Geometrical Diagnostics for Modified Gravitational Theory with the Different Formalisms

Jianbo Lu^*, Mou Xu, Jie Wang, Yan Liu, Zhitong Zhuang

Department of Physics, Liaoning Normal University, Dalian, China.

DOI: 10.4236/jhepgc.2022.84059                

Abstract

Geometrical diagnostic methods were often applied to distinguish the gravitational models. But it is scarce to investigate the differences between the different formalisms of modified gravitational theories (e.g. the metric formalism and the Palatini formalism). In this paper, we discriminate the gravitational theory with the different formalisms by using the geometrical diagnostic methods. For a considered modified theory of gravity (e.g. the f(R) theory or GBD theory), we can see that the difference between the two formalisms is remarkable according to the diagnostic results. And relative to the ΛCDM model, there are more deviations in metric formalism than those in Palatini formalism, according to the {r, s} diagnostic. Given that the GBD (generalized Brans-Dicke theory) is a time-variable Newton gravitational constant (VG) theory, the differences between the VG theory and the constant-G theory are studied. It indicates that the variation of Newton’s gravitational constant could induce notable effects on geometrical quantities (e.g. r, s and q) in both metric formalism and Palatini formalism.

Keywords

Time-Variable Gravitational Constant, Metric Formalism, Palatini Formalism, Geometrical Diagnostic

1. Introduction

For exploring properties of gravity or solving questions in the theory of general relativity (GR) [1] - [6], lots of modified theories of GR have been constructed, e.g. the f(R) theory [7] [8], the f(G) theory [9] [10], the Brans-Dicke (BD) theory [11], and so on [12] - [28]. There is a prior assumption that we have to take in studying modified theories of gravity (MG), i.e. which one or ones of the dynamical variables should be chosen to describe the gravitational interaction? Correspondingly, the modified gravitational theories could be divided into two classifications: the metric formalism [8] [29] and the Palatini formalism [30] [31]. In the metric formalism, the Levi-Civita connection is related to metric, while in the Palatini formalism, the metric and the connection are regarded as independent dynamical variables. Usually, the different field equations are gained in the metric formalism theories and the Palatini formalism theories [8], respectively.

To find the “final” theory of describing the gravitational interaction, it is significant to explore the differences between the different modified theories (or the different formalisms). Then one can test them according to the observational results. For example, gravitational wave astronomy, which was recently started by the famous LIGO detections [32] [33], could be, in principle, fundamental for testing the effective viability of extended theories of gravity. The key point is that some differences between different gravity theories can be found in linearized gravity by analyzing gravitational wave polarizations via the interferometric response functions [34].

In addition, lots of gravitational models have been differentiated via the geometrical diagnostic methods [35] [36] [37] [38] [39]. But studying the distinctions between the different formalisms is scarce by using the so-called geometrical diagnostic. Obviously, which formalism should be chosen preferentially is an important issue, since it can decrease the uncertainty of the theoretical research of gravity. In this paper, we probe the discrepancy between the different formalisms of modified theory, by selecting the f(R) and the generalized Brans-Dicke (GBD) theories as examples. Also, we try to give an answer, i.e. under the observational limits on the Newton gravitational constant G, whether the change of G could lead to the remarkable geometrical effect in the generalized Brans-Dicke theory (a theory with the time-variable Newton gravitational constant). Due to the dynamics of BD field and the coupling between the Brans-Dicke field and the gravitational geometry, we find that the difference of some geometrical quantity (e.g. r, s or q) between the variable-G theory and the constant-G theory could be obvious for both the metric formalism and the Palatini formalism.

The constructions of this paper are as follows. In Section II, we introduce the basic equations for f(R) modified gravitational theory with the different formalisms, and apply the geometrical diagnostics to distinguish the different formalisms. Section III investigates the geometrical diagnostics for the different formalisms of GBD theory. Especially, we study the influences of variable G on the geometrical quantities in this theory. Section IV is the conclusion.

2. The Geometrical Diagnostics for f(R) Theory with the Different Formalisms

1) Basic equations for f(R) modified theory in both the metric formalism and the Palatini formalism

In this section, we show the field equations of f(R) theory in the metric formalism and the Palatini formalism, respectively. f(R) modified gravitational theory is a simple and popular extension relative to GR. In the metric formalism, the action of f(R) theory is denoted by

$S_{met}=S_g\left(g_{\mu v}\right)+S_m\left(g_{\mu v},\Psi \right)={\displaystyle \int {\text{d}}^4}x\sqrt{-g}\left[\frac{1}{16\pi G}f\left(R\right)+L_M\right],$ (1)

where g denotes the determinant of metric $g_{\mu v}$, R is the Ricci scalar, f(R) is an arbitrary function of R, G denotes the Newton gravitational constant, L_m denotes the Lagrangian density of matter, respectively. Using the variation principle, one gets the modified field equation of gravity

