In this review we refer to scalar field models with canonical kinetic energy in Einstein’s gravity as “quintessence models”. Scalar fields are obvious candidates for dark energy, as they are for the inflaton, for many reasons: they are the simplest fields since they lack internal degrees of freedom, do not introduce preferred directions, are typically weakly clustered (as discussed later on), and can easily drive an accelerated expansion. If the kinetic energy has a canonical form, the only degree of freedom is then provided by the field potential (and of course by the initial conditions). The typical requirement is that the potentials are flat enough to lead to the slow-roll inflation today with an energy scale and a mass scale .

Quintessence models are the protoypical DE models [195] and as such are the most studied
ones. Since they have been explored in many reviews of DE, we limit ourselves here to a few
remarks.^{2}

The quintessence model is described by the action

where and is the Ricci scalar and is the matter action. The fluid satisfies the continuity equation The energy-momentum tensor of quintessence is As we have already seen, in a FLRW background, the energy density and the pressure of the field are which give the equation of state In the flat universe, Einstein’s equations give the following equations of motion: where . The variation of the action (1.4.1) with respect to gives where .During radiation or matter dominated epochs, the energy density of the fluid dominates over that of quintessence, i.e., . If the potential is steep so that the condition is always satisfied, the field equation of state is given by from Eq. (1.4.6). In this case the energy density of the field evolves as , which decreases much faster than the background fluid density.

The condition is required to realize the late-time cosmic acceleration, which translates into the condition . Hence the scalar potential needs to be shallow enough for the field to evolve slowly along the potential. This situation is similar to that in inflationary cosmology and it is convenient to introduce the following slow-roll parameters [104]

If the conditions and are satisfied, the evolution of the field is sufficiently slow so that and in Eqs. (1.4.7) and (1.4.9).From Eq. (1.4.9) the deviation of from is given by

where . This shows that is always larger than for a positive potential and energy density. In the slow-roll limit, and , we obtain by neglecting the matter fluid in Eq. (1.4.7), i.e., . The deviation of from is characterized by the slow-roll parameter . It is also possible to consider Eq. (1.4.11) as a prescription for the evolution of the potential given and to reconstruct a potential that gives a desired evolution of the equation of state (subject to ). This was used, for example, in [102].However, in order to study the evolution of the perturbations of a quintessence field it is not even necessary to compute the field evolution explicitly. Rewriting the perturbation equations of the field in terms of the perturbations of the density contrast and the velocity in the conformal Newtonian gauge, one finds [see, e.g., 536, Appendix A] that they correspond precisely to those of a fluid, (1.3.17) and (1.3.18), with and with . The adiabatic sound speed, , is defined in Eq. (1.4.31). The large value of the sound speed , equal to the speed of light, means that quintessence models do not cluster significantly inside the horizon [see 785, 786, and Section 1.8.6 for a detailed analytical discussion of quintessence clustering and its detectability with future probes, for arbitrary ].

Many quintessence potentials have been proposed in the literature. A simple crude classification divides them into two classes, (i) “freezing” models and (ii) “thawing” models [196]. In class (i) the field was rolling along the potential in the past, but the movement gradually slows down after the system enters the phase of cosmic acceleration. The representative potentials that belong to this class are

(i) Freezing models

- ,
- .

The former potential does not possess a minimum and hence the field rolls down the potential toward infinity. This appears, for example, in the fermion condensate model as a dynamical supersymmetry breaking [138]. The latter potential has a minimum at which the field is eventually trapped (corresponding to ). This potential can be constructed in the framework of supergravity [170].

In thawing models (ii) the field (with mass ) has been frozen by Hubble friction (i.e., the term in Eq. (1.4.9)) until recently and then it begins to evolve once drops below . The equation of state of DE is at early times, which is followed by the growth of . The representative potentials that belong to this class are

(ii) Thawing models

- ,
- .

The former potential is similar to that of chaotic inflation () used in the early universe (with [577], while the mass scale is very different. The model with was proposed by [487] in connection with the possibility to allow for negative values of . The universe will collapse in the future if the system enters the region with . The latter potential appears as a potential for the Pseudo-Nambu–Goldstone Boson (PNGB). This was introduced by [370] in response to the first tentative suggestions that the universe may be dominated by the cosmological constant. In this model the field is nearly frozen at the potential maximum during the period in which the field mass is smaller than , but it begins to roll down around the present ().

Potentials can also be classified in several other ways, e.g., on the basis of the existence of special solutions. For instance, tracker solutions have approximately constant and along special attractors. A wide range of initial conditions converge to a common, cosmic evolutionary tracker. Early DE models contain instead solutions in which DE was not negligible even during the last scattering. While in the specific Euclid forecasts section (1.8) we will not explicitly consider these models, it is worthwhile to note that the combination of observations of the CMB and of large scale structure (such as Euclid) can dramatically constrain these models drastically improving the inverse area figure of merit compared to current constraints, as discussed in [467].

In a quintessence model it is the potential energy of a scalar field that leads to the late-time acceleration of the expansion of the universe; the alternative, in which the kinetic energy of the scalar field which dominates, is known as k-essence. Models of k-essence are characterized by an action for the scalar field of the following form

where . The energy density of the scalar field is given by and the pressure is simply . Treating the k-essence scalar as a perfect fluid, this means that k-essence has the equation of state where the subscript indicates a derivative with respect to . Clearly, with a suitably chosen the scalar can have an appropriate equation of state to allow it to act as dark energy.The dynamics of the k-essence field are given by a continuity equation

or equivalently by the scalar equation of motion where For this second order equation of motion to be hyperbolic, and hence physically meaningful, we must impose K-essence was first proposed by [61, 62], where it was also shown that tracking solutions to this equation of motion, which are attractors in the space of solutions, exist during the radiation and matter-dominated eras for k-essence in a similar manner to quintessence.The speed of sound for k-essence fluctuation is

So that whenever the kinetic terms for the scalar field are not linear in , the speed of sound of fluctuations differs from unity. It might appear concerning that superluminal fluctuations are allowed in k-essence models, however it was shown in [71] that this does not lead to any causal paradoxes.

