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Int. J. Math. And Appl., 9(3)(2021), 67–80
ISSN: 2347-1557
Available Online: http://ijmaa.in/Applications•ISSN:234
7-15
57•In
ternationalJo
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l of MathematicsAnd
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International Journal ofMathematics And its Applications
A Simple Way to Estimate the Variation of the
Gravitational Constant as a Function of Redshift in the
Framework of Brans-Dicke Theory
Sudipto Roy1,∗
1 Department of Physics, St. Xavier’s College, Kolkata, West Bengal, India.
Abstract: The objective of the present study is to find theoretically the nature of evolution of the time-varying gravitational constant(G) and its relative time-rate of change (G/G) with respect to the redshift parameter (z). For this purpose, we have used
the field equations of the Brans-Dicke (BD) theory of gravity for a flat universe of zero pressure, with a homogeneous and
isotropic space-time. Our entire formulation is based on four mathematical models constructed with empirical expressionsinvolving the scale factor, BD scalar field and their time derivatives. Substituting these expressions into the field equations,
we have determined the values of the constants associated with these ansatzes. It is clearly evident from these valuesthat the gravitational constant increases as the redshift (z) decreases with time. We have also determined the nature of
variation of the relative time-rate of change of the gravitational constant (G/G). It has been found to be increasing as z
decreases with time. The variation of the gravitational constant and its relative time-rate of change, as functions of theredshift parameter, have been depicted graphically on the basis of the four models discussed in the present article. Based
on their characteristics of variation, we have proposed an empirical relation representing the evolution of the gravitational
constant (G) as a function of time. Using this relation, we have determined the nature of dependence of redshift (z) upontime and represented it graphically. Similar findings have been obtained from studies based on various other methods.
An important feature of the present study is that all its findings have been obtained without solving the field equations.
MSC: 00A79, 83D05, 83F05, 85A40.
Keywords: Cosmology, Gravitational constant, Brans-Dicke theory, Scalar field, Redshift (z).
© JS Publication.
1. Introduction
We have a huge number of astrophysical observations and theoretical findings that motivate us to explore the modified
or alternative theories of general relativity (GR). Modified versions of GR have always been extremely important areas of
research. A number of modified gravity theories have been proposed and extensively studied [1–6]. Most importantly, two
simple modifications to general relativity, which have been very widely studied, are the f(R) theory and the Brans-Dicke
(BD) theory of gravity [7–9].
In order to be sufficiently consistent with Mach’s principle and less dependent on the absolute characteristics of space,
Brans and Dicke proposed an interesting alternative to general relativity in the year 1961 [10, 11]. In this new theory,
there is a clear violation of the strong principle of equivalence, on the basis of which Einstein constructed the theory of
general relativity. They needed to build a framework in which the gravitational constant (G) could be obtained from the
structure and dynamics of the universe. Therefore, a time varying parameter G can be regarded as the Machian outcome
∗ E-mail: [email protected]
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A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory
of an expanding universe. This was achieved by introducing a scalar field parameter (φ) into the variational principle and
consequently in the field equations in such a way that the behaviour of φ is analogous to that of G−1 in Einstein’s theory.
However, the Lagrangian and the action for matter remained unaltered by the modification of the theory of GR in this
manner.
Recent astrophysical observations have given us enough indications that the Newtonian gravitational constant (G) has a
dependence upon time [12–16]. Brans-Dicke (BD) theory allows us to describe cosmological phenomena in terms of a time
varying gravitational constant. Among the scalar-tensor theories, BD theory seems to be a simple one which has been
formulated in accordance with Mach’s principle [17, 18]. Here, a scalar field parameter (φ) can be introduced very naturally
into the theory, which would be behaving in a manner such that φ(t) is proportional to 1/G(t). But it is difficult to account
for the accelerated expansion of the universe, as obtained from observations, with the help of the original BD theory [9, 19–
21]. To be able to interpret an accelerating universe, one had to modify this theory by introducing an exotic component,
i.e., the dark energy of the universe [22].
