New general parametrization of quintessence fields and its observational constraints

To write Eq.(1d) as a set of autonomous equations, we introduce a new set of dimensionless variablesCopeland et al. (1998); Urena-Lopez (2016); Roy and Banerjee (2014b)

As a result, Eq.(1d) is transformed into

where now a prime is the derivative with respect to the number of e-foldings, $N\equiv\ln(a/a_{i})$ and $a_{i}$ is the initial scale factor of the universe. In writing Eqs. (4) we have used the Friedmann constraint (1a) in the form $\Omega_{r}+\Omega_{m}+\Omega_{\phi}=1$, where the density parameters of the different matter fields are defined in the standard way as $\Omega_{j}=\kappa^{2}\rho_{j}/(3H^{2})$. In addition, the total EoS is given by

Equations (4) have been thoroughly used in the literature to study the properties of quintessence potentials, see for instance Refs.Copeland et al. (1998); Bahamonde et al. (2017) and references therein. Their main advantage is the possibility to consider a compact phase space for the variables $x$ and $y$, so that all trajectories and critical points of interest can be studied without the intrinsic difficulties in the standard variables $(\phi,\dot{\phi})$. One main limitation of this approach is that the form of the new potential variables $y_{1}$ and $y_{2}$ have to be calculated individually for each quintessence potential, and in this respect they do not offer a clear advantage over the direct solution of the KG equation (1d), and the solution of the latter still is the popular approach in the numerical studies of quintessence models.

We now introduce the following polar transformations of the variables: $x=\Omega_{\phi}^{1/2}\sin(\theta/2)$ and $y=\Omega_{\phi}^{1/2}\cos(\theta/2)$, where $\Omega_{\phi}=\kappa^{2}\rho_{\phi}/3H^{2}$ is the density parameter associated to the quintessence field and $\theta$ represents an angular degree of freedom. The system of equations (4), after simple manipulations, reduces to

The EoS of the scalar field in terms of the polar variable is $w_{\phi}=p_{\phi}/\rho_{\phi}=(x^{2}-y^{2})/(x^{2}+y^{2})=-\cos\theta$, which tells us of the direct relation between the two variables. Likewise, the ratio of kinetic and potential energies is given by $\tan^{2}\theta=(1/2)\dot{\phi}^{2}/V(\phi)=x^{2}/y^{2}$. Equations (6a) and (6c) are the same for any kind of potential, and it is only Eq. (6b) that changes for different cases because of the presence of $y_{2}$.

To find a solution of Eqs. (6) one needs to close the system of equations, and this can be done whenever the second potential variable $y_{2}$ can be written in terms of the variables $\theta$, $y_{1}$ and $\Omega_{\phi}$. This is equivalent, in the standard approach, to the fixing of the scalar field potential. Then, one possibility is to choose the functional form of the potential $V(\phi)$ and to derive from it the form of $y_{2}$ following the prescriptions in Eqs. (3). In Table 1 we give a list of thawing and freezing quintessence potentials that are very familiar in the current literature and their corresponding closed form in terms of $y_{2}$. For those potentials, the dynamical system (6) becomes an autonomous one upon which we can use the known mathematical tools of such systems Roy and Banerjee (2017). It must be noticed that our classification in thawing and freezing is based upon on the behavior of the solutions as described in the corresponding references, but such classifications cannot be read directly from the final form of $y_{2}$, as also a proper choice of initial conditions must be taken into account. More details can be found in Sec. III below.

One can see that for the examples in Table 1 the functional forms of $y_{2}$ can be expressed in terms of the variables $y$ and $y_{1}$, or more precisely, as a polynomial in terms of the ratio $y_{1}/y$. It is then natural to consider that there exists a more general function of $y_{2}$ in the form

where $\alpha_{i}$ are constant coefficients. As also shown in the examples in Table 1, the constant coefficients $\alpha_{i}$ will then drive the dynamics of the quintessence field, although they will not be directly related to other free parameters in the potential, which are denoted with Latin capital letters in the examples of Table 1¹¹1We discuss in Appendix A a more general approach for the functional form of the ratio $y_{2}/y$..

For completeness, we show in Table 2 the inverse process that can be used upon Eq. (7) to obtain from it different quintessence potentials. The simplest possibility is $\alpha_{j}=0$, for which Eq. (7) can be written as $\partial^{2}_{\phi}V^{1/2}=0$ and then upon integration we obtain $V(\phi)=(A+B\phi)^{2}$, where $A,B$ are integration constants. This is precisely one particular example (with $\alpha_{2}=0$) of Class Ia in Table 2.

