Deconfinement Transition Effects on Cosmological Parameters and Primordial Gravitational Wave Spectrum

The cosmological evolution can be described in terms of directly measurable cosmological scalar parameters (deceleration $q$, jerk $j$, snap $s$, etc...) constructed out of high order derivatives of the scale factor. Their behavior at the critical temperature of the Quantum Chromodynamics (QCD) phase transition in early universe could be a specific tool to study the transition, analogously to the fluctuations of conserved charges in QCD. We analyze the effect of the crossover transition from quarks and gluons to hadrons in early universe on the cosmological scalars and on the gravitational wave spectrum, by using the recent lattice QCD equation of state and including the electroweak degrees of freedom and different models of dark matter. Near the transition the cosmological parameters follow the behavior of QCD trace anomaly and of the speed of sound of the entire system. The effects of deconfinement turn out to be more relevant for the modification of the primordial spectrum of gravitational waves and our complete analysis, based on lattice QCD simulations and on the hadron resonance gas below the critical temperature, refines previous results.


Introduction
Quantum Chromodynamics (QCD) deconfinement phase transition has a interesting role at cosmological level, modifying, for example, the primordial spectrum of the gravitational waves [1][2][3].
Other consequences of the QCD transition show up in the cosmological parameters (deceleration q, jerk j, etc.) which involve the derivatives of the scale factor a(t) [4]. Indeed, the fluctuations of the cosmological parameters with higher order derivatives are strongly enhanced by the phase transition.
This effect is similar to the fluctuations of conserved charges (net baryon-number, net electric charge, net strangeness) evaluated in lattice QCD at finite temperature, which require the calculation of the higher order cumulants, i.e. high order derivatives of the logarithm of the QCD partition function. These fluctuations provide a wealth of information on the properties of strong-interaction matter in the transition region from the low temperature hadronic phase to the quark-gluon plasma phase and, in particular, they can be used to quantify deviations from the hadron resonance gas (HRG) model [5,6].
Previous analyses [4,7,8] considered the evolution of the first cosmological parameters (q, j) and of the energy density fluctuations during the deconfinement transition by a specific parametrization of the QCD equation of state (EoS) and neglecting the dark matter contribution.
In this paper we discuss the behavior of a larger set of cosmological scalars, with higher order derivatives of the scale factor, and take into account the electroweak sector, the strongly interacting sector and different models of dark matter also. Moreover the transition (cross-over) between the quark-gluon phase and the hadronic phase is described by recent lattice QCD EoS [9] and by the HRG [10] below the critical temperature T c ≃ 150 MeV.
Finally, the detailed treatment of the EoS above and below T c permits a refined analysis of the modification of the primordial spectrum of the gravitational waves.
The paper is organized as follows: the definition of the cosmological parameters is given in Sec. I; the relevant degrees of freedom and the role of the different contributions to the to-tal energy density and to the EoS of the whole system are discussed in Sec. II; Sections III and IV contain respectively the results on the speed of sound and on the fluctuations of the cosmological parameters during the deconfinement transition, with different dark matter models; Sec. V is devoted to the modification of the primordial gravitational wave spectrum due to the transition; comments and conclusions are in Sec. VI.

I. COSMOLOGICAL PARAMETERS
The standard model of the cosmological evolution is based on the Friedmann -Lemaître -Robertson -Walker (FLRW) equations and a set of equations of state for the different contributions to the total energy density. By defining the total energy density ε T and the total pressure p T as where are the dark energy contributions and the other terms correspond to strong (s), electroweak (ew) and dark matter (d) sector, the FLRW equations for a flat Universe are given by The cosmological parameters are defined as [11,12] H and their evolution is directly related to the EoS. Indeed, A n can be written as the sum of terms containing the first n − 1 derivatives of the Hubble parameter H, which can be expressed in terms of the w ≡ p T /ε T , of the speed of sound, c 2 s ≡ ∂p T /∂ε T , and its derivatives. For example, the jerk, j, is given by and, by FLRW equations, one haṡ The complete set of relations for various cosmological parameters is given in Appendix A.
The cosmological evolution can be described by the Hubble parameter H, the deceleration q, the jerk j, the snap (s = A 4 ) and the others cosmological parameters since they specify the various terms of the Taylor expansion of the scale factor: In Sec. IV the effect of the QCD deconfinement transition on the cosmological parameters will be analyzed and, as discussed in the introduction, higher order derivatives of a(t) show larger fluctuations.

