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Cosmological particle creation in a hadronic fluid

Acoustic perturbations in an expanding hadronic fluid at temperatures below the chiral transition point represent massless pions propagating in curved spacetime geometry. In comoving coordinates the corresponding analog metric tensor describes a hyperbolic FRW spacetime. We study the analog cosmological particle creation of pions below the critical point of the chiral phase transition. We compare the cosmological creation spectrum with the spectrum of analog Hawking radiation at the analog trapping horizon.


Introduction
Quantum field theory in a curved spacetime predicts that the gravitation field creates particles and antiparticles. The main related phenomena are the Hawking effect in black holes and particle creation due to the cosmological expansion [1]. In the latter case the creation is caused by the time dependence of the background metric which in turn causes a nontrivial time evolution of the ground state similar to what happens to a quantum harmonic oscillator with time dependent frequency [2]. The process of particle creation depends only on the particulars of the metric and particle properties (mass, spin etc.) irrespective of whether the expansion is of cosmological or another origin. In particular, particle creation is expected in time dependent analog gravity systems such as expanding Bose Einstein (BE) condensates [3,4,5,6], Bose and Fermi superfluids [7], expanding hadronic fluids [8,9].
Analog gravity has proved to be useful in studying various physical phenomena [10], e.g., acoustics [11], optics [12], superfluidity [13], black hole accretion [14,15], and hadron fluid [8,9,16]. The purpose of this paper is to study in the framework of analog gravity the phenomenon of cosmological particle creation in a hadronic fluid produced in high energy collision experiments. Strongly interacting matter is described at the fundamental level by a nonabelian gauge theory called quantum chromodynamics (QCD). At low energies, the QCD vacuum is characterized by a non-vanishing expectation value [17]: ψ ψ ≈ (235 MeV) 3 , the so called chiral condensate. This quantity describes the density of quark-antiquark pairs found in the QCD vacuum and its non-vanishing value is a manifestation of chiral symmetry breaking [18]. Our approach is based on the linear sigma model [19] combined with a boost invariant Bjorken type spherical expansion [20]. The linear sigma model serves as an effective model for the low-temperature phase of QCD [21,22]. In the chirally broken phase, i.e., at temperatures below the point of the chiral phase transition, the pions are massless but owing to the finite temperature effects propagate slower than light [23,24,25]. Moreover, the pion velocity approaches zero at the critical temperature.
As in general relativity, a notable manifestation of analog gravity are two effects both having a quantum origin: the Hawking radiation and cosmological particle creation. These two phenomena are similar but appear under different physical conditions. The Hawking thermal radiation is due to the information loss across the apparent horizon whereas the cosmological particle creation generates a quasi-thermal radiation as a result of time variation of the spacetime geometry. The Hawking radiation takes place only if there exists a trapping (or apparent) horizon, whereas the cosmological particle creation takes place with or without a horizon. On the other hand, in any stationary geometry the cosmological particle creation is absent whereas the Hawking radiation is present in a stationary geometry with an event horizon. Besides, if a field theory is conformally invariant there there will be no cosmological particle creation. In contrast, the Hawking radiation is present even in the conformal case as long as a trapping horizon exists.
The analog Hawking effect has been studied in our previous papers [8,9] in the context of an expanding hadronic fluid. We have demonstrated that there exists a region where the flow velocity exceeds the pion velocity and the analog trapped region forms which then causes the Hawking radiation of massless pions. Here we study the effect of cosmological creation of pions in an expanding hadronic fluid in terms of the Bogoliubov transformation [26] and adiabatic expansion [1,27]. For alternative approaches to cosmological particle creation see, e.g., [28,29].
The remainder of the paper is organized as follows. In Sec. 2 we describe the analog model based on the expanding chiral fluid. The cosmological particle creation of pions is studied in Sec. 3 in which we derive the spectrum of the created pions and the time dependence of the particle number. We estimate the temperature by fitting our spectrum to the Planck black body radiation spectrum. In the concluding section, Sec. 4, we summarize our results and discuss physical implications.

Expanding chiral fluid
In this section we describe the hadron fluid in terms of the linear sigma model at finite temperature undergoing a spherically symmetric expansion. Our model is based on a scalar field Lagrangian with spontaneously broken chiral symmetry which we describe in Sec. 2.1. Then in Sec. 2.2 we specify the dynamics of the fluid based on the Bjorken expansion model.

