1. Introduction
During the last few decades Cosmology has evolved from being mainly a theoretical area of physics to become a field supported by high precision observational data. Recent experiments call upon state of the art technology in Astronomy and Astrophysics to provide detailed information on the contents and history of the Universe, which has led to the measurement of parameters that describe our Universe with increasing precision. The standard model of Cosmology is remarkably successful in accounting for the observed features of the Universe. However, a number of fundamental open questions remains at the foundation of the standard model. In particular, we lack a fundamental understanding of the recent acceleration of the Universe [
1,
2]. What is the so-called “dark energy” that is driving the cosmic acceleration? Is it vacuum energy or a dynamical field? Or is the acceleration due to infra-red modifications of Einstein’s theory of General Relativity (GR)? How is structure formation affected in these alternative scenarios? What are the implications of this acceleration for the future of the Universe?
The resolution of these fundamental questions is extremely important for theoretical cosmology. Dark energy models are usually assumed to be responsible for the acceleration of the cosmic expansion in most cosmological studies. However, it is clear that these questions involve not only gravity, but also particle physics. String theory provides a synthesis of these two branches of physics and is widely believed to be moving towards a viable quantum gravity theory. One of the key predictions of string theory is the existence of extra spatial dimensions. In the brane-world scenario, motivated by recent developments in string theory, the observed 3-dimensional universe is embedded in a higher-dimensional spacetime [
3]. The new degrees of freedom belong to the gravitational sector, and can be responsible for the late-time cosmic acceleration [
4,
5]. On the other hand, generalizations of the Einstein-Hilbert Lagrangian, including quadratic Lagrangians which involve second order curvature invariants have also been extensively explored [
6,
7,
8,
9,
10]. These modified theories of gravity not only provide an alternative explanation for the expansion history of the Universe [
11,
12,
13], but they also offer a paradigm fundamentally distinct from the simplest dark energy models of cosmic acceleration [
14], even from those that perfectly mimic the same expansion history. Nevertheless, it has been realized that a large number of modified gravity theories are amenable to a scalar-tensor formulation by means of appropriate metric re-scalings and field redefinitions.
It is, therefore, not surprising that we can think of scalar-tensor gravity theories as a first stepping stone to explore modifications of GR. They have the advantage of apparent simplicity and of a long history of examination. First proposed in its present form by Brans and Dicke for a single scalar field [
15], they have been extensively generalised and have maintained the interest of researchers until the present date. For instance, an extensive field of work has been developed in the cosmological dynamics of scalar-tensor theories. This can be elegantly summarised by carrying out a unified qualitative analysis of the dynamical system for a single scalar field. We also have an established understanding of the observational bounds in these models, where we can use the Parametrized Post-Newtonian formalism to constrain the model parameters. Finally, with a conformal transformation, these theories can be recast as matter interacting scalar fields in General Relativity. In this format, they can still play an important role in dark energy modelling, such as in coupled quintessence models. The consideration of a multi-scalar fields scenario, which can be perceived as the possible reflection of a multi-scalar tensor gravity theory allows for cooperative effects between the fields yielding assisted quintessence.
Indeed, scalar fields are popular building blocks used to construct models of present-day cosmological acceleration. They are appealing because such fields are ubiquitous in theories of high energy physics beyond the standard model and, in particular, are present in theories which include extra spatial dimensions, such as those derived from string theories. Recently, relative to scalar-tensor theory, much work has been invested in the Galileon models and their generalizations [
16]. The latter models allow nonlinear derivative interactions of the scalar field in the Lagrangian and lead to second order field equations, thus removing any ghost-like instabilities. The Lagrangian was first written down by Horndeski in 1974 [
17], which contains four arbitrary functions of the scalar field and its kinetic energy. The form of the Lagrangian is significantly simplified by requiring specific self-tuning properties (though it still has four arbitrary functions), however, the screening is too effective, and will screen curvature from other matter sources as well as from the vacuum energy [
18]. An alternative approach consists of searching for a de Sitter critical point for any kind of material content [
19]. These models might alleviate the cosmological constant problem and can deliver a background dynamics compatible with the latest observational data.
