1.4 Models of dark energy and modified gravity

In this section we review a number of popular models of dynamical DE and MG. This section is more technical than the rest and it is meant to provide a quick but self-contained review of the current research in the theoretical foundations of DE models. The selection of models is of course somewhat arbitrary but we tried to cover the most well-studied cases and those that introduce new and interesting observable phenomena.

1.4.1 Quintessence

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 ρ ≃ 10− 123m4 DE pl and a mass scale −33 m ϕ ≲ 10 eV.

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

∫ [ ] S = d4x√ −-g -1--R + ℒ ϕ + SM , ℒ ϕ = − 1g μν∂ μϕ∂νϕ − V (ϕ), (1.4.1 ) 2κ2 2
where 2 κ = 8πG and R is the Ricci scalar and SM is the matter action. The fluid satisfies the continuity equation
˙ρM + 3H (ρM + pM ) = 0. (1.4.2 )
The energy-momentum tensor of quintessence is
2 δ(√ −-gℒ ϕ) Tμ(ϕν) = − √---------μν---- (1.4.3 ) − g δg [ ] 1-αβ = ∂μϕ∂ νϕ − gμν 2g ∂α ϕ∂βϕ + V (ϕ) . (1.4.4 )
As we have already seen, in a FLRW background, the energy density ρϕ and the pressure pϕ of the field are
0(ϕ) 1 2 1 i(ϕ) 1 2 ρϕ = − T0 = 2ϕ˙ + V (ϕ), pϕ = 3Ti = 2ϕ˙ − V (ϕ), (1.4.5 )
which give the equation of state
pϕ ˙ϕ2 − 2V (ϕ) w ϕ ≡ ---= -----------. (1.4.6 ) ρϕ ˙ϕ2 + 2V (ϕ)
In the flat universe, Einstein’s equations give the following equations of motion:
κ2 [ 1 ] H2 = --- -ϕ˙2 + V (ϕ) + ρM , (1.4.7 ) 3 2 ˙ κ2-( ˙2 ) H = − 2 ϕ + ρM + pM , (1.4.8 )
where 2 κ = 8πG. The variation of the action (1.4.1View Equation) with respect to ϕ gives
ϕ¨+ 3H ˙ϕ + V,ϕ = 0, (1.4.9 )
where V,ϕ ≡ dV∕d ϕ.

During radiation or matter dominated epochs, the energy density ρM of the fluid dominates over that of quintessence, i.e., ρ ≫ ρ M ϕ. If the potential is steep so that the condition ϕ˙2 ∕2 ≫ V (ϕ) is always satisfied, the field equation of state is given by wϕ ≃ 1 from Eq. (1.4.6View Equation). In this case the energy density of the field evolves as ρϕ ∝ a−6, which decreases much faster than the background fluid density.

The condition w ϕ < − 1 ∕3 is required to realize the late-time cosmic acceleration, which translates into the condition ˙2 ϕ < V (ϕ). 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]

( ) 1 V,ϕ 2 V,ϕϕ 𝜖s ≡ --2- --- , ηs ≡ --2- . (1.4.10 ) 2κ V κ V
If the conditions 𝜖s ≪ 1 and |ηs| ≪ 1 are satisfied, the evolution of the field is sufficiently slow so that ϕ˙2 ≪ V (ϕ) and |¨ϕ| ≪ |3H ϕ˙| in Eqs. (1.4.7View Equation) and (1.4.9View Equation).

From Eq. (1.4.9View Equation) the deviation of w ϕ from − 1 is given by

2 1 + w = ------V,ϕ------, (1.4.11 ) ϕ 9H2 (ξs + 1 )2ρ ϕ
where ¨ ˙ ξs ≡ ϕ∕(3H ϕ). This shows that w ϕ is always larger than − 1 for a positive potential and energy density. In the slow-roll limit, |ξs| ≪ 1 and ˙ϕ2∕2 ≪ V (ϕ ), we obtain 1 + w ϕ ≃ 2𝜖s∕3 by neglecting the matter fluid in Eq. (1.4.7View Equation), i.e., 3H2 ≃ κ2V (ϕ). The deviation of w ϕ from − 1 is characterized by the slow-roll parameter 𝜖s. It is also possible to consider Eq. (1.4.11View Equation) as a prescription for the evolution of the potential given w ϕ(z) and to reconstruct a potential that gives a desired evolution of the equation of state (subject to w ∈ [− 1,1]). This was used, for example, in [102Jump To The Next Citation Point].

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 δϕ and the velocity 𝜃ϕ in the conformal Newtonian gauge, one finds [see, e.g., 536Jump To The Next Citation Point, Appendix A] that they correspond precisely to those of a fluid, (1.3.17View Equation) and (1.3.18View Equation), with π = 0 and 2 2 2 2 δp = csδρ + 3aH (cs − ca)(1 + w )ρ𝜃∕k with 2 cs = 1. The adiabatic sound speed, ca, is defined in Eq. (1.4.31View Equation). The large value of the sound speed 2 cs, equal to the speed of light, means that quintessence models do not cluster significantly inside the horizon [see 785Jump To The Next Citation Point, 786Jump To The Next Citation Point, and Section 1.8.6 for a detailed analytical discussion of quintessence clustering and its detectability with future probes, for arbitrary c2 s].

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 w ϕ = − 1). This potential can be constructed in the framework of supergravity [170].

In thawing models (ii) the field (with mass m ϕ) has been frozen by Hubble friction (i.e., the term H ϕ˙ in Eq. (1.4.9View Equation)) until recently and then it begins to evolve once H drops below m ϕ. The equation of state of DE is w ϕ ≃ − 1 at early times, which is followed by the growth of wϕ. The representative potentials that belong to this class are

(ii) Thawing models

The former potential is similar to that of chaotic inflation (n = 2,4) used in the early universe (with V = 0 ) 0 [577Jump To The Next Citation Point], while the mass scale M is very different. The model with n = 1 was proposed by [487] in connection with the possibility to allow for negative values of V(ϕ ). The universe will collapse in the future if the system enters the region with V (ϕ ) < 0. 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 m ϕ is smaller than H, but it begins to roll down around the present (m ϕ ≃ H0).

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 w ϕ and Ω ϕ 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].

