Generalized uncertainty principle and stochastic gravitational wave background spectrum
aa r X i v : . [ g r- q c ] J a n Generalized uncertainty principle and stochastic gravitational wave backgroundspectrum
Mohamed Moussa ∗ Physics Department, Faculty of Science, Benha University, Benha 13518, Egypt
Homa Shababi † Center for Theoretical Physics, College of Physical Science and Technology,Sichuan University, Chengdu 610065, P. R. China
Ahmed Farag Ali ‡ Physics Department, Faculty of Science, Benha University, Benha, 13518, Egypt
This paper concerned with the effect of generalized uncertainty principle (GUP) on the stochasticgravitational wave (SGW) background signal that produced during first order cosmological QCDphase transition in early universe. A modified formula of entropy is used to calculate the temporalevolution of temperature of the universe as a function of the Hubble parameter. The pressure thatresults from the recent lattice calculations, which provides parameterizations of the pressure dueto u, d, s quarks and gluons, with trace anomaly is used to describe the equation of state aroundQCD epoch. A redshift in the peak frequency of SGW at current epoch is calculated. The resultsindicate an increase in the frequency peak due to GUP effect, which improves the ability to detectit. Taking into account bubble wall collisions (BWC) and turbulent magnetohydrodynamics (MHD)as a source of SGW, a fractional energy density is investigated. It is found that the GUP effectweakens the SGW signal generated during QCD phase transition in comparison to its counterpartin the absence of GUP. These results support understanding the cosmological QCD phase transitionand test the effectiveness of the GUP theory.
I. INTRODUCTION
One of the most important physical phenomena that have attracted both attention in theoretical physicists andobservational astrophysicists is the discovery of gravitational waves (GWs). It is known that, from the merger ofblack holes by Laser Interferometer Gravitational-Wave Observatory (LIGO) collaboration, GWs usher in a new erain astronomy and cosmology [1]. LIGO detectors are structured in such a way that they operate in high frequencyrange (10 − HZ), for detecting sources like compact binary inspirals. Also, to identify lower frequency sources(10 − − − Hz (around 10 − Hz). These divisions of experiments indicate that although the strongestgravitational waves are generated by events such as colliding black holes, stellar core collapses (supernovae), coalescingneutron stars and etc, there also be a random background of GWs, which is called stochastic gravitational wave (SGW)background, and characterized by sharp frequency component. Detectors of multi-wavelength can assist observationthose waves, and the results may play a major role in a deeper understanding of the universe.Although low frequencies are difficult to be observed experimentally, modeling their sources is theoretically veryimportant because they can provide vital information about the early stages of the universe [6]. In this regard, we canrefer to the SGW background that contains information about the early universe. It is agreed that long-duration first-order cosmological phase transitions could be a potential source of very low-frequency gravitational wave background[7–10]. According to standard particle physics, there are at least two phase transitions, the electroweak phase transitionat T ∼