$f_R{R}_{\mu v}-\frac{1}{2}f\left(R\right)g_{\mu v}-\left[{\nabla }_{\mu }{\nabla }_v-g_{\mu v}\square \right]f_R=8\pi G{T}_{\mu v},$ (2)

Here $f\left(R\right)=\frac{\partial f\left(R\right)}{\partial R}$, and $\square \equiv {\nabla }^{\mu }{\nabla }_{\mu }$. $R_{\mu \nu }$ and $T_{\mu \nu }$ denote the Ricci tensor and the energy-momentum tensor of matter, respectively. The trace of Equation (2) is

$f_R\left(R\right)R-2f\left(R\right)+3\square f_R=8\pi GT.$ (3)

In the Palatini formalism, the action of f(R) theory read as,

$S_{Pala}=S_g\left(g_{\mu \nu },{\tilde{\Gamma }}_{\mu \nu }^{\lambda }\right)+S_m\left(g_{\mu \nu },\psi \right)={\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left[\frac{1}{16\pi G}f\left(\tilde{R}\right)+L_m\right].$ (4)

Here the metric $g_{\mu \nu }$ and the connection ${\tilde{\Gamma }}_{\mu \nu }^{\lambda }$ are regarded as the independent dynamical variables. $\tilde{R}=g^{\mu \nu }{\tilde{R}}_{\mu \nu }$ and the Ricci tensor ${\tilde{R}}_{\mu \nu }$ is defined by the independent Palatini connection

${\tilde{R}}_{\mu \nu }={\partial }_{\lambda }{\Gamma }_{\mu \nu }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \lambda }^{\lambda }+{\Gamma }_{\lambda \sigma }^{\lambda }{\Gamma }_{\mu \nu }^{\sigma }-{\Gamma }_{\mu \sigma }^{\lambda }{\Gamma }_{\lambda \nu }^{\sigma }$ (5)

Varying the action (4) with respect to $g_{\mu \nu }$, we gain the gravitational field equation in the Palatini formalism

$F\left(\tilde{R}\right){\tilde{R}}_{\mu v}-\frac{1}{2}f\left(\tilde{R}\right)g_{\mu \nu }=\kappa T_{\mu \nu }{\bar{\nabla }}_{\lambda },$ (6)

where $F\left(\tilde{R}\right)=\frac{\partial f\left(\tilde{R}\right)}{\partial \tilde{R}}$. The trace of Equation (6) is

$F\left(\tilde{R}\right)\tilde{R}-2f\left(\tilde{R}\right)=8\pi GT.$ (7)

Varying the action with respect to ${\tilde{\Gamma }}_{\mu \nu }^{\lambda }$ gives

${\tilde{\nabla }}_{\lambda }\left(\sqrt{-g}F\left(\tilde{R}\right)g^{\mu v}\right)=0,$ (8)

where $\tilde{\nabla }$ is the covariant derivative with respect to the Palatini connection. Equation (8) implies that the connection can be represented as the Christoffel symbol associated with the metric $h_{\mu \nu }$ by defining $h_{\mu \nu }=F\left(R\right)g_{\mu \nu }$. Then we arrive at a relation:

${\tilde{\Gamma }}_{\mu \nu }^{\lambda }={\Gamma }_{\mu \nu }^{\lambda }+\frac{1}{2F}\left[-g_{\mu \nu }{\partial }^{\lambda }F+{\delta }_{\nu }^{\lambda }{\partial }_{\mu }F+{\delta }_{\mu }^{\lambda }{\partial }_{\nu }F\right],$ (9)

where ${\Gamma }_{\mu \nu }^{\lambda }$ is the Livi-Civita connection associated with the metric $g_{\mu \nu }$. Thus, by using Equation (5) the Ricci tensor and the Ricci scalar in the Palatini formalism are rewritten as

${\tilde{R}}_{\mu \nu }=R_{\mu \nu }\left(g\right)+\frac{3}{2F^2}{\nabla }_{\mu }F{\nabla }_{\nu }F-\frac{1}{F}{\nabla }_{\mu }{\nabla }_{\nu }F-\frac{1}{2F}{g}_{\mu \nu }{\nabla }_{\sigma }{\nabla }^{\sigma }F,$ (10)

where $R_{\mu \nu }$ denotes the Ricci tensor defining in the metric formalism, and all covariant derivatives are taken with respect to the metric $g_{\mu \nu }$. Combining above equations, the modified Einstein equation in the Palatini-formalism $f\left(\tilde{R}\right)$ theory can be reexpressed as

$G_{\mu v}=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=\frac{\kappa T_{\mu \nu }}{F}+8\pi T_{\mu \nu }^{eff}$ (11)

with $\kappa =8\pi G$, $8\pi T_{\mu \nu }^{eff}=-\frac{1}{2}{g}_{\mu v}\left(\tilde{R}-\frac{f}{F}\right)+\frac{1}{F}\left({\nabla }_{\mu }{\nabla }_v-g_{\mu v}\square \right)F-\frac{3}{2}\frac{1}{F^2}\left[{\nabla }_{\mu }F{\nabla }_{v}F-\frac{1}{2}{g}_{\mu v}{\left(\nabla F\right)}^2\right]$.

