*Derivation of Equation (5)*

Let us show that √ −*g* R can be presented in the form of Equation (5). Using Equation (3) we get an obvious relation

$$
\begin{split} \sqrt{-g}\,\mathcal{R} &= \begin{array}{c} \sqrt{-g}\,\mathcal{R} + \sqrt{-g}\,\mathcal{C} + \sqrt{-g}\,\mathcal{G}^{\mu\nu} \left( \frac{\partial}{\partial\mathbf{x}^{\nu}} \mathcal{K}^{a}{}\_{a\mu} - \frac{\partial}{\partial\mathbf{x}^{a}} \mathcal{K}^{a}{}\_{\nu\mu} + \left\{ \begin{array}{c} \iota\_{\nu\rho} \end{array} \right\} \mathcal{K}^{\rho}{}\_{a\mu} + \left\{ \begin{array}{c} \iota\_{a\mu} \end{array} \right\} \mathcal{K}^{a}{}\_{\nu\rho} \\ & \left( \end{array} \right) . \end{split} \tag{7}
$$

Having replaced the first term in the brackets by the total derivative and by adding the last term in the brackets, we arrive at the expression

$$
\begin{array}{lll}
\sqrt{-g}\,\mathrm{\mathcal{R}} &=& \sqrt{-g}\,\mathrm{\mathcal{R}} + \sqrt{-g}\,\mathrm{\mathcal{C}} + \frac{\partial}{\mathrm{div}} \left(\sqrt{-g}\,\mathrm{g}^{\mu\nu}\,\mathrm{\mathcal{K}^{a}}\_{\mathrm{a}\mu}\right) - \frac{\partial}{\mathrm{div}} \left(\sqrt{-g}\,\mathrm{g}^{\mu\nu}\right)\,\mathrm{\mathcal{K}^{a}}\_{\mathrm{a}\mu} - \sqrt{-g}\,\mathrm{g}^{\mu\nu}\,\left\{\mathrm{\mathcal{C}}\_{\nu\mu}\right\}\,\mathrm{\mathcal{K}^{a}}\_{\mathrm{a}\mu} \\
& - \quad \sqrt{-g}\,\mathrm{g}^{\mu\nu} \left(\frac{\partial}{\partial\mathbf{r}^{a}}\,\mathrm{\mathcal{K}^{a}}\_{\mathrm{v}\mu} + \left\{{}^{a}\_{a\rho}\right\}\,\mathrm{\mathcal{K}^{g}}\_{\mathrm{v}\mu} - \left\{{}^{a}\_{v\rho}\right\}\,\mathrm{\mathcal{K}^{a}}\_{\mathrm{a}\mu} - \left\{{}^{a}\_{a\mu}\right\}\,\mathrm{\mathcal{K}^{a}}\_{\mathrm{v}\nu}\right).
\end{array} \tag{8}
$$

Then, the first term in the brackets we rewrite as follows:

$$
\begin{split}
\sqrt{-\mathsf{g}}\,\mathsf{R} &= \sqrt{-\mathsf{g}}\,\mathsf{R} + \sqrt{-\mathsf{g}}\,\mathsf{C} + \frac{\mathsf{\partial}}{\partial\mathsf{x}^{\mathsf{u}}} \left(\sqrt{-\mathsf{g}}\,\mathsf{g}^{\mu\nu}\,\mathsf{K}^{\mathsf{a}}{}\_{\mu\mu}\right) - \frac{\mathsf{\partial}}{\partial\mathsf{x}^{\mu}} \left(\sqrt{-\mathsf{g}}\,\mathsf{g}^{\mu\nu}\right)\,\mathsf{K}^{\mathsf{a}}{}\_{\mu\mu} - \sqrt{-\mathsf{g}}\,\mathsf{g}^{\mu\nu}\,\left\{{}^{\mathsf{q}}\boldsymbol{\eta}\_{\nu\mu}\right\}\,\mathsf{K}^{\mathsf{a}}{}\_{\mathsf{zq}} \\ &- \sqrt{-\mathsf{g}}\,\mathsf{g}^{\mu\nu} \left(\frac{1}{\sqrt{-\mathsf{g}}}\,\frac{\mathsf{\partial}}{\partial\mathsf{x}^{\mu}} \left(\sqrt{-\mathsf{g}}\,\mathsf{K}^{\mathsf{a}}{}\_{\nu\mu}\right) - \frac{1}{\sqrt{-\mathsf{g}}}\,\frac{\mathsf{\partial}\sqrt{-\mathsf{g}}}{\partial\mathsf{x}^{\mu}}\,\mathcal{K}^{\mathsf{a}}{}\_{\nu\mu} + \left\{{}^{\mathsf{a}}\boldsymbol{a}\boldsymbol{q}\right\}\,\mathcal{K}^{\mathsf{p}}{}\_{\nu\mu} - \left\{{}^{\mathsf{a}}\boldsymbol{q}\right\}\,\mathcal{K}^{\mathsf{a}}{}\_{\mu\mu} - \left\{{}^{\mathsf{q}}\boldsymbol{q}\right\}\,\mathcal{K}^{\mathsf{a}}{}\_{\nu\mu}\right).
\end{split}
$$

