**2.** *f*(*R*)**-Gravity with Torsion**

In *f*(*R*)-gravity with torsion, the (gravitational) dynamical fields are given by a pseudo-Riemannian metric *g* and a metric compatible linear connection Γ, defined on the space-time manifold *M*. The covariant derivative induced by connection Γ is given by:

$$
\nabla\_{\partial\_i} \partial\_j = \Gamma\_{ij}{}^h \partial\_h. \tag{1}
$$

The torsion and Riemann curvature tensors, induced by the dynamical connection Γ, are expressed as:

$$\left(T\_{ij}\right)^{h} = \Gamma\_{ij}\,^{h} - \Gamma\_{ji}\,^{h} \,. \tag{2a}$$

$$\mathcal{R}^{h}\_{\ i ij} = \partial\_{\dot{i}} \Gamma\_{jk}{}^{h} - \partial\_{\dot{j}} \Gamma\_{ik}{}^{h} + \Gamma\_{ip}{}^{h} \Gamma\_{jk}{}^{p} - \Gamma\_{jp}{}^{h} \Gamma\_{ik}{}^{p} \,. \tag{2b}$$

In view of the metric compatibility, the linear connection Γ can be decomposed as [14,15]:

$$
\Gamma\_{\rm ij}{}^h = \tilde{\Gamma}\_{\rm ij}{}^h - K\_{\rm ij}{}^h{}\_{\prime} \tag{3}
$$

where:

$$K\_{ij}{}^h := \frac{1}{2} \left( -T\_{ij}{}^h + T\_j{}^h{}\_i - T\_{ij}{}^h{}\_j \right) \tag{4}$$

is the so-called contorsion tensor, and Γ˜ *<sup>h</sup> ij* is the Levi–Civita connection induced by the metric *g*.

The field equations are obtained by varying an action functional of the form:

$$\mathcal{A}(\mathbf{g}, \Gamma) = \int \left( \sqrt{|\mathbf{g}|} f(\mathbf{R}) + \mathcal{L}\_m \right) \, d\mathbf{s} \tag{5}$$

where *R*(*g*, Γ) = *g ijRij* (with *Rij* := *R h ihj*) denotes the scalar curvature associated with the connection Γ. The field equations result in [18–20]:

$$f'(R)R\_{ij} - \frac{1}{2}f(R)g\_{ij} = \mathcal{T}\_{ij} \tag{6a}$$

and:

$$T\_{ij}{}^h = \frac{1}{2f'} \left(\frac{\partial f'}{\partial \mathbf{x}^p} + \mathcal{S}\_{pq}{}^q\right) \left(\delta\_j^p \delta\_i^h - \delta\_i^p \delta\_j^h\right) + \frac{1}{f'} \mathcal{S}\_{ij}{}^h,\tag{6b}$$

where T*ij* and S *h ij* denote the stress–energy and the spin density tensors, respectively. In Equation (6a), attention must be paid to the order of the indexes, because the Ricci and stress–energy tensors *Rij* and T*ij* are not symmetric, in general.

It is worth noticing that, due to the independence between the metric tensor *gij* and the dynamical linear connection Γ *h ij* , the variation of the action functional (5) with respect to the metric tensor does not generate in Equation (6a) any term containing covariant derivatives of the scalar *f* ′ (*R*) (for details, see [18]); this is a remarkable difference with respect to the purely metric formulation of *f*(*R*)-gravity [2], and it has important consequences: for instance, the theory with torsion is not of fourth derivative order, as is the purely metric *f*(*R*)-theory. Taking the trace of Equation (6a) into account, we get relation:

$$f'(R)R - 2f(R) = \mathcal{T} \tag{7}$$

between the curvature scalar *R* and the trace T of the stress–energy tensor.

