In the framework of Brans-Dicke (BD) theory, the first part of the present study determines the time dependence of BD parameter, energy density and Equation of State (EoS) parameter of the cosmic fluid in a universe expanding with acceleration, preceded by a phase of deceleration. For this purpose, a scale factor has been so chosen that the deceleration parameter, obtained from it, shows a signature flip with time. Considering the dark energy to be responsible for the entire pressure, the time evolution of energy parameters for matter and dark energy and the EoS parameter for dark energy have been determined. A model for an effective interaction term, between matter and dark energy, has been proposed and calculated. Its negative value at the present time indicates conversion of matter into dark energy. Using this term, the time dependence of the rates of change of matter and dark energy has been determined. It is found that the nature of dependence of the scalar field upon the scale factor plays a very important role in governing the time evolution of the cosmological quantities studied here. The present study provides us with a simple way to determine the time evolution of dark energy for a homogeneous and isotropic universe of zero spatial curvature, without involving any self-interaction potential or cosmological constant in the formulation.
This document presents a five dimensional cosmological model with a perfect fluid coupled to a massless scalar field in general relativity. The field equations are solved assuming an equation of state of p=ρ and a relation between the metric potentials of R=kAn, where k and n are constants. The solutions show the scale factors and scalar field as functions of time. The model expands anisotropically with no initial singularity and decelerates similarly to standard cosmology. Physical quantities like density and pressure diverge initially but vanish at later times.
1) The document discusses testing modifications to general relativity (GR) to explain the observed accelerated expansion of the universe. Two representative modified gravity (MG) models are studied: f(R) theories and the Bean-Tangmatitham parametrization.
2) These MG models modify the gravitational potentials in GR through additional parameters that can generate different gravitational behaviors, including those described by GR.
3) The MG parameters are constrained using CosmoSIS, a cosmological parameter estimation code, with data from the Cosmic Microwave Background and weak lensing surveys. Estimates of the MG parameters suggest no deviation from GR predictions.
Using resonant ultrasound spectroscopy (RUS), the author will determine the complete elastic constant matrices of two thermoelectric single crystal samples, Ce.75Fe3CoSb12 and CeFe4Sb12. RUS involves measuring the resonant frequencies of a sample's vibrations, which depend on the sample's elastic constants, shape, orientation, and density. The author aims to obtain the elastic moduli from a single RUS spectrum for each sample. Understanding the elastic properties may help identify better thermoelectric materials by correlating low elastic stiffness with low thermal conductivity and higher thermoelectric efficiency. The author will compute the resonant frequencies using the samples' properties and compare to measurements.
Bianchi type i wet dark universe in bimetric relativityAlexander Decker
This academic article summarizes a document investigating the role of Wet Dark Fluid in a Bianchi Type-I cosmological model within the framework of bimetric theory of relativity. The document proposes a new equation of state for dark energy known as Wet Dark Fluid and uses it to study a Bianchi Type-I universe model in bimetric relativity. It is concluded that the Bianchi Type-I model in bimetric relativity does not accommodate Wet Dark Fluid, as both the Wet Dark Fluid energy density and metric terms are found to be identically zero, resulting in only a vacuum solution.
Two-temperature elasto-thermo diffusive response inside a spherical shell wit...IJERA Editor
The present work deals with the investigation of elasto-thermo diffusion interaction of a homogeneous isotropic
spherical shell in the context of two-temperature generalized theory of thermo-elasticity with diffusion. The
inner and outer boundaries of the spherical shell are free from stress and subjected to a time dependent thermal
stoke. The chemical potential is also assumed to be a function of time on the boundary of the shell. The
governing equations are solved in the Laplace transformation by using aoperator theory. The inverse of the
transformed solution is carried out by applying a method of Bellman et al.. The stress, conductive temperature,
displacement, mass concentration and chemical potential are computed numerically and presented graphically in
a number of figures for copper material. A comparison of the results for three different models two-temperature
Lord Shulman model (2TLS), two-temperature Green Naghdi model (2TGN-III) and two-temperature threephase-
lag model (2T3P) arealso presented for the different types of temperature field (one-temperature and twotemperature).
An exact solution of einstein equations for interior field of an anisotropic ...eSAT Journals
Abstract
In this paper, an anisotropic relativistic fluid sphere with variable density, which decreases along the radius and is maximum at
the centre, is discussed. Spherically symmetric static space-time with spheroidal physical 3-space is considered. The source is an
anisotropic fluid.
The solution is an anisotropic generalization of the solution discussed by Vaidya and Tikekar [1]. The physical three space
constant time has spheroidal solution. The line element of the solution can be expressed in the form Patel and Desai [2]. The
material density is always positive. The solution efficiently matches with Schwarzschild exterior solution across the boundary. It is
shown that the model is physically reasonable by studying the numerical estimates of various parameters. The density vs radial
pressure relation in the interior is discussed numerically. An anisotropy effect on the redshift is also studied numerically.
Key Words: Cosmology, Anisotropic fluid sphere, Radial pressure, Radial density, Relativistic model.
Energy in form of space may solve the dark energy problemPremier Publishers
A review of recent observations suggests a universe that is light weight (matter density is 1/3rd of the critical value), accelerating and flat. This implies the existence of a cosmic Dark Energy that overcomes the gravitational self-attraction force of matter and causes the accelerating expansion. Finding out the cause of expansion and acceleration of the universe is a challenging job in present day cosmology. Cosmological models with different types of dark energy are becoming viable standard models to analyze and simulate experimental data from a number of high red shift supernovae. In this article, physical significance and analytical expression for dark energy related to total energy (or energy density) and matter (or matter density) in the universe is presented. It is assumed that 'space' or 'vacuum' is another form of energy (other form is mass which is related as E = mc2). With this assumption new cosmological equation of state is constructed which is in very good agreement with present observations. Thus energy evolves from matter to radiation to space. It is also predicted that the existence of a fundamental particle with mass less than the mass of a quark is possible.
Exact Solutions of Axially Symmetric Bianchi Type-I Cosmological Model in Lyr...IOSR Journals
In this paper we have obtained axially symmetric Bianchi type-I cosmological models for perfect
fluid distribution in the context of Lyra’s manifold. Exact solutions of the field equations are obtained by
assuming the expansion in the model is proportional to the shear . This leads to the condition
A Bn
where A and B are scale factors and n( 0) is a constant. Some kinematical and physical parameters of the
model have been discussed. The solutions are compatible with recent observations.
This document presents a five dimensional cosmological model with a perfect fluid coupled to a massless scalar field in general relativity. The field equations are solved assuming an equation of state of p=ρ and a relation between the metric potentials of R=kAn, where k and n are constants. The solutions show the scale factors and scalar field as functions of time. The model expands anisotropically with no initial singularity and decelerates similarly to standard cosmology. Physical quantities like density and pressure diverge initially but vanish at later times.
1) The document discusses testing modifications to general relativity (GR) to explain the observed accelerated expansion of the universe. Two representative modified gravity (MG) models are studied: f(R) theories and the Bean-Tangmatitham parametrization.
2) These MG models modify the gravitational potentials in GR through additional parameters that can generate different gravitational behaviors, including those described by GR.
3) The MG parameters are constrained using CosmoSIS, a cosmological parameter estimation code, with data from the Cosmic Microwave Background and weak lensing surveys. Estimates of the MG parameters suggest no deviation from GR predictions.
Using resonant ultrasound spectroscopy (RUS), the author will determine the complete elastic constant matrices of two thermoelectric single crystal samples, Ce.75Fe3CoSb12 and CeFe4Sb12. RUS involves measuring the resonant frequencies of a sample's vibrations, which depend on the sample's elastic constants, shape, orientation, and density. The author aims to obtain the elastic moduli from a single RUS spectrum for each sample. Understanding the elastic properties may help identify better thermoelectric materials by correlating low elastic stiffness with low thermal conductivity and higher thermoelectric efficiency. The author will compute the resonant frequencies using the samples' properties and compare to measurements.
Bianchi type i wet dark universe in bimetric relativityAlexander Decker
This academic article summarizes a document investigating the role of Wet Dark Fluid in a Bianchi Type-I cosmological model within the framework of bimetric theory of relativity. The document proposes a new equation of state for dark energy known as Wet Dark Fluid and uses it to study a Bianchi Type-I universe model in bimetric relativity. It is concluded that the Bianchi Type-I model in bimetric relativity does not accommodate Wet Dark Fluid, as both the Wet Dark Fluid energy density and metric terms are found to be identically zero, resulting in only a vacuum solution.
Two-temperature elasto-thermo diffusive response inside a spherical shell wit...IJERA Editor
The present work deals with the investigation of elasto-thermo diffusion interaction of a homogeneous isotropic
spherical shell in the context of two-temperature generalized theory of thermo-elasticity with diffusion. The
inner and outer boundaries of the spherical shell are free from stress and subjected to a time dependent thermal
stoke. The chemical potential is also assumed to be a function of time on the boundary of the shell. The
governing equations are solved in the Laplace transformation by using aoperator theory. The inverse of the
transformed solution is carried out by applying a method of Bellman et al.. The stress, conductive temperature,
displacement, mass concentration and chemical potential are computed numerically and presented graphically in
a number of figures for copper material. A comparison of the results for three different models two-temperature
Lord Shulman model (2TLS), two-temperature Green Naghdi model (2TGN-III) and two-temperature threephase-
lag model (2T3P) arealso presented for the different types of temperature field (one-temperature and twotemperature).