Since √ <sup>−</sup>*g gµν*{ *ϕ νµ*} and { *α αϕ*} are equal to [69] (see Equation (10.107) and Equation (9.56))

$$\left\{\sqrt{-g}\,g^{\mu\nu}\{^{\rho}\}\_{\nu\mu}\right\} = -\frac{\partial}{\partial x^{\nu}}\left(\sqrt{-g}\,g^{\rho\nu}\right) \quad , \quad \left\{\,^{a}\_{\;\,a\rho}\right\} = \frac{1}{\sqrt{-g}}\frac{\partial\sqrt{-g}}{\partial x^{\rho}} \,, \tag{10}$$

the fourth and fifth terms in the first line in Equation (9) and the second and third terms in the brackets are cancelled out in pairs. This reduces Equation (9) to Equation (5).

Now we may show that the contribution of the fourth term in Equation (5) to the Einstein–Hilbert action reduces to the contribution of the term √ <sup>−</sup>*g gµν* ;*<sup>α</sup>* <sup>K</sup>*<sup>α</sup> νµ*. The contribution of the fourth term in Equation (5) to the Einstein–Hilbert action is defined by the integral

$$\int d^4 \mathbf{x} \sqrt{-g} \, g^{\mu \nu} \left( \frac{1}{\sqrt{-g}} \frac{\partial}{\partial \mathbf{x}^a} (\sqrt{-g} \, \mathcal{K}^a{}\_{\nu \mu}) - \{ ^\{ \mathcal{P}\_{a \mu} \} \, \mathcal{K}^a{}\_{\nu \varphi} - \{ ^\{ \mathcal{K}\_{\nu \varphi} \} \, \mathcal{K}^\varphi } \right). \tag{11}$$

After the integration by parts in the first term we get

$$\oint \sqrt{-g} \, g^{\mu \nu} \mathcal{K}^{a}{}\_{\nu \mu} d\mathcal{S}\_{\mathfrak{a}} - \int d^{4}x \sqrt{-g} \left( \frac{\partial g^{\mu \nu}}{\partial x^{a}} \mathcal{K}^{a}{}\_{\nu \mu} + g^{\mu \nu} \{ \! \!^{\rho} a \!\!\_{a \mu} \} \! \! \!^{\kappa} \!\_{\nu \rho} + g^{\mu \nu} \{ \! \!^{a} a \!\!\_{\nu \rho} \} \! \! \! \!^{\kappa} \!^{\rho} \! \_{a \mu} \right). \tag{12}$$

Having omitted the surface term and having renamed some indices in the second integral in Equation (12), we arrive at the expression

$$-\int d^4x\sqrt{-g}\left(\frac{\partial \mathcal{g}^{\mu\nu}}{\partial x^{\alpha}} + \mathcal{g}^{\rho\nu}\{^{\mu}\_{\rho\alpha}\} + \mathcal{g}^{\nu\rho}\{^{\mu}\_{\rho\alpha}\}\right)\mathcal{K}^{a}\_{\ \nu\mu} \tag{13}$$

where we have used the property of the Christoffel symbols { *µ αρ*} = { *µ ρα*}. The expression in the brackets is the covariant derivative of the metric tensor *g µν* (see, for example, [27,69])

$$\mathcal{g}^{\mu\nu}{}\_{;\mu} = \frac{\partial \mathcal{g}^{\mu\nu}}{\partial \mathfrak{a}^{\mu}} + \mathcal{g}^{\rho\nu} \{^{\mu}{}\_{\rho\mu}\} + \mathcal{g}^{\mu\rho} \{^{\nu}{}\_{\rho\mu}\}. \tag{14}$$

