In the flat FLRW background the component of Eq. (12.10) leads to

where and are the energy densities of non-relativistic matter and radiation, respectively. The cosmological dynamics in dark energy models have been discussed in [458, 165, 383, 188, 633, 430]. We can realize the late-time cosmic acceleration by the existence of a de Sitter point satisfying the condition [458] where and are the Hubble parameter and the GB term at the de Sitter point, respectively. From the stability of the de Sitter point we require the following condition [188]The GB term is given by

where is the effective equation of state. We have and during both radiation and matter domination. The GB term changes its sign from negative to positive during the transition from the matter era () to the de Sitter epoch (). Perturbing Eq. (12.11) about the background radiation and matter dominated solutions, the perturbations in the Hubble parameter involve the mass squared given by [188]. For the stability of background solutions we require that , i.e., . Since the term in Eq. (12.11) is of the order of , this is suppressed relative to for during the radiation and matter dominated epochs. In order to satisfy this condition we require that approaches in the limit . The deviation from the CDM model can be quantified by the following quantity [182] where a prime represents a derivative with respect to . During the radiation and matter eras we have and , respectively, whereas at the de Sitter attractor .The GB term inside and outside a spherically symmetric body (mass and radius ) with a homogeneous density is given by and , respectively ( is a distance from the center of symmetry). In the vicinity of Sun or Earth, is much larger than the present cosmological GB term, . As we move from the interior to the exterior of the star, the GB term crosses 0 from negative to positive. This means that and its derivatives with respect to need to be regular for both negative and positive values of whose amplitudes are much larger than .

The above discussions show that viable models need to obey the following conditions:

- 1.
- and its derivatives , , …are regular.
- 2.
- for all and approaches in the limit .
- 3.
- at the de Sitter point.

A couple of representative models that can satisfy these conditions are [188]

where , and are positive constants. The second derivatives of in terms of for the models (A) and (B) are and , respectively. They are constructed to give rise to the positive for all . Of course other models can be introduced by following the same prescription. These models can pass the constraint of successful expansion history that allows the smooth transition from radiation and matter eras to the accelerated epoch [188, 633]. Although it is possible to have a viable expansion history at the background level, the study of matter density perturbations places tight constraints on these models. We shall address this issue in Section 12.3.4.

In order to discuss cosmological solutions in the low-redshift regime numerically for the models (12.16) and (12.17), it is convenient to introduce the following dimensionless quantities

where . We then obtain the following equations of motion [188] where a prime represents a derivative with respect to . The quantities and can be expressed by and once the model is specified.Figure 12 shows the evolution of and without radiation for the model (12.16) with parameters and . The quantity is much smaller than unity in the deep matter era () and it reaches a maximum value prior to the accelerated epoch. This is followed by the decrease of toward 0, as the solution approaches the de Sitter attractor with . While the maximum value of in this case is of the order of , it is also possible to realize larger maximum values of such as .

For high redshifts the equations become too stiff to be integrated directly. This comes from the fact that, as we go back to the past, the quantity (or ) becomes smaller and smaller. In fact, this also occurs for viable f (R) dark energy models in which decreases rapidly for higher . Here we show an iterative method (known as the “fixed-point” method) [420, 188] that can be used in these cases, provided no singularity is present in the high redshift regime [188]. We define and to be and , where the subscript “0” represents present values. The models (A) and (B) can be written in the form

where and is a function of . The modified Friedmann equation reduces to where (which represents the Hubble parameter in the CDM model). In the following we omit the tilde for simplicity.In Eq. (12.24) there are derivatives of in terms of up to second-order. Then we write Eq. (12.24) in the form

where . At high redshifts () the models (A) and (B) are close to the CDM model, i.e., . As a starting guess we set the solution to be . The first iteration is then , where . The second iteration is , where .If the starting guess is in the basin of a fixed point, will converge to the solution of the equation after the -th iteration. For the convergence we need the following condition

which means that each correction decreases for larger . The following relation is also required to be satisfied: Once the solution begins to converge, one can stop the iteration up to the required/available level of precision. In Figure 13 we plot absolute errors for the model (12.16), which shows that the iterative method can produce solutions accurately in the high-redshift regime. Typically this method stops working when the initial guess is outside the basin of convergence. This happen for low redshifts in which the modifications of gravity come into play. In this regime we just need to integrate Eqs. (12.19) – (12.22) directly.We study local gravity constraints on cosmologically viable models. First of all there is a big difference between and f (R) theories. The vacuum GR solution of a spherically symmetric manifold, the Schwarzschild metric, corresponds to a vanishing Ricci scalar () outside the star. In the presence of non-relativistic matter, approximately equals to the matter density for viable f (R) models.

