1.2 Background evolution

Most of the calculations in this review are performed in the Friedmann–Lemaître–Robertson–Walker (FLRW) metric
2 2 2 2 --dr---- 2 2 2 2 2 ds = − dt + a(t) (1 − kr2 + r d 𝜃 + r sin 𝜃dϕ ), (1.2.1 )
where a(t) is the scale factor and k the spatial curvature. The usual symbols for the Hubble function H = a˙∕a and the density fractions Ωx, where x stands for the component, are employed. We characterize the components with the subscript M or m for matter, γ or r for radiation, b for baryons, K for curvature and Λ for the cosmological constant. Whenever necessary for clarity, we append a subscript 0 to denote the present epoch, e.g., Ω M,0. Sometimes the conformal time ∫ η = dt∕a and the conformal Hubble function ℋ = aH = da∕(ad η) are employed. Unless otherwise stated, we denote with a dot derivatives w.r.t. cosmic time t (and sometimes we employ the dot for derivatives w.r.t. conformal time η) while we use a prime for derivatives with respect to lna.

The energy density due to a cosmological constant with p = − ρ is obviously constant over time. This can easily be seen from the covariant conservation equation ν Tμ;ν = 0 for the homogeneous and isotropic FLRW metric,

˙ρ + 3H (ρ + p) = 0. (1.2.2 )
However, since we also observe radiation with p = ρ∕3 and non-relativistic matter for which p ≈ 0, it is natural to assume that the dark energy is not necessarily limited to a constant energy density, but that it could be dynamical instead.

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’ [954Jump To The Next Citation Point, 754Jump To The Next Citation Point]. In the following, the scalar field will be indicated with ϕ. 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 ϕ are defined as

2 2 ϕ˙- ˙ϕ-- pϕ- ρϕ = 2 + V (ϕ), pϕ = 2 − V (ϕ), w ϕ = ρϕ, (1.2.3 )
and wϕ is called the equation-of-state parameter. Minimally-coupled dark-energy models can allow for attractor solutions [252, 573, 867]: if an attractor exists, depending on the potential V(ϕ ) in which dark energy rolls, the trajectory of the scalar field in the present regime converges to the path given by the attractor, though starting from a wide set of different initial conditions for ϕ and for its first derivative ˙ ϕ. Inverse power law and exponential potentials are typical examples of potential that can lead to attractor solutions. As constraints on w ϕ become tighter [e.g., 526Jump To The Next Citation Point], the allowed range of initial conditions to follow into the attractor solution shrinks, so that minimally-coupled quintessence is actually constrained to have very flat potentials. The flatter the potential, the more minimally-coupled quintessence mimics a cosmological constant, the more it suffers from the same fine-tuning and coincidence problems that affect a ΛCDM scenario [646Jump To The Next Citation Point].

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. (1.2.3View Equation) we can easily see that w ϕ ≥ − 1 as long as ρϕ > 0, i.e., uncoupled canonical scalar field dark energy never crosses wϕ = − 1. 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 w ϕ = − 1 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.

1.2.1 Parametrization of the background evolution

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 w ϕ 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 k-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 [956Jump To The Next Citation Point]. We recall it here because we will refer to this example in Section In this case the equation of state is parametrized as:

w (z) = ------w0------, (1.2.4 ) X 1 + bln(1 + z)
where b is a constant related to the amount of dark energy at early times, i.e.,
3¯w0 b = − --1−ΩX,e-----1−Ωm,0. (1.2.5 ) ln-ΩX,e- + ln -Ωm,0-
Here the subscripts ‘0’ and ‘e’ refer to quantities calculated today or early times, respectively. With regard to the latter parametrization, we note that concrete theoretical and realistic models involving a non-negligible energy component at early times are often accompanied by further important modifications (as in the case of interacting dark energy), not always included in a parametrization of the sole equation of state such as (1.2.4View Equation) (for further details see Section 1.6 on nonlinear aspects of dark energy and modified gravity).

The second approach is to start from a simple expression of w 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 [470Jump To The Next Citation Point, 623, 953] (linear and logarithmic parametrization in z), [229Jump To The Next Citation Point], [584Jump To The Next Citation Point] (linear and power law parametrization in a), [322], [97] (rapidly varying equation of state).

The most common parametrization, widely employed in this review, is the linear equation of state [229Jump To The Next Citation Point, 584Jump To The Next Citation Point]

wX (a) = w0 + wa (1 − a ), (1.2.6 )
where the subscript X refers to the generic dark-energy constituent. While this parametrization is useful as a toy model in comparing the forecasts for different dark-energy projects, it should not be taken as all-encompassing. In general a dark-energy model can introduce further significant terms in the effective wX (z) that cannot be mapped onto the simple form of Eq. (1.2.6View Equation).

An alternative to model-independent constraints is measuring the dark-energy density ρX (z) (or the expansion history H (z)) as a free function of cosmic time [942Jump To The Next Citation Point, 881, 274]. Measuring ρX (z) has advantages over measuring the dark-energy equation of state wX (z) as a free function; ρX (z) is more closely related to observables, hence is more tightly constrained for the same number of redshift bins used [942Jump To The Next Citation Point, 941]. Note that ρX (z) is related to wX (z) as follows [942]:

{ ∫ z ′ } ρX(z)- ′3[1 +-wX-(z-)] ρX(0) = exp 0 dz 1 + z′ . (1.2.7 )
Hence, parametrizing dark energy with wX (z) implicitly assumes that ρX (z) does not change sign in cosmic time. This precludes whole classes of dark-energy models in which ρX(z) becomes negative in the future (“Big Crunch” models, see [943] for an example) [944].

Note that the measurement of ρ (z) X is straightforward once H (z) is measured from baryon acoustic oscillations, and Ωm 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 [534Jump To The Next Citation Point].

Another useful possibility is to adopt the principal component approach [468Jump To The Next Citation Point], which avoids any assumption about the form of w 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 w and ρDE(z)∕ρDE (0) [939Jump To The Next Citation Point].

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