# Van der Waals universe with adiabatic matter creation

## Abstract

A FRWL cosmological model with perfect fluid comprising of van der Waals gas and dust has been studied in the context of dynamical analysis of a three-component autonomous non-linear dynamical system for the particle number density *n*, the Hubble parameter *H*, and the temperature *T*. Perfect fluid isentropic particle creation at rate proportional to an integer power \(\alpha \) of *H* has been incorporated. The existence of a global first integral allows the determination of the temperature evolution law and hence the reduction of the dynamical system to a two-component one. Special attention is paid to the cases of \(\alpha = 2\) and \(\alpha = 4\) and these are illustrated with numerical examples. The global dynamics is comprehensively studied for different choices of the values of the physical parameters of the model. Trajectories in the (*n*, *H*) phase space are identified for which temporary inflationary regime exists.

## 1 Introduction

The acceleration of the cosmic expansion and observational data (Supernovæ Type Ia, Cosmic Microwave Background, Baryon Acoustic Oscillations) are fit best by the current concordance model – the \(\Lambda \)CDM model which incorporates Dark Energy, modelled by the cosmological constant \(\Lambda \), and cold (pressureless) Dark Matter. There are open issues in relation to such model – the so called Cosmological Coincidence Problem: it is known observationally that the present values of the densities of dark energy and dark matter are of the same order of magnitude while, under the \(\Lambda \)CDM model, the dark-energy density is constant and the dark-matter density is proportional to the inverse third power of the scale factor with the ratio of the two densities varying in time from infinity to zero. There are numerous alternative models, not without open issues on their own, which accommodate acceleration of the cosmic expansion: modified gravity theories, inhomogeneous cosmologies, gravitationally induced particle creation models. In the literature, special attention has been gathered by the adiabatic, or isentropic, production [1, 2, 3, 4, 5] of perfect fluid particles in which the specific entropy (entropy per particle) is conserved (with “isentropic” referring to this). There is overall entropy production due to the enlargement of the phase space of the system as the particle number increases. The imposed condition of conserved specific entropy during the production of perfect fluid particles leads to a simple relationship between the particle production rate and particle “creation” pressure. Zimdahl [6] studies cosmological particle production with production rate which depends quadratically on the Hubble rate *H* and confirms the existence of solutions which describe a smooth transition from inflationary to non-inflationary behavior. The present work falls in this category and offers a full dynamical analysis of isentropic perfect fluid particle production rate that depends on \(H^{\alpha }\) with \(\alpha \) being a positive integer. Special attention is paid to the cases of \(\alpha = 2\) and \(\alpha = 4\), but the analysis can be easily extended to any other integer positive values of \(\alpha \), including odd values – due to the second law of thermodynamics, these work in the regime of expansion only [7]. The setting of the proposed model is a flat FRWL Universe with perfect fluid comprising of two fractions: real gas wit van der Waals equation of state and dust and the tools used are those of dynamical systems, see for example, [8, 9, 10, 11, 12], and as those used in the study of *n*–*H*–*T* (where *n* is the particle number density, and *T* is the temperature) dynamical analysis of cosmological quintessence real gas model with a general equation of state [13]. The dynamical variables are again *n*, *H*, and *T*, but due to the existence of a global first integral (in addition to second integrals), the temperature evolution law has been easily determined and the dynamical system reduced to a two component one over the (*n*, *H*) phase space. Inflationary regime with exit from the inflationary behaviour has been identified, both for \(\alpha = 2\) and for \(\alpha = 4\), and full classification of the possible phase-space trajectories, subject to the variation of the several physical parameters of the model, has been provided.

## 2 The model

*p*is the pressure,

*T*is the temperature, \(n = N / V\) – the number of particles

*N*per unit volume

*V*– is the particle number density and

*F*(

*T*) is the term describing two-particle interaction: \(F(T) = A - B/T\), where

*A*and

*B*are positive constants.

^{1}

*a*(

*t*) is the scale factor of the Universe.

*n*(

*t*) and the temperature

*T*(

*t*).

The separate conservation laws stipulate that there would be no exchange between the two components of the Universe.

*s*(i.e. entropy per particle, \(s = S/N\), where

*S*is the total entropy), is given by [3]:

*S*is not conserved due to the enlargement of the phase space resulting from the particle production [3].

