Tracker phantom field and a cosmological constant: dynamics of a composite dark energy model

We consider a spatially flat, homogeneous and isotropic Universe described by the FRW metric filled with barotropic fluids and a phantom scalar field. The Einstein field equations together with the wave equation of the above mentioned Universe are,

where $\kappa^{2}=8\pi G$, $\rho_{j}$ and $p_{j}$ are respectively the energy and pressure density of ordinary matter, a dot denotes derivative with respect to cosmic time $t$, and $H=\dot{a}/a$ is the Hubble parameter, with $a$ the scale factor of the Universe.

The index $j$ runs over all the matter species in the Universe apart from the scalar field (e.g. photons, baryons, etc.), and the perfect fluids are related through the barotropic relation $p_{j}=(\gamma_{j}-1)\rho_{j}$. The barotropic equation of state (EoS) takes the usual values of $\gamma_{j}=4/3$ for a relativistic species, $\gamma_{j}=1$ for a nonrelativistic one, and $\gamma_{j}=0$ for a cosmological constant.

Given our interest to include a cosmological constant in our analysis, we note that the phantom potential can also be written in the form $V(\phi)=V_{0}+V_{1}(\phi)$, where $V_{0}$ is simply a constant term and all the field dependencies in the potential are encoded in the term $V_{1}$. The expressions for the phantom energy density and pressure are, respectively, $\rho_{\phi}=-(1/2)\dot{\phi}^{2}+V_{1}(\phi)+V_{0}$ and $p_{\phi}=-(1/2)\dot{\phi}^{2}-V_{1}(\phi)-V_{0}$. Notice that the dynamics of the phantom field is not modified by the introduction of the constant term $V_{0}$ in the potential (see Eq. (1c)), but the latter only appears in the equations of motion for the Hubble parameter (1a) as an extra cosmological constant.

Under this freedom to include a constant term in the phantom potential, we will refer to $\rho_{\Lambda}$ as the effective density that contains all possible constant terms in the total density, and likewise for the corresponding pressure which satisfies the relation $p_{\Lambda}=-\rho_{\Lambda}$. In line with this, and for simplicity in the notation, hereafter we make the change $V_{1}(\phi)\to V(\phi)$.

To ease the numerical solution of the phantom equation of motion, and inspired by the case of the quintessence field Ureña-López and Gonzalez-Morales (2016); Roy et al. (2018); Ureña López and Roy (2020), we define a new set of hyperbolic polar coordinates in the following form

with which the Klein-Gordon equation (1c) is written as the following dynamical system,

The prime denotes derivative with respect to the number of $e$-foldings $N\equiv\ln(a/a_{i})$, with $a_{i}$ the initial value of the scale factor. Here, $\gamma_{tot}=(p_{tot}+\rho_{tot})/\rho_{tot}$ is the total EoS written in terms of the total pressure $p_{tot}$ and total density $\rho_{tot}$ of all the matter species. In particular, the EoS parameter of the phantom field can be written as $\gamma_{\phi}=(p_{\phi}+\rho_{\phi})/\rho_{\phi}=1-\cosh\theta$.

A note is in turn. In the new variables (2) we assumed that $\Omega_{\phi}$ is positive definite, and in consequence so is the phantom density, $\rho_{\phi}=3H^{2}\Omega_{\phi}/\kappa^{2}>0$. This is not necessarily the case of phantom fields, as for certain cases the energy density can be negative. However, we will consider initial conditions for a radiation dominated Universe, and then $\Omega_{\phi}\to 0^{+}$ at early times, which assures that $\Omega_{\phi}$ will be positive definite for the rest of the evolution.

Now, we are going to consider linear perturbations around the background values of the FRW line element (in the synchronous gauge),

as well as for the scalar field in the form $\phi(\vec{x},t)=\phi(t)+\varphi(\vec{x},t)$. Here, $h_{ij}$ and $\varphi$ are the metric and scalar field perturbations, respectively. The linearized KG equation for the phantom field, for a Fourier mode $\varphi(k,t)$, reads Ratra (1991); Ferreira and Joyce (1997, 1998); Perrotta and Baccigalupi (1999):

where $\bar{h}$ is the trace of the spatial part of the metric perturbation, and $k$ is its comoving wave number.

