⚛️ general relativity

Combining astrometry with pulsar timing: the first joint analysis of very low frequency gravitational waves

This paper presents the first joint analysis of Gaia astrometric data and binary pulsar timing to search for ultra-low frequency continuous gravitational waves, finding no significant detection but establishing the tightest upper limits to date on the strain amplitude in the pHz band while demonstrating the power of combining orthogonal observables to break spatial degeneracies.

Original authors: Lorenzo Filipello, William Beordo, Volodymyr Akhmetov, Mariateresa Crosta, Mario Gilberto Lattanzi

Published 2026-09-21

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CC BY 4.0

Original authors: Lorenzo Filipello, William Beordo, Volodymyr Akhmetov, Mariateresa Crosta, Mario Gilberto Lattanzi

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

For decades, the universe has whispered its deepest secrets through ripples in the fabric of space-time itself. These ripples, known as gravitational waves, are born when massive objects like black holes collide or spiral toward one another. While detectors on Earth have successfully caught the sharp, high-pitched chirps of these collisions, a vast, silent range of frequencies remains largely uncharted. This is the realm of the very slow, the very low, where waves stretch out over years or even millennia. In this quiet zone, scientists suspect they might find the slow, graceful approach of supermassive black holes long before they crash, or perhaps even echoes from the very birth of the cosmos. To hear these faint signals, researchers must look for the subtle, long-term drifts they cause in the positions of distant stars and the ticking of cosmic clocks, rather than the sudden jolts of a collision.

A team of astronomers has now taken the first step in listening to this slow frequency by combining two very different ways of watching the sky. They brought together data from a space telescope that maps the positions of millions of distant galaxies and a network of pulsars—rapidly spinning neutron stars that act as the most precise clocks in the galaxy. By merging these two independent sets of observations, the researchers created a new method to hunt for the continuous, low-frequency hum of gravitational waves. Their goal was to see if the slow drift of these cosmic objects matched the pattern expected from a passing wave, or if the universe remained silent in this specific band.

The researchers began by gathering a massive collection of data. From the European Space Agency's Gaia mission, they used a catalog containing the measured movements of about 1.5 million quasars. These are the bright, active cores of distant galaxies, so far away that their own motion is negligible; any shift in their apparent position is likely due to external forces. Simultaneously, they gathered timing data from eleven binary pulsar systems, where a pulsar orbits a companion star. In these systems, the time it takes for the pulsar to complete an orbit can be measured with extreme precision. If a gravitational wave passes through, it should slightly speed up or slow down the orbit, just as it should nudge the apparent position of the distant quasars.

To make this comparison work, the team had to be incredibly careful to rule out other causes for these movements. The stars and pulsars are not moving in a perfect vacuum; they are influenced by the gravity of our own Milky Way galaxy, the motion of our solar system, and the natural spin of the pulsars themselves. The researchers built a complex model to account for all these known effects, including the uncertain mass and shape of the galaxy's gravitational pull. They also had to filter out the "noise" of the instruments and the quirks of the data collection process. Only after stripping away these known influences could they look for the faint signature of a gravitational wave.

When the team combined the data from the quasars and the pulsars, they found no evidence of the continuous gravitational waves they were hunting. The movements of the stars and the timing of the pulsars were consistent with what would happen if no such waves were present. The statistical analysis showed that the data did not support the existence of a signal in this frequency range. Instead of finding a signal, the researchers used this silence to set a new, stricter boundary on how strong such waves could possibly be. They determined that if these waves exist at the frequency they studied, their strength must be less than a specific, incredibly small value. This limit is tighter than any previous attempt to measure this specific type of wave, improving upon earlier results by about 20 to 30 percent.

The study also looked ahead to the future, using computer simulations to predict what would happen when the next major release of data from the Gaia telescope becomes available. With more years of observation and more precise measurements, the team's models suggest that this combined approach could soon become sensitive enough to detect waves that are even weaker than their current limit. They found that with the upcoming data, the method could potentially break through a significant barrier in sensitivity, opening the door to detecting waves that have so far been invisible. While the current search came up empty, the method itself proved successful, demonstrating that combining the sky-map of distant galaxies with the ticking of cosmic clocks is a powerful way to explore the universe's quietest frequencies.

This work represents a new chapter in the search for gravitational waves, moving beyond the loud crashes of colliding black holes to listen for the slow, steady approach of giants. By proving that two different types of astronomical data can be woven together to create a more sensitive ear, the researchers have paved the way for future discoveries. Even though they did not find the waves this time, they have sharpened the tools needed to find them, bringing us closer to understanding the hidden, slow-motion dance of the universe's most massive objects.

Technical Summary: Combining Astrometry with Pulsar Timing for Very Low Frequency Gravitational Waves

Problem and Motivation
The pHz to sub-nHz gravitational wave (GW) frequency regime ( $10^{-12} \text{ Hz} \leq f_{GW} \leq 10^{-9} \text{ Hz}$ ) remains largely unexplored despite its importance for mapping the early inspiral of supermassive black hole binaries (SMBHBs) and probing early-Universe physics. While Pulsar Timing Arrays (PTAs) have made progress at higher frequencies ( $\sim 1 \text{ yr}^{-1}$ ) and space missions like Planck have constrained inflationary backgrounds at extremely low frequencies, the intermediate pHz band lacks clear observational constraints. Current limits are derived either from astrometric proper motions or pulsar timing separately, but no joint analysis has been performed. This work addresses the need to combine these orthogonal observables to break spatial degeneracies and improve sensitivity to continuous gravitational waves (CWs) in this ultra-low-frequency band.

