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.
- accelerated expansion
- gravitational theory
- dark energy
- observational test
- cosmic growth rate
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
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 , 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 , which induces decoupling of the Galileon field from gravity at small scales via the Vainshtein mechanism . 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. , 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 . 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.  is equivalent to Horndeski’s action under field redefinition . 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.
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 ). 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 , we focus on future observations of the growth rate by Euclid. We adopt the extended DGP model , kinetic gravity braiding model , 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 , 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.
At scales larger than
Dvali and Turner  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
Thus, the independent parameters of the cosmological model are
2.2. Kinetic gravity braiding model
Variation with respect to the metric produces the gravity equations, and variation with respect to the scalar field
and the equation of motion for the scalar field gives
Here, an overdot denotes differentiation with respect to cosmic time
In the kinetic gravity braiding model , the functions in Horndeski’s theory are given as follows:
In the case of the kinetic braiding model using the Hubble parameter as the present epoch
2.3. Galileon model
where the effective dark energy density
and the effective pressure of dark energy
The equation of motion for the scalar field is given by Eq. (16).
For the numerical analysis, we adopt a specific model in which
At early times, we set the initial condition to recover general relativity. Using these initial conditions reduces the Friedmann equations (Eqs. (27) and (28)) to their usual forms: and . This is the cosmological version of the Vainshtein effect , which is a method to recover general relativity below a certain scale. At present, to induce cosmic acceleration, the value of
The energy density parameter of matter at present in this model is defined as . Therefore, in the numerical analysis, the value of
The effective gravitational constant
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
We set the same initial conditions as in the conventional ΛCDM case ( and ) 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
Refs. [48, 49] showed that the growth rate
Euclid  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
The mock data are used to compute the statistical
where is the future observational (mock) data of the growth rate. The theoretical growth rate is computed as Eq. (35). In Ref. , 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 (
In Figure 4, the parameters are fixed by
In Figure 5, the parameters are fixed by
In Figure 6, we plot the probability contours in the (
In Figure 8, we add constraints on Ω
4.2. Kinetic gravity braiding model
In Figure 9, we plot the probability contours in the (
In Figure 11, we add constraints on Ω
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 (
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.
The growth rate
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 in Horndeski’s theory , we compute the linear matter density perturbations for the growth rate. In future work, we will study the model having non-zero function in Horndeski’s theory and investigate the nonlinear effect.
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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