The action functional of our model takes the form

$$S=\int {{d^4}x} \sqrt { - g} \left[ {\frac{1}{2}\left( {\Phi R - \frac{\omega }{\Phi }{\Phi _{,\mu }}{\Phi ^{,\mu }}} \right) - \frac{1}{2}\left( {{\phi _{,\mu }}{\phi ^{,\mu }}+2V(\phi )} \right)+{L_b}\left( {{\psi _b};{g_{\mu \nu }}} \right)+C(\phi )L_{c}^{{(0)}}\left( {{\psi _c};{g_{\mu \nu }}} \right)} \right]$$

1

Where we set \(8\pi G=1\)and \({G_N}\)is the Newton constant, therefore scalar field\(\phi\) is dimensionless. The matter Laglangian consist of two components: one is for the baryon matter which is not coupled to scalar field and another component is dark matter which is coupled to scalar because the weak equivalence principle is valid for the usual baryon matter. Coupling function \(C(\phi )\)represents on explicit direct interaction of scalar with dark matter. \({\psi _b}\) and \({\psi _c}\) represent the collective fields of baryons and dark matters, respectively. The variations of action (1) with respect to \({g^{\mu \nu }}\)and \(\phi\)give following dynamical equations

$$\left( {{R_{\mu \nu }} - \frac{1}{2}{g_{\mu \nu }}R} \right)\phi =C(\phi )T_{{\mu \nu (c)}}^{{(0)}}+{T_{\mu \nu (b)}}+{T_{\mu \nu (\phi )}}$$

2

$$\left( {2\omega (\phi )+3} \right)\square \phi =\left( {C(\phi ) - 2\frac{{dC(\phi )}}{{d\phi }}\phi } \right)T_{c}^{{(0)}}+{T_b} - \frac{{d\omega (\phi )}}{{d\phi }}{(\nabla \phi )^2} - 4V(\phi )+2\frac{{dV(\phi )}}{{d\phi }}\phi$$

3

Where energy-momentum tensor of scalar field is given by

$${T_{\mu \nu (\phi )}}=\frac{{\omega (\phi )}}{\phi }\left( {{\phi _{,\mu }}{\phi _{,\nu }} - \frac{1}{2}{g_{\mu \nu }}{{(\nabla \phi )}^2}} \right)+\left( {{\phi _{,\mu ;\nu }} - {g_{\mu \nu }}\square \phi } \right) - {g_{\mu \nu }}V(\phi )$$

4

Where \(\square \equiv \frac{1}{{\sqrt g }}{\partial _\mu }\left( { - \sqrt g } \right)\), the matter energy-momentum tensor is determined by\(T_{{\mu \nu (c)}}^{{(0)}}=\frac{2}{{\sqrt g }}\frac{{\partial \sqrt g L_{c}^{{(0)}}}}{{\partial {g^{\mu \nu }}}}\)and \(T_{c}^{{(0)}}=T_{{\mu (c)}}^{{(0)}}.\)

Taking covariant derivative of Eq. (2) one finds. The energy-momentum conservation equations for baryon and for cold dark matter + scalar field, respectively, as follows

$$T_{{\mu (c);\nu }}^{\nu }=0$$

4

$$T_{{\mu (c);\nu }}^{\nu }+T_{{\mu (\phi );\nu }}^{\nu }=0$$

5

Where \(T_{{\mu \nu (c)}}^{\nu }=C(\phi )T_{{\mu \nu (c)}}^{{(0)}}\). Because cold dark matter and scalar field exchange the energy, cold dark matter and scalar field do not conserve the energy separately and the conservation equation for dark matter reads [9].

$$T_{{\mu (c);\nu }}^{\nu }=\frac{{d\ln C(\phi )}}{{d\phi }}{T_c}{\phi _{,\mu }}=\frac{{dC(\phi )}}{{d\phi }}T_{c}^{{(0)}}{\phi _{,\mu }}$$

6

Where \({T_c}=T_{{\mu (c)}}^{\mu }\). We apply the Eqs. (2) and (3) to a flat FRW universe where metric is given by

$$d{s^2}= - d{t^2}+{a^2}(t)d{x^2}.$$

7

Matter will be described by perfect fluids for both baryon and cold dark matter. Then dynamical equations derived from Eqs. (2) and (3) read

$$3\phi {H^2}=C(\phi )\rho _{c}^{{(0)}}+{\rho _b}+\frac{1}{2}\frac{{\omega (\phi )}}{\phi }{\dot {\phi }^2}+V(\phi ) - 3H\dot {\phi }$$

8

$$- \phi \dot {H}=\frac{1}{2}\left[ {C(\phi )\rho _{c}^{{(0)}}+{\rho _b}+\frac{{\omega (\phi )}}{\phi }{{\dot {\phi }}^2}+\ddot {\phi } - H\dot {\phi }} \right]$$