100 GeV, which is accompanied by breaking the electroweak symmetry, and the QCD phase transition at T ∼ . P is no longer equal to its value in the radiation dominated epoch [22]. This modification is specified in terms of ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ email: [email protected] trace anomaly which can lead to some exciting cosmic results such as the prediction of Weakly Interacting MassiveParticles (WIMPs), and pure glue lattice QCD calculations [23].Recently, some research has been guided in this direction. The authors in [24], studies the effect of equation ofstate of relativistic particles along with QCD equation of state, that emerged by parameterization of the pressure dueto u, d, s quarks and gluons with the energy density that computed from trace anomaly. It is found that the rate ofexpansion of the universe is decreased, the gravitational wave signal is increased by almost percent and the peakfrequency redshift to current time is changed by percent. In [25], a fractional energy density and a peak frequencyredshift at current time of SGW is investigated using effective QCD equation of state of three chiral quark flavors u, d, s including chemical potential and finite temperature. It is obtained that as the chemical potential increases,the frequency and amplitude of SGW signal that received today, were increased. The effective contribution of thechemical potential of quarks to the wave detection is also discussed.In this paper, we investigate the effect of GUP on the SGW background signal which is emanated from the firstorder cosmological QCD phase transition in early universe. The entropy of photon gas that modified by the GUPis used in the calculation of temporal evolution of the temperature of the universe as a function of the Hubbleparameter. Hubble parameter and the peak frequency of the SGW frequency red shifted with its corresponding valueat current time are calculated. Within GUP framework and taking into account bubble wall collisions and turbulentmagnetohydrodynamics in plasma as a source of SGW at the epoch of phase transition, a fractional energy density ofSGW are investigated. It is found that the GUP effect reduces the SGW signal in comparison to its counterpart inthe absence of GUP. These results can shed light on a better understanding of the cosmological QCD phase transitionand show the importance of the GUP theory. II. GUP AND PHOTONS GAS
A different approaches to quantum gravity such as string theory, non-commutative geometry and black hole physicspredict existence of a minimal measurable length. This in turn will produce an essential modification of the Heisenberguncertainty principle to what so called generalized uncertainty principle (GUP) [26–31]. According to this approach,it is suggested that [32] ∆ x i ∆ p j ≥ ~ δ ij (1 + β [(∆ p ) + h p i ] ) , (1)where β is the GUP parameter and defined as β = β m p c which m p is the Planck mass and β is of the order ofunity. Then, the absolute minimal measurable length is obtained by saturating the above inequality as (∆ x ) min = ~ √ β p β h p i and so (∆ x ) min = ~ √ β = √ β ℓ p for h p i = 0, where ℓ p is the Planck length. It should be noted that(∆ x ) min is not exactly equal to the Planck length but it is at the same order, depending on whether β is biggeror smaller than one. It is known that, the exact value of β is not precisely known but it is extensively investigatedand expected to be obtained in future experiments. The uncertainty relation (1) leads to the following deformedcommutation relation, namely [ x i , p j ] = i ~ δ ij (cid:0) βp (cid:1) , (2)where p = P i p i . Also, from the Jacobi identity, this equation ensures [ x i , x j ] = 0 and [ p i , p j ] = 0. Then, as a resultof Eq. (2), GUP modifies the physical momentum as p i = p i (cid:0) βp i (cid:1) , x i = x i (3)while x i and p i satisfy the usual canonical commutation relations [ x i , p j ] = i ~ δ ij . Thus we can consider p i is themomentum in Planck scale and p i is the momentum at low scale. On the other hand, according to the Liouvilletheorem, the number of quantum states inside phase space should not be changed with time evolution within GUPframework. Thus GUP should adjust the density of state to agree with Liouville theorem, which will have an effecton the thermodynamics properties of quantum systems. This way, the number of quantum states should be modified,such that [33] V (2 π ) Z ∞ d p → V (2 π ) Z ∞ d p (1 + βp ) . (4)Within framework of mentioned GUP, thermodynamic properties of photons gas will be