The unphysical character of dark energy fluids

It is well known that, in the context of general relativity, an unknown kind of matter that must violate the strong energy condition is required to explain the current accelerated phase of expansion of the Universe. This unknown component is called dark energy and is characterized by an equation of state parameter $w=p/\rho<-1/3$. Thermodynamic stability requires that $3w-d\ln |w|/d\ln a\ge0$ and positiveness of entropy that $w\ge-1$. In this paper we proof that we cannot obtain a differentiable function $w(a)$ to represent the dark energy that satisfies these conditions trough the entire history of the Universe.


I. INTRODUCTION
Between the end of the twenty century and the beginning of the twenty one's, a large amount of observational data revealed that the Universe is currently expanding at an accelerated rate [1][2][3][4]. This accelerated expansion can be easily explained if a cosmological constant is added to the Einstein field equations. It can be shown that the zero point energy of all fields filling the Universe acts into the Einstein equations as a cosmological constant [5]. However, the observational constraints on the cosmological constant differs from the theoretical value provided by quantum field theory by at least 60 orders of magnitude [6][7][8]. Although many proposals to solve this problem appeared in the literature [9][10][11][12][13][14][15][16] none of them provide a real conclusive solution to the problem. The lack of a convincing explanation for the cosmological constant problem has led physicists to adopt a pragmatic approach which assume that the cosmological constant is canceled out by some unknown symmetries in nature and that the accelerated expansion is due to some unknown kind of matter. In order to obtain an accelerated expansion of the Universe at present time, the matter content of the Universe must violates the strong energy condition, i. e., where ρ i and p i are, respectively, the energy density and * Electronic address: ronaldocesar@uern.br † Electronic address: edesiobarboza@uern.br ‡ Electronic address: evertonabreu@ufrrj.br § Electronic address: jorge@fisica.ufjf.br the pressure of the i−th component of the matter content of the Universe and c the speed of the light. The violation of the strong energy condition implies that the total pressure must be negative. Since baryonic and cold dark matter are pressureless (p m = 0), and the pressure of relativistic matter is ρ γ c 2 /3, the Universe must contain an additional source term with a pressure sufficiently negative to ensure the validity of (1). This unknown source was dubbed dark energy (DE) (see Ref. [17] for a review). DE is frequently characterized by the equation of state (EoS) parameter w = p/ρc 2 which mimics a cosmological constant if w = −1, a quintessence scalar field if −1 ≤ w ≤ 1 [18], a phantom field if w < −1 [19] and many other forms of exotic matter.
Since most DE models are able to adjust the data seamlessly, it is extremely difficult to decide which of these models is correct, if there is one. The data shows only that the Universe is expanding at an accelerated rate but does not reveal which object causes this acceleration, i.e., if it is DE or something else. However, although unknown, DE is not immune to the laws of physics. Such an exotic fluid must satisfy the bounds imposed by the laws of thermodynamics which have a strong experimental basis. Here we investigate the limits imposed by thermodynamics to a DE fluid. We proof that thermodynamics rule out DE fluids. This paper is organized as follows: Section II summarizes the main results of the thermodynamical of cosmic fluids contained in Ref. [20]; Section III contains the proof that there is no EoS parameter w(a) of DE satisfying the thermodynamical through the entire history of the Universe: there is at least one point of discontinuity at 0 < a(t) < 1 where w(a) is not a differentiable function and where the stability condition fails. Section IV contains our conclusion.

II. THERMODYNAMICS OF THE COSMIC FLUIDS
In this section we will analyze the thermodynamical properties of the function that describes DE cosmic fluids. The objective is to construct, based on these "heat" properties, a general function to analyze afterwards, its mathematical viability.

A. Internal energy and entropy
Let us consider an expanding, homogeneous and isotropic Universe filled by n no interacting perfect fluids. Since all physical distances scale with the same factor a(t), the physical volume of the Universe at a given time is V = a 3 (t)V 0 1 . In such a model the internal energy of the ith fluid component can be written as Assuming a reversible adiabatic expansion, the first law of thermodynamics leads to so-called fluid equation, which expresses the energy-momentum conservation. Assuming that the density is a function of the temperature and volume, i. e., ρ i = ρ i (T i , V ), the fact that dS i is an exact differential implies that [21] d ln or, using Eq. (4) to eliminate w i , Integration of Eq. (6) provides Thus, the internal energy of the ith fluid component (2) can be written as The entropy of the ith fluid component is obtained 1 Here the index 0 will denote the present time value of an observable and we will adopt the convention a 0 = 1 from the Euler relation [22]: where µ i and N i are, respectively, the chemical potential and the number of particles of the ith component. By assuming that the chemical potential is zero we obtain, combining Eqs. (2), (7) and (9), that which shows the direct relation between the entropy and the EoS parameter, w i , for a given component of the cosmic fluid.