Linear sigma model
Consider a linear sigma model at finite temperature in the Minkowski spacetime background. The background medium is a hadronic fluid consisting of predominantly pions. The dynamics of mesons in such a medium is described by an effective Lagrangian with spontaneously broken chiral symmetry [30] where u µ is the velocity of the fluid and g µν is the background metric. The coefficients f and g depend on the local temperature T and on the parameters of the sigma model: the coupling constant λ and the pion decay constant f π , and may be calculated in perturbation theory. The scalar fields σ and π i , where i = 1, 2, 3, represent fluctuations around the expectation values σ and π i , respectively. The expectation values of the pion fields π i are chosen to vanish always whereas the expectation value of the sigma field σ , usually referred to as the chiral condensate, is temperature dependent and vanishes at the critical temperature T c . The scaling and universality analysis [24] yields σ ∼ (T − T c ) β in the vicinity of the critical point. At zero temperature σ is normalized to the pion decay constant, i.e., σ = f π at T = 0. The meson masses depend on temperature and below the chiral transition point are given by The potential U is The temperature dependence of σ is obtained by minimizing the thermodynamic potential Ω = −(T /V ) ln Z with respect to σ at fixed temperature T [22]. The scaling and universality analysis [24] yields σ ∼ (T − T c ) β in the vicinity of the critical point with β = 0.388 for the O(4) universality class [31,32]. Furthermore, the extremum condition was solved numerically at one loop order [9,22] and the value of the critical temperature T c = 183 MeV was found with f π = 92.4 MeV and m σ = 1 GeV as a phenomenological input.
The action corresponding to the Lagrangian (1) may be expressed as [9] where p and q are positive parameters. The constants in (10) are then fixed to (11) we capture the main features: with the values p = 0.1 and q = 3.58 we match the zero temperature limit [23] and we recover the correct critical behavior (10). The variation of the action (4) yields the Klein-Gordon wave equation in curved space where ϕ stands for π i or σ and are the corresponding interaction potentials.

Spherical Bjorken expansion
In order to specify the background metric g µν and the velocity of the fluid we consider a boost invariant Bjorken type spherical expansion [35]. In this model the radial three-velocity in radial coordinates x µ = (t, r, θ, φ) is a simple function v = r/t. Then the four-velocity is given by where τ = √ t 2 − r 2 is the proper time. With the substitution t = τ cosh y, r = τ sinh y, the four-velocity velocity is expressed as u µ = (cosh y, sinh y, 0, 0).
The substitution (15) may be regarded as a coordinate transformation from ordinary radial coordinates to new coordinates (τ, y, θ, φ) in which the flat background metric takes the form and the velocity components become u µ = (1, 0, 0, 0). Thus, the transformation (15) maps the spatially flat Minkowski spacetime into an expanding FRW spacetime with cosmological scale a = τ and negative spatial curvature. The resulting flat spacetime with metric (17) is known in cosmology as the Milne universe [36]. The temperature of the expanding chiral fluid is, to a good approximation, proportional to τ −1 . This follows from the fact that the expanding hadronic matter is dominated by massless pions, and hence, the density of the fluid may be approximated by the density ρ = (π 2 /10)T 4 of an ideal massless pion gas [37]. Using this and energy-momentum conservation one finds where T c is the critical temperature of the chiral transition and τ c may be fixed from the phenomenology of high energy collisions. For example, if we take T c = 0.183 GeV, then a typical value of ρ = 1 GeV/fm 3 at τ ≈ 5 fm [38] is obtained with τ c ≈ 8.2 fm = 41.6 GeV −1 . The physical range of τ is fixed by Eq. (18) since the available temperature ranges between T = 0 and T = T c . Hence, the proper time range is τ c ≤ τ < ∞. In the following we keep τ c unspecified so that physical quantities of dimension of time or length are expressed in units of τ c . In the comoving coordinate frame defined by the coordinate transformation (15) the velocity is u µ = (1, 0, 0, 0) and, as a consequence, the analog metric (5) is diagonal where This metric represents a general form of an FRW spacetime with negative spatial curvature.
Here, the parameters f and c π are functions of the temperature T which in turn is a function of τ . More explicitly, using (11) and (18) we have where z = τ c /τ = T /T c , ν = 0.749, β = 0.388, p = 0.1, and q = 3.58.