investigated. Then we canget the modified entropy of photons due to GUP effect, which is the aim of this section. According to above modifiedphase space, the grand canonical partition function of photons can be written asln Z = − V g π π Z ∞ ln h − e − pT i p dp (1 + βp ) , ≃ − V g π π Z ∞ ln h − e − pT i (1 − βp ) p dp, (5)where g π is the number of degrees of freedom. After some manipulation, we obtainln Z = V g π π T Z ∞ e pT − (cid:20) p − βp (cid:21) dp = 190 V g π π T − βV g π π T . (6)Using the modified partition function, the entropy of photon gas can be written as S = ∂∂T ( T ln Z ) = 245 g π π T − βg π π T . (7)In a good approximation, the entropy expression can be rewritten as S = g π π T π βT . (8)In the limit of β →
0, we can recover standard form S = g π π T . III. MODIFIED STOCHASTIC GRAVITATIONAL WAVE SPECTRUM
It is agreed that the SGW is generated at the epoch of QCD and electroweak phase transition and propagate to thecurrent epoch. Here we interested in SGW that produced at the first order QCD phase transition. To investigate theobservable spectrum of SGW, we will use the assumption that the universe has expanded adiabatically which resultsin the constancy of the entropy per comoving volume, such that ˙ SS = 0. We will use the entropy of photons gas wherethe photons dominate universe more than baryons. Thus, the entropy can be estimated using the entropy of QGPstate in the previous section, i.e. S ∼ a g s π T π βT , (9)where g s is the effective number of relativistic degrees of freedom involved in the entropy density and a is the scalefactor. Employing the adiabatic condition ˙ SS = 0, the formula of time variation of temperature can be obtained dTdt = − H (cid:18) T + 1 g s dg s dT − π βT π βT (cid:19) − , (10)which H is the Hubble parameter. It is clear that the ordinary case is recovered in the limit of β → a ∗ a = exp "Z T T ∗ T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT , (11)where ” ∗ ” refers to quantities at the epoch of phase transition and ” 0 ” denotes quantities at the current time.Based on the fact that the SGW are ultimately decoupled to the rest of the universe, the energy density of the SGWcan satisfy Boltzmann equation ddt (cid:0) ρ gw a (cid:1) = 0. Applying Eq. (11) and Boltzmann equation, the energy density ofSGW at current time reads ρ gw ( T ) = ρ gw ( T ∗ ) (cid:18) a ∗ a (cid:19) , = ρ gw ( T ∗ ) exp "Z T T ∗ T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT . (12)Let us define the density parameter of SGW at phase transition epoch and current time as Ω gw ∗ = ρ gw ( T ∗ ) ρ cr ( T ∗ ) andΩ gw = ρ gw ( T ) ρ cr ( T ) , respectively, where ρ cr is the critical density. So, using Eq. (12), we obtainΩ gw = Ω gw ∗ (cid:18) H ∗ H (cid:19) exp "Z T T ∗ T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT , (13)where (cid:18) H ∗ H (cid:19) = ρ cr ( T ∗ ) ρ cr ( T ) . (14)In order to determine the evaluation of Hubble parameter from phase transition epoch to the current time, we needto use the continuity equation, namely: ˙ ρ t = − Hρ t (cid:18) P t ρ t (cid:19) , (15)where ρ t and P t are the total energy density and total pressure density of the universe at cosmic time t , respectivelyand dot denotes the derivative with respect to cosmic time. Now using Eq. (10), Eq. (15) can be written in terms oftemperature as dρ t ρ t = 3 T (1 + ω eff ) (cid:18) T g s dg s dT − π βT π βT (cid:19) dT, (16)where ω eff = P t ρ t is the effective equation of state parameter which may depend on temperature. The critical energydensity of radiation ρ cr ( T ∗ ), at the epoch of phase transition can be obtained by integration from early time T r , wherethe radiation dominated, to the time of phase transition T ∗ , as ρ cr ( T ∗ ) = ρ r ( T r ) exp "Z T ∗ T r ω eff ) T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT . (17)Then, putting Eq. (17) into Eq. (14), one can get (cid:18) H ∗ H (cid:19) = Ω r ρ r ( T r ) ρ r ( T ) exp "Z T ∗ T r ω eff ) T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT , (18)where Ω r = ρ r ( T ) ρ cr ( T ) is the current value of fractional energy density of radiation with the value almost equal Ω r =8 . × − . Now, applying Boltzmann equation, it can be proved that ρ r ( T r ) ρ r ( T ) = ( a a r ) . Hence, Eq. (18) can berewritten as (cid:18) H ∗ H (cid:19) = Ω r (cid:18) a a r (cid:19) exp "Z T ∗ T r ω eff ) T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT . (19)Substituting Eq. (19) into Eq. (13), we getΩ gw = Ω r Ω gw ∗ exp "Z T r T ∗ T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT × exp "Z T ∗ T r ω eff ) T (cid:18) T g s dg s dT − π βT π βT (cid:19) dT . (20)At this point, we want to determine the functional form of the effective equation of state, ω eff . It is known that forthe ultra-relativistic gas with non-interacting particles, ω eff = . In this case and without GUP effect, the above tworelations will be (cid:18) H ∗ H (cid:19) = Ω r (cid:18) T ∗ T (cid:19) (cid:20) g ( T ∗ ) g ( T ) (cid:21) , ω eff = 13 , β = 0 . (21)Ω gw = Ω r Ω gw ∗ , ω eff = 13 , β = 0 . (22)However, around T ∼ a few hundred MeV, the relation ω eff = deviates because the effect of QCD interactioncomes to play [6]. Thus, we will use QCD equation of state that results from the parametrization of the pressure dueto the strong interactions between u, d, s quarks and gluons, as [34] PT = F ( T ) = 12 (1 + tanh [ c τ ( τ − τ )]) p i + a n τ + b n τ + c n τ a d τ + b d τ + c d τ , (23) QCD phaseideal gas T * ( GeV ) ω eff FIG. 1: The effective equation of state parameteras a function of transition temperature T ∗ without trace anomalywith trace anomaly T * ( GeV ) × × × × H * / H FIG. 2: H ∗ H as a function of transition temperature T ∗ withoutGUP effect β = 0 where τ = TT c with T c = 154 M eV is the QCD transition temperature and p i = π is the ideal gas value of PT forQCD interaction with three massless quarks. It is worth noting that the function tanh x approaches unity for largevalues of x , therefore Eq. (23) approaches the ideal gas law at T ≫ T c , i.e. τ ≫ u, d, s quarks and gluons for all temperatures above 100 M eV , are c τ = 3 . τ = 0 . a n = − . b n = 3 . c n = 0 . a d = − . b d = 0 . c d = − . ω eff , at the temperature range above 100 M eV , using Eq. (23) with the trace of theenergy momentum tensor or called the trace anomaly [35] ρ − PT = T ddT (cid:18) PT (cid:19) . (24)Thus, the effective equation of state with QCD effect is obtained ω eff = Pρ = (cid:20) TF ( T ) dF ( T ) dT + 3 (cid:21) − . (25)To investigate the results more, we have plotted the effective equation of state function versus transition temperatures T ∗ , in Fig. (1). As it is shown in this figure, the trace anomaly matches ideal gas at ∼ GeV . This behaviourindicates that the effect of trace anomaly can not be ignored near QCD transition epoch and it should be taken intoaccount. Then in Fig. (2), we have depicted relative Hubble parameter as a function of transition temperature T ∗ ,in case of β = 0, with trace anomaly, shown with solid curve, and without trace anomaly, shown with dashed curvewhere H ∗ H ∼ T ∗ . This figure implies that, under the influence of QCD, Hubble parameter changes slower than T ∗ before ∼ . GeV and then changes faster at higher temperature till matches with T ∗ -curve at ∼ GeV . We haveused T r = 10 GeV . It is clear that the effect of QCD appears at low temperatures, so we will use equation of stateof trace anomaly in all our calculations, i.e. Eq. (25). The peak frequency of the SGW frequency redshifted to thecorresponding value at current epoch is given by ν peak ν ∗ = a ∗ a , = T T ∗ (cid:20) g s ( T ) g s ( T ∗ ) (cid:21) (cid:20) π βT π βT ∗ (cid:21) − . (26)Fig. (3) shows the frequency received at the current time to that presents at the epoch of transition as a function ofthe transition temperature. It shows that the effect of GUP increases with increasing transition temperature. We haveused the numerical values T = 2 . K = 2 . × − GeV , g s ( T ∗ ) ∈ [33 − ≈
35 and g s ( T ) = 3 .
