Planck scale effects on the stochastic gravitational wave background generated from cosmological hadronization transition: A qualitative study

We reconsider the stochastic gravitational wave background spectrum produced during the first order hadronization process, in presence of ultraviolet cutoffs suggested by the generalized uncertainty principle as a promising signature towards the Planck scale physics. Unlike common perception that the dynamics of QCD phase transition and its phenomenological consequences are highly influenced by the critical temperature, we find that the underlying Planck scale modifications can affect the stochastic gravitational spectrum arising from the QCD transition without a noteworthy change in the relevant critical temperature. Our investigation shows that incorporating the natural cutoffs into MIT bag equation of state and background evolution leads to a growth in the stochastic gravitational power spectrum, while the relevant redshift of the QCD era, remains unaltered. These results have double implications from the point of view of phenomenology. Firstly, it is expected to enhance the chance of detecting the stochastic gravitational signal created by such a transition in future observations. Secondly, it gives a hint on the decoding from the dynamics of QCD phase transition.

We reconsider the stochastic gravitational wave background spectrum produced during the first order hadronization process, in presence of ultraviolet cutoffs suggested by the generalized uncertainty principle as a promising signature towards the Planck scale physics. Unlike common perception that the dynamics of QCD phase transition and its phenomenological consequences are highly influenced by the critical temperature, we find that the underlying Planck scale modifications can affect the stochastic gravitational spectrum arising from the QCD transition without a noteworthy change in the relevant critical temperature. Our investigation shows that incorporating the natural cutoffs into MIT bag equation of state and background evolution leads to a growth in the stochastic gravitational power spectrum, while the relevant redshift of the QCD era, remains unaltered. These results have double implications from the point of view of phenomenology. Firstly, it is expected to enhance the chance of detecting the stochastic gravitational signal created by such a transition in future observations. Secondly, it gives a hint on the decoding from the dynamics of QCD phase transition.

I. INTRODUCTION
The detection of gravitational waves (GWs), as a physical phenomenon generated by some energetic processes in the universe, have become increasingly important recently both for the theoretical physicists and the observational astrophysicists. In the light of technological developments via the construction of some sensitive detectors, the existence of gravitational radiation as a physical reality has been confirmed for repeated occasions. From the experimental point of view, each of technologies depending on their frequency sensitivities, have been designed for a specific purpose. Detectors in LIGO and Virgo scientific collaborations [1,2] have been designed for capture of high frequency (10 − 10 3 HZ) GWs from compact binary inspiral events. For detection of sources with lower frequency (10 −5 − 1 HZ) GWs signals from sources such as Supernovae, the ELISA experiment [3] has been designed. However, there are some setups as SKA [4,5] and PTA [6] which are trying to measure possible GWs generated with frequencies even lower than 10 −5 HZ (in particular around 10 −9 HZ). This category of experiments shows that, although the strongest GWs are produced by catastrophic events such as colliding black holes, stellar core collapses (supernovae) and coalescing neutron stars and so on, there is also the possibility of a random background of GWs, the so called "stochastic gravitational-wave background " (SGWB) without any specific sharp frequency component. Furthermore, this background could be related to extended theories of gravity [7,8] which could give rise to observable effects at cosmological level [9].
Despite the fact that low-frequency SGWBs are hardly detectable empirically, the modeling of their sources are of great theoretical interest in the sense that might give vital information about the very early stages of our universe [10]. Based on the existing literatures, some sources such as soliton stars, cosmic strings and cosmological phase transitions can be considered as SGWB generators, see for instance [11,12] for some good reviews reported recently. Concerning the cosmological sources, first order phase transitions could give rise to the possibility of production of SGWB signals with very low frequency as designated for SKA and PTA experiments [12,13]. According to the standard model of particle physics, the universe, in the course of its evolution, has experienced transitional phases for several times due to spontaneous symmetry breaking. Among these phase transitions, one can point out to the electroweak symmetry breaking via Higgs mechanism and the chiral symmetry breaking which results in quantum chromodynamics (QCD) phase transition [14][15][16]. About the nature of these transitions and the exact critical temperature where they occurred, there is no agreement so far. The main goal in the study of the hadronization process, relevant to strong QCD phase transition, is to reach the equation of state (EoS) governing two different phases, quark-gluon plasma (QGP) and hadronic gas (HG). In relation to the undeniable role of the EoS in direct or sideway investigations of QCD, it has to be emphasized that our knowledge of the different aspects of the system under study is highly affected by the choice of EoS [10,17]. Although some phenomenological models point out that the nature of hadronization process should be a crossover [18][19][20], there are other models for describing the collective flow in heavy ion collisions which suggest EoS in a first order phase transition with a critical temperature around ∼ 0.2 GeV [21][22][23].