In the present study we have done simple mathematical calculations to determine the dependence of the gravitational
constant (G) and its relative time-rate of change (G/G) upon the redshift parameter (z). For this purpose, we have used
the field equations of the Brans-Dicke theory of gravity for a homogeneous and isotropic universe with zero spatial curvature
and zero pressure (dust filled universe). Calculations for the present study have been done on the basis of four models which
are based on four different ansatzes involving the scalar field parameter (φ) and the scale factor (a). It has been shown
in this article that, as the redshift parameter (z) approaches its present value (i.e., zero) both G and G/G increases at a
gradually increasing rate with respect to z. Based on the variation of G and G/G with respect to z, we have proposed an
empirical expression for G to represent its dependence upon time. Using this expression, we have determined the nature of
change of z with respect to time. All findings have been depicted graphically.
2. Brans-Dicke Field Equations
According to the Brans-Dicke theory of gravity, the action is expressed as,
A =1
16π
∫d4x√−g(φR+
ω (φ)
φgµν∂µφ∂νφ+ Lm
)(1)
In equation (1), Lm represents the Lagrangian for matter and g stands for the determinant of the metric tensor gµν . φ
stands for the Brans-Dicke scalar field and R denotes the Ricci scalar. The dimensionless quantity ω is called the Brans-Dicke
parameter. In generalized BD theory, this parameter is regarded as a function of the scalar field (φ).
By the variation of action (A), given by equation (1), following field equations are obtained.
Rµν −1
2Rgµν =
1
φTµν −
ω
φ2
[φ,µφ,ν −
1
2gµνφ,βφ
′β]− 1
φ
[φµ;ν − gµν
∂2φ
∂t2−∇2φ
](2)
∂2φ
∂t2−∇2φ =
1
2ω + 3T (3)
Rµν represents the Ricci tensor and Tµν denotes the energy-momentum tensor. A comma denotes an ordinary derivative
and a semicolon stands for a covariant derivative with respect to xβ , in equations (2) and (3). The energy-momentum tensor
(Tµν) for the constituents of the universe is expressed as,
Tµν = (ρ+ P )uµuν + gµνP (4)
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Sudipto Roy
In the above equations, P represents the isotropic pressure, ρ stands for the energy density, T denotes the trace of the
tensor Tij and uν represents the four-velocity vector. This expression for Tµν in equation (4) is based on an assumption
that the constituents of the expanding universe behave as a perfect fluid. We have gµνuµuν = 1 in a co-moving coordinate
system having uν = (0, 0, 0, 1). The line element for FRW space-time (representing a homogeneous and isotropic universe)
is expressed as,
ds2 = −dt2 + a2 (t)
[dr2
1− sr2 + r2dθ2 + r2sin2θ dξ2]
(5)
In equation (5), a(t) represents the scale factor, t denotes the cosmic time and s stands for the spatial curvature. Three
coordinates of the spherical polar system are represented by r, θ and ξ. The spatial curvature parameter (denoted by s)
represents the closed, flat and open universes respectively according to s = 1, 0,−1. We have taken s = 0 (i.e., flat space)
for all calculations of the present article.
For s = 0, equation (5) takes the following form.
ds2 = −dt2 + a2 (t)[dr2 + r2dθ2 + r2sin2θ dξ2
](6)
In this homogeneous and isotropic space-time expressed by equation (5), the field equations of the Brans-Dicke theory of
gravity are given below. To obtain these equations, one has to combine the equations (2), (3), (4) and (5).
3a2 + s
a2+ 3
aφ
aφ− ωφ2
2φ2=ρ
φ(7)
2a
a+a2 + s
a2+ωφ2
2φ2+ 2
aφ
aφ+φ
φ= −P
φ(8)
φ
φ+ 3
aφ
aφ=ρ− 3P
2ω + 3
1
φ− ω
2ω + 3
φ
φ(9)
The symbols a and a represent respectively the first and second order derivatives of the scale factor (a) with respect to time.
These particular symbols (i.e., dot & double-dot), which have also been used for other cosmological variables in the present
study, represent the same mathematical operations (i.e., 1st & 2nd order differentiation with respect to time, respectively).
For zero spatial curvature (that is flat space where s = 0) and zero pressure (pressure-less dust, P = 0, ρ = ρm), equations
(7)-(9) take the following forms.
