Chinese Physics C Vol. 43, No. 6 (2019) 065101 Time-crystal ground state and production of gravitational waves from QCD phase transition* Andrea Addazi1;1) Antonino Marcianò1;2) Roman Pasechnik2;3) 1Department of Physics &Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China 2Department of Astronomy and Theoretical Physics, Lund University, Lund SE-223 62, Sweden Abstract: We propose a novel mechanism for the production of gravitational waves in the early Universe that origin- ates from the relaxation processes induced by the QCD phase transition. While the energy density of the quark-gluon mean-field is monotonously decaying in real time, its pressure undergoes a series of violent oscillations at the charac- teristic QCD time scales that generate a primordial multi-peaked gravitational waves signal in the radio frequencies’ domain. The signal is an echo of the QCD phase transition that is accessible by planned measurements at the FAST and SKA telescopes. Keywords: primordial gravitational waves, homogeneous gluon condensate, effective Yang-Mills theory, QCD phase transition PACS: 98.80.Cq, 04.30.-w, 11.15.-q DOI: 10.1088/1674-1137/43/6/065101 1 Introduction was initially, although qualitatively, proposed by Witten [1], and then quantitatively re-elaborated in Refs. [9,10]. The GW signal associated to the QCD phase trans- The intriguing possibility that prompt phase trans- ition (QCDPT) cannot be detected in GW terrestrial inter- itions in the early Universe might have imprinted signa- ferometers, such as LIGO/VIRGO [11] and KAGRA tures in the background of gravitational radiation will be [12], and cannot be either measured in future space ex- testable through the next generation of gravitational inter- periments, such as LISA [13], U-DECIGO [14], BBO ferometers. The idea was firstly suggested in Refs. [1-5]. [15], TAIJI [16] and TianQin [17] projects. The GWs fre- Important developments on understanding the primordial quency range of a QCDPT is around 10−8 ÷ 10−9 Hz, gravitational waves (GW) production in the early Uni- which is 5-6 digits lower than the one provided by space verse were made, in particular, in Refs. [6,7]. experiments, and 9-10 digits away from LIGO/VIRGO/ Dynamical effects of quarks confinement in baryons KAGRA limits [10]. Furthermore, QCDPT does not leave and mesons are related to the dimensional-scale transmu- any smoking-gun imprinting in the Cosmic Microwave tation as much as first-order phase transition (FOPT) phe- Background as being sensitive to very low frequency nomena, which are characterized by a dynamically gener- modes (5-6 digits less [10]). ated energy scale ΛQCD ' 200MeV [8]. This suggested the Nonetheless, a nHz phase transition such as a QCDPT possibility that the Quantum Chromodynamics (QCD) can be detected, with high precision, from radio astro- phase transition may generate a GW signal in the hot nomical observation of pulsar timing: the GW back- Early Universe, at a temperature of T ' ΛQCD ' 200MeV. grounds propagating through pulsar systems alter the ra- That a FOPT related to strong interactions may emit GWs dio signal, leaving an imprint that is in principle observ- Received 27 January 2019, Published online 19 April 2019 * A.A and A.M. acknowledge support by the NSFC(11875113), the Shanghai Municipality(KBH1512299), and by Fudan University(JJH1512105). R.P. was par- tially supported by the Swedish Research Council, contract numbers 621-2013-428 and 2016-05996, by CONICYT grant MEC80170112 (Chile), as well as by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme(668679). This work was supported in part by the Ministry of Education, Youth and Sports of the Czech Republic, project LT17018. The work has been performed in the framework of COST Action CA15213 “Theory of hot matter and relativistic heavy-ion collisions” (THOR) 1) E-mail: [email protected] 2) E-mail: [email protected] 3) E-mail: [email protected] Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main- tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub- lishing Ltd 065101-1 Chinese Physics C Vol. 43, No. 6 (2019) 065101 able1). This opens a pathway towards the exciting possib- the main contribution to the energy-momentum tensor ility of testing fundamental particle physics with current trace variation. These pressure kinks inject kinetic en- and future radio astronomy experiments, including FAST ergy into the primordial plasma, inducing turbulence and [20] and SKA [21]. Within previous QCDPT analyses, sound/shock waves in the plasma very efficiently. In ana- the role of possible relaxation phenomena and gravita- logy with the case of a bubble propagating in the plasma, tional back-reactions were completely