2) The geometrical diagnostics for f(R) theory with the different formalisms

In this part, we utilize the f(R) theory with the different formalisms to cosmology. A flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric has a form:

$\text{d}{s}^2=-\text{d}{t}^2+a^2\left(t\right)\text{d}{\stackrel{\to }{x}}^2.$ (12)

Here a is the cosmic scale factor, t is the cosmic time. Inserting the FLRW metric into field equations of f(R) theory with metric formalism and assuming the energy momentum tensor: $T_{\mu \nu }=\left(\rho +p\right)U_{\mu }{U}_{\nu }+p{g}_{\mu \nu }$, we have the cosmological equations [8]

$H^2=\frac{\kappa }{3f_R}\left[\rho +\frac{R{f}_R-f}{2}-3H\stackrel{\dot{}}{R}{f}_{RR}\right],$ (13)

$2\stackrel{\dot{}}{H}+3H^2=-\frac{\kappa }{f_R}\left[p+{\stackrel{\dot{}}{R}}^2{f}_{RRR}+2H\stackrel{\dot{}}{R}{f}_{RR}+\ddot{R}{f}_{RR}+\frac{1}{2}\left(f-R{f}_R\right)\right].$ (14)

Here $H\equiv \frac{\stackrel{\dot{}}{a}}{a}$ is a geometrical quantity, called the Hubble parameter. ρ and p denote the density and pressure of universal matter, respectively. $U_{\mu }$ represents the four-velocity of an observer comoving with the fluid. For solving Equations (13) and (14), we define two dimensionless quantities: $y_H=H^2/m^2-{\left(1+z\right)}^3$ and $y_R=R/m^2-3{\left(1+z\right)}^3$, with the cosmic redshift z = 1/a − 1. Then we receive two differential equations:

${y^{\prime }}_H=-\frac{1}{1+z}\left(\frac{1}{3}{y}_R-4y_H\right),$ (15)

${y^{\prime }}_R=\frac{\left[y_H+{\left(1+z\right)}^3\right]f_R-{\left(1+z\right)}^3-\frac{f_R}{6}\left[y_R+3{\left(1+z\right)}^3\right]+\frac{f}{6m^2}}{\left[y_H+{\left(1+z\right)}^3\right]\left(1+z\right)f_{RR}{m}^2}-9{\left(1+z\right)}^2,$ (16)

where ' denotes the derivative with respect to z. The initial conditions are provided by: ${y_H|}_{z=0}=H_0^2/m^2-1$ and ${y_R|}_{z=0}=6H_0^2\left(1-q_0\right)/m^2-3$, with $m^2={\left(8315\text{Mpc}\right)}^{-2}\left(\frac{{\Omega }_{0m}{h}^2}{0.13}\right)$. According to the observational results, we consider the current dimensionless energy density of matter Ω_0m = 0.27 [40], the current value of deceleration parameter q₀ = −0.63 [41], the current value of dimensionless Hubble constant h = 0.673 ± 0.010 [42], with H₀ = 100 h∙km∙s⁻¹∙Mpc⁻¹.

In the Palatini-f(R) theory, the cosmological equation could be exhibited as [8]:

${\left(H+\frac{{\stackrel{\dot{}}{f}}_R}{2f_R}\right)}^2=\frac{\kappa \left(\rho +3p\right)+f}{6f_R}$ (17)

Combining Equation (17) and the conservation equation, we have

$\stackrel{\dot{}}{R}=-\frac{3H\left[f_R\tilde{R}-2f\right]}{f_R{}_R\tilde{R}-f_R}$ (18)

Also, Equation (17) can be rewritten as

$H^2=\frac{1}{6f_R}\frac{3f-f_R\tilde{R}}{{\left[1-\frac{3f_R{}_R\left(f_R\tilde{R}-2f\right)}{2f_R\left(f_R{}_R\tilde{R}-f_R\right)}\right]}^2}$ (19)

Furthermore, we gain

$\frac{\text{d}\tilde{R}}{\text{d}z}=-\frac{9H_0^2{\Omega }_{m0}{\left(1+z\right)}^2}{f_{RR}\tilde{R}-f_R}$ (20)

$\frac{H^2}{H_0^2}=\frac{1}{6f_R}\frac{3{\Omega }_{m0}{\left(1+z\right)}^3+f/H_0^2}{{\left[1+\frac{9H_0^2{\Omega }_{m0}{\left(1+z\right)}^3{f}_{RR}}{2f_R\left(f_R{}_{R}R-f_R\right)}\right]}^2}$ (21)

Thus, combining with trace equation in Palatini formalism of f(R) theory, we can solve Equation (21).