3a2
a2+ 3
aφ
aφ− ωφ2
2φ2=ρmφ
(10)
2a
a+a2
a2+ωφ2
2φ2+ 2
aφ
aφ+φ
φ= 0 (11)
φ
φ+ 3
aφ
aφ=
ρm2ω + 3
1
φ− ω
2ω + 3
φ
φ(12)
Here a, φ, ω and ρm are respectively the scale factor, scalar field, Brans-Dicke coupling parameter and the density of matter
(dark + baryonic). Combining equations (10) and (11) we get,
4a2
a2+ 2
a
a=ρmφ− 5
a
a
φ
φ− φ
φ(13)
The evolution of the gravitational constant (G = 1/φ) has been determined in the present paper with the help of the four
models discussed below. In each of these models, we have assumed an empirical expression involving the scalar field (φ) and
the scale factor (a). The values of cosmological parameters at the present time (i.e., at t = t0), used for the determination
of constants connected to these empirical relations, are given below.
H0 = 2.39× 10−18sec−1, t0 = 4.13×1017sec, ρm0 = 2.97× 10−27 kg/m3,
G0 = 6.67× 10−11 Nm2kg−2, φ0 = 1/G0 = 1.50× 1010N−1m−2kg2, q0 = −0.55.
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A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory
3. Models for the Scalar Field (φ)
To obtain expressions connected to G in terms of redshift (z), we have proposed four mathematical models based on four
ansatzes that we have assumed in the form of expressions for φ and φ/φ, written in terms of the scale factor (a). Using the
values of some cosmological parameters, obtained from recent observations, we have determined the values of the constants
involved in those ansatzes.
3.1. Model-1
In this model we have assumed the following ansatz for the scalar field (φ).
φ = φ0
(a
a0
)n(14)
Here, φ0 and a0 are the values of φ and a at the present time (i.e., at t = t0) and n is a constant parameter. Using equation
(14) in equation (13), one gets the following equation.
a
a+ (n+ 2)
(a
a
)2
=1
(n+ 2)
ρmφ0
(a
a0
)−n
(15)
Using the definitions of the Hubble parameter(H = a
a
)and deceleration parameter
(q = − aa
a2
), equation (15) can be written
as,
− qH2 (n+ 2) +H2(n+ 2)2 =ρmφ0
(a
a0
)−n
(16)
Putting a = a0, H = H0, q = q0 and ρm = ρm0 (for t = t0) in equation (16) we get,
H20n
2 +(4H2
0 − q0H20
)n+ 4H2
0 − 2q0H20 −
ρm0
φ0= 0 (17)
Equation (17) is quadratic in n. Two roots of this equation are given by,
n1,2 =
−(4H2
0 − q0H20
)±√(
4H20 − q0H2
0
)2 − 4H20
(4H2
0 − 2q0H20 −
ρm0φ0
)2H2
0
(18)
The subscripts 1 and 2 correspond to plus and minus signs respectively in the above equation. Using the values of cosmological
parameters, we get n1 = −1.94 and n2 = −2.61.
According to equation (14), a negative value of the parameter n corresponds to a decrease of the scalar field (φ) with time,
since the scale factor (a) increases with time for an expanding universe. Thus, G (≡ 1/φ) increases with time for negative
values of n. The expression for the gravitational constant, based on equation (14), is given by,
G =1
φ=
1
φ0
(a
a0
)−n
(19)
In terms of the cosmological redshift parameter(z ≡ a0
a− 1), the gravitational constant can be expressed as,
G =(z + 1)n
φ0(20)
Figure 1 shows the variation of G as a function of the redshift parameter (z ), for the two values of n (i.e., n1 and n2)
obtained from equation (18).
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Sudipto Roy
3.2. Model-2
In this model we have assumed the following ansatz for the scalar field (φ).
