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), [266] parametrize the case of slow-rolling fields, [796] study thawing quintessence, [446] and
[232] include non-minimally coupled fields, [817] quintom quintessence, [325] parametrize hilltop
quintessence, [231] extend the quintessence parametrization to a class of
-essence models, [459]
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 [956
]. We recall it here because we will
refer to this example in Section 1.6.1.1. 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
),
[229
], [584
] (linear and power law parametrization in
), [322], [97] (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 [942]:
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 [534
].
Another useful possibility is to adopt the principal component approach [468], 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
[939
].
http://www.livingreviews.org/lrr-2013-6 |
Living Rev. Relativity 16, (2013), 6
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