Brans-Dicke gravity with scalar matter coupling and its cosmological probe

Brans-Dicke gravity with scalar matter coupling and its cosmological probe | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Brans-Dicke gravity with scalar matter coupling and its cosmological probe SongChol Ri, IlMyong Yun, JikSu Kim, Ryong Gwang Kim This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3890256/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this work we investigate the Brans-Dicke gravity with self-interacting potential and potentialess model, the scalar field depends on redshift. But our model includes both potential and coupling, so we put the redshift dependence of the scalar in other form. Making use of CMB, BAO and SNIa observational data, we probe the model and determine model parameters, but our main concern is the dependence of model parameters on the temporal and spatial scales. Therefore we separate the probes for high(CMB) and low (BAO + SN) redshift observational data. gravity scalar 1. Introduction One of the biggest problems in modern cosmology in our opinion, is whetherthe expansion dynamics of the universe is governing by the Einstein tensor gravity or the scalar-tensor one. After the Brans-Dicke scalar-tensor theory was formulated a plethora of works has been dedicated to the test of theory in scalar system and in cosmological scale and to the elucidation of the true implication of the scalar field. Nevertheless for example the Brans-Dicke coupling parameter \(\omega\) is being evaluated with tremendous tension hitherto. The measurement of the frequency shift of radiowaves to and from Cossini spacecraft asit passed near the sun yielded \(\gamma =1+(2.1 \pm 2.3) \times {10^{ - 5}}\) ,which was equivalent to Brans-Dicke parameter, \(\omega >40000\) [ 1 , 2 ]. Observations of Nordtvedt effect using the Lunar Laser Ranging experiment yielded a slightly smaller value \(\omega >1000\) [ 2 , 3 ]. Observations of orbital period derivative of the aircular white dwarf-neutron star binary system PSR J1012 + 5307 yielded \(\omega >1250\) [ 2 ]. On the other hand, the recent estimations of the parameter yield ever negative values. Ref[ 4 ] obtained \(- 407.0<\omega <175.89\) at 95% confidence level. Ref[ 5 ] using SNIa data obtained a best fit value \(\omega = - 1.477\) . Ref[ 6 ] for three different models found \(\omega = - 0.9782\) , \(\omega = - 1.0646\) and \(\omega = - 1.7817\) respectively. Where their explanation of such a wide range of estimation of the parameter \(\omega\) from minus a few hundreds to a few tens of thousand should be found? First, it might be associated with whether the model one used was appropriate for description of the real Universe. Such a requirement makes it necessity construct a model as more inclusive as possible so as to be close to the observed Universe. Second, it may be associated with temporal evolution of the Universe and the difference of the spatial scales for the observational data. As Ref. [ 7 ] has pointed out the cosmological constraint on Brans-Dicke theory is different from the solar systems constraint since they concern very different spatial and temporal scales. For example the solar system experiment concerns smaller scale but curvature there is higher than in cosmological scale and this could lead to screen scalar and as a result the metric gravity (GR) plays the leading role. Perivolaropoulos [ 8 ] considered Brans-Dicke gravity with massive scalar field and obtained a modified expression on PPN parameter \(\gamma\) for that theory. It reads \(\gamma (\omega ,{m_s},r)=\frac{{1 - \frac{{{e^{ - \frac{{\sqrt {2{\phi _0}} }}{{\sqrt {2\omega +3} }}{m_s}r}}}}{{2\omega +3}}}}{{1+\frac{{{e^{ - \frac{{\sqrt {2{\phi _0}} }}{{\sqrt {2\omega +3} }}{m_s}r}}}}{{2\omega +3}}}}.\) Where \({m_s}\) is mass of the massive scalar field, \({\phi _0}\) homogeneous background field of scalar and r- the scale performing the experimental observation. The relation(1) is different from the relation of the conventional Brans-Dicke theory with massless scalar \(\gamma =\frac{{1+\omega }}{{2+\omega }}\) .PPN-parameter \(\gamma (\omega ,{m_s},r)\) expressed by relation (1) shows that parameter \(\gamma\) , in the case of massive scalar, depends not only on the mass of scalar itself, but also on what scale the experiment is performed on. Above-mentioned variety of the Brans-Dicke parameter \(\omega\) may be explained by the relation (1) partly. But the relation (1) was derived under the assumption that the temporal evolution was neglected. However our main concern is the cosmological temporal evolution of the model parameters for long period of the evolution of the Universe. Therefore making use of current observational data we first calculate the model parameters for the high (CMB) and low (BAO + SN) redshift. Sect 2 describes the basic equations of the cosmological model to be investigated. Sect 3 probes the model using observation data and Sect 4 summarizes the results. 2. Basic equation 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 3. Probe of the model with observations In this work, our main concern is focused on the behavior of the Brans-Dicke parameter \(\omega\) with time and spatial scale in the scalar-tensor theory of gravity. Therefore, in the utilization of cosmological observational data we do not mix the data corresponding to different cosmological epochs and spatial scales. For example, Cosmic Microwave Background (CMB) contains the information on the rate of expansion at redshift \(z\sim 1100\) and baryon Acoustic Oscillation (BAO) includes the information at low redshifts \(z \leqslant 1\sim 3\) . The observational data of SNIa come also from the low redshifts. Therefore, in our probes, we separate CMB data from BAO and SNIa data. 3.1 CMB As CMB data, we use the acoustic scale $${l_A}=\pi \frac{{r({z_*})}}{{{r_s}({z_*})}}$$ 31 which is defined in Ref. [ 15 ]. Here the commoving distance \(r(z)\) is defined $$r({z_{}})=\frac{c}{{{H_0}}}\int\limits_{0}^{{{z_*}}} {\frac{{dz^{\prime}}}{{E(z^{\prime})}}}$$ 32 And the commoving sound horizon distance at recombination \(z={z_*} \approx 1091\) is given by $${r_s}({z_*})=\int\limits_{{{z_*}}}^{\infty } {\frac{{{c_s}(z^{\prime})}}{{H(z^{\prime})}}} dz^{\prime}.$$ 33 Where the speed of sound is as follows $${c_s}(z)=c{\left[ {3\left( {1+\frac{{3{\Omega _b}}}{{4{\Omega _\gamma }}}\frac{1}{{1+z}}} \right)} \right]^{ - \frac{1}{2}}}$$ 34 $$E(z)={\raise0.7ex\hbox{${H(z)}$} \!