1.4.2 K-essence

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

∫ S = d4x √ − gp (ϕ,X ), (1.4.12 )
where μν X = (1∕2)g ∇ μϕ∇ νϕ. The energy density of the scalar field is given by
-dp- ρϕ = 2X dX − p, (1.4.13 )
and the pressure is simply p ϕ = p(ϕ,X ). Treating the k-essence scalar as a perfect fluid, this means that k-essence has the equation of state
p-ϕ -----p----- wϕ = ρ ϕ = − p − 2Xp,X , (1.4.14 )
where the subscript ,X indicates a derivative with respect to X. Clearly, with a suitably chosen p 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

˙ρϕ = − 3H (ρϕ + pϕ), (1.4.15 )
or equivalently by the scalar equation of motion
2 G μν∇ ∇ ϕ + 2X -∂-p-- − ∂p- = 0, (1.4.16 ) μ ν ∂X ∂ ϕ ∂ϕ
μν ∂p-- μν ∂2p-- μ ν G = ∂X g + ∂X2 ∇ ϕ∇ ϕ. (1.4.17 )
For this second order equation of motion to be hyperbolic, and hence physically meaningful, we must impose
p,XX- 1 + 2X p, > 0. (1.4.18 ) X
K-essence was first proposed by [61Jump To The Next Citation Point, 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

p, c2s = -------X------. (1.4.19 ) p,X +2Xp,XX
So that whenever the kinetic terms for the scalar field are not linear in X, 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.

1.4.3 A definition of modified gravity

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 [537Jump To The Next Citation Point]. 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 [457Jump To The Next Citation Point, 535].

The simplest example can be discussed by looking at Eqs. (1.3.23View Equation). One can modify gravity and obtain a modified Poisson equation, and therefore Q ⁄= 1, or one can introduce a clustering dark energy (for example a k-essence model with small sound speed) that also induces the same Q ⁄= 1 (see Eq. 1.3.23View Equation). This extends to the anisotropic stress η: 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:

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 γ. 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 Q = η = 1 (see Eq. 1.3.23View Equation) 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 Q, η or on related quantities like the growth index γ.

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.

1.4.4 Coupled dark-energy models

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:

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

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’ [954Jump To The Next Citation Point], 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 Λ-FLRW would evolve independently. The stress-energy tensor μ T ν 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 m of matter fields to depend on a scalar field ϕ via a function m (ϕ) whose choice specifies the interaction. This wide class of cosmological models can be described by the following action:

∫ √ ---[ 1 ] 𝒮 = d4x − g − --∂μϕ∂μϕ − U(ϕ ) − m (ϕ)ψ¯ψ + ℒkin[ψ] , (1.4.20 ) 2
where U(ϕ ) is the potential in which the scalar field ϕ rolls, ψ describes matter fields, and g is defined in the usual way as the determinant of the metric tensor, whose background expression is 2 2 2 2 gμν = diag[− a ,a ,a ,a ].

For a general treatment of background and perturbation equations we refer to [514Jump To The Next Citation Point, 33Jump To The Next Citation Point, 35Jump To The Next Citation Point, 724Jump To The Next Citation Point]. Here the coupling of the dark-energy scalar field to a generic matter component (denoted by index α) is treated as an external source Q (α)μ in the Bianchi identities:

ν ∇ νT(α )μ = Q (α)μ, (1.4.21 )
with the constraint
∑ Q (α)μ = 0. (1.4.22 ) α

The zero component of (1.4.21View Equation) gives the background conservation equations:

dρϕ dϕ ----= − 3ℋ (1 + w ϕ)ρϕ + β(ϕ)---(1 − 3w α)ρα, (1.4.23 ) dη d η d-ρα dϕ- dη = − 3ℋ (1 + wα )ρ α − β(ϕ)d η(1 − 3w α)ρα, (1.4.24 )
for a scalar field ϕ coupled to one single fluid α with a function β (ϕ ), which in general may not be constant. The choice of the mass function m (ϕ ) corresponds to a choice of β(ϕ) and equivalently to a choice of the source Q (α)μ and specifies the strength of the coupling according to the following relations:
∂ lnm (ϕ) Q (ϕ)μ = ---------T α∂μϕ, mα = ¯m α e −β(ϕ)ϕ, (1.4.25 ) ∂ ϕ
where ¯m α is the constant Jordan-frame bare mass. The evolution of dark energy is related to the trace T α and, as a consequence, to density and pressure of the species α. We note that a description of the coupling via an action such as (1.4.20View Equation) 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:

( ) dv α dϕ -dη- + ℋ − β (ϕ)dη- vα − ∇ [Φ α + βϕ] = 0. (1.4.26 )
The Euler equation in cosmic time (dt = adτ) can also be rewritten in the form of an acceleration equation for particles at position r:
&tidle; v˙α = − &tidle;Hv α − ∇ G-αm-α. (1.4.27 ) r
The latter expression explicitly contains all the main ingredients that affect dark-energy interactions:
  1. a fifth force ∇ [Φα + β ϕ] with an effective G&tidle;α = GN [1 + 2β2 (ϕ )] ;
  2. a velocity dependent term ( ) &tidle; ϕ˙ Hv α ≡ H 1 − β (ϕ)H vα
  3. a time-dependent mass for each particle α, evolving according to (1.4.25View Equation).

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. Dark energy and baryons

A coupling between dark energy and baryons is active when the baryon mass is a function of the dark-energy scalar field: mb = mb (ϕ). 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, 304Jump To The Next Citation Point, 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). Dark energy and dark matter

An interaction between dark energy and dark matter (CDM) is active when CDM mass is a function of the dark-energy scalar field: mc = mc (ϕ). 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 β presently allowed by observations (within current CMB data) are within the range 0 < β < 0.06 (at 95% CL for a constant coupling and an exponential potential) [114Jump To The Next Citation Point, 47Jump To The Next Citation Point, 35Jump To The Next Citation Point, 44Jump To The Next Citation Point], or possibly more [539Jump To The Next Citation Point, 531Jump To The Next Citation Point] 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 β (ϕ) [33Jump To The Next Citation Point, 619Jump To The Next Citation Point, 35Jump To The Next Citation Point, 518Jump To The Next Citation Point, 414Jump To The Next Citation Point, 747Jump To The Next Citation Point, 748Jump To The Next Citation Point, 724Jump To The Next Citation Point, 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, 33Jump To The Next Citation Point, 35Jump To The Next Citation Point], therefore affecting CMB and cross-correlation of CMB and LSS [44Jump To The Next Citation Point, 35Jump To The Next Citation Point, 47Jump To The Next Citation Point, 45Jump To The Next Citation Point, 114Jump To The Next Citation Point, 539Jump To The Next Citation Point, 531Jump To The Next Citation Point, 970, 612Jump To The Next Citation Point, 42Jump To The Next Citation Point].