φ = φ0Exp [m (a− a0)] (21)
Here, φ0 and a0 are the values of φ and a at the present time (i.e., at t = t0) and m is a constant parameter. Using equation
(21) in equation (13) and using the definitions of Hubble parameter(H = a
a
)and deceleration parameter
(q = − aa
a2
), one
gets,
H2a2m2 +(5aH2 − qaH2)m+ 4H2 − 2qH2 − ρm
φ= 0 (22)
Replacing the cosmological parameters in equation (22) by their values at the present time (i.e., at t = t0), one gets the
following equation.
H20m
2 +(5H2
0 − q0H20
)m+ 4H2
0 − 2q0H20 −
ρm0
φ0= 0 (23)
In obtaining equation (23) from (22), we have chosen a scale for ‘a’ such that a0 = 1. Equation (23) is quadratic in m. Two
roots of this equation are given by,
m1,2 =
−(5H2
0 − q0H20
)±√(
5H20 − q0H2
0
)2 − 4H20
(4H2
0 − 2q0H20 −
ρm0φ0
)2H2
0
(24)
The subscripts 1 and 2 of m correspond to plus and minus signs respectively in the above equation. Using the values of
cosmological parameters, we get m1 = −1.15 and m2 = −4.40.
According to equation (21), a negative value of the parameter m corresponds to a decrease of the scalar field (φ) with
time, since the scale factor (a) increases with time for an expanding universe. Thus, G (≡ 1/φ) decreases with time due to
negative values of m.
The expression for the gravitational constant, based on equation (21), is given by,
G =1
φ=
1
φ0Exp [−m (a− a0)] (25)
In terms of the cosmological redshift parameter(z = a0
a− 1), the gravitational constant can be expressed as,
G =1
φ0Exp
[m
1 + 1/z
](26)
In obtaining equation (26) from (25), we have chosen a scale for ‘a’ such that a0 = 1. Figure 2 shows the variation of G as
a function of the redshift parameter (z ), for the two values of m (m1 and m2).
3.3. Model-3
For this model we have assumed the following ansatz for φ/φ.
φ
φ= Aak (27)
Here, A and k are constants. Substituting equation (27) into equation (13) and replacing all parameters by their values at
the present time (t = t0) we get,
k =ρ0/φ0 − 5H0A−A2 − 4H0
2 + 2q0H02
AH0(28)
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A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory
Since G = 1/φ, we have GG
= − φφ
= −Aak. Thus, the order of magnitude of A must be the same as that of(GG
)0
obtained
from observations, which is around 10−11 Y r−1 [23]. The sign of A should be chosen to be negative because it has been
obtained from Model-1 and Model-2 that G increases with time.
Taking A = −10−18sec−1 (i.e., −3.15 × 10−11Y r−1) we get k = 7.52 from equation (28). Thus, we have,(GG
)0
= −A =
3.15× 10−11Y r−1, based on a scale for which a0 = 1.
Using equation (27), GG
can be expressed (as a function of z) as,
G
G= − φ
φ= −Aak = −A
(1
z + 1
)k(29)
In obtaining equation (29) from (27), we have chosen a scale for ‘a’ such that a0 = 1. Since A is negative and k is positive,
G/G of the universe must have increased as z has decreased to reach its present value (i.e., z = 0). Figure-3 shows the
variation of G/G as a function of z, as per equation (29).
3.4. Model-4
For this model we have assumed the following ansatz for φ/φ.
φ
φ= B Exp[la] (30)
Here, B and l are constants. Since G = 1/φ, we have GG
= − φφ
= −B Exp[la]. The parameter B must have a negative value
to ensure that G increases with time, in accordance with the findings based on Model-1 and Model-2. Substituting equation
(30) into equation (13) and replacing all parameters by their values at the present time (i.e., t = t0) we get,
4H02 − 2q0H0
2 =ρ0φ0− 5H0Be
l −(Bel
)2−BlH0e
l (31)
Here, one needs to find the values of B and l that satisfy equation (31). To determine these values, we have defined a
function Y (based on equation 31), which is expressed as,
Y = 4H02 − 2q0H0
2 − ρ0φ0
+ 5H0Bel +(Bel
)2+BlH0e
l (31A)
The above expression has been obtained by subtracting the right-hand side of equation (31) from its left-hand side. Ideally
one has to find the combination of values for B and l for which we have Y = 0. Numerically, it would only be possible
to find such values (of B and l) for which Y would be extremely small (i.e., sufficiently smaller than the value of any of
the terms in the expression for Y). Choosing B = −1× 10−18 sec−1 (i.e., −3.15 × 10−11Y r−1), the value of Y has been
found (by trial and error using Microsoft-Excel) to be extremely small (1.98× 10−40) for l = 0.9105. This combination of
values leads to the result of(GG
)0
= −(φφ
)0
= −Bel = 2.48× 10−18Sec−1 (i.e., 7.83× 10−11Y r−1) which is consistent with
astrophysical observations [23]. The scaling for a has been chosen to be such that a0 = 1.