In this review we often make reference to DE and MG models. Although in an increasing number of publications a similar dichotomy is employed, there is currently no consensus on where to draw the line between the two classes. Here we will introduce an operational definition for the purpose of this document.

Roughly speaking, what most people have in mind when talking about standard dark energy are models of minimally-coupled scalar fields with standard kinetic energy in 4-dimensional Einstein gravity, the only functional degree of freedom being the scalar potential. Often, this class of model is referred to simply as “quintessence”. However, when we depart from this picture a simple classification is not easy to draw. One problem is that, as we have seen in the previous sections, both at background and at the perturbation level, different models can have the same observational signatures [537]. This problem is not due to the use of perturbation theory: any modification to Einstein’s equations can be interpreted as standard Einstein gravity with a modified “matter” source, containing an arbitrary mixture of scalars, vectors and tensors [457, 535].

The simplest example can be discussed by looking at Eqs. (1.3.23). One can modify gravity and obtain a modified Poisson equation, and therefore , or one can introduce a clustering dark energy (for example a k-essence model with small sound speed) that also induces the same (see Eq. 1.3.23). This extends to the anisotropic stress : there is in general a one-to-one relation at first order between a fluid with arbitrary equation of state, sound speed, and anisotropic stress and a modification of the Einstein–Hilbert Lagrangian.

Therefore, we could simply abandon any attempt to distinguish between DE and MG, and just analyse different models, comparing their properties and phenomenology. However, there is a possible classification that helps us set targets for the observations, which is often useful in concisely communicating the results of complex arguments. In this review, we will use the following notation:

- Standard dark energy: These are models in which dark energy lives in standard Einstein gravity and does not cluster appreciably on sub-horizon scales. As already noted, the prime example of a standard dark-energy model is a minimally-coupled scalar field with standard kinetic energy, for which the sound speed equals the speed of light.
- Clustering dark energy: In clustering dark-energy models, there is an additional contribution to the Poisson equation due to the dark-energy perturbation, which induces . However, in this class we require , i.e., no extra effective anisotropic stress is induced by the extra dark component. A typical example is a k-essence model with a low sound speed, .
- Explicit modified gravity models: These are models where from the start the Einstein equations are modified, for example scalar-tensor and type theories, Dvali–Gabadadze–Porrati (DGP) as well as interacting dark energy, in which effectively a fifth force is introduced in addition to gravity. Generically they change the clustering and/or induce a non-zero anisotropic stress. Since our definitions are based on the phenomenological parameters, we also add dark-energy models that live in Einstein’s gravity but that have non-vanishing anisotropic stress into this class since they cannot be distinguished by cosmological observations.

Notice that both clustering dark energy and explicit modified gravity models lead to deviations from what is often called ‘general relativity’ (or, like here, standard dark energy) in the literature when constraining extra perturbation parameters like the growth index . For this reason we generically call both of these classes MG models. In other words, in this review we use the simple and by now extremely popular (although admittedly somewhat misleading) expression “modified gravity” to denote models in which gravity is modified and/or dark energy clusters or interacts with other fields. Whenever we feel useful, we will remind the reader of the actual meaning of the expression “modified gravity” in this review.

Therefore, on sub-horizon scales and at first order in perturbation theory our definition of MG is straightforward: models with (see Eq. 1.3.23) are standard DE, otherwise they are MG models. In this sense the definition above is rather convenient: we can use it to quantify, for instance, how well Euclid will distinguish between standard dynamical dark energy and modified gravity by forecasting the errors on or on related quantities like the growth index .

On the other hand, it is clear that this definition is only a practical way to group different models and should not be taken as a fundamental one. We do not try to set a precise threshold on, for instance, how much dark energy should cluster before we call it modified gravity: the boundary between the classes is therefore left undetermined but we think this will not harm the understanding of this document.

A first class of models in which dark energy shows dynamics, in connection with the presence of a fifth force different from gravity, is the case of ‘interacting dark energy’: we consider the possibility that dark energy, seen as a dynamical scalar field, may interact with other components in the universe. This class of models effectively enters in the “explicit modified gravity models” in the classification above, because the gravitational attraction between dark matter particles is modified by the presence of a fifth force. However, we note that the anisotropic stress for DE is still zero in the Einstein frame, while it is, in general, non-zero in the Jordan frame. In some cases (when a universal coupling is present) such an interaction can be explicitly recast in a non-minimal coupling to gravity, after a redefinition of the metric and matter fields (Weyl scaling). We would like to identify whether interactions (couplings) of dark energy with matter fields, neutrinos or gravity itself can affect the universe in an observable way.

In this subsection we give a general description of the following main interacting scenarios:

- couplings between dark energy and baryons;
- couplings between dark energy and dark matter (coupled quintessence);
- couplings between dark energy and neutrinos (growing neutrinos, MaVaNs);
- universal couplings with all species (scalar-tensor theories and ).

In all these cosmologies the coupling introduces a fifth force, in addition to standard gravitational attraction. The presence of a new force, mediated by the DE scalar field (sometimes called the ‘cosmon’ [954], seen as the mediator of a cosmological interaction) has several implications and can significantly modify the process of structure formation. We will discuss cases (2) and (3) in Section 2.

In these scenarios the presence of the additional interaction couples the evolution of components that in the standard -FLRW would evolve independently. The stress-energy tensor of each species is, in general, not conserved – only the total stress-energy tensor is. Usually, at the level of the Lagrangian, the coupling is introduced by allowing the mass of matter fields to depend on a scalar field via a function whose choice specifies the interaction. This wide class of cosmological models can be described by the following action:

where is the potential in which the scalar field rolls, describes matter fields, and is defined in the usual way as the determinant of the metric tensor, whose background expression is .For a general treatment of background and perturbation equations we refer to [514, 33, 35, 724]. Here the coupling of the dark-energy scalar field to a generic matter component (denoted by index ) is treated as an external source in the Bianchi identities:

with the constraintThe zero component of (1.4.21) gives the background conservation equations:

for a scalar field coupled to one single fluid with a function , which in general may not be constant. The choice of the mass function corresponds to a choice of and equivalently to a choice of the source and specifies the strength of the coupling according to the following relations: where is the constant Jordan-frame bare mass. The evolution of dark energy is related to the trace and, as a consequence, to density and pressure of the species . We note that a description of the coupling via an action such as (1.4.20) is originally motivated by the wish to modify GR with an extension such as scalar-tensor theories. In general, one of more couplings can be active [176].As for perturbation equations, it is possible to include the coupling in a modified Euler equation:

The Euler equation in cosmic time () can also be rewritten in the form of an acceleration equation for particles at position : The latter expression explicitly contains all the main ingredients that affect dark-energy interactions:- a fifth force with an effective ;
- a velocity dependent term
- a time-dependent mass for each particle , evolving according to (1.4.25).