The geometrical quantities—statefinder parameters {r, s} are introduced in Ref. [35], which are defined as follows:

$r\equiv \frac{\stackrel{⃛}{a}}{a{H}^3}=\frac{H^{″}{\left(z+1\right)}^2}{H}-\frac{2H^{\prime }\left(z+1\right)}{H}+\frac{{H^{\prime }}^2{\left(z+1\right)}^2}{H^2}+1,$ (22)

$s\equiv \frac{r-1}{3\left(q-\frac{1}{2}\right)}=\frac{\frac{H^{″}{\left(z+1\right)}^2}{H}-\frac{2H^{\prime }\left(z+1\right)}{H}+\frac{{H^{\prime }}^2{\left(z+1\right)}^2}{H^2}}{3\frac{H^{\prime }\left(z+1\right)}{H}-\frac{9}{2}}.$ (23)

Obviously, both r and s are the third-order derivative (the highest) of a with respect to t. Here

$q\equiv \frac{-\ddot{a}}{a{H}^2}$ (24)

is another geometrical quantity, called the deceleration parameter, which is the second derivative of a with respect to t.

Lots of models are diagnosed by using the statefinder parameters [36] [37] [38] [39]. But applying the statefinder diagnose to distinguish the modified gravitational theory with the different formalisms are lack. In this paper, we investigate the differences between the Palatini formalism and the metric formalism by using the diagnostic method. From above, we can find that the gravitational field equation is the fourth-order PDE (partial differential equation) in the metric formalism, while the field equation in the Palatini formalism is the second-order PDE which is easier to solve and interpret [43]. Then the different methods are utilized to solve the cosmological equations numerically for the different formalisms. For plotting pictures, we take a viable model $f\left(R\right)=R-\beta R_s\left(1-{\text{e}}^{-R/R_s}\right)$, which is proposed and developed by Refs. [44] [45] [46]. Here β and R_s are two constants, and could be related by $\beta R_s\simeq 18H_0^2{\Omega }_{0m}$. Some viable conditions on this model can be found in Refs. [45] [46]. This model has an important feature that it owns only one more parameter than the ΛCDM model.

The graphs of {r, s} geometrical diagnostic are plotted in Figure 1 with the metric formalism (left) and the Palatini formalism (right), respectively. The values of model parameter β are taken as [0.9, 1.1, 1.3], and marked by β₁, β₂ and β₃, respectively. Considering that for the most popular ΛCDM model, we have {r, s} = {1, 0}. Hence we could find the deviation of f(R) model in both formalisms from the ΛCDM model, which show that the values of {r, s} in metric formalism are larger (or more deviation) than those in Palatini formalism according to Figure 1. Also, Figure 1 shows that for the same function of f(R), the difference between the metric formalism and the Palatini formalism are notable according to the {r, s} diagnostic. In addition, difference between these two formalisms can be reflected by the values of model parameter β. We can notice that the more larger value of β, the {r, s} pictures are more close to ΛCDM in Palatini formalism, while the opposite results are given in metric formalism. The arrow describes the evolution of universe from the early stage to the late stage.

Using the same model-parameter values with those in Figure 1, Figure 2 depicts geometric diagnostic of {r, q}. We can read that the shapes of {r, q} in

Figure 1. The {r, s} geometrical diagnostic for f(R) model with the metric formalism (left) and the Palatini formalism (right), respectively. The selected values of model parameter β are marked on the pictures.

Figure 2. The {r, q} geometrical diagnostic forf(R) model with the metric formalism (left) and the Palatini formalism (right).

metric formalism are different from those in Palatini formalism. For seeing the effect of q on the {r, q} diagnostic, we illustrate the evolutions of deceleration parameter q for the considered f(R) model. The metric formalism is drawn in Figure 3 (left), while the Palatini formalism is plotted in Figure 3 (right). Given that we have q(z) = 1/2 for the matter dominated universe, then there must be q(z) ≤ 1/2 for any cosmological model. For selected model-parameter values in this paper, we can see that the evolutional curves of q(z) in Palatini formalism are consistent with the requirement: q(z) ≤ 1/2.

