**2. Einstein's Equations in the Einstein–Cartan Gravitational Theory with Chameleon and Matter Fields**

We take the Einstein–Hilbert action of the Einstein–Cartan gravitational theory without chameleon and matter fields in the standard form [27,37,69]:

$$S\_{\rm EH} = \frac{1}{2} M\_{\rm Pl}^2 \int d^4 x \sqrt{-g} \,\mathcal{R}\_{\prime} \tag{1}$$

where *M*Pl = 1/ √ <sup>8</sup>*πG<sup>N</sup>* <sup>=</sup> 2.435 <sup>×</sup> <sup>10</sup><sup>27</sup> eV is the reduced Planck mass, *<sup>G</sup><sup>N</sup>* is the Newtonian gravitational constant [70] and *g* is the determinant of the metric tensor *gµν*. The scalar curvature R is defined by [27,37]

$$\mathcal{R} = \mathcal{g}^{\mu\nu} \mathcal{R}^{a}{}\_{\mu\nu\nu} = \mathcal{g}^{\mu\nu} \left( \frac{\partial}{\partial \mathbf{x}^{\nu}} \Gamma^{a}{}\_{a\mu} - \frac{\partial}{\partial \mathbf{x}^{a}} \Gamma^{a}{}\_{\nu\mu} + \Gamma^{a}{}\_{\nu\rho} \Gamma^{\rho}{}\_{a\mu} - \Gamma^{a}{}\_{a\rho} \Gamma^{\rho}{}\_{\nu\mu} \right) = \mathcal{g}^{\mu\nu} \mathcal{R}\_{\mu\nu} \tag{2}$$

where <sup>R</sup>*<sup>α</sup> µβν* and R*µν* are the Riemann and Ricci tensors in the Einstein–Cartan gravitational theory, respectively. Then, Γ *α µν* is the affine connection

$$
\Gamma^{a}{}\_{\mu\nu} = \left\{ {}^{a}{}\_{\mu\nu} \right\} + \mathcal{K}^{a}{}\_{\mu\nu} = \left\{ {}^{a}{}\_{\mu\nu} \right\} + \mathcal{g}^{a\sigma} \mathcal{K}\_{\sigma\mu\nu} \tag{3}
$$

where { *α µν*} are the Christoffel symbols [69]

$$\{\mathbf{x}^{a}\}\_{\mu\nu} = \frac{1}{2}g^{a\lambda} \left( \frac{\partial \mathbf{g}\_{\lambda\mu}}{\partial \mathbf{x}^{\nu}} + \frac{\partial \mathbf{g}\_{\lambda\nu}}{\partial \mathbf{x}^{\mu}} - \frac{\partial \mathbf{g}\_{\mu\nu}}{\partial \mathbf{x}^{\lambda}} \right) \tag{4}$$

and <sup>K</sup>*σµν* is the contorsion tensor, related to torsion <sup>T</sup>*σµν* by <sup>K</sup>*σµν* <sup>=</sup> <sup>1</sup> 2 (T*σµν* + T*µσν* + T*νσµ*) and T *α µν* = *g ασ*T*σµν* <sup>=</sup> <sup>Γ</sup> *α µν* − <sup>Γ</sup> *α νµ* with the following properties: K*σµν* = −K*νµσ* and T*µν* = −T*νµ* [27,37]. The integrand of the Einstein–Hilbert action Equation (1) can be represented in the following form:

$$\sqrt{-\text{g}}\,\text{R} = \sqrt{-\text{g}}\,\text{R} + \sqrt{-\text{g}}\,\text{C} + \frac{\partial}{\partial\mathbf{x}^{\text{v}}}(\sqrt{-\text{g}}\,\text{g}^{\text{uv}}\mathcal{K}^{\text{u}}{}\_{\text{u\theta}}) - \sqrt{-\text{g}}\,\text{g}^{\text{uv}}\left(\frac{1}{\sqrt{-\text{g}}}\,\frac{\partial}{\partial\mathbf{x}^{\text{d}}}(\sqrt{-\text{g}}\,\mathcal{K}^{\text{u}}{}\_{\text{v\theta}}) - \left\{{}^{\text{e}}\,\_{\text{ap}}\right\}\mathcal{K}^{\text{u}}{}\_{\text{v\theta}} - \left\{{}^{\text{e}}\,\_{\text{v\theta}}\right\}\mathcal{K}^{\text{g}}{}\_{\text{ap}}\right), \tag{5}$$

where we have denoted

$$\mathcal{L} = \mathcal{g}^{\mu\nu} (\mathcal{K}^{\mathcal{q}}{}\_{a\mu} \mathcal{K}^{\mathcal{u}}{}\_{\nu\mathcal{q}} - \mathcal{K}^{\mathcal{u}}{}\_{a\mathcal{q}} \mathcal{K}^{\mathcal{q}}{}\_{\nu\mu}) \tag{6}$$

and *R* is the Ricci scalar curvature of the Einstein gravitational theory, expressed in terms of the Christoffel symbols { *α µν*} [69] only. When removing in Equation (5) the total derivatives and integrating by parts, we delete the third term and transcribe the fourth term into the form √ <sup>−</sup>*g gµν* ;*<sup>α</sup>* <sup>K</sup>*<sup>α</sup> νµ*, where *g µν* ;*<sup>α</sup>* is the covariant derivative of the metric tensor *g µν*, vanishing because of the metricity condition *g µν* ;*<sup>α</sup>* = 0 [69].