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

An alternative approach, also investigated in the literature [619Jump To The Next Citation Point, 916Jump To The Next Citation Point, 915Jump To The Next Citation Point, 613Jump To The Next Citation Point, 387Jump To The Next Citation Point, 388Jump To The Next Citation Point, 193Jump To The Next Citation Point, 794Jump To The Next Citation Point, 192Jump To The Next Citation Point], where the authors consider as a starting point Eq. (1.4.21View Equation): 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 [916Jump To The Next Citation Point]. 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. (1.4.21View Equation). 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, 279Jump To The Next Citation Point], pointing out that coupled dark energy behaves as a fluid with an effective equation of state w ≲ − 1, though staying well defined and without the presence of ghosts [279Jump To The Next Citation Point].

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

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 β 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 ϕ. Such a coupling has been investigated within MaVaNs [356Jump To The Next Citation Point, 714Jump To The Next Citation Point, 135Jump To The Next Citation Point, 12Jump To The Next Citation Point, 952Jump To The Next Citation Point, 280Jump To The Next Citation Point, 874Jump To The Next Citation Point, 856Jump To The Next Citation Point, 139Jump To The Next Citation Point, 178Jump To The Next Citation Point, 177Jump To The Next Citation Point] and more recently within growing neutrino cosmologies [36Jump To The Next Citation Point, 957Jump To The Next Citation Point, 668Jump To The Next Citation Point, 963Jump To The Next Citation Point, 962Jump To The Next Citation Point, 727Jump To The Next Citation Point, 179Jump To The Next Citation Point, 78Jump To The Next Citation Point]. 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 [668Jump To The Next Citation Point, 963Jump To The Next Citation Point, 78] at very large scales only (∼ 100 Mpc and beyond) as well as to signatures in the CMB spectra [727Jump To The Next Citation Point]. For an illustration of observable effects related to this case see Section (2.9) of this report. Scalar-tensor theories

Scalar-tensor theories [954Jump To The Next Citation Point, 471, 472, 276, 216, 217, 955Jump To The Next Citation Point, 912, 722, 354, 146, 764, 721, 797, 646, 725Jump To The Next Citation Point, 726, 205, 54Jump To The Next Citation Point] 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 ϕ is coupled (universally) to all species [954Jump To The Next Citation Point, 608Jump To The Next Citation Point, 936Jump To The Next Citation Point, 351Jump To The Next Citation Point, 724Jump To The Next Citation Point, 219Jump To The Next Citation Point]. In other words, these theories correspond to the case where, in action (1.4.20View Equation), the mass of all species (baryons, dark matter, …) is a function m = m (ϕ) with the same coupling for every species α. Indeed, a description of the coupling via an action such as (1.4.20View Equation) 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], [725Jump To The Next Citation Point] 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 V (ϕ ) 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, 464Jump To The Next Citation Point, 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 (β ∼ 1) 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:

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 w. f(R ) theories, which can be mapped into a subclass of scalar-tensor theories, will be discussed in more detail in Section 1.4.6.

1.4.5 Phantom crossing

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 w. Current limits on the equation of state parameter w = p∕ρ of the dark energy indicate that p ≈ − ρ, and so do not exclude p < − ρ, a region of parameter space often called phantom energy. Even though the region for which w < − 1 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 w < − 1 for the background evolution, there are apparent divergences appearing in the perturbations when a model tries to cross the limit w = − 1. 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 w = − 1 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 w → − 1 the terms in Eqs. (1.3.17View Equation) and (1.3.18View Equation) containing 1∕ (1 + w ) will generally diverge. This can be avoided by replacing 𝜃 with a new variable V defined via V = ρ(1 + w )𝜃. This corresponds to rewriting the 0-i component of the energy momentum tensor as j ikjT0 = V, which avoids problems if j T0 ⁄= 0 when p¯= − ¯ρ. Replacing the time derivatives by a derivative with respect to the logarithm of the scale factor ln a (denoted by a prime), we obtain [599Jump To The Next Citation Point, 450, 536Jump To The Next Citation Point]:

V ( δp ) δ′ = 3 (1 + w )Φ′ −----− 3 ---− w δ (1.4.28 ) Ha ¯ρ ′ -k2-δp- k2-- V = − (1 − 3w )V + Ha ¯ρ + (1 + w )Ha (Ψ − π) . (1.4.29 )
In order to solve Eqs. (1.4.28View Equation) and (1.4.29View Equation) we still need to specify the expressions for δp and π, 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 Φ and Ψ differ, we will set it in this section to zero, π = 0. 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. Parameterizing the pressure perturbation

Barotropic fluids.
We define a fluid to be barotropic if the pressure p depends strictly only on the energy density ρ: p = p(ρ). These fluids have only adiabatic perturbations, so that they are often called adiabatic. We can write their pressure as

| p(ρ) = p(¯ρ + δρ) = p(¯ρ) + dp-|| δρ + O [(δρ)2]. (1.4.30 ) d ρ|¯ρ
Here p(ρ¯) = ¯p is the pressure of the isotropic and homogeneous part of the fluid. The second term in the expansion (1.4.30View Equation) can be re-written as
| dp-|| ˙¯p- -----w˙----- 2 dρ | = ˙¯ρ = w − 3aH (1 + w ) ≡ ca, (1.4.31 ) ¯ρ
where we used the equation of state and the conservation equation for the dark-energy density in the background. We notice that the adiabatic sound speed c2 a will necessarily diverge for any fluid where w crosses − 1. However, for a perfect barotropic fluid the adiabatic sound speed 2 ca turns out to be the physical propagation speed of perturbations. Therefore, it should never be larger than the speed of light – otherwise our theory becomes acausal – and it should never be negative (c2a < 0) – otherwise classical, and possible quantum, instabilities appear. Even worse, the pressure perturbation
( ) 2 -----w˙----- δp = caδρ = w − 3aH (1 + w ) δρ (1.4.32 )
will necessarily diverge if w crosses − 1 and δ ρ ⁄= 0. Even if we find a way to stabilize the pressure perturbation, for instance an equation of state parameter that crosses the − 1 limit with zero slope (w˙), there will always be the problem of a negative speed of sound that prevents these models from being viable dark-energy candidates.