4. According toEq. (11), the function a a r can be determined and used into Eq. (19) as (cid:18) H ∗ H (cid:19) = Ω r (cid:18) T r T (cid:19) (cid:20) g s ( T r ) g s ( T ) (cid:21) (cid:20) π βT r π βT (cid:21) − × g s ( T ∗ ) ω eff ( T ∗ ) g s ( T r ) ω eff ( T r ) exp "Z T ∗ T r (1 + ω eff ) (cid:18) T − π βT π βT (cid:19) dT . (27) (cid:1)= (cid:0)(cid:2) - GeV - (cid:3)(cid:4) - GeV - T * ( GeV ) × - × - × - (cid:5) peak / (cid:6) * FIG. 3: ν peak ν ∗ = a ∗ a as a function of transition temperature T ∗ for some values of β (cid:7)(cid:8) (cid:9)(cid:10) - GeV - (cid:11)(cid:12) - GeV - T * (cid:18) GeV (cid:19) × × × × H * / H FIG. 4: H ∗ H as a function of transition temperature T ∗ using trace anomaly formalism with differentvalues of β (cid:20)(cid:21) - GeV - (cid:22)(cid:23) - GeV - (cid:24)(cid:25) (cid:26) eff (cid:27) T * (cid:28) GeV (cid:29) × - × - × - × - Ω gw /Ω gw * FIG. 5: Ω gw Ω gw ∗ as a function of transition temperature T ∗ usingtrace anomaly formalism with different values of β and the dot-ted dashed line represents the case ω eff = where Ω gw Ω gw ∗ = Ω r Fig. (4) shows the ratio between Hubble parameter at the epoch of transition to its counterpart at current epochas a function of transition time. It is clear that the effect of GUP increases with increasing transition temperatureand leads to a decrease in the ratio with temperature. In other words, for high temperature, if β > β > β it isconcluded that H ∗ H | β < H ∗ H | β < H ∗ H | β . Finally, we can rewrite Eq. (20) asΩ gw Ω gw ∗ = Ω r (cid:18) T r T ∗ (cid:19) (cid:20) g s ( T r ) g s ( T ∗ ) (cid:21) (cid:20) π βT r π βT ∗ (cid:21) − × g s ( T ∗ ) ω eff ( T ∗ ) g s ( T r ) ω eff ( T r ) exp "Z T ∗ T r (1 + ω eff ) (cid:18) T − π βT π βT (cid:19) dT . (28)Using trace anomaly formalism, the relative density parameter with different values of β and applying equationof states of ultra-relativistic non-interacting gas is depicted in Fig. (5) In the case of using equation of state forultra-relativistic gas with non-interacting particles, ω eff = , the two integrations in Eq. (20) will cancel each otherand the relative density parameter will be constant, i.e. Ω gw Ω gw ∗ = Ω r = 8 . × − . This shown in Fig. (5) as astraight line. As the QCD equation of state goes to the equation of state of the ultra-relativistic and non-interactinggas, employing the QCD state equation, the relative density parameter also goes to the relative density parameterby using equation of state of ultra-relativistic and noninteracting gas, at ∼ GeV . This behaviour is clearly shownin Fig. (5). Also, we can notice that the relative Hubble parameter and density parameter ratio decrease due to theGUP effect. The reduction is always proportional to the GUP parameter β as shown in Figs. (4) and (5). We haveused the numerical values T r = 10 GeV and g s ( T r ) = 106. IV. MODIFIED QCD SOURCES OF STOCHASTIC GRAVITATIONAL WAVE
In this section, we turn our attention to the density parameter of gravitational wave at the epoch of transitionΩ gw ∗ = Ω gw ( T ∗ ), which is necessary to define the SGW spectrum through Eq. (28). There are many sources thatcontribute to the SGW background such as solitons and solitons stars [36], cosmic strings and domain walls [37, 38].Here we are interested in the SGW background that is generated due to the first order phase transition in the earlyuniverse. We consider two component sources of bubble percolation that take place after bubble nucleation and itsexpansion in the QCD first order phase transition, i.e, bubble wall collisions (BWC) and shocks in the plasma [39–44],and turbulent magnetohydrodynamics (MHD) in plasma after bubble collision [45].Based on envelope approximation and using numerical simulation, the contribution to the SGW spectrum by bubblecollisions is given by [44, 46, 47]Ω BW Cgw ∗ ( ν ) = (cid:18) H ∗ γ (cid:19) (cid:18) κ b α α (cid:19) (cid:18) . v .