The relative significance of these key ingredients can lead to a variety of potentially observable effects, especially on structure formation. We will recall some of them in the following subsections as well as, in more detail, for two specific couplings in the dark matter section (2.11, 2.9) of this report.

A coupling between dark energy and baryons is active when the baryon mass is a function of the dark-energy scalar field: . Such a coupling is constrained to be very small: main bounds come from tests of the equivalence principle and solar system constraints [130]. More in general, depending on the coupling, bounds on the variation of fundamental constants over cosmological time-scales may have to be considered ([631, 303, 304, 639] and references therein). It is presumably very difficult to have significant cosmological effects due to a coupling to baryons only. However, uncoupled baryons can still play a role in the presence of a coupling to dark matter (see Section 1.6 on nonlinear aspects).

An interaction between dark energy and dark matter (CDM) is active when CDM mass is a function of the dark-energy scalar field: . In this case the coupling is not affected by tests on the equivalence principle and solar-system constraints and can therefore be stronger than the one with baryons. One may argue that dark-matter particles are themselves coupled to baryons, which leads, through quantum corrections, to direct coupling between dark energy and baryons. The strength of such couplings can still be small and was discussed in [304] for the case of neutrino–dark-energy couplings. Also, quantum corrections are often recalled to spoil the flatness of a quintessence potential. However, it may be misleading to calculate quantum corrections up to a cutoff scale, as contributions above the cutoff can possibly compensate terms below the cutoff, as discussed in [958].

Typical values of presently allowed by observations (within current CMB data) are within the range (at 95% CL for a constant coupling and an exponential potential) [114, 47, 35, 44], or possibly more [539, 531] if neutrinos are taken into account or for more realistic time-dependent choices of the coupling. This framework is generally referred to as ‘coupled quintessence’ (CQ). Various choices of couplings have been investigated in literature, including constant and varying [33, 619, 35, 518, 414, 747, 748, 724, 377].

The presence of a coupling (and therefore, of a fifth force acting among dark-matter particles) modifies the background expansion and linear perturbations [34, 33, 35], therefore affecting CMB and cross-correlation of CMB and LSS [44, 35, 47, 45, 114, 539, 531, 970, 612, 42].

Furthermore, structure formation itself is modified [604, 618, 518, 611, 870, 3, 666, 129, 962, 79, 76, 77, 80, 565, 562, 75, 980, 640].

An alternative approach, also investigated in the literature [619, 916, 915, 613, 387, 388, 193, 794, 192], where the authors consider as a starting point Eq. (1.4.21): the coupling is then introduced by choosing directly a covariant stress-energy tensor on the RHS of the equation, treating dark energy as a fluid and in the absence of a starting action. The advantage of this approach is that a good parameterization allows us to investigate several models of dark energy at the same time. Problems connected to instabilities of some parameterizations or to the definition of a physically-motivated speed of sound for the density fluctuations can be found in [916]. It is also possible to both take a covariant form for the coupling and a quintessence dark-energy scalar field, starting again directly from Eq. (1.4.21). This has been done, e.g., in [145], [144]. At the background level only, [235], [237], [302] and [695] have also considered which background constraints can be obtained when starting from a fixed present ratio of dark energy and dark matter. The disadvantage of this approach is that it is not clear how to perturb a coupling that has been defined as a background quantity.

A Yukawa-like interaction was investigated [357, 279], pointing out that coupled dark energy behaves as a fluid with an effective equation of state , though staying well defined and without the presence of ghosts [279].

For an illustration of observable effects related to dark-energy–dark-matter interaction see also Section (2.11) of this report.

A coupling between dark energy and neutrinos can be even stronger than the one with dark matter and as compared to gravitational strength. Typical values of are order 50 – 100 or even more, such that even the small fraction of cosmic energy density in neutrinos can have a substantial influence on the time evolution of the quintessence field. In this scenario neutrino masses change in time, depending on the value of the dark-energy scalar field . Such a coupling has been investigated within MaVaNs [356, 714, 135, 12, 952, 280, 874, 856, 139, 178, 177] and more recently within growing neutrino cosmologies [36, 957, 668, 963, 962, 727, 179, 78]. In this latter case, DE properties are related to the neutrino mass and to a cosmological event, i.e., neutrinos becoming non-relativistic. This leads to the formation of stable neutrino lumps [668, 963, 78] at very large scales only ( 100 Mpc and beyond) as well as to signatures in the CMB spectra [727]. For an illustration of observable effects related to this case see Section (2.9) of this report.

Scalar-tensor theories [954, 471, 472, 276, 216, 217, 955, 912, 722, 354, 146, 764, 721, 797, 646, 725, 726, 205, 54] extend GR by introducing a non-minimal coupling between a scalar field (acting also as dark energy) and the metric tensor (gravity); they are also sometimes referred to as ‘extended quintessence’. We include scalar-tensor theories among ‘interacting cosmologies’ because, via a Weyl transformation, they are equivalent to a GR framework (minimal coupling to gravity) in which the dark-energy scalar field is coupled (universally) to all species [954, 608, 936, 351, 724, 219]. In other words, these theories correspond to the case where, in action (1.4.20), the mass of all species (baryons, dark matter, …) is a function with the same coupling for every species . Indeed, a description of the coupling via an action such as (1.4.20) is originally motivated by extensions of GR such as scalar-tensor theories. Typically the strength of the scalar-mediated interaction is required to be orders of magnitude weaker than gravity ([553], [725] and references therein for recent constraints). It is possible to tune this coupling to be as small as is required – for example by choosing a suitably flat potential for the scalar field. However, this leads back to naturalness and fine-tuning problems.