Non-adiabatic fluids.
To construct a model that can cross the phantom divide, we therefore need to violate the constraint that p is a unique function of ρ. At the level of first-order perturbation theory, this amounts to changing the prescription for δp, which now becomes an arbitrary function of k and t. One way out of this problem is to choose an appropriate gauge where the equations are simple; one choice is, for instance, the rest frame of the fluid where the pressure perturbation reads (in this frame)

ˆδp = ˆc2δˆρ, (1.4.33 ) s
where now the ˆc2s is the speed with which fluctuations in the fluid propagate, i.e., the sound speed. We can write Eq. (1.4.33View Equation), with an appropriate gauge transformation, in a form suitable for the Newtonian frame, i.e., for Eqs. (1.4.28View Equation) and (1.4.29View Equation). We find that the pressure perturbation is given by [347, 112, 214]
( ) V δp = cˆ2sδρ + 3aH (a) ˆc2s − c2a ρ¯--. (1.4.34 ) k2
The problem here is the presence of c2a, which goes to infinity at the crossing and it is impossible that this term stays finite except if V → 0 fast enough or ˙w = 0, but this is not, in general, the case. This divergence appears because for w = − 1 the energy momentum tensor Eq. (1.3.3View Equation) reads: T μν = pgμν. Normally the four-velocity uμ is the time-like eigenvector of the energy-momentum tensor, but now all vectors are eigenvectors. So the problem of fixing a unique rest-frame is no longer well posed. Then, even though the pressure perturbation looks fine for the observer in the rest-frame, because it does not diverge, the badly-defined gauge transformation to the Newtonian frame does, as it also contains 2 ca. Regularizing the divergences

We have seen that neither barotropic fluids nor canonical scalar fields, for which the pressure perturbation is of the type (1.4.34View Equation), 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 but with a constant w on either side of w = − 1. The combination of the two fluids then effectively crosses the phantom divide if we start with wtot > − 1, as the energy density in the fluid with w < − 1 will grow faster, so that this fluid will eventually dominate and we will end up with wtot < − 1.

The perturbations in this scenario were analyzed in detail in [536Jump To The Next Citation Point], 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

δp (k,t) = γ (k,t)δρ(k,t) (1.4.35 )
the quantity γ 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 (1.4.34View Equation) in the Newtonian frame cannot cross w = − 1, 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 [536Jump To The Next Citation Point] found that a good approximation to the quintom model behavior can be found by regularizing the adiabatic sound speed in the gauge transformation with

c2 = w − ----w˙(1 +-w)----- (1.4.36 ) a 3Ha [(1 + w )2 + λ]
where λ is a tunable parameter which determines how close to w = − 1 the regularization kicks in. A value of λ ≈ 1∕1000 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 [33Jump To The Next Citation Point]. The effective dark energy in these models can also cross the phantom divide without divergences [462, 279, 534Jump To The Next Citation Point].

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

ρ˙ + 3H ρ = λ (1.4.37 ) M M ρ˙x + 3H (1 + wx )ρx = − λ, (1.4.38 )
where the subscripts m, x refer to dark matter and dark energy, respectively. In this approximation, the adiabatic sound speed 2 ca reads
˙p w˙ c2a,x = -x-= wx − ----------x----------, (1.4.39 ) ˙ρx 3aH (1 + wx ) + λ∕ρx
which stays finite at crossing as long as λ ⁄= 0.

However in this class of models there are other instabilities arising at the perturbation level regardless of the coupling used, [cf. 916Jump To The Next Citation Point]. A word on perturbations when w = − 1

Although a cosmological constant has w = − 1 and no perturbations, the converse is not automatically true: w = − 1 does not necessarily imply that there are no perturbations. It is only when we set from the beginning (in the calculation):

p = − ρ (1.4.40 ) δp = − δρ (1.4.41 ) π = 0, (1.4.42 )
i.e., T μν ∝ gμν, that we have as a solution δ = V = 0.

For instance, if we set w = − 1 and δp = γδρ (where γ can be a generic function) in Eqs. (1.4.28View Equation) and (1.4.29View Equation) we have δ ⁄= 0 and V ⁄= 0. However, the solutions are decaying modes due to the 1 − a (1 − 3w) V 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 π then the Eqs. (1.4.28View Equation) – (1.4.29View Equation) will have a source term that will influence the growth of δ and V in the same way as Ψ does (just because they appear in the same way). The (1 + w ) term in front of π should not worry us as we can always define the anisotropic stress through

( ) ˆ ˆ 1- i ρ (1 + w)π = − kikj − 3 δij Π j, (1.4.43 )
where i Πj ⁄= 0 when i ⁄= j is the real traceless part of the energy momentum tensor, probably the quantity we need to look at: as in the case of V = (1 + w )𝜃, there is no need for Π ∝ (1 + w )π to vanish when w = − 1.

It is also interesting to notice that when w = − 1 the perturbation equations tell us that dark-energy perturbations are not influenced through Ψ and ′ Φ (see Eq. (1.4.28View Equation) and (1.4.29View Equation)). Since Φ and Ψ are the quantities directly entering the metric, they must remain finite, and even much smaller than 1 for perturbation theory to hold. Since, in the absence of direct couplings, the dark energy only feels the other constituents through the terms (1 + w )Ψ and (1 + w )Φ ′, it decouples completely in the limit w = − 1 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 [537Jump To The Next Citation Point], 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.

1.4.6 f(R) gravity

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 f(R ) gravity in which the 4-dimensional action is given by some generic function f(R) of the Ricci scalar R (for an introduction see, e.g., [49Jump To The Next Citation Point]):

∫ -1-- 4 √ --- S = 2κ2 d x − gf (R) + Sm (gμν,Ψm ), (1.4.44 )
where as usual 2 κ = 8 πG, and Sm is a matter action with matter fields Ψm. Here G 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 f (R) 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 [954Jump To The Next Citation Point, 608, 936, 351, 724Jump To The Next Citation Point, 219], action (1.4.44View Equation) can, therefore, be related to the action (1.4.20View Equation) shown previously. Here we describe f (R ) in the Jordan frame: the matter fields in Sm obey standard conservation equations and, therefore, the metric gμν corresponds to the physical frame (which here is the Jordan frame).

There are two approaches to deriving field equations from the action (1.4.44View Equation).

In GR we have f(R ) = R − 2 Λ and F (R ) = 1, so that the term □F (R ) in Eq. (1.4.46View Equation) vanishes. In this case both the metric and the Palatini formalisms give the relation 2 2 R = − κ T = κ (ρ − 3P), which means that the Ricci scalar R is directly determined by the matter (the trace T).

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

The de Sitter point corresponds to a vacuum solution at which the Ricci scalar is constant. Since □F (R) = 0 at this point, we get

F(R )R − 2f (R) = 0, (1.4.50 )
which holds for both the metric and the Palatini formalisms. Since the model f(R) = αR2 satisfies this condition, it possesses an exact de Sitter solution [862Jump To The Next Citation Point].

It is important to realize that the dynamics of f (R) 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 f(R ) = R + αR2 can be responsible for inflation in the early universe [862]. This comes from the fact that the presence of the quadratic term 2 αR gives rise to an asymptotically exact de Sitter solution. Inflation ends when the term αR2 becomes smaller than the linear term R. Since the term αR2 is negligibly small relative to R at the present epoch, this model is not suitable to realizing the present cosmic acceleration.