On the other hand, even for the vacuum exterior of the Schwarzschild metric, the GB term has a non-vanishing value [178, 185], where is the Schwarzschild radius of the object. In the regime the models (A) and (B) have a correction term of the order plus a cosmological constant term . Since does not vanish even in the vacuum, the correction term can be much smaller than 1 even in the absence of non-relativistic matter. If matter is present, this gives rise to the contribution of the order of to the GB term. The ratio of the matter contribution to the vacuum value is estimated as

At the surface of Sun (radius and mass ), the density drops down rapidly from the order to the order . If we take the value we have (where we have used ). Taking the value leads to a much smaller ratio: . The matter density approaches a constant value around the distance from the center of Sun. Even at this distance we have , which means that the matter contribution to the GB term can be neglected in the solar system we are interested in.In order to discuss the effect of the correction term on the Schwarzschild metric, we introduce a dimensionless parameter

where is the scale of the GB term in the solar system. Since is of the order of the Hubble parameter , the parameter for the Sun is approximately given by . We can then decompose the vacuum equations in the form where is the Einstein tensor and Here is defined by .We introduce the following ansatz for the metric

where the functions and are expanded as power series in , as Then we can solve Eq. (12.30) as follows. At zero-th order the equations read which leads to the usual Schwarzschild solution, . At linear order one has Since and are known, one can solve the differential equations for and . This process can be iterated order by order in .For the model (A) introduced in (12.16), we obtain the following differential equations for and [185]:

where . The solutions to these equations areHere we have neglected the contribution coming from the homogeneous solution, as this would correspond to an order renormalization contribution to the mass of the system. Although , the term in only contributes by a factor of order . Since the largest contributions to and correspond to those proportional to , which are different from the Schwarzschild–de Sitter contribution (which grows as ). Hence the model (12.16) gives rise to the corrections larger than that in the cosmological constant case by a factor of . Since is very small, the contributions to the solar-system experiments due to this modification are too weak to be detected. The strongest bound comes from the shift of the perihelion of Mercury, which gives the very mild bound [185]. For the model (12.17) the constraint is even weaker, . In other words, the corrections look similar to the Schwarzschild–de Sitter metric on which only very weak bounds can be placed.

In the following we shall discuss ghost conditions for the action (12.9). For simplicity let us consider the vacuum case () in the FLRW background. The action (12.9) can be expanded at second order in perturbations for the perturbed metric (6.1), as we have done for the action (6.2) in Section 7.4. Before doing so, we introduce the gauge-invariant perturbed quantity

This quantity completely describes the dynamics of all the scalar perturbations. Note that for the gauge choice one has . Integrating out all the auxiliary fields, we obtain the second-order perturbed action [186] where we have defined Recall that has been introduced in Eq. (12.15).In order to avoid that the scalar mode becomes a ghost, one requires that , i.e.

This relation is dynamical, as one requires to know how and its derivatives change in time. Therefore whatever is, the propagating scalar mode can still become a ghost. If and , then and hence the ghost does not appear. The quantity characterizes the speed of propagation for the scalar mode, which is again dependent on the dynamics. For any GB theory, one can give initial conditions of and such that becomes negative. This instability, if present, governs the high momentum modes in Fourier space, which corresponds to an Ultra-Violet (UV) instability. In order to avoid this UV instability in the vacuum, we require that the effective equation of state satisfies . At the de Sitter point () the speed is time-independent and reduces to the speed of light ().Suppose that the scalar mode does not have a ghost mode, i.e., . Making the field redefinition and , the action (12.41) can be written as

where a prime represents a derivative with respect to and . In order to realize the positive mass squared (), the condition needs to be satisfied in the regime (analogous to the condition in metric f (R) gravity).

In the presence of matter, other degrees of freedom appear in the action. Let us take into account a perfect fluid with the barotropic equation of state . It can be proved that, for small scales (i.e., for large momenta ) in Fourier space, there are two different propagation speeds given by [182]

The first result is expected, as it corresponds to the matter propagation speed. Meanwhile the presence of matter gives rise to a correction term to in Eq. (12.43). This latter result is due to the fact that the background equations of motion are different between the two cases. Recall that for viable models one has at high redshifts. Since the background evolution is approximately given by and , it follows that

Hence the UV instability can be avoided for . During the radiation era () and the matter era (), the large momentum modes are unstable. In particular this leads to the violent growth of matter density perturbations incompatible with the observations of large-scale structure [383, 182]. The onset of the negative instability can be characterized by the condition [182] As long as we can always find a wavenumber satisfying the condition (12.49). For those scales linear perturbation theory breaks down, and in principle one should look for all higher-order contributions. Hence the background solutions cannot be trusted any longer, at least for small scales, which makes the theory unpredictable. In the same regime, one can easily see that the scalar mode is not a ghost, as Eq. (12.44) is satisfied (see Figure 12). Therefore the instability is purely classical. This kind of UV instability sets serious problems for any theory, including gravity.

We shall also discuss more general theories given by Eq. (12.8), i.e.

where we do not take into account the matter term here. It is clear that this function allows more freedom with respect to the background cosmological evolutionThe second-order action for perturbations is given by

where we have introduced the gauge-invariant field with and . The forms of , and are given explicitly in [186].The quantity vanishes either on the de Sitter solution or for those theories satisfying

If , then the modes with high momenta have a very different propagation. Indeed the frequency becomes -dependent, that is [186] If , then a violent instability arises. If , then these modes propagate with a group velocity This result implies that the superluminal propagation is always present in these theories, and the speed is scale-dependent. On the other hand, when , this behavior is not present at all. Therefore, there is a physical property by which different modifications of gravity can be distinguished. The presence of an extra matter scalar field does not change this regime at high [185], because the Laplacian of the gravitational field is not modified by the field coupled to gravity in the form .http://www.livingreviews.org/lrr-2010-3 |
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