II. THE EQUATION OF STATE IN THE EARLY UNIVERSE
Early Universe was a hot and dense plasma and during the cosmological evolution the number of degrees of freedom changed due to various phase transitions (see Fig. 1.1 of ref. [1]). Since we are interested in the effect of the deconfinement phase transition on the cosmological parameters and on the spectrum of gravitational waves, we consider the temperature T in the range 70 MeV < T < 400 MeV. In this section the number of degrees of freedom and the equations of state for strong, electroweak and dark matter sectors will be discussed.

A. a. Strong and electroweak sectors
The QCD deconfinement transition rapidly reduces the number of the strongly interacting degrees of freedom, g s . However, lattice QCD simulations indicates that the transition is not so sharp and is indeed a cross-over between a system of quarks and gluons and a hadron gas [9,13,14]. The (pseudo) critical temperature turns out to be T c ∼ 150 − 160 M eV by the analysis of chiral susceptibility.
Starting from the lattice QCD partition function, one defines the trace anomaly Θ µµ (T ) as the derivative with respect to the lattice spacing a l [9] and one evaluates all other thermodynamical quantities, i.e. the pressure the energy density the entropy density s and the speed of sound where C V is the specific heat. The pressure obtained by the HotQCD collaboration can be parametrized as follows [9]: c t a n b n c n d n 3.8706 -8. 7704 3.9200 0 0.3419 where and t = T /T c , T c = 154 M eV , p id = 95/180π 2 is the ideal gas value of p/T 4 for massless 3-flavor QCD and the other parameters are summarized in Table I.
In the temperature region T < T c all thermodynamic quantities are well described by the hadron resonance gas (HRG) model where the gran canonical partition function can be expressed as a sum of one-particle partition functions Z 1 i over all hadrons and resonances [16]. If m max is the maximum mass one includes, the trace anomaly can be written as a sum over all particles species with mass m i ≤ m max [6,9], where η i = −1(+1) for bosons (fermions), K 1 is the modified Bessel function, d i are the degeneracy factors. The trace anomaly for the HRG has been parametrized as [10] ε − 3 P T 4 with a i given in Table II and, by interpolation with lattice data, the pressure is given by where p lattice l is the pressure at T l = 130 M eV and The electroweak sector is included as a relativistic gas of massless particles, i.e.
where g ew = 14.45 is the effective number of electroweak degrees of freedom [4].

B. b. Dark matter sector
In previous analyses of the evolution of the cosmological parameters the dark matter contribution has been neglected and we now evaluate its role in the EoS of the complete system by considering the following different models (see Appendix B for details): • Cold Dark Matter: for pressure, p cdm , and energy density, ε cdm , of the cold dark matter one has By considering the isentropic condition a 3 s = a 3 0 s 0 , and indicating with ǫ and p the non-dark-matter contributions to the energy density and to the pressure, one obtains (see Appendix B 1) where T 0 is the present temperature, Ω b0 and Ω cdm0 the present baryon and cold dark matter density respectively.
Since we study the cosmological evolution for temperatures such that K T ≫ 1, and the cold dark matter term can be neglected.
• General Barotropic case: a barotropic equation of state is a linear relation between energy density, ǫ b , and pressure, p b , with w b = const. Using again the isentropic evolution, one gets, as in the previous case, an equation connecting the energy of the dark sector to the energy of all other sectors. This equation can be analytically solved for a few values of w b (w b = 0 gives the cold dark matter) and, in general, one needs numerical methods. We study the value w b = 1/3, i.e. a massless dark matter, useful to estimate the maximum contribution of the dark sector, although excluded by phenomenological analyses [15].
• Polytropic case and Chaplygin gas: for a polytropic gas the pressure, p p , and the energy density, ǫ p , are related by where γ is the polytropic index. In Appendix B 3 one shows that, in a cosmological contest, such a case is equivalent to a generalized Chaplygin gas with index α = −γ, i.e. the energy density is given by with w 0 p ≡ p 0 p /ε 0 p and a 0 , p 0 p and ε 0 p evaluated at some reference time. In particular, by choosing a 0 at the present time, in the early universe (a ≪ a 0 ) one has to study three different cases: a) γ = 1 (see Eq. (27)), which is the general barotropic case; 2) γ < 1, then and, by Eqs. (13) and (27), the entropy density turns out to be where A, B and C constants, i.e. with the same behavior of the cold dark matter; 3) γ > 1 is physically meaningful only for w 0 p ≤ 0 and in such a case (see Eq. (28)) and the resulting entropy density is constant and negligible.

