*3.2. The Friedmann–Einstein (Equation Equation (28)) as the First Integral of the Friedmann–Einstein Equation (Equation (29))*

It is well–known that without the chameleon field and for the conformal factor *f*(*a*) = 1 the Friedmann–Einstein differential equation for *a*˙ <sup>2</sup>/*a* 2 is the first integral of the Friedmann–Einstein differential equation for *a*¨/*a* [69]. However, such a property of Equation (28) with the chameleon field and the conformal factor *f*(*a*) 6= 1 to be the first integral of Equation (29) has not so far been investigated and proven in the literature. In order to prove that Equation (28) is the first integral of Equation (29) with the contributions of the chameleon field and the conformal factor *f*(*a*) 6= 1, we rewrite Equation (28) as follows:

$$\frac{d^2}{a^2} = \frac{1}{3M\_{\text{Pl}}^2} \left(\rho\_{\text{ch}} + \rho\_r f + \rho\_m\right),\tag{47}$$

where *<sup>ρ</sup>*ch <sup>=</sup> *<sup>ρ</sup><sup>φ</sup>* <sup>+</sup> *<sup>ρ</sup>m*(*<sup>f</sup>* <sup>−</sup> <sup>1</sup>) = <sup>1</sup> 2 *φ*˙ <sup>2</sup> + *V*eff(*φ*) is the chameleon field density, given by Equation (30) with the replacement *V*(*φ*) → *V*eff(*φ*) (see Equation (32)). In order to find *ρ*ch as a function of the expansion parameter *a* we use Equation (31) and transcribe it into the form

$$a\frac{d}{da}\rho\_{\rm ch}(a) + 6\rho\_{\rm ch}(a) = 6V\_{\rm eff}(a)\_{\prime} \tag{48}$$

where we have denoted *V*eff(*φ*) = *V*eff(*a*), assuming that *φ* is a function of *a*; i.e., *φ* = *φ*(*a*). As a function of the expansion parameter *a*, the effective potential *V*eff(*a*) is given by

$$V\_{\rm eff}(a) = V(a) + \rho\_m(a)(f(a) - 1),\tag{49}$$

where *V*(*a*) = *V*(*φ*) = *V*(*ϕ*(*a*)) with the additive contribution of the relic dark energy density, induced by torsion, and *f*(*a*) = *e βϕ*(*a*)/*M*Pl . The solution to Equation (48) is equal to

$$\rho\_{\rm ch}(a) = \frac{\mathbb{C}\_{\Phi}}{a^6} + \frac{6}{a^6} \int a^5 V\_{\rm eff}(a) da,\tag{50}$$

where the term *Cφ*/*a* 6 corresponds to the contribution of the kinetic term of a scalar field [87]. The integration constant *Cφ*, we define as follows: *C<sup>φ</sup>* = 3*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω*φa* 6 0 , where Ω*<sup>φ</sup>* is the integration constant, having the meaning of a relative density of a scalar field at time *t*<sup>0</sup> = 1/H<sup>0</sup> [70]. As a result, Equation (47) takes the form

$$\frac{d^2}{a^2} = \frac{1}{3M\_{\text{Pl}}^2} \left( \rho\_{\text{ch}}(a) + \rho\_r(a)f(a) + \rho\_m(a) \right),\tag{51}$$

where in the right-hand-side (r.h.s.) all densities and the conformal factor are functions of the expansion parameter *a*. Further, it is convenient to rewrite Equation (29) as follows:

$$\frac{d\mathbf{l}}{a} = -2\frac{d^2}{a^2} + \frac{1}{3M\_{\text{Pl}}^2} \,\rho\_r(a)f(a) + \frac{1}{2M\_{\text{Pl}}^2} \,\rho\_m(a)f(a) + \frac{1}{M\_{\text{Pl}}^2} \, V(a),\tag{52}$$

where we have used Equation (51). Since the second derivative *a*¨ of the expansion parameter *a* with respect to time can be given by

$$\vec{u} = \frac{1}{2} \frac{d\vec{u}^2}{da} \,' \,. \tag{53}$$

one may transcribe Equation (47) into the form

$$a\frac{d}{da}d^2 + 4d^2 = \frac{2}{3M\_{\text{Pl}}^2} \left. a^2 \rho\_r(a)f(a) + \frac{1}{M\_{\text{Pl}}^2} \left. a^2 \rho\_m(a)f(a) + \frac{2}{M\_{\text{Pl}}^2} \left. a^2 V(a) \right. \tag{54}$$