In the most general case, Eq. (7) can be written as a differential equation by means of the definitions of the variables $y$, $y_{1}$ and $y_{2}$ in Eqs. (3). Hence,

where the derivatives are calculated with respect to the dimensionless variable $\kappa\phi$. Using the auxiliary function $\lambda=y_{1}/y=-2\sqrt{6}\,\partial_{\kappa\phi}V^{1/2}/V^{1/2}$, we can write Eq. (8) in the form

Thus, the inverse process to find the quintessence potential if the dynamical parameters $\alpha_{j}$ are given consists in the integration of the fundamental equation (9). In general terms, Eq. (9) can be integrated by the method of partial fraction decomposition, for which we require first to find the roots of the polynomial on the right hand side. Once a solution is found for the auxiliary function $\lambda=\lambda(\kappa\phi)$, the corresponding quintessence potential is obtained from $V(\phi)=\exp\left[-\lambda(\kappa\phi)/\sqrt{6}\right]$.

The cases in Table 2 are those that correspond to the quadratic expansion ($\alpha_{j}=0$ for $j\geq 3$) in Eq. (9). It can be verified that there is a direct correspondence between the general cases in Table 2 with the particular examples shown in Table 1, as long as the constants $A$, $B$ and $C$ are adjusted accordingly. From the numerical point of view, the most general form of the potential is obtained from $\alpha_{j}\neq 0$, and all other forms should be a subclass of this. But analytically this is not achievable as the integration scheme is different for different choices of $\alpha_{j}$, and this is why we find it more natural to classify the quintessence potentials in the four classes shown in Table 2.

We said before that the $\alpha$-parameters are the only dynamical ones, and then their allowed values suggest natural classifications of the potentials in general classes. We hereafter dub them dynamical parameters. Apart from this, there will be constants of integration ($A$, $B$ and $C$ in the examples of Table 2), which are then redundant from the dynamical point of view and do not have any influence of the behavior of the field solutions, except in the set up of the initial conditions. We will refer to them as passive parameters.

To briefly illustrate the difference with respect to the dynamical ones, let us consider the well known example in Class Ia which is the quadratic potential $V(\phi)=(m^{2}_{\phi}/2)\phi^{2}$ ($A=0$, $B=m_{\phi}/\sqrt{2}$ and $\alpha_{2}=0$), where $m_{\phi}$ is the mass of the scalar fieldUrena-Lopez (2016). It can be shown that for this case $y_{1i}=2\sqrt{2}B/H_{i}=2m_{\phi}/H_{i}$, where $H_{i}$ is the initial value of the Hubble parameter (see also Sec. IV below). The initial values of the other two dynamical variables, $\theta_{i}$ and $\Omega_{\phi,i}$ must be fixed by taking additional considerations, like the expected contribution of the quintessence field at the present time. Thus, the value of parameter $B$ plays a role in the set up of the initial conditions of the field variables, even though it does not affect at all their evolution and dynamics. More about the free parameters in the potentials, both dynamical and passive, is discussed in the sections below.

As the universe is dominated by radiation the total EoS simply is $\omega_{tot}=1/3$, and due to the smallness of $\theta$ we can use the following approximations: $\sin\theta\simeq\theta$ and $\cos\theta\simeq 1$, so that also $w_{\phi}\simeq-1$. Hence, Eqs. (6) reduce to

The growing solution of Eqs. (10), within the radiation domination era, are given by

where a subindex $r$ denote the solution during radiation domination and a subindex $i$ denote the initial value of the corresponding variable. In addition, we also find that $y_{1}=5\theta$, which is just the attractor solution for these variables during radiation domination.

As the universe is dominated by matter we now consider that $\omega_{tot}=0$, and after using the same approximations as in Eqs. (10), Eqs. (6) now become

After solving Eqs. (12) we obtain the matter dominated solutions

Here, a subindex $m$ denote the solution during matter domination and a subindex $eq$ denote the initial value of the corresponding variable at the time of radiation-matter equality. In contrast to the previous radiation dominated case, we are not neglecting the decaying solution in Eq. (13) as it will be required below to handle the transition between the two cosmological eras.