A promising alternative to explain the late-time cosmic acceleration is to assume that at large scales Einstein’s theory of GR breaks down, and a more general action describes the gravitational field. Thus, one may generalize the Einstein-Hilbert action by including second order curvature invariants such as
,
,
,
, etc. Some of the physical motivations for these modifications of gravity were inspired on effective models raised in string theory, which indeed may lead to the possibility of a more realistic representation of the gravitational fields near curvature singularities [
20]. Moreover, the quantization of fields in curved space-times tell us that the high-energy completion of the Einstein-Hilbert Lagrangian of GR involves higher-order terms on the curvature invariants above [
21]. This is in agreement with the results provided from the point of view of treating GR as an effective field theory [
22]. Among these extensions of GR the so-called
gravity has drawn much attention over the last years, since it can reproduce late-time acceleration and in spite of containing higher order derivatives, it is free of the Ostrogradsky instability, as can be shown by its equivalence with scalar-tensor theories (for a review on
gravity see Refs. [
6,
7,
8,
9]). Moreover,
gravities have been also proposed as solutions for the inflationary paradigm [
23], where the so-called Starobinsky model is a successful proposal, since it satisfies the latest constraints released by Planck [
24]. In addition, the equivalence of
gravities to some class of scalar-tensor theories has provided an extension of the so-called chameleon mechanism to
gravity, leading to some viable extensions of GR that pass the solar system constraints [
25,
26]. Other alternative formulations for these extensions of GR have been considered in the literature, namely, the Palatini formalism, where metric and affine connection are regarded as independent degrees of freedom, which yields an interesting phenomenology for Cosmology [
27]; and the metric-affine formalism, where the matter part of the action now depends and is varied with respect to the connection [
28]. Recently, a novel approach to modified theories of gravity was proposed that consists of adding to the Einstein-Hilbert Lagrangian an
term constructed
a la Palatini [
29]. It was shown that the theory can pass the Solar System observational constraints even if the scalar field is very light. This implies the existence of a long-range scalar field, which is able to modify the cosmological and galactic dynamics, but leaves the Solar System unaffected.
Note that these modified theories of gravity are focussed on extensions of the curvature-based Einstein-Hilbert action. Nevertheless, one could equally well modify gravity starting from its torsion-based formulation and, in particular, from the Teleparallel Equivalent of General Relativity (TEGR) [
30]. The interesting point is that although GR is completely equivalent with TEGR at the level of equations, their modifications (for instance
and
gravities, where
T is the torsion) are not equivalent and they correspond to different classes of gravitational modifications. Hence,
gravity has novel and interesting cosmological implications, capable of describing inflation, the late-time acceleration, large scale structure, bouncing solutions, non-minimal couplings to matter, etc. [
31,
32,
33].
Another gravitational modification that has recently attracted much interest is the massive gravity paradigm, where instead of introducing new scalar degrees of freedom, such as in
gravity, it modifies the graviton itself. Massive gravity is a well-defined theoretical problem on its own and has important cosmological motivations, namely, if gravity is massive it will be weaker at large scales and thus one can obtain the late-time cosmic acceleration. Fierz and Pauli presented the first linear massive gravity. However, it was shown to suffer from the van Dam-Veltman-Zakharov (vDVZ) discontinuity [
34,
35], namely the massless limit of the results do not yield the massless theory, namely, GR. The incorporation of nonlinear terms cured the problem but introduced the Boulware-Deser (BD) ghost. This fundamental problem puzzled physicists until recently, where a specific nonlinear extension of massive gravity was proposed by de Rham, Gabadadze and Tolley (dRGT), in which the BD ghost is eliminated by a Hamiltonian constraint [
16]. This new nonlinear massive gravity has interesting cosmological implications, for instance, it can give rise to inflation, late-time acceleration [
16]. However, the basic versions of this theory exhibit instabilities at the perturbative level, and thus suitable extensions are necessary. These could be anisotropic versions,
extensions, bigravity generalizations, partially-massive constructions. The crucial issue is whether one can construct a massive gravity and cosmology that can be consistent as an alternative to dark energy or other models of modified gravity, and whether this theory is in agreement with high-precision cosmological data, such as the growth-index or the tensor-to-scalar ratio, remains to be explored in detail.