Since a late-time acceleration requires modification for small R, models of the type n f (R) = R − α∕R (α > 0,n > 0) 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 f,RR [230Jump To The Next Citation Point, 319Jump To The Next Citation Point, 852, 697Jump To The Next Citation Point, 355Jump To The Next Citation Point]. Moreover a standard matter epoch is not present because of a large coupling between the Ricci scalar and the non-relativistic matter [43Jump To The Next Citation Point].

Then, we can ask what are the conditions for the viability of f(R) 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.

For example, the model n f(R ) = R − α ∕R (α > 0, n > 0) does not satisfy the condition (ii).

Below we list some viable f (R) models that satisfy the above conditions.

(A ) f (R ) = R − μRc(R ∕Rc )p with 0 < p < 1, μ,Rc > 0, (1.4.51 ) (R ∕Rc )2n (B ) f(R ) = R − μRc -------2n----- with n,μ, Rc > 0, (1.4.52 ) (R[ ∕Rc ) + 1 ] (C ) f (R ) = R − μR 1 − (1 + R2∕R2 )−n with n, μ,R > 0, (1.4.53 ) c c c (D ) f (R ) = R − μRctanh (R ∕Rc) with μ, Rc > 0. (1.4.54 )
The models (A), (B), (C), and (D) have been proposed in [39Jump To The Next Citation Point], [456Jump To The Next Citation Point], [864Jump To The Next Citation Point], and [904Jump To The Next Citation Point], 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 p needs to be close to 0 to satisfy the condition (iii). In models (B) and (C) the function f(R ) asymptotically behaves as 2 2 −n f (R ) → R − μRc [1 − (R ∕R c) ] for R ≫ Rc and hence the condition (iii) can be satisfied even for n = 𝒪(1). In model (D) the function f (R ) rapidly approaches f(R ) → R − μRc in the region R ≫ R c. These models satisfy f(R = 0) = 0, so the cosmological constant vanishes in the flat spacetime.

Let us consider the cosmological dynamics of f (R ) gravity in the metric formalism. It is possible to carry out a general analysis without specifying the form of f(R ). In the flat FLRW spacetime the Ricci scalar is given by

R = 6(2H2 + H˙), (1.4.55 )
where H is the Hubble parameter. As a matter action Sm we take into account non-relativistic matter and radiation, which satisfy the usual conservation equations ˙ρm + 3H ρm = 0 and ρ˙ + 4H ρ = 0 r r respectively. From Eqs. (1.4.45View Equation) and (1.4.46View Equation) we obtain the following equations
2 2 3F H = κ (ρm + ρr) + (F R − f)∕2 − 3H F˙, (1.4.56 ) − 2F H˙ = κ2 [ρm + (4∕3)ρr] + F¨ − H F˙. (1.4.57 )
We introduce the dimensionless variables:
F˙ f R κ2ρr x1 ≡ − ----, x2 ≡ − ----2-, x3 ≡ ---2-, x4 ≡ -----2, (1.4.58 ) HF 6FH 6H 3F H
together with the following quantities
2 κ--ρm- Ωm ≡ 3F H2 = 1 − x1 − x2 − x3 − x4, Ωr ≡ x4, ΩDE ≡ x1 + x2 + x3. (1.4.59 )
It is straightforward to derive the following differential equations [39Jump To The Next Citation Point]:
′ 2 x1 = − 1 − x3 − 3x2 + x1 − x1x3 + x4, (1.4.60 ) x′ = x1x3- − x (2x − 4 − x ), (1.4.61 ) 2 m 2 3 1 ′ x1x3- x3 = − m − 2x3(x3 − 2), (1.4.62 ) x′ = − 2x x + x x , (1.4.63 ) 4 3 4 1 4
where the prime denotes d∕d lna and
d lnF Rf,RR m ≡ ------ = ------ , (1.4.64 ) d lnR f,R d-lnf- Rf,R- x3- r ≡ − d ln R = − f = x2. (1.4.65 )
From Eq. (1.4.65View Equation) one can express R as a function of x3∕x2. Since m is a function of R, it follows that m is a function of r, i.e., m = m (r). The ΛCDM model, f (R ) = R − 2Λ, corresponds to m = 0. Hence the quantity m characterizes the deviation from the ΛCDM model. Note also that the model, f(R ) = αR1+m − 2Λ, gives a constant value of m. The analysis using Eqs. (1.4.60View Equation) – (1.4.63View Equation) is sufficiently general in the sense that the form of f(R ) does not need to be specified.

The effective equation of state of the system (i.e., ptot∕ρtot) is

we ff = − 1(2x3 − 1). (1.4.66 ) 3

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 m (r) positive and close to ΛCDM, i.e., m ≥ 0, 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 &tidle; δF ≡ δF ∕F, and the new variable x5 ≡ aH satisfies

x′5 = (x3 − 1)x5. (1.4.69 )
The perturbation δF can be written as δF = f,RRδR and, therefore, &tidle; δF = m δR∕R. These equations can be integrated numerically to derive the behavior of δm at all scales. However, at sub-Hubble scales they can be simplified and the following expression for the two MG functions Q, η of Eq. (1.3.23View Equation) can be obtained:
k2 Q = 1 − ----2--2----2- 3 (a M + k ) -----2k2----- η = 1 − 3a2M 2 + 4k2 (1.4.70 )
1 M 2 = ------. (1.4.71 ) 3f,RR
Note that in the ΛCDM limit f,RR → 0 and Q, η → 1.

These relations can be straightforwardly generalized. In [287] the perturbation equations for the f (R ) Lagrangian have been extended to include coupled scalar fields and their kinetic energy X ≡ − ϕ,μϕμ∕2, resulting in a f(R, ϕ,X )-theory. In the slightly simplified case in which f (R, ϕ, X ) = f1(R, ϕ) + f2(ϕ,X ), with arbitrary functions f1,2, one obtains

1 (1 + 2r1)(f,X + 2r2) + 2F 2,ϕ∕F Q = − ---------------------------2---, F (1 + 3r1)(f,X + 2r2) + 3F ,ϕ∕F (1 + 2r1)(f,X + 2r2) + 2F 2,ϕ∕F η = -------------------------2---, (1.4.72 ) (1 + 4r1)(f,X + 2r2) + 4F ,ϕ∕F
where the notation f,X or F,ϕ denote differentiation wrt X or ϕ, respectively, and where k2 m- r1 ≡ a2 R and a2 2 r2 ≡ k2M ϕ, M ϕ = − f,ϕϕ∕2 being the scalar field effective mass. In the same paper [287] an extra term proportional to X □ ϕ in the Lagrangian is also taken into account.

Euclid forecasts for the f(R ) models will be presented in Section 1.8.7.

