To answer the question posed in the previous section, one has to think of what could possibly act as a counterpart to the super-exponential expansion of the universe caused by the phantom energy³. In this work, the attention is turned to a phenomenon called the Schwinger effect. This phenomenon, first derived by Julian Schwinger in 1951, is the production of a particle-antiparticle pair out of the vacuum in the presence of a strong electric field [28]. The spectrum of the produced pairs is given by the formula [29] [30]

d n S ( p ) d 3 p → ≈ θ ( p z ) θ ( q e E x 0 − p z ) exp ( − π ( p T 2 + m 2 ) q e E ) , (7)

with p T the transverse momentum, x 0 the time component of the spacetime 4-vector, x μ = ( x 0 , x → ) , m the mass, qe the charge of each of the produced particles of the pair and E the electric field.

It is easy to observe that one should create a very strong electric field, i.e. q e E ≫ ( p T 2 + m 2 ) , in order for the exponential factor in Equation (7) not to suppress the pair production from the vacuum. Such large electric fields are difficult to be produced in the laboratories which is why the Schwinger effect has not yet been observed in Nature. However, one may still try to think how this effect could potentially play a role on a cosmological scale. To make this connection, the horizon temperature T h for the case of de Sitter spacetime is defined [31] [32] [33].

T h : = κ h 2 π = H 2 π , (8)

where κ h is the acceleration of gravity at the surface of de Sitter horizon and is proportional to the Hubble rate H in the case of de Sitter spacetime. Connecting the relation (8) with the Unruh temperature of a uniformly accelerated observer [34], one can find the following relation between the gravitational acceleration k h of a particle of mass m, charge qe and transverse momentum p T with the electric field E [35] [36] ⁴

κ h = q e E p T 2 + m 2 . (9)

The combination of the Equations (8) and (9) results in

q e E = 2 π T h p T 2 + m 2 = 2 π H p T 2 + m 2 . (10)

One may calculate the contribution of the gravitational Schwinger effect to the energy density. According to the definition of the energy density, this is as following

ρ S = ∫ d 3 p →   n S ( p ) p → 2 + m 2 , (11)

Then using the relations (7) and (10), (11) becomes

ρ S = ∫ 0 ∞ d ( p T 2 ) ∫ 0 T h p T 2 + m 2 x 0 d p z p T 2 + p z 2 + m 2 exp ( − p T 2 + m 2 2 T h ) . (12)

Assuming light-produced particles ( m ≈ 0 ) (12) becomes

ρ S ≈ C S H 4 , (13)

with C S > 0 the respective coefficient. This is a radiation term, which is expected given the light-produced particles and whose energy density scales like T 4 according to the Stefan-Boltzmann Law. However, in the case of the gravitational production of particles, the coefficient does not have to be the same as in the Stefan-Boltzmann Law.⁵

Considering the contributions of the phantom energy and the Schwinger effect, the Friedmann Equation (2) becomes

H 2 = 8 π G 3 ( ρ D E + ρ S ) = 8 π G 3 ( ρ D E + C S H 4 ) . (14)

Simultaneously, considering (13) and that ρ ˙ S = d ρ S d t ~ H 4 H − 1 = H 5 , the continuity Equation (6) becomes

ρ ˙ D E − 3 H | w D E + 1 | ρ D E + C ˜ S H 5 = 0   , (15)

where it is assumed that w D E < − 1 is a constant. Also C ˜ S is a constant function of C S and w D E ⁶.

The analysis above has been done in the context of an almost de Sitter space, although w D E < − 1 . This is a reasonable assumption because, even if w D E < − 1 , still it should be close to −1, according to the observations, as discussed in paragraph (1). Additionally, there is no concrete and universally accepted definition of the surface gravity for general curved spacetimes [37], so (8) could not be used for the general case of w D E < − 1 . Also for our model we assume that there no catastrophic instabilities, which can be justified in the context of [5], as mentioned in the paragraph (1).

It is noted that the phantom fluid does not evolve independently from the Schwinger pair production, so the continuity Equation (6) does not hold independently for each component. One can see that the Schwinger effect contributes with a term proportional to H⁴ and H⁵ in (14) and (15) respectively. These terms become important to higher values of H, therefore later in the evolution of the universe if it is dominated by a phantom substance. Even though at later times the solutions of (14) and (15) start deviating from the de Sitter case, one expects gravitational particle production to occur whenever there is an event horizon (as with Hawking radiation in the case of backholes [31]). Therefore, one would expect at least a radiation term proportional to H⁴ and a source term proportional to H⁵ in the Friedmann and continuity equations as in (14) and (15), independent of whether the spacetime is described by a metric close to de Sitter or not.

