In the theory of Brans-Dicke (BD) scalar-tensor gravity, different models have been proposed for different potential functional types, usually by setting the scalar field to the power function form of the cosmic distance factor (power exponent is constant). However, in previous studies, the observational limitation in the solar system range of the coupling parameter (BD coupling parameter) between the BD scalar field and gravity is obtained from negative values to several thousand. In particular, previous studies investigating holographic scalar field models using cosmological holographic principle, in which the BD scalar field is placed in the form of a power function of the cosmic distance factor, have considered the problem of accelerated expansion of the late universe and cosmological coincidence problem, but none of the models have been fully explained. However, the use of a model with the coupling strength of dark matter and dark energy as a non-constant variable requires parameterization, provided that previous studies have been conducted to address the problem of Hubble constant tension and cosmological coincidence. Therefore, in this study, we construct a scalar field model that generalizes the exponent in the expression of the scalar field to a function of the redshift using well-known Jassal-Bagla-Padmanabhan (JBP) parameterization in cosmology, and obtain a cosmological constraint on the BD coupling parameter, in the context of the theory of BD scalar-tensor gravity, not the Einstein’s theory of tensor gravity, and show that the accelerating expansion of the late Universe and cosmological coincidence problem are well explained by the proposed model. First, we solve the continuity equation under the assumption that there is no interaction between matter and the scalar field, and obtain an analytical expression for the energy density of the mass and the BD scalar field. Then, using 580 supernova observations, the best-fitting values of the proposed model parameters and the range of values of the parameters at different confidence levels are obtained by minimizing the chi-square function. We have obtained the result that the scalar field decreases, the BD coupling parameter increases, and the current value of BD coupling parameter is around -1.746, from past to present. Also, the coincidence parameter now converges to 1, so that the problem of cosmic coincidence problem is alleviated by the proposed model, and the rate of change of the gravitational constant is considered.
Ri et al. (Mon,) studied this question.