Again, the perturbed KG equation (5) can be transformed into a dynamical system by using the following change of variables Ureña-López and Gonzalez-Morales (2016); Cedeño et al. (2017),

where $\beta$ and $\vartheta$ are the new variables introduced related to the evolution of the scalar field perturbation. With another set of variables defined through: $\delta_{0}=e^{\beta}\sinh(\theta/2+\vartheta/2)$ and $\delta_{1}=e^{\beta}\cosh(\theta/2+\vartheta/2)$, the perturbed KG equation  (5) is transformed into the dynamical system (see Appendix A),

where $k_{J}^{2}\equiv a^{2}H^{2}y_{1}$ is the (squared) Jeans wave number, and

In writing Eqs. (7) we have used the relation $\partial^{2}_{\phi}V=H^{2}(y^{2}_{1}/4-yy_{2}/2)$ in Eq. (5). Similarly to the case of scalar fields studied in Cedeño et al. (2017); Linares Cedeño et al. (2021), the variable $\delta_{0}$ is the phantom density contrast, as from Eqs. (2) and (6) we find that $\delta\rho_{\phi}/\rho_{\phi}=(-\dot{\varphi}\dot{\phi}+\varphi\partial_{\phi}V)/\rho_{\phi}=\delta_{0}$.

Likewise, there is a Jeans wave number $k_{J}$ for the phantom density perturbations that only involves the function $y_{1}$ Ureña-López (2016); Cedeño et al. (2017); Linares Cedeño et al. (2021). In the cases we will explore one expects that $y_{1}\lesssim\mathcal{O}(1)$, and then the associated Jeans scale length will be equal or larger than the Hubble horizon, $k^{-1}_{J}\gtrsim 1/H$, which in general suggests that phantom perturbations will be suppressed in sub-horizon scales. It must be noticed that there is another scale involved in the evolution of the density perturbations, $k^{2}_{eff}$, which means that tachyonic effects will appear in phantom perturbations whenever $k^{2}_{eff}<0$ Cedeño et al. (2017); Linares Cedeño et al. (2021), but this will depend on the chosen potential and the behavior of the combined variable $y_{2}\Omega_{\phi}/y$. In general, phantom density perturbations are negligible, but we will include them in our study for completeness.

To calculate the solutions of physical interest, in this section we start with the equations of the critical values $\theta_{c}$, $y_{1c}$ and $\Omega_{\phi c}$ as obtained from the dynamical system (3), namely

From Eq. (10a) we obtain the condition $y_{1c}=-3\sinh\theta_{c}$, which is common to all possible critical points from Eqs. (10), and which will be also explicitly assumed in the analysis below for the phantom tracker and phantom dominated solutions in the following sections. Furthermore, if we consider Eq. (9), then we get from Eq. (10b) either that $\sinh{\theta_{c}}=0$, or

It is customary in the literature to classify the critical points that appear in the phantom equations of motion, in our case from Eqs. (10) and (11). The first critical point is the so-called fluid domination, for which $\Omega_{\phi c}=0$. One straightforward solution is $\sinh{\theta_{c}}=0$, which means that the phantom EoS takes the critical value $\gamma_{\phi c}=-1$. In contrast to the quintessence case, this time there is not kinetic dominated solution. Another possible solution under the condition $\Omega_{\phi c}=0$ is the tracker solution, but that is studied in more detail in Sec. III.2 below.

One final note is that there are not scaling solutions for phantom fields, in which the phantom EoS takes on the same values as that of the background dominant component $\gamma_{\phi}=\gamma_{tot}$, unless the background component is the cosmological constant or a phantom-like component too.

Let us first consider the case $\alpha_{0}=0=\alpha_{1}$, for which we obtain from Eq. (11) that the critical condition for the hyperbolic variable is $\sinh^{2}(\theta_{\phi,c}/2)=\gamma_{tot}/4\alpha_{2}$. In terms of the phantom EoS, the latter condition reads

Notice that in Eq. (12) we must choose positive definite values for $\alpha_{2}$ so that $\gamma_{\phi,c}<0$. Moreover, a quick comparison with previous studies confirms that Eq. (12) is the tracker condition for phantom fields.