\mathord{\left/ {\vphantom {{H(z)} {{H_0}}}}\right.\kern-0pt}\!\lower0.7ex\hbox{${{H_0}}$}}$$ . The seven year WMAP observations give \({\Omega _\gamma }=2.469 \times {10^{ - 5}}{h^{ - 2}}\) and \({\Omega _\gamma }=0.02258\) [ 16 ]. In our analysis, the used acoustic scales \({l_A}({z_*})\) are given in Table 1 . Table 1 Observed acoustic scale \({l_A}\) Reference 301.57 [ 17 ] 302.06 [ 18 , 19 ] 302.35 [ 20 ] 3.2 BAO As the second observation data BAO, we use BAO distance ratio $$d(z)=\frac{{{r_s}({z_d})}}{{{D_\nu }(z)}}$$ 35 where \({r_s}({z_d})\) is the commoving sound horizon distance at the drag epoch \(z={z_d}=1021\) and the so-called lilaton scale \({D_\nu }\) is defined by $${D_\nu }(z)={\left[ {{{\left( {\int {\frac{{dz^{\prime}}}{{E(z^{\prime})}}} } \right)}^2}\frac{z}{{E(z)}}} \right]^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt}\!\lower0.7ex\hbox{$3$}}}}\frac{C}{{{H_0}}}.$$ 36 In Table 2 , the BAO distance ratios are presented which is cited from Ref. [ 21 ]. Table 2 Observed BAO distance ratio Redshift, z BAO distance ratio \(d(z)=\frac{{{r_s}({z_d})}}{{{D_\nu }(z)}}\) 0.106 0.336 \(\pm\) 0.015 0.20 0.1905 \(\pm 0.0061\) 0.275 0.1390 \(\pm 0.0037\) 0.278 0.1389 \(\pm\) 0.0043 0.314 0.1239 \(\pm\) 0.0033 0.35 0.1126 \(\pm\) 0.0022 0.44 0.0916 \(\pm\) 0.0071 0.57 0.0732 \(\pm\) 0.0012 0.60 0.0726 \(\pm\) 0.0034 0.73 0.0592 \(\pm\) 0.0032 2.34 0.0320 \(\pm\) 0.0007 3.3 SNIa The third observation data are SNIa Union 2.The observed quantity is the distance modulus determines through the luminosity distance $$\mu (z)=5\lg {D_L}(z)+42.38 - 5\lg h - \frac{{15}}{4}\lg \Phi (z)$$ 37 where \({H_0}=100h{\raise0.7ex\hbox{${km}$} \!\mathord{\left/ {\vphantom {{km} {Mpc}}}\right.\kern-0pt}\!\lower0.7ex\hbox{${Mpc}$}}\) and \({D_L}(z)\) is dimensionless luminosity distance defined by relation $${D_L}(z)=(1+z)\int\limits_{0}^{z} {\frac{{dz^{\prime}}}{{E(z^{\prime})}}} .$$ 38 In Eq. a term containing scalar field is associated with fact that the peak luminosity of SNIa varies like \(L\sim {G^{ - \frac{3}{2}}}\) and corresponding absolute magnitude of SNIa evolves like $$M - {M_0}=\frac{{15}}{4}\lg \frac{G}{{{G_0}}}$$ 39 In the Brans-Dicke theory of gravity, \(\phi \sim {G^{ - 1}}\) and the scalar field term in Eq. ( 1 ) stands for this fact. We use “Union 2” data for SNIa. 4 Conclusion As we can see in Table 3, the coupling less model yields negative value of \(\alpha\) and \(\omega\) in the coupling less case he relation $$\alpha =\frac{1}{{1+\omega }}.$$ 40 Is satisfied, so we can estimate the parameter \(\omega\) from the value of \(\alpha\) , immediately. The values of \(\omega\) are − 7.2406 and − 10.4118 for CMB and BAO + SN, respectively. To evade the ghost field, the parameter \(\omega\) must satisfy \(\omega > - \frac{3}{2}\) . Therefore, the above values \(\omega < - \frac{3}{2}\) are thought to be unphysical and we delete them. This unphysical result are considered to rise from the assumption for the scalar-matter coupling to be absent. Hence, we recalculate the parameters of our model including the scalar matter coupling.Result yields a positive value of \(\alpha\) and positive one of \(\omega\) . It is \(\omega =59000\) .The scalar-matter coupling parameter is negative \(\delta = - 0.0099\) .Our estimation of the parameter \(\omega\) is more than the Cassini-experiment’s result. On the other hand, the fact that the scalar-matter coupling, parameter \(\delta\) is negative, implies a flow of energy from matter to scalar as the Eq. ( 16 ) shows. Our probe is preliminary because the above results are relying on the combined data of CMB + BAO + SN. The separate probes for the high and low redshift data will give more interesting results. Declarations Author Contribution SongChol Ri: Supervision. IlMyong Yun: Software. JikSu Kim: Conceptualization, Methodology.RyongGwang Kim: Writing – original draft, Funding acquisition. References Berttoti, B., et al.: Nature, 425, 374 Alsing, J., et al.: arXiv:1182.4903[gr-gc] Williams, J.G., et al.: Int. J. Mod. Phy. D 13567 (2004) Li, Y. C., et al.: Phy. Rev. D 88, 084053 (2013) Fabris, J. C., et al.: Grav. Cosmol., 12, 49 (2001) Hrycyna, O., et al.: arkiv:1404.7112 [astro-ph.CO] Avilez, A. and Skordis, C.: arkiv:1303.4330 [astro-ph.CO] Perivolaropoulos, L.: Phys. Rev. D 81, 047501 (2010) Jik Su, Kim, et al.: Phys. Rev. D 96, 043507 (2017) Brans, C., et al.: Phys. Rev. D 124, 925 (1961) Weinberg, S.: Gravitation and cosmology, (1972) Amendola, L., et al.: Phys. Rev. D 74, 023525 (2006) Majerotto, E., et al.: arXiv:astro-ph/0410543 Amendola, L.: Phys. Rev. D 60, 043501 (1999) Bond, J., et al.: MNRAS, 291, L 33 (1997) Komatsu, E., et al.: Ap. J. Suppl. 192, 18 (2011) Ade, P. A. R., et al.: [Planck Collaboration] Planck (2014) Wang, Y., et al.: Phys. Rev. D 85, 023517 (2012) Wang, A., et al.: Phys. Rev. D 88, 043522 (2013) Bennet, C. L., et al.: [WMAP Collaboration] Ap. J. Suppl. 208, 20 (2013) Zhang, Y.: arXiv:1411.5522 [astro-ph.CO] Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3890256","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":268826824,"identity":"26527045-5d49-44b8-b217-7006f2656aa3","order_by":0,"name":"SongChol Ri","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"SongChol","middleName":"","lastName":"Ri","suffix":""},{"id":268826825,"identity":"feee7d0a-203b-4986-b628-01a4f278459a","order_by":1,"name":"IlMyong Yun","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"IlMyong","middleName":"","lastName":"Yun","suffix":""},{"id":268826826,"identity":"05c06b1c-1c7a-45cb-90ff-c175ae03e7bd","order_by":2,"name":"JikSu Kim","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"JikSu","middleName":"","lastName":"Kim","suffix":""},{"id":268826827,"identity":"dd295972-4560-4f7e-a28a-3309159fe6ef","order_by":3,"name":"Ryong Gwang Kim","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVRIiWNgGAWjYBACCWYQaWAjx8/egCKRQEBLQZqxZM8BRoSmA/i0gMkPhxM33EggUotkO3faxy8Gh40lZz5//uDnHgZ5fgbeg48/MKTl4dIizcy7ebaMQbocv3SOYWPPMwbDmQ18yQYHGHKKcWmRA2phljCwNpacncPYwHPgf4LBAR4ziQMMFYkN+LUwJ264efxh4x+gH4BazH/g0wJyGOMHA2eg9xkMm3kgWsyA3s/BqUWyGWgLgwEokHMMZ8scAPqlmcdY4oxBGk6/SJw/u5nxxx9QVB5/8PHNAWCIsfcYfqioSMYZYiDAzIPKBREGeCIGCBh/YBPFq2UUjIJRMApGFAAAYB9VmkRUXvMAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Ryong","middleName":"Gwang","lastName":"Kim","suffix":""}],"badges":[],"createdAt":"2024-01-23 