In Sections 1.4.6 and 1.4.7 we will discuss in more detail a number of ways in which new scalar degrees of freedom can naturally couple to standard model fields, while still being in agreement with observations. We mention here only that the presence of chameleon mechanisms [171, 672, 670, 172, 464, 173, 282] can, for example, modify the coupling depending on the environment. In this way, a small (screened) coupling in high-density regions, in agreement with observations, is still compatible with a bigger coupling () active in low density regions. In other words, a dynamical mechanism ensures that the effects of the coupling are screened in laboratory and solar system tests of gravity.

Typical effects of scalar-tensor theories on CMB and structure formation include:

- enhanced ISW [725, 391, 980];
- violation of the equivalence principle: extended objects such as galaxies do not all fall at the same rate [45, 464].

However, it is important to remark that screening mechanisms are meant to protect the scalar field in high-density regions (and therefore allow for bigger couplings in low density environments) but they do not address problems related to self-acceleration of the DE scalar field, which still usually require some fine-tuning to match present observations on . theories, which can be mapped into a subclass of scalar-tensor theories, will be discussed in more detail in Section 1.4.6.

In this section we pay attention to the evolution of the perturbations of a general dark-energy fluid with an evolving equation of state parameter . Current limits on the equation of state parameter of the dark energy indicate that , and so do not exclude , a region of parameter space often called phantom energy. Even though the region for which may be unphysical at the quantum level, it is still important to probe it, not least to test for coupled dark energy and alternative theories of gravity or higher dimensional models that can give rise to an effective or apparent phantom energy.

Although there is no problem in considering for the background evolution, there are apparent divergences appearing in the perturbations when a model tries to cross the limit . This is a potential headache for experiments like Euclid that directly probe the perturbations through measurements of the galaxy clustering and weak lensing. To analyze the Euclid data, we need to be able to consider models that cross the phantom divide at the level of first-order perturbations (since the only dark-energy model that has no perturbations at all is the cosmological constant).

However, at the level of cosmological first-order perturbation theory, there is no fundamental limitation that prevents an effective fluid from crossing the phantom divide.

As the terms in Eqs. (1.3.17) and (1.3.18) containing will generally diverge. This can be avoided by replacing with a new variable defined via . This corresponds to rewriting the - component of the energy momentum tensor as , which avoids problems if when . Replacing the time derivatives by a derivative with respect to the logarithm of the scale factor (denoted by a prime), we obtain [599, 450, 536]:

In order to solve Eqs. (1.4.28) and (1.4.29) we still need to specify the expressions for and , quantities that characterize the physical, intrinsic nature of the dark-energy fluid at first order in perturbation theory. While in general the anisotropic stress plays an important role as it gives a measure of how the gravitational potentials and differ, we will set it in this section to zero, . Therefore, we will focus on the form of the pressure perturbation. There are two important special cases: barotropic fluids, which have no internal degrees of freedom and for which the pressure perturbation is fixed by the evolution of the average pressure, and non-adiabatic fluids like, e.g., scalar fields for which internal degrees of freedom can change the pressure perturbation.

We have seen that neither barotropic fluids nor canonical scalar fields, for which the pressure perturbation is of the type (1.4.34), can cross the phantom divide. However, there is a simple model [called the quintom model 360, 451] consisting of two fluids of the same type as in the previous Section 1.4.5.1 but with a constant on either side of . The combination of the two fluids then effectively crosses the phantom divide if we start with , as the energy density in the fluid with will grow faster, so that this fluid will eventually dominate and we will end up with .

The perturbations in this scenario were analyzed in detail in [536], where it was shown that in addition to the rest-frame contribution, one also has relative and non-adiabatic perturbations. All these contributions apparently diverge at the crossing, but their sum stays finite. When parameterizing the perturbations in the Newtonian gauge as

the quantity will, in general, have a complicated time and scale dependence. The conclusion of the analysis is that indeed single canonical scalar fields with pressure perturbations of the type (1.4.34) in the Newtonian frame cannot cross , but that this is not the most general case. More general models have a priori no problem crossing the phantom divide, at least not with the classical stability of the perturbations.Kunz and Sapone [536] found that a good approximation to the quintom model behavior can be found by regularizing the adiabatic sound speed in the gauge transformation with

where is a tunable parameter which determines how close to the regularization kicks in. A value of should work reasonably well. However, the final results are not too sensitive on the detailed regularization prescription.This result appears also related to the behavior found for coupled dark-energy models (originally introduced to solve the coincidence problem) where dark matter and dark energy interact not only through gravity [33]. The effective dark energy in these models can also cross the phantom divide without divergences [462, 279, 534].

The idea is to insert (by hand) a term in the continuity equations of the two fluids

where the subscripts refer to dark matter and dark energy, respectively. In this approximation, the adiabatic sound speed reads which stays finite at crossing as long as .However in this class of models there are other instabilities arising at the perturbation level regardless of the coupling used, [cf. 916].

Although a cosmological constant has and no perturbations, the converse is not automatically true: does not necessarily imply that there are no perturbations. It is only when we set from the beginning (in the calculation):

i.e., , that we have as a solution .For instance, if we set and (where can be a generic function) in Eqs. (1.4.28) and (1.4.29) we have and . However, the solutions are decaying modes due to the term so they are not important at late times; but it is interesting to notice that they are in general not zero.

As another example, if we have a non-zero anisotropic stress then the Eqs. (1.4.28) – (1.4.29) will have a source term that will influence the growth of and in the same way as does (just because they appear in the same way). The term in front of should not worry us as we can always define the anisotropic stress through

where when is the real traceless part of the energy momentum tensor, probably the quantity we need to look at: as in the case of , there is no need for to vanish when .It is also interesting to notice that when the perturbation equations tell us that dark-energy perturbations are not influenced through and (see Eq. (1.4.28) and (1.4.29)). Since and are the quantities directly entering the metric, they must remain finite, and even much smaller than for perturbation theory to hold. Since, in the absence of direct couplings, the dark energy only feels the other constituents through the terms and , it decouples completely in the limit and just evolves on its own. But its perturbations still enter the Poisson equation and so the dark matter perturbation will feel the effects of the dark-energy perturbations.

