Modified Gravity Theories: Distinguishing from ΛCDM Model

The method and probability of distinguishing between the Λ cold dark matter ( Λ CDM) model and modified gravity are studied from future observations for the growth rate of cosmic structure (Euclid redshift survey). We compare the mock observational data to the theoretical cosmic growth rate by modified gravity models, including the extended Dvali – Gabadadze – Porrati (DGP) model, kinetic gravity braiding model, and Galileon model. In the original DGP model, the growth rate f σ 8 is suppressed in comparison with that in the Λ CDM model in the setting of the same value of the today ’ s energy density of matter 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 f σ 8 is enhanced in comparison with the Λ CDM model 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 (Euclid), the compatible value of Ω m ; 0 differs according to the model. Fur- thermore, Ω m ; 0 can be stringently constrained. Thus, we find the Λ CDM model is distinguishable from modified gravity by combining the growth rate data of Euclid with other observations.


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 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 singlefield 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.
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.

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 = AE1). The choice of ε = þ1 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 = À1 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 = þ1).
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 Ω m;0 . The effective gravitational constant of the extended DGP model is given in order to interpolate between ΛCDM and the original DGP model. The effective gravitational constant is as follows. 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.

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][33][34][35]47] where Here, K and G i (i = 3, 4, 5) are functions of the scalar field φ and its kinetic energy X ¼ À∂ μ φ∂ μ φ=2 with the partial derivatives G i;X ∂G i =∂X. R is the Ricci scalar, and G μν is the Einstein tensor. The above Lagrangian was first derived by Horndeski in a different form Ref. [32]. This Lagrangian (Eqs. (8)- (12)) is equivalent to that derived by Horndeski [35]. The total action is then given by where g represents a determinant of the metric g μν , and L m is the Lagrangian of non-relativistic matter.
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 K Kðφ;XÞ, G G 3 ðφ;XÞ, F 2 M 2 pl G 4 ðφ;XÞ, and assuming G 5 ðφ;XÞ ¼ 0 for Friedmann-Robertson-Walker spacetime, the gravity equations give and the equation of motion for the scalar field gives Here, an overdot denotes differentiation with respect to cosmic time t, and H ¼ _ a=a is the Hubble expansion rate. Note that we use the partial derivative notation K ;X ∂K=∂X and K ;XX ∂ 2 K=∂X 2 and a similar notation for other variables. Also, 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: Kðφ;XÞ ¼ ÀX; ð17Þ M pl is the reduced Planck mass related to Newton's gravitational constant by M pl ¼ 1= ffiffiffiffiffiffiffiffiffi 8πG p , and 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 Here, we used the attractor condition. Although the background evolution for large n approaches the ΛCDM model, the growth history of matter density perturbations is different due to the timedependent effective gravitational constant.

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 ξðφÞ is a function of 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, ξðφÞð∇φÞ 2 □ φ. Thus, 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 φ≃M 2 pl =2 to recover general relativity. Using these initial conditions reduces the Friedmann equations (Eqs. (27) and (28)) to their usual forms: pl . This is the cosmological version of the Vainshtein effect [23], which is a method to recover general relativity below a certain scale. At present, to induce cosmic acceleration, the value of r c must be fine tuned.
The energy density parameter of matter at present in this model is defined as Ω m;0 ¼ ρ m;0 =3H 2 0 φ 0 . Therefore, in the numerical analysis, the value of 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.

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 and _ δ ≈ _ a) because we trace the difference between the evolution of the matter perturbations in modified gravity and the evolution in the ΛCDM model. From the evolution equation, we numerically obtain the growth factor 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.

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 f ¼ Ω m γ . Mock data of the cosmic growth rate are built based on the 1σ 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 1σ 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.
The mock data are used to compute the statistical χ 2 function. The χ 2 function for the growth rate is defined as where f obs ðz i Þ is the future observational (mock) data of the growth rate. The theoretical growth rate f theory ðz i Þ is computed as Eq. (35). In Ref. [50], constraints on neutrino masses are estimated based on future observations of the growth rate of the cosmic structure from the Euclid redshift survey.
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. Here, z represents the redshift and σ f g represents the 1σ marginalized errors of the growth rates.

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 1σ (68.3%) and 2σ (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 À1 Mpc scale (where h is the normalized Hubble parameter H 0 ¼ 100 hkmsec À1 Mpc À1 ). In 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,   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 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively. We demonstrate why Ω m,0 is positively correlated with α in Figure 7.
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.

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 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively.   [46] to the data in 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.

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 1σ (68.3%) and 2σ (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.

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 G 5 ðφ; XÞ in Horndeski's theory G 5 ðφ; XÞ ¼ 0, we compute the linear matter density perturbations for the growth rate. In future work, we will study the model having non-zero function G 5 ðφ; XÞ in Horndeski's theory and investigate the nonlinear effect.