An exact solution of einstein equations for interior field of an anisotropic ...eSAT Journals
Abstract
In this paper, an anisotropic relativistic fluid sphere with variable density, which decreases along the radius and is maximum at
the centre, is discussed. Spherically symmetric static space-time with spheroidal physical 3-space is considered. The source is an
anisotropic fluid.
The solution is an anisotropic generalization of the solution discussed by Vaidya and Tikekar [1]. The physical three space
constant time has spheroidal solution. The line element of the solution can be expressed in the form Patel and Desai [2]. The
material density is always positive. The solution efficiently matches with Schwarzschild exterior solution across the boundary. It is
shown that the model is physically reasonable by studying the numerical estimates of various parameters. The density vs radial
pressure relation in the interior is discussed numerically. An anisotropy effect on the redshift is also studied numerically.
Key Words: Cosmology, Anisotropic fluid sphere, Radial pressure, Radial density, Relativistic model.
Energy in form of space may solve the dark energy problemPremier Publishers
A review of recent observations suggests a universe that is light weight (matter density is 1/3rd of the critical value), accelerating and flat. This implies the existence of a cosmic Dark Energy that overcomes the gravitational self-attraction force of matter and causes the accelerating expansion. Finding out the cause of expansion and acceleration of the universe is a challenging job in present day cosmology. Cosmological models with different types of dark energy are becoming viable standard models to analyze and simulate experimental data from a number of high red shift supernovae. In this article, physical significance and analytical expression for dark energy related to total energy (or energy density) and matter (or matter density) in the universe is presented. It is assumed that 'space' or 'vacuum' is another form of energy (other form is mass which is related as E = mc2). With this assumption new cosmological equation of state is constructed which is in very good agreement with present observations. Thus energy evolves from matter to radiation to space. It is also predicted that the existence of a fundamental particle with mass less than the mass of a quark is possible.
Exact Solutions of Axially Symmetric Bianchi Type-I Cosmological Model in Lyr...IOSR Journals
In this paper we have obtained axially symmetric Bianchi type-I cosmological models for perfect
fluid distribution in the context of Lyra’s manifold. Exact solutions of the field equations are obtained by
assuming the expansion in the model is proportional to the shear . This leads to the condition
A Bn
where A and B are scale factors and n( 0) is a constant. Some kinematical and physical parameters of the
model have been discussed. The solutions are compatible with recent observations.
This document summarizes a physics research project on models of cosmological inflation. The project analytically and numerically evaluates single field models, multifield hybrid models, and f(R) models arising from modified gravity. It calculates the dynamics of these models and makes theoretical predictions for observables like the scalar spectral index and tensor-to-scalar ratio. These predictions are compared to experimental data to determine which models produce viable inflationary theories. The project reproduces known single field results and explores higher-order f(R) terms, showing they can only be a small correction to Starobinsky inflation. It also evaluates a multifield hybrid potential and how two fields can combine to produce inflation.
This document presents a study of five dimensional string cosmological models with bulk viscosity in general relativity. It investigates three cases of the field equations and determines solutions for each. For case I, the solution reduces the model to Minkowski spacetime. For case II, the solution yields a model that expands from a big bang and stops expanding at a finite time, with the volume increasing over time. The energy density is only due to bulk viscosity and Lyra geometry. For case III, three specific solutions are presented: one that reduces to a flat 5D spacetime; one that contracts/expands depending on constants; and one that follows power law inflation. Physical properties like expansion, shear, and density are also analyzed for each case.
Holographic Dark Energy in LRS Bianchi Type-II Space Timeinventionjournals
The document summarizes research on holographic dark energy in an anisotropic cosmological model. Key points:
- The study investigates holographic dark energy in a LRS Bianchi type-II spacetime using two assumptions: a law of variation for the Hubble parameter and a relation between metric coefficients.
- The law of variation generates power-law and exponential forms for the average scale factor in terms of cosmic time.
- For n not equal to 0, the holographic dark energy EOS parameter behaves like quintessence; for n equal to 0 it behaves like a cosmological constant.
- Solutions to the field equations are obtained for both cases of n not equal to
This document discusses energy-size reduction relationships in comminution processes. It outlines three main theories proposed to explain the relationship between energy input and size reduction: Rittinger's law, Kick's law, and Bond's law. The author proposes a new general form for the energy-size reduction relationship that is a linear combination of Kick-type and Rittinger-type laws. Empirical data from various studies is used to validate that this combined law provides a better fit than previous generalized laws, particularly in intermediate size ranges. The combined law also approximates to Kick-type and Rittinger-type laws at size extremes as previously observed. This combined law may have applications in computer simulations of comminution processes and mill
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
MHD Natural Convection Flow of an incompressible electrically conducting visc...IJERA Editor
We consider a two-dimensional MHD natural convection flow of an incompressible viscous and electrically
conducting fluid through porous medium past a vertical impermeable flat plate is considered in presence of a
uniform transverse magnetic field. The governing equations of velocity and temperature fields with appropriate
boundary conditions are solved by the ordinary differential equations by introducing appropriate coordinate
transformations. We solve that ordinary differential equations and find the velocity profiles, temperature profile,
the skin friction and nusselt number. The effects of Grashof number (Gr), Hartmann number (M) and Prandtl
number (Pr), Darcy parameter (D-1) on velocity profiles and temperature profiles are shown graphically.
On the Logical Origin of the Laws Governing the Fundamental Forces of Nature:...Ramin (A.) Zahedi
- The document presents a new algebraic-axiomatic matrix approach to derive the fundamental laws of nature from basic axioms.
- It is shown that the laws governing fundamental forces like gravity and electromagnetism can be derived as general covariant massive field equations from axioms assuming rational relativistic energy-momentum and linearizing energy-momentum relations.
- The derived field equations include a torsion field that generates the invariant mass of each field and represent generalized forms of Einstein, Maxwell, Yang-Mills, Dirac, and Rarita-Schwinger equations.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Investigation of mgx sr1 xo mixed alloy under high pressureAlexander Decker
This document reports on research investigating the structural and mechanical properties of the mixed compound MgxSr1-xO under high pressure. An extended interaction potential model was applied that included zero point energy effects to predict phase transition pressures and volume collapses. The model parameters were determined using equilibrium conditions and followed Vegard's law varying linearly with concentration. The results for the mixed compounds were in fair agreement with experimental data generated by applying Vegard's law to the endpoint members MgO and SrO.
Bulk Viscous String Cloud with Strange Quark Matter in Self Creation Cosmologyiosrjce
We have investigated strange quark matter attached to the string cloud with bulk viscous fluid for
higher dimensional Friedman-Robertson-Walker (FRW) universe in Self Creation Cosmology (SCC). We have
obtained zero string density (휌푠 = 0) in this model. Our solutions agree with the studies of Reddy, SahooMishra
and Mohanty et al. in four dimensions.
This document summarizes and generalizes the concept of potential vorticity (PV) in fluid mechanics using the mathematical tools of differential geometry. It defines a PV 3-form field A in spacetime that unifies the PV scalar and 3D PV flux vector into a single object. Tensor operations are used to prove conservation theorems that generalize previous results. An analogy is drawn between the generalized structure of equations for fluid mechanics and electromagnetism. Finally, it indicates how the framework enables a linear inversion scheme with higher-order derivatives to solve for flow variables.
This paper develops a mathematical model to analyze the kinetic energy present in a stable bathtub vortex. It uses Burgers' vortex model to describe the azimuthal velocity field and derives an expression for kinetic energy by integrating over the cylindrical tank volume. The model shows that kinetic energy increases linearly with tank height and Reynolds number. It also indicates that a turbine placed at 20% of the tank radius would optimally capture energy, as this corresponds to the peak kinetic energy region away from the vortex core. The paper concludes by discussing opportunities to improve the model by incorporating more realistic fluid effects and accounting for turbine properties.
This paper proposes a maximum likelihood estimation method for determining the parameters of a modified Gutenberg-Richter magnitude-frequency relation proposed by Lomnitz-Adler and Lomnitz. The relation is a double exponential function with parameters a and B. The maximum likelihood method provides equations that allow iterative determination of parameter a from earthquake catalog data. Parameter B can then be estimated from the data and value of a. The method is demonstrated on a Chinese earthquake catalog, obtaining values of a = 0.10 and B = 10.34.
The document discusses groups and graphs in probability theory. It provides background on the probability that a group element fixes a set and related work applying this concept to graph theory, specifically generalized conjugacy class graphs and orbit graphs. The main results are:
1) For a dihedral group of order 2n acting regularly on the set of commuting pairs of elements, the probability that a group element fixes a set is 1/n.
2) Under this action, the generalized conjugacy class graph of the dihedral group has n isolated vertices and is empty.
3) The orbit graph vertices correspond to the n orbits of sizes 1 or 2n, with an edge between vertices if their corresponding orbits are conjugate.