Thus, the contribution of the fourth term to the Einstein–Hilbert action is proportional to the integral

$$\int d^4x \sqrt{-g} \,\mathcal{g}^{\mu\nu} \left( \frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^a} (\sqrt{-g} \,\mathcal{K}^a{}\_{\nu\mu}) - \left\{ ^\rho \boldsymbol{a}\_{\mu\mu} \right\} \mathcal{K}^a{}\_{\nu\rho} - \left\{ ^\mu \boldsymbol{a}\_{\rho} \right\} \mathcal{K}^\rho{}\_{a\mu} \right) = - \int d^4x \sqrt{-g} \,\mathcal{g}^{\mu\nu} \,\_\mu \mathcal{K}^a{}\_{\nu\mu}.\tag{15}$$

This confirms our assertion concerning a vanishing contribution of the fourth term in Equation (5) to the Einstein–Hilbert action in case of the metricity condition *g µν* ;*<sup>α</sup>* = 0 [69].

Since it has been shown in [39] that C = −2Λ*C*, where <sup>Λ</sup>*<sup>C</sup>* is the cosmological constant [69,71,72] (see also [15]) or the relic dark energy density, the Einstein–Hilbert action Equation (1) of the Einstein–Cartan gravitational theory with the scalar curvature Equation (2) can be represented in the following form [39]:

$$S\_{\rm EH} = \frac{1}{2} M\_{\rm Pl}^2 \int d^4 x \sqrt{-g} \left(\mathcal{R} - 2\Lambda\_{\rm C}\right). \tag{16}$$

As has been shown in [39], the same result is valid for the Poincaré gauge gravitaitonal theory [73–77] (see also [31–34]). Using Equation (11) the action of the Einstein–Cartan gravitational theory with torsion, chameleon fields and matter fields we take in the form [39]

$$S\_{\rm EH} = \frac{1}{2} M\_{\rm Pl}^2 \int d^4 \mathbf{x} \sqrt{-\mathbf{g}} \, \mathbb{R} + \int d^4 \mathbf{x} \sqrt{-\mathbf{g}} \, \mathcal{L}[\phi] + \int d^4 \mathbf{x} \sqrt{-\mathbf{g}} \, \mathcal{L}\_{\rm m}[\tilde{\mathbf{g}}],\tag{17}$$

where L[*φ*] is the Lagrangian of the chameleon field

$$\mathcal{L}[\phi] = \frac{1}{2} \mathcal{g}^{\mu \nu} \partial\_{\mu} \phi \partial\_{\nu} \phi - V(\phi) \tag{18}$$

and *V*(*φ*) is the potential of the chameleon self-interaction. In Equation (17), following Khoury and Weltman [1], we have included additively the cosmological constant Λ*<sup>C</sup>* in the form of the relic dark energy density *ρ*<sup>Λ</sup> = *M*<sup>2</sup> PlΛ*<sup>C</sup>* into the potential *V*(*φ*) of the chameleon field self-interaction; i.e., *V*(*φ*) = *ρ*<sup>Λ</sup> + Φ(*φ*). This implies that the chameleon field has no relation to the origin of the cosmological constant or the relic dark energy density. It can only evolve above the relic background of the dark energy, caused by torsion.