We matched the approximate solutions (11) and (13) at the time of radiation-matter equality $a_{eq}=\Omega_{r0}/\Omega_{m0}$ so that we can find a solution at matter domination that carries information about the initial conditions set up in radiation domination. From Eqs. (11) we find the values of the variables at radiation-matter equality: $\theta_{eq}=\theta_{i}(a_{eq}/a_{i})^{2}$, $y_{1eq}=5\theta_{eq}=y_{1i}(a_{eq}/a_{i})^{2}$ and $\Omega_{\phi eq}=\Omega_{\phi i}(a_{eq}/a_{i})^{4}$, which we substitute in Eqs. (13) to obtain

Equations (14) can be used to estimate the initial conditions on the dynamical variables from the present values of $\theta$ and $\Omega_{\phi}$. We will assume that the matter domination solutions (14) are valid up to the present time. This is not correct from a formal point of view, but we have verified the appropriateness of Eqs. (14) by a direct comparison with numerical solutions, see Sec. IV below. Hence, by taking $a=1$ in Eqs. (14), the initial conditions for the quintessence dynamical variables can be estimated from

where we have only taken the leading term in Eq. (14a) for the expression of $\theta_{i}$. The same procedure applied to Eq. (14b) gives $y_{1i}=5\theta_{i}$, which is exactly the attractor solution expected during radiation domination, see Eqs. (11).

We learn from the first of Eqs. (15), because of the direct relation of the polar variable $\theta$ to the scalar field EoS, that the present value of the latter $w_{\phi 0}$ is an output value that is directly determined by the initial value $\theta_{i}$. This would be in agreement with the standard field approach to quintessence, in which one has to try different initial conditions, for both $\phi_{i}$ and $\dot{\phi}_{i}$, to explore a given range of values for $w_{\phi 0}$. The main difficulty in the latter is that there is not a straightforward relation between the pair $(\phi_{i},\dot{\phi}_{i})$ and the present value $w_{\phi 0}$, and then the search of initial conditions for a proper sampling of $w_{\phi 0}$ must be done differently for each potential $V(\phi)$. One big advantage of our approach in this respect is that we can avoid such a hassle and use generic initial conditions for all cases, irrespectively of the particular form of the potential.

On the other hand, the relation between the present and initial values of the scalar field density parameter $\Omega_{\phi}$ is just the one that is obtained for a cosmological constant; this is hardly surprising as one assumption was that the scalar field EoS was close to $-1$ for most of the evolution of the Universe.

One final quantity of interest is the ratio $y_{1}/y$ at the present time; from the solutions presented above we find that

Its present value is basically set up by the initial conditions, or equivalently, as suggested by the last equality in Eq. (16), by the present values of the quintessence parameters. Hence, the ratio $y_{1}/y$ should remain small for most of the evolution of the Universe if the quintessence field is to be the dark energy, i.e. if $1+w_{\phi 0}\sim 0$. This reinforces our assumption that it is just enough to consider an expansion up to the second order in Eq. (7), and then the dynamics of the quintessence fields will be represented, in general, by the values of the first three coefficients $\alpha_{0}$, $\alpha_{1}$ and $\alpha_{2}$.

We explain here the correspondence, within our approach, between quintessence and the cosmological constant. Let us start with the simplest possibility which is $y_{2}=0$, in terms of the expansion of $y_{2}$ in Eq. (7) this also means that: $\alpha_{0}=\alpha_{1}=\alpha_{2}=0$. As shown in Table 2 this case corresponds to the parabolic potential $V(\phi)=(A+B\phi)^{2}$, where $A$ and $B$ are integration constants. The potential has its minimum at $\phi_{c}=-A/B$, and then the mass of the quintessence field is simply given by $m^{2}_{\phi}\equiv\partial^{2}_{\phi}V(\phi_{c})=2B^{2}$. Thus, parameter $B$ gives us information about the mass scale of the quintessence field, whereas parameter $A$ tells us about the displacement of the minimum away from the origin at $\phi=0$. Moreover, there is now a straightforward interpretation of one of the potential parameters: $y_{1}=2\sqrt{2}B/H$, see Eq. (3b), and then we find that at all times $y_{1}=2m_{\phi}/H$. The parameter $A$ is left undetermined as it plays no role in the dynamics of the quintessence field.