III. THE COSMOLOGICAL DECONFINE-MENT TRANSITION
By previous parametrization of the EoS of strongly interacting, electroweak and dark matter sectors, we now analyze the full Eos in the range 70 MeV ≤ T ≤ 400 MeV by interpolating the lattice data and the HRG results at The results for the w = p T /ε T and for the speed of sound c 2 s are summarized in Fig. 1, where the continuous curves indicate the speed of sound and the dashed lines the value of w. The blue lines give the results for the strong sector, the red ones contain the electroweak sector and the black curves take into account the massless dark matter also. Cold dark matter is not included since its contribution is negligible.
The arrows indicate the temperature of the transition, defined as the temperature at the  minimum of the speed of sound, which goes from the T s t = 147 MeV for strongly interaction only, to T ew t = 158 MeV adding the electroweak sector, to T d t = 157 MeV including the massless dark sector.
The relation between the temperature and the cosmological time is which is numerically solved (with T 0 = 500 MeV and t 0 = 1 µs [4]). In Figure 2 we have shown how the temperature decrease in the different cases previously discussed and in the pure radiation era (red dotted line). The transition time is reduced by adding more and more sectors:  Fig. 1 for the colors legend) as a function of time.
speed of sound as a function of the cosmological time. For the whole system, after about 100µs the values of w and c 2 s come back to be that ones of a radiation dominated era.

IV. EVOLUTION OF THE COSMOLOGI-CAL PARAMETERS
The results in the previous sections are the starting point to study the behavior of the cosmological parameters during the deconfinement transition. Since the cosmological parameters can depend on the higher order derivatives of the Hubble parameter, i.e. on the higher order derivative of the thermodynamical quantities, it could be possible that some effects show up near the critical temperature [4].
We have analyze three different cases: strong sector only (blue curves in the figures); strong plus electroweak sector (red curves); the whole system, including the three EoS of section 2b for the dark matter (black lines). In all figures, the arrows indicate the transition time.
In Figure 4 and in Figure 5 are respectively depicted the time behavior of the scale factor a(t) (normalized to the value at 400 MeV, a * ) and of H(t). The final result is essentially independent on the specific setting.
In Figures 6, 7, 8, 9 and 10 the time evolution of q, j, s, A 5 and A 6 is plotted. As expected the parameters with high order derivative show larger deviations from the typical values of a radiation dominated era. However once the transi-  tion is over, the Universe is again dominated by radiation.