1.4.7 Massive gravity and higher-dimensional models

Instead of introducing new scalar degrees of freedom such as in f(R ) 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 327Jump To The Next Citation Point, 328Jump To The Next Citation Point, 58Jump To The Next Citation Point, 330Jump To The Next Citation Point], 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 below). Self-acceleration

The DGP model is one of the important infrared (IR) modified theories of gravity. From a four-dimensional point of view this corresponds effectively to a theory in which the graviton acquires a soft mass m. In this braneworld model our visible universe is confined to a brane of four dimensions embedded into a five-dimensional bulk. At small distances, the four-dimensional gravity is recovered due to an intrinsic Einstein–Hilbert term sourced by the brane curvature causing a gravitational force law that scales as r− 2. At large scales the gravitational force law asymptotes to an r−3 behavior. The cross over scale rc = m −1 is given by the ratio of the Planck masses in four (M4) and five (M5) dimensions. One can study perturbations around flat spacetime and compute the gravitational exchange amplitude between two conserved sources, which does not reduce to the GR result even in the limit m→ 0. However, the successful implementation of the Vainshtein mechanism for decoupling the additional modes from gravitational dynamics at sub-cosmological scales makes these theories still very attractive [913Jump To The Next Citation Point]. Hereby, the Vainshtein effect is realized through the nonlinear interactions of the helicity-0 mode π, as will be explained in further detail below. Thus, this vDVZ discontinuity does not appear close to an astrophysical source where the π field becomes nonlinear and these nonlinear effects of π restore predictions to those of GR. This is most easily understood in the limit where M4, M5 → ∞ and m → 0 while keeping the strong coupling scale Λ = (M4m2 )1∕3 fixed. This allows us to treat the usual helicity-2 mode of gravity linearly while treating the helicity-0 mode π nonlinearly. The resulting effective action is then

1 2 ℒπ = 3 π□ π − --3(∂π) □ π, (1.4.73 ) Λ
where interactions already become important at the scale Λ ≪ MPl [593].

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]

H2 ± mH = 8πG--ρ, (1.4.74 ) 3
such that at higher energies one recovers the usual four-dimensional behavior, H2 ∼ ρ, 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 H = const = m even in the absence of any cosmological constant ρ Λ = 0 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.
Even though the DGP model is interesting for several reasons like giving the Vainshtein effect a chance to work, the self-acceleration solution unfortunately introduces extra ghost states as outlined above. However, it has been generalized to a “Galileon” model, which can be considered as an effective field theory for the helicity-0 field π. Galileon models are invariant under shifts of the field π and shifts of the gradients of π (known as the Galileon symmetry), meaning that a Galileon model is invariant under the transformation

μ π → π + c + vμx , (1.4.75 )
for arbitrary constant c and v μ. In induced gravity braneworld models, this symmetry is naturally inherited from the five-dimensional Poincaré invariance [295Jump To The Next Citation Point]. The Galileon theory relies strongly on this symmetry to constrain the possible structure of the effective π Lagrangian, and insisting that the effective field theory for π bears no ghost-like instabilities further restricts the possibilities [686Jump To The Next Citation Point]. It can be shown that there exist only five derivative interactions, which preserve the Galilean symmetry without introducing ghosts. These interactions are symbolically of the form ℒ(π1)= π and ℒ (nπ) = (∂π )2(∂∂ π)n−2, for n = 2,...5. A general Galileon Lagrangian can be constructed as a linear combination of these Lagrangian operators. The effective action for the DGP scalar (1.4.73View Equation) can be seen to be a combination of (2) ℒ π and (3) ℒ π. Such interactions have been shown to naturally arise from Lovelock invariants in the bulk of generalized braneworld models [295]. However, the Galileon does not necessarily require a higher-dimensional origin and can be consistently treated as a four-dimensional effective field theory. As shown in [686Jump To The Next Citation Point], such theories can allow for self-accelerating de Sitter solutions without any ghosts, unlike in the DGP model. In the presence of compact sources, these solutions can support spherically-symmetric, Vainshtein-like nonlinear perturbations that are also stable against small fluctuations. However, this is constrained to the subset of the third-order Galileon, which contains only (1) ℒ π, (2) ℒπ and (3) ℒπ [669].

The Galileon terms described above form a subset of the “generalized Galileons”. A generalized Galileon model allows nonlinear derivative interactions of the scalar field π 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. (1.4.75View Equation). 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 (n) ℒ π by an arbitrary scalar function of the scalar π 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.

The idea behind degravitation is to modify gravity in the IR, such that the vacuum energy could have a weaker effect on the geometry, and therefore reconcile a natural value for the vacuum energy as expected from particle physics with the observed late-time acceleration. Such modifications of gravity typically arise in models of massive gravity [327, 328, 58, 330], i.e., where gravity is mediated by a massive spin-2 field. The extra-dimensional DGP scenario presented previously, represents a specific model of soft mass gravity, where gravity weakens down at large distance, with a force law going as 1∕r. Nevertheless, this weakening is too weak to achieve degravitation and tackle the cosmological constant problem. However, an obvious way out is to extend the DGP model to higher dimensions, thereby diluting gravity more efficiently at large distances. This is achieved in models of cascading gravity, as is presented below. An alternative to cascading gravity is to work directly with theories of constant mass gravity (hard mass graviton).

Cascading gravity.
Cascading gravity is an explicit realization of the idea of degravitation, where gravity behaves as a high-pass filter, allowing sources with characteristic wavelength (in space and in time) shorter than a characteristic scale rc to behave as expected from GR, but weakening the effect of sources with longer wavelengths. This could explain why a large cosmological constant does not backreact as much as anticipated from standard GR. Since the DGP model does not modify gravity enough in the IR, “cascading gravity” relies on the presence of at least two infinite extra dimensions, while our world is confined on a four-dimensional brane [293Jump To The Next Citation Point]. Similarly as in DGP, four-dimensional gravity is recovered at short distances thanks to an induced Einstein–Hilbert term on the brane with associated Planck scale M 4. The brane we live in is then embedded in a five-dimensional brane, which bears a five-dimensional Planck scale M5, itself embedded in six dimensions (with Planck scale M6). From a four-dimensional perspective, the relevant scales are the 5d and 6d masses m4 = M 35∕M 42 and m5 = M 64∕M 35, which characterize the transition from the 4d to 5d and 5d to 6d behavior respectively. Such theories embedded in more-than-one extra dimensions involve at least one additional scalar field that typically enters as a ghost. This ghost is independent of the ghost present in the self-accelerating branch of DGP but is completely generic to any codimension-two and higher framework with brane localized kinetic terms. However, there are two ways to cure the ghost, both of which are natural when considering a realistic higher codimensional scenario, namely smoothing out the brane, or including a brane tension [293, 290, 294].

