Equivalence between Scalar-Tensor theories and $f(R)$-gravity: From the action to Cosmological Perturbations

In this paper we calculate the field equations for Scalar-Tensor from a variational principle, in which we have taken into account the Gibbons-York-Hawking type boundary term. We do the same for the theories $f(R)$, following, Guarnizo et al. Then, we review the equivalences between both theories in the metric formalism. Thus, starting from the perturbations under conformal-Newtonian gauge for Scalar-Tensor theories, we find the perturbations for $f(R)$ gravity under the equivalences in the same gauge. Working with two specific models of $f(R)$, we explore the equivalences between the theories. Further, we show the perturbations for both theories under the sub-horizon approach. Finally, we show how to calculate the cosmological perturbations using the package xPand.


I. INTRODUCTION
Recent observations of the CMB show that the universe is in accelerated expansion [2,3]. The broadly used model is the Λ-Cold-Dark-Matter (ΛCDM). However, this model introduces an exotic term of energy, called Dark Energy (DE), associated to the cosmological constant term Λ. Assuming that the theory of general relativity (GR) is not entirely correct at cosmological scales, it is possible that a cosmological constant term is not necessary to explain the accelerated expansion of the universe. The alternative theories to the Einstein's proposal are known as modified gravity theories (MG). One set of these theories is known as Scalar-Tensor gravity theories (ST) [4,5], where the gravitational action in these theories, in addition to the metric, to contain a scalar field which intervenes in the generation of the space-time curvature, associated to the metric. This scalar field is not directly coupled to the matter and, therefore, the matter responds only to the metric. It should be noted that the Brans-Dicke theory (BD), [6] proposed by C.H. Brans y R.H. Dicke in 1961, is a particular case of theories ST, where the parameter ω(φ) is independent of the scalar field. Another type of generalization to GR are the theories of gravity f (R) [7,8], where the lagrangian of Einstein-Hilbert is generalized, replacing the scalar curvature R by a more general function of it, f (R). The gravitational field in this theory is represented by the metric like GR does. The equivalence between theories ST and f (R) has been studied e.g., in [9][10][11][12]. It is, starting from the ST action, without the kinetic term of the scalar field, we arrive at the action of the gravity theories f (R). In this paper, in addition to the above, we show these equivalences for the field equations, the Friedmann equations of the homogeneous and isotropic universe and the Friedmann's scalar perturbations under the conformal-Newtonian gauge. The paper is organized as following: in the section II we get the field equations for RG, ST and f (R) theories starting from the variational principle, taking into account the Gibbons-York-Hawking (GYH) boundary term type, for every of the above theories. It is found that the consideration to obtain the field equations for ST, under the equivalence of the theories, to coincide to the f (R) condition. In the section III the equivalence between ST and f (R) for the actions and the field equations of the theories is shown. In the section IV the Friedmann equations for the background universe (homogeneous and isotropic) are calculated. Besides, we calculate the perturbed Friedmann equations under the conformal-Newtonian gauge, for ST, and the ones for f (R), using the equivalence between the theories. Then, we show how to construct the potential for the Hu-Sawicki and Starobisnky f (R) models, in order to calculate the Friedmann equations for the background and perturbed universe in these models for the two formalisms. Inmediately, we perform the sub-horizon approach to the perturbations, for both theories, and we show that they can not be calculated using the equivalences, due to the parameter ω = 1 for ST. Finally, in the section V we show the conclusions. In the appendix B we show how the perturbations were calculated under the package xPand from software Mathematica. Throughout the review, we adopt natural units 8πG = c = 1, here G is Newton's gravitational constant and c is speed of light. Have a metric signature (− + ++). Small latin indices a, b, . . . assume the values 0 to 3, while greek indices α, β, . . . assume the values 1,2,3.

II. FIELD EQUATIONS AND VARIATIONAL PRINCIPLES
This section shows how the field equations, through a variational principle for the theories GR, ST and f (R) are found; taking into account in all of these theories the boundary term type GYH. .

