## 1. Introduction

Cosmological observations, including type Ia supernovae (SNIa) [1, 2], cosmic microwave background (CMB) anisotropies, and baryon acoustic oscillations (BAO), indicate that the universe is undergoing an accelerated phase of expansion. This late-time acceleration is one of the biggest mysteries in current cosmology. The standard explanation is that this acceleration is caused by dark energy [3–6]. This would mean that a large part of components in the universe is unknown. The cosmological constant is a candidate of dark energy. To explain the late-time accelerated expansion of the universe, the cosmological constant must be a very small value. However, its value is not compatible with a prediction from particle physics, and it has fine-tuning and coincidence problems.

An alternative explanation for the current acceleration of the universe is to modify general relativity to be a more general theory of gravity at a long-distance scale. Several modified gravity theories have been studied, such as *f*(*R*) gravity (for reviews, see, e.g., [7]), scalar-tensor theories [8–10], and the Dvali–Gabadadze–Porrati (DGP) braneworld model [11–13].

Furthermore, as an alternative to general relativity, Galileon gravity models have been proposed [14–22]. These models are built by introducing a scalar field with a self-interaction whose Lagrangian, which is invariant under Galileon symmetry

Galileon theory has been covariantized and studied in curved backgrounds [24, 25]. Although Galileon symmetry cannot be maintained in the case that the theory is covariantized, it is possible to preserve the equation of motion at second order, which means that the theory does not raise ghost-like instabilities. Galileon gravity induces self-accelerated expansion of the current universe. Thus, inflation models inspired by the Galileon gravity theory have been studied [26–28]. In Ref. [29], the parameters of the generalized Galileon cosmology were constrained from the observational data of SNIa, CMB, and BAO. The evolution of matter density perturbations for Galileon cosmology has also been investigated [16–18, 30, 31].

Almost 40 years ago, Horndeski derived the action of most general scalar-tensor theories with second-order equations of motion [32]. His theory received much attention as an extension of covariant Galileons [14, 24, 25, 33]. One can show that the four-dimensional action of generalized Galileons derived by Deffayet et al. [34] is equivalent to Horndeski’s action under field redefinition [35]. Because Horndeski’s theory contains all modified gravity models and single-field inflation models with one scalar degree of freedom as specific cases, considerable attention has been paid to various aspects of Horndeski’s theory and its importance in cosmology.

Recently, more general modified gravity theories have been studied, including Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories [36, 37] and eXtended Galileon with 3-space covariance (XG3) [38].

In this chapter, the probability of distinguishing between the Λ cold dark matter (ΛCDM) model and modified gravity is studied by using future observations for the growth rate of cosmic structure (e.g., Euclid redshift survey [39]). We computed the growth rate of matter density perturbations in modified gravity and compared it with mock observational data. Whereas the background expansion history in modified gravity is almost identical to that of dark energy models, the evolution of matter density perturbations of modified gravity is different from that of dark energy models. Thus, it is important to study the growth history of perturbations to distinguish modified gravity from models based on the cosmological constant or dark energy.

Although past observations of the growth rate of matter density perturbations have been used to study modified gravity [40], we focus on future observations of the growth rate by Euclid. We adopt the extended DGP model [41], kinetic gravity braiding model [30], and Galileon model [16, 17] as modified gravity models. The kinetic gravity braiding model and the Galileon model are specific aspects of Horndeski’s theory.

This chapter is organized as follows. In the next section, we present the background evolution and the effective gravitational constant in modified gravity models. In Section 3, we describe the theoretical computations and the mock observational data of the growth rate of matter density perturbations. In Section 4, we study the probability of distinguishing between the ΛCDM model and modified gravity by comparing the predicted cosmic growth rate by models to the mock observational data. Finally, conclusions are given in Section 5.

## 2. Modified gravity models

### 2.1. Extended DGP model

In the DGP model [11], it is assumed that we live on a 4D brane embedded in a 5D Minkowski bulk. Matter is trapped on the 4D brane, and only gravity experiences the 5D Minkowski bulk.

The action is

where quantities of the 4D brane and the 5D Minkowski bulk are represented with subscripts (4) and (5), respectively. *M* is the Planck mass, and *L*_{m} is the Lagrangian of matter confined on the 4D brane. The transition between 4D and 5D gravity occurs at the crossover scale *r*_{c}.

At scales larger than *r*_{c}, gravity appears in 5D. At scales smaller than *r*_{c}, gravity is effectively bound to the brane, and 4D Newtonian dynamics is recovered to a good approximation. *r*_{c} is a parameter in this model, which has the unit of length [42].