Although this situation may seem contrived, it might be that the acceleration of the universe is just an observed effect as a consequence of a modified theory of gravity. As was shown in [537], any modified gravity theory can be described as an effective fluid both at background and at perturbation level; in such a situation it is imperative to describe its perturbations properly as this effective fluid may manifest unexpected behavior.

In parallel to models with extra degrees of freedom in the matter sector, such as interacting quintessence (and k-essence, not treated here), another promising approach to the late-time acceleration enigma is to modify the left-hand side of the Einstein equations and invoke new degrees of freedom, belonging this time to the gravitational sector itself. One of the simplest and most popular extensions of GR and a known example of modified gravity models is the gravity in which the 4-dimensional action is given by some generic function of the Ricci scalar (for an introduction see, e.g., [49]):

where as usual , and is a matter action with matter fields . Here is a bare gravitational constant: we will see that the observed value will in general be different. As mentioned in the previously, it is possible to show that theories can be mapped into a subset of scalar-tensor theories and, therefore, to a class of interacting scalar field dark-energy models universally coupled to all species. When seen in the Einstein frame [954, 608, 936, 351, 724, 219], action (1.4.44) can, therefore, be related to the action (1.4.20) shown previously. Here we describe in the Jordan frame: the matter fields in obey standard conservation equations and, therefore, the metric corresponds to the physical frame (which here is the Jordan frame).There are two approaches to deriving field equations from the action (1.4.44).

- (I) The metric formalism
The first approach is the metric formalism in which the connections are the usual connections defined in terms of the metric . The field equations can be obtained by varying the action (1.4.44) with respect to :

where (we also use the notation ), and is the matter energy-momentum tensor. The trace of Eq. (1.4.45) is given by where . Here and are the energy density and the pressure of the matter, respectively. - (II) The Palatini formalism
The second approach is the Palatini formalism, where and are treated as independent variables. Varying the action (1.4.44) with respect to gives

where is the Ricci tensor corresponding to the connections . In general this is different from the Ricci tensor corresponding to the metric connections. Taking the trace of Eq. (1.4.47), we obtain where is directly related to . Taking the variation of the action (1.4.44) with respect to the connection, and using Eq. (1.4.47), we find

In GR we have and , so that the term in Eq. (1.4.46) vanishes. In this case both the metric and the Palatini formalisms give the relation , which means that the Ricci scalar is directly determined by the matter (the trace ).

In modified gravity models where is a function of , the term does not vanish in Eq. (1.4.46). This means that, in the metric formalism, there is a propagating scalar degree of freedom, . The trace equation (1.4.46) governs the dynamics of the scalar field – dubbed “scalaron” [862]. In the Palatini formalism the kinetic term is not present in Eq. (1.4.48), which means that the scalar-field degree of freedom does not propagate freely [32, 563, 567, 566].

The de Sitter point corresponds to a vacuum solution at which the Ricci scalar is constant. Since at this point, we get

which holds for both the metric and the Palatini formalisms. Since the model satisfies this condition, it possesses an exact de Sitter solution [862].It is important to realize that the dynamics of dark-energy models is different depending on the two formalisms. Here we confine ourselves to the metric case only.

Already in the early 1980s it was known that the model can be responsible for inflation in the early universe [862]. This comes from the fact that the presence of the quadratic term gives rise to an asymptotically exact de Sitter solution. Inflation ends when the term becomes smaller than the linear term . Since the term is negligibly small relative to at the present epoch, this model is not suitable to realizing the present cosmic acceleration.

Since a late-time acceleration requires modification for small , models of the type () were proposed as a candidate for dark energy [204, 212, 687]. While the late-time cosmic acceleration is possible in these models, it has become clear that they do not satisfy local gravity constraints because of the instability associated with negative values of [230, 319, 852, 697, 355]. Moreover a standard matter epoch is not present because of a large coupling between the Ricci scalar and the non-relativistic matter [43].

Then, we can ask what are the conditions for the viability of dark-energy models in the metric formalism. In the following we first present such conditions and then explain step by step why they are required.

- (i) for , where is the Ricci scalar at the present epoch. Strictly
speaking, if the final attractor is a de Sitter point with the Ricci scalar , then the
condition needs to hold for .
This is required to avoid a negative effective gravitational constant.

- (ii) for .
This is required for consistency with local gravity tests [319, 697, 355, 683], for the presence of the matter-dominated epoch [43, 39], and for the stability of cosmological perturbations [213, 849, 110, 358].

- (iii) for .
This is required for consistency with local gravity tests [48, 456, 864, 53, 904] and for the presence of the matter-dominated epoch [39].

- (iv) at .
This is required for the stability of the late-time de Sitter point [678, 39].

For example, the model (, ) does not satisfy the condition (ii).

Below we list some viable models that satisfy the above conditions.

The models (A), (B), (C), and (D) have been proposed in [39], [456], [864], and [904], respectively. A model similar to (D) has been also proposed in [53], while a generalized model encompassing (B) and (C) has been studied in [660]. In model (A), the power needs to be close to 0 to satisfy the condition (iii). In models (B) and (C) the function asymptotically behaves as for and hence the condition (iii) can be satisfied even for . In model (D) the function rapidly approaches in the region . These models satisfy , so the cosmological constant vanishes in the flat spacetime.Let us consider the cosmological dynamics of gravity in the metric formalism. It is possible to carry out a general analysis without specifying the form of . In the flat FLRW spacetime the Ricci scalar is given by

where is the Hubble parameter. As a matter action we take into account non-relativistic matter and radiation, which satisfy the usual conservation equations and respectively. From Eqs. (1.4.45) and (1.4.46) we obtain the following equations We introduce the dimensionless variables: together with the following quantities It is straightforward to derive the following differential equations [39]: where the prime denotes and From Eq. (1.4.65) one can express as a function of . Since is a function of , it follows that is a function of , i.e., . The CDM model, , corresponds to . Hence the quantity characterizes the deviation from the CDM model. Note also that the model, , gives a constant value of . The analysis using Eqs. (1.4.60) – (1.4.63) is sufficiently general in the sense that the form of does not need to be specified.The effective equation of state of the system (i.e., ) is

The dynamics of the full system can be investigated by analyzing the stability properties of the critical phase-space points as in, e.g., [39]. The general conclusions is that only models with a characteristic function positive and close to CDM, i.e., , are cosmologically viable. That is, only for these models one finds a sequence of a long decelerated matter epoch followed by a stable accelerated attractor.