The potentials that exhibit the tracker behavior according to Eq. (12) are of the power-law form $V(\phi)=M^{4-p}\phi^{p}$, where $p=2/(1+2\alpha_{2})$. In contrast to the quintessence in which the tracker potentials are of the inverse-power law type, this time the tracker condition is achieved for $0<p<2$ (corresponding to $0<\alpha_{2}<\infty$), which means that the phantom field evolves away from the minimum of the potential while in the tracker regime.

As argued in Ureña López and Roy (2020), the tracker condition (12) is of wider applicability if $(\alpha_{0},\,\alpha_{1})\neq 0$, as long as $\Omega_{\phi c}$ is negligible, which is generically expected at early times. Moreover, if we can write $y_{2}=yf(y_{1}/y)$, where $f$ is an arbitrary function of its argument, then the critical condition (11) reads

In writing Eq. (13) we have used $y_{1c}/y_{c}=3\sinh\theta_{c}/[\Omega^{1/2}_{\phi c}\cosh(\theta_{c}/2)]=3\sqrt{2}\sinh(\theta_{c}/2)/\Omega^{1/2}_{\phi c}$. Thus, the tracker solution exists whenever the following condition is satisfied,

where $g(x)$ would be the resultant function after the limit operation. The tracker equation derived from Eq. (13) under the result (14) would simply read: $9\gamma_{tot}+g(\sinh(\theta_{c}/2))=0$. Any valid solution of the latter equation should be considered a generalized tracker solution for the phantom field.

Let us turn our attention to phantom dominated solutions at late times; these solutions are characterised by the conditions $\Omega_{\phi c}=1$ and $\gamma_{tot}=\gamma_{\phi c}$. Our main interest here are the phantom dominated solutions that are related to the tracker solutions at early times.

A small note is in turn. The phantom EoS, given by $\gamma_{\phi}=-2\sinh^{2}(\theta_{c}/2)$, is the same irrespective of the sign of $\theta$, but because $\gamma_{\phi}\leq 0$, $\theta$ does not cross the zero value and then one needs to choose either the negative or positive branch of the hyperbolic sine. For convenience, we will hereafter choose the negative branch, $\theta\leq 0$, which also allows for the potential variable $y_{1}$ to be positive definite.

Recalling that the first option for a critical value is $\sinh\theta_{c}=0$, we find that one possible asymptotic value of the phantom EoS is $\gamma_{\phi c}=0$, for which the phantom density is dominated by its potential part $V(\phi)$. This means that at late times the phantom field approaches the behavior of a cosmological constant.

Another possibility arises from the solution of Eq. (11), which for the aforementioned conditions of phantom domination reads

The critical solutions of Eq. (15) will depend on the values of the active parameters $\alpha$. For the particular case of purely tracker solutions, $\alpha_{0}=0=\alpha_{1}$ the only critical solution possible is again $\theta_{c}=0$, and then $\gamma_{\phi c}=0$, which means that the phantom density will asymptotically behave as a cosmological constant.

We now study the conditions for Eq. (15) to have at least one negative solution, that is, $\theta_{c}<0$, under the tracker condition $\alpha_{2}>0$. Let us start with $\alpha_{1}=0$, for which the solution of Eq. (15) is

In the case $\alpha_{1}\neq 0$, the general solution of Eq. (15) can be written in the form

where $\Delta=1-2\alpha_{0}(1+2\alpha_{2})/\alpha^{2}_{1}$. Notice that we require $\Delta\geq 0$ to have real valued solutions of Eq. (16b).

We first consider the case $\alpha_{0}<0$. The latter implies that $\Delta>1$, which then assures the existence of at least one negative solution of Eq. (16b), irrespective of the value of $\alpha_{1}$. In other words, a negative value of $\alpha_{0}$ assures the existence of a phantom EoS at late times for tracker potentials. Next, we take the case $\alpha_{0}\geq 0$, such that $0\leq\Delta\leq 1$. There will be at least one negative solution of Eq. (16b), and then again a phantom EoS, if $\alpha_{1}<0$.