Under spatial homogeneity and isotropy, a Friedmann-like equation is obtained on the brane [43, 44]:

where *ρ* represents the total fluid energy density on the 4D brane. The DGP model has two branches (*ε* = ±1). The choice of *ε* = +1 is called the self-accelerating branch. In this branch, the accelerated expansion of the universe is induced without dark energy, since the Hubble parameter comes close to a constant *H* = 1/*r*_{c} as time passes. By contrast, *ε* = −1 is the normal branch. In this case, the expansion cannot accelerate without a dark energy component. Therefore, in the following, we adopt the self-accelerating branch (*ε* = +1).

The original DGP model, however, is plagued by the ghost problem [45] and is incompatible with cosmological observations [46].

Dvali and Turner [41] phenomenologically extended the Friedmann-like equation of the DGP model (Eq. (3)). This model interpolates between the original DGP model and the ΛCDM model by adding the parameter *α*. The modified Friedmann-like equation is

For *α* = 1, this is equivalent to the original DGP Friedmann-like equation, whereas *α* = 0 leads to an expansion history identical to ΛCDM cosmology. This is important for distinguishing the ΛCDM model from the original DGP model between *α* = 0 and 1. In the extended DGP model, the crossover scale *r*_{c} can be expressed as follows:

Thus, the independent parameters of the cosmological model are *α* and today’s energy density parameter of matter

where

*G*_{eff} / *G* is the effective gravitational constant normalized to Newton’s gravitational constant, and an overdot represents differentiation with respect to cosmic time *t*.

### 2.2. Kinetic gravity braiding model

The kinetic gravity braiding model [30] is proposed as an alternative to the dark energy model. One can say that the kinetic gravity braiding model is a specific aspect of Horndeski’s theory [32].

The most general four-dimensional scalar-tensor theories keeping the field equations of motion at second order are described by the Lagrangian [32–35, 47]

where

Here, *K* and *G*_{i} (*i* = 3, 4, 5) are functions of the scalar field *φ* and its kinetic energy *R* is the Ricci scalar, and

where *g* represents a determinant of the metric

Variation with respect to the metric produces the gravity equations, and variation with respect to the scalar field *φ* yields the equation of motion. By using the notation

and the equation of motion for the scalar field gives

(16) |

Here, an overdot denotes differentiation with respect to cosmic time *t*, and *ρ*_{m} and *ρ*_{r} are the energy densities of matter and radiation, respectively, and *p*_{r} is the pressure of the radiation.

In the kinetic gravity braiding model [30], the functions in Horndeski’s theory are given as follows:

*M*_{pl} is the reduced Planck mass related to Newton’s gravitational constant by *r*_{c} is called the crossover scale in the DGP model [42]. The kinetic braiding model we study is characterized by parameter *n* in Eq. (18). For *n* = 1, this corresponds to Deffayet’s Galileon cosmological model [22]. For *n* equal to infinity, the background expansion of the universe of the kinetic braiding model approaches that of the ΛCDM model. This helps distinguish the kinetic braiding model from the ΛCDM model.

In the case of the kinetic braiding model using the Hubble parameter as the present epoch *H*_{0}, the crossover scale *r*_{c} is given by

where Ω_{r,0} is the density parameter of the radiation at the present time. Thus, the independent parameters of the cosmological model are *n* and Ω_{m,0}. The effective gravitational constant normalized to Newton’s gravitational constant *G*_{eff} / *G* of the kinetic braiding model is given by

where Ω_{m} is the matter energy density parameter defined as *n* approaches the ΛCDM model, the growth history of matter density perturbations is different due to the time-dependent effective gravitational constant.

### 2.3. Galileon model

The Galileon gravity model is proposed as an alternative to the dark energy model. It is thought that the Galileon model studied in Refs. [16, 17] is a specific aspect of Horndeski’s theory [32].

In the Galileon model [16, 17], the functions in the Lagrangian (Eqs. (8)–(12)) of the Horndeski’s theory are given as follows:

where *ω* is the Brans–Dicke parameter and *φ*.

In this case, the Friedmann-like equations, Eqs. (14) and (15), can be written in the following forms, respectively:

where the effective dark energy density *ρ*_{φ} is defined as

and the effective pressure of dark energy *p*_{φ} is

The equation of motion for the scalar field is given by Eq. (16).

For the numerical analysis, we adopt a specific model in which

where *r*_{c} is the crossover scale [15]. This Galileon model extends the Brans–Dicke theory by adding the self-interaction term, *ω* of this model is not exactly the same as the original Brans–Dicke parameter. The evolution of matter density perturbations of this model was computed in Refs. [16, 17].

At early times, we set the initial condition *r*_{c} must be fine tuned.