9

$$\left( {\ddot {\phi }+3H\dot {\phi }} \right)\frac{{\omega (\phi )}}{\phi }=3(\dot {H}+2{H^2}) - \frac{{d\omega (\phi )}}{{d\phi }}\frac{{{{\dot {\phi }}^2}}}{{2\phi }}+\frac{{\omega (\phi )}}{2}{\left( {\frac{{\dot {\phi }}}{\phi }} \right)^2} - \frac{{dV(\phi )}}{{d\phi }} - \frac{{dC(\phi )}}{{d\phi }}\rho _{c}^{{(0)}}$$

10

Where superscript (0) stands for the quantities corresponding to the couplingless Laglangian \(L_{c}^{{(0)}}\)for cold dark matter in action (1).In the original Brans-Dicke theory of gravity without potential \(V(\phi )\), scalar field evolves according to relation [10, 11].

$$\phi (z)={(1+z)^{ - \frac{1}{{1+\omega }}}}$$

11

Our model (1) however, is different from the original Brans-Dicke theory, in that is additionally contains a self-interacting potential \(V(\phi )\)and the scalar-matter coupling \(C(\phi )L_{c}^{{(0)}}\). Therefore we take the evolution of scalar field, generalizing Eq. (12) as following form

$$\phi (z)={(1+z)^{ - \alpha (z)}}.$$

12

Where \(\alpha (z)\) is generally a function of redshift to bedetermined. In modern cosmology, dark energy as a scalar field coupled with cold dark matter comes into fashion, as it seems to resolve so called “coincidence” problem [12, 13, 14]. Therefore we include in our model (1) such a coupling. And then as we mentioned in Sect. 1, the mass of scalar in the scalar-tensor gravity, if any plays important role in behavior of Brans-Dicke parameter \(\omega\)[8, 2], we consider self-interacting. Potential which determines the mass of scalar field \(m_{s}^{2}\). As Ref. [9] has shown the conservation equation of dark matter from Eq. (7)

$${\dot {\rho }_c}(\phi )+3H{\rho _c}(\phi )=\frac{{d\ln C(\phi )}}{{dt}}{\rho _c}(\phi )=\delta H{\rho _c}(\phi )$$

13

yields

$$C\sim {a^\delta } \Rightarrow C={C_0}{(1+z)^{ - \delta }}.$$

14

Where \(\delta\) is a constant and \({C_0}\) is value of coupling function at \(z=0\). Therefore matter densities evolves as follows

$${\rho _c}=C\rho _{c}^{{(0)}}={\rho _{{c_0}}}{(1+z)^{3 - \delta }}$$

15

$${\rho _b}={\rho _{{b_0}}}{(1+z)^3}$$

16

The conservation equation for scalar field is expressed as follows [9]

$${\dot {\rho }_c}(\phi )+3H\left( {{\rho _\phi }+{P_\phi }} \right)=\frac{{d\ln C(\phi )}}{{d\phi }}\dot {\phi }{\rho _c}$$

17

Substituting relation (16) into r. h. s. of Eq. (17), can obtain the energy density of scalar field resolving Eq. (17) with respect to \({\rho _\phi }\)where \({P_\phi }={\omega _\phi }{\rho _\phi }\)and \({\omega _\phi }\)is equation of state parameter of scalar field. It reads

$${\rho _\phi }={\rho _{{\phi _0}}}{(1+z)^{3(1+{\omega _\phi })}}+\frac{{\delta {\rho _{{c_0}}}\left[ {{{\left( {1+z} \right)}^{3(1+{\omega _\phi })}} - {{(1+z)}^{3 - \delta }}} \right]}}{{\delta +3{\omega _\phi }}}.$$

18

Then Friedman equation

$$3{H^2}\phi =C(\phi )\rho _{c}^{{(0)}}+{\rho _b}+{\rho _\phi }.$$

19

Making use of Eqs. (16), (17) and (19), is written as follows

$${H^2}(z)=H_{0}^{2}{(1+z)^\alpha }\left\{ {\left( {1 - {\Omega _c} - {\Omega _b}} \right){{\left( {1+z} \right)}^{3(1+{\omega _\phi })}}+{\Omega _b}{{(1+z)}^3}+{\Omega _c}\left[ {\frac{{3{\omega _\phi }}}{{\delta +3{\omega _\phi }}}{{(1+z)}^{3 - \delta }}+\frac{\delta }{{\delta +3{\omega _\phi }}}{{(1+z)}^{3(1+{\omega _\phi })}}} \right]} \right\}.$$

20

Here, the parameters \({\Omega _c}\)and \({\Omega _b}\)are defined as follows

$$\left. {\begin{array}{*{20}{c}} {{\Omega _c} \equiv \frac{{{\rho _{{c_0}}}}}{{3{\phi _0}H_{0}^{2}}}} \\ {{\Omega _b} \equiv \frac{{{\rho _{{b_0}}}}}{{3{\phi _0}H_{0}^{2}}}} \end{array}} \right\}.$$

