*6.2. Theoretical Interpretations*

We now investigate possible theoretical explanations for our results, focusing particular attention on modified gravity models. We first discuss a general scalar-tensor formulation and, then, we concentrate our attention on the so-called metric *f*(*R*) gravity.

#### 6.2.1. The Scalar Tensor Theory of Gravity

The action of the scalar tensor theories (STTs) of gravity is given by *S* = *S JF* + *S<sup>m</sup>* [370–374] with the Jordan Frame (JF) action

$$S^{\rm IF} = \frac{1}{16\pi} \int d^4x \sqrt{-\tilde{\mathfrak{F}}} \left[ \Phi^2 \tilde{\mathcal{R}} + 4\omega(\Phi) \tilde{\mathfrak{F}}^{\mu\nu} \partial\_\mu \Phi \partial\_\nu \Phi - 4\tilde{V}(\Phi) \right],\tag{12}$$

where *R*˜ is the Ricci scalar obtained with the physical metric *g*˜*µν*, while the matter fields Ψ*<sup>m</sup>* couple to the metric tensor *g*˜*µν* and not to Φ, i.e., *S<sup>m</sup>* = *Sm*[Ψ*m*, *g*˜*µν*].

In this Section we adopt natural units such that *c* = 1 and *G* = 1. Different STTs follow with the appropriate choice of the two functions *ω*(Φ) and *V*˜(Φ): e.g., the Brans–Dicke (BD) theory [375–378] can be obtained for *ω*(Φ) = *ω* (const.) and *V*˜(Φ) = 0, while the metric *f*(*R*) gravity, discussed in the next subsection, would correspond to *ω* ≡ 0.

The action *S JF* can be rewritten in the Einstein Frame (EF), where one defines *<sup>g</sup>*˜*µν* <sup>≡</sup> *A* 2 (*ϕ*)*gµν*, <sup>Φ</sup><sup>2</sup> <sup>≡</sup> <sup>8</sup>*πM*<sup>2</sup> ∗*A* −2 (*ϕ*), *V*(*ϕ*) ≡ *A* 4 (*ϕ*) 4*π <sup>V</sup>*˜(Φ), *<sup>γ</sup>*(*ϕ*) <sup>≡</sup> *d* log *A*(*ϕ*) *dϕ* , and *γ* 2 (*ϕ*) = 1 4*ω*(Φ)+6 , to get

$$S^{EF} = \frac{M\_\*^2}{2} \int d^4 \mathbf{x} \sqrt{-\mathbf{g}} \left[ \mathcal{R} + \mathcal{g}^{\mu \nu} \partial\_\mu \varphi \partial\_\nu \varphi - \frac{2}{M\_\*^2} V(\varphi) \right]. \tag{13}$$

Matter is coupled to *ϕ* only through a purely metric coupling, *S<sup>m</sup>* = *Sm*[Ψ*m*, *A* 2 (*ϕ*)*gµν*] and *M*<sup>∗</sup> is the Planck mass.

The physical quantities in the Jordan and Einstein frame are related by *dτ*˜ = *A*(*ϕ*)*dτ*, *a*˜ = *A*(*ϕ*)*a*, *ρ*˜ = *A*(*ϕ*) <sup>−</sup>4*ρ*, *p*˜ = *A*(*ϕ*) <sup>−</sup><sup>4</sup> *p*, where *τ* is the synchronous time variable. Defining *<sup>N</sup>* <sup>≡</sup> log *<sup>a</sup> a*0 , *λ* ≡ *V*(*ϕ*) *ρ* , *w* ≡ *p ρ* , and *ϕ* 0 = *dϕ dN* = *a dϕ da* , the combination of cosmological equations allows to write the equation for *ϕ* in the form (for a flat Friedmann–Robertson– Walker geometry) [379]

$$\frac{2}{3} \frac{1+\lambda}{1-\varrho^{f^2}/6} \left. \varrho'' + [(1-w) + 2\lambda] \varrho' \right| = -\sqrt{2} \left. \gamma(\varrho) \left( 1 - 3w \right) - 2 \lambda \frac{V\_{\varphi}(\varrho)}{V} \right. \tag{14}$$

Moreover, the Jordan- and Einstein-frame Hubble parameters, *<sup>H</sup>*˜ <sup>≡</sup> *<sup>d</sup>* log *<sup>a</sup>*˜/*dτ*˜ and *<sup>H</sup>* <sup>≡</sup> *d* log *a*/*dτ*, respectively, are related as

$$
\tilde{H} = \frac{1 + \gamma(\varphi) \,\varphi'}{A(\varphi)} \, H. \tag{15}
$$