The perturbation equations have been derived in, e.g., [473, 907]. Neglecting the contribution of radiation one has

where , and the new variable satisfies

The perturbation can be written as and, therefore, . These equations can be integrated numerically to derive the behavior of at all scales. However, at sub-Hubble scales they can be simplified and the following expression for the two MG functions of Eq. (1.3.23) can be obtained: where Note that in the CDM limit and .These relations can be straightforwardly generalized. In [287] the perturbation equations for the Lagrangian have been extended to include coupled scalar fields and their kinetic energy , resulting in a -theory. In the slightly simplified case in which , with arbitrary functions , one obtains

where the notation or denote differentiation wrt or , respectively, and where and , being the scalar field effective mass. In the same paper [287] an extra term proportional to in the Lagrangian is also taken into account.Euclid forecasts for the models will be presented in Section 1.8.7.

Instead of introducing new scalar degrees of freedom such as in theories, another philosophy in modifying gravity is to modify the graviton itself. In this case the new degrees of freedom belong to the gravitational sector itself; examples include massive gravity and higher-dimensional frameworks, such as the Dvali–Gabadadze–Porrati (DGP) model [326] and its extensions. The new degrees of freedom can be responsible for a late-time speed-up of the universe, as is summarized below for a choice of selected models. We note here that while such self-accelerating solutions are interesting in their own right, they do not tackle the old cosmological constant problem: why the observed cosmological constant is so much smaller than expected in the first place. Instead of answering this question directly, an alternative approach is the idea of degravitation [see 327, 328, 58, 330], where the cosmological constant could be as large as expected from standard field theory, but would simply gravitate very little (see the paragraph in Section 1.4.7.1 below).

Furthermore, in this model, one can recover an interesting range of cosmologies, in particular a modified Friedmann equation with a self-accelerating solution. The Einstein equations thus obtained reduce to the following modified Friedmann equation in a homogeneous and isotropic metric [298]

such that at higher energies one recovers the usual four-dimensional behavior, , while at later time corrections from the extra dimensions kick in. As is clear in this Friedmann equation, this braneworld scenario holds two branches of cosmological solutions with distinct properties. The self-accelerating branch (minus sign) allows for a de Sitter behavior even in the absence of any cosmological constant and as such it has attracted a lot of attention. Unfortunately, this branch suffers from a ghost-like instability. The normal branch (the plus sign) instead slows the expansion rate but is stable. In this case a cosmological constant is still required for late-time acceleration, but it provides significant intuition for the study of degravitation.

The Galileon terms described above form a subset of the “generalized Galileons”. A generalized Galileon model allows nonlinear derivative interactions of the scalar field in the Lagrangian while insisting that the equations of motion remain at most second order in derivatives, thus removing any ghost-like instabilities. However, unlike the pure Galileon models, generalized Galileons do not impose the symmetry of Eq. (1.4.75). These theories were first written down by Horndeski [445] and later rediscoved by Deffayet et al. [300]. They are a linear combination of Lagrangians constructed by multiplying the Galileon Lagrangians by an arbitrary scalar function of the scalar and its first derivatives. Just like the Galileon, generalized Galileons can give rise to cosmological acceleration and to Vainshtein screening. However, as they lack the Galileon symmetry these theories are not protected from quantum corrections. Many other theories can also be found within the spectrum of generalized Galileon models, including k-essence.

When properly taking into account the issue associated with the ghost, such models give rise to a theory of massive gravity (soft mass graviton) composed of one helicity-2 mode, helicity-1 modes that decouple and 2 helicity-0 modes. In order for this theory to be consistent with standard GR in four dimensions, both helicity-0 modes should decouple from the theory. As in DGP, this decoupling does not happen in a trivial way, and relies on a phenomenon of strong coupling. Close enough to any source, both scalar modes are strongly coupled and therefore freeze.

The resulting theory appears as a theory of a massless spin-2 field in four-dimensions, in other words as GR. If and for , the respective Vainshtein scale or strong coupling scale, i.e., the distance from the source within which each mode is strongly coupled is , where . Around a source , one recovers four-dimensional gravity for , five-dimensional gravity for and finally six-dimensional gravity at larger distances .

Recent progress has shown that theories of hard massive gravity can be free of any ghost-like pathologies in the decoupling limit where and keeping the scale fixed [291, 292]. The absence of pathologies in the decoupling limit does not guarantee the stability of massive gravity on cosmological backgrounds, but provides at least a good framework to understand the implications of a small graviton mass. Unlike a massless spin-2 field, which only bears two polarizations, a massive one bears five of them, namely two helicity-2 modes, two helicity-1 modes which decouple, and one helicity-0 mode (denoted as ). As in the braneworld models presented previously, this helicity-0 mode behaves as a scalar field with specific derivative interactions of the form

Here, denotes the canonically-normalized (rescaled by ) tensor field perturbation (helicity-2 mode), while and are respectively, linear, quadratic and cubic in the helicity-0 mode . Importantly, they are all transverse (for instance, ). Not only do these interactions automatically satisfy the Bianchi identity, as they should to preserve diffeomorphism invariance, but they are also at most second order in time derivatives. Hence, the interactions (1.4.76) are linear in the helicity-2 mode, and are free of any ghost-like pathologies. Therefore, such interactions are very similar in spirit to the Galileon ones, and bear the same internal symmetry (1.4.75), and present very similar physical properties. When is absent, one can indeed recover an Einstein frame picture for which the interactions are of the form where is an arbitrary constant and matter fields do not couple to the metric but to . Here again, the recovery of GR in the UV is possible via a strong coupling phenomena, where the interactions for are already important at the scale , well before the interactions for the usual helicity-2 mode. This strong coupling, as well as the peculiar coupling to matter sources, have distinguishable features in cosmology as is explained below [11, 478].