From Equation (7), it is seen that if the trace T is constant, so *R* is. Of course, the same conclusion holds when T*ij* = 0. In such circumstances, the field equations of *f*(*R*)-gravity with torsion are seen to amount to the ones of Einstein–Cartan theory with (or without) cosmological constant if spin is present, or the ones of Einstein theory with (or without) cosmological constant in the absence of spin. This holds in general, with the exception of the particular case T = 0 and *f*(*R*) = *αR* 2 . In such a case, indeed, Equation (7) is a trivial identity, and it does not impose any restriction on the scalar curvature *R*.

Therefore, from now on, we shall systematically suppose that T*ij* is not zero and T is not constant, as well as that the relation (7) is invertible. In this way, the curvature scalar *R* can be thought as a suitable function of T , namely:

$$R = R(\mathcal{T}).\tag{8}$$

The relation (8) plays a crucial role for the formulation of *f*(*R*)-gravity with torsion presented in this paper, as well as in our previous works. About this, it is worth noticing that the trace equation (7) gives rise to an algebraic or transcendental relation between the curvature scalar and the stress–energy tensor trace, but it is not a differential relation (unlike what happens in the purely metric formulation of *f*(*R*)-gravity). Therefore, the Dini theorem is generally applicable, and the relation (8) can be (almost always) supposed to locally exist. This allows us to express the torsion as a function of the matter fields and, therefore, to separate purely metric contributions from torsional ones within the Einstein-like equations, exactly as it happens in ECSK theory.

Defining the scalar field:

$$\varphi(\mathcal{T}) := f'(\mathcal{R}(\mathcal{T})),\tag{9}$$

we can rewrite Equation (6a) in the equivalent form:

$$\mathcal{R}\_{\text{ij}} - \frac{1}{2} \mathcal{R} \mathfrak{g}\_{\text{ij}} = \frac{1}{\varrho} \mathcal{T}\_{\text{ij}} + \frac{1}{2\varrho} \left( f(\mathcal{R}(\mathcal{T})) - f'(\mathcal{R}(\mathcal{T})) \mathcal{R}(\mathcal{T}) \right) \mathfrak{g}\_{\text{ij}}.\tag{10a}$$

$$T\_{ij}{}^h = \frac{1}{2\varrho} \left(\frac{\partial \varrho}{\partial \mathbf{x}^p} + \mathcal{S}\_{pq}{}^q\right) \left(\delta\_j^p \delta\_i^h - \delta\_i^p \delta\_j^h\right) + \frac{1}{\varrho} \mathcal{S}\_{ij}{}^h\,\mathrm{\,} \tag{10b}$$

which will be used in the following discussion. Making use of Equations (3), (4) and (10b), we can express the contorsion tensor as:

$$\mathcal{K}\_{ij}{}^h = \hat{\mathcal{K}}\_{ij}{}^h + \hat{\mathcal{S}}\_{ij}{}^h{}\_{\prime} \tag{11a}$$

$$\hat{S}\_{ij}{}^h := \frac{1}{2\varrho} \left( -\mathcal{S}\_{ij}{}^h + \mathcal{S}\_j{}^h{}\_i - \mathcal{S}\_{\,ij}{}^h{}\_j \right) . \tag{11b}$$

$$\mathcal{K}\_{ij}{}^h := -\hat{T}\_j \delta\_i^h + \hat{T}\_p \mathbf{g}^{ph} \mathbf{g}\_{ij\prime} \tag{11c}$$

$$
\hat{T}\_j := \frac{1}{2\varrho} \left( \frac{\partial \varrho}{\partial x^j} + \mathcal{S}\_{jk}{}^k \right). \tag{11d}
$$

Introducing the so-called torsion vector *T<sup>i</sup>* := *T j ij* , we also mention the conservation laws [42]:

$$T\nabla\_a T^{\text{ai}} + T\_a T^{\text{ri}} - T\_{ca} T^{\text{ica}} - \frac{1}{2} \mathcal{S}\_{\text{sta}} \mathcal{R}^{\text{stai}} = 0,\tag{12a}$$

$$\nabla\_{\hbar} \mathcal{S}^{ij\hbar} + T\_{\hbar} \mathcal{S}^{ij\hbar} + \mathcal{T}^{ij} - \mathcal{T}^{ji} = \mathbf{0},\tag{12b}$$

which have to be satisfied by the stress–energy and spin density tensors of the matter fields. In particular, we recall that Equation (12b) amount to the antisymmetric part of the Einstein-like Equation (10a).