The matter fields and the radiation [78,79] are described by the Lagrangian L*m*[*g*˜*µν*]. The interactions of the matter fields and radiation with the chameleon field are expressed in terms of the metric tensor *g*˜*µν* in the Jordan frame [1,2,80], which is conformally related to Einstein's frame metric tensor *gµν* by *g*˜*µν* = *f* <sup>2</sup> *gµν* (or *g*˜ *µν* = *f* <sup>−</sup><sup>2</sup> *g µν*) and √ −*g*˜ = *f* 4 √ −*g* with *f* = *e βφ*/*M*Pl , where *β* is the chameleon–matter coupling constant [1,2]. The factor *f* = *e βφ*/*M*Pl can be interpreted also as a conformal coupling to matter fields and radiation [80] (see also [1,2,81]). For simplicity we have set the chameleon–photon coupling constant *β<sup>γ</sup>* [79] to be equal to the chameleon–matter coupling constant *β*.

By varying the action of Equation (17) with respect to the metric tensor *δg µν* (see, for example, [69]), we arrive at Einstein's equations, modified by the contribution of the chameleon field. We get

$$R\_{\mu\nu} - \frac{1}{2}g\_{\mu\nu}R = -\frac{1}{M\_{\text{Pl}}^2} \left( f^2 \mathcal{T}\_{\mu\nu}^{(m)} + T\_{\mu\nu}^{(\phi)} \right),\tag{19}$$

where *Rµν* is the Ricci tensor [69]; *T*˜ (*m*) *µν* and *T* (*φ*) *µν* are the matter (with radiation, which we treat as a radiative fluid [82–86]) and chameleon energy–momentum tensors, respectively, determined by

$$\begin{split} \tilde{T}^{(\mu)}\_{\mu\nu} &= \quad \frac{2}{\sqrt{-\mathfrak{F}}} \frac{\delta}{\delta \hat{\mathcal{g}}^{\mu\nu}} \Big( \sqrt{-\mathfrak{F}} \mathcal{L}[\mathfrak{g}] \Big) = \left( \mathfrak{p} + \mathfrak{p} \right) \mathfrak{u}\_{\mu} \mathfrak{u}\_{\nu} - \mathfrak{p} \, \mathfrak{g}\_{\mu\nu} \\\ T^{(\mathfrak{g})}\_{\mu\nu} &= \quad \frac{2}{\sqrt{-\mathfrak{g}}} \frac{\delta}{\delta \hat{\mathcal{g}}^{\mu\nu}} \Big( \sqrt{-\mathfrak{g}} \mathcal{L}[\mathfrak{g}] \Big) = \frac{\partial \mathfrak{g}}{\partial \mathbf{x}^{\mu}} \frac{\partial \mathfrak{g}}{\partial \mathbf{x}^{\nu}} - \mathfrak{g}\_{\mu\nu} \left( \frac{1}{2} \mathcal{g}^{\lambda\rho} \frac{\partial \mathfrak{g}}{\partial \mathbf{x}^{\lambda}} \frac{\partial \mathfrak{g}}{\partial \mathbf{x}^{\rho}} - V(\mathfrak{g}) \right). \end{split} \tag{20}$$

The factor *f* <sup>2</sup> appears in front of *T*˜ (*m*) *µν* because of the relation

$$\frac{2}{\sqrt{-\mathcal{g}}} \frac{\delta}{\delta \mathcal{g}^{\mu \nu}} \left( \sqrt{-\mathcal{g}} \mathcal{L}\_{\mathfrak{m}}[\mathfrak{g}] \right) = \frac{\sqrt{-\mathcal{g}}}{\sqrt{-\mathcal{g}}} \frac{\delta \mathcal{g}^{\lambda \rho}}{\delta \mathcal{g}^{\mu \nu}} \, \tilde{T}^{(m)}\_{\lambda \rho} = f^2 \, \tilde{T}^{(m)}\_{\mu \nu} \,. \tag{21}$$

where we have used that

$$\frac{\sqrt{-\mathcal{S}}}{\sqrt{-\mathcal{S}}} = f^4 \quad , \quad \frac{\delta \mathfrak{g}^{\lambda \rho}}{\delta g^{\mu \nu}} = f^{-2} \frac{1}{2} (g^{\lambda}{}\_{\mu} g^{\rho}{}\_{\nu} + g^{\lambda}{}\_{\nu} g^{\rho}{}\_{\mu}) , \tag{22}$$

since *g*˜ *λρ* = *f* <sup>−</sup><sup>2</sup> *g λρ* [80] and *T*˜ (*m*) *µν* = *T*˜ (*m*) *νµ* . Then, the quantities *ρ*˜, *p*˜ and *u*˜*<sup>µ</sup>* in the Jordan frame are related to the quantities *ρ*, *p* and *u<sup>µ</sup>* in Einstein's frame as [80]

$$\mathfrak{p} = f^{-3}\mathfrak{p} \quad , \quad \mathfrak{p} = f^{-3}p \quad , \quad \mathfrak{u}\_{\mu} = f \, \mathfrak{u}\_{\mu} \quad , \quad \mathfrak{u}^{\mu} = f^{-1} \, \mathfrak{u}^{\mu} \,. \tag{23}$$

