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Ultra-low-frequency gravitational waves from cosmological and astrophysical processes


Gravitational waves at ultra-low frequencies (100 nHz) are key to understanding the assembly and evolution of astrophysical black hole binaries with masses ~106–109M at low redshifts1,2,3. These gravitational waves also offer a unique window into a wide variety of cosmological processes4,5,6,7,8,9,10,11. Pulsar timing arrays12,13,14 are beginning to measure15 this stochastic signal at ~1–100 nHz and the combination of data from several arrays16,17,18,19 is expected to confirm a detection in the next few years20. The dominant physical processes generating gravitational radiation at nHz frequencies are still uncertain. Pulsar timing array observations alone are currently unable21 to distinguish a binary black hole astrophysical foreground22 from a cosmological background due to, say, a first-order phase transition at a temperature ~1–100 MeV in a weakly interacting dark sector8,9,10,11. This letter explores the extent to which incorporating integrated bounds on the ultra-low-frequency gravitational wave spectrum from any combination of cosmic microwave background23,24, big bang nucleosynethesis25,26 or astrometric27,28 observations can help to break this degeneracy.

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Fig. 1: Combined constraints on GW backgrounds and foregrounds.
Fig. 2: Posteriors on the PT-model parameters.
Fig. 3: Combined constraints on power-law stochastic GW signals.
Fig. 4: Combined measurements of the GW energy densities in the background and foreground.

Data availability

The data used for the analyses presented is publicly available from:

Code availability

The code used for the analyses presented here, including that to make all figures, is available at, which uses code from


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A.V. acknowledges the support of the Royal Society and Wolfson Foundation. We thank W. Ratzinger and K. Schmitz for useful discussions on the analysis.

Author information




Both C.J.M. and A.V. jointly conceived this study. C.J.M. performed the numerical calculations. Both C.J.M. and A.V. jointly analysed and interpreted the results and wrote the manuscript.

Corresponding author

Correspondence to Christopher J. Moore.

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The authors declare no competing interests.

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Peer review informationNature Astronomy thanks Lijing Shao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Moore, C.J., Vecchio, A. Ultra-low-frequency gravitational waves from cosmological and astrophysical processes. Nat Astron (2021).

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