V. MODIFICATION OF THE PRIMOR-DIAL SPECTRUM OF THE GRAVITA-TIONAL WAVES
According to previous results, the fluctuations of the cosmological parameters in the whole system (strong, electroweak and dark matter sectors) are limited to a short time interval of about  100 µs. Therefore the deconfinement transition turns out to be more relevant in the modification of the primordial spectrum of the gravitational waves, proposed in [2], that will be recalled in this section and reevaluated on the basis of the detailed description of the transition in Sec. III.   During the inflation era the wavelengths of the quantum fluctuations are stretched to scales greater than the casually connected region and the fluctuations of the metric tensor result in a background of stochastic gravitational waves [17].
In the transverse traceless (TT) gauge, tensor perturbations h ij of the FLRW metric satisfy the linearized equation of motion where ";" indicates the covariant derivative, and the corresponding Fourier modes take the form where λ = (+, ×) are the two polarization states and e λ ij is the symmetric polarization tensor (e ii = 0, k i e ij = 0). In conformal time, η, the equation of motion for the perturbations reads [3] h ′′ k,λ (η) + 2 where d/dη is denoted by prime " ′ ". By defining µ kλ = a h kλ , Eq. (36) can be written as Two different regimes are physically relevant and correspond to fluctuations well inside the Hubble horizon or well outside the horizon. Since a ′′ /a ∼ (a H) 2 , when k ≫ a H the wavelength is smaller than the horizon: this is the subhorizon regime.
In this case Eq. (37) is the harmonic oscillator, hence, µ k (η) ∼ e ikη and for the perturbation one obtains which implies that the amplitude decreases in time. In the superhorizon regime, i.e. for k ≪ a H, Eq. (37) has two independent solutions: a decaying mode µ k ∼ a −2 , which we neglect, and µ k ∼ a that leads to that is the amplitudes are almost frozen, being outside the casually connected region. Therefore, during the inflation era the amplitudes are stretched to size larger than the horizon, where they remain constant, but when inflation ends the comoving Hubble horizon (a H) −1 grows in time and each mode crosses the horizon and reenters inside the casually connected region when the wavelength is comparable to the horizon size, i.e. k = a H. In this case, a general solution of Eq. (37) can be written introducing a factor depending on mode's amplitude in superhorizon regime and a transfer function, T k (η), as where h prim k,λ is the amplitude when the mode left the horizon during the inflationary period and T k (η) describes the evolution of the gravitational wave after it crosses the horizon. In radiation dominated universe, the solution reads where j 0 (x) is the spherical Bessel function [17]. Let us define the power spectrum of gravitational waves. The energy density is given by and in k space the spatial average reads (43) Moreover, one assumes that the primordial gravitational waves are unpolarized, that is |h ′ k,+ (η)| 2 = |h ′ k,× (η)| 2 . Using Eq. (40), we can write the energy density as where ∆ 2 k,prim is the primordial amplitude which in de Sitter inflation turns out to be H dS and M P l being the Hubble constant in de Sitter inflation and the Planck mass, respectively. The logarithmic energy density is defined as dε h /d ln k and the fractional energy density is given by where ε c is the critical energy density.
From Friedman equations (5) we finally get A gravitational wave of mode k has frequency f = 2πk/a. Because of redshift, once a wave crosses the horizon its frequency decreases. From the definition of fractional energy density follows that Ω decreases as a −4 H −2 , since gravitational waves are decoupled from the rest of the Universe and ε c ∼ H 2 from Friedman equation. For waves that reentered at a certain time η, the fractional energy density today is and the frequency today is f 0 = 2πk/a 0 , where a 0 and H 0 are the scale factor and the Hubble parameter today. The evolution of the h k modes and of the crossing condition, k = a H, are controlled by the scale factor a and the modification of the spectrum of gravitational waves from the primordial one depends on the content of matter in the epoch they reenter the horizon. As previously shown (see Fig. 3), during the QCD transition the speed of sound c 2 s is strongly modified since the Universe stands no longer in a pure radiation era and, correspondingly, the primordial gravitational waves cross the horizon near that transition time at different rates.
By lattice QCD simulations, the HRG model and including the electroweak and dark matter sectors, we now discuss a detailed analysis of this effect by numerical integration of Eqs. (36)-(37), improving previous analysis [2,3].
It is more useful to write Eq. (36) for the transfer function as a function of the temperature, that is (see Appendix C for details) In order to integrate numerically Eq. (49), we set boundary conditions at high temperature, such as 10 4 MeV, where the modes h k are given by the radiation era solutions (Eq. (41)). In Figure 11 the numerical results for different values of k are reported.
Waves with higher frequencies cross the horizon earlier and waves that reentered at T ∼ 150 MeV have frequencies of about 10 −7 Hz, the typical frequency f * of waves from the QCD transition. The effects of the transition are expected to be impressed in the fractional energy density Ω and, in particular, one computes [2] the quantity Ω(f )/Ω(f ≪ f * ), that is the fractional energy density of the gravitational waves with respect to the same quantity evaluated for waves that do not encounter the transition (f being a fixed frequency much lower than f * ).
From Eq. (48), this quantity evaluated today is   The redshift factor gives the shape of the step and the final result is showed in Fig. 12. The size of the step is about 38%, larger than previous results [2,3]. In particular, in [2] the step size was ≃ 30%. Figure 13 shows a direct comparison between the latter result and our evaluation.

VI. COMMENTS AND CONCLUSIONS
The fluctuations of conserved charges at the deconfinement transition are a clear signature of the different behavior between a quark-gluon plasma and a hadron resonance gas model, but their detection in relativistic heavy ion collisions is difficult. The fluctuations of the cosmological parameters at the QCD transition have, in principle, the same physical basis, i.e. they originate from the combined effect of the equation of state and of the calculation of higher order derivatives of the relevant physical parameters, that is, in early Universe, the scale factor.
We have shown, by a complete treatment of the thermodynamics of the whole system (strong, electroweak and dark matter contributions), that after about 100µs the cosmological parameters return to the typical values of a radiation dominated era, i.e. to their values before the transition, and that this result remains valid also for cosmological scalars involving higher order derivatives of the scale factor (see Figs. [6][7][8][9][10].
Moreover the dark matter contribution turns out to be negligible, independently on the considered model and, as shown in ref. [7], the effects of the QCD transition on the density fluctuations are small. Therefore the possible signature of the deconfinement transition in early Universe is restricted to the modification of the primordial gravitational wave spectrum.
By using the recent lattice QCD simulation data and the HRG below T c to describe the transition, one evaluates the fraction of energy den-sity of gravitational waves with respect to waves that do not encounter the QCD transition. A difference of about 10% is observed with respect to previous analyses [2]. and, by the FLRW equations and by the isentropic expansion condition, one obtains where C V is the specific heat.