Study on a Parameterized Scalar Field Model

Song Chol Ri; Ryong Gwang Kim; Il Guk Kim; Jik Su Kim

doi:doi:10.11648/j.ajpa.20261401.12

Research Article |

| Peer-Reviewed

Study on a Parameterized Scalar Field Model

Song Chol Ri

, Ryong Gwang Kim

, Il Guk Kim^*

, Jik Su Kim

Published in American Journal of Physics and Applications (Volume 14, Issue 1)

Received: 22 September 2025 Accepted: 6 November 2025 Published: 16 March 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

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.

Published in	American Journal of Physics and Applications (Volume 14, Issue 1)
DOI	10.11648/j.ajpa.20261401.12
Page(s)	12-17
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Scalar, Coupling, (JBP) Parameterization

1. Introduction

The so-called dark energy that contributes to the cosmological observations including supernovae, cosmic microwave background radiation, baryon acoustic oscillation, and Hubble parameters, suggest the accelerated expansion of the late universe.

[1-7]

Within the scope of Einstein’s tensor theory of gravity, several dark energy models have been proposed, including the cosmological constant model, which often encounter difficulties such as fine-tuning problem and cosmological coincidence problem.

[8-11]

The Brans-Dicke (BD) scalar theory of gravity is the first historical attempt to extend Einstein's theory of gravity.

[12]

In this theory, the Newton's gravitational constant is assumed to vary with time, and instead of the gravitational constant, a scalar field is introduced in the action term, where the coupling parameter characterizing the coupling with gravity plays an important role and is called the BD parameter. Because of the importance of this BD parameter, several studies have been made to constraints the use of different observations with different models. Observations of Nordtvedt effect using the Lunar Laser Ranging experiment yielded a slightly smaller value

. Observations of orbital period derivative of the aircular white dwarf-neutron star binary system PSR J1012+5307 yielded

[13]

On the other hand, the recent estimations of the parameter yield ever negative values and

at 95% confidence level and a best fit value

by using SNIa data.

[13, 14]

Where their explanation of such a wide range of estimation of the parameter

from minus a few hundreds to a few tens of thousand should be found? On the other hand, parameterization of cosmological quantities such as the parameters of the state equation, the holographic constant in the holographic dark energy model, and the dark matter-dark energy coupling, has shown in several studies the possibility of solving the unsolved problem such as the problem of cosmological coincidence.

[15-18]

For the universal parameterization in cosmology, we can include Jassal-Bagla-Padmanabhan (JBP) parameterization, Ma-Zhang parameterization, and Wetterich parameterizations.

[18-21]

Therefore, we will parameterize the BD parameter using JBP parameterization, find the optimal values and ranges of the model parameters using supernova observations, and investigate the parameterized BD scalar gravity theory with cosmic evolution by considering the variation of the scalar field with the redshift. The structure of the paper is as follows. In Section 2, model validation using the basic equations and observational data of the parameterized BD scalar gravity theory is presented, and in Section 3, the relationship between scalar fields and cosmic evolution is investigated.

2. Basic Equation

The action for the BD theory is given by

(1)

where

is a BD scalar field which couples with gravity,

denotes the Ricci scalar curvature,

is a coupling parameter between gravity and scalar field called BD parameter and

represents the matter lagrangian density.

[12]

In this paper, we consider a flat Friedmann-Robertson-Walker (FRW) metric given by the line element.

(2)

where

denotes the scale factor of the Universe.

[12]

The variation of the action (1) with respect to the metric tensor

for the line element (2) with the energy-tensor yield the following Friedman equation.

(3)

where

is a time-dependent scalar field called BD scalar field which coupled with gravity,

denotes the Hubble parameter,

and

are the energy density of the matter and scalar field. In the original BD theory of gravity without potential, scalar field and energy density for the scalar field evolves according to relation.

[12]

(4)

(5)

where

represents the redshift and the over dot denotes the derivative with respect to the cosmic time

. We generalize the Eq (4) and the scalar field as follows:

(6)

where

is generally a function of redshift to be determined as following

(7)

in which

and

are a dimensionless constants. It is well-known Jassal-Bagla-Padmanabhan (JBP) parameterization.

[18]

It is obvious that the

case is reduced to the origin scalar field with constant BD coupling parameter. The continuity equations for matter and scalar field, without interaction between them can be written as follows

(8)

(9)

where

is the equation of state parameter of scalar field.

[18]

From Eq (8) and Eq (9) the energy density for the matter and BD scalar field is given by

(10)

(11)

where

and

are constants. Using Eq (3), (10), (11), we find that

(12)

where

In this paper, we constraint observationally the model, using by 580 supernovae observations.