A. Field Equations in GR
The Einstein field equations (EFEs) can be deduced through a variational principle. We give a detailed review following [13][14][15]. The action for GR is where the first term is known as the Einstein-Hilbert action, d 4 x √ −g is the element of invariant volume and R is the Ricci scalar. The second term is the matter action defined by where ψ denotes the matter fields. The variation of the action (1) with respect to g ab takes the form Given the variation of the Ricci scalar we get where The second integral of the equation (5) is a divergence term. Thus, we can use the Gauss-Stokes theorem where ∂M its the boundary of a hypervolume on M, h is the determinant of the induced metric, n d is the unit normal vector to ∂M, ǫ is +1 if ∂M is timelike and −1 if ∂M is spacelike (it is assumed that ∂M is nowhere null). Coordinates x a are used for the finite region M and y a for the boundary ∂M.
In the equation (6) (8) where it has been imposed that the variation of the metric tensor is null in the boundary, i.e., Found the equation (8), the vector V d = g ed V e is calculated at the boundary Now we evaluate the term n d V d ∂M , using for this then where we use the antisymmetric part of ǫn a n b , with ǫ = n d n d = ±1. To the fact δg ab = 0 in the boundary we have h ab ∂ b δg da = 0, we get The variation of the action (5) takes the form The above equation shows that fixing δg ab = 0 on ∂M there is an additional boundary term. It could be argued that both the variation of the metric and its first derivative vanish in the boundary, i.e., δg ab = 0 and ∂ c δg ab = 0 in ∂M. Although this last argument leads directly to Einstein field equations, it implies to fix two conditions in the boundary. To avoid this, a boundary term is introduced, the Gibbons-York-Hawking (GYH) boundary term, that allows to have a well defined variational problem only fixing the variation of the metric in the boundary, δg ab ∂M = 0 [16,17]. This term is where K is the trace of extrinsic curvature. The variation of the GYH action is where δh ab = 0 in the boundary ∂M.
Using the definition of the extrinsic curvature [14] the trace is given by where we have used the equation (11). Taking into account (8), δK is calculated on the boundary The variation (16) gives This term to cancel with the second integral of (14) (the boundary term contribution). Hence we have The variation of the action (2) takes the form Defining the stress-energy tensor by then Imposing that the total variations to remain invariant with respect to δg ab , i.e., Finally, we get which are the Einstein field equations.

B. Field Equations in ST gravity
Scalar-Tensor theories of gravity belong to the MG theories, where a function of scalar field φ is non-minimal coupling to the Ricci scalar R. The action in the so-called Jordan Frame is [18] where S (m) is the action (2) describing ordinary matter (any form of matter different from the scalar field φ), ω is a parameter that is a function of the scalar field φ. Notice that the matter is not directly coupled to φ, in the sense that the Lagrangian density L (m) does not depend on φ, but the scalar field is directly coupled to the Ricci scalar R. The scalar field potential V (φ) constitutes a natural generalization of the cosmological constant [5]. From the action of ST theories of gravity, the BD's action can be gotten by [18] where ω 0 is a constant, and the potential is rescaled by a factor 16π. The ST field equations can be obtained from a variational principle. The variation of the action (27) with respect to δg ab gives Taking into account the equation (4), we get Let us write the second integral in the following way The term in parentheses is given by (e.g. see [1]) Using the above relation and the fact about the metric compatibility (∇ c g ab = 0), the term (31) yields where the D'Alembert operator definition has been used, i.e.
≡ ∇ d ∇ d . It allow us to define the next quantities to express the integral above in a different way The quantities M c and N c allow us to write the equation (33) as (for details view A) Thus, the variation of the action (30) takes the form where the Gauss-Stokes theorem (7) has been used in the boundary term. Evaluating the terms M c and N c at the boundary, we have (38) and Using (11) we compute the following terms that appear in the integrals (37) and where we have used the facts that n c h ac = 0, ǫ 2 = 1 and the tangential derivative h bf ∂ f (δg ab ) to vanish (e.g., see [14]). The variation of the action (37) takes the form As previously mentioned for GR, the last integral can be vanished arguing that, in addition to the variation of the metric δg ab , its first derivative ∂ c δg ab to vanish in the bpundary. Instead of, we use the boundary term type GYH for ST theories [19,20] S (ST ) The variation of this term with respect to δg ab is Taking into account (19), the above equation gives Thus, we can see that the term type GYH cancels with the second integral of the equation (42). Finally, using (24), the variation of the action of ST theories yields Imposing that this variation becomes stationary we get which are the field equations in the metric formalism of ST theories of gravity.
Since the action (27) it depends on the metric as the scalar field φ, the variation of the action (27) with respect to δφ is calculated Allow us to write, Now, the second term in the integral we can write it as Thus, the variation gives we define the following quantity for can be expressed diferently the above integral The covariant derivative of L c is the second term in (51) takes the form Using the Gauss-Stokes theorem (7) at the divergence term, we have Imposing that the variation of the scalar field in the boundary vanishes we can see that the Gauss-Stokes term cancels-off. Now, the variation of the term type GYH for ST theories(43) with respect to δφ yields because the imposition (56), the above term vanishes. Wherewith, the variation of the action (54) gives Imposing that this variation become stationary we have which are the field equation for the scalar field in ST theories of gravity.
As a natural extension of GR and higher order theories, f (R) theories emerge, which consider an arbitrary function of the Ricci scalar.
where f (R) is a non-linear analytical function of the Ricci scalar and S (m) is given by (2). In the paper ( [1]), shows how the field equations are obtained taking into account the boundary term type GYH for f (R). Here show the main results found there. The variation of the action with respect to δg ab is where the terms H c and I c are given by and Here f R = df dR . Using the Gauss-Stokes theorem to the divergence term in the variation and evaluating the terms n c H c and n c I c at the boundary, we have The boundary term type GYH for f (R) is [20] S f (R) The variation of the above action gives The second term of the above equation cancels the boundary term of the equation (65), but in addition needs to impose δR = 0 in the boundary to obtain the field equations [1].
We will see in more detail the equivalences between theories of gravity ST and f (R) in the next section, but it should be noted that for f (R) an additional imposition was made on δR, which in the equivalence is δφ in ST.
Taking into account the variation of the matter action (24) and imposing that the variation for f (R) theories becomes stationary thus, we have which are the field equations for f (R) theories.