07:29:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3890256/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3890256/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":50177630,"identity":"343ed3a8-7057-404b-8a9a-b41343e4d845","added_by":"auto","created_at":"2024-01-25 17:04:30","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":247210,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3890256/v1/a8ca8e58-0525-4f3b-9cdc-b6b79cb472eb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Brans-Dicke gravity with scalar matter coupling and its cosmological probe","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eOne of the biggest problems in modern cosmology in our opinion, is whetherthe expansion dynamics of the universe is governing by the Einstein tensor gravity or the scalar-tensor one. After the Brans-Dicke scalar-tensor theory was formulated a plethora of works has been dedicated to the test of theory in scalar system and in cosmological scale and to the elucidation of the true implication of the scalar field. Nevertheless for example the Brans-Dicke coupling parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003eis being evaluated with tremendous tension hitherto. The measurement of the frequency shift of radiowaves to and from Cossini spacecraft asit passed near the sun yielded \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma =1+(2.1 \\pm 2.3) \\times {10^{ - 5}}\\)\u003c/span\u003e\u003c/span\u003e,which was equivalent to Brans-Dicke parameter, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega \u0026gt;40000\\)\u003c/span\u003e\u003c/span\u003e[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Observations of Nordtvedt effect using the Lunar Laser Ranging experiment yielded a slightly smaller value \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega \u0026gt;1000\\)\u003c/span\u003e\u003c/span\u003e[\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Observations of orbital period derivative of the aircular white dwarf-neutron star binary system PSR J1012\u0026thinsp;+\u0026thinsp;5307 yielded \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega \u0026gt;1250\\)\u003c/span\u003e\u003c/span\u003e[\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. On the other hand, the recent estimations of the parameter yield ever negative values. Ref[\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] obtained \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(- 407.0\u0026lt;\\omega \u0026lt;175.89\\)\u003c/span\u003e\u003c/span\u003eat 95% confidence level. Ref[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] using SNIa data obtained a best fit value \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega = - 1.477\\)\u003c/span\u003e\u003c/span\u003e. Ref[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] for three different models found \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega = - 0.9782\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega = - 1.0646\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega = - 1.7817\\)\u003c/span\u003e\u003c/span\u003e respectively. Where their explanation of such a wide range of estimation of the parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003efrom minus a few hundreds to a few tens of thousand should be found?\u003c/p\u003e \u003cp\u003eFirst, it might be associated with whether the model one used was appropriate for description of the real Universe. Such a requirement makes it necessity construct a model as more inclusive as possible so as to be close to the observed Universe. Second, it may be associated with temporal evolution of the Universe and the difference of the spatial scales for the observational data. As Ref. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] has pointed out the cosmological constraint on Brans-Dicke theory is different from the solar systems constraint since they concern very different spatial and temporal scales. For example the solar system experiment concerns smaller scale but curvature there is higher than in cosmological scale and this could lead to screen scalar and as a result the metric gravity (GR) plays the leading role. Perivolaropoulos [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] considered Brans-Dicke gravity with massive scalar field and obtained a modified expression on PPN parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003efor that theory. It reads\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\gamma (\\omega ,{m_s},r)=\\frac{{1 - \\frac{{{e^{ - \\frac{{\\sqrt {2{\\phi _0}} }}{{\\sqrt {2\\omega +3} }}{m_s}r}}}}{{2\\omega +3}}}}{{1+\\frac{{{e^{ - \\frac{{\\sqrt {2{\\phi _0}} }}{{\\sqrt {2\\omega +3} }}{m_s}r}}}}{{2\\omega +3}}}}.\\)\u003c/span\u003e \u003c/span\u003e \u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({m_s}\\)\u003c/span\u003e\u003c/span\u003eis mass of the massive scalar field, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\phi _0}\\)\u003c/span\u003e\u003c/span\u003ehomogeneous background field of scalar and r- the scale performing the experimental observation. The relation(1) is different from the relation of the conventional Brans-Dicke theory with massless scalar \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma =\\frac{{1+\\omega }}{{2+\\omega }}\\)\u003c/span\u003e\u003c/span\u003e.PPN-parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma (\\omega ,{m_s},r)\\)\u003c/span\u003e\u003c/span\u003eexpressed by relation (1) shows that parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e, in the case of massive scalar, depends not only on the mass of scalar itself, but also on what scale the experiment is performed on. Above-mentioned variety of the Brans-Dicke parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003emay be explained by the relation (1) partly. But the relation (1) was derived under the assumption that the temporal evolution was neglected. However our main concern is the cosmological temporal evolution of the model parameters for long period of the evolution of the Universe. Therefore making use of current observational data we first calculate the model parameters for the high (CMB) and low (BAO\u0026thinsp;+\u0026thinsp;SN) redshift. Sect 2 describes the basic equations of the cosmological model to be investigated. Sect 3 probes the model using observation data and Sect 4 summarizes the results.