B. Heat capacity
The classical thermodynamical definition of a fluid's heat capacity C i is [23], where dT i is the fluid temperature increase due to an absorbed heat dQ i = T i dS i . The heat capacity of a fluid will differ depending on whether the fluid is heated at constant volume or at constant pressure. From the first law of thermodynamics, Eq. (3), at constant volume, Eq. (11) becomes where is the fluid's heat capacity at constant volume. From Eq. (8), it is easy to show that for any component of the Universe. Now, from the enthalpy definition, the first law of thermodynamics can be written as Therefore, at constant pressure, Eq. (11) becomes where is the fluid's heat capacity at constant pressure. Since

the enthalpy Eq. (15) becomes
and, from Eqs. (8) and (5), we have that which shows that we can write a compact relation between the heat capacities at constant pressure and constant volume, something like C pi = Ω(w i )C iV , where Ω(w i ) is a function of the EoS parameter, defined in the last equation.

C. Compressibility and expansibility
By considering the volume as function of temperature and pressure, we have that 2 where is the thermal expansibility and is the isothermal compressibility. The thermal expansibility measures the thermal volume expansion at constant pressure and isothermal compressibility measures the relative modification of the volume together with the increasing pressure at fixed temperature.
Analogously to the isothermal compressibility, we can define the adiabatic compressibility κ Si if, instead of temperature, the entropy is kept fixed. It can be shown that the isothermal compressibility and the isothermal expansibility are related by and that the ratio between the adiabatic and the isothermal compressibilities are equal to the ratio between the heat capacities at constant volume and at constant pres-sure, i.e., Notice that p i V = w i C iV T i and using Eq. (5) we obtain From (24) is easy to show that and from the above equation and (25) we have and, differently from Eq. (10), that shows a direct relation between the entropy and the EoS parameter w i , Eq.
(28) shows a much more intricate relation between the adiabatic compressibility and w i Substituting Eq. (26) into Eqs. (27) and (28) we have and which can be used to determine the constraints on cosmic fluids EoS parameter w i . We will define some of these constraints just below.

D. Constraints on cosmic fluids
Thermodynamical stability requires that C iV , C pi , κ Si , κ Ti ≥ 0 simultaneously.
Conversely, these quantities are all negative simultaneously if the stability cannot be obtained. From Eq. (14) it is clear that C iV ≥ 0 so that the cosmic fluids satisfies the stability conditions. Moreover, C iV , C pi , κ Si , and κ Ti are related by so that C pi ≥ C iV and κ Ti ≥ κ Si . From Eqs. (29) and (30) it is easy to see that the above conditions are satis-fied only if the fluid EoS parameter obeys the constraint Along with the above constraint, the positiveness of entropy implies that It is obvious that if w i is constant, thermodynamical stability implies that w i ≥ 0 which rule out all negative pressure fluids with a constant EoS parameter. In order to accelerate the Universe at present time is also required by (1) that w(a → 1) < −1/3.

III. THERMODYNAMICAL INVIABILITY OF DARK ENERGY FLUIDS
In this section we will investigate the existence of functions w(a) that satisfies the following conditions: Conditions i) and ii) are thermodynamical constraints on all cosmic fluids physically acceptable, and condition iii) is required to accelerate the Universe at present time. Below we will investigate the existence of differentiable functions, except an a finite set, i. e., functions w such that the set {t; w is not differentiable at t} is finite, whose derivative is continuous for t sufficiently small and that satisfies the condition i), ii) and iii) above. We will show that there is no function w(a) of C 1 class satisfying the conditions i), ii) and iii). In our proof, we will use the follow theorems (see [24] for the proofs): Proposition 1 Let w : (0, +∞) → R be a function and suppose that there is δ > 0 such that w is differentiable and its derivative is continuous at (0, δ) with If there is t 0 ∈ (0, δ) such that w(t 0 ) < 0 then lim t→0 w(t) = −∞.
Proof 1 Let δ > 0 with w differentiable and with a continuous derivative at (0, δ). Suppose that there is t 0 ∈ (0, δ) such that w(t 0 ) < 0. Let c < 0 such that w(t 0 ) < c < 0. w is continuous at (0, δ), since it is differentiable in this interval and, therefore, we can suppose that at t ∈ (t 0 − , t 0 + ) we have w(t) < c for with t 0 + < δ. We affirm that w(t) ≤ c for all t ∈ (0, t 0 + ). In fact, otherwise we would take t = sup {t ∈ (0, t 0 + ); w(t) > c} and we would have . and, by supremum definition, that at t ∈ (t, t 0 + ), in this interval. But, by the Theorem 2, there would exist s ∈ (t, t 0 ) such that and that would be contradictory.
Proof 2 If there were t 0 ∈ (0, δ) such that w(t 0 ) < 0, then by the previous proposition we wold have that lim s→0 w(s) = −∞. However from hypothesis 2 above this could not occur.
Corollary 2 Let w : (0, +∞) → R be a differentiable function at (0, δ) and with continuous derivative in this interval for some δ < 1. Additionally, suppose that the set {t ∈ (0, 1); w non differentiable at t} is finite and that Then w is discontinuous at some point t ∈ (0, 1).