[7]

In this case the distance module of supernova is defined as

(13)

where

is a dimensionless luminosity distance of the supernova,

is the Hubble parameter in units of 100kms ^-1Mpc^-1. We construct the

function as follows

(14)

where

and

are the theoretical distance module, the observational distance module and error of the observation for the

of the supernova, respectively. Minimizing the

function parameters of the model are determined.

3. Conclusion

The obtained best fitting value of parameters and the contours of the parameters at the different confidence levels are shown in Figure 1.

Download: Download full-size image

Figure 1. The contours of the parameters (

) at the different confidence levels (

From Eq (3) and Eq (7), we get

(15)

The variation of the BD parameter

for the best fitting value of the parameters (

) with redshift

is shown in Figure 2.

Download: Download full-size image

Figure 2. The variation of the BD parameter with redshift (

It can be seen from Figure 2 that the BD parameter shows an increasing trend from past to present. The present time, i.e.

correspond to the BD parameter

. A variation of the scalar field

for the best fitting value of the parameters (

) with redshift

is shown in Figure 3.

Download: Download full-size image

Figure 3. The variation of the scalar field with redshift (

From the Figure 3, we can see that the scalar field decreases from the past to the present. It is also found that the BD scalar field has decreased rapidly in the past and has become almost constant recently. This shows that the BD scalar theory of gravity is true in the past description of cosmic evolution, but now the Einstein’s tensor theory is valid. This is in agreement with the previous results.

[22-24]

This means that the scalar field plays a fundamental role in cosmic evolution. On the other hand, using the Eq (3)- Eq (6) we get the energy density for the BD scalar field as follows

(16)

(17)

Substituting Eq (3) and Eq (9) to the Eq (15), the energy density of the BD scalar field is rewritten

(18)

From Eq (3) and (16) the energy density ratio called the coincidence parameter

is written as follows

(19)

From Eqs. (17) and (19), it can be seen that when the red shift z goes to zero, i.e., the consistency parameter at the present time becomes

(20)

(21)

This is in good agreement with previous studies

[25, 26]

. Therefore, we conclude that the late time accelerating universe is explained by the model with parameterized BD parameter. On the other hand, the rate of change of

obtained by our model is

From the above expression, we can see that the rate of change of

at the current point is only related to

, which is

where we have taken

. Using the same observation as ranging and Doppler data from the Mars Global Surveyor, Mars Odyssey, Mars Reconnaissance Orbiter, and other spacecraft, Ref.

[27]

obtain a strong observational constraint:

In addition, Ref.

[28]

use big bang nucleosynthesis to obtain the following restrictions:

Our result are much different from this. Using other cosmological observations such as the Hubble parameter, we can obtain a more reliable constraint on

In the present model, the cosmic coincidence problem has been alleviated more effectively. Future work on varying BD parameter will be more interesting.

Abbreviations

BD	Brans-Dicke
JBP	Jassal-Bagla-Padmanabhan

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Funding

This research received no external funding.

