A.A and A.M. acknowledge support by the NSFC(11875113), the Shanghai Municipality(KBH1512299), and by Fudan University(JJH1512105). R.P. was partially 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)

^{3}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 Publishing Ltd

We propose a novel mechanism for the production of gravitational waves in the early Universe that originates 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 characteristic 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.

Article funded by SCOAP^{3}

The intriguing possibility that prompt phase transitions in the early Universe might have imprinted signatures in the background of gravitational radiation will be testable through the next generation of gravitational interferometers. The idea was firstly suggested in Refs. [

Dynamical effects of quarks confinement in baryons and mesons are related to the dimensional-scale transmutation as much as first-order phase transition (FOPT) phenomena, which are characterized by a dynamically generated energy scale

The GW signal associated to the QCD phase transition (QCDPT) cannot be detected in GW terrestrial interferometers, such as LIGO/VIRGO [

Nonetheless, a nHz phase transition such as a QCDPT can be detected, with high precision, from radio astronomical observation of pulsar timing: the GW backgrounds propagating through pulsar systems alter the radio signal, leaving an imprint that is in principle observable^{①}

Gravitational radiation from FOPTs detectable in a radio-astronomy experiment may also be originated from Warm Dark Matter models (see e.g. Ref. [

In this letter, we study in detail the possible effects of the gluon condensate relaxation phenomena. We analyze the non-linear field equations for the gluonic condensate, coupled to the Einstein equations, in a Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological background. Our formalism is based upon the effective Einstein–Yang-Mills equations of motion for real-time evolution of the homogeneous gluon condensate in expanding Universe initially formulated in Ref. [^{②}

The original implementation of this idea was criticized in Refs. [

During the relaxation stage, a new characteristic feature is the produced GW signal. While the energy-density part of the energy momentum tensor does not exhibit so violent transitions, the condensate pressure provides the main contribution to the energy-momentum tensor trace variation. These pressure kinks inject kinetic energy into the primordial plasma, inducing turbulence and sound/shock waves in the plasma very efficiently. In analogy with the case of a bubble propagating in the plasma, the gravitational radiation is emitted from magnetohydrodynamical (MHD) turbulence and sound waves. From our numerical simulations, which we compare with semi-analytical estimates, we show that such gravitational background signal can be tested in future radio observatories form pulsar timing effects. The spectrum that is predicted not only lies within the SKA sensitivity, but it further displays very peculiar features of the shape form that cannot be reproduced in any other known mechanism. In other words, time crystallization of the QCD medium during the relaxation phase can be tested in close future, which implies a radical reconsideration of our picture of QCD confinement itself from the prospective of dynamical cosmological evolution.

A standard static domain-wall can be easily obtained from a scalar field theory that is

As is well-known, for the standard domain-walls the translational invariance is spontaneously broken, being the barrier localized in a

Intriguingly and exotically, one may consider a kink profile that, despite of being localized in a space direction, is also localized in time. A new domain wall extended in three spatial dimension but localized in a time lapse, which we dub as a chronon, may correspond to this solution. By just replacing the

In the case of the gluon condensate field equation coupled to gravity, in a FLRW cosmological background one can decompose the gluonic field in a classical background field

where

where

This shows that the moduli field is massless, according to the Nambu-Goldstone theorem.

When the gravitational dynamics is taken into account, and the scale factor time-dependence is considered, a more complicated time pattern for the space-like domain walls arises— see Appendix for technical details. In this latter case, time-translation is not only broken down to a

The general coupled field equations of the gluon field with gravity reads

where

In order to account for thermal bath effects, we have to consider the thermal loop correction to the classical equations. The leading order corrections are proportional to

Solving the dynamical system of equations specified above, at

(color online) The gravitational waves spectrum is displayed for different efficiency factors, in comparison with the FAST sensitivity curve [

(color online) The gravitational waves spectrum is displayed for different efficiency factors, in comparison with FAST sensitivity curve [

We provide below simple semi-analytic estimates, which are nonetheless in agreement with our numerical analysis.

The red-shift due to the expansion of the gravitational background must be taken into account while comparing the GW signals

The order of magnitude of the GW energy density today, denoted as

where

which is the Hubble contribution in the radiation dominated epoch.

The sound and turbulence spectrum induced by the spiky pressure kinks is in general very complicated. However, it will display a characteristic series of peaks, related to the pressure peaks. The magnitude of these GW peaks can be estimated very easily, thanks to semi-analytical estimates. The turbulence GW peaks are described by

where the sum is over the number of peaks that contribute significantly to the GW spectrum, and

is the ratio between the energy density

The sound waves spectrum is characterized by the expressions

We estimated the

Assuming the efficiency factors

We have shown that the relaxation dynamics of the gluon condensate close to the QCD phase transition behaves as a time-crystal within a time range of

We discuss in this section how a gauge-invariant description of spatially homogeneous isotropic Yang-Mills (YM) condensates that depend only on time can be obtained. For this purpose, it is most useful to work in the ghost-free temporal (Hamilton or Weyl) gauge, fixed

which is the basis of the Hamiltonian formulation. In this gauge, the asymptotic states of the