Quantum field theory predicts that the universe underwent, in its early stages, a series of symmetry breaking phase transitions, some of which may have led to the formation of topological defects. Different types of defects may be formed depending on the (non-trivial) topology of the vacuum manifold or the type of symmetry being broken. For instance, Domain Walls—which are surfaces that separate domains with different vacuum expectation values—may arise due to the breaking of a discrete symmetry, whenever the vacuum manifold is disconnected. Line-like defects, or Cosmic strings, are formed if the vacuum is not simply connected or, equivalently, if it contains unshrinkable loops. This type of vacuum manifold results, in general, from the breaking of an axial symmetry. Moreover, if the vacuum manifold contains unshrinkable surfaces, the field might develop non-trivial configurations corresponding to point-like defects, known as Monopoles. The spontaneous symmetry breaking of more complex symmetry groups may lead to the formation of textures, delocalized topological defects which are unstable to collapse. The production of topological defects networks as remnants of symmetry breaking phase transitions is thus predicted in several grand unified scenarios and in several models of inflation. Moreover, recent developments in the braneworld realization of string theory suggest that its fundamental objects—p-dimensional D-branes and fundamental strings—may play the cosmological role of topological defects.
Topological defects networks, although formed in the early universe, may in most instances survive throughout the cosmological history and leave a variety of imprints on different observational probes. The observational consequences of topological defect networks can be very diverse, depending both on the type of defects formed and on the evolution of the universe after they are generated. Although the possibility of a significant contribution to the dark energy budget has been ruled out both dynamically [
36] and observationally [
37], light domain walls may leave behind interesting astrophysical and cosmological signatures. For instance, they may be associated to spatial variations of the fundamental couplings of nature (see, e.g., [
38]). On the other hand, cosmic strings may contribute significantly to small-scale cosmological perturbations and have consequently been suggested to have significant impact on the formation of ultracompact minihalos [
39], globular clusters [
40], super-massive black holes [
41] and to provide a significant contribution to the reionization history of the Universe [
42]. Both cosmic strings and domain walls may be responsible for significant contributions to two of the most significant observational probes: the temperature and polarization anisotropies of the cosmic microwave background and the stochastic gravitational wave background. This fact—alongside the possibility of testing string theory through the study of topological defects—greatly motivates the interest on the astrophysical and cosmological signatures of topological defects.
An extremely important aspect of modern cosmology is the synergy between theory and observations. Dark energy models and modified gravity affect the geometry of the universe and cosmological structure formation, impacting the background expansion and leaving an imprint on the statistical properties of the large-scale structure. There are a number of well-established probes of cosmic evolution, such as type Ia supernovae, baryon acoustic oscillations (BAO), weak gravitational lensing, galaxy clustering and galaxy clusters properties [
43]. Different methods measure different observables, probing expansion and structure formation in different and often complementary ways and have different systematic effects. In particular, joint analyses with Cosmic Microwave Background (CMB) data are helpful in breaking degeneracies by constraining the standard cosmological parameters. Indeed, CMB has revolutionized the way we perceive the Universe. The information encoded in its temperature and polarization maps provides one of the strongest evidences in favour of the hot Big-Bang theory and has enabled ways to constrain cosmological models with unprecedented accuracy [
44].
The CMB also encodes additional information about the growth of cosmological structure and the dynamics of the Universe through the secondary CMB anisotropies. These are originated by physical effects acting on the CMB after decoupling [
45], such as the integrated Sachs-Wolfe effect and the Sunyaev-Zel’dovich (SZ) effect, manifest respectively on the largest and arc-minute scales of the CMB. In this review, we will discuss in some detail both a well-established acceleration probe (weak lensing) and a few promising ones related to galaxy cluster properties and the SZ effect. Galaxy clusters are the largest gravitationally bound objects in the Universe and are among the latest bound structures forming in the Universe. For this reason, their number density is highly sensitive to the details of structure formation as well as to cosmological background parameters and cluster abundance is a well established cosmological probe. The Sunyaev-Zel’dovich effect [
46,
47] is the scattering of CMB photons by electrons in hot reservoirs of ionized gas in the Universe, such as galaxy clusters. In particular, SZ galaxy cluster counts, profiles, scaling relations, angular power spectra and induced spectral distortions are promising probes to confront model predictions with observations. Weak gravitational lensing [
48] describes the deflection of light by gravity in the weak regime. Its angular power spectrum is a direct measure of the statistical properties of gravity and matter on cosmological scales. Weak lensing, together with galaxy clustering, is the core method of the forthcoming Euclid mission to map the dark universe. Euclid [
49] will provide us with weak lensing measurements of unprecedented precision. To obtain high-precision and high-accuracy constraints on dark energy or modified gravity properties, with both weak lensing and SZ clusters, non-linearities on structure formation must be taken into account. While linear covariant perturbations equations may be evolved with Boltzmann codes, non-linearities require dedicated N-body cosmological simulations [
50]. There currently exist a number of simulations for various modified gravity and dark energy models, together with a set of formulas that fit a non-linear power spectrum from a linear one. Hydrodynamic simulations, commonly used in cluster studies, are increasingly needed in weak lensing applications to model various baryonic effects on lensing observables, such as supernova and AGN feedback, star formation or radiative cooling [
51].