In summary, one consequence of our choice $\theta\leq 0$ is that the cosmological constant case ($\gamma_{\phi}=0$) is the only asymptotic solution available for the phantom EoS if both conditions $\alpha_{0}\geq 0$ and $\alpha_{1}\geq 0$ are satisfied, as for such conditions there are not negative solutions of $\theta_{c}$ from Eq. (15). In all other cases, as long as $\Delta\geq 0$, the phantom EoS remains below the phantom divide ($\gamma_{\phi}<0$) and its asymptotic value is given by the negative solution of Eqs. (16).

Here we study purely phantom solutions, and then $\rho_{\Lambda}=0$; we label this case as $\phi$. Typical examples for the behavior of the phantom EoS are shown in Fig. 1 for the fixed value $\alpha_{2}=5$, together with different combinations of the other active parameters $\alpha_{0}$ and $\alpha_{1}$. Other relevant parameters, like the present density contributions of the different matter species, were fixed to the values reported by the Planck collaboration (see their Table 1) Aghanim et al. (2020a).

In the top panel of Fig. 1, it can be seen that all solutions maintain their tracker behavior at early times, as seen from the values of the phantom EoS during the radiation and matter domination epochs, which are $-17/15$ and $-11/10$ (dashed black lines), respectively. The evolution of the solutions from radiation to matter domination for the different examples are so identical that they are not distinguishable in the plot. Recalling that the initial conditions are set up at $a_{i}=10^{-14}$, this indicates that the tracker condition (12) is a stable solution of the background evolution at early times.

As for late times, a better view of the evolution of the phantom EoS is provided in the bottom panel of Fig. 1, in terms of the phase space $(w_{\phi},w^{\prime}_{\phi})$, where $w^{\prime}_{\phi}=\sinh\theta(3\sinh\theta+y_{1})$. All solutions depart from the tracker point at radiation domination $(-17/15,0)$ (left black dot), and evolve towards that at matter domination $(-11/10,0)$ (right black dot), while tracking the background and with identical evolutionary paths. Again, for the cases in which $\alpha_{0}>0$ and $\alpha_{1}>0$, see Sec. III.3 above, the curves are deflected away from the second tracker point and the phantom EoS evolves towards the cosmological constant point at $(-1,0)$.

For all other cases, the asymptotic values of the phantom EoS are also indicated by dots in the bottom panel of Fig. 1, with the same color as that of the corresponding evolution curve. The coordinates of the asymptotic points (brown, purple, grey, and light blue dots) were obtained from the solutions of Eqs. (15).

To show the influence of phantom density perturbations in models of phantom DE, we show in the top panels of Fig. 3 the two-point temperature power spectrum $\mathcal{C}^{TT}_{\ell}$ of the cosmic microwave background (CMB) and the mass power spectrum (MPS) of linear density perturbations $P(k)$, for the same numerical examples shown in Fig. 1. In comparison with the standard case with $\Lambda$ as DE, we see that there are noticeable changes, specially for the CMB spectrum, but only at large scales and for the most extreme phantom values of the DE EoS.

We now turn our attention to the case in which both the cosmological constant and the phantom field are part of the DE budget, a case we label as $\phi+\Lambda$. For a comparison with the phantom case in the previous section, we show in Fig. 2 the evolution of the density parameters $\Omega_{\phi}$ and $\Omega_{\Lambda}$ at recent times, together with the phase space of the phantom EoS for the same triplets (and colors) $(\alpha_{0},\alpha_{1},\alpha_{2})$ as in Fig. 1.

For all plots, we chose $\Omega_{\phi}=1.0$. As the present density contributions of the different matter species are fixed to the values reported by the Planck collaboration, the present value of $\Omega_{\Lambda}$ was adjusted so as to fulfill the Friedmann constraint for a flat Universe. For this reason the contribution of the cosmological constant is in general negative, see the top panel of Fig. 2.

Notice that the corresponding behavior of the phantom EoS, as shown in the middle and bottom panels of Fig. 2, is qualitatively the same as in the standard phantom case in Fig. 1, the only difference being that the present EoS seems to reach more negative values than in the phantom-only case.