The goal now is to solve the Equations (14) and (15) simultaneously. First of all, the Equation (14) has two solutions in H²

H 2 = 3 16 π G C S ( 1 ± 1 − 256 π 2 G 2 C S 9 ρ D E ) . (16)

In (16), only the solution with the relative minus sign is kept, since this is the one that reduces to H 2 ≈ 8 π G 3 ρ D E for small H, where the H⁴ term coming from the Schwinger effect in (14) is negligible⁷. Additionally, this solution has an upper value for ρ D E , which is ρ D E , m a x = 9 256 π 2 G 2 C S . This leads to a maximum value of H², which is H m a x 2 = 3 16 π G C S ⁸. Setting also R D E ≡ ρ D E / ρ D E , m a x , the solution with the relative minus sign in Equation (16) becomes

H 2 = H m a x 2 ( 1 − 1 − R D E ) . (17)

Using the Equation (17), the continuity Equation (15) becomes

d R D E d t = 6 | 1 + w D E | H m a x ( 1 − 1 − R D E ) 1 / 2 ( R D E + 1 − R D E − 1 ) . (18)

Integrating the Equation (18) and using the fact that R D E ( t m a x ) = 1 results in

ρ D E ρ D E , m a x = 1 − [ 1 − 1 ( 1 + 3 2 | 1 + w D E | H m a x ( t m a x − t ) ) 2 ] 2 . (19)

Putting (19) into (16) leads to

H H m a x = 1 1 + 3 2 | 1 + w D E | H m a x ( t m a x − t ) . (20)

From Figure 1 and Figure 2 one can see that the energy density and Hubble rate do not reach infinite values in a finite amount of time, as it would happen

in the case of a Big Rip scenario. On the contrary, they reach the maximum values ρ m a x and H m a x respectively as t → t m a x and therefore the Big Rip singularity is avoided. This happens thanks to the ever increasing rate of Schwinger pair-production that an observer inside a causal horizon of radius r H ~ H − 1 would observe as the Hubble rate increases.

It is also important to have a qualitative understanding of the evolution of the horizon entropy in the scenario discussed in this paper. Its calculation is quite straightforward. Using the analogy between the thermodynamics of black holes and cosmological horizons, the Generalised Second Law (GSL) for black holes [38] [39] [40] extended to de Sitter horizons is [32] [33]

Δ ( S o u t s i d e + S H ) ≥ 0, (21)

S H ∝ A H ∝ r H 2 ∝ H − 2 , (22)

where S H is the horizon entropy, S o u t s i d e is the entropy outside the horizon, A H is the area of the de Sitter horizon and r H is the Hubble radius⁹.

The relations (21) and (22) have been proven to hold true also in the case of accelerated horizons and are independent of whether the horizon area A H increases or decreases, as long as the rate of increase of the entropy outside the horizon S o u t s i d e outweighs the rate of decrease of the horizon entropy S H [41].

In the case analysed in this paper, where the universe is dominated by the phantom fluid, the horizon entropy decreases because of the increase of the Hubble rate, until it reaches asympotically a minimum constant value S H , m i n as t → t m a x which is

S H , m i n ∝ H m a x − 2 ∝ C S   . (23)

One observes that the entropy does not reach a zero value thanks to the particle-pair production because of the Schwinger effect. When S H reaches its minimum value S H , m i n ~ O ( C S ) in the far-future, the universe enters into a de Sitter phase of constant and high Hubble rate H m a x and energy density ρ m a x . The fact that the horizon entropy decreases is not a problem as long as the causal patch of an observer is not an isolated system and the rate of increase of the entropy outside the horizon outweighs the rate of decrease of the horizon entropy, as discussed above¹⁰.

Using (17) and (23) results in the following evolution of the horizon entropy (22) (Figure 3)

S H S H , m i n = ( 1 + 3 2 | 1 + w D E | H m a x ( t m a x − t ) ) 2 . (24)

The combination of the results of paragraphs (2.1) and (2.2) leads to the following conclusion: If the universe is dominated by a phantom fluid with w D E < − 1 and if the gravitational Schwinger pair-production takes place, the Big Rip singularity could be avoided in the far-future. In particular, because the ρ = ρ D E , m a x and H = H m a x are solutions to the Equations (14) and (15), the energy density and the Hubble rate would remain constant after they reach these values in the far-future, according to the analysis in this paper, and the universe could start a phase of de Sitter inflation. Then, by taking one more step of speculation, the phantom substance could decay through an unknown hypothetical mechanism, which would lead to the reheating of the universe and to a new Big Bang; thus the beginning of a new cycle of the universe. Of course, the specific characteristics of the decay of the phantom fluid to matter and radiation would have to be understood and this is beyond the scope of this paper¹¹. This idea can be seen graphically in Figure 4.