\u003c/p\u003e"},{"header":"2. Basic equation","content":"\u003cp\u003eThe action functional of our model takes the form\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$S=\\int {{d^4}x} \\sqrt { - g} \\left[ {\\frac{1}{2}\\left( {\\Phi R - \\frac{\\omega }{\\Phi }{\\Phi _{,\\mu }}{\\Phi ^{,\\mu }}} \\right) - \\frac{1}{2}\\left( {{\\phi _{,\\mu }}{\\phi ^{,\\mu }}+2V(\\phi )} \\right)+{L_b}\\left( {{\\psi _b};{g_{\\mu \\nu }}} \\right)+C(\\phi )L_{c}^{{(0)}}\\left( {{\\psi _c};{g_{\\mu \\nu }}} \\right)} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere we set \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(8\\pi G=1\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G_N}\\)\u003c/span\u003e\u003c/span\u003eis the Newton constant, therefore scalar field\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\phi\\)\u003c/span\u003e\u003c/span\u003e 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C(\\phi )\\)\u003c/span\u003e\u003c/span\u003erepresents on explicit direct interaction of scalar with dark matter. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\psi _b}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\psi _c}\\)\u003c/span\u003e\u003c/span\u003e represent the collective fields of baryons and dark matters, respectively. The variations of action (1) with respect to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g^{\\mu \\nu }}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\phi\\)\u003c/span\u003e\u003c/span\u003egive following dynamical equations\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\left( {{R_{\\mu \\nu }} - \\frac{1}{2}{g_{\\mu \\nu }}R} \\right)\\phi =C(\\phi )T_{{\\mu \\nu (c)}}^{{(0)}}+{T_{\\mu \\nu (b)}}+{T_{\\mu \\nu (\\phi )}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\left( {2\\omega (\\phi )+3} \\right)\\square \\phi =\\left( {C(\\phi ) - 2\\frac{{dC(\\phi )}}{{d\\phi }}\\phi } \\right)T_{c}^{{(0)}}+{T_b} - \\frac{{d\\omega (\\phi )}}{{d\\phi }}{(\\nabla \\phi )^2} - 4V(\\phi )+2\\frac{{dV(\\phi )}}{{d\\phi }}\\phi$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere energy-momentum tensor of scalar field is given by\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${T_{\\mu \\nu (\\phi )}}=\\frac{{\\omega (\\phi )}}{\\phi }\\left( {{\\phi _{,\\mu }}{\\phi _{,\\nu }} - \\frac{1}{2}{g_{\\mu \\nu }}{{(\\nabla \\phi )}^2}} \\right)+\\left( {{\\phi _{,\\mu ;\\nu }} - {g_{\\mu \\nu }}\\square \\phi } \\right) - {g_{\\mu \\nu }}V(\\phi )$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\square \\equiv \\frac{1}{{\\sqrt g }}{\\partial _\\mu }\\left( { - \\sqrt g } \\right)\\)\u003c/span\u003e\u003c/span\u003e, the matter energy-momentum tensor is determined by\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(T_{{\\mu \\nu (c)}}^{{(0)}}=\\frac{2}{{\\sqrt g }}\\frac{{\\partial \\sqrt g L_{c}^{{(0)}}}}{{\\partial {g^{\\mu \\nu }}}}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(T_{c}^{{(0)}}=T_{{\\mu (c)}}^{{(0)}}.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eTaking covariant derivative of Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) one finds. The energy-momentum conservation equations for baryon and for cold dark matter\u0026thinsp;+\u0026thinsp;scalar field, respectively, as follows\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$T_{{\\mu (c);\\nu }}^{\\nu }=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$T_{{\\mu (c);\\nu }}^{\\nu }+T_{{\\mu (\\phi );\\nu }}^{\\nu }=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(T_{{\\mu \\nu (c)}}^{\\nu }=C(\\phi )T_{{\\mu \\nu (c)}}^{{(0)}}\\)\u003c/span\u003e\u003c/span\u003e. 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 [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$T_{{\\mu (c);\\nu }}^{\\nu }=\\frac{{d\\ln C(\\phi )}}{{d\\phi }}{T_c}{\\phi _{,\\mu }}=\\frac{{dC(\\phi )}}{{d\\phi }}T_{c}^{{(0)}}{\\phi _{,\\mu }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T_c}=T_{{\\mu (c)}}^{\\mu }\\)\u003c/span\u003e\u003c/span\u003e. We apply the Eqs.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) to a flat FRW universe where metric is given by\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$d{s^2}= - d{t^2}+{a^2}(t)d{x^2}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMatter will be described by perfect fluids for both baryon and cold dark matter. Then dynamical equations derived from Eqs.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) read\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$3\\phi {H^2}=C(\\phi )\\rho _{c}^{{(0)}}+{\\rho _b}+\\frac{1}{2}\\frac{{\\omega (\\phi )}}{\\phi }{\\dot {\\phi }^2}+V(\\phi ) - 3H\\dot {\\phi }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$- \\phi \\dot {H}=\\frac{1}{2}\\left[ {C(\\phi )\\rho _{c}^{{(0)}}+{\\rho _b}+\\frac{{\\omega (\\phi )}}{\\phi }{{\\dot {\\phi }}^2}+\\ddot {\\phi } - H\\dot {\\phi }} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\left( {\\ddot {\\phi }+3H\\dot {\\phi }} \\right)\\frac{{\\omega (\\phi )}}{\\phi }=3(\\dot {H}+2{H^2}) - \\frac{{d\\omega (\\phi )}}{{d\\phi }}\\frac{{{{\\dot {\\phi }}^2}}}{{2\\phi }}+\\frac{{\\omega (\\phi )}}{2}{\\left( {\\frac{{\\dot {\\phi }}}{\\phi }} \\right)^2} - \\frac{{dV(\\phi )}}{{d\\phi }} - \\frac{{dC(\\phi )}}{{d\\phi }}\\rho _{c}^{{(0)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere superscript (0) stands for the quantities corresponding to the couplingless Laglangian \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(L_{c}^{{(0)}}\\)\u003c/span\u003e\u003c/span\u003efor cold dark matter in action (1).In the original Brans-Dicke theory of gravity without potential \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V(\\phi )\\)\u003c/span\u003e\u003c/span\u003e, scalar field evolves according to relation [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e].\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\phi (z)={(1+z)^{ - \\frac{1}{{1+\\omega }}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eOur model (1) however, is different from the original Brans-Dicke theory, in that is additionally contains a self-interacting potential \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V(\\phi )\\)\u003c/span\u003e\u003c/span\u003eand the scalar-matter coupling \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C(\\phi )L_{c}^{{(0)}}\\)\u003c/span\u003e\u003c/span\u003e. Therefore we take the evolution of scalar field, generalizing Eq.