The energy density parameter of matter at present in this model is defined as *r*_{c} is fine tuned so that Ω_{m,0} becomes an assumed value. Thus, the independent parameters of the cosmological model are *ω* and Ω_{m,0}. For the Galileon model specified by Eqs. (23)–(26) and (31), the effective gravitational constant is given by

where

The effective gravitational constant *G*_{eff} is close to Newton’s constant *G* at early times, but increases at later times.

## 3. Cosmic growth rate

### 3.1. Density perturbations

Under the quasistatic approximation on sub-horizon scales, the evolution equation for cold dark matter over-density *δ* in linear theory is given by

where *G*_{eff} represents the effective gravitational constant of the modified gravity models described in the previous section.

We set the same initial conditions as in the conventional ΛCDM case (*δ* / *a* for modified gravity models. The linear growth rate is written as

where *δ* is the matter density fluctuations and *a* is the scale factor. The growth rate can be parameterized by the growth index *γ*, defined by

Refs. [48, 49] showed that the growth rate *f* in the Galileon model specified by Eqs. (23)–(26) and (31) is enhanced compared with the ΛCDM model for the same value of Ω_{m,0} due to enhancement of the effective gravitational constant.

### 3.2. Euclid

Euclid [39] is a European Space Agency mission that is prepared for a launch at the end of 2020. The aim of Euclid is to study the origin of the accelerated expansion of the universe. Euclid will investigate the distance-redshift relationship and the evolution of cosmic structures by measuring shapes and redshifts of galaxies and the distribution of clusters of galaxies over a large part of the sky. Its main subject of research is the nature of dark energy. However, Euclid will cover topics including cosmology, galaxy evolution, and planetary research.

In this study, Euclid parameters are adopted as the growth rate observations. The growth rate can be parameterized by using the growth index *γ*, defined by *σ* marginalized errors of the growth rate by Euclid. These data are listed in Table 4 in the paper by Amendola et al. [39]. **Table 1** shows the 1*σ* marginalized errors for the cosmic growth rates with respect to each redshift in accordance with Table 4 in [39]. In **Figure 1**, the mock data of the cosmic growth rate used in this study are plotted.

Experiment | z | |
---|---|---|

Euclid [39] | 0.7 | 0.011 |

0.8 | 0.010 | |

0.9 | 0.009 | |

1.0 | 0.009 | |

1.1 | 0.009 | |

1.2 | 0.009 | |

1.3 | 0.010 | |

1.4 | 0.010 | |

1.5 | 0.011 | |

1.6 | 0.012 | |

1.7 | 0.014 | |

1.8 | 0.014 | |

1.9 | 0.017 | |

2.0 | 0.023 |

Here, *z* represents the redshift and *σ* marginalized errors of the growth rates.

The mock data are used to compute the statistical *χ*^{2} function. The *χ*^{2} function for the growth rate is defined as

where

The estimated errors from the observational technology of Euclid are known, but the center value of future observations is not known. Therefore, the purpose of this study is not to validate the ΛCDM model or modified gravity but to find ways and probabilities to distinguish between the ΛCDM model and modified gravity.

## 4. Comparison with observations

### 4.1. Extended DGP model

In **Figure 2**, we plot the probability contours in the (*α, σ*_{8})-plane in the extended DGP model from the observational (mock) data of the cosmic growth rate by Euclid. The blue (dark) and light blue (light) contours show the 1*σ* (68.3%) and 2*σ* (95.0%) confidence limits, respectively. Part of *α* = 0 for the horizontal axis corresponds to the ΛCDM model, and part of *α* = 1 corresponds to the original DGP model. *σ*_{8} is the root mean square (rms) amplitude of over-density at the comoving 8 *h*^{−1} Mpc scale (where *h* is the normalized Hubble parameter **Figures 3**–**5**, we demonstrate why *σ*_{8} is stringently constrained.

We plot *fσ*_{8} (the product of growth rate and *σ*_{8}) in the extended DGP model as a function of redshift *z* for various values of the energy density parameter of matter at the present Ω_{m,0} in **Figures 3**–**5**. In **Figure 3**, the parameters are fixed by *α* = 1, *σ*_{8} = 0.6. For the various values of Ω_{m,0}, the theoretical curves seem to revolve around the dashed circle. Hence, the value of *σ*_{8} = 0.6 is incompatible with the observational (mock) data.

In **Figure 4**, the parameters are fixed by *α* = 1, *σ*_{8} = 1.0. For the various values of Ω_{m,0}, the theoretical curves seem to revolve around the dashed circle. Hence, the value of *σ*_{8} = 1.0 is incompatible with the observational (mock) data.

In **Figure 5**, the parameters are fixed by *α* = 1, *σ*_{8} = 0.855. For the various values of Ω_{m,0}, although the theoretical curves seem to revolve around the dashed circle, some theoretical curves are comparatively close to the observational (mock) data. Hence, the value of *σ*_{8} = 0.855 is compatible with the observational (mock) data in the original DGP model (*α* = 1).

