The energy density due to a cosmological constant with is obviously constant over time. This can easily be seen from the covariant conservation equation for the homogeneous and isotropic FLRW metric,
One of the simplest models that explicitly realizes such a dynamical dark energy scenario is described by a minimally-coupled canonical scalar field evolving in a given potential. For this reason, the very concept of dynamical dark energy is often associated with this scenario, and in this context it is called ‘quintessence’ [954, 754]. In the following, the scalar field will be indicated with . Although in this simplest framework the dark energy does not interact with other species and influences spacetime only through its energy density and pressure, this is not the only possibility and we will encounter more general models later on. The homogeneous energy density and pressure of the scalar field are defined as
However, when GR is modified or when an interaction with other species is active, dark energy may very well have a non-negligible contribution at early times. Therefore, it is important, already at the background level, to understand the best way to characterize the main features of the evolution of quintessence and dark energy in general, pointing out which parameterizations are more suitable and which ranges of parameters are of interest to disentangle quintessence or modified gravity from a cosmological constant scenario.
In the following we briefly discuss how to describe the cosmic expansion rate in terms of a small number of parameters. This will set the stage for the more detailed cases discussed in the subsequent sections. Even within specific physical models it is often convenient to reduce the information to a few phenomenological parameters.
Two important points are left for later: from Eq. (1.2.3) we can easily see that as long as , i.e., uncoupled canonical scalar field dark energy never crosses . However, this is not necessarily the case for non-canonical scalar fields or for cases where GR is modified. We postpone to Section 1.4.5 the discussion of how to parametrize this ‘phantom crossing’ to avoid singularities, as it also requires the study of perturbations.
The second deferred part on the background expansion concerns a basic statistical question: what is a sensible precision target for a measurement of dark energy, e.g., of its equation of state? In other words, how close to should we go before we can be satisfied and declare that dark energy is the cosmological constant? We will address this question in Section 1.5.
If one wants to parametrize the equation of state of dark energy, two general approaches are possible. The first is to start from a set of dark-energy models given by the theory and to find parameters describing their as accurately as possible. Only later can one try and include as many theoretical models as possible in a single parametrization. In the context of scalar-field dark-energy models (to be discussed in Section 1.4.1),  parametrize the case of slow-rolling fields,  study thawing quintessence,  and  include non-minimally coupled fields,  quintom quintessence,  parametrize hilltop quintessence,  extend the quintessence parametrization to a class of -essence models,  study a common parametrization for quintessence and phantom fields. Another convenient way to parametrize the presence of a non-negligible homogeneous dark energy component at early times (usually labeled as EDE) was presented in . We recall it here because we will refer to this example in Section 22.214.171.124. In this case the equation of state is parametrized as:
The second approach is to start from a simple expression of without assuming any specific dark-energy model (but still checking afterwards whether known theoretical dark-energy models can be represented). This is what has been done by [470, 623, 953] (linear and logarithmic parametrization in ), ,  (linear and power law parametrization in ), ,  (rapidly varying equation of state).
The most common parametrization, widely employed in this review, is the linear equation of state [229, 584]
An alternative to model-independent constraints is measuring the dark-energy density (or the expansion history ) as a free function of cosmic time [942, 881, 274]. Measuring has advantages over measuring the dark-energy equation of state as a free function; is more closely related to observables, hence is more tightly constrained for the same number of redshift bins used [942, 941]. Note that is related to as follows :
Note that the measurement of is straightforward once is measured from baryon acoustic oscillations, and is constrained tightly by the combined data from galaxy clustering, weak lensing, and cosmic microwave background data – although strictly speaking this requires a choice of perturbation evolution for the dark energy as well, and in addition one that is not degenerate with the evolution of dark matter perturbations; see .
Another useful possibility is to adopt the principal component approach , which avoids any assumption about the form of and assumes it to be constant or linear in redshift bins, then derives which combination of parameters is best constrained by each experiment.
For a cross-check of the results using more complicated parameterizations, one can use simple polynomial parameterizations of and .
Living Rev. Relativity 16, (2013), 6
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