Let us in addition impose $B=0$, which also means that $y_{1}=0$ for the whole evolution. Equation (4b) is identically satisfied, whereas Eq. (4a) provides the solution $\tan(\theta/2)=\tan(\theta_{i}/2)\,(a/a_{i})^{-3}$; we find that $\theta\to 0$ as $(a/a_{i})\to\infty$, and then also that $w_{\phi}\to-1$ at late times. This case corresponds to the case $V(\phi)=\mathrm{const.}$, that is, to the so-called skater models discussed inLinder (2005); Sahlen et al. (2007). Skater models then belong to our Class Ia of quintessence potentials under the condition $y_{1}=0$.²²2The evolution equation for the variable $\theta$ in this case simply is $\theta^{\prime}=-3\sin\theta$, which in terms of the EoS can be rewritten as $w^{\prime}_{\phi}=-3(1-w^{2}_{\phi})$. The foregoing equation is exactly the one for skater modelsLinder (2005).

We now revise the case of a constant EoS $w_{\phi}=w_{\phi 0}$. Although this can be seen as the simplest generalization of the cosmological constant, from Eqs. (6) we find that a constant EoS in the quintessence case could be obtained if, apart from $y_{1}=0$, we also impose $\theta_{i}=-\cos^{-1}(-w_{\phi 0})$ and $\theta^{\prime}=0$. However, the latter condition cannot be sustained if $\theta\neq 0$, see Eq. (6a), and then we get the same situation for skater models described in the above paragraph in which $\theta\to 0$. That is, the quintessence EoS cannot remain constant and must evolve towards the value $w_{\phi}\to-1$. These same calculations show that the only consistent conditions for a constant EoS are $\theta=0$ and $\theta^{\prime}=0$, which correspond exactly to the cosmological constant case.

The morale from this discussion is then twofold. First, it is not possible to find a quintessence solution that emulates a constant EoS for DE apart from the cosmological constant case. And second, in terms of the parameters in our approach, the cosmological constant is just the null hypothesis ($\theta=0$ and $y_{1}=0$), and then any deviation from the null value of the dynamical variables and parameters will be a measurement of the preference of the data catalogs on the quintessence models.

One of the primary cosmological parameters in the studies of DE is the EoS of the DE field. In our approach, the EoS is, through the relation $w_{\phi}=-\cos\theta$, also one of the dynamical variables to describe the evolution of the quintessence field. Here we will discuss the influence of the dynamical parameters $\alpha$ and in doing so we will determine the behavior of the EoS under general quintessence potentials.

We show in Fig. 1 the plots of $1+w_{\phi}$ as a function of the redshift $z$ for different values of the dynamical parameters $\alpha_{0},\alpha_{1}$ and $\alpha_{2}$, respectively. The plots are for the quantity $1+\omega_{\phi}$ instead of $\omega_{\phi}$ as we are interested about the deviations of the quintessence EoS from the cosmological constant case. But also because at present we can make the approximation $1+w_{\phi 0}\simeq\theta^{2}_{0}/2$, and we see that it is variable $\theta$ that provides such deviations. The numerical solutions are grouped according to the Class in Table 2 they belong to and for the indicated values of the dynamical parameters.

From these plots, one can clearly see that the variation of the EoS is more sensitive to $\alpha_{2}$, and less sensitive to the variation of $\alpha_{0}$. This is just the expected result as $\alpha_{2}$ is the partner coefficient of $y^{2}_{1}/y^{2}$ in the series expansion (11). It is interesting to note from Fig. 1 that a desired value of EoS of the dark energy can be obtained for a wide range of $\alpha$ parameters. Recent cosmological observations can only constrain the present value of the EOS, but unless there is any constraint on the evolution of the EOS it will not be possible to choose from the solutions for different $\alpha$ parameters. Hence, our expectation is that the statistical analysis using cosmological observations in the next section will not be able to constrain the $\alpha$ parameters. Additionally, we also see that the curves show a monotonic growth if the dynamical parameters are positive, but the curves develop a bump (ie a maximum appears) if the parameters are negative enough. We can only speculate that this latter effect seems to be an indication for the possible appearance of oscillations in the evolution of the EoS, but we will leave this topic for a future study.

