1 Introduction

The discovery of the accelerating expansion of the universe [1, 2] is one of the most intriguing and puzzling questions in cosmology today [3,4,5,6,7]. There are several attempts to address this question and these include: the cosmological constant, quintessence field [8,9,10,11], a brane cosmology scenario [12, 13] and k-essence models [14,15,16,17,18,19,20,21]. Among the former proposals the cosmological constant is the simplest choice, although it needs a fine tuning value. There are different approaches that try to model the late-time accelerated expansion of the universe including modified gravity at large distances via extra dimensions or including a dark energy component that allows accelerating universes by means of a kind of energy that does not fulfill the strong energy condition [22].

Scalar fields models with non-canonical kinetic terms have been proposed as an alternative to describe the dark energy component of the universe. The k-essence type Lagrangians were introduced in several contexts, for example, as a possible model for inflation [14, 23]. Later on k-essence models were analyzed as alternative descriptions of the characteristics of dark energy and as possible mechanisms of unifying dark energy and dark matter [24]. Purely kinetic k-essence models [25,26,27,28] are, in some sense, as simple as quintessence because they depend only on a single function F by means of the expression for the density Lagrangian \(\mathcal{{L}} = F(X)\), where the kinetic term is X. It is very interesting to note that the form of the purely k-essence Lagrangian appears in superconducting membranes where it is constructed by means of a spontaneously symmetry breaking of an underlying field theory [29,30,31,32]. The scalar field is related to the U(1) symmetry breaking corresponding to the electromagnetic field and it is responsible for the existence of permanent currents. This kind of fields has been studied for superconducting strings and walls [33,34,35]. Following the idea of Rubakov and Shaposhnikov [36], who postulated that our universe can be viewed as a topological defect, we want to introduce the k-essence field in the form of a superconducting membrane field. In this proposal we can start with a 5-dimensional field theory and follow a similar pattern for the formation of a superconducting membrane. We postulate that our universe can be viewed as a topological defect and the k-essence field emerges in a similar fashion as a scalar field from a spontaneously symmetry breaking of the underlying field theory. It is worth to mention that several equations of state have been studied including \(\rho = p + \mathrm{constant}\) [29] and the so-called permanently transonic or Nielsen model for superconducting strings and membranes of the form \(\rho p =-\mathrm{constant}\) [31, 37, 38]. It is very important to note that this equation of state corresponds to a k-essence Chaplygin gas model [39,40,41].

It has been argued, in the context of superconducting extended objects [33], that a polynomial type equation of state of the formFootnote 1 \(\rho = d + b p + cp^2\) (where d, b, and c are constants related to the mass of the charge carriers and the Kibble mass) is a better equation of state in order to describe some properties of this kind of objects. As a first approach we would like to study this kind of equation of state. However, it remains to develop the complete construction of a 5-dimensional field theory and to obtain the possible modifications that can arise from higher dimensions. In addition, we can consider this kind of equation of state as an expansion in Taylor series of \(\rho (p)\) around an specific value of the pressure. Inspired by these ideas we study in this work a generalized polynomial equation of state in the context of k-essence models giving an additional physical meaning for the scalar field. In this paper we also analyze the adiabatic speed of sound and restrict the allowed values for the parameters included in the equation of state in order to satisfy stability and positive energy and, besides, we find the region for a subluminal speed of sound. Also we study the evolution of the luminosity distance \(\mathrm{d}_{L}\) with redshift z, comparing (normalizing) it with the \(\Lambda \)CDM model and also applying the Alcock–Paczyński test.

This paper is organized as follows. In Sect. 2 we present the essential characteristics of purely kinetic k-essence fields. In Sect. 3 we analyze the polynomial type equation of state and establish the physical allowed region in the space of parameters. In Sect. 4 we study some cosmological properties of the parameters of the model. Finally, in the last section we give our conclusions.

