The cosmic jerk parameter in f(R) gravity

We derive the expression for the jerk parameter in $f(R)$ gravity. We use the Palatini variational principle and the field equations in the Einstein conformal gauge. For the particular case $f(R)=R-\frac{\alpha^2}{3R}$, the predicted value of the jerk parameter agrees with the SNLS SNIa and X-ray galaxy cluster distance data but does not with the SNIa gold sample data.


Introduction
A particular class of alternative theories of gravity that has recently attracted a lot of interest is that of the f (R) gravity models, in which the gravitational Lagrangian is a function of the curvature scalar R [1] It has been shown that current cosmic acceleration may originate from the addition of a term R −1 to the Einstein-Hilbert Lagrangian R [2].
As in general relativity, f (R) gravity theories obtain the field equations by varying the total action for both the field and matter. In this work we use the metric-affine (Palatini) variational principle, according to which the metric and connection are considered as geometrically independent quantities, and the action is varied with respect to both of them [3]. The other one is the metric (Einstein-Hilbert) variational principle, according to which the action is varied with respect to the metric tensor g µν , and the affine connection coefficients are the Christoffel symbols of g µν . Both approaches give the same result only if we use the standard Einstein-Hilbert action [4]. The field equations in the Palatini formalism are second-order differential equations, while for metric theories they are fourth-order. Another remarkable property of the metricaffine approach is that the field equations in vacuum reduce to the standard Einstein equations of general relativity with a cosmological constant [4].
One can show that f (R) theories of gravitation are conformally equivalent to the Einstein theory of the gravitational field interacting with additional matter fields, if the action for matter does not depend on connection [3,5]. This can be done by means of a Legendre transformation, which in classical mechanics replaces the Lagrangian of a mechanical system with the Helmholtz Lagrangian. For f (R) gravity, the scalar degree of freedom due to the occurrence of nonlinear second-order terms in the Lagrangian is transformed into an auxiliary scalar field φ [5]. The set of variables (g µν , φ) is commonly called the Jordan conformal gauge. In the Jordan gauge, the connection is metric incompatible unless f (R) = R. The compatibility can be restored by a certain conformal transformation of the metric: g µν → h µν = f ′ (R)g µν . The new set (h µν , φ) is called the Einstein conformal gauge, and we will regard the metric in this gauge as physical.
f (R) gravity models have been compared with cosmological observations by several authors [6,7] and the problem of viability of these models is still open (see [8] and references therein). Current SNIa observations provide the data on the time evolution of the deceleration parameter q in the form of q = q(z), where z is the redshift [9]. The extraction of the information from these data depends, however, on assumed parametrization of q(z) [10]. For small values of z such a dependence can be linear, q(z) = q 0 +q 1 z [9], but its validity should fail at z ∼ 1. A convenient method to describe models close to ΛCDM is based on the cosmic jerk parameter j, a dimensionless third derivative of the scale factor with respect to the cosmic time [11,12]. A deceleration-to-acceleration transition occurs for models with a positive value of j 0 and negative q 0 . Flat ΛCDM models have a constant jerk j = 1.
In this work we derive the general expression for the jerk parameter in f (R) gravity. We use the field equations in the Palatini formalism and the Einstein conformal gauge [13]. We find the current value of this parameter for the case 2,7] and compare it with recent cosmological data [10].

Palatini variation in f (R) gravity
The action for f (R) gravity in the original (Jordan) gauge with the metricg µν is given by [13] Here, √ −gf (R) is a Lagrangian density that depends on the curvature scalar R = R µν (Γ λ ρ σ )g µν , S m is the action for matter represented symbolically by ψ and independent of the connection, and κ = 8πG c 4 . Tildes indicate quantities calculated in the Jordan gauge.
Variation of the action S J with respect tog µν yields the field equations where the dynamical energy-momentum tensor of matter is generated by the Jordan metric tensor: and the prime denotes the derivative of a function with respect to its variable. From variation of S J with respect to the connection Γ ρ µ ν it follows that this connection is given by the Christoffel symbols of the conformally transformed metric [5] g µν = f ′ (R)g µν .
The metric g µν defines the Einstein gauge, in which the connection is metric compatible.
The action (1) is dynamically equivalent to the following Helmholtz action [5,13]: where p is a new scalar variable. The function φ(p) is determined by From Eqs. (4) and (6) it follows that where R = R µν (Γ λ ρ σ )g µν is the Riemannian curvature scalar of the metric g µν .
In the Einstein gauge, the action (5) becomes the standard Einstein-Hilbert action of general relativity with an additional scalar field: Choosing φ (which is the curvature scalar in the Jordan gauge) as the scalar variable leads to where V (φ) is the effective potential Variation of the action (9) with respect to g µν yields the equations of the gravitational field in the Einstein gauge [13]: while variation with respect to φ reproduces (7). Eqs. (7) and (11) give from which we obtain φ = φ(T ). Substituting φ into the field equations (11) leads to a relation between the Ricci tensor and the energy-momentum tensor. Such a relation is in general nonlinear and depends on the form of the function f (R).

