Twinlike models for parametrized dark energy

We study cosmological models involving a single real scalar field that has an equation of state parameter which evolves with cosmic time. We highlight some common parametrizations for the equation of state as a function of redshift in the context of twinlike theories. The procedure is used to introduce different models that have the same acceleration parameter, with the very same energy densities and pressure in flat spacetime.


Introduction
Cosmological observations of Type Ia Supernovae and Cosmic Microwave Background suggest that the Universe had started to accelerate its expansion at the present epoch [1][2][3][4][5][6]. The standard explanation refers to an exotic component which has positive energy density and negative pressure, known as "dark energy" (DE). A variety of theoretical models has been proposed to explain this acceleration. The most natural and simplest model for DE is the Λ CDM model, containing a mixture of cosmological constant Λ and cold dark matter (CDM), for which the equation of state parameter is ω = −1 [7][8][9]. However, this model suffers from two major problems, namely, fine-tuning and cosmological coincidence problems [9][10][11].
The same idea was applied in the context of Friedmann-Robertson-Walker (FRW) cosmology [35], in a previous paper. In the present work, we extend this view and consider some popular DE parametrizations for canonical and tachyonic scalar field models. We find that the two models present the very same acceleration parameter, with the same energy density, and we name them twinlike models.
The basic concepts are presented in Sec. 2. In Sec. 3 we investigated the twin nature of the standard and tachyonic models. In Sec. 4 we present some illustrations. The paper ends with a summary in Sec. 5.

Einstein equations
In order to investigate this proposal, we present some basic theoretical considerations. The action for a universe with spacetime curvature R, filled with a scalar field φ and containing matter, is given by where we have made 4πG = c = 1, X = 1 2 ∂ µ φ ∂ µ φ , and S m is the action of the matter.
The metric representing a homogeneous, isotropic and spatially flat universe is the FRW metric ds 2 = dt 2 − a 2 (t) dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 (2) where a(t) is the scale factor of the universe, r is the radial coordinate and dΩ 2 = dθ 2 + sin 2 θ dφ 2 describes the arXiv:2003.14351v1 [astro-ph.CO] 31 Mar 2020 angular portion of the metric. In this scenario, the Einstein equations are where ρ φ and p φ are respectively energy density and pressure of the scalar field φ , ρ m represents the energy density of the matter component of the universe, H =ȧ/a denotes Hubble parameter, and an overdot indicates differentiation with respect to time t. The conservation of the scalar field and matter is represented respectively by the equations of continuitẏ and the cosmic acceleration parameter is given by Rewriting the equations in terms of redshift z = a 0 a − 1, from (5) and (6), we obtain where ω φ = p φ /ρ φ is the dark energy EoS parameter and the subscript 0 indicates the present epoch. The Friedmann equations then takes the form where Ω m0 = 2ρ m0 3H 2 0 and are the density parameters of the matter and scalar field, respectively, at the present epoch. The acceleration parameter is also rewritten as 3 The twinlike models

Standard case
If the scalar field (dark energy) is described by the standard dynamics, we have where V (φ ) is the potential of the scalar field. In this case, energy density and pressure are given by and the scalar field evolves as follows From (14) and (15), we express the potential, and we write an equation for the scalar field, both in terms of redshift z.

Tachyonic modified case
Let us now consider the scalar field described by tachyonic dynamics. We change the model as follows where U(φ ) and f (φ ) are functions to be determined. Energy density and pressure are now given by and the scalar field obeys From (20) and (21), in terms of redshift z, we obtain and

The twin nature
In order to get to twinlike models, we need to make the appropriate choice for f (z). In this case we consider So, the modified potential takes the form In both cases (standard and modified), we have the same energy density, given by (8), and the same pressure. The scalar field is also the same in both cases, being The acceleration parameter also has the same form, given by (12), and the Friedmann equations have the same evolution in both cases. The models are twin.

Cosmological constant
As a first example we take ω φ (z) = ω 0 , a cosmological constante, in the limit −1 < ω 0 < − 1 3 . In this situation, the energy density of scalar field is written as So the potentials of standard and modified cases are, respectively, The evolution of the Hubble parameter with the redshift is given by the Friedmann equation And the acceleration parameter is The density parameters of the matter and scalar field are respectively With the help of Eq. (28), we can solve Eq. (27) for the scalar field analytically. The result is Figure 1(a) shows the plot of φ (z). Equations (29) and (30) express V and U as functions of z. It is very difficult to work with these potentials in terms of φ . Figure 1(b) shows the plot of V (φ ) and U(φ ) from numerical results. The V and U curves are clearly distinct, but the twin nature is shown in the graph of the acceleration parameter q, which is the same for both models. The plot of q(z) in Figure 1(c) shows the transition from a decelerating to an accelerating regime as z decreases. The evolutions of Ω φ and Ω m are showns in Figure 1(d). Note that Ω φ starts dominating over Ω m at around z ∼ 0.4.

Linear parametrization
As a second example we now consider ω φ (z) = ω 0 + ω 1 z [36,37]. In this case, the energy density of scalar field take the form So the potentials of standard and modified cases are, respectively,

38) The Friedmann equation is given by
The density parameters of the matter and scalar field are, respectively, (42) Figure 2 shows the plots of (a) φ (z), (b) V (φ ) and U(φ ) from numerical results. The plot of q(z) in Figure 2(c) shows also the transition from a decelerating to an accelerating regime as z decreases. The evolutions of Ω φ and Ω m are shown in Figure 2(d), and Ω φ starts dominating over Ω m at around z ∼ 0.4.

Chevallier-Polarski-Linder (CPL) parametrization
The CPL parametrization [19,20,38] is characterized by The energy density of scalar field is The potentials of standard and modified cases are, respectively, The Friedmann equation is The acceleration parameter is given by The density parameters of the matter and scalar field are, respectively, (50) Figure 3 shows the plots of (a) φ (z), (b) V (φ ) and U(φ ), (c) q(z), (d) Ω m and Ω φ . Once again, the distinction between the models is evidenced in the graphs of V and U, as well as the twin nature of these models requires that the curve of the acceleration parameter q be the same for both cases.

Barboza-Alcaniz (BA) parametrization
The last example is the BA parametrization, proposed by Barboza and Alcaniz [16,39], which is represented by In this case, the energy density of scalar field is The potentials of standard and modified cases are, respectively, The Hubble parameter evolves as follows The acceleration parameter is then The density parameters of the matter and scalar field are, respectively, (58) Figure 4 shows the plots of (a) φ (z), (b) V (φ ) and U(φ ), (c) q(z), (d) Ω m and Ω φ . Discussion similar to previous ones can be performed around these graphs. The transition between Ω m and Ω φ occurs again around z ∼ 0.4.

Summary and conclusions
In this work we studied the presence of twinlike models in FRW cosmology driven by a single real scalar field, in flat spacetime, from the study of some common parametrizations for the equation of state parameter ω(z). We showed that, regardless of the choice of ω(z), it is always possible to have models driven by standard and tachyonic dynamics with the same acceleration parameter, the same energy density and the same pressure.