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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

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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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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)

71


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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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.

74

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

75


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

76

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

77


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

78

Sudipto Roy

Figure 8. Plot of redshift (z) versus time, for m = m2, based on Model-2

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