3. The Geometrical Diagnostics for GBD Theory with the Different Formalisms

1) Basic equations for GBD modified theory in both the metric formalism and the Palatini formalism

Some problems on f(R) theory have been found in some reference [8], such as the inconsistent problem of γ between the theoretical value and the observational value (here γ is the parametrized post-Newtonian parameter). The f(R) theory can be extended by considering some methods, such as the f(G) theory (adding the higher-order terms) [9] [10], the GBD theory [47] [48] [49], etc. In this part, we apply the geometric diagnostic methods to distinguish the different formalisms of GBD modified theory. In addition, given that the time-variable gravitational constant G have been investigated in some theoretical and observational issues [50] - [57], we explore the effects of time-variable G in the modified theory with the different formalisms.

The action of system in the metric-formalism GBD theory is written as

$\begin{array}{c}S=S_g\left(g_{\mu \nu },\varphi \right)+S_{\varphi }\left(g_{\mu \nu },\varphi \right)+S_m\left(g_{\mu \nu },{\psi }_m\right)\\ =\frac{1}{2}{\displaystyle \int L_T}{\text{d}}^{4}x=\frac{1}{2}{\displaystyle \int {\text{d}}^{4}x\sqrt{-g}\left[\varphi f\left(R\right)-\frac{\omega }{2\varphi }{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi +\frac{16\pi }{c^4}{L}_m\right]}.\end{array}$ (25)

Using the variational principle, in the metric-formalism GBD theory we obtain the gravitational field equation and the BD scalar field equation as follows

$\begin{array}{l}\varphi \left[f_R{R}_{\mu \nu }-\frac{1}{2}f\left(R\right)g_{\mu \nu }\right]-\left({\nabla }_{\mu }{\nabla }_{\nu }-g_{\mu \nu }\square \right)\left(\varphi f_R\right)\\ +\frac{1}{2}\frac{\omega }{\varphi }{g}_{\mu \nu }{\partial }_{\sigma }\varphi {\partial }^{\sigma }\varphi -\frac{\omega }{\varphi }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi =8\pi T_{\mu \nu }.\end{array}$ (26)

Figure 3. The evolutions of deceleration parameter q for f(R) model with the metric formalism (left) and the Palatini formalism (right).

$f\left(R\right)+2\omega \frac{\square \varphi }{\varphi }-\frac{\omega }{{\varphi }^2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi =0.$ (27)

where $\varphi$ is the BD scalar field, ω is the coupling constant, ${\nabla }_{\mu }$ is the covariant derivative associated with the Levi-Civita connection of the metric. The trace of Equation (26) is

$f_{R}R-2f\left(R\right)+\frac{3\square \left(\varphi f_R\right)}{\varphi }+\frac{\omega }{{\varphi }^2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi =\frac{8\pi T}{\varphi }.$ (28)

The action of GBD theory in the Palatini formalism read as,

$S_p=S_g\left(g_{\mu \nu },{\tilde{\Gamma }}_{\mu \nu }^{\lambda },\varphi \right)+S_{\varphi }\left(g_{\mu \nu },\varphi \right)+S_m\left(g_{\mu \nu },\psi \right)=\frac{1}{2}{\displaystyle \int {\text{d}}^{4}x}{L}_T,$ (29)

with the total Lagrange quantity $L_T=\sqrt{-g}\left[\varphi f\left(\tilde{R}\right)-\frac{\omega }{\varphi }{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi +\frac{16\pi }{c^4}{L}_M\right]$. In the Palatini formalism, varying the action (29) with respect to g_µν and $\varphi$, we gain two field equations as follows

$\varphi F\left(\tilde{R}\right){\tilde{R}}_{\mu \nu }-\frac{1}{2}\varphi f\left(\tilde{R}\right)g_{\mu \nu }-\frac{\omega }{\varphi }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +\frac{1}{2}{g}_{\mu \nu }\frac{\omega }{\varphi }{\partial }_{\sigma }\varphi {\partial }^{\sigma }\varphi =8\pi T_{\mu \nu }$ (30)

$\frac{2\omega }{\varphi }\square \varphi -\frac{\omega }{{\varphi }^2}{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +f\left(\tilde{R}\right)=0$ (31)

The trace of Equation (30) is

$F\left(\tilde{R}\right)\tilde{R}-2f\left(\tilde{R}\right)+\frac{\omega }{{\varphi }^2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi =\frac{8\pi T}{\varphi }.$ (32)

Varying the action with respect to ${\tilde{\Gamma }}_{\mu \nu }^{\lambda }$ gives

${\tilde{\nabla }}_{\lambda }\left(\sqrt{-g}\varphi F\left(\tilde{R}\right)g^{\mu \nu }\right)=0,$ (33)

where $\tilde{\nabla }$ is the covariant derivative with respect to the Palatini connection.