2 Purely kinetic k-essence

The k-essence model was proposed originally in [14,15,16] as a model to describe the accelerated expansion of the universe within the context of inflation. The k-essence field became later a candidate to dark energy [17,18,19], and as such it is defined as a scalar field \(\phi \) with non-linear kinetic energy terms in X according to the action

$$\begin{aligned} S = \int \mathrm{d}^4 x \sqrt{-g}F(\phi , X), \end{aligned}$$
(1)

where \(X = \frac{1}{2}\partial _\mu \phi \partial ^\mu \phi \). Furthermore, in this work we focus attention on purely kinetic k-essence models described by the action

$$\begin{aligned} S = \int \mathrm{d}^4 x \sqrt{-g}F(X). \end{aligned}$$
(2)

The equation of motion for the field \(\phi \) is obtained in the usual way from the last action and is written in the form

$$\begin{aligned} \nabla _\mu J^\mu = 0, \end{aligned}$$
(3)

where the conserved current is [31, 42]

$$\begin{aligned} J^\mu = F_X g^{\mu \nu }\partial _\nu \phi , \end{aligned}$$
(4)

where \(F_{X} \equiv \frac{\mathrm{d}F}{\mathrm{d}X}\). The conservation of the current is equivalent to the conservation of the energy-momentum tensor [43] of the k-essence field,

$$\begin{aligned} T^{\mu \nu } = F_X \partial ^\mu \phi \partial ^\nu \phi - g^{\mu \nu }F. \end{aligned}$$
(5)

We consider the Friedmann–Robertson–Walker (FRW) metric

$$\begin{aligned} \mathrm{d}s^2 = \mathrm{d}t^2 - a^2(t)\mathrm{d}\overrightarrow{x}^2, \end{aligned}$$
(6)

and we taken the field \(\phi \) to be smooth, thus \(X=\frac{1}{2}\dot{\phi }^2\), i.e. X is positive. We have a comoving perfect fluid energy-momentum tensor where the k-essence energy density \(\rho \) and the pressure p are given by

$$\begin{aligned} \rho = 2XF_X - F \end{aligned}$$
(7)

and

$$\begin{aligned} p = F(X). \end{aligned}$$
(8)

The conserved current as a result gives the following relation between the X parameter and the scale factor a:

$$\begin{aligned} XF_X ^2 = k_{0}a^{-6}, \end{aligned}$$
(9)

where \(k_{0}\) is a constant of integration. This solution was derived for the first time by Chimento in [21]. Given a form of F(X), Eq. (9) gives a relation between X and a, then the evolution of the equation of state parameter w and the sound speed \(c^{2}_{s}\) as a function of the scale factor a. The equation of state parameter w in k-essence models has the form

$$\begin{aligned} w = \frac{p}{\rho }= \frac{p}{2Xp_X - p}, \end{aligned}$$
(10)

and the adiabatic speed of sound is given by

$$\begin{aligned} c_s ^2 \equiv \frac{\partial p/\partial X}{\partial \rho /\partial X} = \frac{F_X}{2XF_{XX} + F_X} = \frac{F^{2}_X}{(XF^{2}_{X})_{X}} \end{aligned}$$
(11)

with \(F_{XX} \equiv \mathrm{d}^{2}F/\mathrm{d}X^{2}\). In order to satisfy the stability condition \(c_s ^2 \ge 0\), the last equation imposes the relation \(\frac{\mathrm{d}a}{\mathrm{d}X} < 0\), after using Eq. (9).

3 The model

We consider a generic equation of state of the form

$$\begin{aligned} \rho = d + bp + cp^2, \end{aligned}$$
(12)

where d, b, and c are constants and their range of validity will be determined by means of physical requirements. The form of the last equation of state is inspired by the study of superconducting membranes (see [31]). It is interesting to note here that an analogous equation of the form \(p = \lambda + \delta \rho + \kappa \rho ^2\) has been introduced in [44], was furthermore studied in [45,46,47], and subsequently generalized in [48]. A general review of different fluids models with such a generalized equation of state is given in [49]. The relevant physical quantities \(p, \rho , w\), and \(c_s ^2\) can all be computed using Eqs. (7) and (8) together with Eqs. (10) and (11). In order to obtain the form of F(x) we use Eqs. (7) and (12) so that the equation we have to integrate is the following:

$$\begin{aligned} 2X \frac{\mathrm{d}p}{\mathrm{d}X} = d + (b + 1)p + cp^{2}. \end{aligned}$$
(13)

If \(\Delta \equiv 4cd -(b + 1)^{2}\) the integration of Eq. (13) leads to three different solutions depending on \(\Delta < 0\), \(\Delta > 0\), or \(\Delta = 0\).