Given the important phenomenological role of EoS, here, we want to revisit the relevant SGWB power spectrum arising from de-confinement to confinement first order transition in the light of Planck scale modified EoS released in the MIT bag phenomenological model [24] where the chemical potential is set to be zero. In particular, we utilize the proposed "generalized uncertainty principle" (GUP) which via the following deformation predicts two natural ultraviolet (UV) cutoffs as a minimal physical length (x min ≈ α 0 ℓ p ) as well as a maximum physical momentum (p max ≈ Mpc α0 ) in contrary to the standard quantum commutator relation governing the micro-scale particles i.e. Heisenberg uncertainty principle (HUP) [25][26][27]. It is clear that the above deformed commutator relation is different from HUP via Planck scale characteristic parameter α = α0 Mpc = α0ℓp , where α 0 , M p , l p denote a dimensionless constant, Planck mass and Planck length, respectively. By setting c = 1 (speed of light), one finds α = α0 Ep which E p is Planck energy ≈ 10 19 GeV. Due to the importance of UV cutoffs in the vicinity of Planck scale, quantum gravity (QG) models impose the order of magnitude of unity and subsequently 10 −19 GeV −1 for α 0 and α, respectively. By involving QG considerations in a common physical framework, so far researchers were not able to derive upper bound better than α 0 < 10 10 (equivalent to α < 10 −9 GeV −1 ), [28]. See also [29][30][31][32][33][34] for other upper bounds obtained further in different contexts. However, unlike the prevailing notion that, in the context of today's technology, it is impossible to probe the Planck scale physics with ideal resolution, in Refs. [35][36][37], by employing an opto-mechanical setup, surprisingly it has been shown that this ideal is achievable via a simple table-top experiment. This is a success for QG in the sense that α 0 = 1 ( or α = 10 −19 GeV −1 ) is no longer merely a theoretical ideal out of reach of today's technology. We note also that in Ref. [38], the authors by considering the Planck scale physics into the gravitational bar detectors, have been succeeded in extracting a solid constraint for possible Planck-scale corrections on the ground-level energy of an oscillator.
This idea that GUP can affect the dynamics of QCD phase transition comes back to the fact that GUP, by modifying the fundamental commutator bracket between position and momentum operators, leads to some modifications in the Hamiltonian of physical systems [28]. Therefore, GUP, by imposing its contribution within the thermodynamical quantities relevant to QGP and hadron phases, is expected to affect the hadronization process. Besides, studies carried out so far, confirm the point that the natural cutoff approaches make remarkable contributions in cosmological and astrophysical systems, for instance we can mention the Refs. [39][40][41][42]. In this paper, the QGP phase includes two massless quark flavors with zero chemical potential which at a HG-QGP phase thermodynamic equilibrium is affected by the underlying GUP modifications. As a result, the relevant thermodynamic quantities are derived in the presence of the above mentioned natural UV cutoffs. Finally, by means of the effective degrees of freedom and MIT bag pressure the confinement hadronic phase shall be distinct from the de-confined QGP. So, as discussed in section II, for each phase we deal with the GUPmodified EoS(s), separately. We emphasize that, for the present paper, our main focus is on the relevant EoS of QGP phase, because we are interest in the SGWB from phase transition above the critical temperature. Then, by using the GUP-modified QCD EoS derived in section II, we calculate the modified general expression of SGWB power spectrum in section III. The discussion is expanded to section IV by introducing some QCD sources which during first order transition production of SGWB can be involved. Finally, we provide a summary of the present work along with conclusions in section V.

II. QUARK-GLUON EOS IN GUP MODEL INCLUDING MINIMAL LENGTH AND MAXIMAL MOMENTUM INVARIANT CUTOFFS
Let us introduce UV cutoffs modified thermodynamics quantities such as pressure P , energy density ρ and entropy density s for QGP and Hadronic Gas HG phases. Concerning the GUP model at hand, for a massless particle as Pion in HG state, the standard dispersion relation E 2 (k) = k 2 modifies as [43][44][45][46] for energies close to Planck scale 1 . Given that the GUP affects the measure of integral, for a large volume of all possible Pion gas the standard partition function extends to In Eq. (3) the factor of g π refers to the number of degrees of freedom in the final HG state which is full of Pion gas.
