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Gravitational waves from two scalar fields unifying the dark sector with inflation

Orlando Luongo1,2,3,4,5,*, Tommaso Mengoni1,2,4,†, and Paulo M. Sá6,7,‡

  • *Contact author: orlando.luongo@unicam.it
  • †Contact author: tommaso.mengoni@unicam.it
  • ‡Contact author: pmsa@ualg.pt
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Phys. Rev. D 113, 023541 – Published 27 January, 2026

DOI: https://doi.org/10.1103/pj2r-gnd6

Abstract

We investigate the gravitational wave background predicted by a two-scalar-field cosmological model that aims to unify primordial inflation with the dark sector, namely, late-time dark energy and dark matter, in a single and self-consistent theoretical framework. The model is constructed from an action inspired by several extensions of general relativity and string-inspired scenarios and features a nonminimal interaction between the two scalar fields, while both remain minimally coupled to gravity. In this context, we derive the gravitational wave energy spectrum over wavelengths ranging from today’s Hubble horizon to those at the end of inflation. We employ the continuous Bogoliubov coefficient formalism, originally introduced to describe particle creation in an expanding Universe, in analogy to the well-established mechanism of gravitational particle production and, in particular, generalized to gravitons. Using this method, which enables an accurate description of graviton creation across all cosmological epochs, we find that inflation provides the dominant gravitational wave contribution, while subdominant features arise at the inflation-radiation, radiation-matter, and matter-dark energy transitions, i.e., epochs naturally encoded inside our scalar field picture. The resulting energy density spectrum is thus compared with the sensitivity curves of the planned next-generation ground- and space-based gravitational wave observatories. The comparison identifies frequency bands where the predicted signal could be probed, providing those windows associated with potentially detectable signals, bounded by our analyses. Consequences of our recipe are thus compared with numerical outcomes and the corresponding physical properties are discussed in detail.

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Phys. Rev. D 113, 023541– Published 27 January, 2026
  • Received 1 October 2025
  • Accepted 9 January 2026
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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.

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