III. EQUIVALENCE BETWEEN ST AND f (R) THEORIES
The equivalence between ST and f (R) theories has been broadly studied, e.g., in [9][10][11][12]. In this paper shows the equivalence between the actions and the field equations, but as we will see in the next section, in addition we will show them in the cosmological perturbations. We start from the following ST action without a kinetic term in the scalar field donde φ has been included as an auxiliary field. when f φφ = 0 in the above action, we can set Thus, the action (70) takes the form If φ = R, we have and we recover the action (61). Moreover, the variation with respect to φ of the above action gives The action (70) corresponds to the action (27) of ST theories with the parameter ω(φ) = 0. If we start with the field equations f (R) (rewriting the equations (69) for to include the Einstein tensor G ab ) Taking into account (74) in the above field equations, we get where it has been used (72), with the potential rescaled by 1 2 . The above equations are the field equations (48) for ST theories with the parameter ω(φ) = 0.

IV. COSMOLOGICAL PERTURBATIONS
In this section we study the Friedmann equations in a homogeneous and isotropic universe with the metric Friedmann-Lemaître-Robertson-Walker (FLRW) as the background metric for the ST and f (R) theories. Then we calculate the linear cosmological perturbations under conformal-Newtonian gauge for the theories above mentioned. Note that the equations found by f (R) theories for both the background and the perturbed ones were found under the equivalence relations with the ST theories.

A. Background Universe
Consider a statistically spatially homogeneous and isotropic universe with the spatially flat FLRW metric as background The energy conservation is where is the stress-energy tensor for perfect fluid. whit this, the energy conservation giveṡ Here, p is the fluid pressure, ρ the energy density y u a is the four-velocity of the fundamental observers. The Friedmann equations for the evolution of the background in ST theories are [18] and where H ≡ȧ (η) a(η) . The equation for the evolution of the scalar field is To obtain the Friedmann equations for BD theory, must be taking into account the relations (28) in the friedmann equation for ST theories. From the equivalence relation (74), we have where f 3 . Replacing (72) and the above relations in the equations (82) and (83) with the parameter ω(φ) = 0, we come to Friedmann equations for the f (R) theories and As mentioned above, one of the motivations for MG theories, is to explain the accelerating expansion of the universe. For ST theories, given a potential V (φ) [21,22] we can get a universe in accelerating expansion, while for f (R) theories, the same function is responsible for achieve it [23,24].

B. Equivalence between Cosmological Perturbations in ST and f (R) gravity
Scalar metric perturbations in the conformal-Newtonian gauge are describes by the line element [25] where Φ and Ψ are the so-called Bardeen potentials [26]. To find the linear perturbations of ST theories, the field equations (48) are perturbed, taking into account the metric (89). Here, δφ represents the perturbation of the scalar field. The perturbed Friedmann equations are (For more details on how the perturbations are calculated, see, e.g., [27]. In the appendix B it is shown as calculated with the package xPand from software Mathematica) which is the time-time component. we shall refer to the background quantities with the overbar. Now, the spacespace component is To find the relationship between Bardeen potentials and anisotropic pressure, we take the off-diagonal part, after having calculated the trace of the above equation If there is no anisotropic pressure, i.e., if Π = 0, the two potentials can be related to each other as Forf = 1, it implies that Φ = Ψ, which corresponds to the case of GR in the absence of anisotropic pressure. The perturbed equation of the evolution of the scalar field (60) is (see B) where δR is (for more details, e.g., see [28]) where the potential has been rescaled by 1 2 . Thus, the perturbed Friedmann equations of f (R) theories for time-time and space-space components take the form and The off-diagonal part, after having calculated the trace of the above equation For Π = 0, we take the relations between potentials Taking f (R) = R, we get the relations of GR Φ = Ψ is absence of anisotropic pressure.

