Recently, strong evidence for a gravitational-wave background has been reported by collaborations of pulsar timing arrays (PTAs). In the framework of scalar-induced gravitational waves, we concurrently investigate the second- and third-order gravitational waves by jointly analyzing PTA data alongside big-bang nucleosynthesis and cosmic microwave background datasets. We determine the primordial curvature spectral amplitude as $0.021<A_{\zeta }<0.085$ and the spectral peak frequency as ${10}^{-7.3}\mathrm{Hz}<f_{*}<{10}^{-6.3}\mathrm{Hz}$ at a 95% confidence interval, pointing towards a mass range for primordial black holes of ${10}^{-4.5}{M}_{\odot }<m_{\mathrm{PBH}}<{10}^{-2.5}{M}_{\odot }$. Our findings suggest that third-order gravitational waves contribute more significantly to the integrated energy density than the second-order ones when $A_{\zeta }\gtrsim 0.06$. Furthermore, we expect future PTA projects to validate these findings and provide robust means to investigate the genesis and evolution of the universe, especially inflation.

• Received 19 September 2023
• Accepted 23 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013207

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society