2) The geometrical diagnostics for GBD theory with the different formalisms

For a flat FLRW universe, using Equations (26) and (27) we can derive the evolutional equations of background universe in the metric-GBD theory as,

$3f_R{H}^2=\frac{8\pi {\rho }_m}{\varphi }+\frac{f_{R}R-f\left(R\right)}{2}-3H{\stackrel{\dot{}}{f}}_R+\frac{1}{2}\omega {\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^2-3H{f}_R\frac{\stackrel{\dot{}}{\varphi }}{\varphi },$ (34)

$-2f_R\stackrel{\dot{}}{H}=\frac{8\pi }{\varphi }\left({\rho }_m+p_m\right)+{\ddot{f}}_R-H{\stackrel{\dot{}}{f}}_R+\omega {\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^2-H{f}_R\frac{\stackrel{\dot{}}{\varphi }}{\varphi }+f_R\frac{\ddot{\varphi }}{\varphi }+2\frac{\stackrel{\dot{}}{\varphi }}{\varphi }{\stackrel{\dot{}}{f}}_R,$ (35)

$f\left(R\right)-\omega {\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^2+2\omega \frac{\ddot{\varphi }}{\varphi }+6\omega H\frac{\stackrel{\dot{}}{\varphi }}{\varphi }=0.$ (36)

Here $R=6\left(2H^2+\stackrel{\dot{}}{H}\right)$, and “dot” denotes the derivative with respect to cosmic time t. For $\varphi$ = constant, Equations (34) - (36) are reduced to the cases of f(R) theory.

Using Equations (30) - (33), we can get the evolutional equations of background universe in the Palatini-GBD theory,

$\begin{array}{c}3F{H}^2=\frac{8\pi \left(\rho +3p\right)}{2\varphi }+\frac{1}{2}f+\omega {\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^2-\frac{3}{4}{\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^{2}F-\frac{3}{2}\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\stackrel{\dot{}}{F}\\ -\frac{3}{4}\frac{{\stackrel{\dot{}}{F}}^2}{F}-3HF\frac{\stackrel{\dot{}}{\varphi }}{\varphi }-3H\stackrel{\dot{}}{F},\end{array}$ (37)

$\begin{array}{c}-2F\stackrel{\dot{}}{H}=\frac{8\pi \left(\rho +p\right)}{\varphi }+\omega {\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^2-\frac{3}{2}{\left(\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right)}^{2}F-3\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\stackrel{\dot{}}{F}-\frac{3}{2}\frac{{\stackrel{\dot{}}{F}}^2}{F}\\ +\frac{\ddot{\varphi }}{\varphi }F+2\stackrel{\dot{}}{F}\frac{\stackrel{\dot{}}{\varphi }}{\varphi }+\ddot{F}-HF\frac{\stackrel{\dot{}}{\varphi }}{\varphi }-H\stackrel{\dot{}}{F}\end{array}$ (38)

$-f\left(\tilde{R}\right)-\frac{\omega }{{\varphi }^2}{\stackrel{\dot{}}{\varphi }}^2+2\omega \frac{\ddot{\varphi }}{\varphi }+6\omega H\frac{\stackrel{\dot{}}{\varphi }}{\varphi }=0$ (39)

In the following, we solve the cosmological equations of GBD theory with the different formalisms. For solving Equations (34) - (36) in the framework of metric formalism, we define the dimensionless variables: $y_H=H^2/m^2-{\left(1+z\right)}^3$, $y_R=R/m^2-3{\left(1+z\right)}^3$, $y_{\varphi }=\varphi /{\varphi }_0$, ${y^{\prime }}_{\varphi }={\varphi }^{\prime }/{\varphi }_0$. Then Equations (34) - (36) provide the differential equations for $\left\{y_H,y_R,y_{\varphi },{y^{\prime }}_{\varphi }\right\}$ as follows

${y^{\prime }}_H=-\frac{1}{1+z}\left(\frac{1}{3}{y}_R-4y_H\right),$ (40)

$\begin{array}{c}{y^{\prime }}_R=\frac{1}{\left[y_H+{\left(z+1\right)}^3\right]\left(z+1\right)m^2{f}_{RR}}\{\left[y_H+{\left(z+1\right)}^3\right]f_R\begin{array}{c}\\ \end{array}\\ -\left[y_R+3{\left(z+1\right)}^3\right]\frac{f_R}{6}+\frac{f}{6m^2}-\frac{\omega }{6}{\left(z+1\right)}^2\left[y_H+{\left(z+1\right)}^3\right]{\left(\frac{{y^{\prime }}_{\varphi }}{y_{\varphi }}\right)}^2\\ -\left[y_H+{\left(z+1\right)}^3\right]\left(z+1\right)f_R\frac{{y^{\prime }}_{\varphi }}{y_{\varphi }}-\frac{{\left(z+1\right)}^3}{\varphi }\}-9{\left(z+1\right)}^2\end{array}$ (41)