Choosing the change of variables as x = k T √ 1 − 2αk and by neglecting O(α 2 ) terms, after some manipulation the partition function (3) reads as By a Taylor series expansion of (1 + αxT ) 3 (1 − αxT ) 3 around zero in above integral, we come to the final integral form of the partition function as follows Now by using the relation between partition function and grand canonical potential, ln z B = −Ω/T , one gets which can be represented for the pressure as P = − Ω V . Here, Γ(n) = (n − 1)! is the gamma function and I(0) (±) n denote the Bose and Fermi integrals defined as Given the fact that by fixing y = 0, then I(0) with Riemann zeta function, ζ(n) = ∞ j=1 j (−n) which is valid for integer (n ≥ 2), finally in the hadronic phase we obtain the following UV cutoffs modified expression for the pressure of a massless ideal HG. Now, using trace anomaly relation ρ−3p and also thermodynamic formula s = dP dT [50], the relevant energy density and entropy density can be obtained as respectively. The above three thermodynamic equations are valid for the HG state and also are extendable to the QGP state by regarding the bag constant B and converting the HG number of degree of freedom g π to its counterpart in QGP phase, g QGP . As a result, the counterpart of the above thermodynamic quantities in QGP phase read as where τ = Tc T with T c as critical temperature and also It is important to know that by computing the entropy density also one can find the effect of modified dispersion relation (2) on the bulk to shear viscosity ratio as an extractable quantity in relativistic heavy ion collisions [51]. Note that in a first order phase transition, some energy will be released in the medium. Therefore, the bag constant indeed plays the role of laten heat released during transition and also enters the pressure of the de-confined phase which its value can be calculated via Gibbs phase equilibrium condition P HG (T c ) = P QGP (T c ) i.e. the existence of a critical temperature T c in which the pressures in the two phases are equal [52]. The phenomenological fits of light hadron properties as well as low energy hadron spectroscopy address an explicit constraint on the bag constant as B ∈ (10 −4 − 16 × 10 −4 ) GeV [53][54][55][56][57]. Now, using constraints governed on bag constant B as well as the Planck scale characteristic parameter α (as previously discussed) we will be able to determine the allowed range of the critical temperature T c in the presence of QG effects. We find that T c can be one of possible values in interval 0.07 − 0.15 GeV, as can be seen in Fig. 1 1/4 [55][56][57] no changes are observed. Therefore, it can be said that taking QG considerations via incorporation of some UV natural cutoffs (in particular as minimal length and maximal momentum) in nature, has no effect on the relevant critical temperature of hadronization process in early universe. It is clear that by ignoring the UV cutoffs modifications i.e. by setting lim α→0 , then Eqs. (12) and (13) recover their standard forms T 4 c . It should be noted that both phases involved in the hadronization transition in early universe contain effectively massless electro-weak (EW) particles which should be taken in number of degrees of freedom. In de-confined QGP state it includes quark (g q ), gluon (g g ) and massless EW particles (g EW ) contributions which with assumption of two flavors, three colors and fixing QCD coupling constant at α S = 0.5 − 0.6, we will have g QGP = 35 ± 2 [58]. In the final HG phase, aside from three light bosons (Pions, π ± and (π 0 )), the presence of heavier hadrons may contributes at < T c which regardless of them one finds for the hadronic degrees of freedom g π ∼ 3.