21

Where \({\rho _{{c_0}}}\)and \({\rho _{{b_0}}}\)are current values of dark matter and baryon matter densities. In Eq. (21), parameters to be determined are \({\Omega _c},{\Omega _b},{\omega _\phi },\alpha\)and\(\delta\). Our concern, however, is focused on the determination of the Brans-Dicke parameter\(\omega\), so we must find another equation containing the parameter\(\omega\). To that end, we use scalar field Eq. (11), but Eq. (11) includes potential function, we utilize Friedman Eq. (9) containing \(V(\phi )\). Differentiating Eq. (9) with respect to \(\phi\)we obtain expression.

$$\frac{{dV(\phi )}}{{d\phi }}=3{H^2}+3{H^2}\left[ {\alpha +(1+z)\ln (1+z)\frac{{d\alpha }}{{dz}}} \right] - \frac{\omega }{2}{H^2}{\left[ {\alpha +(1+z)\ln (1+z)\frac{{d\alpha }}{{dz}}} \right]^2} - \frac{{dC(\phi )}}{{d\phi }}\rho _{c}^{{(0)}}$$

21

Here we used a relation derived from Eq. (13)

$$\dot {\phi }=\phi H\left[ {\alpha +(1+z)\ln (1+z)\frac{{d\alpha }}{{dz}}} \right]$$

22

Where \(\frac{d}{{dt}}= - (1+z)H\frac{d}{{dz}}\)is used. Substituting the expression (22) into Eq. (11) for the scalar field we obtain a differential equation on the Hubble parameter \({H^2}(z)\)as follows

$$\begin{gathered} \frac{1}{2}\left\{ {2 - \omega (1+z)\left[ {\alpha +(1+z)\ln (1+z)\alpha ^{\prime}} \right]} \right\}\frac{{d{H^2}(z)}}{{dz}}= \hfill \\ =\left\{ {\omega (1+z)\left[ {2\alpha ^{\prime}+\alpha ^{\prime}\ln (1+z)+\alpha ^{\prime\prime}\ln (1+z)} \right]+3 - 3(\omega +1)\left[ {\alpha +(1+z)\ln (1+z)\alpha ^{\prime}} \right]+\frac{1}{2}\omega ^{\prime}(1+z)\left[ {\alpha +(1+z)\ln (1+z)\alpha ^{\prime}} \right]} \right\}{H^2}(z). \hfill \\ \end{gathered}$$

23

Where \(\alpha ^{\prime}=\frac{{d\alpha }}{{dz}}\)and\(\omega ^{\prime}=\frac{{d\omega }}{{dz}}\). Above, we assumed the parameter \(\omega\)to be a function of redshift. If we assume the parameters \(\alpha\)and\(\omega\) to be constants, then Eq. (23) reduces to a simple equation after integration.

$${H^2}(z)=H_{0}^{2}{(1+z)^{\frac{{6(1 - \alpha - \omega \alpha )}}{{3 - \omega \alpha }}}}$$

24

As the power in Eq. (24) is a constant, putting

$$A \equiv \frac{{1 - \alpha - \omega \alpha }}{{3 - \omega \alpha }}.$$

25

We can get the Brans-Dicke parameter as follows

$$\omega =\frac{{1 - \alpha - 3A}}{{\alpha (1 - A)}}$$

26

if the constant A is known. In the deriving Eq. (25), we can use the slow-rall approximation \(\ddot {\phi }<<2H\dot {\phi }\). Then the scalar field equation becomes

$$\frac{3}{2}(1+z)\frac{{d{H^2}(z)}}{{dz}}=\left\{ {3 - \left[ {3+3\omega - \omega ^{\prime}(1+z)} \right]\left[ {\alpha +(1+z)\ln (1+z)\alpha ^{\prime}} \right]+\omega {{\left[ {\alpha +(1+z)\ln (1+z)\alpha ^{\prime}} \right]}^2}} \right\}{H^2}(z)$$

27

In the case of constant \(\alpha\)and\(\omega\), the Eq. (27), after integration reduces to a simple equation.

$${H^2}(z)=H_{0}^{2}{(1+z)^{2\left[ {(1 - \alpha - \omega \alpha )+\frac{1}{3}\omega {\alpha ^2}} \right]}}$$

28

Putting

$$B \equiv (1 - \alpha - \omega \alpha )+\frac{{\omega {\alpha ^2}}}{3}.$$

29

The unknown parameter \(\omega\)is a determined through

$$\omega =\frac{{(1 - \alpha ) - B}}{{\alpha (1 - \frac{\alpha }{3})}}.$$

30