark matter hypothesis. The authors emphasize that the method provides a rapid characterization of candidates, including chirp mass estimates, which is essential for follow-up studies. The work lays the groundwork for systematic searches in upcoming observing runs (e.g., O4) and suggests that future multi-detector analyses could further enhance sensitivity by distinguishing local noise glitches from genuine astrophysical signals.