The ΛCDM framework provides a very good fit to various datasets, but it contains some open issues [
52]. As an example, there are inconsistencies between probes, such as the tension between CMB primary signal (Planck) and weak lensing (CFHTLenS) [
53], as well as problems with the interpretation of large-scale CMB measurements (the so-called CMB anomalies) [
54]. Alternatives to ΛCDM or deviations to General Relativity are usually confronted with data using one of two approaches: model selection or parameterizations. In model selection a specific model is analyzed and its parameters constrained. Such analyses have a narrower scope but may be better physically motivated. Parameterizations are good working tools and are helpful in highlighting a particular feature and in ruling out larger classes of models, however they must be carefully defined in a consistent way. Parameterizations are commonly applied to the dark energy equation of state and to deviations from General Relativity. An example of the latter is the gravitational slip, which provides an unambiguous signature of modified gravity, and can be estimated combining weak lensing measurements of the lensing potential with galaxy clustering measurements of the Newtonian potential. Model parameters in both of the approaches are usually estimated with Monte Carlo techniques, while the viability of different models may be compared using various information criteria.
Besides model testing, cosmological data is also useful to test foundational assumptions, such as the (statistical) cosmological principle and the inflationary paradigm. The common understanding is that cosmological structures are the result of primordial density fluctuations that grew under gravitational instability collapse. These primordial density perturbations would have originated during the inflationary phase in the early universe. Most single field slow-roll inflationary models produce nearly Gaussian distributed perturbations, with very weak possible deviations from Gaussianity at a level beyond detection [
55,
56,
57]. However, non-Gaussianities may arise in inflationary models in which any of the conditions leading to the slow-roll dynamics fail [
58], such as the curvaton scenario [
59,
60,
61], the ekpyrotic inflacionary scenario [
62,
63], vector field populated inflation [
64,
65,
66] and multi-field inflation [
67,
68,
69,
70]. Tests of non-Gaussianity are thus a way to discriminate between inflation models and to test the different proposed mechanisms for the generation of primordial density perturbations. Likewise, the assumption of statistical homogeneity may be tested. Locally, matter is distributed according to a pattern of alternate overdense regions and underdense regions. Since averaging inhomogeneities in the matter density distribution yields a homogeneous description of the Universe, then the apparent homogeneity of the cosmological parameters could also result from the averaging of inhomogeneities in the cosmological parameters, which would reflect the inhomogeneities in the density distribution. The theoretical setup closest to this reasoning is that of backreaction models, where the angular variations in the parameters could also source a repulsive force and potentially emulate cosmic acceleration. Hence the reasoning is to look for these inhomogeneities, not in depth, but rather across the sky and then to use an adequate toy model to compute the magnitude of the acceleration derived from angular variations of the parameters compared to the acceleration driven by a cosmological constant [
71].
This work will focus on all of the above-mentioned topics. More specifically, this article is outlined in the following manner: In
Section 2, we present scalar-tensor theories, and in
Section 3, we consider Horndeski theories and the self-tuning properties. In
Section 4,
modified theories of gravity and extensions are reviewed. In
Section 5, an extensive review on topological defects is carried out. The following sections are dedicated to observational cosmology. In particular, in
Section 6, cosmological tests with galaxy clusters at CMB frequencies are presented and in
Section 7, gravitational lensing will be explored. In
Section 8, the angular distribution of cosmological parameters as a measurement of spacetime inhomogeneities will be presented. Finally, in
Section 9 we conclude.