neglected. Note, the gravitational radiation is emitted from magnetohydro- after the QCDPT, the prompt and violent relaxation ef- dynamical (MHD) turbulence and sound waves. From our fects around the QCD vacuum energy state are expected, numerical simulations, which we compare with semi-ana- which retains a broad analogy with the reheating mechan- lytical estimates, we show that such gravitational back- ism in inflationary models. ground signal can be tested in future radio observatories In this letter, we study in detail the possible effects of form pulsar timing effects. The spectrum that is pre- the gluon condensate relaxation phenomena. We analyze dicted not only lies within the SKA sensitivity, but it fur- the non-linear field equations for the gluonic condensate, ther displays very peculiar features of the shape form that coupled to the Einstein equations, in a Friedmann- cannot be reproduced in any other known mechanism. In Lemaître-Robertson-Walker (FLRW) cosmological back- other words, time crystallization of the QCD medium ground. Our formalism is based upon the effective Ein- during the relaxation phase can be tested in close future, stein–Yang-Mills equations of motion for real-time evol- which implies a radical reconsideration of our picture of ution of the homogeneous gluon condensate in expand- QCD confinement itself from the prospective of dynamic- ing Universe initially formulated in Ref. [22]. During the al cosmological evolution. relaxation phase, a surprising non-equilibrium phenomen- on arises: the gluonic condensate field violently oscil- 2 Space-like domain walls from T-breaking lates during the relaxation phase, inducing fast oscilla- tions of the energy-momentum tensor trace for a transi- ent time of τ ' 10 ÷ 20Λ−1 . The oscillating solution is a A standard static domain-wall can be easily obtained QCD from a scalar field theory that is Z invariant. With a classical non-perturbative solution of the Yang-Mills 2 simple sombrero-like Higgs potential, Z can be spontan- field equations coupled to the Einstein field equations. 2 eously broken when the scalar field rolls down to one of The emergence of spikes, localized in a characteristic the two possible minima ϕ = v. These internal field QCD time lapse ∆t ' Λ−1 , and extended in the space di- vac QCD configurations can be localized in the space direction z as mensions, reveals the presence of a ordered pattern of kink profiles. The kink profile interpolates the two min- space-like soliton/domain-walls solutions. We dub these ima, namely, ϕ(z = −∞) = −v and ϕ(z = 1) = v. A domain- new solutions chronons. At large cosmological times wall configuration, as a xy-plane orthogonal to the z-dir- > Λ−1 t 20 QCD, the spikes' periodicity disappears, and the en- ection, is achieved through the kink profile transition re- ergy density approaches the QCD vacuum energy minim- gion, and its characteristic thickness in z-direction is dir- um. The time-ordered classical solution that we found is a ectly related to the kink shape. For a λϕ4 theory with a time crystal, i.e. a periodic classical solution spontan- sombrero potential, one can find a simple analytic kink eously breaking time invariance down to a discrete time h λv i − ϕ = p − shift symmetry T : t ! t + nΛ 1 , n denoting a natural solution, specified by (z) vtanh (z z0) , with z0 n QCD 2 number. The concept of time crystal has been first pro- being the kink center. posed by Wilczek in Refs. [23,24] within the context of As is well-known, for the standard domain-walls the superconductors and superfluids physics2). For a review translational invariance is spontaneously broken, being of time crystals, see e.g. Ref. [27]. The experimental dis- the barrier localized in a z0 point. This corresponds to the covery of time crystals was achieved in Ref. [28]. The appearance of a Nambu-Goldstone modulus boson spontaneous symmetry breaking of T-invariance from the z0(t; x;y), localized on the surface of the domain wall, as a localization of chronons is associated to the appearance low energy-excitation of its surface in the z-direction. of Nambu-Goldstone bosons, as time-like moduli excita- Intriguingly and exotically, one may consider a kink tions over the classical background. profile that, despite of being localized in a space direc- During the relaxation stage, a new characteristic fea- tion, is also localized in time. A new domain wall exten- ture is the produced GW signal. While the energy-dens- ded in three spatial dimension but localized in a time ity part of the energy momentum tensor does not exhibit lapse, which we dub as a chronon, may correspond to this so violent transitions, the condensate pressure provides solution.