This gives *T*˜ (*m*) *µν* = *f* <sup>−</sup>1*T* (*m*) *µν* . By plugging Equation (20) with *T*˜ (*m*) *µν* = *f* <sup>−</sup>1*T* (*m*) *µν* into Equation (19), we arrive at Einstein's equations

$$R\_{\mu\nu} - \frac{1}{2}g\_{\mu\nu}R = -\frac{1}{M\_{\text{Pl}}^2}T\_{\mu\nu} \tag{24}$$

where *Tµν* is the total energy–momentum tensor equal to

$$T\_{\mu\nu} = \left(\left(\rho + p\right)u\_{\mu}u\_{\nu} - p\,g\_{\mu\nu}\right)e^{\frac{\rho}{2}\phi/M\eta} + \left(\frac{\partial\phi}{\partial x^{\mu}}\frac{\partial\phi}{\partial x^{\nu}} - g\_{\mu\nu}\left(g^{\lambda\rho}\frac{1}{2}\frac{\partial\phi}{\partial x^{\lambda}}\frac{\partial\phi}{\partial x^{\rho}} - V(\phi)\right)\right),\tag{25}$$

where the contribution of torsion *T* (tor) *µν* = *ρ*Λ*gµν* = *M*<sup>2</sup> PlΛ*Cgµν* [39] is included additively in the potential *V*(*φ*) of the self-interactions of the chameleon field. Below we analyze the Einstein equations (Equation (24)) in the cold dark matter (CDM) model [70] in the Friedmann flat spacetime with the line element [69,70]

$$ds^2 = g\_{\mu\nu}(\mathbf{x})d\mathbf{x}^{\mu}d\mathbf{x}^{\nu} = dt^2 + a^2(t)\,\eta\_{ij}dx^idx^j,\tag{26}$$

where *g*00(*x*) = 1 and *gij*(*x*) = *a* 2 (*t*) *ηij* with *ηij* = −*δij*. Then, *a*(*t*) is the expansion parameter of the Universe's evolution [69]. The Christoffel symbols { *α µν*}, the components of the Ricci tensor *Rµν* and the scalar curvature *R* are equal to [69]

$$\begin{array}{rcl} \{\}^{0}\_{00}\} &=& \{\}^{0}\_{0j}\} = \{\}^{i}\_{00}\} = \{\,^{i}\_{kj}\} = 0, \; \{^{0}\_{kj}\} = -a\overline{a}\,\eta\_{kj} \; \; \left\{^{i}\_{0j}\right\} = \frac{\dot{a}}{a}\delta^{i}{}\_{j},\\ R\_{00} &=& 3\frac{\ddot{a}}{a} \; \; \; R\_{0j} = 0 \; \; \; R\_{ij} = \left(\frac{\ddot{a}}{a} + 2\frac{\dot{a}^{2}}{a^{2}}\right) \; \; \; R = 6\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^{2}}{a^{2}}\right) . \end{array} \tag{27}$$

where *η <sup>i</sup>*ℓ*η*ℓ*<sup>j</sup>* = *δ i <sup>j</sup>* and *a*˙ and *a*¨ are first and second derivatives with respect to time.