3. Horndeski Theories and Self-tuning
In the previous sections we have considered the simplest scalar-tensor theories of gravity, which are given by the action (
1). Those theories can be re-written as a scalar field, which is coupled to the geometry through a term
in the action, with a canonical kinetic term and a potential. This is:
where
. Nevertheless,
there are more things in heaven and earth for scalar-tensor theories of gravity. K-essence models [
133] assume a scalar field minimally coupled to gravity (
) with an action that is not necessarily the sum of a kinetic term and a potential but a more general function
. One can go further and consider interaction terms in the Lagrangian containing second order derivatives of the scalar field, as in the kinetic braiding models [
134] that have a scalar field Lagrangian given by
. These models can be stable because their field equations are only second order. However, theories with Lagrangians containing second order derivatives generically have field equations which contain higher than second order derivatives. Therefore, such theories usually propagate an extra (ghostly) degree of freedom which means that they are affected by the Ostrogradski instability [
135]. In fact, in 1974, Horndeski found the most general scalar-tensor action leading to second order equations of motion [
17]. Nevertheless, Horndeski work passed mostly unnoticed until Deffayet et al. [
136] rediscovered his theory when generalizing the covariantized version [
137] of the galileons models [
138].
The Horndeski Lagrangian can be written as [
17]
where
and
are arbitrary functions, satisfying
. Brans–Dicke theory, k-essence, kinetic braiding and many other models can be seen as particular cases of the most general Horndeski family. Thus, although the Horndeski Lagrangian restricts the number of stable scalar-tensor theories. Actually it has been shown that there are still stable theories beyond the Horndeski Lagrangian [
139,
140,
141]. Those theories have second order equations of motion only when one particular gauge is fixed.
On the other hand, the vacuum energy also gravitates in modified theories of gravity. Thus, the cosmological constant problem [
142,
143,
144] is still present if, as Weinberg suggested, the extra degree of freedom could screen only a given value of that constant [
142]. In this context, the
fab four models are based in the observation by Charmousis et al. [
18,
145] that Weinberg’s no-go theorem can be avoided by relaxing one of the assumptions, that is allowing the field to have a non-trivial temporal dependence once the cosmological constant has been screened. Their dynamical screening is based in requiring Minkowski to be a critical point of the dynamics and, when the critical point is an attractor, it may alleviate the cosmological constant problem. Nevertheless, these models are forced by construction to decelerate its expansion while approaching the Minkowski final state. Thus, a late time accelerating cosmology does not naturally arise in this scenario. When considering the compatibility of dynamical screening the vacuum energy with a late time phase of accelerated expansion, the concept of self-adjustment was extended to non-Minkowskian final states [
19]. As we will now show in detail, a scalar field self-tuning to de Sitter can lead to very promising scenarios from a phenomenological point of view [
146,
147]. In addition, it may alleviate the cosmological constant problem if the field is able to screen the vacuum energy before the material content in a particular model.
3.1. Dynamical Screening
In a cosmological context, one can consider a FLRW geometry to express the Horndeski Lagrangian in the minisuperspace. Once the dependence on higher derivatives is integrated by parts the point-like Lagrangian takes the simple form [
18]
V is the spatial integral of the volume element,
is the Hubble expansion rate, and an over-dot represents a derivative with respect the cosmic time
t. The functions
are given by
where
and
are written in terms of the Horndeski free functions [
18]. Taking into account Equation (
40), the Hamiltonian density can be written as
Assuming that the matter content, described by
, is minimally coupled with the geometry and non-interacting with the field, the modified Friedmann equation is given by
Now, one can follow a similar argument as that presented in reference [
18] for screening to Minkwoski, the only difference being that we require self-tuning to a more general given on-shell solution with
(where the subscript “on” means evaluated
on-shell-in-a), as done in reference [
19]. Thus, assuming that the field is continuous in any phase transition that changes the value of the vacuum energy and noting that the screening of any value of the vacuum energy would lead to the screening of any material content, one has to require
to be approached dynamical in order to have acceptable cosmological solutions. That is, we want
to be an attractor solution, although this particular adjustment mechanism only ensures that it is a critical point. As it is subtly discussed in reference [
18], this requirement leads to the following three conditions:
The field equation evaluated at the critical point has to be trivially satisfied leading the value of the field free to screen. This implies that the minisuperspace Lagrangian density at the critical point takes the form
The modified Freedman equation evaluated once screening has taken place has to depend on
to absorb possible discontinuities of the cosmological constant appearing on the r. h. s. of this equation. Thus, taking into account Equations (
43) and (
44) (and its
-derivative), it leads to
The full scalar equation of motion must depend on
to allow for a non-trivial cosmological dynamics before screening. This leads to a condition equivalent to (
45) if
.