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 r ≪ m5 and for m6 ≤ m5, the respective Vainshtein scale or strong coupling scale, i.e., the distance from the source M within which each mode is strongly coupled is 3 2 2 ri = M ∕m iM 4, where i = 5,6. Around a source M, one recovers four-dimensional gravity for r ≪ r5, five-dimensional gravity for r5 ≪ r ≪ r6 and finally six-dimensional gravity at larger distances r ≫ r 6.

Massive gravity.
While laboratory experiments, solar systems tests and cosmological observations have all been in complete agreement with GR for almost a century now, these bounds do not eliminate the possibility for the graviton to bear a small hard mass m ≲ 6.10−32 eV [400]. The question of whether or not gravity could be mediated by a hard-mass graviton is not only a purely fundamental but an abstract one. Since the degravitation mechanism is also expected to be present if the graviton bears a hard mass, such models can play an important role for late-time cosmology, and more precisely when the age of the universe becomes on the order of the graviton Compton wavelength.

Recent progress has shown that theories of hard massive gravity can be free of any ghost-like pathologies in the decoupling limit where MPl → ∞ and m → 0 keeping the scale Λ33 = MPlm2 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 π). As in the braneworld models presented previously, this helicity-0 mode behaves as a scalar field with specific derivative interactions of the form

( ) μν (1) -1- (2) -1- (3) ℒ π = h X μν + Λ33X μν + Λ63X μν . (1.4.76 )
Here, hμν denotes the canonically-normalized (rescaled by Mpl) tensor field perturbation (helicity-2 mode), while Xμ(1ν),X (2μ)ν , and X (μ3)ν are respectively, linear, quadratic and cubic in the helicity-0 mode π. Importantly, they are all transverse (for instance, (1) X μν ∝ ημν□ π − ∂μ∂νπ). 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 (1.4.76View Equation) 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 (1.4.75View Equation), and present very similar physical properties. When (3) X μν is absent, one can indeed recover an Einstein frame picture for which the interactions are of the form
2 2 M-Pl√ --- 3- -3β- 2 -β-- 2( 2 2) ℒ = 2 − gR + 2 π□ π + 2Λ33(∂π) □ π + 2Λ63(∂π ) (∂α∂βπ ) − (□ π) + ℒmat [ψ, &tidle;gμν], (1.4.77 )
where β is an arbitrary constant and matter fields ψ do not couple to the metric gμν but to &tidle;gμν = gμν + πη μν +-β3∂μπ∂νπ Λ 3. Here again, the recovery of GR in the UV is possible via a strong coupling phenomena, where the interactions for π are already important at the scale Λ3 ≪ MPl, 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 [11Jump To The Next Citation Point, 478Jump To The Next Citation Point]. Observations

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 [913Jump To The Next Citation Point]. This Vainshtein mechanism is similar in spirit but different in practice to the chameleon and symmetron mechanisms presented in detail in the next Sections and 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. Screening mechanisms

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, 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.

Density dependent masses: The chameleon.
The chameleon [499] is the archetypal model of a scalar field with a mass that depends on its environment, becoming heavy in dense environments and light in diffuse ones. The ingredients for construction of a chameleon model are a conformal coupling between the scalar field and the matter fields of the standard model, and a potential for the scalar field, which includes relevant self-interaction terms.

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

Veff(ϕ ) = V (ϕ) + ρA (ϕ), (1.4.78 )
where V (ϕ ) is the bare potential, ρ the local energy density and A (ϕ) the conformal coupling function. For suitable choices of A (ϕ ) and V (ϕ) the effective potential has a minimum and the position of the minimum depends on ρ. Self-interaction terms in V (ϕ) ensure that the mass of the field in this minimum also depends on ρ 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. Density dependent couplings

The Vainshtein Mechanism.
In models such as DGP and the Galileon, the effects of the scalar field are screened by the Vainshtein mechanism [913, 299]. This occurs when nonlinear, higher-derivative operators are present in the Lagrangian for a scalar field, arranged in such a way that the equations of motion for the field are still second order, such as the interactions presented in Eq. (1.4.73View Equation). In the presence of a massive source the nonlinear terms force the suppression of the scalar force in the vicinity of a massive object. The radius within which the scalar force is suppressed is known as the Vainshtein radius. As an example in the DGP model the Vainshtein radius around a massive object of mass M is

( M )1∕3 1 r⋆ ∼ ------- --, (1.4.79 ) 4 πMPl Λ
where Λ is the strong coupling scale introduced in section For the Sun, if m ∼ 10− 33 eV, or in other words, Λ −1 = 1000 km, then the Vainshtein radius is r ∼ 102 pc ⋆.

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.

The Symmetron.
The symmetron model [436] is in many ways similar to the chameleon model discussed above. It requires a conformal coupling between the scalar field and the standard model and a potential of a certain form. In the presence of non-relativistic matter this leads to an effective potential for the scalar field

1-( -ρ-- 2) 2 1- 4 Veff(ϕ) = − 2 M 2 − μ ϕ + 4λ ϕ , (1.4.80 )
where M, μ and λ are parameters of the model, and ρ is the local energy density. In sufficiently dense environments, ρ > μ2M 2, the field sits in a minimum at the origin. As the local density drops the symmetry of the field is spontaneously broken and the field falls into one of the two new minima with a non-zero vacuum expectation value. In high-density symmetry-restoring environments, the scalar field vacuum expectation value should be near zero and fluctuations of the field should not couple to matter. Thus, the symmetron force in the exterior of a massive object is suppressed because the field does not couple to the core of the object.

The Olive–Pospelov model.
The Olive–Pospelov model [696] again uses a scalar conformally coupled to matter. In this construction both the coupling function and the scalar field potential are chosen to have quadratic minima. If the background field takes the value that minimizes the coupling function, then fluctuations of the scalar field decouple from matter. In non-relativistic environments the scalar field feels an effective potential, which is a combinations of these two functions. In high-density environments the field is very close to the value that minimizes the form of the coupling function. In low-density environments the field relaxes to the minimum of the bare potential. Thus, the interactions of the scalar field are suppressed in dense environments.

1.4.8 Einstein Aether and its generalizations

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:

( ( ) ) |∇ Ψ | ∇ ⋅ μ --a-- ∇ Ψ = 4πG ρ, (1.4.81 ) 0
where Ψ is the gravitational potential, a0 is a number with dimensions Length− 1 and ρ is the baryonic matter density. The number a0 is determined by looking at the dynamics of visible matter in galaxies [783Jump To The Next Citation Point]. The function μ (x ) would simply be equal to unity in Newtonian gravity. In MOND, the functional form is only fixed at its limits: μ → 1 as x → ∞ and μ → x as x → 0.