All models of modified gravity presented in this section have in common the presence of at least one additional helicity-0 degree of freedom that is not an arbitrary scalar, but descends from a full-fledged spin-two field. As such it has no potential and enters the Lagrangian via very specific derivative terms fixed by symmetries. However, tests of gravity severely constrain the presence of additional scalar degrees of freedom. As is well known, in theories of massive gravity the helicity-0 mode can evade fifth-force constraints in the vicinity of matter if the helicity-0 mode interactions are important enough to freeze out the field fluctuations [913]. This Vainshtein mechanism is similar in spirit but different in practice to the chameleon and symmetron mechanisms presented in detail in the next Sections 1.4.7.3 and 1.4.7.4. One key difference relies on the presence of derivative interactions rather than a specific potential. So, rather than becoming massive in dense regions, in the Vainshtein mechanism the helicity-0 mode becomes weakly coupled to matter (and light, i.e., sources in general) at high energy. This screening of scalar mode can yet have distinct signatures in cosmology and in particular for structure formation.

While quintessence introduces a new degree of freedom to explain the late-time acceleration of the universe, the idea behind modified gravity is instead to tackle the core of the cosmological constant problem and its tuning issues as well as screening any fifth forces that would come from the introduction of extra degrees of freedom. As mentioned in Section 1.4.4.1, the strength with which these new degrees of freedom can couple to the fields of the standard model is very tightly constrained by searches for fifth forces and violations of the weak equivalence principle. Typically the strength of the scalar mediated interaction is required to be orders of magnitude weaker than gravity. It is possible to tune this coupling to be as small as is required, leading however to additional naturalness problems. Here we discuss in more detail a number of ways in which new scalar degrees of freedom can naturally couple to standard model fields, whilst still being in agreement with observations, because a dynamical mechanism ensures that their effects are screened in laboratory and solar system tests of gravity. This is done by making some property of the field dependent on the background environment under consideration. These models typically fall into two classes; either the field becomes massive in a dense environment so that the scalar force is suppressed because the Compton wavelength of the interaction is small, or the coupling to matter becomes weaker in dense environments to ensure that the effects of the scalar are suppressed. Both types of behavior require the presence of nonlinearities.

In the presence of non-relativistic matter these two pieces conspire to give rise to an effective potential for the scalar field

where is the bare potential, the local energy density and the conformal coupling function. For suitable choices of and the effective potential has a minimum and the position of the minimum depends on . Self-interaction terms in ensure that the mass of the field in this minimum also depends on so that the field becomes more massive in denser environments.The environmental dependence of the mass of the field allows the chameleon to avoid the constraints of fifth-force experiments through what is known as the thin-shell effect. If a dense object is embedded in a diffuse background the chameleon is massive inside the object. There, its Compton wavelength is small. If the Compton wavelength is smaller than the size of the object, then the scalar mediated force felt by an observer at infinity is sourced, not by the entire object, but instead only by a thin shell of matter (of depth the Compton wavelength) at the surface. This leads to a natural suppression of the force without the need to fine tune the coupling constant.

Inside the Vainshtein radius, when the nonlinear, higher-derivative terms become important they cause the kinetic terms for scalar fluctuations to become large. This can be interpreted as a relative weakening of the coupling between the scalar field and matter. In this way the strength of the interaction is suppressed in the vicinity of massive objects.

In 1983 it was suggested by Milgrom [659] that the emerging evidence for the presence of dark matter in galaxies could follow from a modification either to how ‘baryonic’ matter responded to the Newtonian gravitational field it created or to how the gravitational field was related to the baryonic matter density. Collectively these ideas are referred to as MOdified Newtonian Dynamics (MOND). By way of illustration, MOND may be considered as a modification to the non-relativistic Poisson equation:

where is the gravitational potential, is a number with dimensions Length and is the baryonic matter density. The number is determined by looking at the dynamics of visible matter in galaxies [783]. The function would simply be equal to unity in Newtonian gravity. In MOND, the functional form is only fixed at its limits: as and as .We are naturally interested in a relativistic version of such a proposal. The building block is the perturbed spacetime metric already introduced in Eq. 1.3.8

A simple approach is to introduce a dynamical clock field, which we will call . If it has solutions aligned with the time-like coordinate then it will be sensitive to . The dynamical nature of the field implies that it should have an action that will contain gradients of the field and thus potentially scalars formed from gradients of , as we seek. A family of covariant actions for the clock field is as follows [988]: where with The quantity is a number with dimensions of length, the are dimensionless constants, the Lagrange multiplier field enforces the unit-timelike constraint on , and is a function. These models have been termed Generalized Einstein-Aether (GEA) theories, emphasizing the coexistence of general covariance and a ‘preferred’ state of rest in the model, i.e., keeping time with .Indeed, when the geometry is of the form (1.4.82), anisotropic stresses are negligible and is aligned with the flow of time , then one can find appropriate values of the and such that is dominated by a term equal to . This influence then leads to a modification to the time-time component of Einstein’s equations: instead of reducing to Poisson’s equation, one recovers an equation of the form (1.4.81). Therefore the models are successful covariant realizations of MOND.

Interestingly, in the FLRW limit , the time-time component of Einstein’s equations in the GEA model becomes a modified Friedmann equation:

where the function is related to and its derivatives with respect to . The dynamics in galaxies prefer a value on the order the Hubble parameter today [783] and so one typically gets a modification to the background expansion with a characteristic scale , i.e., the scale associated with modified gravity models that produce dark-energy effects. Ultimately the GEA model is a phenomenological one and as such there currently lack deeper reasons to favor any particular form of . However, one may gain insight into the possible solutions of (1.4.85) by looking at simple forms for . In [991] the monomial case was considered where the kinetic index was allowed to vary. Solutions with accelerated expansion were found that could mimic dark energy.Returning to the original motivation behind the theory, the next step is to look at the theory on cosmological scales and see whether the GEA models are realistic alternatives to dark matter. As emphasized, the additional structure in spacetime is dynamical and so possesses independent degrees of freedom. As the model is assumed to be uncoupled to other matter, the gravitational field equations would regard the influence of these degrees of freedom as a type of dark matter (possibly coupled non-minimally to gravity, and not necessarily ‘cold’).