On the issue of the equivalence of bulk viscosity and matter creation, Calvaõ et al. [4] and Lima et al. [5] argue that the matter creation process, as described by Prigogine [3], can generate the same dynamic behavior as a FRWL universe with bulk viscosity, while the models being quite different from a thermodynamic point of view. Brevik et al. [17] conclude that creation and viscosity concepts do not describe one and the same physical process – it is shown that viscous and creation universes can develop dynamically in the same manner, but the thermodynamic requirement for their identification is violated. The dynamic pressure \(\Pi \) in case of bulk viscosity is given by \(\Pi = - 3 \zeta H\), where \(\zeta \) is the bulk viscosity co-efficient [4, 5, 17], while in the case of matter creation processes, similar arguments lead to \(\Pi = - \alpha n \Gamma / (3H)\), where \(\alpha \) is a phenomenological co-efficient, called creation co-efficient, and it is closely related to the creation process – see [4, 5, 17] and the references therein. The adiabaticity of the fluid, namely, the conservation of the specific entropy, \({\dot{s}} = 0\), leads to the dependence on time of the creation co-efficient \(\alpha \): one gets \(\alpha = (\rho + p)/n\) – see [17] – and with this, \(\Pi = - \alpha n \Gamma / (3H)\) becomes the same as (11).

The dynamical equation (8), multiplied across by \(a^3\), reads off as \(dN/dt = (d/dt) (a^3 n) = a^3 n \Gamma \). Differentiating separately \(N = n a^3\) with respect to time, using \({\dot{a}} = a H\) and (8) gives \({\dot{N}} = 3 \beta N H^\alpha \). Also, from \(s = S/N = \) const, one gets \({\dot{S}} / S = {\dot{N}} / N = 3 \beta H^\alpha \). Thus, the constant \(\beta \) will be taken as positive and \(\alpha \) will be taken as a positive even integer. In the analysis, \(\alpha \) and \(\beta \) will be considered as parameters of the model.

*n*and

*T*, the time evolution of the energy density is:

In the absence of particle creation or annihilation (i.e. when \(\beta = 0\)), the above reduces to the well known form given in [13, 18, 19].

*A*and

*B*are both set to zero. This gives \(\phi (T) = m_0 + (3/2) T\). Namely, the energy density \(\rho \), the number density

*n*, and the temperature

*T*of the van der Waals gas are related via

*H*.

Equation (27) and its solution are the same as the ones encountered in the case of absence of matter creation or annihilation [13].

## 3 Analysis

*global*first integral given by:

*second integral*\(K(\vec {x}) = 0\) of an autonomous dynamical system of the type \(\dot{\vec {x}}(t) = \vec {F}[\vec {x}(t)]\) is defined by \((d/dt) K(\vec {x}) = \mu (\vec {x}) K(\vec {x})\). It is as an invariant, but only on a restricted subset, given by its zero level set [20]. As no trajectory can cross a hyper-surface defined by a second integral, the second integrals “fragment” the phase space into regions with separate dynamics (yet governed by the same dynamical system). For the two-component dynamical system, the ordinate \(n = 0\) is one such invariant manifold because \((d/dt) n = [3H(\beta H^{\alpha -1}-1)] n\). Similarly, the curve defined by \(3H^2 - \rho = 3H^2 - n[m_0 + (3/2)T] + Bn^2 = 0\) is another second integral because \((d/dt)(3H^2 - \rho ) = -3H(3H^2 - \rho ).\) It will be called a

*separatrix*– see Fig. 1.

There is a value \(\tau _0\) of \(\tau _0\) for which the separatrix \(3H^2 - n[m_0 + (3/2)T] + Bn^2 = 3H^2 - n[m_0 + (3/2) \, \tau \, n^{2/3} \, e^{2An/3}] + Bn^2 = 0\) is tangent to the *n*-axis at point, say \(n_0\) (see Fig. 1). Both \(\tau _0\) and \(n_0\) can be determined as follows. When \(\tau = \tau _0\), the separatrix has a minimum at \(n_0\) and that minimum is 0. Thus, \((3/2) \, \tau _0 \, n_0^{2/3} \, e^{2An_0/3} = Bn_0 - m\) and \((d/dn) \left[ n [m_0 + (3/2) \, \tau \, n^{2/3} \, e^{2An/3}] - Bn^2\right] _{n=n_0, \tau = \tau _0} = 0\) with solutions \(n_0 = [ 2 m_0 A + B + (4 m_0^2 A^2 + 20 m_0 A B + B^2)^{1/2} \, ] / (4AB)\) and \(\tau _0 = (2/3) (Bn_0 - m_0) n_0^{-2/3} \, e^{-2An_0/3}\). The energy density \(\rho [n, T(n)] {=} n[m_0 + (3/2) \, \tau \, n^{2/3} \, e^{2An/3}] - Bn^2 > 0\) may become negative over a certain range of *n*, depending on the choice of initial conditions, namely, depending on \(\tau \). Such trajectories would temporarily violate the weak energy condition and, as this is admissible in phantom cosmology models [21], the validity of the model will not be restricted by this.