\u0026nbsp;(\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e12\u003c/span\u003e) as following form\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\phi (z)={(1+z)^{ - \\alpha (z)}}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha (z)\\)\u003c/span\u003e\u003c/span\u003e 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 \u0026ldquo;coincidence\u0026rdquo; problem [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003e[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], we consider self-interacting. Potential which determines the mass of scalar field \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m_{s}^{2}\\)\u003c/span\u003e\u003c/span\u003e. As Ref. [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] has shown the conservation equation of dark matter from Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e7\u003c/span\u003e)\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$${\\dot {\\rho }_c}(\\phi )+3H{\\rho _c}(\\phi )=\\frac{{d\\ln C(\\phi )}}{{dt}}{\\rho _c}(\\phi )=\\delta H{\\rho _c}(\\phi )$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eyields\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$C\\sim {a^\\delta } \\Rightarrow C={C_0}{(1+z)^{ - \\delta }}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e is a constant and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C_0}\\)\u003c/span\u003e\u003c/span\u003e is value of coupling function at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z=0\\)\u003c/span\u003e\u003c/span\u003e. Therefore matter densities evolves as follows\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$${\\rho _c}=C\\rho _{c}^{{(0)}}={\\rho _{{c_0}}}{(1+z)^{3 - \\delta }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$${\\rho _b}={\\rho _{{b_0}}}{(1+z)^3}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe conservation equation for scalar field is expressed as follows [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$${\\dot {\\rho }_c}(\\phi )+3H\\left( {{\\rho _\\phi }+{P_\\phi }} \\right)=\\frac{{d\\ln C(\\phi )}}{{d\\phi }}\\dot {\\phi }{\\rho _c}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting relation (16) into r. h. s. of Eq.\u0026nbsp;(\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e17\u003c/span\u003e), can obtain the energy density of scalar field resolving Eq.\u0026nbsp;(\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e17\u003c/span\u003e) with respect to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _\\phi }\\)\u003c/span\u003e\u003c/span\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P_\\phi }={\\omega _\\phi }{\\rho _\\phi }\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega _\\phi }\\)\u003c/span\u003e\u003c/span\u003eis equation of state parameter of scalar field. It reads\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$${\\rho _\\phi }={\\rho _{{\\phi _0}}}{(1+z)^{3(1+{\\omega _\\phi })}}+\\frac{{\\delta {\\rho _{{c_0}}}\\left[ {{{\\left( {1+z} \\right)}^{3(1+{\\omega _\\phi })}} - {{(1+z)}^{3 - \\delta }}} \\right]}}{{\\delta +3{\\omega _\\phi }}}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThen Friedman equation\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$$3{H^2}\\phi =C(\\phi )\\rho _{c}^{{(0)}}+{\\rho _b}+{\\rho _\\phi }.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMaking use of Eqs.\u0026nbsp;(\u003cspan refid=\"Equ17\" class=\"InternalRef\"\u003e16\u003c/span\u003e), (\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e17\u003c/span\u003e) and (19), is written as follows\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$${H^2}(z)=H_{0}^{2}{(1+z)^\\alpha }\\left\\{ {\\left( {1 - {\\Omega _c} - {\\Omega _b}} \\right){{\\left( {1+z} \\right)}^{3(1+{\\omega _\\phi })}}+{\\Omega _b}{{(1+z)}^3}+{\\Omega _c}\\left[ {\\frac{{3{\\omega _\\phi }}}{{\\delta +3{\\omega _\\phi }}}{{(1+z)}^{3 - \\delta }}+\\frac{\\delta }{{\\delta +3{\\omega _\\phi }}}{{(1+z)}^{3(1+{\\omega _\\phi })}}} \\right]} \\right\\}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, the parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Omega _c}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Omega _b}\\)\u003c/span\u003e\u003c/span\u003eare defined as follows\u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e\n$$\\left. {\\begin{array}{*{20}{c}} {{\\Omega _c} \\equiv \\frac{{{\\rho _{{c_0}}}}}{{3{\\phi _0}H_{0}^{2}}}} \\\\ {{\\Omega _b} \\equiv \\frac{{{\\rho _{{b_0}}}}}{{3{\\phi _0}H_{0}^{2}}}} \\end{array}} \\right\\}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _{{c_0}}}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _{{b_0}}}\\)\u003c/span\u003e\u003c/span\u003eare current values of dark matter and baryon matter densities. In Eq.\u0026nbsp;(\u003cspan refid=\"Equ23\" class=\"InternalRef\"\u003e21\u003c/span\u003e), parameters to be determined are \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Omega _c},{\\Omega _b},{\\omega _\\phi },\\alpha\\)\u003c/span\u003e\u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e. Our concern, however, is focused on the determination of the Brans-Dicke parameter\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003e, so we must find another equation containing the parameter\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003e. To that end, we use scalar field Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e11\u003c/span\u003e), but Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e11\u003c/span\u003e) includes potential function, we utilize Friedman Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e9\u003c/span\u003e) containing \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V(\\phi )\\)\u003c/span\u003e\u003c/span\u003e. Differentiating Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e9\u003c/span\u003e) with respect to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\phi\\)\u003c/span\u003e\u003c/span\u003ewe obtain expression.