### **3. Friedmann–Einstein Equations of the Universe's Evolution**

In Friedmann spacetime, Einstein's equations (Equation (24)) define the equations of the Universe's evolution, which are usually called Friedmann's equations (or the Friedmann–Einstein equations) [69]. They are given by

$$\frac{d^2}{d^2} = \frac{1}{3M\_{\text{Pl}}^2} \left( \rho\_{\phi} + (\rho\_r + \rho\_m) f(\phi) \right) \tag{28}$$

and

$$\frac{d\ddot{t}}{dt} = -\frac{1}{6M\_{\text{Pl}}^2} \left(\rho\_{\phi} + 3p\_{\phi} + (\rho\_r + 3p\_r)f(\phi) + \rho\_m f(\phi)\right),\tag{29}$$

where *ρ<sup>r</sup>* and *ρ<sup>m</sup>* are the radiation and matter densities. The scalar field *φ* couples to radiation and matter densities through the conformal factor *f*(*φ*) = *e βφ*/*M*Pl . Then, the radiation density *ρ<sup>r</sup>* and pressure *p<sup>r</sup>* are related by the equation of state *p<sup>r</sup>* = *ρr*/3 [69]. For the description of matter we use the cold dark matter (CDM) model with the pressureless dark and baryon matter [70]. The scalar field density *ρ<sup>φ</sup>* and pressure *p<sup>φ</sup>* are equal to

$$p\_{\Phi} = \frac{1}{2}\dot{\phi}^2 + V(\phi) \quad , \quad p\_{\Phi} = \frac{1}{2}\dot{\phi}^2 - V(\phi). \tag{30}$$

Varying the action Equation (17) with respect to the scalar field *φ* and its derivative one gets the equation of motion for the scalar field [81]. In Friedmann spacetime it reads

$$
\ddot{\phi} + 3\frac{d}{a}\phi + \frac{dV\_{\rm eff}(\phi)}{d\phi} = 0,
\tag{31}
$$

where *V*eff(*φ*) is the effective potential given by

$$V\_{\rm eff}(\phi) = V(\phi) + \rho\_m (f(\phi) - 1). \tag{32}$$

The contribution of the radiation density comes into the effective potential in the form (*ρ<sup>r</sup>* − 3*pr*) *f*(*φ*) − 1 . As for the equation of state *p<sup>r</sup>* = *ρr*/3, such a contribution vanishes. Thus, through the interaction with matter density *ρ<sup>m</sup>* the scalar field can acquire a non-vanishing mass if the effective potential *V*eff(*φ*) obeys the constraints

$$\frac{d V\_{\rm eff}(\phi)}{d\phi}\Big|\_{\phi=\phi\_{\rm min}} = 0 \quad , \quad \frac{d^2 V\_{\rm eff}(\phi)}{d\phi^2}\Big|\_{\phi=\phi\_{\rm min}} > 0,\tag{33}$$

i.e., the effective potential *V*eff(*φ*) possesses a minimum at *φ* = *φ*min. An important role for a dependence of a chameleon field mass on a density of an environment is the conformal factor *f*(*φ*) and its deviation from unity.

## *3.1. Bianchi Identity, Conservation of Total Energy–Momentum Tensor and Conformal Factor*

By using Equation (27) and taking into account that in the Friedmann flat spacetime the non-vanishing components of the Einstein tensor *G µν* = *R µν* <sup>−</sup> <sup>1</sup> 2 *g µνR* are equal to

$$G^{00} = -3\frac{d^2}{a^2} \quad , \quad G^{ij} = \left(-2\frac{d}{a} - \frac{d^2}{a^2}\right) \mathbf{g}^{ij} \tag{34}$$

one may show that Einstein's tensor *Gµν* obeys the Bianchi identity [69]

$$\mathcal{G}^{\mu\nu}{}\_{;\mu} = \frac{1}{\sqrt{-\mathcal{g}}} \frac{\partial}{\partial \mathbf{x}^{\rho}} \left( \sqrt{-\mathcal{g}} \, \mathcal{G}^{\rho\nu} \right) + \Gamma^{\nu}{}\_{\mu\rho} \mathcal{G}^{\mu\rho} = \mathbf{0},\tag{35}$$

where *G µν* ;*<sup>µ</sup>* is a covariant divergence and Γ *ν µρ* = { *ν µρ*} are the Christoffel symbols [69]. As a result, the covariant divergence of the total energy–momentum tensor *T µν* ;*<sup>µ</sup>* should also vanish

$$T^{\mu\nu}\_{\ \ j\mu} = \frac{1}{\sqrt{-g}} \frac{\partial}{\partial \mathbf{x}^{\rho}} \left(\sqrt{-g} \, T^{\rho\nu}\right) + \Gamma^{\nu}\_{\ \mu\rho} T^{\mu\rho} = 0. \tag{36}$$