Proof 3
Suppose that there is a w, differentiable at (0, δ) with continuous derivative in this interval, that it satisfies i), ii) and iii) and that w is continuous at (0, 1). From where w 0 is given by hypothesis iii). From Theorem 1, there is a t 1 ∈ (t 0 , 1) such that w(t 1 ) = d. Let us define The set A is not empty, given that t 1 ∈ A. Let t = sup A (sup A is the supremum of the set A). Using the continuity condition and iii) there is > 0 such that for all t ∈ (1 − , 1), w(t) < d. This imply that t ≤ 1 − < 1. For all t ∈ (t, 1) we have w(t) < d < 0, in particular, by i) Let t 1 , t 2 , ..., t n be the points of the interval (t, 1) such that w is not differentiable. Let us suppose, without lost of generality, that t 0 = t < t 1 < t 2 <, ..., < t n < 1 = t n+1 . Note that, by using Eq. (34), the function w is an increasing one at (t i , t i+1 ) for all i = 0, 1, 2, ..., n + 1. Therefore, for all s, t ∈ (t i , t i+1 ) we have w(s) < w(t) if s < t. From the continuity of w at t i we have and from the continuity of w at t i+1 we have for all i = 0, 1, 2, ...n. By the inequalities in Eq. (35), the supremum definition and the continuity of w, we have This inequality is in contradiction with the inequality in Eq. (33). This result proofs the corollary.
Corollary 3 There is no differentiable function w : (0, +∞) → R with continuous derivative at (0, δ) for some δ > 0 which satisfy: Proof 4 This corollary is a straightforward consequence of the previous corollary and of the fact that every differentiable function is continuous.
This result proofs our assertion that there is no function of C 1 class that explain the present time accelerated expansion of the Universe and at same time satisfies the thermodynamical bounds. It should be noted that if the continuity condition is relaxed, it is possible to build functions w(a) that fulfill the conditions i), ii) and iii) above except on a finite set of points where w(a) is not differentiable. However, physically speaking, we have no reason to think that w is not differentiable. 3 . Also, in these discontinuity points the stability condition fails. Thermodynamics do not allows such exceptions.
Nevertheless, it is interesting for the sake of completeness obtain an example of a differentiable function except from a set of finite points that satisfies the conditions i), ii) and iii). As we have seen, in this case we should look for a positive function for t sufficiently small and with at least one discontinuity point in the interval (0, 1). In the next proposition we will show a function with these characteristics.  Proof 5 The function w is decreasing at (0, e d ). In this case, and given that w > 0 in this interval for all a ∈ (0, e d ). If a > e d , then Therefore the function w satisfies the condition 1. Note that e d < 1 and that w is continuous at 1. Note that the discontinuity point of the functions shown in the previous proposition occurs at the point e d such that e d ∈ (0, 1), since d < 0.
Corollary 4 Let us consider 0 > w 0 > −1 and d = (w 0 +1)/(3w 0 ). Let us define the function w : (0, +∞) → R such as This EoS parameter mimics a quintessence scalar field (−1 ≤ w(a) ≤ 1), but the stability condition fails at just an point: e d . This is the closest definition we can obtain to a DE fluid that mimics a quintessence scalar field since the thermodynamical bounds implies that any DE EoS parameter will have at least one discontinuity point at (0, 1). Figure 1 shows the curves from the above function for w 0 = −1/3, −2/3 and −20/21. As we can see, the discontinuity approaches of a = 1 as w 0 approaches of −1. In fact, as will be show below, even if we allow that the EoS parameter w to be differentiable except on a finite number of points, it is impossible to build a quintessential like EoS which mimics a cosmological constant at present time (w(a → 1) → −1) and satisfies the thermodynamical constraints.
Proposition 3 There is no function w : (0, +∞) → R, such that 1 is an isolated point from the set of points in which w is differentiable and that satisfies: 3. −1 ≤ w(a) ≤ 1 for all a > 0.
Proof 7 Suppose by contradiction that there is a function that fulfill all conditions described in the proposition 3. Let J be an interval centered at 1 such that w is differentiable at J \ {1}. It is possible to choose this interval since 1 is a isolated point from the set of points in which w is differentiable. Given that lim a→1 w(a) = −1 then we can suppose, without lost of generality, that w < 0 at