As for the monotonic growing behavior at late times that is found for positive values of the dynamical parameters, it can be fit by the following expression,

where $w_{1}$ and $\gamma$ are free parameters. Notice that $w_{\phi 0}$ is explicitly present in Eq. (17) to ensure that the actual value of the EoS is obtained at $a=1$. The second term on the rhs of Eq. (17) corresponds to the expected behavior during matter domination: from the leading solution in Eq. (14a), and together with Eq. (15), we obtain that $(1+w_{\phi})_{m}\simeq(1/2)\theta^{2}_{m}\simeq(1/2)\theta^{2}_{0}a^{3}$. Hence, for those cases in which this approximation is good enough until the present time we expect that $w_{1}\sim 1$ and $\gamma\sim 0$. Any difference with respect to these values will signal the transition from matter to quintessence domination and of the presence of the dynamical parameters $\alpha$.

The results from a least-squares fitting of the parametrization (17) to the numerical solutions obtained from CLASS in some selected cases are shown in Table 3 and in Fig. 2, in all examples we considered the scale factor $a$ in the range $[0.1:1]$. It can be verified that the fits are indeed very good in all cases as the standard errors around the obtained values of the parameters are $\lesssim 1\%$. Not surprisingly, it is consistently found that $\gamma\gtrsim 0$, which indicates that the EoS accelerates its growth from $-1$ as the quintessence field starts to dominate the matter budget.

Equation (17) can be compared with other parameterizations of the dark energy EoS, like the famous Chevalier-Polarski-Linder one: $w=w_{0}+w_{1}(1-a)$, which is clearly inappropriate to describe the evolution of the quintessence models in this work. There exist other parameterizations, see for instanceJaber and de la Macorra (2017); de la Macorra (2016); Akarsu et al. (2015); Escamilla-Rivera et al. (2016); Zhao et al. (2017); Jaime et al. (2018) and references therein, but they usually have a more complicated form than Eq. (17). Although they may serve to test more complicated DE models, they are certainly not the best options to test one DE model as simple as quintessence.

Like in the case of the CPL parametrization, notice that Eq. (17) uses the present value of the EoS as an explicit parameter, but one clear advantage of our approach is that we are parameterizing the underlying dynamical variable $\theta$, and then we are recovering the right behavior of the EoS at early times. Notwithstanding this, we will not pursue a study of the dynamics represented by the parametrized EoS (17) because of the obvious degeneracies with the dynamical parameters: one can see from Table 3 that different combinations of the $\alpha$’s will result in similar values of the free parameters $\gamma$ and $w_{1}$. Also, our parameterization (17) is only valid for redshifts $z\lesssim 10$, as for larger redshifts we need to take into account the full solutions for radiation domination and the radiation-matter transition, see Sec. III above. All of this makes any reconstruction of the quintessence potential from the EoS parametrization fruitless, and then it is more convenient to work directly with the dynamical variables $\alpha$ extracted from the potentials.

We use an amended version of the Boltzmann code CLASS Lesgourgues (2011) and the Monte Carlo code Monte Python Brinckmann and Lesgourgues (2018); Audren et al. (2013). Amendments to CLASS includes those necessary for MontePhyton to be able to sample the parameters that we describe next. There are 6 parameters that we want to constrain: $\theta_{0},y_{10},\Omega_{\phi 0},\alpha_{0},\alpha_{1},\alpha_{2}$, but only 5 of them are required as input parameters, namely: $\Omega_{\phi 0},\theta_{0},\alpha_{0},\alpha_{1},\alpha_{2}$, because the value of $y_{10}$ is to be inferred from the full numerical evolution. It must be stressed out that as we sample the values of $\theta_{0}$ we will also be sampling the present values of the quintessence EoS $w_{\phi 0}$ through the relation $w_{\phi 0}=-\cos\theta_{0}$. In practice, the present EoS is then an input value and we will have full control of its sampling, which is another advantage of our method and variables over the standard approach to quintessence fields.

As in many other instances, we still need to finely tune the initial values of the dynamical variables at the beginning of every numerical run. For that, we write $y_{1i}=5\theta_{i}$, $\theta_{i}=P\times$Eq. (15a) and $\Omega_{\phi i}=Q\times$Eq. (15b), where the values of $P$ and $Q$ are adjusted with the shooting method already implemented within CLASS for scalar field models. A few iterations of the shooting routine are enough to find the correct values of $\theta_{i}$, $y_{1i}$ and $\Omega_{\phi i}$ that lead to the desired $\Omega_{\phi 0}$ and $w_{\phi 0}$ with a very high precision; in all instances it has been found that $P,Q=\mathcal{O}(1)$, which indicates that Eqs. (15) are good approximations to the required initial conditions. Here we only consider the background dynamics of the quintessence fields and leave the study of their linear perturbations for a future work.