As for the phantom perturbations, we also show in the bottom row of Fig. 3 the temperature anisotropies and the MPS for the same cases shown in Fig. 2. There is an enhancement of the power at large scales in the two observables, which seems to be an effect of the larger contribution of the phantom field to the DE budget, and also of the respective smaller influence of the (negative) cosmological constant.

The obtained constraints on the models are shown in Fig. 4, with their detailed values listed in Table 1. Also, the models considered were labeled as: $\Lambda$ (the cosmological constant), $\phi$ (phantom DE) and $\phi+\Lambda$ (phantom and a cosmological constant). For the latter two, we have two further sub-cases: the purely tracker solution labeled as $\phi+\alpha_{2}$, and the generalized one $\phi+\alpha$’s.

In Fig. 4 we show the confidence regions for the Hubble parameter $H_{0}$, the physical matter density $\Omega_{M}h^{2}$, and the effective DE equation of state $\gamma_{eff}$ at the present time. We first note that the obtained values of $H_{0}$ and $\Omega_{M}h^{2}$ are the same for the phantom models, which is expected from the strong constraints imposed by the compressed Planck likelihood on these parameters, even in the presence of other late-time observations.

Note that there is a noticeable shift in the central values of both parameters as compared to the case of $\Lambda$, which is an effect that only appears when late-time observations are included in the analysis. However, the shift in the Hubble parameter in the phantom models is far from solving the Hubble tension with the SH0ES measurement.

The effective barotropic EoS of the whole DE budget is explicitly defined as:

where the index $j$ only runs through the DE components in the model. In our case, given that by definition $\gamma_{\Lambda}=0$, we find that $\gamma_{eff}=\gamma_{\phi}\Omega_{\phi}/(\Omega_{\phi}+\Omega_{\Lambda})$.

Clearly, if $\Omega_{\phi}=0$ ($\Omega_{\Lambda}=0$), i.e., if $\Lambda$ ($\phi$) is the only DE component then the effective DE EoS simply is $\gamma_{eff}=0$ ($\gamma_{eff}=\gamma_{\phi}$). More generally, if $\Omega_{\phi}>0$ and $\Omega_{\phi}+\Omega_{\Lambda}>0$, a negative value of $\gamma_{eff}$ would indicate a preference of the observations for a phantom-like DE component. This seems to be precisely the case as inferred from the values in Table 1: quite consistently $\gamma_{eff}<0$ at $1-\sigma$. Moreover, the value of the effective DE EoS is practically the same in the presence of the phantom component, irrespective of the model and the form and combination of the DE components, and just a little bit below the phantom divide: $\gamma_{eff}\simeq-0.045$.

In Fig. 5 we show the constraints on the active parameters $\alpha$ of the phantom potential, see Eq. (9). The overall result is that, independently of the DE model with the phantom field and $\Lambda$, the values of $\alpha_{0}$ and $\alpha_{1}$ are completely unconstrained, which means that their inclusion does not make any difference in the fitting to the data, and the latter does not seem to support any added complexity on the phantom models.

Another consequence of the unconstrained values of $\alpha_{0}$ and $\alpha_{1}$ is that the ultimate fate of the Universe under the phantom models remains unknown, as any of the late-time values of the EoS discussed in Sec. III.3 is equally likely. In general, the big or little rip solutions cannot be discarded under the models studied here.

Interestingly, the active parameter $\alpha_{2}$, which also controls the tracker properties of the phantom model, appears to be constrained by the data at around $\alpha_{2}\simeq 8.7$. This suggests that the tracker values of the phantom EoS are $\gamma_{\phi,c}\simeq-0.077$ ($\gamma_{\phi,c}\simeq-0.056$) during the radiation (matter) domination era. Although the deviation from the phantom divide is small, it remains to be studied why the data seem to prefer such negative values at early times.

To assess whether the observations have a preference for any of the model variations studied here, we first compute for each one the difference in the value of $\chi^{2}_{min}$ with respect to $\Lambda$, which curiously enough is the same for all models with phantom: $\Delta\chi^{2}_{min}=\chi^{2}_{\phi}-\chi^{2}_{\Lambda}=-5$. This indicates that the quality of the fit increases a bit with the inclusion of $\phi$, irrespective of the presence of $\Lambda$ and of the active parameters $\alpha$’s.