Five Dimensional Bianchi Type-V Space-Time in f (R,T) Theory of GravitywIJERA Editor
We study the spatially homogeneous anisotropic Bianchi type-V universe in f(R,T) theory of gravity, where R is
the Ricci scalar and T is the trace of the energy-momentum tensor. We assume the variation law of mean
Hubble parameter and constant deceleration parameter to find two different five dimensional exact solutions of
the modified field equations. The first solution yields a singular model for n 0 while the second gives a nonsingular
model for n 0. The physical quantities are discussed for both models in future evolution of the
universe.
Quantum approach to fast protein folding timepacao1
This document proposes a quantum approach to modeling protein folding that provides a general framework without artificial hypotheses. It formulates protein folding as a quantum walk on a definite graph based on a lattice model of protein structure. Evaluating the folding time using the mean first passage time, it finds that the folding time via this quantum approach is much shorter than that obtained via classical random walks. This quantum strategy is expected to provide new insights into theoretical studies of protein folding.
This document summarizes a numerical study of the structure and thermodynamics of colloidal suspensions using the variational method and integral equation theory. The interactions between colloid particles are modeled using either a Yukawa or Sogami potential. Results from the integral equation theory using a Sogami potential are found to be in good agreement with Monte Carlo simulation results and experimental data. The variational method and integral equation theory are used to calculate structural properties like the pair correlation function and thermodynamic properties.
This document discusses nonlinear electrodynamics theories in the context of black hole solutions to the Einstein field equations. It presents a generalization of the Christodoulou-Ruffini mass formula for charged black holes in the weak field limit of nonlinear electrodynamics theories. The paper proves that the outer horizon of black holes never decreases under reversible transformations, and that such transformations are equivalent to solutions with a constant horizon for asymptotically flat black hole solutions of nonlinear theories. It then uses this result to decompose the total mass-energy of black holes in nonlinear theories in terms of irreducible and extractable quantities.
Five-Dimensional Cosmological Model with Time-Dependent G and Lamda for Const...IOSR Journals
In this paper we have consider five-dimensional cosmological model in presence of perfect fluid
source with time dependent G and .The Einstein field equations are solvable with the help of constant
deceleration parameter. Physical and kinematical properties of this model are investigated. It has been shown
that the solutions are comparable with recent observations. The behavior of gravitational constant,
cosmological constant, density, critical density and pressure is discussed for dust, radiation dominated and stiff matter of the Universe. It is also examined the behavior of gravitational constant and cosmological constant for expansion law and exponential law for stiff matter
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...IOSR Journals
An anisotropic Bianchi type-III cosmological model is investigated in a Saez-Ballester scalar-tensor theory of gravitation. Three different time-dependent skewness parameters along spatial directions are introduced to represent the deviation of pressure from isotropy. To get deterministic solutions of the field equations, we choose variation law of scale factor 𝑆= (𝑡𝑟𝑒𝑡)1𝑙 which yields a time-dependent deceleration parameter (DP) representing a model that generates a transition of the universe from the early decelerating phase to the present accelerating phase. Some physical and geometrical properties of the model are also discussed.
This document summarizes a physics research project on models of cosmological inflation. The project analytically and numerically evaluates single field models, multifield hybrid models, and f(R) models arising from modified gravity. It calculates the dynamics of these models and makes theoretical predictions for observables like the scalar spectral index and tensor-to-scalar ratio. These predictions are compared to experimental data to determine which models produce viable inflationary theories. The project reproduces known single field results and explores higher-order f(R) terms, showing they can only be a small correction to Starobinsky inflation. It also evaluates a multifield hybrid potential and how two fields can combine to produce inflation.
This document presents a study of five dimensional string cosmological models with bulk viscosity in general relativity. It investigates three cases of the field equations and determines solutions for each. For case I, the solution reduces the model to Minkowski spacetime. For case II, the solution yields a model that expands from a big bang and stops expanding at a finite time, with the volume increasing over time. The energy density is only due to bulk viscosity and Lyra geometry. For case III, three specific solutions are presented: one that reduces to a flat 5D spacetime; one that contracts/expands depending on constants; and one that follows power law inflation. Physical properties like expansion, shear, and density are also analyzed for each case.
Holographic Dark Energy in LRS Bianchi Type-II Space Timeinventionjournals
The document summarizes research on holographic dark energy in an anisotropic cosmological model. Key points:
- The study investigates holographic dark energy in a LRS Bianchi type-II spacetime using two assumptions: a law of variation for the Hubble parameter and a relation between metric coefficients.
- The law of variation generates power-law and exponential forms for the average scale factor in terms of cosmic time.
- For n not equal to 0, the holographic dark energy EOS parameter behaves like quintessence; for n equal to 0 it behaves like a cosmological constant.
- Solutions to the field equations are obtained for both cases of n not equal to
This document discusses energy-size reduction relationships in comminution processes. It outlines three main theories proposed to explain the relationship between energy input and size reduction: Rittinger's law, Kick's law, and Bond's law. The author proposes a new general form for the energy-size reduction relationship that is a linear combination of Kick-type and Rittinger-type laws. Empirical data from various studies is used to validate that this combined law provides a better fit than previous generalized laws, particularly in intermediate size ranges. The combined law also approximates to Kick-type and Rittinger-type laws at size extremes as previously observed. This combined law may have applications in computer simulations of comminution processes and mill
Effect of Mass Transfer and Hall Current on Unsteady MHD Flow with Thermal Di...IJERA Editor
The paper investigated the effect of mass transfer and Hall current on unsteady MHD flow with Thermal Diffusivity of a viscoelastic fluid in a porous medium. The resultant equations have been solved analytically. The velocity, temperature and concentration distributions are derived, and their profiles for various physical parameters are shown through graphs. The coefficient of Skin friction, Nusselt number and Sherwood number at the plate are derived and their numerical values for various physical parameters are presented through tables. The influence of various parameters such as the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter, Hartmann number, Hall parameter, and the frequency of oscillation on the flow field are discussed. It is seen that, the velocity increases with the increase in Gc, Gr, m and K, and it decreases with increase in Sc,M, n and Pr, temperature decreases with increase in Pr and n, Also, the concentration decreases with the increase in Sc and n.
MHD Natural Convection Flow of an incompressible electrically conducting visc...IJERA Editor
We consider a two-dimensional MHD natural convection flow of an incompressible viscous and electrically
conducting fluid through porous medium past a vertical impermeable flat plate is considered in presence of a
uniform transverse magnetic field. The governing equations of velocity and temperature fields with appropriate
boundary conditions are solved by the ordinary differential equations by introducing appropriate coordinate
transformations. We solve that ordinary differential equations and find the velocity profiles, temperature profile,
the skin friction and nusselt number. The effects of Grashof number (Gr), Hartmann number (M) and Prandtl
number (Pr), Darcy parameter (D-1) on velocity profiles and temperature profiles are shown graphically.
On the Logical Origin of the Laws Governing the Fundamental Forces of Nature:...Ramin (A.) Zahedi
- The document presents a new algebraic-axiomatic matrix approach to derive the fundamental laws of nature from basic axioms.
- It is shown that the laws governing fundamental forces like gravity and electromagnetism can be derived as general covariant massive field equations from axioms assuming rational relativistic energy-momentum and linearizing energy-momentum relations.
- The derived field equations include a torsion field that generates the invariant mass of each field and represent generalized forms of Einstein, Maxwell, Yang-Mills, Dirac, and Rarita-Schwinger equations.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Investigation of mgx sr1 xo mixed alloy under high pressureAlexander Decker
This document reports on research investigating the structural and mechanical properties of the mixed compound MgxSr1-xO under high pressure. An extended interaction potential model was applied that included zero point energy effects to predict phase transition pressures and volume collapses. The model parameters were determined using equilibrium conditions and followed Vegard's law varying linearly with concentration. The results for the mixed compounds were in fair agreement with experimental data generated by applying Vegard's law to the endpoint members MgO and SrO.
Bulk Viscous String Cloud with Strange Quark Matter in Self Creation Cosmologyiosrjce
We have investigated strange quark matter attached to the string cloud with bulk viscous fluid for
higher dimensional Friedman-Robertson-Walker (FRW) universe in Self Creation Cosmology (SCC). We have
obtained zero string density (휌푠 = 0) in this model. Our solutions agree with the studies of Reddy, SahooMishra
and Mohanty et al. in four dimensions.
This document summarizes and generalizes the concept of potential vorticity (PV) in fluid mechanics using the mathematical tools of differential geometry. It defines a PV 3-form field A in spacetime that unifies the PV scalar and 3D PV flux vector into a single object. Tensor operations are used to prove conservation theorems that generalize previous results. An analogy is drawn between the generalized structure of equations for fluid mechanics and electromagnetism. Finally, it indicates how the framework enables a linear inversion scheme with higher-order derivatives to solve for flow variables.
This paper develops a mathematical model to analyze the kinetic energy present in a stable bathtub vortex. It uses Burgers' vortex model to describe the azimuthal velocity field and derives an expression for kinetic energy by integrating over the cylindrical tank volume. The model shows that kinetic energy increases linearly with tank height and Reynolds number. It also indicates that a turbine placed at 20% of the tank radius would optimally capture energy, as this corresponds to the peak kinetic energy region away from the vortex core. The paper concludes by discussing opportunities to improve the model by incorporating more realistic fluid effects and accounting for turbine properties.