III. SGW SPECTRUM MODIFIED WITH GUP-CHARACTERISTIC PARAMETER α
In this section we discuss the SGW spectrum generated from the epoch of cosmological QCD phase transition influenced by the QG effects towards today epoch. Indeed, we investigate the effect of Planck scale physics on the estimation of the current observable SGWB. To describe the evolution of the universe it is useful to track a conserved quantity as entropy which in cosmology is more informative than energy. The entropy of the universe is dominated by the entropy of the photon bath since there are far more photons than baryons in the universe. Any entropy production from non-equilibrium processes is therefore completely insignificant relative to the total entropy. With a good approximation one can treat the expansion of the universe as adiabatic (ṡ/s = 0 ), so that the total entropy stays constant even beyond equilibrium. Using in which η 1 = g s π 2 90 , η 2 = 24gsζ(5) and g s is the effective number of degrees of freedom in entropy calculation. Thereupon, using the adiabatic conditionṡ/s = 0 , we can obtain the following expression for time variation of temperature which in the limit α → 0, one recovers the the standard form as By integrating Eq. (19) we arrive at which by taking the Boltzmann equation 3 as d dt (ρ gw a 4 ) = 0 , then the energy density of the GWs at the today epoch is given as Here and also in what follows, the signs " * " and " 0 " refer to the quantities at the epochs of phase transition and today respectively. Now by definition of the density parameter of the GWs at today and phase transition epoch as Ω gw = ρgw(T0) ρcr(T0) and Ω gw * = ρgw(T * ) ρcr(T * ) respectively, we have Using Eq. (19), then the energy conservation equationρ + 3Hρ(1 + w) = 0 can be rewritten as where by integrating it between two intervals, radiation dominated epoch in some early time with relevant quantities ρ(T r ) and T r until phase transition epoch, one comes to the following expression for the critical energy density at the time of phase transition. Now by introducing the radiation density parameter Ω r0 = ρr(T0) ρcr(T0) which can be interpreted as fractional energy density of radiation in today epoch with observational value of ≃ 8.5 × 10 −5 along with the above equation we have where finally by inserting into (23), the GW spectrum observed today in the presence of Planck scale corrections is acquired as Now by using the relation between the scale factor and redshift as a0 a * = 1 + z = ν * ν0 , the peak frequency of the gravitational wave red shifted to current epoch is given by where One can show that by discarding the α terms, the standard form of above equation can be recovered In order to see the effect of UV cutoffs on the behavior of the relevant redshift z of GW spectrum, in Fig. 2 (left panel) using Eqs. (28) and (30)  This is reasonable in the sense that in the previous section it was shown that the relevant critical temperature of QCD transition, has not been affected due to the addition of α-terms into EOS. Namely, corrections generated by the underlying GUP model does not move the location (red shift) of phase transition relative to today observer. In what follows, using Eq. (21) we reexpress Eq. (27) as By focusing on the form of the above equation written in terms of critical temperature 4 , we notice a tangible change in the fractional energy density of the SGWB due to application of the UV cutoffs' modifications to the MIT bag EOS, see Fig.2 (middle and right panels). As is clear, the addition of Planck scale characteristic parameter α into the MIT bag EOS, leads to a reduction in SGWB signal arising from QCD phase transition. Interestingly, we found that the deviations caused by UV cutoffs for values very close to α = 10 −19 GeV −1 , are highly dependent on the value specified for T r . The middle panel of Fig. 2 is depicted with fixed values T r = 10 8 GeV, T 0 = 0.2348×10 −3 GeV, g s (T r ) = 106, g s (T * ) ∈ [33−37], g s (T 0 ) = 3.4 with α = 2 × 10 −10 , 5 × 10 −10 , 10 × 10 −10 GeV −1 from black meshed towards red meshed line respectively, while in the right panel we have set T r = 10 16 GeV with α = 1 × 10 −19 , 2 × 10 −10 , 5 × 10 −9 GeV −1 with the same other numerical values. We see that by increasing the value fixed for T r , the possibility to see deviations arising from Planck scale modification increases for α parameter very close to what is expected from theory.

IV. QCD SOURCES OF SGW
Since our focus in this paper is on the SGWB sourced by the first order hadronization process, in this section we consider two significant components involved in strong QCD phase transition which have important role in production of the SGWB. Those two components are: "bubble collisions" which create some shocks in the plasma medium [59][60][61][62] and "Magnetohydrodynamic turbulence" (MHDT) which may be produced after bubble collision in the plasma [63,64]. In order to calculate the contribution of the SGWB spectrum produced by the bubble collisions which the observer receives today, based on an envelope approximation the following expression has been obtained in Ref. [62]  where Here, u denotes the wall velocity and κ b is the fraction of the latent heat relevant to first order phase transition which residues on the bubble wall. Also δ and β −1 represent the ratio of the vacuum energy density released in the phase transition relative to that of the radiation and the time duration of the phase transition, respectively. The function S b (ν) has been released via fitting simulation data analytically [62,65] which its role is the parameterization of the the spectral shape of the SGWB.