In doing a full sampling of the dynamical parameters $\alpha_{0},\alpha_{1},\alpha_{2}$, we will also be sampling the general form of the potentials shown in Table 2. This way we expect to be able to impose constrains on the dynamical parameters but not on the passive ones of the potential $V(\phi)$. As explained before, these other parameters are related and can obtained from the dynamical variables $\theta_{0}$, $y_{10}$ and $\Omega_{\phi 0}$, although this would have to be done case by case for each one of the potentials in Table 2. For purposes of generality, we will focus on the constraints to the dynamical parameters and consider only two examples of constraints on passive parameters.

We use two data sets that are sensitive to the background quantities: (i) the SDSS-II/SNLS3 Joint Light-curve Analysis (JLA) supernova dataBetoule et al. (2014) and (ii) BAO measurements (Barionic acoustic oscillations), in this case the following datasets are included in the likelihood: 2dFGS,MGS, DR11 LOWZ and DR11 CMASS Anderson et al. (2014). We imposed a Planck2015 prior on the the baryonic and cold dark matter contributionAdam et al. (2016); Aghanim et al. (2016); Ade et al. (2016b, c, d): $\omega_{b}=0.02230\pm 0.00014$ and $\omega_{cdm}=0.1188\pm 0.0010$; whereas for the scalar field parameter we used flat priors in the range $-20<\alpha_{0}<20$, $-5<\alpha_{1}<5$ and $-2<\alpha_{2}<2$. The total set of parameters being sampled are: $\omega_{b}$, $\omega_{b}$, $H_{0}$, and the scalar field contribution $\Omega_{\phi 0}$ is set by the closure relation for the given $\theta_{0}$, $\alpha_{0}$, $\alpha_{1}$ and $\alpha_{2}$; whereas the set of derived parameters is: $\Omega_{m}$, $\Omega_{\phi}$, $w_{\phi}$ and $y_{1}$.

The general constraints on the parameters of the quintessence models are shown in Fig. 3, where the 1D and 2D posterior distributions are represented in a triangle plot;it is also shown the Mean Likelihood Estimate (MELE) (dashed lines), which is another output from the Montepython code. In what follows we report our results using the median values of the 1D Posterior (not the mean likelihood) plus/minus a confidence interval, which is defined as the range containing 90% of the samples. This particular choice is because some of the posterior parameters are not Gaussian. All of our analyses achieved a convergence ratio (Gelman-Rubin criteria) of $R-1\approx 0.005$ for the standard cosmological parameters, although for some of the scalar field parameters the best convergence ratio we got did not go below $R-1\approx 0.01$. Notice that all parameters are well constrained to some region of the parameter space, in both the posteriors and the MELE, except for the dynamical parameters $\alpha$ whose posterior distributions are plainly flat along the full prior range. This means that the data sets considered does not show any preference for a particular quintessence potential.

The present contribution of the quintessence field to the matter budget results in $\Omega_{\phi 0}=0.719^{+0.015}_{-0.015}$, which is in agreement with previous studiesAde et al. (2016a). Similarly, we find that the EoS $1+w_{\phi 0}<0.107$ and that $0<y_{1}<2.24$ (95% C.L.), values that are in agreement with the cosmological constant value $w_{\phi}=-1$ and $y_{1}=0$. These results together show that the quintessence models revolve around the cosmological constant values.

We now go back to the flat posteriors of the dynamical parameters $\alpha$. It means that a solution of the quintessence field compatible with the observational dataset can always be found for any value of the dynamical parameters, and the reason behind such result is that the initial conditions of the quintessence variables can be finely tuned accordingly to compensate for any $\alpha\neq 0$. For instance, for larger values of any of the dynamical parameters we can start the field evolution closer to the cosmological constant case, so that initially $w_{\phi}\to-1$ as much as necessary.

Moreover, the flat posteriors in Fig. 3 also imply that there is not clear preference for any of the classes of potentials in Table 2. Given this situation, it may be reasonable to just consider the most economic possibility which is Class Ia in Table 2: $V(\phi)=(A+B\phi)^{2}$. As discussed in Sec. II.4, one actually recover the quadratic potential if $A=0$ and $B=m_{\phi}/\sqrt{2}$, and then we can say that for practical purposes no quintessence potential can fit the data any better than the quadratic potential.