This paper proposes a maximum likelihood estimation method for determining the parameters of a modified Gutenberg-Richter magnitude-frequency relation proposed by Lomnitz-Adler and Lomnitz. The relation is a double exponential function with parameters a and B. The maximum likelihood method provides equations that allow iterative determination of parameter a from earthquake catalog data. Parameter B can then be estimated from the data and value of a. The method is demonstrated on a Chinese earthquake catalog, obtaining values of a = 0.10 and B = 10.34.
The document discusses groups and graphs in probability theory. It provides background on the probability that a group element fixes a set and related work applying this concept to graph theory, specifically generalized conjugacy class graphs and orbit graphs. The main results are:
1) For a dihedral group of order 2n acting regularly on the set of commuting pairs of elements, the probability that a group element fixes a set is 1/n.
2) Under this action, the generalized conjugacy class graph of the dihedral group has n isolated vertices and is empty.
3) The orbit graph vertices correspond to the n orbits of sizes 1 or 2n, with an edge between vertices if their corresponding orbits are conjugate.
Five Dimensional Bianchi Type-V Space-Time in f (R,T) Theory of GravitywIJERA Editor
We study the spatially homogeneous anisotropic Bianchi type-V universe in f(R,T) theory of gravity, where R is
the Ricci scalar and T is the trace of the energy-momentum tensor. We assume the variation law of mean
Hubble parameter and constant deceleration parameter to find two different five dimensional exact solutions of
the modified field equations. The first solution yields a singular model for n 0 while the second gives a nonsingular
model for n 0. The physical quantities are discussed for both models in future evolution of the
universe.
Quantum approach to fast protein folding timepacao1
This document proposes a quantum approach to modeling protein folding that provides a general framework without artificial hypotheses. It formulates protein folding as a quantum walk on a definite graph based on a lattice model of protein structure. Evaluating the folding time using the mean first passage time, it finds that the folding time via this quantum approach is much shorter than that obtained via classical random walks. This quantum strategy is expected to provide new insights into theoretical studies of protein folding.
This document summarizes a numerical study of the structure and thermodynamics of colloidal suspensions using the variational method and integral equation theory. The interactions between colloid particles are modeled using either a Yukawa or Sogami potential. Results from the integral equation theory using a Sogami potential are found to be in good agreement with Monte Carlo simulation results and experimental data. The variational method and integral equation theory are used to calculate structural properties like the pair correlation function and thermodynamic properties.
This document discusses nonlinear electrodynamics theories in the context of black hole solutions to the Einstein field equations. It presents a generalization of the Christodoulou-Ruffini mass formula for charged black holes in the weak field limit of nonlinear electrodynamics theories. The paper proves that the outer horizon of black holes never decreases under reversible transformations, and that such transformations are equivalent to solutions with a constant horizon for asymptotically flat black hole solutions of nonlinear theories. It then uses this result to decompose the total mass-energy of black holes in nonlinear theories in terms of irreducible and extractable quantities.
Five-Dimensional Cosmological Model with Time-Dependent G and Lamda for Const...IOSR Journals
In this paper we have consider five-dimensional cosmological model in presence of perfect fluid
source with time dependent G and .The Einstein field equations are solvable with the help of constant
deceleration parameter. Physical and kinematical properties of this model are investigated. It has been shown
that the solutions are comparable with recent observations. The behavior of gravitational constant,
cosmological constant, density, critical density and pressure is discussed for dust, radiation dominated and stiff matter of the Universe. It is also examined the behavior of gravitational constant and cosmological constant for expansion law and exponential law for stiff matter
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...IOSR Journals
An anisotropic Bianchi type-III cosmological model is investigated in a Saez-Ballester scalar-tensor theory of gravitation. Three different time-dependent skewness parameters along spatial directions are introduced to represent the deviation of pressure from isotropy. To get deterministic solutions of the field equations, we choose variation law of scale factor 𝑆= (𝑡𝑟𝑒𝑡)1𝑙 which yields a time-dependent deceleration parameter (DP) representing a model that generates a transition of the universe from the early decelerating phase to the present accelerating phase. Some physical and geometrical properties of the model are also discussed.
Structure and transport coefficients of liquid Argon and neon using molecular...IOSR Journals
Molecular dynamics simulation was employed to deduce the dynamics property distribution function of Argon
and Neon liquid. With the use of a Lennnard-Jones pair potential model, an inter-atomic interaction function was observed
between pair of particles in a system of many particles, which indicates that the pair distribution function determines the
structures of liquid Argon. This distribution effect regarding the liquid structure of Lennard-Jones potential was strongly
affected such that its viscosity depends on density distribution of the model. The radial distribution function, g(r) agrees well
with the experimental data used. Our results regarding Argon and Neon show that their signatures are quite different at
each temperature, such that their corresponding viscosity is not consistent. Two sharps turning points are more
prominent in Argon, one at temperature of 83.88 Kelvin (K) with viscosity of -0.548 Pascal second (Pa-s) and the
other at temperature of 215.64 K with viscosity of -0.228 Pa-s.
In Argon and Neon liquid, temperature and density are inversely and directly proportional to diffusion
coefficient, in that order. This characteristic suggests that the observed non linearity could result from the non
uniform thermal expansion in liquid Argon and Neon, which are between the temperature range of 21.98 K and
239.52 K.
Natural frequencies of magnetoelastic longitudinal wave propagationAlexander Decker
This document summarizes research on the natural frequencies of longitudinal wave propagation in an orthotropic circular cylinder permeated by a magnetic field. It begins by introducing the topic and providing background. It then presents the basic equations that govern magnetoelastic wave propagation. Specifically, it outlines the equations of motion for an elastic solid in a magnetic field, including the Lorentz force term. It also presents Maxwell's equations in the quasi-static approximation. The document derives a frequency equation for this problem and describes numerically solving for the natural frequencies of the first three modes for different material properties and cylinder geometries.
Study of the various aspects of interacting dark energy.pptxssuser2cf036
1. The document describes a research proposal on studying various aspects of interacting dark energy under modified gravity theories.
2. The objectives of the proposed research include studying thermodynamic quantities in generalized gravity theories, validity of generalized second law of thermodynamics in different expansions of the universe, interacting dark energy models in loop quantum cosmology and fractional action cosmology, and dark energy reconstruction in modified gravity theories.
3. The proposal discusses approaches to account for late-time cosmic acceleration through dark energy and modified gravity theories and reviews relevant literature.
The document discusses Einstein's field equations and Heisenberg's uncertainty principle. It begins by providing background on Einstein's field equations, which relate the geometry of spacetime to the distribution of mass and energy within it. It then discusses some key mathematical aspects of the field equations, including their nonlinear partial differential form. Finally, it notes that the field equations can be consolidated with Heisenberg's uncertainty principle to provide a unified description of gravity and quantum mechanics.
This document presents a study of new Bianchi type-IX cosmological models with a binary mixture of anisotropic dark energy and perfect fluid. The models are analyzed using the statefinder diagnostic pair (sr,). Field equations are derived for the metric and energy-momentum tensors. Assuming the energy conservation equations vanish separately and a special law for the Hubble parameter, the field equations yield power law and exponential type solutions. The statefinder parameters are used to characterize different phases of the universe, and properties of the models such as anisotropy and expansion rates are discussed.
A new universal formula for atoms, planets, and galaxiesIOSR Journals
In this paper a new universal formula about the rotation velocity distribution of atoms, planets, and galaxies is presented. It is based on a new general formula based on the relativistic Schwarzschild/Minkowski metric, where it has been possible to obtain expressions for the rotation velocity - and mass distribution versus the distance to the atomic nucleus, planet system centre, and galactic centre. A mathematical proof of this new formula is also given. This formula is divided into a Keplerian(general relativity)-and a relativistic(special relativity) part. For the atomic-and planet systems the Keplerian distribution is followed, which is also in accordance with observations.
According to the rotation velocity distribution of the galaxies the rotation velocity increases very rapidly from the centre and reaches a plateau which is constant out to a great distance from the centre. This is in accordance with observations and is also in accordance with the main structure of rotation velocity versus distance from different galaxy measurements.
Computer simulations were also performed to establish and verify the rotation velocity distributions in the atomic – planetary- and galaxy system, according to this paper. These computer simulations are in accordance with observations in two and three dimensions. It was also possible to study the matching percentage in these calculations showing a much higher matching percentage between theoretical and observational values by this new formula.
This document summarizes a study on the structure and thermodynamics of colloid solutions interacting through Yukawa or Lu-Marlow potentials. The researchers used an expression described by Lu and Marlow that accounts for finite particle size. They calculated structure factors using a variational method based on the Gibbs-Bogoliubov inequality. The theoretical structure factors obtained were found to be in good agreement with experimental data, justifying the interest in the Lu-Marlow potential. They also used a reference system of hard spheres and the variational method to estimate thermodynamic properties of the colloid solutions.