\u003cdiv id=\"Equ23\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ23\" name=\"EquationSource\"\u003e\n$$\\frac{{dV(\\phi )}}{{d\\phi }}=3{H^2}+3{H^2}\\left[ {\\alpha +(1+z)\\ln (1+z)\\frac{{d\\alpha }}{{dz}}} \\right] - \\frac{\\omega }{2}{H^2}{\\left[ {\\alpha +(1+z)\\ln (1+z)\\frac{{d\\alpha }}{{dz}}} \\right]^2} - \\frac{{dC(\\phi )}}{{d\\phi }}\\rho _{c}^{{(0)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere we used a relation derived from Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e13\u003c/span\u003e)\u003cdiv id=\"Equ24\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ24\" name=\"EquationSource\"\u003e\n$$\\dot {\\phi }=\\phi H\\left[ {\\alpha +(1+z)\\ln (1+z)\\frac{{d\\alpha }}{{dz}}} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{d}{{dt}}= - (1+z)H\\frac{d}{{dz}}\\)\u003c/span\u003e\u003c/span\u003eis used. Substituting the expression (22) into Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e11\u003c/span\u003e) for the scalar field we obtain a differential equation on the Hubble parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({H^2}(z)\\)\u003c/span\u003e\u003c/span\u003eas follows\u003cdiv id=\"Equ25\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ25\" name=\"EquationSource\"\u003e\n$$\\begin{gathered} \\frac{1}{2}\\left\\{ {2 - \\omega (1+z)\\left[ {\\alpha +(1+z)\\ln (1+z)\\alpha ^{\\prime}} \\right]} \\right\\}\\frac{{d{H^2}(z)}}{{dz}}= \\hfill \\\\ =\\left\\{ {\\omega (1+z)\\left[ {2\\alpha ^{\\prime}+\\alpha ^{\\prime}\\ln (1+z)+\\alpha ^{\\prime\\prime}\\ln (1+z)} \\right]+3 - 3(\\omega +1)\\left[ {\\alpha +(1+z)\\ln (1+z)\\alpha ^{\\prime}} \\right]+\\frac{1}{2}\\omega ^{\\prime}(1+z)\\left[ {\\alpha +(1+z)\\ln (1+z)\\alpha ^{\\prime}} \\right]} \\right\\}{H^2}(z). \\hfill \\\\ \\end{gathered}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha ^{\\prime}=\\frac{{d\\alpha }}{{dz}}\\)\u003c/span\u003e\u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega ^{\\prime}=\\frac{{d\\omega }}{{dz}}\\)\u003c/span\u003e\u003c/span\u003e. Above, we assumed the parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003eto be a function of redshift. If we assume the parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003e to be constants, then Eq.\u0026nbsp;(\u003cspan refid=\"Equ25\" class=\"InternalRef\"\u003e23\u003c/span\u003e) reduces to a simple equation after integration.\u003cdiv id=\"Equ26\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ26\" name=\"EquationSource\"\u003e\n$${H^2}(z)=H_{0}^{2}{(1+z)^{\\frac{{6(1 - \\alpha - \\omega \\alpha )}}{{3 - \\omega \\alpha }}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAs the power in Eq.\u0026nbsp;(\u003cspan refid=\"Equ26\" class=\"InternalRef\"\u003e24\u003c/span\u003e) is a constant, putting\u003cdiv id=\"Equ27\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ27\" name=\"EquationSource\"\u003e\n$$A \\equiv \\frac{{1 - \\alpha - \\omega \\alpha }}{{3 - \\omega \\alpha }}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWe can get the Brans-Dicke parameter as follows\u003cdiv id=\"Equ28\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ28\" name=\"EquationSource\"\u003e\n$$\\omega =\\frac{{1 - \\alpha - 3A}}{{\\alpha (1 - A)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eif the constant A is known. In the deriving Eq.\u0026nbsp;(\u003cspan refid=\"Equ27\" class=\"InternalRef\"\u003e25\u003c/span\u003e), we can use the slow-rall approximation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\ddot {\\phi }\u0026lt;\u0026lt;2H\\dot {\\phi }\\)\u003c/span\u003e\u003c/span\u003e. Then the scalar field equation becomes\u003cdiv id=\"Equ29\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ29\" name=\"EquationSource\"\u003e\n$$\\frac{3}{2}(1+z)\\frac{{d{H^2}(z)}}{{dz}}=\\left\\{ {3 - \\left[ {3+3\\omega - \\omega ^{\\prime}(1+z)} \\right]\\left[ {\\alpha +(1+z)\\ln (1+z)\\alpha ^{\\prime}} \\right]+\\omega {{\\left[ {\\alpha +(1+z)\\ln (1+z)\\alpha ^{\\prime}} \\right]}^2}} \\right\\}{H^2}(z)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the case of constant \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003e, the Eq.\u0026nbsp;(\u003cspan refid=\"Equ29\" class=\"InternalRef\"\u003e27\u003c/span\u003e), after integration reduces to a simple equation.\u003cdiv id=\"Equ30\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ30\" name=\"EquationSource\"\u003e\n$${H^2}(z)=H_{0}^{2}{(1+z)^{2\\left[ {(1 - \\alpha - \\omega \\alpha )+\\frac{1}{3}\\omega {\\alpha ^2}} \\right]}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e28\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ePutting\u003cdiv id=\"Equ31\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ31\" name=\"EquationSource\"\u003e\n$$B \\equiv (1 - \\alpha - \\omega \\alpha )+\\frac{{\\omega {\\alpha ^2}}}{3}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e29\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe unknown parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003eis a determined through\u003cdiv id=\"Equ32\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ32\" name=\"EquationSource\"\u003e\n$$\\omega =\\frac{{(1 - \\alpha ) - B}}{{\\alpha (1 - \\frac{\\alpha }{3})}}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e30\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"3. Probe of the model with observations","content":"\u003cp\u003eIn this work, our main concern is focused on the behavior of the Brans-Dicke parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003ewith time and spatial scale in the scalar-tensor theory of gravity. Therefore, in the utilization of cosmological observational data we do not mix the data corresponding to different cosmological epochs and spatial scales. For example, Cosmic Microwave Background (CMB) contains the information on the rate of expansion at redshift \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z\\sim 1100\\)\u003c/span\u003e\u003c/span\u003eand baryon Acoustic Oscillation (BAO) includes the information at low redshifts\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z \\leqslant 1\\sim 3\\)\u003c/span\u003e\u003c/span\u003e. The observational data of SNIa come also from the low redshifts. Therefore, in our probes, we separate CMB data from BAO and SNIa data.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 CMB\u003c/h2\u003e \u003cp\u003eAs CMB data, we use the acoustic scale\u003cdiv id=\"Equ33\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ33\" name=\"EquationSource\"\u003e\n$${l_A}=\\pi \\frac{{r({z_*})}}{{{r_s}({z_*})}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e31\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhich is defined in Ref. [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. Here the commoving distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(r(z)\\)\u003c/span\u003e\u003c/span\u003eis defined\u003cdiv id=\"Equ34\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ34\" name=\"EquationSource\"\u003e\n$$r({z_{}})=\\frac{c}{{{H_0}}}\\int\\limits_{0}^{{{z_*}}} {\\frac{{dz^{\\prime}}}{{E(z^{\\prime})}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e32\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAnd the commoving sound horizon distance at recombination \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z={z_*} \\approx 1091\\)\u003c/span\u003e\u003c/span\u003e is given by\u003cdiv id=\"Equ35\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ35\" name=\"EquationSource\"\u003e\n$${r_s}({z_*})=\\int\\limits_{{{z_*}}}^{\\infty } {\\frac{{{c_s}(z^{\\prime})}}{{H(z^{\\prime})}}} dz^{\\prime}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e33\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere the speed of sound is as follows\u003cdiv id=\"Equ36\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ36\" name=\"EquationSource\"\u003e\n$${c_s}(z)=c{\\left[ {3\\left( {1+\\frac{{3{\\Omega _b}}}{{4{\\Omega _\\gamma }}}\\frac{1}{{1+z}}} \\right)} \\right]^{ - \\frac{1}{2}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e34\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$E(z)={\\raise0.7ex\\hbox{${H(z)}$} \\!