Due to time-dependence only Equation (31) takes the form

$$\frac{1}{\sqrt{-\mathcal{g}}} \frac{\partial}{\partial t} \left( \sqrt{-\mathcal{g}} \, T^{00} \right) + \Gamma^{0}\_{\,\,ij} T^{ij} = 0,\tag{37}$$

where we have taken into account Equation (27). Using the non-vanishing components of the total energy momentum tensor

$$T^{00} = \rho\_{\Phi} + (\rho\_r + \rho\_m)f(\phi) \quad , \quad T^{ij} = -\left(p\_{\Phi} + p\_{l}f(\phi)\right)g^{ij} \tag{38}$$

we transcribe Equation (32) into the form

$$\frac{d}{dt}\left(\rho\_{\phi} + \left(\rho\_{r} + \rho\_{\text{\textquotedblleft}}\right)f(\phi)\right) + 3\frac{d}{a}\left(\rho\_{\phi} + p\_{\phi} + \left(\rho\_{r} + p\_{r}\right)f(\phi) + \rho\_{\text{\textquotedblleft}}f(\phi)\right) = 0. \tag{39}$$

Since Equation (31) can be rewritten as follows:

$$\frac{d}{dt}\left(\rho\_{\phi} + \rho\_{m}f(\phi)\right) = \frac{d}{dt}\rho\_{m} - \Im \frac{d}{a}\left(\rho\_{\phi} + p\_{\phi}\right),\tag{40}$$

we may remove the contribution of the chameleon field in Equation (39). As result, we get

$$\frac{d}{dt}\left(\rho\_r f(\phi) + \rho\_m\right) + \frac{d}{a}\left(4\rho\_r f(\phi) + 3\rho\_m f(\phi)\right) = 0,\tag{41}$$

where we have used the equation of state *p<sup>r</sup>* = *ρr*/3 [69]. Due to independence of radiation and matter densities, Equation (41) can be split into evolution equations of the radiation and matter densities:

$$\begin{aligned} \frac{d}{dt} \left( \rho\_r f(\phi) \right) + 4 \frac{d}{a} \left( \rho\_r f(\phi) \right) &= \quad 0, \\\frac{d}{dt} \rho\_m + 3 \frac{d}{a} \rho\_m f(\phi) &= \quad 0. \end{aligned} \tag{42}$$

For the standard dependence of the radiation and matter densities on the expansion parameter *a*(*t*) [69],

$$
\rho\_r = 3M\_{\rm Pl}^2 \mathcal{H}\_0^2 \Omega\_r \frac{a\_0^4}{a^4}, \quad \rho\_m = 3M\_{\rm Pl}^2 \mathcal{H}\_0^2 \Omega\_m \frac{a\_0^3}{a^3}, \tag{43}
$$

where *<sup>a</sup>*0, H<sup>0</sup> <sup>=</sup> 1.438(11) <sup>×</sup> <sup>10</sup>−<sup>33</sup> eV, <sup>Ω</sup>*<sup>r</sup>* and <sup>Ω</sup>*<sup>m</sup>* are the expansion parameter, the Hubble rate and the relative radiation and matter densities at our time *t*<sup>0</sup> = 1/H<sup>0</sup> [70], the equations for the radiation and matter densities Equation (42) are satisfied identically for *f*(*φ*) = 1.