In the case that the spin density tensor is zero, separating the Levi–Civita terms from the torsional ones, we can rewrite the Einstein-like field Equation (10a) in the form [18]:

$$\begin{split} \tilde{\mathcal{R}}\_{ij} - \frac{1}{2} \tilde{\mathcal{R}} g\_{ij} &= \frac{1}{\varrho} \mathcal{T}\_{ij} + \frac{1}{\varrho^2} \left( -\frac{3}{2} \frac{\partial \varrho}{\partial \mathbf{x}^i} \frac{\partial \varrho}{\partial \mathbf{x}^j} + \varrho \tilde{\nabla}\_j \frac{\partial \varrho}{\partial \mathbf{x}^i} + \frac{3}{4} \frac{\partial \varrho}{\partial \mathbf{x}^h} \frac{\partial \varrho}{\partial \mathbf{x}^k} \mathbf{g}^{hk} g\_{ij} \\ & - \varrho \tilde{\nabla}^h \frac{\partial \varrho}{\partial \mathbf{x}^h} \mathbf{g}\_{ij} - V(\varrho) \mathbf{g}\_{ij} \right), \end{split} \tag{13}$$

where the effective potential for the scalar field *ϕ*:

$$V(\varphi) := \frac{1}{4} \left[ \varphi F^{-1}((f')^{-1}(\varphi)) + \varphi^2 (f')^{-1}(\varphi) \right],\tag{14}$$

has been introduced. In Equation (13), *R*˜ *ij*, *<sup>R</sup>*˜, and <sup>∇</sup>˜ denote, respectively, the Ricci tensor, the scalar curvature, and the covariant derivative associated with the Levi–Civita connection of the dynamical metric *gij*.

The Einstein-like Equation (13) (together with Equation (9)) are deducible from a scalar-tensor theory with Brans–Dicke parameter *ω*<sup>0</sup> = −3/2. This can be seen by recalling the action functional of a (purely metric) scalar-tensor theory:

$$\mathcal{A}(\mathbf{g}, \boldsymbol{\varrho}) = \int \left[ \sqrt{|\mathbf{g}|} \left( \boldsymbol{\varrho} \tilde{\mathbf{R}} - \frac{\omega\_0}{\varrho} \boldsymbol{\varrho}\_i \boldsymbol{\varrho}^i - \boldsymbol{\mathcal{U}}(\boldsymbol{\varrho}) \right) + \mathcal{L}\_m \right] \, d\mathbf{s}, \tag{15}$$

where *ϕ* is the scalar field, *ϕ<sup>i</sup>* := *∂ϕ ∂x i* and *U*(*ϕ*) is the potential of *ϕ*, L*m*(*gij*, *ψ*) is the matter Lagrangian, function of the metric and some other matter fields *ψ*, and *ω*<sup>0</sup> is the so called Brans–Dicke parameter. By varying (15) with respect to the metric tensor and the scalar field, one gets the field equations:

$$\tilde{\mathcal{R}}\_{\text{ij}} - \frac{1}{2} \tilde{\mathcal{R}} g\_{\text{ij}} = \frac{1}{\varrho} \mathcal{T}\_{\text{ij}} + \frac{\omega\_0}{\varrho^2} \left( \varrho\_{\text{i}} \varrho\_{\text{j}} - \frac{1}{2} \varrho\_{\text{h}} \varrho^{\text{h}} g\_{\text{ij}} \right) + \frac{1}{\varrho} \left( \mathfrak{T}\_{\text{j}} \varrho\_{\text{i}} - \mathfrak{T}\_{\text{h}} \varrho^{\text{h}} g\_{\text{ij}} \right) - \frac{\mathcal{U}}{2\varrho} g\_{\text{ij}} \tag{16}$$