\\mathord{\\left/ {\\vphantom {{H(z)} {{H_0}}}}\\right.\\kern-0pt}\\!\\lower0.7ex\\hbox{${{H_0}}$}}$$\u003c/div\u003e\u003c/div\u003e.\u003c/p\u003e \u003cp\u003eThe seven year WMAP observations give \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Omega _\\gamma }=2.469 \\times {10^{ - 5}}{h^{ - 2}}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Omega _\\gamma }=0.02258\\)\u003c/span\u003e\u003c/span\u003e[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. In our analysis, the used acoustic scales \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({l_A}({z_*})\\)\u003c/span\u003e\u003c/span\u003eare given in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eObserved acoustic scale\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({l_A}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e301.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e302.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e302.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 BAO\u003c/h2\u003e \u003cp\u003eAs the second observation data BAO, we use BAO distance ratio\u003cdiv id=\"Equ37\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ37\" name=\"EquationSource\"\u003e\n$$d(z)=\\frac{{{r_s}({z_d})}}{{{D_\\nu }(z)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e35\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({r_s}({z_d})\\)\u003c/span\u003e\u003c/span\u003eis the commoving sound horizon distance at the drag epoch \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z={z_d}=1021\\)\u003c/span\u003e\u003c/span\u003eand the so-called lilaton scale \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D_\\nu }\\)\u003c/span\u003e\u003c/span\u003eis defined by\u003cdiv id=\"Equ38\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ38\" name=\"EquationSource\"\u003e\n$${D_\\nu }(z)={\\left[ {{{\\left( {\\int {\\frac{{dz^{\\prime}}}{{E(z^{\\prime})}}} } \\right)}^2}\\frac{z}{{E(z)}}} \\right]^{{\\raise0.7ex\\hbox{$1$} \\!\\mathord{\\left/ {\\vphantom {1 3}}\\right.\\kern-0pt}\\!\\lower0.7ex\\hbox{$3$}}}}\\frac{C}{{{H_0}}}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e36\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn Table \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the BAO distance ratios are presented which is cited from Ref. [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eObserved BAO distance ratio\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRedshift, z\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBAO distance ratio\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d(z)=\\frac{{{r_s}({z_d})}}{{{D_\\nu }(z)}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.106\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.336\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.015\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.1905\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm 0.0061\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.275\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.1390\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm 0.0037\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.278\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.1389\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0043\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.314\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.1239\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0033\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.1126\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0022\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0916\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0071\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0732\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0012\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0726\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0034\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0592\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0032\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0320\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pm\\)\u003c/span\u003e\u003c/span\u003e0.0007\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 SNIa\u003c/h2\u003e \u003cp\u003eThe third observation data are SNIa Union 2.The observed quantity is the distance modulus determines through the luminosity distance\u003cdiv id=\"Equ39\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ39\" name=\"EquationSource\"\u003e\n$$\\mu (z)=5\\lg {D_L}(z)+42.38 - 5\\lg h - \\frac{{15}}{4}\\lg \\Phi (z)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e37\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({H_0}=100h{\\raise0.7ex\\hbox{${km}$} \\!\\mathord{\\left/ {\\vphantom {{km} {Mpc}}}\\right.\\kern-0pt}\\!\\lower0.7ex\\hbox{${Mpc}$}}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D_L}(z)\\)\u003c/span\u003e\u003c/span\u003eis dimensionless luminosity distance defined by relation\u003cdiv id=\"Equ40\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ40\" name=\"EquationSource\"\u003e\n$${D_L}(z)=(1+z)\\int\\limits_{0}^{z} {\\frac{{dz^{\\prime}}}{{E(z^{\\prime})}}} .$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e38\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn Eq. a term containing scalar field is associated with fact that the peak luminosity of SNIa varies like \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(L\\sim {G^{ - \\frac{3}{2}}}\\)\u003c/span\u003e\u003c/span\u003eand corresponding absolute magnitude of SNIa evolves like\u003cdiv id=\"Equ41\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ41\" name=\"EquationSource\"\u003e\n$$M - {M_0}=\\frac{{15}}{4}\\lg \\frac{G}{{{G_0}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e39\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the Brans-Dicke theory of gravity, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\phi \\sim {G^{ - 1}}\\)\u003c/span\u003e\u003c/span\u003eand the scalar field term in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) stands for this fact. We use \u0026ldquo;Union 2\u0026rdquo; data for SNIa.