Thus, if the radiation and matter densities depend on the expansion parameter *a* as *ρr*(*a*) ∼ *a* −4 and *ρm*(*a*) ∼ *a* −3 , local conservation of the total energy–momentum in the Universe can be fulfilled if and only if the conformal factor *f*(*φ*), relating Einstein's and Jordan's frames and defining the chameleon–matter coupling, is equal to unity; i.e., *f*(*φ*) = 1. However, in this case there is no influence of the chameleon field on the evolution of the radiation and matter densities and a dependence of the chameleon field mass on a density of its environment. In turn, for *f*(*φ*) 6= 1 the evolution equations (Equation (37)) admit some exact solutions. It is convenient to search these solutions independently of the expansion parameter *a*. Treating the conformal factor *f*(*φ*) as a function of the expansion parameter *a*, i.e., setting *f*(*φ*) = *f*(*a*) 6= 1, the solutions to Equation (37) can be given by

$$\begin{array}{rcl}\rho\_{\mathcal{I}}(a) &=& \rho\_{r0}\frac{a\_0^4}{a^4}\frac{f(a\_0)}{f(a)},\\\rho\_{\mathcal{m}}(a) &=& \rho\_{\mathcal{m}0}\frac{a\_0^3}{a^3}\exp\left(3\int\_a^{a\_0}\frac{f(a')-1}{a'}da'\right),\end{array} \tag{44}$$

where *ρr*<sup>0</sup> = 3*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω*<sup>r</sup>* and *<sup>ρ</sup>m*<sup>0</sup> = 3*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω*<sup>m</sup>* are the radiation and matter densities at out time *t*<sup>0</sup> = 1/H<sup>0</sup> and *a*(*t*0) = *a*0, i.e., in the era of the late-time acceleration of the Universe's expansion or the dark energy–dominated era. The integration constants of the first order differential equations (Equation (35)) are fixed by the conditions *ρr*(*a*0) = *ρr*<sup>0</sup> and *ρm*(*a*0) = *ρm*0, respectively [69,70]. According to the solutions (Equation (44)), the chameleon field has an influence on the evolution of the radiation and matter densities.

As an example of the conformal factor we may use *f* = *e βϕ*(*a*)/*M*Pl [1,2], where *ϕ*(*a*) is the chameleon field as a function of the expansion parameter *a* and the solution to Equation (31), i.e., *φ*(*t*) = *ϕ*(*a*). Keeping the linear order contributions in the *βϕ*(*a*)/*M*Pl expansion we get

$$\rho\_r(a) \quad = \quad \rho\_{r0} \frac{a\_0^4}{a^4} \left( 1 + \frac{\beta}{M\_{\rm Pl}} (\varphi(a\_0) - \varphi(a)) \right),$$

$$\rho\_m(a) \quad = \quad \rho\_{m0} \frac{a\_0^3}{a^3} \left( 1 + 3 \frac{\beta}{M\_{\rm Pl}} \int\_a^{a\_0} \varphi(a') \frac{da'}{a'} \right). \tag{45}$$

Thus, the deviations of the radiation and matter densities from their standard behavior *ρr*(*a*) ∼ *a* −4 and *ρm*(*a*) ∼ *a* <sup>−</sup><sup>3</sup> are given by

$$
\delta\rho\_r(a) = \begin{array}{c c c} \frac{\beta}{M\_{\text{Pl}}} \rho\_{r0} \frac{a\_0^4}{a^4} \left(\varphi(a\_0) - \varphi(a)\right), \\ \delta\rho\_m(a) = \Im \frac{\beta}{M\_{\text{Pl}}} \rho\_{m0} \frac{a\_0^3}{a^3} \int\_a^{a\_0} \varphi(a') \frac{da'}{a'}. \end{array} \tag{46}$$

Some observations of deviations of the radiation and matter densities in the Universe from their standard form might, in principle, evidence an existence of the chameleon field. Nevertheless, we have to emphasize that the contributions of the conformal factor to the radiation and matter densities at our time are not practically observable. It is seen from the solutions (Equation (44)) that the conformal factor affects the evolution of the radiation and matter densities during the radiation and matter-dominated eras only. Of course, an influence of the chameleon field evolution on the distribution of the radiation density might seem rather questionable, since the evolution equation (Equation (37)) defines an evolution of the product *ρ<sup>r</sup> f*(*φ*), where one may hardly separate *ρ<sup>r</sup>* from *f*(*φ*). By introducing an effective radiation density *ρ* (eff) *<sup>r</sup>* = *ρ<sup>r</sup> f*(*φ*) we obtain a canonical radiation density Equation (43), where the contribution of *f*(*φ*) at *a* = *a*<sup>0</sup> is hidden very likely in Ω*<sup>r</sup>* .