Creation of pions in analog cosmology
The particles associated to quantum fluctuations of the chiral field are pions and sigma mesons. Since the chiral fluid is expanding and the particles experience an effective time dependent metric, the pions and sigma mesons will be created during the expansion in complete analogy with the standard cosmological particle creation. As a consequence, the pion and sigma-mesons numbers will not be conserved and a vacuum state generally evolves into a multi-particle state. In Sec. 3.1 we review the standard procedure of canonical quantization of scalar fields in an FRW geometry and the derivation of Bogoliubov coefficients. In Sec.
3.2 we deal with the particle interpretation ambiguity in a time dependent geometry. Next, we solve the Klein-Gordon equation with the help of the WKB ansatz and in Sec. 3.3 we presents the numerical results.

Canonical quantization
The effective action (4) with (3) is of the ϕ 4 type. However, as we are primarily interested in the effects of cosmological particle creation we shall in the following disregard the selfinteraction terms in the potential. In other words, we do not consider particle production caused by the self interaction although this effect may be significant [39]. Hence, we can split the action (4) into a sum of the actions for each field: where ϕ stands for π i or σ. Here we have introduced a time dependent effective mass defined by where m stands for m π or m σ as given by (2). For completeness, we have included the non-minimal coupling term of the scalar fields to the effective scalar curvature R. This term is required by renormalization in curved spacetime because, even if the renormalized ξ = 0, loop corrections would induce a nonminimal coupling term of this type [39,40]. As we shall see, this term introduces another type of criticality in addition to the critical behavior owing to the above mentioned chiral symmetry breaking and restoration at finite temperature. The field equation derived from (23) is the free Klein-Gordon equation in curved spacetime with metric (19).

Schrödinger representation
In the canonical quantization formalism we expand the field ϕ where the time independent particle creation and annihilation operators (the "Schrödinger picture") satisfy the commutation relations and ϕ J (x) are solutions to (25) labeled by a collective index J. In spherical coordinates J ≡ {k, l, m}, where l is the angular momentum, m its projection, and k is the magnitude of the comoving momentum related to the physical momentum as p = k/a. Note that in the coordinate frame (τ, y, θ, φ) with metric (19) k is dimensionless. Thus, the physical energy of a particle is where m eff is defined by (24). In a hyperbolic space the large volume limit V → ∞ can be applied and the sum over discrete momentum is replaced by an integral over continuous k. Then the sum over J in (26) can be written as [26] As usual, we can separate the time and space dependence using Then, the functions χ k and Φ J satisfy respectively. Here and from here on the prime denotes a partial derivative with respect to τ . The differential operator ∆ is the Laplace-Beltrami operator on the three-dimensional space with line element The metric dependent factor on the righthand side of (30) is introduced in order to get rid of the first order derivative in the equation for χ. The time dependent function Ω is given by where R is the Ricci scalar, and with mass m = m(τ ) generally depending on τ by way of its temperature dependence. The spacetime defined by the metric (19) with (21) and (22) has a curvature singularity since the Ricci scalar diverges at τ c as The solutions to (32) is known and the explicit form of Φ J (x) may be found in [26]. Here we only use the following properties Using these properties, the sum over l and m in (29) may be carried out in the case of spherical symmetry. First, using (38) with J = K we rewrite (29) as Then, by (39) we obtain where V denotes the comoving proper volume To solve (31) we first impose the condition which we can do because the left-hand side, the Wronskian, is a constant of motion of (31). Next we make use of the WKB ansatz which automatically meets the condition (43), where the positive function W (τ ) satisfies A solution to (45) may be expressed as where the series {W (n) } is obtained from the adiabatic expansion [1] The contribution ω (i) (τ ) containing derivatives of a, b, and ω with respect to τ of order i may be found iteratively at each adiabatic order i starting with ω (0) (τ ) = ω(τ ). Using (47) we define the adiabatic modes of order n as ϕ (n) where χ (n)