We are naturally interested in a relativistic version of such a proposal. The building block is the perturbed spacetime metric already introduced in Eq. 1.3.8View Equation

ds2 = − (1 + 2 Ψ)dt2 + (1 − 2Φ)a2(t)(dR2 + R2d Ω2 ). (1.4.82 )
A simple approach is to introduce a dynamical clock field, which we will call μ A. If it has solutions aligned with the time-like coordinate μ t then it will be sensitive to Ψ. 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 Ψ, as we seek. A family of covariant actions for the clock field is as follows [988]:
∫ [ ] I[gab,Aa, λ] = --1--- d4x √ − g -1F (K ) + λ(AaAa + 1 ) , 16πG ℓ2
2 μνγδ K = ℓ K ∇ μAν∇ γA δ (1.4.83 )
μνγδ μγ νδ μν γδ μδ νδ K = c1g g + c2g g + c3g g . (1.4.84 )
The quantity ℓ is a number with dimensions of length, the cA are dimensionless constants, the Lagrange multiplier field λ enforces the unit-timelike constraint on Aa, and F 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 A μ.

Indeed, when the geometry is of the form (1.4.82View Equation), anisotropic stresses are negligible and Aμ is aligned with the flow of time tμ, then one can find appropriate values of the c A and ℓ such that K is dominated by a term equal to 2 2 |∇ Ψ |∕a 0. 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 (1.4.81View Equation). Therefore the models are successful covariant realizations of MOND.

Interestingly, in the FLRW limit Φ, Ψ → 0, the time-time component of Einstein’s equations in the GEA model becomes a modified Friedmann equation:

( ) H2 2 8πG ρ β -2- H = ------, (1.4.85 ) a0 3
where the function β is related to F and its derivatives with respect to K. The dynamics in galaxies prefer a value a0 on the order the Hubble parameter today H0 [783] and so one typically gets a modification to the background expansion with a characteristic scale H 0, 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 F. However, one may gain insight into the possible solutions of (1.4.85View Equation) by looking at simple forms for F. In [991Jump To The Next Citation Point] the monomial case F ∝ Knae was considered where the kinetic index nae 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 [989Jump To The Next Citation Point, 564] and [991Jump To The Next Citation Point]. 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 w (z) 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 Φ and Ψ. 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.

1.4.9 The Tensor-Vector-Scalar theory of gravity

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 &tidle;gab (with connection ∇&tidle;a) that we refer to as the geometric metric, the vector field A a and the scalar field ϕ. The action for all matter fields, uses a single physical metric g ab (with connection ∇a). The two metrics are related via an algebraic, disformal relation [116] as

gab = e− 2ϕ &tidle;gab − 2 sinh (2 ϕ)AaAb. (1.4.86 )
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.,
&tidle;gabAaAb = AaAa = − 1. (1.4.87 )
The theory is based on an action S, which is split as S = S &tidle;g + SA + Sϕ + Sm where
1 ∫ ∘ --- S &tidle;g = ------ d4x − &tidle;gR&tidle;, (1.4.88 ) 16πG
where &tidle;g and R&tidle; are the determinant and scalar curvature of &tidle;gμν respectively and G is the bare gravitational constant,
1 ∫ 4 ∘ --- [ ab a ] SA = − ------ d x − &tidle;g KF Fab − 2λ(AaA + 1) , (1.4.89 ) 32πG
where Fab = ∇aAb − ∇bAa leads to a Maxwellian kinetic term and λ is a Lagrange multiplier ensuring the unit-timelike constraint on Aa and K is a dimensionless constant (note that indices on Fab are raised using the geometric metric, i.e., F a = &tidle;gacF b cb) and
1 ∫ ∘ --- [ ] Sϕ = − ------ d4x − &tidle;g μˆgab&tidle;∇a ϕ∇&tidle;b ϕ + V (μ) , (1.4.90 ) 16πG
where μ is a non-dynamical dimensionless scalar field, ˆgab = &tidle;gab − AaAb and V (μ ) is an arbitrary function that typically depends on a scale ℓB. The matter is coupled only to the physical metric gab and defines the matter stress-energy tensor Tab through 1∫ 4 √ --- ab δSm = − 2 d x − gTabδg. The TeVeS action can be written entirely in the physical frame [987, 840Jump To The Next Citation Point] 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

3 &tidle;H2 = 8πGe −2ϕ(ρϕ + ρ) , (1.4.91 )
where &tidle; H is the Hubble rate in terms of the geometric scale factor, ρ 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
2ϕ ( ) ρ = -e---- μdV--+ V (1.4.92 ) ϕ 16πG dμ
Exact analytical and numerical solutions with the Bekenstein free function have been found in [841Jump To The Next Citation Point] and in [318Jump To The Next Citation Point]. 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 ρϕ is subdominant and slowly-rolling and 4∕5 ϕ ∝ a. [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 [61Jump To The Next Citation Point] 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 H0 ∼ 70 km ∕s∕Mpc), we have to supplement the absence of CDM with something else. Possibilities include the scalar field itself, massive neutrinos [841Jump To The Next Citation Point, 364Jump To The Next Citation Point] 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.

View Image

Figure 1: Left: the cosmic microwave background angular power spectrum l(l + 1)Cl ∕(2π) for TeVeS (solid) and ΛCDM (dotted) with WMAP 5-year data [689]. Right: the matter power spectrum P (k) for TeVeS (solid) and ΛCDM (dotted) plotted with SDSS data.

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 [841Jump To The Next Citation Point]. 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 P(k ) in TeVeS looks quite similar to that in ΛCDM. Thus TeVeS can produce matter power spectra that cannot be distinguished from ΛCDM by current observations. One would have to turn to other observables to distinguish the two models. The power spectra for TeVeS and ΛCDM are plotted on the right panel of Figure 1View Image. [318Jump To The Next Citation Point] provided an analytical explanation of the growth of structure seen numerically by [841Jump To The Next Citation Point] and found that the growth in TeVeS is due to the vector field perturbation.

It is premature to claim (as in [843, 855Jump To The Next Citation Point]) that only a theory with CDM can fit CMB observations; a prime example to the contrary is the EBI theory [83Jump To The Next Citation Point]. Nevertheless, in the case of TeVeS [841Jump To The Next Citation Point] 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 ΛCDM model (see the left panel of Figure 1View Image). The CMB seems to put TeVeS into trouble, at least for the Bekenstein free function. The result of [318Jump To The Next Citation Point] has a further direct consequence. The difference Φ − Ψ, sometimes named the gravitational slip (see Section 1.3.2), has additional contributions coming from the perturbed vector field α. Since the vector field is required to grow in order to drive structure formation, it will inevitably lead to a growing Φ − Ψ. If the difference Φ − Ψ can be measured observationally, it can provide a substantial test of TeVeS that can distinguish TeVeS from ΛCDM.

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