universe Article Gravitational Waves from the Cosmological Quark-Hadron Phase Transition Revisited Pauline Lerambert-Potin and José Antonio de Freitas Pacheco *   Citation: Lerambert-Potin, P.; de Freitas Pacheco, J.A. Gravitational Waves from the Cosmological Quark-Hadron Phase Transition Revisited. Universe 2021, 7, 304. https://doi.org/10.3390/ universe7080304 Academic Editor: Panos Christakoglou Received: 21 July 2021 Accepted: 13 August 2021 Published: 16 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Observatoire de la Côte d’Azur, Laboratoire Lagrange, Université Côte d’Azur, CEDEX 4, 06304 Nice, France; pauline.lerambert-potin@etu.univ-cotedazur.fr * Correspondence: pacheco@oca.eu Abstract: The recent claim by the NANOGrav collaboration of a possible detection of an isotropic gravitational wave background stimulated a series of investigations searching for the origin of such a signal. The QCD phase transition appears as a natural candidate and in this paper the gravitational spectrum generated during the conversion of quarks into hadrons is calculated. Here, contrary to recent studies, equations of state for the quark-gluon plasma issued from the lattice approach were adopted. The duration of the transition, an important parameter affecting the amplitude of the gravitational wave spectrum, was estimated self-consistently with the dynamics of the universe controlled by the Einstein equations. The gravitational signal generated during the transition peaks around 0.28 μHz with amplitude of h 2 0 Ω gw ≈ 7.6 × 10 -11 , being unable to explain the claimed NANOGrav signal. However, the expected QCD gravitational wave background could be detected by the planned spatial interferometer Big Bang Observer in its advanced version for frequencies above 1.0 mHz. This possible detection assumes that algorithms recently proposed will be able to disentangle the cosmological signal from that expected for the astrophysical background generated by black hole binaries. Keywords: QCD phase transition; equation of state of quark matter; cosmological gravitational wave background 1. Introduction The last scattering surface situated around z crit ∼ 1100 represents a boundary beyond which the universe is opaque to the electromagnetic radiation. In other words, physical processes occurring at redshift higher than z crit cannot be probed by using photons as messengers. However, current fundamental physical theories predict a series of important processes that should have occurred in the primitive universe, like the electroweak or the quark-hadron phase transitions and a putative inflation period necessary to explain, for instance, the observed homogeneity of the cosmic microwave background (CMB). If the early universe is opaque to the electromagnetic radiation, how could the afore- mentioned process be probed? Fortunately, the investigation of those events is possible because gravitational waves are simultaneously generated, producing a stochastic cos- mological background characterized by a specific spectrum, the smoking-gun of each considered process. These waves interact weakly with matter and, consequently, contrary to photons, may reach present observers with a strong redshifted spectrum. Gravitational waves were first detected on 14 September 2015 by the laser interferom- eters LIGO [1], representing a breakthrough on the experimental basis of General Relativity. The signal was produced by the merger of two massive stellar black holes and subsequent detections indicate fusion of binaries involving a pair of black holes (the majority of the cases), a pair of neutron stars [2] or even pairs constituted by a black hole or a neutron star, cases of the sources GW200105 and GW200115. The gravitational wave emission of all possible astrophysical sources along the history of the universe produces also a stochastic background [3], which despite its own interest will interfere with the detection Universe 2021, 7, 304. https://doi.org/10.3390/universe7080304 https://www.mdpi.com/journal/universe