Heisenberg representation
In the following we assume the adiabatic invariance of the particle number in each mode of the field ϕ, i.e., we require that the particle number in each mode should be constant [41] in the limit of an infinitely slow expansion. To meet this requirement we expand the field operator ϕ(x) and its derivative ϕ (x) in terms of the time dependent operators (the "Heisenberg picture") a J (τ ) whereφ andW (τ ) are conveniently chosen smooth functions of ω, a, and b, and their derivatives, such thatW at some initial τ = τ 0 . In addition, for a given k the functionW (τ ) should coincide with W (0) (τ ) at lowest adiabatic order, i.e., we requireW (τ ) = ω(τ ) at lowest adiabatic order. The functionW (τ ) is, obviously, not unique but the adiabatic expansion (47) offers a natural choice -each adiabatic mode W (n) (τ ) of order n satisfies the above criteria. However, the lowest order mode W (0) is unacceptable as it leads to UV divergence in particle production rate [42,43]. We shall shortly discuss this point in more detail. The simplest, and perhaps the most natural choice isW (τ ) ≡ W (1) (τ ) = ω(τ ). Another choice,W (τ ) ≡ Ω(τ ), which meets the above criteria also seems natural since Ω appears in the harmonic oscillator equation (31) as the (time-dependent) frequency. However, in contrast to W (1) (τ ), the function Ω(τ ) contains derivative terms of second order but not all such terms that appear in the next order adiabatic mode W (2) (τ ). Hence, the choicẽ W (τ ) ≡ Ω(τ ) is incomplete and inconsistent from the adiabatic expansion point of view. Another problem withW (τ ) ≡ Ω(τ ), as we shall shortly demonstrate, is that it yields a nonvanishing particle creation rate in the conformal case.

Bogoliubov transformation
The time dependent operators a J (τ ) and a † J (τ ) are related to a J (τ 0 ) ≡ a J and a † J (τ 0 ) ≡ a † J via the Bogoliubov transformation [26,44] a J (τ ) = α k (τ )a J + β * k (τ )a †J ϑ J , where the coefficients satisfy The conjugate labelJ is defined so that and ϑ J is a phase with property ϑJ = ϑ * J .

Particle interpretation
As is well known, there exist an intrinsic ambiguity of the particle interpretation in spacetimes with time dependent metric, in particular in an FRW spacetime [27]. In this section we present a simple demonstration of this ambiguity and the prescription how to remove it. At τ 0 the vacuum state vector | is defined as the state which is annihilated by the operator a J , i.e., a J | = 0. A one particle state with quantum numbers J is defined using the creation operator a † J acting on the vacuum, i.e., a † J | = |J , so that in the coordinate representation we define x|J = ϕ J (τ 0 , x).
In the Heisenberg picture the state | is time independent whereas a J (τ ) and a † J (τ ) evolve with τ according to (55) and (56) so a J (τ )| = 0 (66) for τ = τ 0 . In the Schrödinger picture, the vacuum state | evolves into a new state vector | τ which represents the vacuum with respect to a J (τ ) such that where the one particle state in the coordinate representation is defined as From (55) and (67) it follows a J | τ = 0, so the state vector | τ at late times is different from the state vector | containing no particles at an early time τ 0 and hence, there is no unambiguous unique Heisenberg state which can be identified as the vacuum state. The total number of particles with quantum number J created at the time τ is where the last two equations follow from (55) and (56). Then, the particle number density is the total number divided by the physical volume a 3 V , i.e., where we have exploited spherical symmetry and used (42) to replace the sum by an integral. Thus, the occupation number of created particles is equal to the square of the magnitude of the Bogoliubov coefficient β k . The square of the Bogoliubov coefficient may be obtained directly from (64), and by way of (44) and (53) may be conveniently expressed in terms of W andW The above mentioned ambiguity is in the choice ofχ(τ ) orW (τ ) which one has to fix in order to evaluate the righthand side of (72). As we discussed previously, a natural choice would be an adiabatic mode χ (n) . In this case, to maintain consistency with the adiabatic expansion, we must keep only the terms up to the adiabatic order n [43] in the derivatives W = W (n) in (72). For example, consider n = 0 and n = 1. It turns out that ω (1) = 0 in the adiabatic expansion (47) so W (1) = W (0) = ω and W (1) = ω , but W (0) must be set to zero. Furthermore, according to the adiabatic regularization prescription, the choice of n is dictated by the asymptotic ultraviolet (UV) behavior of the integrand in (71): one must use the minimal n such that |β k | → 0 faster then 1/k as k → ∞ [43]. For an FRW metric of the general form (19) the integral (71) is UV divergent for n = 0 apart from some special cases discussed in detail by Fulling [42]. For example, it is easy to verify that such a special case is realized if b = a in (19), which corresponds to the conformal form of a spatially hyperbolic metric. In a more general case, i.e., if b = a the integral (71) is UV divergent for n = 0 and converges for n ≥ 1. We shall therefore work with n = 1 and check the convergence explicitly. Hence, the final expression which will be used in our numerical calculations is where ω(τ ) is given by (36) and W (τ ) is a solution to (45) that satisfies the initial conditions at conveniently chosen τ = τ 0 .