3.1 The case \(\Delta <0\)

For this case the following expression for p is found:

$$\begin{aligned} F(X) = p = \frac{1}{2c}\left( \frac{\sqrt{-\Delta } - b - 1 + \tilde{c}(\sqrt{-\Delta } + b + 1)X^{\alpha }}{1 - \tilde{c}X^{\alpha }}\right) \end{aligned}$$
(14)

where \(\tilde{c}\) is an integration constant, and we choose it to be \(\tilde{c} = 1\). Defining \(\alpha \) through the expression \(2\alpha = \sqrt{-\Delta }\), and also \(\beta = \alpha /c\) and \(\gamma = (b+1)/2c\), Eq. (14) finally becomes

$$\begin{aligned} p = -\beta -\gamma + \frac{2\beta }{1 - X^{\alpha }}. \end{aligned}$$
(15)

In the particular case where \(\alpha = 1\) our model coincides with one of the models introduced by Carter and Peter in [33] where they provide a realistic representation of the macroscopic dynamical behavior of Witten-type (superconducting) vortex defects [29].

The energy density could be written in the form

$$\begin{aligned} \rho = \frac{1}{2c}\left( \frac{(2\alpha + b + 1)X^{2\alpha } + (8\alpha ^2 - 2b - 2)X^\alpha + b + 1 - 2\alpha }{(1 - X^\alpha )^2}\right) . \end{aligned}$$
(16)

The equation of state parameter w is found to be

$$\begin{aligned} w = \frac{-(2\alpha + b + 1)X^{2\alpha } + 2(b + 1)X^\alpha + 2\alpha - b - 1}{(2\alpha + b + 1)X^{2\alpha } + (8\alpha ^2 - 2b - 2)X^\alpha + b + 1 - 2\alpha }. \end{aligned}$$
(17)

It is worth to mention that the kinetic k-essence Lagrangian has the scaling property \(\mathcal{{L}} \rightarrow \kappa \mathcal{{L}}\), which leaves the equation of state parameter and the speed of sound unchanged, and as a consequence it gives only two relevant parameters (for example, b and cd).

The expression for the scale factor a as a function of the parameter X is found from Eq. (9) and is given by

$$\begin{aligned} a = \frac{k^{1/6}c^{1/3}}{(2 \alpha ^{2})^{1/3}} \cdot \frac{(1 - X^\alpha )^{2/3}}{X^{(2\alpha - 1)/6}}. \end{aligned}$$
(18)

Taking \(\alpha = 1\) and defining \(r \equiv \frac{k^{1/6}c^{1/3}}{(2 \alpha ^{2})^{1/3}}\) we have finally

$$\begin{aligned} r^{-3/2} \cdot a^{3/2} X^{1/4} + X = 1. \end{aligned}$$
(19)

Our aim is to solve X as a function of the scale factor a and r. Equation (19) admits four solutions, two of them are real, and from these we consider the one physically acceptable.

In Fig. 1 we present the relation between the scale factor a and the parameter X for \(\alpha = 1\). In order to satisfy the stability condition \(\mathrm{d}a/\mathrm{d}X < 0\) the values of X must be restricted to \(0< X < 1\). The following correspondence is observed: when \(X \rightarrow 1\) then \(a \rightarrow 0\) and when \(X \rightarrow 0\) then \(a \rightarrow \infty \) if \(\alpha > \frac{1}{2}\).

Fig. 1
figure 1

Scale factor as a function of X for \(\alpha = 1\). For simplicity, we have chosen the numerical pre-factor in Eq. (18) equal to 1

Full size image
Fig. 2
figure 2

Equation of state parameter w as a function of a for different values of r. All curves correspond to \(\alpha =1\) and \(b=3\)

Full size image

In Fig. 2 we present the evolution of the equation of state parameter w as a function of the scale factor a for different values of r but with a same value for the parameter b and for \(\alpha = 1\). It is observed that at the beginning of its evolution the universe has a dust-type dominant component, furthermore it evolves into radiation (\(w=1/3\)), then again has a dust-type (matter) component, and finally it tends asymptotically to a cosmological constant (\(w=-1\)). On the other hand, in Fig. 3 it is represented the evolution of w as a function of the scale factor a for different values of b with a same value for r. In Fig. 4 the evolution of the state parameter w is shown as a function of a for various values for \(\alpha \) and the parameter b.