${y^{\prime }}_{\varphi }={\varphi }^{\prime }/{\varphi }_0,$ (42)

$\begin{array}{c}{y^{″}}_{\varphi }=\frac{y_{\varphi }}{2\omega \left[y_H+{\left(z+1\right)}^3\right]{\left(z+1\right)}^2}\{\frac{f}{m^2}+\omega \left[y_H+{\left(z+1\right)}^3\right]{\left(z+1\right)}^2{\left(\frac{{y^{\prime }}_{\varphi }}{y_{\varphi }}\right)}^2\\ +\omega \left(z+1\right)\frac{{y^{\prime }}_{\varphi }}{y_{\varphi }}\left[\frac{1}{3}{y}_R-4y_H-3{\left(z+1\right)}^3\right]+4\omega \left(z+1\right)\frac{{y^{\prime }}_{\varphi }}{y_{\varphi }}\left[y_H+{\left(z+1\right)}^3\right]\}\end{array}$ (43)

To solve above differential equations, the initial conditions are selected as respectively: ${y_{\varphi }|}_{z=0}=1$, ${{y^{\prime }}_{\varphi }|}_{z=0}=0.01$. The initial conditions of $y_H$ and $y_R$ are taken the same values with those in f(R) theory. The value of initial condition ${{y^{\prime }}_{\varphi }|}_{z=0}$ can be indicated by the following observations. For example, the limits on the variation of G can be exhibited by: $\left|\frac{\stackrel{\dot{}}{G}}{G}\right|=\left|\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right|\le 4.1\times 10{\text{y}}^{-\text{1}}$ from Pulsating white dwarf G117-B15A [51], −4 × 10 y⁻¹ ≤ $\frac{\stackrel{\dot{}}{\varphi }}{\varphi }$ ≤ 2.5 × 10⁻¹⁰ y⁻¹ from Nonradial pulsations of white dwarfs [52], $\left|\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right|$ ≤ 2.3 × 10⁻¹¹ y⁻¹ from Millisecond pulsar PSR J0437-4715 [53], $\left|\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right|$ ≤ 10⁻¹¹ y⁻¹ from Type-Ia supernovae [54], $\frac{\stackrel{\dot{}}{\varphi }}{\varphi }$ = (0.6 ± 0.42) × 10⁻¹² y⁻¹ from Neutron star masses [55], $\left|\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right|$ ≤ 1.6 × 10⁻¹² y⁻¹ from Helioseismology [56], and $\frac{\stackrel{\dot{}}{\varphi }}{\varphi }$ = (4 ± 9) × 10⁻¹³ y⁻¹ from Lunar laser ranging experiment [57], etc. Taking a stringent bound $\left|\frac{\stackrel{\dot{}}{\varphi }}{\varphi }\right|$ ≤ 10⁻¹² y⁻¹, we can calculate to limit $\left|{y^{\prime }}_{\varphi }\left(z=0\right)\right|\le 0.015$ by using the center value H₀ = 67.3 kms⁻¹∙Mpc⁻¹ = 6.87 × 10⁻¹¹ y⁻¹. Here we take ${y^{\prime }}_{\varphi }\left(z=0\right)=0.01$ as an initial condition in Equation (42). Then the pictures of geometrical diagnostic are illustrated in Figure 4 and Figure 5 for the metric formalism of GBD theory. For the Palatini formalism of GBD theory, using the trace equation in GBD theory we have

$\stackrel{\dot{}}{\tilde{R}}=\frac{-8\pi \left({\stackrel{\dot{}}{\rho }}_m\varphi -\stackrel{\dot{}}{\varphi }{\rho }_m\right)\varphi +\omega \left(2\ddot{\varphi }\stackrel{\dot{}}{\varphi }\varphi -2{\stackrel{\dot{}}{\varphi }}^3\right)}{{\varphi }^3\left(f_{\tilde{R}\tilde{R}}R-f_{\tilde{R}}\right)}.$ (44)

Changing the variable from t to z, with $\frac{\text{d}}{\text{d}t}=-H\left(1+z\right)\frac{\text{d}}{\text{d}z}$, we receive $\stackrel{\dot{}}{\varphi }={\varphi }^{\prime }\left(-\left(1+z\right)H\right)$, and $\ddot{\varphi }={\varphi }^{″}{H}^2{\left(z+1\right)}^2+{\varphi }^{\prime }{H}^2\left(z+1\right)+{\varphi }^{\prime }{H}^{\prime }H{\left(z+1\right)}^2$. Using the definition of geometrical quantities, we can plot the diagnostic pictures of {r, s} and {r, q} in the Palatini formalism of GBD theory.