*L*for the two-component dynamical system (29)–(30) is given by:

To determine the value of \({\widetilde{\tau }},\,\) for which \(T(n^*) = {\widetilde{\tau }} n^{*^{2/3}} e^{2An^*/3}\) is tangent to \(T^*(n^*) = Bn^*/(An^* + 1)\), and to also determine the point \(\widetilde{n^*}\) from the \(n^*\)-axis where these two curves are tangent to each other, consider the following. At point \(\widetilde{n^*}\), the two curves intersect, i.e. \({\widetilde{\tau }} \, {\widetilde{n}}^{*^{2/3}} e^{2A{\widetilde{n}}^*/3} = B{\widetilde{n}}^*/(A{\widetilde{n}}* + 1)\), and, also, the tangents to the two curves coincide, i.e. \([(d/dn^*) T(n^*)]_{(n^* = {\widetilde{n}}^*, \tau = {\widetilde{\tau }})} = [(d/dn^*) T^*(n^*)]_{(n^* = {\widetilde{n}}^*, \tau = {\widetilde{\tau }})} \). From these two simultaneous equations, one immediately determines that \({\widetilde{n}}^* = (\sqrt{3/2}-1)/A\) and that \({\widetilde{\tau }} = B \, {\widetilde{n}}^{*^{1/3}} e^{-2A{\widetilde{n}}^*/3} \, (1+A{\widetilde{n}}^*)^{-1} = \sqrt{2/3} (\sqrt{3/2}-1)^{1/3} e^{2/3-\sqrt{2/3}} A^{-1/3} B\). (For the numerical example considered, one has \({\widetilde{n}}^* = 22.47\) and \({\widetilde{\tau }} = 19.84\).)

Given that to the left of \({\widetilde{n}}^*\) one has \(L_{21}^* > 0\), the eigenvalues will have positive real parts (Fig. 2b). Such critical points are unstable and the trajectories near them are unwinding spirals (Figs. 4, 5).

When \(N_0^*< n^* < {\widetilde{n}}^* = (\sqrt{3/2} - 1)/A = 22.47\) and \(\beta < \sqrt{8B} / m_0 = 0.09\), the eigenvalues are both real and positive (Fig. 2b). The critical points are unstable nodes (Figs. 4, 5). When \(n^* > (\sqrt{3/2} - 1)/A\), the eigenvalues are both real – one positive and one negative (Fig. 2b) and one has saddles (Figs. 4, 5).

When \(\beta > \sqrt{8B} / m_0 = 0.09\), the eigenvalues \(\lambda _{1,2}^*\) are both real and positive for \(0< n^* < {\widetilde{n}}^* = (\sqrt{3/2} - 1)/A = 22.47\) (Fig. 2c). These critical points are unstable nodes (Fig. 6). And, finally, for \(n^* > {\widetilde{n}}^* = (\sqrt{3/2} - 1)/A\), the eigenvalues are both real with \(\lambda _1^*\) being positive and \(\lambda _2^*\) – negative (Fig. 2c). Such critical points are saddles (Fig. 6).

*L*. At the critical point \((n^*, H^* = 0)\), it is not zero when \(\alpha = 2\) and zero when \(\alpha > 2\). Consider next the \(\alpha = 4\) dynamical system and denote the stability matrix by \(L^{(4)}\) in this case. One has \( L^{(4)^*}_{22} = 0\) and the eigenvalues at the critical points \((n^*, H^* =0)\) are given by

For values of \(n^*\) above \({\widetilde{n}}^* = (\sqrt{3/2} - 1)/A\), the eigenvalues are purely real: \(\lambda _{1,2}^{(4) *} = \pm q\).

For \(\tau > {\widetilde{\tau }}\), the curves \(T(n^*)\) and \(T^*(n^*)\) intersect only at the origin, thus critical points \((n^*, H^* = 0)\) do not exists (see Fig. 2a).

