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The fingerprint of a cosmos swirling with gravitational waves.

The fingerprint of a cosmos swirling with gravitational waves

28 November 2023

Recent data from pulsar-timing array experiments are consistent with the Hellings–Downs curve, a unique signature of gravitational radiation.

Three binary supermassive black holes emit gravitational waves into space.
Black hole binaries emit gravitational radiation in this illustration. The gravitational waves from objects such as these interfere with each other as they ripple through spacetime. Credit: Olena Shmahalo/NANOGrav

Earlier this year, researchers from several pulsar timing array (PTA) experiments around the world published evidence for a gravitational-wave background—an all-sky “hum” of gravitational waves at nanohertz frequencies. The waves, which have frequencies that are nearly 10 orders of magnitude lower than the waves detected at ground-based interferometers like the Laser Interferometer Gravitational-Wave Observatory, may originate from binary supermassive black holes that have orbital periods of years to decades.

To observe gravitational waves with such long periods, PTAs such as the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), European Pulsar Timing Array (EPTA), Indian Pulsar Timing Array (InPTA), Parkes Pulsar Timing Array (PPTA), and Chinese Pulsar Timing Array take advantage of the incredible rotational stability of millisecond-period pulsars. The arrival times of radio pulses emitted from those cosmic lighthouses can be modeled to submicrosecond timing precision over decades of observation. Deviations in the arrival times of the pulses indicate the presence of gravitational waves transiting the Earth-to-pulsar line of sight.

By 2020, three independent PTA collaborations had reported the presence of a stochastic signal in their data that produced timing deviations, an expected first sign of a gravitational-wave background signal slowly emerging from noise. On its own, however, the signal was not enough for researchers to identify it as the product of gravitational waves. Terrestrial and astrophysical factors, such as drifts in global clock and timing standards and errors in referencing observations to the solar system’s center of mass, could produce such a stochastic signal.

The Green Bank Telescope in West Virginia, surrounded by fall foliage.
The 100-meter Green Bank Telescope in West Virginia, shown here in 2006, has been tracking subtle timing variations in millisecond pulsars for nearly two decades. Credit: NSF/GBO/Mike Holstine

It would take several more years to find the next crucial evidence in the PTA data: a distinctive pattern of correlations between the timing deviations of different pulsars. The existence of those Hellings–Downs correlations, named after the researchers at NASA’s Jet Propulsion Laboratory who in 1983 proposed using interpulsar correlations to search for gravitational waves, gives us and other PTA experimenters confidence that the timing deviations stem from a churning current of vast ripples in spacetime.

Sifting through the noise

Gravitational waves induce a redshift z in the arrival rate of a pulsar’s pulses according to

z ( t , Ω̂ ) = 1 2 i j 1 + Ω̂ · Δ h i j

where Δ h i j is the difference in the gravitational-wave metric between the pulsar (known as the pulsar term) and Earth (the Earth term), is the position vector of the pulsar, and Ω̂ is the position vector of the gravitational-wave source. The gravitational wave–induced redshift in the arrival rate of pulses depends on the gravitational wave both at the pulsar and at Earth because the wave induces a cumulative effect all along the path of the pulse, from emission to reception.

For a single supermassive binary black hole with a circular orbit, the gravitational-wave signal is relatively simple. The Earth term is periodic and essentially non-evolving, with a period that is half of the black holes’ orbital period. The pulsar terms are also periodic but with periods that do not necessarily match that of the Earth term. The pulsar terms are records of the binary’s behavior from when the gravitational wavefront first passed each pulsar, and the binary may evolve significantly over the light-travel time from the pulsars to Earth.

The signal for a gravitational-wave background of astrophysical origin, which could come from a population of supermassive binary black holes, is much more complicated. The waves from each binary all add together incoherently, producing a seemingly random-in-time signal.

A map and a graph that show how a gravitational-wave signal from a binary supermassive black hole would manifest on the timing of three pulsars.
Response of a pulsar-timing array to gravitational waves from a supermassive black hole binary. The gravitational-wave source is located at the center of the map, and the stars represent pulsars. The two pulsars in the bright regions (yellow and blue) have signals that are correlated with each other and anticorrelated with the signals from the red pulsar in the dark region. Credit: Adapted from an image by Jeffrey S. Hazboun

The key insight of Ron Hellings and George Downs four decades ago was that a collection of pulsars would have a common component of the gravitational-wave signals imprinted in them, and that by computing the correlations between pulsars, those signals could be extracted from the noise. Every pulsar sees a signal with the same Earth-term frequency, but the amplitudes and phases depend on the pulsars’ sky locations relative to the source. The result is a stochastic signal of timing deviations with a power spectrum that encodes the demographics and dynamics of the source population and a correlation that depends on the angular separation of the pulsars. That pattern has come to be known as the Hellings–Downs curve.

A pair of pulsars have a maximum positive correlation if they are close together on the sky and become anticorrelated if they are approximately 90 degrees apart. By convention, the Hellings–Downs curve is normalized to have a maximum correlation of 0.5, to reflect the fact that only half the signal—the Earth term—for each pulsar is correlated with other pulsars. The pulsar term is uncorrelated because each pulsar is probing the gravitational-wave background at a different point in space and at a different time.

Evidence emerges

The 15-year data set of NANOGrav, the collaboration to which we belong, consists of timing data for 68-millisecond pulsars. After analyzing more than 2200 pairs of those pulsars, the team found evidence for interpulsar correlations that are consistent with the Hellings–Downs curve at 3–4 sigma significance; the odds of noise alone producing the pattern are approximately between 1 in 1000 and 1 in 20 000.

It is the first time that PTAs have found evidence for Hellings–Downs correlations. Earlier work had come up short because the magnitudes of the gravitational wave–induced correlations are small; as a result, the evidence for interpulsar correlations lags behind evidence for the first “heads-up” sign of a common stochastic signal.

A graph showing the relationship between the correlation of pulsar timing variations and the separation angle between those pulsars.
Results from the NANOGrav 15-year data set. The blue points show correlations between the timing measurements of 67 pulsars (2211 pulsar pairs), binned as a function of angular separation. The black dashed line is the expected Hellings–Downs correlation predicted by general relativity. Credit: Adapted from G. Agazie et al., Astrophys. J. Lett. 951, L8 (2023)

This major milestone for PTAs marks the beginning of an exciting new phase of gravitational-wave astrophysics. A recent paper by the International Pulsar Timing Array (IPTA) found that the independent results from different PTA collaborations were consistent with one another. And work is underway to combine data from NANOGrav, the EPTA, the InPTA, and the PPTA.

As we continue to observe pulsars, and even add more pulsars to the array, our sensitivity to gravitational waves will climb. In concert with improved modeling of noise processes, increasing sensitivity will allow us to better characterize the spectrum, intensity variations across the sky, and origins of the gravitational-wave background signal.

Data from PTAs may also lead to observations of gravitational waves from potential sources beyond supermassive black holes, including echoes of physics from the very early universe. The Hellings–Downs correlation pattern is specific to gravitational waves with the plus and cross polarizations predicted by general relativity. Other polarizations that are allowed in some alternative theories of gravity would induce different interpulsar correlations. For those reasons and many others, the astrophysics community is excited about the promising future of PTA science.

Sarah Vigeland is an assistant professor of physics at the University of Wisconsin–Milwaukee and the chair of the NANOGrav Gravitational Wave Detection Working Group. Stephen Taylor is an assistant professor of physics and astronomy at Vanderbilt University and the chair of the NANOGrav collaboration.

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