For our purpose, we consider *A*(*ϕ*) = *A*0*e c*1*ϕ*+*c*2*ϕ* <sup>2</sup>/2, which implies *<sup>γ</sup>*(*ϕ*) = *<sup>c</sup>*<sup>1</sup> + *<sup>c</sup>*2*ϕ*, where *c*1,2 are constants. Under the following conditions *ϕ* <sup>00</sup>/*ϕ* 1, *ϕ* <sup>0</sup> <sup>2</sup>/*ϕ* <sup>2</sup> 1, and *Vϕ*(*ϕ*) *<sup>ϕ</sup>V<sup>ρ</sup>* 1, the solution of Equation (14) is *ϕ*(*z*) = *C*(1 + *z*) *<sup>K</sup>* <sup>−</sup> *c*1 *c*2 , where *K* = <sup>1</sup>−3*<sup>w</sup>* 1+*w* √ 2 *c*2, and *C* is an integration constant. We are looking for solutions such that *H* = *f*(*ϕ*)*H*˜ <sup>0</sup>, so that *H*˜ = *H*˜ 0 (1+*z*) *η* , where *H*˜ <sup>0</sup> is constant. These relations and (15) allow to derive *f*(*ϕ*) (the expression of *f*(*ϕ*) is quite involved, and in the case in which *c*1,2 1, it is a polynomial in *ϕ*). The scalar field Φ in the (physical) JF can be cast in the form Φ(*z*) = Φ0(1 + *z*) *K*˜ , where Φ<sup>0</sup> ≡ √ 8*πM*<sup>∗</sup> *A*0 h 1 − *C c*<sup>1</sup> − *Cc*<sup>2</sup> *c*1 i, *<sup>K</sup>*˜ <sup>=</sup> <sup>−</sup> *KC*(*c*1+*Cc*2) 1−*C c*1+ *Cc*2 2 , and *z* < 1 has been used (note:

*<sup>K</sup>*˜ is positive for *<sup>c</sup>*<sup>1</sup> or *<sup>c</sup>*<sup>2</sup> negative). The scalar field <sup>Φ</sup> reduces to *<sup>φ</sup>* for <sup>Φ</sup><sup>0</sup> <sup>→</sup> <sup>1</sup> and *<sup>K</sup>*˜ <sup>→</sup> <sup>2</sup>*η*. From the Friedmann Equation [379]

$$\left(\frac{d}{a}\right)^2 = \frac{1}{3M\_\*^2} \left[\rho + \frac{M\_\*^2}{2}\phi^2 + V(\varphi)\right],\tag{16}$$

with *ρ* given by matter (*ρ* = *ρ*0*m*/*a* <sup>3</sup> = *ρ*0*m*(1 + *z*) 3 ), and *c*1,2 1, one infers the effective potential

$$\frac{\tilde{V}}{3m^2} = \frac{4\pi M\_\*^2}{A\_0^2} \left[ f\_0^2 - \frac{1}{\Omega\_{0m}} \left( \frac{\Phi}{\Phi\_0} \right)^{\frac{2}{2\eta}} - \frac{\mathcal{C}^2 K^2 \varphi\_0^2}{6\Omega\_{0m}} \left( \frac{\Phi}{\Phi\_0} \right)^{\frac{K-\eta}{\eta}} \right],\tag{17}$$

where we recall that Ω0*<sup>m</sup>* = *ρ*0*m*/*ρcr*, *ρcr* = 3*M*<sup>2</sup> ∗*H*˜ <sup>2</sup> 0 , *f*<sup>0</sup> = *f*(*ϕ* = 0), and *m*<sup>2</sup> = Ω0*mH*˜ <sup>2</sup> 0 . For redshift 0 6 *z* < 0.3, to which we are interested, the scalar field varies slowly with *z*, Φ ∼ Φ0, so that the effective potential behaves like a cosmological constant. We see how the proposed scalar-tensor formulation has the right degrees of freedom to reproduce, in the JF, the required behavior of the (physical) trend of *H*0(*z*). In the next subsection, we analyze a sub-case of the general paradigm discussed above, which leads to the well-known *f*(*R*) gravity, which is among the most popular modified gravity formulations.