Using equation (30), GG
can be expressed (as a function of z) as,
G
G= −B Exp [la] = −B Exp
[l
z + 1
](32)
Since l is positive and B is negative in equation (32), G/G of the universe must have increased as z has decreased to reach
its present value (i.e., z = 0). Figure-4 shows the variation of G/G as a function of z, as per equation (32).
According to Model-1 and Model-2, the gravitational constant (G) increases with time, causing G/G to have a positive
value. For the sake of consistency with these findings we have chosen both A and B (of equations 27 & 30 respectively) to
have negative values while determining the values of other constants (k and l) connected to Models-3 & 4.
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Sudipto Roy
4. Time Dependence of the Gravitational Constant (G)
It has been obtained from Models 1-4 that both G and G/G increase with time. Based on these behaviours, we propose the
following empirical relation for G.
G ≡ 1
φ= α Exp (βtγ) (33)
Here, α, β and γ are constants with positive values. Using equation (33), the expression for G/G is obtained as,
G
G≡ − φ
φ= βγtγ−1 (34)
It is clearly evident from equations (33) and (34) that, both G and G/G should be increasing with time if we have γ > 1.
4.1. Time Dependence of Redshift (z) from Model-1
Using equation (33) in equation (14) we get,
a = a0[αφ0 Exp(βtγ)]−1/n (35)
Subjecting equation (35) to the condition that a = a0 at t = t0, we get,
α =Exp(−βt0γ)
φ0(36)
Equation (36) is a relation between the parameters α, β and γ. Using equation (36) in (33) we get,
G ≡ 1
φ=
1
φ0Exp [β (tγ − t0γ)] (37)
Using equation (36) in (35) we get,
a = a0Exp
[−βn
(tγ − t0γ)
](38)
Using equation (38), the expression for the redshift (z) can be written as,
z =a0a− 1 = Exp
[β
n(tγ − t0γ)
]− 1 (39)
Equation (39) shows how the redshift (z) varies with time, based on Model-1 and the empirical relation for G expressed by
equation (33). In Figures 1-4 we have plotted everything as a function of z and here we have its time dependence. The
values of the parameter n, to be used in equation (39), should be in accordance with equation (18). According to a study by
G. K. Goswami, the signature flip of the deceleration parameter took place at z = 0.6818 when the age of the universe was
nearly 7.2371 × 109 years, i.e., at around t = 0.55t0 where t0 is the present age of the universe (t0 = 13.0847 × 109 years)
[24]. For each of the two values of n, the values of the parameters (β and γ) can be so chosen that we get z = 0.6818 at
t = 0.55t0 from equation (39). For n = n1, these values are: β = 5.16 × 10−18 and γ = 1.1. For n = n2, these values are:
β = 6.94× 10−18 and γ = 1.1.
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A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory
4.2. Time Dependence of Redshift (z) from Model-2
Using equation (33) in equation (21) we get,
a = a0 −βtγ + ln (αφ0)
m(40)
Subjecting equation (40) to the condition that a = a0 at t = t0, we get,
α =Exp(−βt0γ)
φ0(41)
The expression for α, given by equation (41), is the same as equation (36) which was obtained by combining equation (33)
with Model-1. Substituting equation (41) in (33) we get,
G ≡ 1
φ=
1
φ0Exp [β (tγ − t0γ)] (42)
The expression for G, given by equation (42), is identical to equation (37) obtained by combining equation (33) with Model-1.