Quantization of the Orbital Motion of a Mass In The Presence Of Einstein’s Gr...IOSR Journals
This document presents a theoretical model to explain the quantized nature of planetary orbits around the sun based on Einstein's general theory of relativity. It derives expressions for the Lagrangian and Hamiltonian of a unit mass moving in the curved spacetime around a massive body. Using these, it writes a Schrodinger-like quantum equation that includes variation of the wavefunction with respect to a curve parameter representing spacetime curvature. The solutions reveal the quantized energy levels and orbits determined by quantum numbers. When applied to planetary orbits, the model estimates distances from the sun in good agreement with observed values.
Non-Darcy Convective Heat and Mass Transfer Flow in a Vertical Channel with C...IJERA Editor
In this paper, We made an attempt to study thermo-diffusion and dissipation effect on non-Darcy convective
heat and Mass transfer flow of a viscous fluid through a porous medium in a vertical channel with Radiation and
heat sources. The governing equations of flow, heat and mass transfer are solved by using regular perturbation
method with δ, the porosity parameter as a perturbation parameter. The velocity, temperature, concentration,
shear stress and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter.
This document discusses temperature dependence of collective proton frequency width and collective proton mode components in paraelectric phase of potassium dihydrogen phosphate (KH2PO4). It modifies an existing Hamiltonian model by including lattice anharmonicity up to fourth order. Correlation functions are evaluated using Green's functions and Dyson's equation. Expressions are derived for collective proton frequency width, collective phonon mode frequency, and temperature dependence of collective proton mode components. Theoretical results are found to agree reasonably well with experimental data.
Characteristics of shock reflection in the dual solution domainSaif al-din ali
This document summarizes a numerical study that investigates the use of laser energy deposition to induce transitions between regular and Mach shock reflections in supersonic flow over dual wedge configurations. The study validated its numerical approach by comparing results to an experiment involving laser deposition in front of a sphere. Simulations then examined how varying the position and amount of laser energy deposition could influence transition characteristics in the dual solution domain over wedge configurations. Key findings included how transition time and occurrence depended on deposition parameters and position relative to the shock waves and wedges.
Characteristics of shock reflection in the dual solution domainSaif al-din ali
This document summarizes a numerical study that investigated the effects of laser energy deposition on shock reflection transitions in supersonic flow. The study used computational fluid dynamics simulations to model laser energy being deposited in front of symmetrical wedges, creating a dual solution domain where different shock reflection patterns can occur. The results showed that laser energy deposition could induce transitions between regular and Mach shock reflections, and that the transition characteristics depended on the location and amount of energy deposited. Depositing more energy required more time for transition, and transition did not occur above a certain energy level or when depositing on the centerline.
International journal of engineering and mathematical modelling vol2 no1_2015_2IJEMM
This document summarizes the homogenization of Maxwell's equations for electromagnetic wave propagation through complex, periodically heterogeneous media. It introduces the heterogeneous and homogenized problems, defines the relevant function spaces, and proves the existence and uniqueness of solutions. Most importantly, it derives the effective homogenized material parameters through a two-scale convergence approach, expressing them in terms of the microscale material properties and solutions to local cell problems. Numerical implementation of the homogenized problem and example simulations are also briefly mentioned.
Effect of Rotation on a Layer of Micro-Polar Ferromagnetic Dusty Fluid Heated...IJERA Editor
This paper deals with the theoretical investigation of effect of rotation on micro-polar ferromagnetic dusty fluid
layer heated from below in a porous medium. Linear stability analysis and normal mode analysis methods are
used to find an exact solution for a flat micro-polar ferromagnetic fluid layer contained between two free
boundaries . In case of stationary convection, the effect of various parameters like medium permeability
parameter, non-buoyancy magnetization parameter, micro-polar coupling parameter, spin-diffusion parameter,
micro-polar heat conduction parameter, dust particles parameter and rotation parameter has been analyzed and
results are depicted graphically. In the absence of dust particles, rotation, micro-viscous effect and micro-inertia,
the sufficient condition is obtained for non-oscillatory modes
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
Similar to Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in Brans-Dicke Gravity (20)
Gainful Human Empowerment Reform in Managerial EngineeringIJASRD Journal
This paper describes the corporate-wide approach to gainful human empowerment reform at engineering. Gainful policy is a part of gainful human empowerment reform. The gainful human empowerment reform is a culture-specific human empowerment reform style that is prevalent in managerial engineering. Gainful human empowerment reform, referring to the internal systematic approach of the organization’s human empowerment reform to strive for daintiness performance excellence, and gainful policy referring to all those measures through which one creates and strengthens confidence and trust in outsiders, especially customers, towards the organization’s abilities and products. The daintiness managers are those who inspire followers to transcend their own self-interests, and who are capable of having a profound and extraordinary effect on their followers. The paper reviews the daintiness strategy implementation, strategic control, daintiness metrics, and daintiness channels.
A Study on Quality of Life and Social Support Breast Cancer Women in Greater ...IJASRD Journal
This document summarizes a research paper about gainful human empowerment reform in managerial engineering. It discusses gainful policy and human empowerment reform, focusing on creating trust and confidence. It also reviews strategic control, metrics, and communication channels used in gainful human empowerment reform. The paper examines how reducing information asymmetry and improving board development can increase strategic involvement. It concludes that investing in director development affects gainful human empowerment reform strategy.
Road Accident Analysis and Prevention in Nigeria: Experimental and Numerical ...IJASRD Journal
This paper empirically analysed road accident and its prevention in Nigeria. Data for road traffic crashes trend was sourced from Federal Road Safety Corps in Nigeria from 1960 - 2017. The data was tested for stationarity using Augmented Dickey Fuller (ADF) test, while the co-integration was conducted using Johansen’s methodology. Least Square estimate was employed for the empirical analysis. The results show that there is long run equilibrium relationship between total number of casualties, total number of fatal cases and total number of minor cases of accidents in Nigeria. The results show that there is positive and significant relationship between fatal cases, severe cases and total number of casualties, while minor cases have negative and significant relationship with total number of casualties. The study therefore recommends that government should invest massively in road transportation infrastructure in order to repair dilapidated roads, expand narrow roads and construct new ones. Government should legislate and enforce installation of speed limit devices for all vehicles operating on Nigerian roads to reduce reckless speeding on the highways which will definitely reduce total number of accidents and casualties on Nigerian roads.
Leadership by Humanistic Approach in Workplace ManagementIJASRD Journal
The form and structure of an organization's human resources system can affect employee motivation levels in several ways. Organizations can adopt various systems type leadership humanistic approach empowerment practices to enhance employee satisfaction. The systems type leadership revolution moves recording and analysis activities that were traditionally professional performance lines of activities focused to high operational content. The scientific and systems type leadership progress, growth and internationalization of markets, processors are processes in which the accounting profession plays a leading role of humanistic approach in workplace management. This paper considers the humanistic approach in workplace management. The strategic importance of workers is discussed and their interaction, as an asset, with other important organization assets. The basic methodologies for workers are then explained and their limitations are considered.
An Experimental Investigation on Normal Concrete using Rice Husk Ash, Dolomit...IJASRD Journal
Generally concrete is good in compression and weak in tension. The fibre in concrete generally increases both the compression and tension in concrete cement is the most important constituent material, since it binds the aggregate and resists the atmosphere action. Since the production of Portland cement clinker is an energy intensive process a partial substitution of clinker by mineral like dolomite and abundantly available agriculture wastes like Rice husk ash obviously represents considerable energy servings and reduction of Co2 emission. In this project work banana fibre are added with 2% by volume of cement, dolomite powder and Rice husk ash is partially replaced by 5.0%, 7.5% and 10% by the volume of cement. The concrete was determined for mechanical properties like compressive strength, split tensile strength and flexural strength they are also tested and studied.
A Novel Implementation for Radio Resource Allocation on Mobile Access Based o...IJASRD Journal
The proposed technology can reduce network costs by allowing optimized use of radio resources in mobile wireless networks. The system provides reliable and high-capacity channels, while the wireless part allows user mobility. Solving the problem involves an attempt to determine an optimal distribution of radio resources to the RAUs (radio allocation unit) and network cells can be split or merged dynamically to optimize this distribution as a function of the fluctuation of demands.
The Internet of Things (IoT) can be able to incorporate a large number of different end systems and it have the ability to transfer data over internet without human interaction. One of the main concerns with our environment has been solid waste management which in addition to disturbing the balance of the environment. it also has adverse effects on the health of the people in the society. The detection, monitoring and management of wastes is one of the primary problems. The traditional way of manually monitoring the wastes in garbage bins is the complex process and also it utilizes more human effort and time. This is an advanced method in waste management since it is automated. This IoT garbage monitoring system is a very innovative system which will help to prevent overflowing of wastes from the bin and keep cities clean. This system monitors the garbage bins and informs about the level of garbage collected in garbage bins. The data is transfer through internet to the control room.
IoT Based Accident Avoidance and Detection System for Two WheelersIJASRD Journal
This document describes an IoT-based accident avoidance and detection system for two-wheelers. The system uses an accelerometer to sense vehicle tilt and activate supporting wheels to prevent falls. A vibration sensor senses accident severity. An IoT modem then sends this data to an Android app, which notifies the nearest medical center and friends via text with the accident location. This allows for timely medical aid. The system can also monitor speed and trigger an alarm if speed exceeds a threshold, helping avoid accidents. The goal is to reduce accident fatalities by facilitating immediate emergency response.