The solution to Equation (54) amounts to

$$d^2 = \frac{\mathbb{C}}{a^4} + \frac{2}{3M\_{\text{Pl}}^2} \frac{1}{a^4} \int a^5 \rho\_r(a) f(a) da + \frac{1}{M\_{\text{Pl}}^2} \frac{1}{a^4} \int a^5 \rho\_m(a) f(a) da + \frac{2}{M\_{\text{Pl}}^2} \frac{1}{a^4} \int a^5 V(a) da,\tag{55}$$

*Universe* **2020**, *6*, 221

where *C* is the integration constant. Dividing both sides of Equation (55) by *a* <sup>2</sup> we arrive at the equation

$$\frac{d^2}{d^2} = \frac{1}{3M\_{\text{Pl}}^2} \left( \frac{\mathbb{C}\_\phi}{a^6} + \frac{6}{a^6} \int a^5 \mathcal{U}(a) da + \frac{2}{a^6} \int a^5 \rho\_r(a) f(a) da + \frac{3}{a^6} \int a^5 \rho\_m(a) f(a) da \right), \tag{56}$$

where we have set *C<sup>φ</sup>* = 3*M*<sup>2</sup> Pl*<sup>C</sup>* = <sup>3</sup>*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω*φ*. Thus, Equation (51) is the first integral of Equation (29). Making a replacement *V*(*a*) = *V*eff(*a*) − *ρm*(*a*)(*f*(*a*) − 1) we arrive at the expression

$$\frac{d^2}{d^2} = \frac{1}{3M\_{\rm Pl}^2} \left( \rho\_{\rm ch}(a) + \frac{2}{a^6} \int a^5 \rho\_r(a) f(a) da + \frac{6}{a^6} \int a^5 \rho\_{\rm w}(a) da - \frac{3}{a^6} \int a^5 \rho\_{\rm w}(a) f(a) da \right). \tag{57}$$

Since the radiation and matter densities as functions of *a* obey the equations

$$a\frac{d}{da}\left(\rho\_r(a)f(a)\right) = -4\left(\rho\_r(a)f(a)\right),$$

$$a\frac{d}{da}\rho\_m(a) = -3\rho\_m(a)f(a) \tag{58}$$

and that *ρr*(*a*)*f*(*a*) = *ρr*<sup>0</sup> *f*(*a*0)*a* 4 0 /*a* 4 (see Equation (44)), we transcribe the right–hand–side (r.h.s.) of Equation (57) into the form

$$\frac{\dot{a}^2}{a^2} = \frac{1}{3M\_{\text{Pl}}^2} \left(\rho\_{\text{ch}}(a) + \rho\_r(a)f(a) + \frac{1}{a^6} \int \frac{d}{da} \left(a^6 \rho\_{\text{fl}}(a)\right) da\right) = \frac{1}{3M\_{\text{Pl}}^2} \left(\rho\_{\text{ch}}(a) + \rho\_r(a)f(a) + \rho\_{\text{fl}}(a)\right), \text{ (59)}$$

This proves that Equation (28) is the first integral of Equation (29) if the total energy–momentum is locally conserved. The evolution of the chameleon field density *ρ*ch(*a*) independently of the expansion parameter *a* is defined by Equation (50), which we rewrite as follows:

$$
\rho\_{\rm ch}(a) = \rho\_{\Lambda} + \frac{\mathbb{C}\_{\Phi}}{a^6} + \frac{6}{a^6} \int a^5 \Phi(a) da + \frac{6}{a^6} \int a^5 \rho\_{\rm m}(a) \left( f(a) - 1 \right) da,\tag{60}
$$

where *ρ*<sup>Λ</sup> = *M*<sup>2</sup> PlΛ*C*. and the third term in Equation (60) is the model-dependent part of the potential of the self-interaction of the chameleon field *V*(*φ*) = *ρ*<sup>Λ</sup> + Φ(*φ*) [4,10,16,88], taken as a function of the expansion parameter *a*, i.e., Φ(*φ*) = Φ(*a*). Such a chameleon field density may affect the acceleration of the Universe's expansion. Setting *f*(*a*) = 1 in Equation (60) we get

$$
\rho\_{\rm ch}(a) = \rho\_{\Lambda} + \frac{\mathbb{C}\_{\Phi}}{a^6} + \frac{6}{a^6} \int a^5 \Phi(a) da,\tag{61}
$$

where the second and the last terms might still provide an acceleration of the Universe's expansion additional to that caused by the first term *ρ*Λ, which is induced by torsion [39].