Fig. 3
figure 3

Equation of state parameter w as a function of the scale factor a for different values of b. All curves correspond to \(\alpha =1\) and \(r=0.145\)

Full size image
Fig. 4
figure 4

Equation of state parameter w as a function of the scale factor a. All curves correspond to \(r=0.15\)

Full size image
Fig. 5
figure 5

Adiabatic speed of sound \(c^{2}_{s}\) as a function of the scale factor a for \(r=0.15\)

Full size image

From definition (11) the expression for the adiabatic sound speed is found to have the form

$$\begin{aligned} c_s ^2 = \frac{1 - X^\alpha }{2\alpha - 1 + (2\alpha + 1)X^\alpha }. \end{aligned}$$
(20)

We would like to satisfy the physical requirement

$$\begin{aligned} 0\le c_s ^2 \le 1. \end{aligned}$$
(21)

The stability condition \(c^{2}_{s}\ge 0\) implies \(\frac{1}{2} \le \alpha \) and the positive energy requirement gives \(b + 1\ge 2\alpha \). The subluminal sound speed condition produces the inequality \(1\le \alpha \). In fact, there is an open debate about the possible existence of acceptable physical systems with superluminal speed of sound [50,51,52,53,54]. The positive energy condition is satisfied in this model if \(dc > 0\).

The evolution of the sound speed is represented in Fig. 5 and we can appreciate that it tends to a constant. In fact, when \(2\alpha \ne b + 1\), the speed of sound takes the value \(c_s ^2 = \frac{1}{2\alpha -1}\) when the scale factor is very large. When the condition \(2\alpha = b + 1\) is fulfilled the evolution of the equation of state parameter does not exhibit a cosmological constant behavior, but for large values of the scale factor a it reaches the value \(w = \frac{1}{b}\).

The evolution of the scale factor can be determined by the Friedmann equation,

$$\begin{aligned} \frac{H^{2}}{H^{2}_{0}} = \frac{\Omega _m}{a^3} + \Omega _{(k-essence)}, \end{aligned}$$
(22)

where \(H_{0}\) represents the Hubble parameter at the current time \(t_0\), \(\Omega _m\) is the parameter representing the contribution from matter (dust), and we represent the dark energy component of the universe with the parameter \(\Omega _{(k-essence)} \equiv \rho /\rho _{c}\) where \(\rho \) is given by Eq. (16) and \(\rho _{c} = 3H^{2}_{0}/8 \pi G\) is the usual critical density. By means of the normalization condition \(\Omega _{m} + \Omega ^{(0)}_{(k-essence)} = 1\) we can determine the parameter c in terms of the other free parameters of the model. As a result we have b and r as free parameters, the latter being defined as in Eq. (19).

Fig. 6
figure 6

Evolution of the scale factor for \(r=0.325\). The dashed line corresponds to \(X\ge 1\) and the solid line to \(X \le 1\)

Full size image

The evolution of the scale factor is shown in Fig. 6 for the cases \(X\ge 1\) and \(X \le 1\). Near the big bang singularity it is expected that the scale factor evolves mainly as for a universe with dust. Furthermore, when the scale factor is large the right hand side of the Friedmann equation tends to a constant \(\Omega _{(k-essence)} = \frac{(1-\Omega _m)}{R_0} (b-1)\) for \(X \le 1\) where \(R_0 = 2c\rho (X_0)\) and \(X_0\) is obtained from the value of X that produces \(a=1\) in Eq. (18). For \(X \ge 1\) there is a similar behavior with the corresponding expression \(\Omega _{(k-essence)} = \frac{(1-\Omega _m)}{R_0}(b+3)\) but the value for \(R_0\) is different from the case \(X \le 1\). It is interesting to note that the evolution of the scale factor is similar to both cases but for the branch \(X \ge 1\) the stability condition \(\mathrm{d}a/\mathrm{d}X <0\) is not satisfied. In fact, for this case the speed of sound takes the value \(c_s ^2 = -\frac{1}{2\alpha +1}\) when the scale factor is very large.