Internet of things (IoT) has started growing due to the developments in RFID, sensors, communication technologies and Internet protocols. The basic aim is to connect sensors in a straight line devoid of human association in the vicinity of help in creation of new types applications. The revolution in mobile, Internet in addition to machine-to-machine (M2M) technologies preserve is seen when the primary step towards IoT. Internet of Things (IoT) is forwarding the association among the objects (the “things”) and the backend systems passing through the Internet. Every object can become connected and smart. As a result, the background is monitored via sensing the background furthermore the data are transferred toward cloud with internet connectivity of the things. The monitoring environment comes closer in spite of distance. There are several platforms that provide services such as data analytics, business analytics, notification, etc. Each platform provides a specific service In this expose, we examine a delegate section of these platforms, together open- source and proprietary, on the source of their capability to convene the potential of dissimilar IoT users. The evaluation focuses on how prepared and functional these platforms are for IoT ecology players, other than on the peculiarities of the essential scientific layers. This paper mainly aims to provide a summary of different platforms and stages for IOT. This paper would help the IoT users to decide their best platform to suit their application.
Realization of Direct Digital Synthesis in Cordic AlgorithmIJASRD Journal
Nowadays the modern communication system and software defined radio-based applications needs Trans receiver consisting of fully programmable circuit which performs modulation and demodulation process. The method which does not need memory for realizing modulators and demodulators is CORDIC algorithm. The CORDIC algorithm is a versatile algorithm which calculates only adder and shifter operations instead of using multiplier. So, this algorithm is mostly used for VLSI and digital signal processing. The main concept used in this project is Direct Digital Synthesis (DDS) which generates the analog waveform in digital format based on CORDIC algorithm approach .This paper focuses on analysis and simulation of Binary phase shift keying (BPSK), Binary amplitude shift keying (BASK), Binary frequency shift keying (BFSK), Quadrature phase shift keying (QPSK) modulation scheme using DDS based on CORDIC algorithm instead of ROM look up table which greatly reduce the number of slices and no of look up tables. The whole simulation is done on Modelsim and Xilinx-ISE using Verilog descriptive language and these modulation schemes are implemented on Spartan-3 FPGA kit.
Virtual Reality Based Requirement Distributions for Disaster Affected PeoplesIJASRD Journal
The general aim of the development of virtual reality technology for automation applications at the IRF is to provide the framework for Projective Virtual Reality which allows users to “project” their actions in the virtual world into the real world. Based on this virtual reality requirement distribution for disaster affected people system is proposed. In this system, it displays the basic need of affected people. Through this the concerned authority will know their requirement and distribute their need. The proposed system gets user’s height and change buttons’ vertical position. In addition, users can change a button label and type.
Implementation of Low Complex Universal Filtered MulticarrierIJASRD Journal
In 5G technology for enhancing the high speed data process the Filter Bank Multicarrier (FBMC), Universal Filtered multicarrier (UFMC), and Generalized Frequency Division Multiplexing (GFDM) techniques are used in effective manner. The FIR filter plays an important role in 5G mobile communication technology. In this paper, the hardware complexity reduced by using the FIR filter. In previous technique, 73 multipliers are required to the filtering process. Here to reduce the number of multipliers by using the multiplexers. The 73 multipliers to be replaced with the 5 number of 16:1 multiplexers, 5 multipliers and 4 registers. The Multiple Constant Multiplication (MCM) scheme is also presented for the block implementation FIR filters. Reducing the memory usage for using the less number of multipliers. Use the less number multipliers the difficulties are to be reduced. The overall implementation has a result of 42% reduction in hardware complexity.
Uncompressed Video Streaming in Wireless Channel without Interpolation using ...IJASRD Journal
This document summarizes an article that proposes methods for uncompressed video streaming over wireless channels using search algorithms without interpolation. It introduces full search and logarithmic search algorithms to estimate pixel motion for video transmission. The full search algorithm provides optimal estimation but is computationally expensive, while the logarithmic search algorithm reduces complexity. Experimental results show the full search algorithm achieves higher peak signal-to-noise ratio than existing methods. The proposed methods improve video quality for wireless transmission without increasing computational complexity.
Massive MIMO-Based 5G Networks: Energy Harvesting Base Stations with Minimum ...IJASRD Journal
The degree of CSI available to Transmitter and Receiver is influenced by the capacity of MIMO (Multiple Input and Multiple Output). The maximizing Energy Efficiency (EE) is to optimum transmission strategy for multiple user Massive MIMO system are to be optimized in radio frequency energy harvesting network. The grid energy permits requite for the changeability and intermittent the harvest energy. Hence, the quality of service constraint has to been solved under the problem of power grid expenditure reduction. In hybrid Massive MIMO system focuses on Energy efficient maximization where Massive MIMO employs where there are two other promising 5G technologies: assorted networks and millimeter wave. For achieving larger Energy Efficiency gains multiple opportunities open up than with conservative Massive MIMO systems. A sarcastic psychoanalysis of the Energy Efficient development approach considering combination Massive MIMO scheme permits as to verify various open research tribulation it will immensely help users in using energy-efficient 5G deployments.
Voice Based Search Engine for Visually Impairment PeoplesIJASRD Journal
World Wide Web (WWW) is unexpectedly emerging because the accepted records supply for our society. The WWW is normally reachable the usage of an internet-browsing package from a networked pc. The layout of facts on the net is visually orientated. The reliance on visible presentation locations excessive cognitive demands on a person to function this sort of system. The interaction might also sometimes require the whole attention of a consumer. The design of information presentation at the web is predominately visible-oriented. This presentation technique requires most, if no longer all, of the person’s attention and imposes significant cognitive load on a user. This technique isn't always sensible, in particular for the visually impaired persons. The awareness of this challenge is to develop a prototype which supports net browsing the use of a speech-based interface, e.g. A telephone, and to degree its effectiveness. The command input and the delivery of web contents are totally in voice. Audio icons are constructed into the prototype so that users will have higher knowledge of the original shape/purpose of a web page. Navigation and manage commands are available to decorate the net browsing enjoy. The effectiveness of this prototype is evaluated in a consumer take a look at involving both generally sighted and visually impaired humans. Voice browsers allow human beings to get right of entry to the Web the usage of speech synthesis, pre-recorded audio, and speech reputation. This may be supplemented via keypads and small presentations. Voice may also be supplied as an accessory to standard computing device browsers with high resolution graphical presentations, presenting an on hand alternative to the use of the keyboard or screen, as an instance in cars in which palms/eyes unfastened operation is crucial. Voice interplay can get away the bodily obstacles on keypads and shows as cell devices turn out to be ever smaller. The browser will have an integrated textual content extraction engine that inspects the content of the page to construct a structured illustration. The inner nodes of the structure constitute diverse tiers of abstraction of the content. This enables in easy and bendy navigation of the page so that it will hastily home into gadgets of interest.
Developing countries like India needs a significant improvement in infrastructure such as Roads or Highways. An express high way, for physical improvement, is one of them. Regardless, we experience a long queue at each toll plazas on expressway which wastes a lot of experience time, fuel and surges of co2. In order to check the issue basic in light of the present manual toll gathering system, Automated Toll Collection structure is exceedingly required. In this venture, we will look at the separate data from the client and it will be prepared for toll gathering framework, to make progressively productive and perfect. Beginning at now, at each toll both the vehicle needs to stop for paying the toll. We have proposed a structure that would pay the toll subsequently and diminish the line at the toll corner. For this purpose we have used the RFID tags to identify the user and so that the amount is automatically deducted from the users wallet. SMS is send to the user after deducting the amount form the user wallet. In addition to this, we have used the GPS to find the user location.
Secure Medical Data Computation using Virtual_ID Authentication and File Swap...IJASRD Journal
PHR provides users with a great deal in leakage of sensitive information. However, securing the sensitive medical data also brings very serious security problems, especially for the data security which is stored in the medical cloud data. Once the data is leaked to a third party, then the data privacy has become a major problem, mainly such as authentication, availability of data, confidentiality etc., and which is to be taken into consideration very effectively. An authentication scheme based on virtual smartcard using hashing function for medical data is proposed to solve the problem of in which the illegal users access the resources of servers. Here, we also maintain PHR sensitive data in the cloud by using file swapping concept. Once the user access the data. The user data will be swapped into different places by file swapping concept, so the file will be more secured and no one can hack or theft our data.
Secure Video Processing using ROI Extraction and ECC EncryptionIJASRD Journal
Video encryption is the process of encrypting videos to hide the object in video for secure video sharing. To hide privacy from sensitive areas in video frame, ROI extraction techniques are proposed. Lightweight encryption algorithm is performed on every ROI to make the privacy sensitive information. To implement ECC encryption for encryption of extracted ROI. Provide secure video communication between sender and receiver.