For \(\tau < {\widetilde{\tau }}\), the curves \(T(n^*)\) and \(T^*(n^*)\) intersect, except at the origin, at points \(\nu _{1,2}^*\) (see Fig. 2a again) and the intersection points \(\nu _{1,2}^*\) are on either side of \({\widetilde{n}}^*\). Thus, at \(n^* = \nu _1^*\), the eigenvalues \(\lambda _{1,2}^{(4) *}\) are purely imaginary while, at \(n^* = \nu _2^*\), they are purely real (with opposite signs) and the corresponding critical points are saddles.

The behaviour of the trajectories near the critical points \((n^*, H^* = 0)\) for which the eigenvalues are purely imaginary, namely, for \(n^* < (\sqrt{3/2} - 1)/A\), are studied with the help of centre-manifold theory [22] in the Appendix. One finds that all critical points with purely imaginary eigenvalues are unstable – the trajectories near them are unwinding spirals [22] – see Fig. 7a, c.

*H*not higher than 3, reduce to \({\dot{n}} = - 3nH\) and \({\dot{H}} = - (3/2) H^2\). The solutions are:

*n*(

*t*) will increase with time. Similarly,

*H*will decay to zero (\(H \simeq 1/t\)) for trajectories in the upper half-plane or

*H*will decrease with time for trajectories in the lower half-plane.

The origin will attract trajectories from the upper half-plane and repel those from the lower half-plane. There are other critical points for the \(\alpha \ge 2\) dynamical system \({\dot{n}} = - 3 n H (1 - \beta H^{\alpha - 1}), \,\,\) \({\dot{H}} = - (3/2) H^2 - (1/2) (1 - \beta H^{\alpha - 1}) p[n, T(n)] + (1/2) \beta H^{\alpha - 1} \rho [n, T(n)]\).

*H*yields:

The function \(T^{**}(n^{**})\) has a minimum at \(\beta ^{\frac{1}{1-\alpha }}\sqrt{3/B}\). When \(\beta \) equals \(\beta _0 = (12B/m_0^2)^{\frac{\alpha -1}{2}}\), this minimum will occur at \(n_0^{**}\) from the \(n^{**}\)-axis: \(n_0^{**} = \beta ^{\frac{1}{1-\alpha }}\sqrt{3/B} = m_0/(2B)\). For values of \(\beta < \beta _0\), the graph of \(T^{**}(n^{**})\) is entirely above the \(n^{**}\)-axis, while for \(\beta > \beta _0\), the function \(T^{**}(n^{**})\) has zeros given by \(\nu _{1,2}^{**} = [m_0/(2B)] \, [ 1 \pm (1 - 12Bm_0^{-2} \beta ^{\frac{2}{1-\alpha }})^{\frac{1}{2}} ]\) – see Fig. 3a. When \(\alpha = 2\), for the numerical example considered one has \(\beta _0 = 0.1095\), while for \(\alpha = 4,\) the corresponding value is \(\beta _0 = 0.0013\).

*L*at the critical points \((n^{**}, H^{**}= \beta ^{\frac{1}{1-\alpha }})\) are: \(L^{**}_{11} = 0, \,\, L^{**}_{12} = 3 (\alpha - 1) n^{**}\),

Four curves \(T^{**}(n^{**})\) with different \(\beta \) are shown on Fig. 3c, together with the curve \(Q(n^{**})\) which starts at point \((0, -2m_0/5)\), crosses the \(n^{**}\)-axis at \(n_0^{**} = m_0/(2B)\) and has a horizontal asymptote at 2*B* / *A*. When \(\beta > \beta _0= (12B/m_0^2)^{(\alpha - 1)/2}\), the curve \(T^{**}(n^{**})\), marked with **(i)** on Fig. 3c, intersects the \(n^{**}\)-axis at points \(\nu _{1,2}^{**}\). The \(n^{**}\)-coordinates of the intersection point of \(T^{**}(n^{**})\) with the curve \(Q(n^{**})\) are \(\sigma _{1,2}^{**}\). As, while negative, \(T^{**}(n^{**})\) cannot intersect the strictly positive \(T(n^{**})\), no critical points \((n^{**}, H^{**} = \beta ^{1/(1-\alpha )})\) can exist for \(T^{**}(n^{**}) < 0\). Thus, stable critical points for \(\beta > \beta _0\) exist in the interval \(\nu _{2}^{**}< n^{**} < \sigma _2^{**}\) – where the non-negative \(T^{**}(n^{**})\) is smaller than \(Q(n^{**})\). When \(\beta = \beta _0\), the curve \(T^{**}(n^{**})\), marked with **(ii)** on Fig. 3c, is tangent to the \(n^{**}\)-axis at point \(\chi _{1}^{**} = n_0^{**}\) – the point at which \(Q(n^{**})\) crosses the abscissa. This curve intersects the curve \(Q(n^{**})\) further – at point \(\chi _{2}^{**}\). Critical points for which \(\chi _{1}^{**}< n^{**} < \chi _2^{**}\) are stable.