Figure 4. The {r, s} geometrical diagnostic for GBD theory with the metric formalism (left) and the Palatini formalism (right).

Figure 5. The {r, q} geometrical diagnostic for GBD theory with the metric formalism (left) and the Palatini formalism (right).

In this part, we distinguish the different formalisms of GBD modified gravitational theory. For comparison, we take the same f(R) function as above. From Figure 4 and Figure 5, we can see that the difference between the metric formalism and the Palatini formalism are still conspicuous according to the {r, s} and {r, q} diagnostics, respectively. In the GBD theory, the evolutions of deceleration parameter q are plotted in Figure 6, which exhibit the different evolutions between two formalisms. According to Figure 6, for case of β = 1.1 in Palatini formalism, it is satisfied with the requirement of q(z) ≤ 1/2. And for this case we have z_T = 0.11 (at where q = 0) for metric formalism, and z_T = 1.09 for Palatini formalism. is called the transition redshift, which describes the universal expansion from deceleration toacceleration. Obviously, the significantly different values of z_T are indicated in these two formalisms.

In order to explore the effects of time-variable G, we also compare the f(R) theory (G is a constant) with the GBD theory (G is a variable quantity) according to the results of geometrical diagnostics. Some results could be exhibited as follows: 1) According to Figure 1 (left) and Figure 4 (left), in the metric formalism the variation of G affects obviously on the values of r (and s). For example, for case of β = 1.3, the value of r vary from 0 to −38.55 for f(R) theory, and from 0 to −16.54 for GBD theory, respectively; the largest value of s vary from 2.88 (corresponding to GBD) to 4.61 (corresponding to f(R) theory). Obviously, for the influence of variable G, the values of r (or s) in GBD are smaller than those in f(R) theory, which indicates that the variation of Newton gravitational constant can induce remarkable effects on geometrical quantities for the existence of BD scalar field (including the terms of its dynamics and the coupling between the BD field and the f(R) in the action). 2) In the Palatini formalism (see Figure 1 (right) and Figure 4 (right)), for β = 1.1 case the difference between the variable-G theory (GBD) and the constant-G theory (f(R)) are not obvious, while for other cases the shapes of r-s curves are different. 3) According to Figure 3 and Figure 6, in the metric formalism we find that the values of q decay more early (about z ~ 2) in f(R) theory than that (about z ~ 0.8) in GBD theory; while in the Palatini formalism, the difference between the GBD theory and the f(R) theory are more reflected at the earlier stage of universe (higher redshift).

Figure 6. The evolutions of a geometrical quantity-deceleration parameter q for GBD theory with the metric formalism (left) and the Palatini formalism (right).

4. Conclusions

Several observational and theoretical motivations require us to investigate the modified theories of GR. In modified gravitational theory, there exist the metric formalism and the Palatini formalism. In these two formalisms, the dynamical variables are considered to be different. Geometrical-diagnostic methods were often applied to distinguish the gravitational models. But it is scarce to investigate the differences between the different formalisms (e.g. the metric formalism and the Palatini formalism) in the modified theories of gravity. In this paper, we discriminate the different formalisms of the modified gravity by using the diagnostic methods. For considered modified theories of gravity (including the f(R) theory and the GBD theory), we can see that the difference between the two formalisms is notable according to the geometrical diagnostics. And relative to the ΛCDM model, there are more deviations in metric formalism than those in Palatini formalism, according to the {r, s} diagnostic.

Given that the GBD is a time-variable Newton gravitational constant theory, the differences between the variable-G theory and the constant-G theory are studied by using the diagnostic methods. According to the observational limits on G, we plot some pictures on the geometrical quantities. For the influence of variable G, the values of r (or s) in metric-formalism GBD are smaller than those in f(R) theory, which indicates that the variation of Newton’s gravitational constant can induce remarkable effects on geometrical quantities for the existence of BD scalar field. In the Palatini formalism, the shapes of r-s curves between the GBD theory and the f(R) theory could be obviously different, depending on the values of model parameter β. In addition, in the metric formalism we can find that the values of q decay more early (about z ~ 2) in f(R) theory than that (about z ~ 0.8) in GBD theory; while in the Palatini formalism, the difference between the f(R) theory and the GBD theory are more reflected at the higher-redshift universe. In summary, according to our study, the effects of variable G could be found in both the metric formalism and the Palatini formalism, by testing the geometrical quantities (e.g. r, s andq).

Acknowledgements

The research work is supported by the National Natural Science Foundation of China (Grant number: 12175095), and supported by LiaoNing Revitalization Talents Program (Grant number: XLYC2007047).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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