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Conclusion","content":"\u003cp\u003eAs we can see in Table\u0026nbsp;3, the coupling less model yields negative value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003ein the coupling less case he relation\u003cdiv id=\"Equ42\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ42\" name=\"EquationSource\"\u003e\n$$\\alpha =\\frac{1}{{1+\\omega }}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e40\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIs satisfied, so we can estimate the parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003efrom the value of\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003e, immediately. The values of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003eare \u0026minus;\u0026thinsp;7.2406 and \u0026minus;\u0026thinsp;10.4118 for CMB and BAO\u0026thinsp;+\u0026thinsp;SN, respectively. To evade the ghost field, the parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003emust satisfy\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega \u0026gt; - \\frac{3}{2}\\)\u003c/span\u003e\u003c/span\u003e. Therefore, the above values \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega \u0026lt; - \\frac{3}{2}\\)\u003c/span\u003e\u003c/span\u003eare thought to be unphysical and we delete them. This unphysical result are considered to rise from the assumption for the scalar-matter coupling to be absent. Hence, we recalculate the parameters of our model including the scalar matter coupling.Result yields a positive value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003eand positive one of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003e. It is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega =59000\\)\u003c/span\u003e\u003c/span\u003e.The scalar-matter coupling parameter is negative \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta = - 0.0099\\)\u003c/span\u003e\u003c/span\u003e.Our estimation of the parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega\\)\u003c/span\u003e\u003c/span\u003eis more than the Cassini-experiment\u0026rsquo;s result. On the other hand, the fact that the scalar-matter coupling, parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003eis negative, implies a flow of energy from matter to scalar as the Eq.\u0026nbsp;(\u003cspan refid=\"Equ17\" class=\"InternalRef\"\u003e16\u003c/span\u003e) shows. Our probe is preliminary because the above results are relying on the combined data of CMB\u0026thinsp;+\u0026thinsp;BAO\u0026thinsp;+\u0026thinsp;SN. The separate probes for the high and low redshift data will give more interesting results.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eSongChol Ri: Supervision. IlMyong Yun: Software. JikSu Kim: Conceptualization, Methodology.RyongGwang Kim: Writing \u0026ndash; original draft, Funding acquisition.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eBerttoti, B., et al.: Nature, 425, 374\u003c/li\u003e\n\u003cli\u003eAlsing, J., et al.: arXiv:1182.4903[gr-gc]\u003c/li\u003e\n\u003cli\u003eWilliams, J.G., et al.: Int. J. Mod. Phy. D 13567 (2004)\u003c/li\u003e\n\u003cli\u003eLi, Y. C., et al.: Phy. Rev. D 88, 084053 (2013)\u003c/li\u003e\n\u003cli\u003eFabris, J. C., et al.: Grav. Cosmol., 12, 49 (2001)\u003c/li\u003e\n\u003cli\u003eHrycyna, O., et al.: arkiv:1404.7112 [astro-ph.CO]\u003c/li\u003e\n\u003cli\u003eAvilez, A. and Skordis, C.: arkiv:1303.4330 [astro-ph.CO]\u003c/li\u003e\n\u003cli\u003ePerivolaropoulos, L.: Phys. Rev. D 81, 047501 (2010)\u003c/li\u003e\n\u003cli\u003eJik Su, Kim, et al.: Phys. Rev. D 96, 043507 (2017)\u003c/li\u003e\n\u003cli\u003eBrans, C., et al.: Phys. Rev. D 124, 925 (1961)\u003c/li\u003e\n\u003cli\u003eWeinberg, S.: Gravitation and cosmology, (1972)\u003c/li\u003e\n\u003cli\u003eAmendola, L., et al.: Phys. Rev. D 74, 023525 (2006)\u003c/li\u003e\n\u003cli\u003eMajerotto, E., et al.: arXiv:astro-ph/0410543\u003c/li\u003e\n\u003cli\u003eAmendola, L.: Phys. Rev. D 60, 043501 (1999)\u003c/li\u003e\n\u003cli\u003eBond, J., et al.: MNRAS, 291, L 33 (1997)\u003c/li\u003e\n\u003cli\u003eKomatsu, E., et al.: Ap. J. Suppl. 192, 18 (2011)\u003c/li\u003e\n\u003cli\u003eAde, P. A. R., et al.: [Planck Collaboration] Planck (2014)\u003c/li\u003e\n\u003cli\u003eWang, Y., et al.: Phys. Rev. D 85, 023517 (2012)\u003c/li\u003e\n\u003cli\u003eWang, A., et al.: Phys. Rev. D 88, 043522 (2013)\u003c/li\u003e\n\u003cli\u003eBennet, C. L., et al.: [WMAP Collaboration] Ap. J. Suppl. 208, 20 (2013)\u003c/li\u003e\n\u003cli\u003eZhang, Y.: arXiv:1411.5522 [astro-ph.CO]\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"gravity, scalar","lastPublishedDoi":"10.21203/rs.3.rs-3890256/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3890256/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this work we investigate the Brans-Dicke gravity with self-interacting potential and potentialess model, the scalar field depends on redshift. But our model includes both potential and coupling, so we put the redshift dependence of the scalar in other form. Making use of CMB, BAO and SNIa observational data, we probe the model and determine model parameters, but our main concern is the dependence of model parameters on the temporal and spatial scales. Therefore we separate the probes for high(CMB) and low (BAO\u0026thinsp;+\u0026thinsp;SN) redshift observational data.\u003c/p\u003e","manuscriptTitle":"Brans-Dicke gravity with scalar matter coupling and its cosmological probe","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-25 16:40:17","doi":"10.21203/rs.3.rs-3890256/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"7b37c741-9d2d-4ed7-b0db-50ac4081d8b3","owner":[],"postedDate":"January 25th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-01-25T16:40:19+00:00","versionOfRecord":[],"versionCreatedAt":"2024-01-25 16:40:17","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3890256","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3890256","identity":"rs-3890256","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}