In the

one employs the polar decomposition (symmetric gauge) [

into an orthogonal matrix

where the YM condensate is positively definite

In the absence of gravity, the spatially homogeneous isotropic part

which can be integrated analytically [

From the group-theoretical viewpoint, the separation into spatially homogeneous and inhomogeneous components (A4) in the Minkowski spacetime has certain similarities with an analogical procedure in the conventional QCD instanton theory in Euclidean spacetime [

Notice furthermore that the homogeneous YM condensate can be introduced for every gauge group, which contains at least one

The effective QCD energy-momentum tensor that includes also quarks is expressed by relations that are similar to ones derived for the case of pure gluodynamics. The only difference between the two expressions arises in the effective beta-function coefficient

in which

From a phenomenological point of view, we may deploy reasonable assumptions, and imagine non-perturbative quantum-wave (hadron) fluctuations to occur at the same space-time scales as quantum topological fluctuations. This implies that they should satisfy a functional relation in analogy to (A7). The operator relation between quark and gluon fluctuations can be then established in terms of the trace of the quark energy-momentum tensor, once the vacuum average

The characteristic topological instanton-type contribution to the energy density of the QCD vacuum, here denoted with

This expression is due to the gluons and the light sea

with obvious meaning of the labels. From this latter, and considering the expression of the operator energy-momentum tensor of the gluon field

one can recover in the one-loop approximation the phenomenologically motivated complete QCD energy-momentum tensor

with

Now we come to an analysis of the equations of motion for physical time evolution of the homogeneous YM condensate in the cosmological environment. For this purpose, we first consider the effective QCD theory in the one-loop approximation, as in Appendix A.2. As was shown in Ref. [

We take a simplistic approach assuming that before the QCD transition epoch the gluon-field energy density is dominated by positively-valued chromoelectric components, while negatively-valued chromomagnetic components are negligibly small. For convenience, in what follows we re-label the corresponding energy-momentum tensor of the homogeneous YM condensate as

By the variational principle, one obtains the EYM system of operator equations of motion in a non-trivial spacetime

Notice that from now on, in all the derivations we will perform a rescaling of the gluon condensate, namely

In what follows, we work in the flat FLRW conformal metric, characterized by the relations

the comoving time being defined in

An additional coefficient 1/2 appears in front of the QCD coupling constant—which has been absorbed into the definition of the gluon field—as compared to the

In full analogy to the

which yields two distinct cases

One of these cases corresponding to

and thus has been considered as a potential driver of early-time acceleration epochs in cosmology.

The second solution, characterized by

Provided such a negative-energy vacuum solution is stable, a positive CC-term should be present to compensate this negative contribution in order to comply with cosmological observations. One may notice, however, that the corresponding negative energy density for

Such a compensation mechanism grossly reduces or eliminates the QCD vacuum effect on the macroscopic late-time Universe expansion. Indeed, under the conditions specified in Ref. [

Consequently, such a relatively slow macroscopic evolution of the Universe for

In the present Universe, for which we pick

corresponding to (B5) and (B6) solutions. Thus, the cosmological evolution of the gluon field in its ground state can be interpreted as a regular sequence of quantum tunneling transitions through the “time-barriers” represented by the regular singularities in the quantum vacuum solution of the effective YM theory. In this sense, the homogeneous gluon condensate in Minkowski spacetime is analogous to the topological condensate in the instanton theory of the QCD vacuum in Euclidean spacetime. This latter can be interpreted in terms of spatially-inhomogeneous gluon field fluctuations, which are induced by the quantum tunneling of the field through topological (spatial) barriers between different classical vacua.

It is worth noticing that the well-known't Hooft-Polyakov monopole [

As was noticed above, the compensation may not be exact at the early stages of the Universe evolution, and may be fulfilled only at asymptotically large times and on the average. Notice furthermore that the macroscopic evolution of the present Universe, in this case, is only affected by a small remnant of the above cancellation, the observable cosmological constant term. The latter can be generated e.g. by an uncompensated quantum-gravity correction to the ground state energy in QCD, which may explain both its absolute value and sign, as was argued in Ref. [

We now consider a deviation from the exact partial solution given by Eq. (B8), and study the general solution of the equations of motion (B2) and (B3), numerically. Let us first choose the subset of initial conditions satisfying

For this choice of initial conditions,

(color online) We illustrate, as functions of the physical time

where

in dimensionless units, and the corresponding solution for the logarithm of the scale factor is given in

Although the amplitude of the condensate

(color online) We show the homogeneous QCD condensate amplitude oscillations

The same quantities have also been studied in the opposite case of initial conditions, i.e. for

In Section B.3, the system of equations (B2) and (B3) was investigated numerically in the general case. We assumed arbitrary initial conditions and found the universal asymptotics corresponding to the (partial) exact analytic solutions with minimal energy

One could expect that the oscillatory behavior of the

The corresponding period in physical time reads approximately

which is close to the numerical result discussed above. Notice that the period of YM condensate oscillations corresponding to the negative-energy solution (B6)

The conformal time derivative of the gluon condensate energy density for

is negative in the initial moment of time

As will be shown analytically below, the scale factor approaches the de-Sitter solution at late times

Under the viable hypothesis of co-existence of chromoelectric and chromomagnetic condensates around the QCD phase transition epoch, their initial values at

Now let us consider the oscillations of the trace

and its relaxation time is the same as for