24 + v (cid:19) S BW C ( ν ) , (29)where γ − is the time duration of phase transition, κ b is the fraction of the latent heat of the phase transition depositedon the bubble wall, α is the ratio of the vacuum energy density released in the phase transition to that of the radiationand v is the wall velocity. Then, parameterization of the SGW spectrum is given by the function S BW C ( ν ), which isdetermined by fitting simulation and analytically data, as S BW C ( ν ) = 3 . (cid:16) νν BWC (cid:17) . . (cid:16) νν BWC (cid:17) . , (30) ν BW C = 62 γ − v + 100 v (cid:18) a ∗ a (cid:19) . (31)It is worth noting that the frequency ν BW C is the today peak frequency of the SGW that generated by BWCmechanism during phase transition. At the epoch of phase transition, the kinetic and magnetic Reynolds numberof cosmic fluid are increased [48], which motivate the bubbles to produce magnetohydrodynamical turbulence in thefully ionized plasm. Then assuming a presence of Kolmogorov-type turbulence, as suggested in [49], the contributionto SGW spectrum form magnetohydrodynamical turbulence can be given as [48, 50]Ω
MHDgw ∗ ( ν ) = (cid:18) H ∗ γ (cid:19) (cid:18) κ MHD α α (cid:19) v S MHD ( ν ) , (32)where κ MHD is the fraction of latent heat that converted into turbulence, and the formula for the spectrum is givenby S MHD ( ν ) = (cid:16) νν MHD (cid:17) (cid:16) νν MHD (cid:17) (cid:20) π νH ∗ (cid:16) a ∗ a (cid:17) − (cid:21) , (33) ν MHD = 7 γ v (cid:18) a ∗ a (cid:19) , (34)where ν MHD is today’s peak frequency of the SGW generated by MHD at the epoch of phase transition. Althoughparameters α and κ play substantial roles in defining the peak position and amplitude of the SGW signal, there isstill no surefire way to find κ . Following [24], we assume v = 0 . κ b α α = κ MHD α α = 0 . γ = nH ∗ = 5 H ∗ and we usethe following substitutions [24, 51] H ∗ = s π m p ρ ( T ∗ ) , T ∗ = T c , (35)where ρ ( T ∗ ) = T ∗ dF ( T ∗ ) dT ∗ + 3 T ∗ F ( T ∗ ) . (36)Now, using the above equations, we obtainΩ BW Cgw ∗ ( ν ) = 10 − . v .
24 + v . (cid:16) νν BWC (cid:17) . . (cid:16) νν BWC (cid:17) . , (37) β (cid:30) β (cid:31) - GeV - × - × - ! × - × - × - " ν ( H ) × - × $ × % × & ’ gw h FIG. 6: Net contribution to the SGW due to bub-ble wall collision and magnetohydrodynamic tur-bulence β = - GeV - β = - GeV - β = T * ( GeV - ) × - × - × - * × - + ν t,./1 ( ) FIG. 7: Today’s net peak frequency of SGW signal arising fromBWC and MHD at the epoch of phase transition Ω MHDgw ∗ ( ν ) = 0 . v (cid:16) νν MHD (cid:17) (cid:16) νν MHD (cid:17) (cid:20) πν q m p π ρ ( T ∗ ) (cid:16) a ∗ a (cid:17) − (cid:21) , (38)where ν BW C = 310180 − v + 100 v s π m p ρ ( T ∗ ) (cid:18) a ∗ a (cid:19) , (39) ν MHD = 354 v s π m p ρ ( T ∗ ) (cid:18) a ∗ a (cid:19) . (40)Then, we use Ω gw ∗ h = (Ω BW Cgw ∗ ( ν ) + Ω MHDgw ∗ ( ν )) h into Eq. (28) and take into account Eq. (11). Using the abovementioned numerical values of the parameters, we estimate the SGW spectrum Ω gw h , as shown in Fig. (6). It isclear that using QCD equation of state with trace anomaly causes the SGW signal becomes weak with increasingfrequency. This effect is enhanced by the presence of quantum gravity effect, where the SGW signal weakens morecompared to normal case. Using Eqs. (39) and (40) we obtain the total peak frequency of the SGW signal that canbe measured today generated from BWC and MHD mechanism at the epoch of phase transition as ν total = ν BW C + ν MHD , = (cid:18) − v + 100 v + 354 v (cid:19) T T ∗ (cid:20) g s ( T ) g s ( T ∗ ) (cid:21) (cid:20) π βT π βT ∗ (cid:21) − s π m p ρ ( T ∗ ) . (41)Fig. (7) shows the influence of GUP on the peak signal of SGW that can be measured today as a function ofthe transition temperature. This effect becomes significant as the transfer temperature increases or as the GUPparameter increasing. In other words, at high limit T ∗ , if β < β it is concluded that for fixed temperatures, ν total ( β ) < ν total ( β ). Also, at this limit, the effects of GUP increases the peak of frequency i.e., ν β =0 total < ν β =0 total .These differences become dominant with increasing temperature.It is useful to compare the our result of Ω gw ∗ h in the presence of GUP effect with the possible sensitivities ofinterferometers designed to probe the stochastic background, such as LIGO-VIRGO or LISA satellites. It is found in[52] the sensitivity is bounded by the value 10 − as set by LIGO-VIRGO. It is also found that old LISA configurationover a year can can detect a white-noise stochastic background at the level of 10 − [53]. It is clear that these sensitivesvalues would set a stringent abound on the GUP parameter β which may signify an intermediate length scale betweenPlanck scale and electroweak scale which may be consistent with that set by the electroweak scale. The stochasticbackground sensitivity in that sense may have interesting implications with minimal length theories. We hope tostudy this in details in the near future. V. CONCLUSION
In this paper, we have studied the effects of GUP on the SGW background signal generated at the quark-hadronphase transition, which corresponds to the cosmological first order phase transition. It was agreed that such acosmological phase transition occurred at the early universe within QCD scale about t ∼ − s after the big bang attemperature T ∼ . GeV and the Hubble radius in order 10 ∼ Km , which involving a mass about M ⊙ .Within GUP framework, the modified thermodynamical properties of photons gas is investigated. Using the re-sulting modified entropy, the temporal evolution of the universe temperature calculated as a function of the Hubbleparameter. Due to the effects of QCD interaction, at epoch of phase transition, QCD equation of state should be em-ployed. A pressure due to strong interactions among massless u, d, s quarks and gluons, with energy density resultingfrom trace anomaly, was used to formulate the QCD equation of state. Within GUP framework and QCD equationof state, evolution of Hubble parameter and the energy density parameter of stochastic gravitational wave from theepoch of phase transition till current time was investigated. It is found that the presence of GUP leads to decrease inrelative Hubble parameter and energy density parameter ratio, and the reduction is always proportional to the valueof GUP parameter. A redshift in the peak frequency of the SGW at current epoch was obtained. Results showedthat GUP effects caused an increase in the frequency peak which can lead to the better detection of the SGW signal.There are a various mechanism that can produce stochastic gravitational wave at epoch of transition. In this paperwe were interested in BWC and MHD as a sources of the gravitational waves. Within GUP framework, the modifiedexpressions of BWC and MHD, that contribute in the SGW spectrum, were calculated. It is found that the SGWsignal generated during QCD phase transition became weaker in comparison to its counterpart for β = 0. Today’snet peak frequency of SGW signal, which was produced from BWC and MHD at the epoch of phase transition, isinvestigated. In the presence of GUP, it is found that the frequency of SGW signal is increased in comparison withoriginal case, and the growth in the frequency depends on the GUP parameter. We also found that the sensitivitiesof interferometers designed to probe the stochastic background could set a stringent bound on the GUP parameter.These results could shed light on increasing the chance of detecting the stochastic gravitational signal created by sucha transition in future observations. Moreover, they can lead to decoding of the dynamics of QCD phase transition atthe early universe. [1] Virgo and LIGO Scientific collaborations, B.P. Abbott et al., Phys. Rev. Lett. (2016) 061102.[2] A. Klein, et al., Phys. Rev. D (2016) 024003.[3] M. Kramer, D. 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