An Approach to Detect and Avoid Social Engineering and Phasing Attack in Soci...IJASRD Journal
Digital physical frameworks are the key advancement driver for some spaces, for example, car, flight, mechanical procedure control, and industrial facility mechanization. Be that as it may, their interconnection possibly gives enemies simple access to delicate information, code, and setups. In the event that aggressors gain control, material harm or even damage to individuals must be normal. To neutralize information burglary, framework control and digital assaults, security instruments must be implanted in the digital physical framework. The social building assault layouts are changed over to social designing assault situations by populating the format with the two subjects and articles from genuine precedents while as yet keeping up the point by point stream of the assault as gave in the format. Social Engineering by E-Mail is by a wide margin the most intensely utilized vector of assault, trailed by assaults beginning from sites. The aggressor in this way misuses the set up trust by requesting that consent utilize the organization's remote system office to send an email. A social designer can likewise join mechanical intends to accomplish the assault goals. The heuristic-based discovery method examines and separates phishing site includes and recognizes phishing locales utilizing that data .Based on the robotized examination of the record in the informal organization, you can construct suppositions about the power of correspondence between clients. In view of this data, it is conceivable to compute the likelihood of achievement of a multistep social building assault from the client to the client in digital physical/digital social framework. Furthermore, the proposed social designing assault layouts can likewise be utilized to create social building mindfulness material.
Effect of HHO Gas as Fuel Additive on the Exhaust Emissions of Internal Combu...IJASRD Journal
The use of carbon-base fuels by diesel engines release gases such as CO, NOx, SO2, and THCs which constitute a major source of environmental pollution. These gases when released into the atmosphere result in the formation of acid rain and cause greenhouse effect. In the present study, an experiment was carried out with and without the use of HHO gas. The concentrations of the emitted gases were measured using an E8500 Plus gas analyzer. The results showed that the average concentrations of carbon monoxide (CO), total unburnt hydrocarbons (THCs), oxides of nitrogen (NOx) and sulphur dioxide (SO2) emitted from the combustion chamber of the test engine was decreased by 68.8%, 35.2%, 16.4% and 97.9% respectively when a mixture of HHO gas/petrol/air was used instead of petrol/air mixture in an internal combustion engine. However, the average concentration of oxygen (O2) gas increased by 1.7%.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
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2. Roy et al., (2019)
45 Volume 06, Issue 04, Version I, Apr’ 2019
I. INTRODUCTION
The accelerated expansion of the universe is one of the most interesting and important
phenomena in the field of cosmology that have been obtained through astrophysical
observations in recent times [1-5]. An exotic form of energy, with a negative pressure, has been
found to be responsible for the accelerated expansion of the universe. This energy is known
as dark energy (DE). In the fields of physics and astronomy, an extensive research is now
taking place, throughout the world, on the nature and dynamics of DE. A number of models
have been proposed to explain this accelerated expansion of the universe, following a phase
of deceleration. In most of the models, DE is represented by cosmological constant Λ [6].
The present article is based on Brans-Dicke (BD) theory of gravitation. The BD theory
is characterized by a scalar field (𝜙) and a dimensionless coupling parameter (𝜔) that govern
the dynamics of space-time geometry. It can be regarded as a natural extension of the general
theory of relativity which is obtained in the limit of an infinite 𝜔 and a constant value of 𝜙 [7].
The BD theory of gravity can be regarded as one of the most important theories, among all
prevalent alternative theories of gravitation, which have very successfully explained the early
and late time behaviours of the universe and solved the problems of inflation [8]. As an
extension of the original BD theory, a generalized version was proposed, where ω is regarded
as a function of the scalar field 𝜙 [9-11]. Several models regarding the expanding universe have
been formulated on the basis of this theory [7, 12-15].
In the present study, BD field equations (for a spatially flat, homogeneous and
isotropic space-time) have been used to determine time dependence of the coupling parameter
(𝜔), equation of state (EoS) parameter (𝛾) and energy density (𝜌). Time dependence of 𝜔 was
previously studied by many groups, using scale factors that have power law dependence upon
time [7, 14, 15]. These scale factors lead to time independent deceleration parameters that are
not consistent with the interpretations of recent astrophysical observations. According to
astrophysical observations, the universe has made a transition from a phase of decelerated
expansion to a phase of accelerated expansion, implying that the deceleration parameter has
a dependence upon time [2, 16]. Taking this fact into consideration, we have used a scale factor
(𝑎) that leads to a time dependent deceleration parameter which changes sign from positive
to its present negative value. For simplicity, we have assumed power law dependence of 𝜔
upon 𝜙 and also of 𝜙 upon 𝑎. Time dependence of 𝜌 and the EoS parameter (𝛾), in the present
study, are consistent with the results obtained from some recent studies based on general
relativity with anisotropic space-time [17, 18]. Two theoretical models have been proposed to
determine the time variation of density parameters (Ω) for matter and dark energy. The
results from this model are in agreement with those obtained from studies based on
completely different premises [19, 20]. An effective interaction term (𝑄 𝑒𝑓𝑓) between matter and
dark energy has been calculated here and its time dependence has been studied. Its time
variation is found to be consistent with the findings of some other studies [21, 22]. In terms of
𝑄 𝑒𝑓𝑓 we have determined the time rate of change of densities for matter and dark energy.
II. METRIC AND FIELD EQUATIONS
The action, in the generalized Brans-Dicke theory, is expressed as,
𝒜 =
1
16𝜋
∫ 𝑑4
𝑥√−𝑔 (𝜙𝑅 +
𝜔(𝜙)
𝜙
𝑔 𝜇𝜈
𝜕 𝜇 𝜙𝜕 𝜈 𝜙 + ℒ 𝑚) (1)
3. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 46
In equation (1), 𝑔 is the determinant of the metric tensor 𝑔 𝜇𝜈
, Lm is the Lagrangian for
matter, 𝑅 denotes the Ricci scalar, 𝜑 is the Brans-Dicke scalar field and 𝜔 is a dimensionless
coupling parameter. This parameter is regarded as a function of 𝜑 in generalized BD theory.
The variation of action in equation (1) leads to the following field equations.
𝑅 𝜇𝜈 −
1
2
𝑅𝑔 𝜇𝜈 =
1
𝜙
𝑇𝜇𝜈 −
𝜔
𝜙2 [𝜙,𝜇 𝜙,𝜈 −
1
2
𝑔 𝜇𝜈 𝜙,𝛽 𝜙,𝛽
] −
1
𝜙
[𝜙 𝜇;𝜈 − 𝑔 𝜇𝜈 {
𝜕2 𝜙
𝜕𝑡2 − ∇2
ϕ}] (2)
𝜕2 𝜙
𝜕𝑡2 − ∇2
ϕ =
1
2𝜔+3
𝑇 (3)
𝑅 𝜇𝜈 is the Ricci tensor and 𝑇𝜇𝜈 is the energy-momentum tensor. In the above equations,
a semicolon stands for a covariant derivative and a comma denotes an ordinary derivative
with respect to 𝑥 𝛽
.
The energy-momentum tensor (𝑇𝜇𝜈) for the cosmic constituents is given by,
𝑇𝜇𝜈 = (𝜌 + 𝑝)𝑢 𝜇 𝑢 𝜈 + 𝑔 𝜇𝜈 𝑝 (4)
Here, 𝜌 denotes the energy density, 𝑝 is the isotropic pressure, 𝑢 𝜈 is the four-velocity
vector and T is the trace of 𝑇𝑗
𝑖
. This form of the energy-momentum tensor is based on an
assumption that the total matter-energy content of the expanding universe is a perfect fluid.
In a co-moving coordinate system, 𝑢 𝜈
= (0,0,0,1), having the characteristic of 𝑔 𝜇𝜈 𝑢 𝜇
𝑢 𝜈
= 1.
The line element for a homogeneous and isotropic universe, in Friedmann-Robertson-
Walker (FRW) cosmology, can be expressed as,
𝑑𝑠2 = −𝑑𝑡2 + 𝑎2(𝑡) [
𝑑𝑟2
1−𝑘𝑟2 + 𝑟2 𝑑𝜃2 + 𝑟2 𝑠𝑖𝑛2 𝜃𝑑𝜉2] (5)
Where 𝑎(𝑡) denotes the scale factor, 𝑡 is the cosmic time, 𝑘 is the spatial curvature
parameter. Three spherical polar coordinates are denoted by 𝑟, 𝜃 and 𝜉 respectively. Here k
describes the closed, flat and open universes corresponding to k = 1, 0, −1 respectively.
The following three equations are the field equations of BD theory for a universe filled
with a perfect fluid, in the geometry of FRW space-time expressed by equation (5). These
equations have been obtained from the field equations (2) and (3) in which 𝑇𝜇𝜈 from equation
(4) has been incorporated and the components of the metric tensor 𝑔 𝜇𝜈 have been taken from
the line element expressed by equation (5).
3
𝑎̇2+𝑘
𝑎2 + 3
𝑎̇
𝑎
𝜙̇
𝜙
−
𝜔(𝜙)
2
𝜙̇ 2
𝜙2 =
𝜌
𝜙
(6)
2
𝑎̈
𝑎
+
𝑎̇2+𝑘
𝑎2 +
𝜔(𝜙)
2
𝜙̇ 2
𝜙2 + 2
𝑎̇
𝑎
𝜙̇
𝜙
+
𝜙̈
𝜙
= −
𝛾𝜌
𝜙
(7)
𝜙̈
𝜙
+ 3
𝑎̇ 𝜙̇
𝑎𝜙
=
𝜌
𝜙
(1−3𝛾)
2𝜔+3
−
𝜙̇
𝜙
𝜔̇
2𝜔+3
(8)
In these field equations, 𝛾(≡ 𝑃/𝜌) is the equation of state (EoS) parameter for the
cosmic fluid, which has been treated as a function of time in the present study.