We now turn our attention to the passive parameters in the quintessence potentials. In contrast to dynamical parameters, we shall argue that the passive ones can be subjected to observational constraints. It must be noticed that passive parameters, in their role as integration constants in Table 2, can only be determined if we fix either the initial or the final conditions in the solutions of the equations of motion. In the cosmological context we are interested in the final conditions as it is necessary to adjust the parameters to recover the present values of different observables.

Taking as a reference the quadratic potential again, for which the mass of the scalar field $m_{\phi}$ is a passive parameter, we show in the top panel of Fig. 4 the posteriors of different cosmological quantities. The fit indicates that $\Omega_{m0}=0.304^{+0.018}_{-0.016}$, $\Omega_{\phi 0}=0.695^{+0.016}_{-0.018}$, $1+w_{\phi 0}<0.129$, and $m_{\phi}<1.36\times 10^{-33}\,\mathrm{eV}$ (95% C.L.). It can be seen that the preferred value of the EoS is close to $-1$, and that the scalar field mass $m_{\phi}$ has an upper bound. The latter constraint can be easily understood if we recall that the field mass can be calculated from the expression $m_{\phi}=(1/2)y_{10}H_{0}$, and then any bounds on the scalar field mass are directly obtained from those on the present values $y_{10}$ and $H_{0}$.

Other cases, though, are not as clear as the quadratic one. Let us consider the axion potential, which we write as $V(\phi)=m^{2}_{\phi}f^{2}_{\phi}\left[1+\cos(\phi/f_{\phi})\right]$, where $m_{\phi}$ is the mass of the axion field and $f_{\phi}$ is its so-called decay constant. The axion potential belongs to Class IIa with $\alpha_{0}=3/(\kappa^{2}f^{2}_{\phi})$ (together with $\alpha_{1}=0=\alpha_{2}$), which indicates that, according to our classification, $f_{\phi}$ is a dynamical parameter, whereas the combination $m^{2}_{\phi}f^{2}_{\phi}$ forms a passive one. The immediate result is that $f_{\phi}$ cannot be constrained from cosmological observations, which is at odds with previous results in the literature (see (Marsh, 2016) and references therein). Our interpretation here is that previous studies were not able to sample all possible values of $f_{\phi}$, mostly because of the intrinsic difficulties in solving the quintessence field equations in their normal form, see Eq. (1d). Smaller values of $f_{\phi}$ require the field to start closer to the top of the potential, and this is a tough numerical task even in our approach.

As for the passive parameter in the axion potential, it can be shown that it can be determined fromUrena-Lopez (2016)

The passive parameter of the quintessence potential cannot be written solely in terms of the cosmological observables ($H_{0}$) and the dynamical variables ($w_{\phi 0}$, $y_{10}$, $\rho_{\phi 0}$), and then it cannot be clearly constrained because of the presence of the decay constant $f_{\phi}$. Notice however that if $f_{\phi}\ll 1$ (in appropriate units), which corresponds to $\alpha_{0}\to\infty$, then $m^{2}_{\phi}f^{2}_{\phi}\simeq\rho_{\phi 0}(1-w_{\phi 0})/2$, where $\rho_{\phi 0}$ is the present quintessence density, and in this limit there appears an upper bound for the passive parameter basically inherited from the one on $\rho_{\phi 0}$. Likewise, if $f_{\phi}\gg 1$, corresponding to $\alpha_{0}\to 0$, we find that $m_{\phi}f_{\phi}\simeq(y_{10}H_{0}/2)f_{\phi}$, which shows that the passive parameter in this limit will be unbounded from above and that $m_{\phi}\simeq(y_{10}H_{0}/2)$, which is exactly the result obtained for the quadratic potential.

The posterior distributions for the axion case are shown in the bottom panel of Fig. 4. The fit indicates that $\Omega_{m0}=0.288^{+0.037}_{-0.035}$, $\Omega_{\phi 0}=0.712^{+0.035}_{-0.037}$, $1+w_{\phi 0}<0.154$, and $m^{2}_{\phi}f^{2}_{\phi}<3.66\times 10^{-10}\,\mathrm{eV}^{4}$ (95% C.L.). Apart from the upper bound for the passive parameter $m_{\phi}f_{\phi}$, the preferred values of the other parameters are similar to those of the quadratic potential (top panel in Fig. 4) and also to those of the general case shown in Fig. 3. Hence, the study of particular cases does not provide stronger bounds for the cosmological parameters.