and:

$$\frac{2\omega\_0}{\varrho}\nabla\_h\varphi^h + \tilde{\mathcal{R}} - \frac{\omega\_0}{\varrho^2}\varphi\_h\varphi^h - \mathcal{U}' = 0,\tag{17}$$

where T*ij* := − 1 p |*g*| *δ*L*<sup>m</sup> δg ij* and *U* ′ := *dU dϕ* . By inserting the trace of Equation (16) into Equation (17), one gets the equation:

$$(2\omega\_0 + 3)\,\,\widetilde{\nabla}\_h \varphi^h = \mathcal{T} + \varrho \mathcal{U}' - 2\mathcal{U}.\tag{18}$$

A direct comparison immediately shows that, for *ω*<sup>0</sup> := − 3 2 and *U*(*ϕ*) := 2 *ϕ V*(*ϕ*) (where *V*(*ϕ*) is defined in Equation (14)), Equation (16) becomes formally identical to the Einstein-like Equation (13) for *f*(*R*)-gravity with torsion. Moreover, in such a circumstance, Equation (18) reduces to the algebraic equation:

$$\mathcal{T} + \mathcal{2}V'(\varphi) - \frac{6}{\varphi}V(\varphi) = 0,\tag{19}$$

relating the matter trace T to the scalar field *ϕ*. In particular, it is easily seen that, under the condition *f* ′′ 6= 0, Equation (19) represents exactly the inverse relation of (9), namely:

$$\mathcal{T} + 2V'(\varphi) - \frac{6}{\varphi}V(\varphi) = 0 \qquad \Longleftrightarrow \qquad \mathcal{T} = \mathcal{F}^{-1}((f')^{-1}(\varphi)),\tag{20}$$

being *F* −1 (*X*) = *f* ′ (*X*)*X* − 2 *f*(*X*). The conclusion follows that, when the matter Lagrangian does not depend on the dynamical connection (the dynamical connection does not couple with matter), *f*(*R*)-gravity with torsion is dynamically equivalent to a scalar-tensor theory with a Brans–Dicke parameter *ω*<sup>0</sup> = − 3 2 .

For later use, we also notice that field equations (13) can be simplified by rewriting them the Einstein frame. In fact, performing the conformal transformation:

$$
\mathfrak{F}\_{\rm ij} = \mathfrak{q} \mathfrak{g}\_{\rm ij}.\tag{21}
$$

Equation (13) assumes the simpler form (see for example [18,43]):

$$
\tilde{\mathcal{R}}\_{\dot{i}\dot{j}} - \frac{1}{2} \tilde{\mathcal{R}} \mathfrak{g}\_{\dot{i}\dot{j}} = \frac{1}{\varrho} \mathcal{T}\_{\dot{i}\dot{j}} - \frac{1}{\varrho^3} V(\varrho) \mathfrak{g}\_{\dot{i}\dot{j}}.\tag{22}
$$

where *R*¯ *ij* and *R*¯ are, respectively, the Ricci tensor and the curvature scalar induced by the conformal metric *g*¯*ij*.

The relationships between the the conservation laws existing in the Jordan and those holding in the Einstein frame are clarified by the following results [44,45]:

**Proposition 1.** *Equations* (13)*,* (14)*,* (19) *imply the standard conservation laws* <sup>∇</sup>˜ *<sup>j</sup>*T*ij* <sup>=</sup> <sup>0</sup>*.*

**Proposition 2.** *The condition* <sup>∇</sup>˜ *<sup>j</sup>*T*ij* <sup>=</sup> <sup>0</sup> *is equivalent to the condition* <sup>∇</sup>¯ *<sup>j</sup>*T¯ *ij* = 0*, where* T¯ *ij* := 1 *ϕ* T*ij* − 1 *ϕ*3 *<sup>V</sup>*(*ϕ*)*g*¯*ij and* <sup>∇</sup>¯ *denotes the covariant derivative associated to the conformal metric <sup>g</sup>*¯*ij.*