It is important to know if the k-essence model presents finite-time future singularities. The finite-time future singularities can be of four types [49]. Type I singularity (big rip): For a finite time \(t \rightarrow t_s\) the scale factor, the pressure and the energy density of the universe diverge. It is possible to consider the case when \(\rho _\mathrm{eff}\) and \(p_\mathrm{eff}\) are finite at \(t=t_s\) where \(\rho _\mathrm{eff}\), \(p_\mathrm{eff}\) includes the contribution of all types of matter and energy. Type II (sudden singularity): For a finite time \(t \rightarrow t_s\) only the pressure diverges and the \(a\rightarrow a_s\), \(\rho _\mathrm{eff}\rightarrow \rho _s \). Type III singularity: For a finite time \(t \rightarrow t_s\) the pressure and the energy density of the universe diverge but the scale factor remain finite \(a\rightarrow a_s\). Type IV: For a finite time \(t \rightarrow t_s\) and \(a\rightarrow a_s\); the pressure and the energy density have finite values but higher derivatives of H become infinite.

In order to study a possible finite-time future singularity we write an expression of the scale factor in terms of the pressure using Eqs. (15) and (18)

$$\begin{aligned} a= r (2\beta )^{2/3} \frac{(p+\beta + \gamma )^{-(2\alpha +1)/6\alpha }}{(p-\beta + \gamma )^{(2\alpha -1)/6\alpha }}. \end{aligned}$$
(23)

In the limit \(p \rightarrow \infty \) the scale factor tends to \(a \sim \frac{1}{p^{2/3}} \rightarrow 0\). Besides, it is needed an infinite time to obtain an infinite scale factor. The former results show the existence of a big bang singularity and the absence of finite-time future singularities of types I, II, and III. Furthermore, near the big bang singularity (where \(p \rightarrow \infty \) and \(X=1\)) we have \(\rho \sim p^2\), and from \(a \sim \frac{1}{p^{2/3}}\) it is obtained \(\rho \sim \frac{1}{a^3}\), so that the scale factor behaves as \( a \sim t^{2/3}\), which represents the evolution of a universe dominated by non-relativistic matter at early times. It is important to mention that the same evolution near the big bang singularity is obtained for all the cases analyzed below. From the relation \(\dot{H} = -\frac{3}{2} (\Omega _m +\Omega _k)\) and Eqs. (15) and (16) we observe that it diverges only at \(X=1\), which corresponds to \(a=0\). We do not observe additional divergences where \(p \rightarrow \infty \)

3.2 The case \(\Delta =0\)

For this case the expression for the pressure is given by

$$\begin{aligned} p = \frac{1}{2c} \left( - b - 1 - \frac{4}{\ln X} \right) . \end{aligned}$$
(24)

The energy density is

$$\begin{aligned} \rho = \frac{1}{2c}\left( b+1 + \frac{2}{\ln X} + \frac{8}{(\ln X)^2} \right) . \end{aligned}$$
(25)

The equation of state parameter is written in the form

$$\begin{aligned} w = \frac{\left( - b - 1 - \frac{4}{\ln X} \right) }{\left( b+1 + \frac{2}{\ln X} + \frac{8}{(\ln X)^2}\right) }. \end{aligned}$$
(26)

The relation between the scale factor a and the parameter X is given by

$$\begin{aligned} a = \left( \frac{kc^2}{4}\right) ^{1/6} X^{1/6} (\ln X)^{2/3}. \end{aligned}$$
(27)

In Fig. 7 we present the relation between the scale factor a and the parameter X. In order to satisfy the stability condition \(\mathrm{d}a/\mathrm{d}X < 0\) the values of X must be restricted to \(X_m< X < 1\) where \(X_m\) gives the maximum value of the scale factor in the region \(0< X < 1\). The following correspondence is observed: when \(X \rightarrow 1\) then \(a \rightarrow 0\) and when \(X \rightarrow 0\) then \(a \rightarrow 0\).