The results
Instead of solving equation (31), for our purpose it is convenient to solve equation (45) for the WKB function W (τ ). With the help of the substitution from (45) we obtain the differential equation where the function Ω(τ ) is defined in (34) with (36). Before proceeding to solve (76) for the analog cosmological model defined by the metric (19) with (21) and (22), it is useful to study three important examples which may be solved analytically. Consider first a conformally invariant field theory, i.e., for m = 0 and ξ = 1/6, in which case there should be no particle creation, as has been argued on general grounds [41]. Indeed, in this case, as may easily be verified, the function is a solution to (76) for arbitrary a(τ ) and b(τ ). This yields W c (τ ) = ω(τ ) and ω (n) = 0 at all adiabatic orders n > 0. Then, with the choiceW = W (n) = ω, by virtue of (73) we find |β k | = 0 and hence no particle creation as expected. In contrast, on account of (72) the choiceW = Ω would generally yield |β k | = 0 and hence an unphysical prediction of particle creation for a conformally invariant field. Second, consider the asymptotic future. In the limit τ → ∞ our system approaches the zero temperature regime and the spacetime described by the analog metric (19) approaches the Milne universe with a = τ and b = 1. It is therefore instructive to compare the analog cosmological particle creation with that of the Milne universe. In particular, in the Milne universe there should be no creation of massless particles since the scalar field satisfies the conformally invariant wave equation. Indeed, for m = 0 the asymptotic solution to (76) gives |β k | 2 = 0 and hence, there is no creation of massless particles as in the previous case. Third, it is worth analyzing the solution to (76) in the critical regime t ≡ τ − τ c → 0. In that regime Ω 2 αt −2 where Equation (76) then simplifies to and may be solved analytically in the limit t → 0. The behavior of the solution in that limit depends crucially on the value of the non-minimal coupling constant ξ. We find three distinct solutions depending on α y =      (α − 1/4) −1/4 t 1/2 , for α > 1/4, y 1 t 1/2− √ 1−4α/2 , for 0 < α < 1/4, where y 1 and y 2 are arbitrary real constants. Clearly, y → 0 for α > 0 and y → ∞ for α < 0. From the definition (79) it follows is the critical non-minimal coupling which takes the value ξ c 0.28322 for the O(4) critical exponents. Hence, the WKB function W (τ ) goes to zero at the critical point if ξ < ξ c and diverges if ξ > ξ c . Note that the critical coupling ξ c > ξ conf = 1/6 contrary to what one would expect since the original sigma model becomes conformally invariant at the critical point. Curiously, ξ c would be equal to ξ conf = 1/6 if the critical exponent ν were equal to zero, in which case the pion velocity, as given by (10), would not necessarily vanish at the critical point.
Since there is no creation of pions in the limit τ → ∞, it is natural to choose as the initial state the vacuum state vector | 0 at some large τ 0 and evolve equation (45)  and, according to (65), the one particle state |J is represented by where the function χ The results of the numerical calculations are presented in Figs by the asymptotic solution (81) in the vicinity of the critical point. Using these numerical solutions we calculate the square of the Bogoliubov coefficients |β k | 2 as given by (73). In Fig. 2. We present |β k | 2 as a function of k for a fixed τ /τ c = 1.12 and various couplings ξ.
Assuming that we are in the adiabatic regime, the spectrum of created particles should be nearly thermal [3]. More precisely, the occupation number |β k | 2 as a function of k for a fixed τ is expected to be close to the Planck spectral distribution where E is the particle energy defined by (28) and T P is a function of τ which we call the Planck temperature. We now assume that the created pions are massless in which case E = k/a. To extract T P it is convenient to use the energy density distribution function [37] f which we have normalized so that it depends only on the dimensionless variable x ≡ k/(aT P ).
For an ideal massless boson gas  with a maximum at Using (86) we can express f (k) as and plot the righthand side as a function of k for various τ (Fig. 3). Then, from the position of the maximum k max (τ ) we obtain In Fig. 4 we plot the Planck distribution temperature as a function of τ for vanishing non-minimal coupling constant ξ together with the Hawking temperature T H of thermal pions emitted at the apparent horizon. For comparison we plot in the same figure the temperature of the fluid versus τ as given by equation (18) with T c = 7.625 τ −1 c . The functional dependence of T H is calculated following the prescription of our previous papers [8,9]. As shown in [8] the Hawking temperature diverges near the critical point as The temperature T P seems to diverge at the critical point in a similar way and vanishes in the limit τ → ∞ corresponding to the zero background temperature of the hadronic fluid.
In contrast, the analog Hawking temperature vanishes at τ = τ max = 1.1002 τ c at which the analog trapping horizon ceases to exist [9].