The jerk parameter in f (R) gravity
The jerk parameter in cosmology is defined as [11,12] j =ȧ where a is the cosmic scale factor, H is the Hubble parameter, and the dot denotes differentiation with respect to the cosmic time. This parameter appears in the fourth term of a Taylor expansion of the scale factor around a 0 : where the subscript 0 denotes the present value. We can rewrite Eq. (13) as where q is the deceleration parameter. For a flat ΛCDM model j = 1 [10]. 1 From the gravitational field equations (11) applied to a flat Robertson-Walker universe with dust we can derive the φ-dependence of the Hubble parameter [13] and the deceleration parameter [7] We also have the expression for the time dependence of φ: Combining Eqs. (16-18) and usingq =φq ′ (φ) leads tȯ From Eq. (15) we finally obtain We now examine the case f (R) = R − α 2 3R , where α is a constant, which is a possible explanation of current cosmic acceleration [2]. In this model the present value of φ is φ 0 = (−1.05±0.01)α, where α = (7.35 +1.12 −1.17 )×10 −52 m −2 [7]. We do not need to know the exact value of α since it does not affect nondimensional cosmological parameters. Substituting φ 0 into (20) gives This value does not overlap with the value j = 2.16 +0.81 −0.75 , obtained from the combination of three kinematical data sets: the gold sample of type Ia supernovae [9], the SNIa data from the SNLS project [14], and the X-ray galaxy cluster distance measurements [10]. The origin of this disagreement could come from the assumption of constant jerk used there. However, two of the three data sets separately are consistent with the f (R) = R − α 2 3R model: the SNLS SNIa set gives j = 1.32 +1.37 −1.21 and the cluster set gives j = 0.51 +2.55 −2.00 , and it is the gold sample data that yields larger j = 2.75 +1.22 −1.10 [10]. 2 In the f (R) = R − α 2 3R model the deceleration-to-acceleration transition occurred at φ t = − 5/3α [7]. The cosmic jerk parameter at this moment can be found from Eq. (20): This value shows that the jerk parameter in f (R) gravity changes significantly between the deceleration-to-acceleration transition and now, indicating the departure of f (R) gravity models from ΛCDM. It would be interesting to generalize the kinematical approach of [10] to time dependent jerk and compare 2 The value q 0 = −0.81 ± 0.14 found in [10] from the combined three data sets agrees with q 0 = −0.67 +0.06 −0.03 derived in the f (R) = R − α 2 3R model [7]. Each set separately agrees with our model as well.
the results with f (R) gravity models. More constraints on these models could also be provided by non-dimensional parameters containing higher derivatives of the scale factor, such as the snap parameter s =ä aH 4 [12].

Summary
We derived the expression for the cosmic jerk parameter in f (R) gravity formulated in the Einstein gauge. We used the Palatini variational principle to obtain the field equations and apply them to a flat, homogeneous, and isotropic universe filled with dust. The value of the jerk parameter for the particular case f (R) = R − α 2 3R does not overlap with the value obtained from cosmological data of the SNIa gold sample, but is consistent with the values obtained from more recent SNLS SNIa data and the X-ray galaxy cluster data [10]. Therefore, Palatini f (R) models in the Einstein gauge, including the case f (R) = R − α 2 3R , provide a possible explanation of current cosmic acceleration. Further observations should give stronger constraints on j and on f (R) gravity.