Fig. 7
figure 7

Scale factor as a function of X. For simplicity, we have chosen the numerical pre-factor in Eq. (27) equal to 1

Full size image
Fig. 8
figure 8

Derivative of the scale factor on the scale factor itself. The dashed line corresponds to \(X\le X_m\), the solid line to \(X_m \le X \le 1\)

Full size image

The evolution of the scale factor can be obtained for the Friedmann equation for the regions \(0 \le X \le X_m\), \(X_m \le X \le 1\) and \(X \ge 1\). It is very important to obtain the behavior of the velocity of the scale factor in order to clarify its evolution because of the presence of a maximum value of scale factor \(a_\mathrm{max} = a(X_m)\). There exists the possibility of a big bounce or a big brake behavior (see for example [55]) where at \(t=t_h\) the evolution of the scale factor comes to stop in \(a(t_h) = a_{*}\) where \(\dot{a}(t_h) = 0\) due to an infinite negative acceleration \(\ddot{a} (t_h) = -\infty \). The velocity of the scale factor itself is shown in Fig. 8 where there is no value for the scale factor that gives \(\dot{a} = 0\) and we do not have a big bounce evolution or a big brake singularity. Then an acceptable possibility for the evolution of the scale factor is that the universe begins with a finite value for \(a=a_\mathrm{max}\) and starts to collapse in a finite time for the case of \(X \le 1\). For \(X\ge 1\) the universe begins from a big bang singularity and the scale factor can take an infinite value. In fact, when the scale factor is large the density parameter approaches a constant value \(\Omega _{(k-essence)} = \frac{(1-\Omega _m)}{R_0} (b+1)\). However, for this case the stability condition is not satisfied because \(\mathrm{d}a/\mathrm{d}X >0\). The evolution of the scale factor is shown in Fig. 9. A possible finite-time future singularity can be found if we write the scale factor in terms of the pressure using Eqs. (24) and (27) as follows:

$$\begin{aligned} a= 4^{2/3} r_1 \frac{e^{-\frac{2}{3(2cp +b+1)}}}{(2cp +b+1)^{2/3}} , \end{aligned}$$
(28)

where \(r_1= \left( \frac{kc^2}{4}\right) ^{1/6}\). In the limit \(p \rightarrow \infty \) the scale factor tends to \(a \sim \frac{1}{p^{2/3}} \rightarrow 0\) and it corresponds to a big bang singularity and finite-time future singularities of the types I, II, and III are not present.

Fig. 9
figure 9

Evolution of the scale factor. The dashed line corresponds to \(X\le X_m\), the solid line to \(X_m \le X \le 1\). The dotted line corresponds to \(X \ge 1\)

Full size image

3.3 The case \(\Delta >0\)

The expression for the pressure becomes

$$\begin{aligned} p = \frac{1}{2c}\left( - b - 1 + 2\beta \tan \left( \frac{\beta \ln X}{2}\right) \right) , \end{aligned}$$
(29)

where \(2\beta = \sqrt{\Delta }\). The energy density is given by

$$\begin{aligned} \rho = \frac{1}{2c}\left( \beta \sec ^2 \left( \frac{\beta \ln X}{2}\right) - 2\beta \tan \left( \frac{\beta \ln X}{2} \right) + b + 1 \right) . \end{aligned}$$
(30)

The equation for the state parameter w is

$$\begin{aligned} w = \frac{- b - 1 + 2\beta \tan \left( \frac{\beta \ln X}{2}\right) }{ \beta \sec ^2 \left( \frac{\beta \ln X}{2}\right) - 2\beta \tan \left( \frac{\beta \ln X}{2} \right) + b + 1}. \end{aligned}$$
(31)

For this case the expression of the scale factor a as a function of the parameter X is found to be

$$\begin{aligned} a = \left( \frac{4kc^2}{\beta ^4 }\right) ^{1/6} X^{1/6} \cos ^{2/3}\left( \frac{\beta }{2}\ln X\right) . \end{aligned}$$
(32)
Fig. 10
figure 10

Scale factor as a function of X. For simplicity, we have chosen the numerical pre-factor in Eq. (32) equal to 1