Because of high kinetic and magnetic Reynolds numbers of cosmic fluid during hadronization process, it is expected that in the perfectly ionized plasma medium MHDT could be produced due to percolation of the bubbles [66]. Supposing the Kolmogorov-type turbulences as raised in [67], the contribution to the SGW is obtain as [66,68] with the parameterized function relevant to the spectral shape of the SGW spectrum as Here, H * = (a * /a 0 )H * and ν mhd = 7β 4u a * a0 , which is the peak frequency red shifted to today observer. Also κ mhdt denotes the fraction of latent heat energy moved into the turbulence.
Between the Hubble parameter H * and the frequency of GW corresponding to the moment of phase transition ν * there is a direct relation so that a change in H * will affect the ν * . By taking the Planck scale physics into account via the underlying UV natural cutoffs, the Hubble parameter deviates from the standard form as follows [69,70] where after combining energy conservation equation and GUP modified EOS (13) the following condition guarantees that the Hubble parameter in the time of phase transition to be physically meaningful This condition leads to a (T c , α) parameter space as Fig. 3 in which just negative values of α are acceptable. Despite that the negative sign of α seems to be unexpected at first glance, we have to encounter with it as a suggestion in a fundamental level. It may be interesting for the reader that such a condition also has been derived in other earlier studies, see [71][72][73] for instance. In Fig. 4, we have depicted the pure SGWB spectrum arising from bubble collision (left panel) and MDHT (right panel) for different values of α < 0 and other involved parameters. In Fig. 4 we have taken only the value β = 5H * . However, for larger values, the overall behavior of the plots is the same too. It is worth noticing that most of the QCD phase transition models suffer from a lack of accurate determination of the duration of phase transition. In Fig. 4, unlike Fig. 2, an increment in the relevant signal of GW due to incorporation of the Planck scale physics in the background evolution can be seen obviously. More technically, the natural UV cutoffs by affecting H * make the transition period longer which subsequently results in increment of the amplitude of GW signal. The reader may think that there is a contradiction between the results obtained here and in the previous section. This is not actually the case and to achieve a unified outcome we should recall the fact that in the previous section (see Fig. 2) we have set positive values for Planck scale characteristic parameter α, while here constraint on Fig. 3 imposes us to consider just the negative values of this parameter. This means that the outcome of Fig. 2 in the case of setting α < 0 generally is in agreement with Fig. 4, (see Fig. 5), of course with no change in the behavior of red shift).

V. SUMMARY AND CONCLUSIONS
There are different possible cosmological sources such as solitons, cosmic strings and phase transitions, involved in the SGWB spectrum, that could be potentially detected by the present interferometric experiments. We have focused on the SGWB arising from the first order QCD phase transition in a framework where QG effects are present. In particular, we have used a version of GUP with the minimal length and maximum momentum cutoffs marked by a characteristic parameter α on extracting the thermodynamics of ideal quark-gluon plasma (QGP) including two massless quark flavors at equilibrium in hadronization process without chemical potential. Using MIT bag equation of state, modified by GUP, we have followed the calculation of the SGWB produced in the period of QCD transition to detect possible Planck scale contributions. We found that, due to the lack of change in the critical temperature for transition via the GUP contribution, in the redshift of the modified SGWB spectrum at QCD era, deviations are not observed (see left panel of Fig. 2). However, our calculations represent a tangible change as a drop in the fractional energy density of the SGWB due the UV cutoffs into MIT bag EOS, (see middle and right panels in Fig. 2). Indeed, here the energy density at period of transition is an important quantity.
Finally, we have considered two significant process: "bubble collisions" and "Magnetohydrodynamic turbulence " (MDHT), involved in the first order QCD transition, which contribute in SGWB. Here, the Hubble parameter, during transition, plays a key role, having a physical meaning due to GUP modifications: it imposes the condition α < 0 (see Fig. 3). It is possible to notice an increase in the amplitude of GWSB without relevant changes in redshift (see Fig. 4). This paradox can be solved by assuming the condition α < 0 at fundamental level since, by replacing it without changing the redshift plot, the middle and right panels in Fig. 2 convert to Fig. 5 which generally is in agreement with Fig. 4.
To conclude, we have revisited the SGWB power spectrum, generated via first order QCD phase transition in presence of GUP-characteristic parameter α with negative signature. We have derived it without changing the transition temperature and subsequently the redshift of QCD era, by slowing down the background evolution which leads to an increase in amplitude of SGWB. This means that the probability of SGWB signal detection, due to QCD phase transition, could becomes significant for future interferometric experiments. These results have also the potentiality to open up a new window for decrypting the dynamics of QCD transition via gravitational radiation physics.