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The method and probability of distinguishing between the Λ cold dark matter (ΛCDM) 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 fσ8 is suppressed in comparison with that in the ΛCDM model in the setting of the same value of the today’s energy density of matter Ωm,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 fσ8 is enhanced in comparison with the ΛCDM model in the same value of Ωm,0, due to enhancement of the effective gravitational constant. For the cosmic growth rate data from the future observation (Euclid), the compatible value of Ωm,0 differs according to the model. Furthermore, Ωm,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.

Keywords

accelerated expansion

gravitational theory

dark energy

observational test

cosmic growth rate

chapter and author info

Author

Koichi Hirano*

Department of Primary Education, Tsuru University, Yamanashi, Japan

*Address all correspondence to: k_hirano@tsuru.ac.jp

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 ∂μφ→∂μφ+bμ, keeps the field equation of motion as a second-order differential equation. This avoids presenting a new degree of freedom, and perturbation of the theory is free from ghost or instability problems. The simplest term of the self-interaction is □φ(∇φ)2, which induces decoupling of the Galileon field from gravity at small scales via the Vainshtein mechanism [23]. Therefore, the Galileon theory recovers general relativity at scales around the high-density region, as is not inconsistent with solar system experiments.

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.

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}.

rc=M(4)22M(5)3.E2

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]:

H2=8πG3ρ+εHrc,E3

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

H2=8πG3ρ+Hαrc2−α.E4

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:

rc=(1−Ωm,0)1α−2H0−1.E5

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

GeffG=1+13β,E6

where

β≡1−2(rcH)2−αα[1+13(2−α)H˙H2].E7

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]

Here, K and G_{i} (i = 3, 4, 5) are functions of the scalar field φ and its kinetic energy X=−∂μφ∂μφ/2with the partial derivatives Gi,X≡∂Gi/∂X. R is the Ricci scalar, and Gμνis 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

S=∫d4x−g(L+Lm),E13

where g represents a determinant of the metric gμν, and Lmis 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≡K(φ,X), G≡G3(φ,X), F≡2Mpl2G4(φ,X), and assuming G5(φ,X)=0for Friedmann–Robertson–Walker spacetime, the gravity equations give

Here, an overdot denotes differentiation with respect to cosmic time t, and H=a˙/ais the Hubble expansion rate. Note that we use the partial derivative notation K,X≡∂K/∂Xand K,XX≡∂2K/∂X2and 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:

K(φ,X)=−X,E17

G3(φ,X)=Mpl(rc2Mpl2X)n,E18

G4(φ,X)=Mpl22,E19

G5(φ,X)=0.E20

M_{pl} is the reduced Planck mass related to Newton’s gravitational constant by Mpl=1/8πG, 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

rc=(2n−13n)1/2n[16(1−Ωm,0−Ωr,0)](2n−1)/4nH0−1,E21

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

GeffG=2n+3nΩm−ΩmΩm(5n−Ωm),E22

where Ω_{m} is the matter energy density parameter defined as Ωm=ρm/3Mpl2H2. 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 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:

K(φ,X)=2ωφX,E23

G3(φ,X)=2ξ(φ)X,E24

G4(φ,X)=φ,E25

G5(φ,X)=0,E26

where ω is the Brans–Dicke parameter and ξ(φ)is a function of φ.

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

3H2=1Mpl2(ρm+ρr+ρφ),E27

−3H2−2H˙=1Mpl2(pr+pφ),E28

where the effective dark energy density ρ_{φ} is defined as

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

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

ξ(φ)=rc2φ2,E31

where r_{c} is the crossover scale [15]. This Galileon model extends the Brans–Dicke theory by adding the self-interaction term, ξ(φ)(∇φ)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≃Mpl2/2to recover general relativity. Using these initial conditions reduces the Friedmann equations (Eqs. (27) and (28)) to their usual forms: 3H2=(ρm+ρr)/Mpl2and −3H2−2H˙=pr/Mpl2. 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 Ωm,0=ρm,0/3H02φ0. 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

Geff=116πφ[1+(1+ξ(φ)φ˙2)2J],E32

where

J≡3+2ω+φ2ξ(φ)[4φ¨φ−2φ˙2φ2+8Hφ˙φ−φ2ξ(φ)φ˙4φ4].E33

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

δ¨+2Hδ˙−4πGeffρδ≃0,E34

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≈aand 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

f=dlnδdlna.E35

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

f=Ωmγ.E36

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 f=Ωmγ. 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.

1σ marginalized errors for the growth rates in each redshift bin based on Table 4 in the study by Amendola et al. [39].

Here, z represents the redshift and σfgrepresents the 1σ 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

χf2=∑i=114(ftheory(zi)−fobs(zi))2σfg(zi)2E37

where fobs(zi)is the future observational (mock) data of the growth rate. The theoretical growth rate ftheory(zi)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.

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 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^{−1} Mpc scale (where h is the normalized Hubble parameter H0=100hkmsec−1Mpc−1). 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, 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 1σ (68.3%) and 2σ (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 1σ (68.3%) and 2σ (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 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.

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 G5(φ,X)in Horndeski’s theory G5(φ,X)=0, we compute the linear matter density perturbations for the growth rate. In future work, we will study the model having non-zero function G5(φ,X)in Horndeski’s theory and investigate the nonlinear effect.

Acknowledgments

This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Grant Number 25400264).

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