Although there are more free parameters in the phantom models than in the standard $\Lambda$ case (see the number of extra parameters $k$ in Table 1) this does not mean that such more general models should be discarded. From a strict Bayesian point of view, for a proper judgment, one must take into account whether the data is able to constrain the extra parameters. This is the case of the active parameters $\alpha_{0}$ and $\alpha_{1}$: being unconstrained by the data, the latter does not provide evidence in favour or against the models containing them Trotta (2008).

To have a more Bayesian assessment, we also show in Table 1 the Bayes factors of the phantom models with respect to the $\Lambda$ case, such that $\ln B_{\phi\Lambda}=\ln\mathcal{Z}_{\phi}-\ln\mathcal{Z}_{\Lambda}$, where $\mathcal{Z}$ represents the Bayesian evidence. For the calculation of the latter, for each model, we relied on the code MCEvidence Heavens et al. (2017a, b), which only requires the chains we generated with Monte Python. We see that consistently $\ln B_{\phi\Lambda}>2$, which means that there is Definite/Positive evidence, under the considered set of observations, in favor of the presence of a phantom DE component.

Another question that we are interested in is whether data indicates any joint contribution from both the phantom and $\Lambda$ components. To try an answer, we take advantage of the above fact that two of the active parameters are unconstrained and then focus on the models with $\alpha_{0}=0=\alpha_{1}$, which in turn makes it easier to find the numerical solutions of the phantom models.

The results are shown in Fig. 6, for the parameters $\Omega_{\Lambda}$, $\Omega_{M}h^{2}$ and $\Omega_{\phi}$. The variation in the phantom component $\Omega_{\phi}$ was extended to the range $[0.1:2]$, with the contribution of $\Omega_{\Lambda}$ inferred from the Friedmann constraint. This case is called as extended-$\phi+\Lambda+\alpha_{2}$ in Table 1.

The interesting case is the combined presence of the phantom field $\phi$ and $\Lambda$ as DE components (blue contours): the confidence regions seem to suggest a preference for a lower value of $\Omega_{\Lambda}$, even a negative one. In contrast, there is a preference for large values of the phantom contribution, this time of the order of unity for the density parameter, $\Omega_{\phi}\simeq 1$.

However, probably more interesting is that the value inferred for the $\Lambda$-only case (green contour) appears to be located in a low likelihood region when compared with the results of the combination $\phi+\Lambda+\alpha_{2}$ (orange contour). Correspondingly, the result for $\Omega_{\phi}$ of the phantom-only case (blue contour) is located within the region of maximum likelihood suggested by the extended case $\phi+\Lambda+\alpha_{2}$. As seen from Table 1, the Bayes factor with respect to the model $\Lambda$, $\ln B_{\phi\Lambda}\simeq+2$, again reinforces our previous result that the data favors the presence of a phantom component in the DE budget.

In other words, the conclusions from the Bayes factor appear to be conservative with respect to the parameter estimation shown in Fig. 6: even though we were unable to try the null value $\Omega_{\phi}=0$ because of numerical limitations, such value seems to be ruled out at $95\%$ confidence level. Hence, the moderate rejection of the model $\Lambda$-only comes from the penalisation the Bayes factor puts on the extended model $\phi+\Lambda+\alpha_{2}$ for using prior values of $\Omega_{\phi}$ that yield very low likelihood Wagenmakers et al. (2010); Trotta (2008); Nesseris and Garcia-Bellido (2013); Jenkins and Peacock (2011); Efstathiou (2008).

Taken together: the fit improvement, the conservative rejection from the Bayes factor and the informative posteriors in Fig. 6, lead us to conclude that the data seem to rule out a significant contribution of a positive $\Lambda$ in our models; rather, the data seem to prefer the phantom-only model. It is still possible to consider a contribution from a negative $\Lambda$, although none of our aforementioned tests, not even together, gives us decisive hints about such possibility.¹¹1In Appendix B we revise the odds of the models in terms of the so-called Savage-Dickey density ratio, which illustrates the interplay of the posterior and the prior of $\Omega_{\phi}$ on the calculation of the Bayes factor in our models. For comparison, we do the same in Appendix C for the case of a fluid model with a constant EoS accompanying $\Lambda$ as a DE component.