Conclusions
We have investigated the cosmological creation of pions in an expanding hadronic fluid in the regime near the critical point of the chiral phase transition. In our approach we have disregarded a possible particle production caused by the self interaction potential of the scalar field. Besides, we have assumed that the created pions are of zero or very light mass and we have neglected the creation of much heavier sigma mesons. The production rate has been calculated using the adiabatic regularization prescription according to which the Bogoliubov coefficients are expressed in terms of the WKB function W (τ ) and its first order adiabatic approximation W (1) (τ ) = ω(τ ). The function W (τ ) has been computed by solving equation (45) numerically. We have analyzed more closely the solution in the limit when τ approaches the critical value. It turns out that the behavior of the solution in that limit depends crucially on the value of the non-minimal coupling constant ξ. We have shown that there exist a certain critical value ξ c larger than the conformal value ξ = 1/6 such that W (τ ) goes to zero at the critical point for ξ < ξ c and diverges for ξ > ξ c . We have calculated the cosmological production rate as a function of the proper time for various masses and various nonminimal coupling constant ξ. The production rate of massless pions shows a strong dependence on ξ and vanishes for ξ = 1/6 as it should. By fitting the production rate to the Planck black body radiation spectrum we have extracted the temperature of the produced pion gas. We use the time dependence of the thus obtained Planck temperature to compare the analog Hawking effect with the analog cosmological particle creation. As we have already mentioned, these two effects, although being of similar quantum origin, are quite distinct physical phenomena that appear under different physical conditions. Compared with the analog Hawking radiation of pions at the trapping horizon, the spectrum of the cosmological radiation shows a similar behavior near the critical point. The temperature of the cosmologically created pions T P diverges at the critical point roughly in the same way as the analog Hawking temperature T H . However, as the proper time increases, the Planck temperature vanishes asymptotically whereas the analog Hawking temperature vanishes at a finite proper time of the order 1.1 τ c when the analog trapping horizon disappears.
Our results could not be easily confronted with observations. First of all, we are dealing with exact spherical symmetry, whereas in most high energy collisions the symmetry is axial involving a transverse expansion superimposed on a longitudinal boost invariant expansion. Second, the cosmologically created and Hawking radiated pions could not be easily distinguished from the background pions produced directly from the quark-gluon plasma (QGP). Nevertheless, we can draw a qualitative post-collision picture as follows.
The high temperature (QGP) produced in the collision expands and cools down until the temperature is as low as the deconfinement temperature of the order of T dec T c = 183. Then, a hadronic fluid mainly consisting of pions forms and expands further according to the Bjorken model with proper time related to the background fluid temperature through the relation (18), where τ −1 c 24 MeV. Immediately below T c = 183 the cosmological creation and the Hawking radiation take place. Initially, both the Hawking and Planck temperatures exceed the background fluid temperature T by a factor of 2 or more. As a consequence, a considerable fraction of the pion gas will be briefly "reheated" but, according to Fig. 1, will quickly cool down during the subsequent expansion. Whereas the Hawking radiation stops at τ = τ max when the temperature of the fluid is of the order 0.9 T c , the cosmological creation continues up to the thermal freezout. The thermal (or kinetic) freezout takes place soon after the so called chemical freezout which is very close or equal to the QCD deconfinement transition [45]. The kinetic freezout temperature depends on the collision energy [46] and is roughly between 0.7 and 0.9 T c , which corresponds to the proper time interval (1.1, 1.4). From Fig. 4 it is evident that the influence of the cosmological production and the Hawking radiation will be more pronounced if the thermal freezout is closer to the critical temperature.