A particular Lagrangian which satisfies conditions can be written as
This Lagrangian (which is a trivial generalization of that presented in reference [
18]) has a critical point at the solution
by construction [
19]. In order to obtian some information about the
’s, we note that, as it has been explicitly proven in [
18] (it must be noted that such a proof is independent of the particular
), two Horndeski theories which self-tune to
are related through a total derivative of a function
. Therefore, one can consider
with
given by Equation (
46). As this equation has to be valid during the whole evolution, we can equate equal powers of
H to obtain
which can be combined to yield [
18,
19]
3.2. Requiring the Existence of a de Sitter Critical Point:
In the first place, we consider
. In this case the dependence on the scale factor
a of the
’s vanishes, as can be easily noted from Equation (
41), and the point-like Lagrangian (
40) is independent of
’s. For
, therefore, Equation (
49) is equal to the
. In the second place, we demand
, this leads to [
19]
As the l. h. s. of this equation is independent of
a, the r. h. s. should also be independent of
a for any value of
, which allows us to obtain
. Thus, we have
Therefore, there are three different kind of terms which can appear in the Lagrangian. These are: (i)
-terms linear on
; (ii)
-terms with non-linear dependence on
, which contribution has to vanish in the on-shell point-like Lagrangian; and, (iii) terms that are not able to self-tune because they contribute through total derivatives or terms multiplied by
k in the Lagrangian. Thus, the point-like Lagrangian takes the form [
19]
with
and
to be given in the following discussion.
3.2.1. Linear Terms: “The Magnificent Seven”
In order to satisfy Equation (
51) considering only terms linear on
, it is enough to have
with the potentials
and
fulfilling the constraint
The Lagrangian of the linear terms is
with the potentials satisfying condition (
54). As there are a total of eight functions and only one constraint, there are effectively only seven free functions which we coined “the magnificent seven”. The field equation for these models is
where we have divided all the equations by
for simplicity and we require that the term in the square bracket does not vanish. On the other hand, the Hamiltonian density is
When only this kind of terms are present the modified Friedmann equation is
. These models are able to screen any material content dynamically as long as there is at least one
with
. The models with only
and
potentials and those with only
’s would not spoil the screening of other models when combined with them, but they are not able to self-tune by themselves [
19]. It is worthy to note that a non-self-tuning Einstein–Hilbert term is contained in these models and can be explicitly written redefining
as
.
3.2.2. Non-linear Terms
We now consider terms with an arbitrary dependence on
and
. These terms have a minisuperspace Lagrangian given by
The functions
are, however, not arbitrary. Taking into account Equation (
51), the contribution of these terms has to vanish when
is evaluated at the critical point. Therefore, they have to satisfy the condition
The Hamiltonian density of this models is
Taking into account condition (
59) and its derivatives, it can be seen that
depends on
; thus, these models are always able to self-tune. If only these nonlinear terms are present, the modified Friedmann equation is
, and
otherwise.
On the other hand, taking into account Lagrangian (
58) the field equation can be written as
with
.
3.3. Cosmology of the Models
3.3.1. Example of a Linear Model
The field equation and the Friedmann equation read [
146]
where a prime represents derivative with respect to
.
As an example of a linear model let us consider the three potentials
,
and
. The constraint equation imposes
, and then
For
,
therefore we need:
, during a matter domination epoch, and
, for a radiation domination epoch. For example, the choice of the potentials,
, and
, give us the desired behaviour, as shown in
Figure 3. The de Sitter evolution is attained when
.
Unfortunately, the contribution of the field at early times is too large to satisfy current constraints.
3.3.2. Example of a Non-Linear Model
We will restrict the analysis to the shift-symmetric cases, which means no dependence on
, and make use of the convenient redefinition
. Under these assumptions we obtain the equations of motion [
147],
where
,
,
,
,
,
, are non-trivial functions of
and
H, and the average equation of state parameter of matter fluids is
Let us consider a case involving the three potentials
,
and
such that
We can obtain a model with
, such that,
and
, which is compatible with current limits and moreover, has a negligible dark energy contribution at early times. The evolution of the energy densities is illustrated in
Figure 4.
3.4. Summary
In this section we have considered a subclass of the Horndeski Lagrangian that leads to a late-time de Sitter evolution of the Universe and that may provide a mechanism to alleviate the cosmological constant problem. We have presented examples of a linear and a non-linear model. In particular, the shift-symmetric non-linear models, the de Sitter critical point is indeed an attractor. We can, thus, understand the current accelerated expansion of our Universe as the result of the dynamical approach of the field to the critical point. The model presented satisfies current observational bounds on the background evolution and appear very promising.