The possibility that the model may then be a viable alternative to the dark sector in background cosmology and linear cosmological perturbations has been explored in depth in [989, 564] and [991]. As an alternative to dark matter, it was found that the GEA models could replicate some but not all of the following features of cold dark matter: influence on background dynamics of the universe; negligible sound speed of perturbations; growth rate of dark matter ‘overdensity’; absence of anisotropic stress and contribution to the cosmological Poisson equation; effective minimal coupling to the gravitational field. When compared to the data from large scale structure and the CMB, the model fared significantly less well than the Concordance Model and so is excluded. If one relaxes the requirement that the vector field be responsible for the effects of cosmological dark matter, one can look at the model as one responsible only for the effects of dark energy. It was found [991] that the current most stringent constraints on the model’s success as dark energy were from constraints on the size of large scale CMB anisotropy. Specifically, possible variation in of the ‘dark energy’ along with new degrees of freedom sourcing anisotropic stress in the perturbations was found to lead to new, non-standard time variation of the potentials and . These time variations source large scale anisotropies via the integrated Sachs–Wolfe effect, and the parameter space of the model is constrained in avoiding the effect becoming too pronounced.

In spite of this, given the status of current experimental bounds it is conceivable that a more successful alternative to the dark sector may share some of these points of departure from the Concordance Model and yet fare significantly better at the level of the background and linear perturbations.

Another proposal for a theory of modified gravity arising from Milgrom’s observation is the Tensor-Vector-Scalar theory of gravity, or TeVeS. TeVeS theory is bimetric with two frames: the “geometric frame” for the gravitational fields, and the “physical frame”, for the matter fields. The three gravitational fields are the metric (with connection ) that we refer to as the geometric metric, the vector field and the scalar field . The action for all matter fields, uses a single physical metric (with connection ). The two metrics are related via an algebraic, disformal relation [116] as

Just like in the Generalized Einstein-Aether theories, the vector field is further enforced to be unit-timelike with respect to the geometric metric, i.e., The theory is based on an action , which is split as where where and are the determinant and scalar curvature of respectively and is the bare gravitational constant, where leads to a Maxwellian kinetic term and is a Lagrange multiplier ensuring the unit-timelike constraint on and is a dimensionless constant (note that indices on are raised using the geometric metric, i.e., ) and where is a non-dynamical dimensionless scalar field, and is an arbitrary function that typically depends on a scale . The matter is coupled only to the physical metric and defines the matter stress-energy tensor through . The TeVeS action can be written entirely in the physical frame [987, 840] or in a diagonal frame [840] where the scalar and vector fields decouple.In a Friedmann universe, the cosmological evolution is governed by the Friedmann equation

where is the Hubble rate in terms of the geometric scale factor, is the physical matter density that obeys the energy conservation equation with respect to the physical metric and where the scalar field energy density is Exact analytical and numerical solutions with the Bekenstein free function have been found in [841] and in [318]. It turns out that energy density tracks the matter fluid energy density. The ratio of the energy density of the scalar field to that of ordinary matter is approximately constant, so that the scalar field exactly tracks the matter dynamics. In realistic situations, the radiation era tracker is almost never realized, as has been noted by Dodelson and Liguori, but rather is subdominant and slowly-rolling and . [157] studied more general free functions which have the Bekenstein function as a special case and found a whole range of behavior, from tracking and accelerated expansion to finite time singularities. [309] have studied cases where the cosmological TeVeS equations lead to inflationary/accelerated expansion solutions.Although no further studies of accelerated expansion in TeVeS have been performed, it is very plausible that certain choices of function will inevitably lead to acceleration. It is easy to see that the scalar field action has the same form as a k-essence/k-inflation [61] action which has been considered as a candidate theory for acceleration. It is unknown in general whether this has similar features as the uncoupled k-essence, although Zhao’s study indicates that this a promising research direction [984].

In TeVeS, cold dark matter is absent. Therefore, in order to get acceptable values for the physical Hubble constant today (i.e., around ), we have to supplement the absence of CDM with something else. Possibilities include the scalar field itself, massive neutrinos [841, 364] and a cosmological constant. At the same time, one has to get the right angular diameter distance to recombination [364]. These two requirements can place severe constraints on the allowed free functions.

Until TeVeS was proposed and studied in detail, MOND-type theories were assumed to be fatally flawed: their lack of a dark matter component would necessarily prevent the formation of large-scale structure compatible with current observational data. In the case of an Einstein universe, it is well known that, since baryons are coupled to photons before recombination they do not have enough time to grow into structures on their own. In particular, on scales smaller than the diffusion damping scale perturbations in such a universe are exponentially damped due to the Silk-damping effect. CDM solves all of these problems because it does not couple to photons and therefore can start creating potential wells early on, into which the baryons fall.

TeVeS contains two additional fields, which change the structure of the equations significantly. The first study of TeVeS predictions for large-scale structure observations was conducted in [841]. They found that TeVeS can indeed form large-scale structure compatible with observations depending on the choice of TeVeS parameters in the free function. In fact the form of the matter power spectrum in TeVeS looks quite similar to that in CDM. Thus TeVeS can produce matter power spectra that cannot be distinguished from CDM by current observations. One would have to turn to other observables to distinguish the two models. The power spectra for TeVeS and CDM are plotted on the right panel of Figure 1. [318] provided an analytical explanation of the growth of structure seen numerically by [841] and found that the growth in TeVeS is due to the vector field perturbation.

It is premature to claim (as in [843, 855]) that only a theory with CDM can fit CMB observations; a prime example to the contrary is the EBI theory [83]. Nevertheless, in the case of TeVeS [841] numerically solved the linear Boltzmann equation in the case of TeVeS and calculated the CMB angular power spectrum for TeVeS. By using initial conditions close to adiabatic the spectrum thus found provides very poor fit as compared to the CDM model (see the left panel of Figure 1). The CMB seems to put TeVeS into trouble, at least for the Bekenstein free function. The result of [318] has a further direct consequence. The difference , sometimes named the gravitational slip (see Section 1.3.2), has additional contributions coming from the perturbed vector field . Since the vector field is required to grow in order to drive structure formation, it will inevitably lead to a growing . If the difference can be measured observationally, it can provide a substantial test of TeVeS that can distinguish TeVeS from CDM.

Living Rev. Relativity 16, (2013), 6
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