C. Hu-Sawicki and Starobinsky models
Below are shown two examples from the equivalences between both f (R) and ST theories. The first is Hu-Sawicki model [29], which is important since it is able of reproducing the accelerated expansion of the universe [30][31][32] besides to satisfice the tests of the solar system [29]. Although the reconstruction of the potential has already been studied, we show the equivalences, in the Friedmann equations of the background and the perturbed ones. Now, this model is given by [33] where Λ is a constant energy scale whose value coincides with the measured value Λ = Λ obs = 3H 2 0 Ω Λ and ǫ ≪ is a small positive deformation parameter. We note that the derivative of f (R) is From the equivalence (74), we have Thus, rewriting the function (102) in terms of the scalar field, gives Using the equivalence in the potential (72). The potential for the Hu-Sawicki model is Once the potential is obtained, we calculate the Friedmann equations in the background in terms of the scalar field, which are (107) and From the relation (74), the Friedmann equations in the formalism f (R) take the form and The perturbed Friedmann equations (time-time and space-space components) in terms of the scalar field are and where we have used Now, the Friedmann equations in the formalism f (R), using the equivalence relations, take the form and where has been used the following relation The second model to discuss is the Starobinsky model [34], which is a cosmic inflation model. Whose perturbations in the inflationary era were first discussed by Mukhanok and Starobinsky himself [35,36]. His predictions agree with the recent CMB data [37]. For more discussions on this model, see e.g., [38]. The Starobinsky model is given by where the constant M has mass dimenssion. Performing the same above procedure for to construct the potencial, we start from From the equivalence (74), we have Thus, the potential gives where the potential has been rescaled by 1 2 . Found the potential, we calculate the Friedmann equations in terms of the scalar field and From the relation (74), we obtain the Friedmann equations for the f (R) formalism Now, the perturbed Friedmann equations in terms of the scalar field are and Once obtained the above perturbations, we find the cosmological perturbations in the formalism f (R), which are and D. Sub-horizon Approximation in ST and f (R) theories The perturbed energy conservation equations are [28] and where δ = δρ ρ is the perturbation of the relative energy density and v is the perturbation of velocity. The above equations in the matter domain, i.e., w = 0 (do not confuse with the parameter ω of ST theories) and taking Π = 0, takes the form in the space Fourier as where k is the wave number. Taking the sub-horizon approach, i.e., ∂ ∂η ∼ H ≪ k. we get To obtain the Poisson type equation in the sub-horizon approach for ST theories, the perturbed time-time component (90) in the Fouries space is taken To find δφ of the above equation, we takeω = 1 in the equation (94), thus Applying the sub-horizon approach to the term δR (95), we get replacing δR in the equation (134), gives Using the relation (93) in the above equation It can be seen that the perturbations of the scalar field in ST gravity theories do not depend on the wave number k in the sub-horizon approach. Replacing the above expression and using (93) in (133), we get Poisson type equation where is the gravitational effective constant for ST theories. The linear evolution of matter density perturbations and scalar field perturbations in sub-horizon approach in the framework of ST theory of gravity can be written as follows where it has been used (132). For more details about the density linear perturbations see [39][40][41].
To calculate the perturbations in f (R) theory under the same approach, we can not start from the equivalences ST theories, since, we take ω = 1 to find δφ. Therefore, to calculate it, we start of the perturbed time-time component (98) replacing the relation (101) in the above equation, we take Using the equation (135), we get This is the Poisson equation in the Fourier space in f (R) theories, where is the gravitational effective constant for f (R) theories. The linear evolution of matter density perturbations in sub-horizon approach for f (R) theories of gravity is where has ben used (132). For more detail about the linear density perturbations in f (R) theories see [42][43][44].

V. SUMMARY
In this section we present the results found in the paper • We found the field equations in the metric formalism of the ST theories using the variational principle, Where, the Gibbons-York-Hawking boundary term type was used to make no further assumptions about variations of the metric δg ab at the boundary. Furthermore, to obtain the field equation for the scalar field, we need impose, δφ = 0 at the boundary. Following [1], were found the field equations for f (R), where it was shown that in addition to δg ab , we have to impose that δR = 0 at the boundary. Which, under the equivalence between the ST and f (R) theories there is agreement.
• We obtained the Friedmann equations of the background and the perturbed ones under the conformal-Newtonian gauge for the ST theories. Then, we find the Friedmann equations of the background and the perturbed ones for f (R) theories using the equivalences between both theories. a split 3 + 1 of the manifold background is made. In this case we use the flat FLRW metric In3 :=SetSlicing[g, n, h, cd, {"|", ∇}, "FLFlat"] The order of the perturbation is 1, so