References

[1]	Riess, A. G., Filippenko. A. V., Challis, P. et al. OBSERVATIONAL EVIDENCE FROM SUPERNOVAE FOR AN ACCELERATING UNIVERSE AND A COSMOLOGICAL CONSTANT. THE ASTRONOMICAL JOURNAL. 1998, 1009-1038.
[2]	PERLMUTTER, S., ALDERING, G., GOLDHABER, G. et al. MEASUREMENTS OF AND FROM 42 HIGH-REDSHIFT SUPERNOVAE. THE ASTROPHYSICAL JOURNAL. 1999, 517, 565-586.
[3]	Komatsu, E., Smith, K. M., Dunkley J. et al. SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: COSMOLOGICAL INTERPRETATION. The Astrophysical Journal Supplement Series. 2011, 192, 18 (47 pp).
[4]	Abazajian, K., Adelman-McCarthy, J. K., Agueros, M. A. et al. THE SECOND DATA RELEASE OF THE SLOAN DIGITAL SKY SURVEY. The Astronomical Journal. 2004, 128, 502–512.
[5]	Tegmark, M., et al. Cosmological parameters from SDSS and WMAP. Phys. Rev. D. 2004, 69, 103501. https://doi.org/10.1103/PhysRevD.69.103501
[6]	Geach, J. E., Cimatti, A., Percival, W. et al. Empirical Hα emitter count predictions for dark energy surveys. Mon. Not. R. Astron. Soc. 2010, 402, 1330–1338. https://doi.org/10.1111/j.1365-2966.2009.15977.x
[7]	Ade, P. A. R., Aghanim, N., Armitage-Caplan, C. et al. Planck 2013 results. XV. CMB power spectra and likelihood. Astronomy & Astrophysics. 2014, 571, A15. https://doi.org/10.1051/0004-6361/201321573
[8]	Caldwell, R. R., Dave, R., Steinhardt, P. J. Cosmological Imprint of an Energy Component with General Equation of State. Physical Review letters. 1998, 80, 8, 1582-1585.
[9]	Caldwell, R. R. A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Physics Letters B. 2002, 545, 23-29.
[10]	Kamenshchik, A., Moschella, U., Pasquier, V. An alternative to quintessence. Physics Letters B. 2001, 511, 265–268.
[11]	Avelino, P. P., Bolejko, K., Lewis, G. F. Nonlinear Chaplygin gas cosmologies. PHYSICAL REVIEW D. 2014, 89, 103004, https://doi.org/10.1103/PhysRevD.89.103004
[12]	Peracaula, J. S., Gomez-Valent, A. et al. Brans-Dicke Gravity with a cosmological Constant Smoothes cut CDM Tensions. The Astrophysical Journal Letters. 2019, 886, L6, https://doi.org/10.3847/2041-8213/ab53e9
[13]	Alsing, J., Berti, E., Will, C. M., Zaglauer, H. Gravitational radiation from compact binary systems in the massive Brans-Dicke theory of gravity. PHYSICAL REVIEW D. 2012, 85, 064041, https://doi.org/10.1103/PhysRevD.85.064041
[14]	WILLIAMS, J. G., TURYSHEV, S. G. IMPROVING LLR TESTS OF GRAVITATIONAL THEORY. International Journal of Modern Physics D, 2004, 13. 3, 567-582.
[15]	Li, Y. C., Wu, F. Q., Chen, X. Constraints on the Brans-Dicke gravity theory with the Planck data. PHYSICAL REVIEW D. 2013, 88, 084053, https://doi.org/10.1103/PhysRevD.88.084053
[16]	Lip, S. Z. W. Interacting cosmological fluids and the coincidence problem. PHYSICAL REVIEW D. 2011, 83, 023528, https://doi.org/10.1103/PhysRevD.83.023528
[17]	Chevallier, M., et al. Accelerating Universes with Scaling Dark Matter. International Journal of Modern Physics. 2001, 10, 213.
[18]	Ghaffari, S., et al. Holographic dark energy model in the GDP braneworld with time varying holographic parameter. New Astronomy. 2019, 67, 76-84.
[19]	Oliveira, A. M., S. Velten, H. E., Fabris, J. C., Salako, I. G. Newtonian view of general relativistic stars. The European Physical journal C. 2014, 74, 3170, https://doi.org/10.1140/epjc/s10052-014-3170-2
[20]	Bousso, R., Freivogel, B., Leichenauer, S., Rosenhaus, V. A Geometric Solution to the Coincidence Problem and the Size of the Landscape as the Origin of Hierarchy. Physical Review Letters. 2011, 106, 101301, https://doi.org/10.1103/PhysRevLett.106.101301
[21]	Frion, E., Almeida, C. R. Affine quantization of the Brans-Dicke theory: Smooth bouncing and the equivalence between the Einstein and Jordan frames. PHYSICAL REVIEW D. 2019, 99, 023524, https://doi.org/10.1103/PhysRevD.99.023524
[22]	Zucca, A., Pogosian, L., Silvestri, A. Generalized Brans-Dicke theories in light of evolving dark energy. PHYSICAL REVIEW D. 2020, 101, 043518, https://doi.org/10.1103/PhysRevD.101.043518
[23]	Papantonopoulo, E., Vlachos, C. Wormhole solutions in modified Brans-Dicke theory. PHYSICAL REVIEW D. 2020, 101, 064025, https://doi.org/10.1103/PhysRevD.101.064025
[24]	Lu, J., Wu, Y., Yang, W. et al. The generalized Brans-Dicke theory and its cosmology. Eur. Phys. J. Plus. 2019 134, 318. https://doi.org/10.1140/epjp/i2019-12684-0
[25]	Xu, L. Constraints on the holographic dark energy model from type Ia supernovae, WMAP7, baryon acoustic oscillation, and redshift-space distortion. PHYSICAL REVIEW D. 2013, 87, 043525. https://doi.org/10.1103/PhysRevD.87.043525
[26]	Pitjeva, E. V. Updated IAA RAS Planetary Ephemerides—EPM2011 and Their Use in Scientific Research. Astronomicheskii Vestnik. 2013, 47, 5, 419–435, https://doi.org/10.1134/S0038094613040059
[27]	Bambi, C., Giannotti, M., Villante, F. L. Response of primordial abundances to a general modification of GN and/or of the early universe expansion rate. PHYSICAL REVIEW D, 2005, 71, 123524, https://doi.org/10.1103/PhysRevD.71.123524
[28]	Bronnikov, K. A., Kononogov, S. A. Variations in the Gravitational Constant in General Theories of Gravitation. Measurement Techniques, 2015, 57, 11 https://doi.org/10.1007/s11018-015-0615-4