Using equation (41) in (40) we get,
a = a0 −β (tγ − t0γ)
m(43)
Using equation (43), the expression for the redshift (z) can be written as,
z =a0a− 1 =
ma0ma0 − β (tγ − t0γ)
− 1 (44)
Equation (44) expresses the redshift (z) as a function of time, based on Model-2 and the empirical relation for G expressed
by equation (33). In Figures 1-4 we have plotted everything as a function of z and here we have its time dependence. The
values of the parameter m, to be used in equation (44), should be in accordance with equation (24). According to a study
by G. K. Goswami, the signature flip of the deceleration parameter took place at z = 0.6818 when the age of the universe
was nearly 7.2371× 109 years, i.e., at around t = 0.55 t0 where t0 is the present age of the universe (with t0 = 13.0847× 109
years) [24]. For each of the two values of m, the values of the parameters (β and γ) can be so chosen that we get z = 0.6818
at t = 0.55t0 from equation (44). For m = m1, these values are: β = 2.38 × 10−18 and γ = 1.1. For n = n2, these values
are: β = 9.12× 10−18 and γ = 1.1.
5. Results and Discussion
Figure 1 shows the variation of the ratio G/G0 as a function of the redshift parameter (z) for two values of the parameter
n, denoted by n1 and n2, obtained from equation (18) of Model-1. These two values are, n1 = −1.94 and n2 = −2.61. As z
approaches its present value (i.e., z = 0), both graphs show a rise in G/G0. The curve for n = n2 has a steeper rise (as z
decreases), in comparison to the other curve, in the regions closer to the present time (i.e., z = 0). The ratio G/G0 has a
higher value for n = n1 over the entire range of z values.
Figure 2 depicts the evolution of the ratio G/G0 as a function of the redshift parameter (z) for two values of the parameter
m, denoted by m1 and m2, obtained from equation (24) of Model-2. These two values are, m1 = −1.15 and m2 = −4.40.
As z approaches its present value (i.e., z = 0), both graphs show a rise in G/G0. The curve for m = m2 has a steeper rise
(as z decreases), in comparison to the other curve, in the regions closer to the present time (i.e., z = 0). The ratio G/G0
has a higher value for m = m1 over the entire range of z values.
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Sudipto Roy
Based on our calculations of Model-1 and Model-2, the first two figures show that the gravitational constant (G) increases
with time (as z decreases with time) with a gradually increasing rate with respect to z. It would require a more rigorous
study, based on astrophysical observations, to determine which of the two values of each of the two parameters (m & n)
would lead to a more accurate finding regarding the time variation of the gravitational constant. The theoretical validity of
the quadratic equations (equations (17) & (23)) of the first two models (from which n & m have been determined) lies in
the correctness of the assumption that the solutions of the field equations will lead exactly to the presently observed values
of the cosmological parameters (H0, q0, φ0, ρm0) at t = t0. Without solving the field equations, the correctness of this
assumption cannot be fully judged.
Figure 3 shows the dependence of G/G upon the redshift parameter (z), based on Model-3. As z approaches its present
value (i.e., z = 0), G/G is found to rise at a gradually increasing rate with respect to z. Figure 4 shows the plot of G/G as
a function of the redshift parameter (z), based on Model-4. As z approaches its present value (i.e., z = 0), G/G is found to
increase at a gradually increasing rate with respect to z. Since the redshift (z) decreases with time in an expanding universe,
the relative time-rate of change of G increases with time as per Model-3 and Model-4, as evident from Figures 3 & 4.
Figures 3 & 4 (which are based on Models-3 & 4 respectively) show qualitatively similar behaviours with regard to the
variation of G/G as a function of the redshift parameter (z). A different set of values for the parameters (A, k, B, l), chosen
in accordance with equations (28) and (31), would change these behaviours but qualitatively they would remain the same.
For both plots, G/G has positive values due to the fact that A and B were chosen to have negative values while determining
the values of the parameters k and l respectively, to be consistent with the results obtained from Models-1 & 2 where G
has been shown to be increasing with time. This increasing trend of the gravitational constant with time is quite consistent
with the findings of some other recent studies based on completely different theoretical models [25, 26].