In **Figure 6**, we plot the probability contours in the (*α*, Ω_{m,0})-plane in the extended DGP model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1*σ* (68.3%) and 2*σ* (95.0%) confidence limits, respectively. We demonstrate why Ω_{m,0} is positively correlated with *α* in **Figure 7**.

We plot *fσ*_{8} in the extended DGP model as a function of redshift *z* in **Figure 7**. The red (solid) line is the theoretical curve for the best-fit parameter in the ΛCDM model (*α* = 0, Ω_{m,0} = 0.257, *σ*_{8} = 0.803). In the case of changing only *α* =1, the growth rate *fσ*_{8} is suppressed due to suppression of the effective gravitational constant (green (dashed) line: *α* = 1, Ω_{m,0} = 0.257, *σ*_{8} = 0.803). For *α* = 1, by tuning the value of Ω_{m,0} and *σ*_{8}, the theoretical curve is compatible with the observational (mock) data again (blue (dotted) line: *α* = 1, Ω_{m,0} = 0.395, *σ*_{8} = 0.855).

In **Figure 8**, we add constraints on Ω_{m,0} for the extended DGP model from the combination of CMB, BAO, and SNIa data (black solid lines) [46] to the probability contours in the (*α*, Ω_{m,0})-plane by the growth rate (mock) data by Euclid of **Figure 6**.

Because Ω_{m,0} is stringently constrained by the cosmic growth rate data from Euclid, we find the ΛCDM model is distinguishable from the original DGP model by combining the growth rate data of Euclid with other observations.

### 4.2. Kinetic gravity braiding model

In **Figure 9**, we plot the probability contours in the (*n*, Ω_{m,0})-plane in the kinetic gravity braiding model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1*σ* (68.3%) and 2*σ* (95.0%) confidence limits, respectively.

We plot *fσ*_{8} in the kinetic gravity braiding model as a function of redshift *z* in **Figure 10**. The red (solid) line is the theoretical curve for the best-fit parameter in the ΛCDM model (Ω_{m,0} = 0.257, *σ*_{8} = 0.803). In the case of the kinetic gravity braiding model for *n* = 1, the growth rate *fσ*_{8} is enhanced due to enhancement of the effective gravitational constant (Green (dashed) line: *n* = 1, Ω_{m,0} = 0.257, *σ*_{8} = 0.803). For *n* = 100, by tuning the value of Ω_{m,0} and *σ*_{8}, the theoretical curve is compatible with the observational (mock) data again (blue (dotted) line: *n* = 100, Ω_{m,0} = 0.196, *σ*_{8} = 0.820).

In **Figure 11**, we add constraints on Ω_{m,0} for the kinetic gravity braiding model from CMB (black dashed lines) and from SNIa (black solid lines) [30], respectively, to the probability contours in the (*n*, Ω_{m,0})-plane by the growth rate (mock) data by Euclid of **Figure 9**.

In the kinetic gravity braiding model, the allowed parameter region obtained by using only the growth rate data does not overlap with the allowed parameter region obtained from CMB or from SNIa data.

### 4.3. Galileon model

In **Figure 12**, we plot the probability contours in the (*ω*, Ω_{m,0})-plane in the Galileon model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1*σ* (68.3%) and 2*σ* (95.0%) confidence limits, respectively. We also plot the constraints on Ω_{m,0} for the Galileon model from the combination of CMB, BAO, and SNIa (black solid lines) [49].

In the Galileon model, the allowed parameter region obtained by using only the growth rate data do not overlap at all with the allowed parameter region obtained from the combination of CMB, BAO, and SNIa data.

## 5. Conclusions

The growth rate *fσ*_{8} in the original DGP model is suppressed in comparison with that in the ΛCDM case in the setting of the same value of Ω_{m,0} due to suppression of the effective gravitational constant. In the case of the kinetic gravity braiding model and the Galileon model, the growth rate *fσ*_{8} is enhanced in comparison with the ΛCDM case in the same value of Ω_{m,0} due to enhancement of the effective gravitational constant. For the cosmic growth rate data from the future observation, compatible values of Ω_{m,0} differ according to the model. Furthermore, values of Ω_{m,0} can be stringently constrained. Thus, we find the ΛCDM model is distinguishable from modified gravity by combining the growth rate data of Euclid with other observations.

The estimated errors from the observational technology of Euclid are known, but the center value of future observations is not known. If the center value of the cosmic growth rate of future observations is different from that of this chapter, the valid model can differ from that of this chapter. However, the methods in this chapter are useful for distinguishing between the ΛCDM model and modified gravity.

In this chapter, assuming the function