Full size image
Fig. 11
figure 11

Derivative of the scale factor on the scale factor itself. The dashed line corresponds to \(X \ge X_{mj}=3.54\), the solid line to \(X_{j-1} \le X \le X_{mj}\)

Full size image

In Fig. 10 we present the dependence of the scale factor a on the parameter X and observe a wild oscillatory behavior of the scale factor when \(X \rightarrow 0\), which corresponds to \(a \rightarrow 0\). In order to satisfy the stability condition \(\mathrm{d}a/\mathrm{d}X < 0\) the values of X must be restricted to \(X_{mj}< X < X_{j}\) where \(X_{mj}\) gives the maximum value of the scale factor in the region \(X_{j-1}< X < X_{j} \) and \(a(X_{j-1, j}) =0\). Near the singularity when \(X \rightarrow 0\) the maximum value for the scale factor is very small. In order to study the possible existence of a big bounce scenario or a big brake singularity we find the dependence of the derivative of the scale factor on the scale factor itself. Figure 11 shows the dependence of the velocity of the scale factor on the scale factor where there is no value that gives \(\dot{a} = 0\) and there is not a big bounce evolution or a big brake singularity. In a similar way to the case before, an acceptable possibility for the evolution of the scale factor is that the universe begins with a finite value for \(a=a_{mi}\) and starts to collapse in a finite time for the case of \(X_{mi} \ge X \ge X_{i} \). This behavior cannot adequately describe the evolution of the universe. A possible finite-time future singularity can be determined if we write the scale factor as a function of the pressure as follows:

$$\begin{aligned} a= (2\beta )^{(2/3)}r_2 \frac{e^{-\frac{1}{3 \beta }\arctan \left( \frac{2pc + b + 1}{2\beta } \right) }}{(4\beta ^2 + (2cp +b+1)^2)^{1/3}}, \end{aligned}$$
(33)

where \(r_2 =\left( \frac{4kc^2}{\beta ^4 }\right) ^{1/6}\). In the limit \(p \rightarrow \infty \) the scale factor tends to \(a \sim \frac{1}{p^{2/3}} \rightarrow 0\), and this corresponds to a big bang singularity and besides there are not finite-time future singularities of the types I, II, and III.

Finally, in Fig. 12 we present the physical allowed regions for the parameters cd and b of the model. It is possible to include higher exponents in the polynomial equation of state, but there is not a general analytical expression for the pressure and the energy.

Fig. 12
figure 12

Allowed physical regions for the parameters cd and b. The dotted line corresponds to points for which \(\Delta = 0\), the full line for points where \(c^{2}_{s} = \infty \), and the dashed line for points where \(c^{2}_{s} = 1\)

Full size image

4 Cosmological analysis of the parameters

The evidence for the late time accelerated expansion of the universe came from Type Ia supernova observations [1, 2] and two important tests to constrain the parameters of dark energy models in general to the current observations are the luminosity distance and the Alcock–Paczyński test [56]. The luminosity distance is calculated from

$$\begin{aligned} \mathrm{d}_L = (1 + z) \int _0 ^z H_0 \frac{\mathrm{d}x}{H(x)}, \end{aligned}$$
(34)

where \(H(t)= \dot{a}/a\) is the Hubble parameter that satisfies the Friedmann equation Eq. (22) By means of the normalization condition \(\Omega _{m} + \Omega ^{(0)}_{(k-essence)} = 1\) we can determine the parameter c in terms of the other free parameters of the model. As a result we have b and r as free parameters, the latter being defined as in Eq. (19). In Figs. 13 and 14 we compare the luminosity distance from our model with the \(\Lambda \)CDM model.