Figures 5-8 show the variation of the redshift parameter (z) as a function of time. Figures 5 & 6 are based on Model-1 and
Figures 7 & 8 are based on Model-2. According to a recent study by G. K. Goswami, in the framework of Brans-Dicke theory,
the change of phase of the universe from decelerated expansion to accelerated expansion took place around z = 0.6818 when
the age of the universe was nearly 7.2371 × 109 years, its present age being 13.0847 × 109 years [24]. According to this
information regarding cosmic expansion, the value of z should be around 0.6818 at t = 0.55t0. Tuning the parameters (β &
γ) we have obtained this result, as shown by Figures 5-8. These plots are in qualitative agreement with the findings of a
recent study in the framework of Brans-Dicke theory [27].
In Figures 1-4, variation of everything has been shown as a function of redshift (z). In Figures 5-8, we have shown the time
dependence of redshift (z). From these two sets of figures, one can obtain information regarding the time variation of G and
G/G.
6. Conclusions
In the present article, the variation of the gravitational constant (G) as a function of the redshift parameter (z) has been
obtained from Brans-Dicke field equations without actually solving those equations. Unlike other studies in this field, we
have not derived the expression for the scale factor (from the field equations) as a function of time to find the nature of time
dependence of G. We have only used four ansatzes (equations (14), (21), (27), (30)), involving the scalar field (φ) and the
scale factor (a), to obtain all results.
The parameters, n and m, in Models-1 & 2 respectively, govern the change of the scalar field (φ) as a function of the scale
factor (a). Two values of each of these parameters have been found to be negative, indicating a reduction in the value of
the scalar field (φ) as the scale factor increases with time, implying a rise in the value of G (≡ 1/φ) with time. Thus, an
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A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory
important finding of the present study is that the gravitational constant increases with time, which is consistent with the
results of studies carried out by methods quite different from the present one [25, 26].
The relative time-rate of change of the gravitational constant (i.e., G/G) has been found to be increasing with time from the
Models-3 & 4, indicating a faster rise in G with time compared with the rate of increase of G. Figures 4-8 depict the time
evolution of redshift (z). Using these figures along with the Figures 1-4, one can find the nature of time dependence of G and
G/G. One may also use the equations (37) or (42) to find the time dependence of G, as per Models-1 & 2. As an extension of
this work, one may think of combining equation (34) with equations (27) and (30) (of Models-3 & 4 respectively) to obtain
time-dependent expressions for the scale factor and thereby calculate the density of matter (ρm) and other parameters from
the field equations.
One may also think about carrying out a study on the time dependence of the Hubble parameter (H = a/a) and deceleration
parameter (q = −aa/a2), using the expressions for the scale factor (a) (given by equations (38) & (43)) derived in the present
article. For a greater accuracy of results, one may consider using a new set of ansatzes, as part of a future project to determine
the characteristics of G and G/G.
The novel aspect of the present study is that the nature of evolution of the gravitational constant and its relative time-rate
of change has been obtained through calculations which are much simpler in comparison with other recent studies in the
same field.
This article provides one with a sufficiently simple mathematical method to determine the characteristics of these cosmological
parameters and this method can be improved in a number of ways by further explorations of the same kind into the field.
Acknowledgment
The author of this article is very much thankful to all researchers whose works have enriched him immensely and inspired
him to carry out the studies for the present work. The author would also like to thank, very sincerely, the editor and the
reviewers of the journal for their constructive comments and suggestions.
Figure 1. Plots of G/G0 versus redshift (z) for two values of n, based on Model-1
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Sudipto Roy
Figure 2. Plots of G/G0 versus redshift (z) for two values of m, based on Model-2
Figure 3. Plot of G/G versus redshift (z), based on Model-3
Figure 4. Plot of G/G versus redshift (z), based on Model-4
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A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory
Figure 5. Plot of redshift (z) versus time, for n = n1, based on Model-1
Figure 6. Plot of redshift (z) versus time, for n = n2, based on Model-1
Figure 7. Plot of redshift (z) versus time, for m = m1, based on Model-2
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Figure 8. Plot of redshift (z) versus time, for m = m2, based on Model-2
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