Methodology
The authors present the first joint data analysis framework combining real astrometric proper motions with binary pulsar timing data. The methodology employs a Bayesian model selection and upper limit estimation framework using Dynamic Nested Sampling and Markov Chain Monte Carlo (MCMC) techniques.

Datasets:
- Astrometry: The analysis utilizes $\sim 1.5 \times 10^6$ quasars from the Gaia Celestial Reference Frame 3 (CRF3) catalog (DR3). To mitigate computational costs and handle correlations, the data is compressed via isolatitude pixelization, preserving the covariance of proper motion components ( $\mu_\alpha, \mu_\delta$ ). Sources with high goodness-of-fit parameters (RUWE $\leq 1.4$ ) and total proper motion amplitudes $> 1 \text{ mas/yr}$ are excluded to minimize stellar contamination.
- Pulsar Timing: The dataset comprises 11 high-precision binary pulsars from the ATNF catalog. Selection criteria include well-defined parallax measurements ( $\sigma_\varpi/\varpi < 10\%$ ) and distances $< 1 \text{ kpc}$ to reduce degeneracies related to phase multiplicity and Galactic acceleration uncertainties.
Theoretical Framework:
- The analysis focuses on secular effects induced by GWs with periods longer than the observation time.
- Pulsar Timing: The observable is the orbital period derivative ( $\dot{P}_b$ ). The model accounts for the Shklovskii effect, intrinsic orbital decay, and Galactic acceleration. Crucially, the uncertainty in the Galactic potential is propagated via Monte Carlo simulations to prevent spurious detections from modeling errors.
- Astrometry: The observable is the apparent proper motion ( $\mu_{GW}$ ) of quasars. The model includes intrinsic motion, secular aberration ("glide"), reference frame rotation, and systematic correlations.
- Systematics Modeling: To prevent the GW signal from absorbing residual systematic power, the authors implement a Vector Spherical Harmonics (VSH) decomposition for the quasar proper motions up to the octupole order ( $\ell=3$ ).
Statistical Approach:
- A joint likelihood function ( $\mathcal{L}_{joint} = \mathcal{L}_{PTA} \times \mathcal{L}_{Gaia}$ ) is constructed.
- Signal detection is performed using Bayes Factors (BF). Due to high parameter dimensionality and systematic noise, the Savage-Dickey Density Ratio (SDDR) is used as a robust countercheck for nested models (comparing a model with a GW signal against one without).
- Upper limits are derived from the 95th quantile of the marginalized posterior distribution for the GW strain amplitude ( $h_0$ ).

Key Results

Signal Detection: The joint analysis finds no statistically significant evidence for a continuous GW signal. The combined Bayes factor is $\ln \mathcal{B}_{joint} = -0.42 \pm 0.03$ , indicating a preference for the null hypothesis. Individual analyses of the pulsar and astrometric datasets also yielded no significant detection.
Upper Limits: In the absence of a detection, the study sets the tightest constraints to date on CW strain in the pHz band. At a reference frequency of $f_{ref} = 4 \times 10^{-10} \text{ Hz}$ , the 95% upper limit is $h_0 \leq 6.4 \times 10^{-11}$ .
Improvement: The combined dataset achieves full sky coverage and improves the upper limits by 20%–30% compared to analyses using pulsar timing data alone.
Directional Sensitivity: The directional sensitivity maps show upper limits ranging from $h_0 \leq 2.94 \times 10^{-11}$ in the most sensitive regions to $h_0 \leq 2.65 \times 10^{-10}$ in less constrained areas, reflecting the spatial distribution of the quasars and the antenna pattern of the pulsar binaries.

Significance and Future Outlook
The paper demonstrates that combining orthogonal observables (line-of-sight acceleration from pulsars and transverse proper motion from quasars) successfully breaks spatial degeneracies intrinsic to isolated searches. The rigorous modeling of systematics, particularly Galactic acceleration uncertainties and VSH decomposition of quasar motions, is highlighted as critical for avoiding false positives.

Looking forward, the authors forecast that the upcoming Gaia Data Release 4 (DR4), with an extended temporal baseline of 5.5 years, will reduce proper motion uncertainties by a factor of $\sim 2.7$ (scaling as $T^{-3/2}$ ). Injection-recovery simulations suggest that a joint analysis with Gaia DR4 data could break the $h_0 < 10^{-11}$ barrier for sub-nHz CWs. Furthermore, the inclusion of single-epoch astrometric time series in DR4 will enable direct searches for periodic GW signals, extending the frequency coverage from pHz to mHz. The authors also note that the future Square Kilometre Array (SKA) will significantly enhance the number of observed millisecond binary pulsars, particularly toward the Galactic bulge, further improving the isotropy and sensitivity of the joint analysis.

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