Fig. 13
figure 13

Luminosity distance for \(r=0.5\) and for different values of b. It can be observed that for large values of b the curves approach \(\Lambda \)CDM

Full size image
Fig. 14
figure 14

Luminosity distance for \(r=0.325\). It can be observed that the model curves are very close to \(\Lambda \)CDM

Full size image
Fig. 15
figure 15

\(\mathrm{d}_L\)(KCDM)/\(\mathrm{d}_L\)(\(\Lambda \)CDM) for \(r=0.5\) and different values of b. We can observe that the difference between the models is very small

Full size image
Fig. 16
figure 16

\(\mathrm{d}_L\)(KCDM)/\(\mathrm{d}_L\)(\(\Lambda \)CDM) for \(r=0.325\) and different values of b

Full size image

From these figures we conclude that the model approaches the \(\Lambda \)CDM model for large values of b and for small values of the parameter r. A more suitable way for comparing the two models is by means of the quotient \(\mathrm{d}_L\)(KCDM)/\(\mathrm{d}_L\)(\(\Lambda \)CDM), where KCDM stands for cold dark matter with k-essence as the agent (dark energy) producing the observed late-time acceleration of the universe.

Some results are shown in Figs. 15 and 16. These results clearly confirm the fact that for larger values of b and lower values for the parameter r the model mimics \(\Lambda \)CDM. It is interesting to note that a similar analysis leading to similar results can be done for the case \(X\ge 1\).

Fig. 17
figure 17

Alcock–Paczyński test for \(r=0.5\)

Full size image
Fig. 18
figure 18

Alcock–Paczyński test for \(r=0.325\)

Full size image

Another useful tool to compare the predictions of our model with standard cosmology with cosmological constant is the Alcock–Paczyński test [56]. This test is independent of evolutionary effects and is a very sensitive estimator for dark energy. Results using this test are presented in Figs. 17, 18, and 19 where \(\mathrm{d}_c = \mathrm{d}_L/(1+z)\) for a flat geometry.

Fig. 19
figure 19

Alcock–Paczyński test for \(r=0.145\). This figure shows that for larger values of b and small values of r the model curves approach the \(\Lambda \)CDM model

Full size image

In this test we can conclude that our model is successful in describing dark energy at this level. However, other tests should be applied in order to constrain the values of the free parameters of the model. These tests may allow direct comparison with supernova Ia observations and Hubble parameter results. This latter task is under current investigation.

5 Conclusions

In this paper we have considered a nonlinear equation of state for a k-essence field inspired by superconducting topological defects. In this proposal the dark energy content of the universe comes from a symmetry breaking mechanism of the kind of process that is at the origin of cosmic topological defects. We have constrained the range of valid parameters of the model demanding stability (the nonexistence of imaginary adiabatic speed of sound), the condition of positive values for the energy density and, moreover, we have found the regions for superluminal and subluminal adiabatic speed of sound. It is important to mention that the velocity of the perturbations tends asymptotically to a constant depending on the parameters used in the model and it can be arbitrarily small. We have shown the evolution for the equation of state parameter with respect to the scale factor and, for certain cases, the k-essence field behaves like radiation; then it evolves toward dust and finally it goes asymptotically to cosmological constant. This is an example of a solution with k-essence that tracks the equation of state of the dominant type of matter. Besides, we have studied the evolution of the scale factor for all the three cases considered in the paper where only the case for \(\Delta < 0\) has an acceptable physically evolution for the scale factor. Additionally, we have not found finite-time future singularities. For large pressure, near the big bang singularity, the universe behaves as if it were matter dominated (We thank the anonymous referee for pointing this out to us). We also considered the evolution of the luminosity distance \(\mathrm{d}_{L}\) with respect to redshift, and for this purpose we compared the predictions of our model to those of standard cosmology with a dark energy content (\(\Lambda \)CDM). The proposed model has b and r as free parameters and the general behavior consists in the fact that for large values of b and small ones for r the model is similar to the \(\Lambda \)CDM model. However, there is a specific case when the k-essence field has a positive value for the equation of state parameter for large values of the scale factor and it does not evolve into a cosmological constant. Since we consider the evolution of the universe from a z-dependent equation of state parameter w(z) the luminosity distance \(\mathrm{d}_{L}(z)\) was also compared (normalized) with that for the standard cosmology \(\Lambda \)CDM using the Alcock–Pacziński test. As stated before, other tests should be applied in order to constrain the values of the free parameters of the model. These tests may allow for a direct comparison with supernova Ia observations and Hubble parameters values. The latter task is under current investigation. We believe that other kinds of superconducting membrane Lagrangians deserve physical exploration within the context of purely kinetic k-essence models.