When \(\beta _Q< \beta < \beta _0\), curve \(T^{**}(n^{**})\), marked with **(iii)** on Fig. 3c, never intersects the \(n^{**}\)-axis. It intersects the curve \(Q(n^{**})\) at points with \(n^{**}\) coordinates given by \(\xi _{1,2}^{**}\). Critical points with \(\xi _{1}^{**}< n^{**} < \xi _2^{**}\) are stable. Finally, when \(\beta < \beta _Q\), curve \(T^{**}(n^{**})\), marked with **(iv)** on Fig. 3c, never intersects the \(n^{**}\)-axis or the curve \(Q(n^{**})\). There are no stable critical points in this case.

For the dynamical system in the case of \(\alpha = 2\), three sub-cases are considered: \(\beta = 0.02\) (Fig. 4), \(\beta = 0.05\) (Fig. 5), and \(\beta = 0.1\) (Fig. 6). With these, all qualitatively different possibilities are analyzed. The case of \(\alpha = 4\) is similar – see Fig. 7 where some representative cases are shown. The two Tables at the end should also be considered as all possibilities for the model parameters are summarized there and references are given to the corresponding figures.

Many of the trajectories exhibit inflationary regime (Figs. 4, 5, 6, 7). This happens in the upper half-plane (\(H > 0\)) and while *H* is increasing (\({\dot{H}} > 0\)), thus \(\ddot{a} > 0\). The un-physical trajectories that diverge to \((n \rightarrow \infty , H \rightarrow \infty )\) have eternal inflation, while all other trajectories with inflation, after exiting their inflationary regimes, either extinguish at the origin \((n \rightarrow 0, H \rightarrow 0)\) in infinite time (Big Freeze); or at a stable critical point in infinite time; or diverge to a Big Crunch: \((n \rightarrow \infty , H \rightarrow -\infty )\).

## 4 Conclusions

A cosmological model with two matter components – dust and gas with van der Waals equation of state has been examined. In addition, the model includes a particle production term, proportional to a constant power, \(\alpha \), of the Hubble parameter *H*. Models with \(\alpha =2\) and \(\alpha = 4\) are studied in detail. However, the presented analysis can easily be extended to an arbitrary integer \(\alpha \) (the special case of \(\alpha =1\) deserves a special attention and will be provided elsewhere).

*n*, the Hubble parameter

*H*and the temperature

*T*. This system admits a global first integral, which explicitly gives

*T*as a function of

*n*and one of the van der Waals gas parameters. Hence, the system is reduced to a two-component one: in the two dimensional

*n*–

*H*phase space. The system exhibits a complex behavior which is influenced by the presence of the several model parameters. This behaviour is examined in detail using the phase-plane analysis for all possible parameter choices. The two second integrals of the system are represented by curves which separate the phase space into domains which can not be crossed by the trajectories. The full classification of the critical points is presented in the two provided tables. It is shown that the critical points can not be reached in a finite time (the stable critical points can only be reached for \(t\rightarrow \infty \), the unstable critical points can be reached only for \(t \rightarrow -\infty .\)). The critical points provide important information about the large-time behaviour of the system. This includes both the distant future (\(t \rightarrow \infty \)) or the distant past (\(t \rightarrow -\infty \)). For example, considering trajectories which end at the origin, i.e. \((n,H) \rightarrow (0,0)\) as \(t \rightarrow \infty \), from (8) one has \({\dot{n}}=-3nH\) asymptotically when \(\alpha \ge 2\) and taking into account (10), it follows that \(d \rho _d / dn = \rho _d / n,\) or \(\rho _d = C n \) for some constant

*C*. Then

Finally, sets of initial values can be identified for which the corresponding trajectories exhibit inflationary behavior.

## Footnotes

- 1.
To aid the analysis, a numerical example is presented in this work. It is for van der Waals gas, the parameters of which are \(A = 1/100\) and \(B = 10\).

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