Cite This Article

Plain Text BibTeX RIS

APA Style

Ri, S. C., Kim, R. G., Kim, I. G., Kim, J. S. (2026). Study on a Parameterized Scalar Field Model. American Journal of Physics and Applications, 14(1), 12-17. https://doi.org/10.11648/j.ajpa.20261401.12

Copy | Download

ACS Style

Ri, S. C.; Kim, R. G.; Kim, I. G.; Kim, J. S. Study on a Parameterized Scalar Field Model. Am. J. Phys. Appl. 2026, 14(1), 12-17. doi: 10.11648/j.ajpa.20261401.12

Copy | Download

AMA Style

Ri SC, Kim RG, Kim IG, Kim JS. Study on a Parameterized Scalar Field Model. Am J Phys Appl. 2026;14(1):12-17. doi: 10.11648/j.ajpa.20261401.12

Copy | Download

@article{10.11648/j.ajpa.20261401.12,
  author = {Song Chol Ri and Ryong Gwang Kim and Il Guk Kim and Jik Su Kim},
  title = {Study on a Parameterized Scalar Field Model},
  journal = {American Journal of Physics and Applications},
  volume = {14},
  number = {1},
  pages = {12-17},
  doi = {10.11648/j.ajpa.20261401.12},
  url = {https://doi.org/10.11648/j.ajpa.20261401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20261401.12},
  abstract = {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.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Study on a Parameterized Scalar Field Model
AU  - Song Chol Ri
AU  - Ryong Gwang Kim
AU  - Il Guk Kim
AU  - Jik Su Kim
Y1  - 2026/03/16
PY  - 2026
N1  - https://doi.org/10.11648/j.ajpa.20261401.12
DO  - 10.11648/j.ajpa.20261401.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 12
EP  - 17
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20261401.12
AB  - 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.
VL  - 14
IS  - 1
ER  -

Copy | Download

Author Information

Song Chol Ri

Department of Physics, Sariwon Ri Kye Sun University of Education, Sariwon, Democratic People’s Republic of Korea

http://orcid.org/0009-0006-6160-3990
Ryong Gwang Kim

Pyongyang Astronomical Observatory, Pyongyang, Democratic People’s Republic of Korea

http://orcid.org/0009-0004-0390-6219
Il Guk Kim

Pyongyang Astronomical Observatory, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0007-0940-2478
Jik Su Kim

Pyongyang Astronomical Observatory, Pyongyang, Democratic People’s Republic of Korea

http://orcid.org/0009-0002-9153-7489

Download PDF

Submit an Article

Plain Text BibTeX RIS

APA Style

Ri, S. C., Kim, R. G., Kim, I. G., Kim, J. S. (2026). Study on a Parameterized Scalar Field Model. American Journal of Physics and Applications, 14(1), 12-17. https://doi.org/10.11648/j.ajpa.20261401.12

Copy | Download

ACS Style

Ri, S. C.; Kim, R. G.; Kim, I. G.; Kim, J. S. Study on a Parameterized Scalar Field Model. Am. J. Phys. Appl. 2026, 14(1), 12-17. doi: 10.11648/j.ajpa.20261401.12

Copy | Download

AMA Style

Ri SC, Kim RG, Kim IG, Kim JS. Study on a Parameterized Scalar Field Model. Am J Phys Appl. 2026;14(1):12-17. doi: 10.11648/j.ajpa.20261401.12

Copy | Download

@article{10.11648/j.ajpa.20261401.12,
  author = {Song Chol Ri and Ryong Gwang Kim and Il Guk Kim and Jik Su Kim},
  title = {Study on a Parameterized Scalar Field Model},
  journal = {American Journal of Physics and Applications},
  volume = {14},
  number = {1},
  pages = {12-17},
  doi = {10.11648/j.ajpa.20261401.12},
  url = {https://doi.org/10.11648/j.ajpa.20261401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20261401.12},
  abstract = {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.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Study on a Parameterized Scalar Field Model
AU  - Song Chol Ri
AU  - Ryong Gwang Kim
AU  - Il Guk Kim
AU  - Jik Su Kim
Y1  - 2026/03/16
PY  - 2026
N1  - https://doi.org/10.11648/j.ajpa.20261401.12
DO  - 10.11648/j.ajpa.20261401.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 12
EP  - 17
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20261401.12
AB  - 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.
VL  - 14
IS  - 1
ER  -

Copy | Download