III. THE THEORETICAL MODEL
Combining equations (6), (7) and (8), for zero spatial curvature (𝑘 = 0), one obtains,
𝜔̇ + (2
𝜙̈
𝜙̇ + 6
𝑎̇
𝑎
−
𝜙̇
𝜙
) 𝜔 − 6 (
𝑎̇2
𝑎2 +
𝑎̈
𝑎
)
𝜙
𝜙̇ = 0 (9)
In the present study we have assumed the following ansatzes for 𝜙 and 𝜔. These
parameters have been assumed to have power-law relations with 𝑎 and 𝜑 respectively.
𝜙 = 𝜙0 (
𝑎
𝑎0
)
𝑛
(10)
4. Roy et al., (2019)
47 Volume 06, Issue 04, Version I, Apr’ 2019
𝜔 = 𝜔0 (
𝜙
𝜙0
)
𝑚
(11)
The expression of 𝜙 in equation (10) has been taken from some earlier studies in this
regard [12, 16, 23, 24]. The reason for choosing the above empirical relation for 𝜔 is the fact that
this is regarded as a function of the scalar field in the generalized Brans-Dicke theory [10, 11].
Combining equation (10) with (11), one gets,
𝜔 = 𝜔0 (
𝑎
𝑎0
)
𝑚𝑛
(12)
Using the expressions (10) and (11) in equation (9), one obtains,
𝑚𝜔𝑛 + (𝑛 + 4 − 2𝑞)𝜔 −
6
𝑛
(1 − 𝑞) = 0 (13)
Here, 𝑞 (≡ −𝑎̈ 𝑎/𝑎̇2) is the deceleration parameter.
Writing 𝜔 = 𝜔0 and 𝑞 = 𝑞0 (i.e., their values at 𝑡 = 𝑡0) in equation (13), we get,
𝑚 =
6(1−𝑞0)
𝑛2 𝜔0
−
𝑛+4−2𝑞0
𝑛
(14)
Combining equation (6) with (10) and taking 𝑘 = 0 (for flat space), one gets,
𝜌 = 𝜙𝐻2
(3 + 3𝑛 −
𝜔
2
𝑛2
) (15)
Here, 𝐻 (≡ 𝑎̇/𝑎) is the Hubble parameter.
Replacing all parameters in equation (15) by their values at 𝑡 = 𝑡0, one obtains,
𝜔0 =
2
𝑛2 𝐻0
2 (3𝐻0
2
+ 3𝑛𝐻0
2
− 𝜌0/𝜙0) (16)
Using equation (16) in equation (14) one gets,
𝑚 =
3(1−𝑞0)𝐻0
2
3𝐻0
2+3𝑛𝐻0
2−𝜌0/𝜙0
−
𝑛+4−2𝑞0
𝑛
(17)
Equations (16) and (17) show that both 𝜔0 and m depend upon the parameter n, which
determines the time dependence of 𝜙, as per equation (10).
Using equations (10) and (15) in (7), for 𝑘 = 0 (zero spatial curvature) and writing
𝑎̈
𝑎
=
−𝑞𝐻2
one obtains the following expression of the equation of state parameter.
𝛾 =
2𝑞−1−0.5𝜔𝑛2−𝑛−𝑛2+𝑛𝑞
3+3𝑛−0.5𝜔𝑛2 (18)
The value of the equation of state parameter at the present time is therefore given by,
𝛾0 =
2𝑞0−1−0.5𝜔0 𝑛2−𝑛−𝑛2+𝑛𝑞0
3+3𝑛−0.5𝜔0 𝑛2 (19)
To use the expression of 𝜔 (eqn. 11), the values of 𝜔0 and 𝑚 should be taken from
equations (16) and (17) respectively. The value of 𝜔, required for the expressions of 𝜌 and 𝛾,
in equations (15) and (18) respectively, should then be taken from equation (11). The value of
𝜙, required for equation (15), should be taken from equation (10).
To determine the time dependence of several cosmological parameters (𝜙, 𝜔, 𝜌, 𝛾),
following empirical expression of the scale factor has been used in the present model.
𝑎 = 𝑎0 𝐸𝑥𝑝[𝛼{(𝑡/𝑡0) 𝛽
− 1}] (20)
This scale factor has been so chosen that it generates a deceleration parameter
(𝑞 = −𝑎̈ 𝑎/𝑎̇2) which changes sign from positive to negative, as a function of time. This
signature flip indicates a transition of the universe from a phase of decelerated expansion to
a phase of accelerated expansion, in accordance with several recent studies based on
astrophysical observations [1, 2]. We have taken 𝑎0 = 1 for all calculations.
Here 𝛼, 𝛽 should have the same sign to ensure an increase of the scale factor with time.
Using equation (20), the Hubble parameter (𝐻) and the deceleration parameter (𝑞) are
obtained as,
5. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 48
𝐻 =
𝑎̇
𝑎
=
𝛼𝛽
𝑡0
(
𝑡
𝑡0
)
𝛽−1
(21)
𝑞 = −
𝑎̈ 𝑎
𝑎̇2 = −1 +
1−𝛽
𝛼𝛽
(
𝑡
𝑡0
)
−𝛽
(22)
For 0 < 𝛽 < 1 and 𝛼 > 0 one finds that, 𝑞 → +∞ as 𝑡 → 0 and 𝑞 → −1 as 𝑡 → ∞, showing
clearly a signature flip of 𝑞 with time.
Taking 𝐻 = 𝐻0 and 𝑞 = 𝑞0, at 𝑡 = 𝑡0, in equations (21) and (22), the values of the
constants 𝛼 and 𝛽 are obtained as,
𝛼 =
𝐻0 𝑡0
1−𝐻0 𝑡0(1+𝑞0)
(23)
𝛽 = 1 − 𝐻0 𝑡0(1 + 𝑞0) (24)
It is often necessary to find the evolution of a cosmological quantity as a function of
the redshift parameter (z) where 𝑧 =
𝑎0
𝑎
− 1. Using equation (20) one gets the following
expression as a relation between redshift parameter and time.
𝑧 =
𝑎0
𝑎
− 1 = 𝐸𝑥𝑝[−𝛼{(𝑡/𝑡0) 𝛽
− 1}] − 1 (25)
Using equations (20-24), one can determine the time dependence of 𝜙, 𝜔, 𝜌 and 𝛾 from
(10), (12), (15) and (18) respectively.
The values of different cosmological parameters used in this article are:
𝐻0 =
72
𝐾𝑚
𝑠
𝑀𝑝𝑐
= 2.33 × 10−18
𝑠𝑒𝑐−1
, 𝑞0 = −0.55, 𝜌0 = 9.9 × 10−27
𝐾𝑔𝑚−3
, Ω 𝐷0=0.7
𝜑0 =
1
𝐺0
= 1.498 × 1010
𝐾𝑔𝑠2
𝑚−3
, 𝑡0 = 1.4 × 1010
𝑌𝑒𝑎𝑟𝑠 = 4.42 × 1017
𝑠
IV. DETERMINATION OF DENSITY PARAMETERS
The total pressure (𝑃) of the entire matter-energy content of the universe is
contributed by dark energy because, the whole matter content (dark matter + baryonic
matter) is regarded as pressure less dust [12, 25]. Thus, we can write,
𝑃 = 𝛾𝜌 = 𝛾 𝐷 𝜌 𝐷 (26)
Here, 𝛾 𝐷 is the EoS parameter for dark energy and 𝜌 𝐷 is the density of dark energy.
Using equation (26), the density parameter for dark energy is given by,
Ω 𝐷 =
𝜌 𝐷
𝜌
=
𝛾
𝛾 𝐷
(27)
Let us assume the following empirical expression for the EoS parameter of dark
energy.
𝛾 𝐷 = 𝛾 𝑓(𝑡) (28)
Here 𝑓(𝑡) is a function of time for which we have chosen two empirical forms in the
following two models. The reason for assuming the above ansatz is the remarkable similarity
of the nature of time evolution of the EoS parameter for dark energy with that for the total
matter-energy content of the universe [17, 18, 26]. Here 𝑓(𝑡) is assumed to be a function that
connects 𝛾 𝐷 to 𝛾.
4.1 Model - 1
We propose the following form of 𝑓(𝑡) for model-1 of density parameter calculation.
𝑓(𝑡) = 𝐴 (
𝑡
𝑡0
)
𝜇
(29)
Here 𝐴 and 𝜇 are dimensionless constants.
6. Roy et al., (2019)
49 Volume 06, Issue 04, Version I, Apr’ 2019
Using equations (28) and (29) in (27) we get,
Ω 𝐷 =
𝜌 𝐷
𝜌
=
𝛾
𝛾 𝐷
= 𝐴−1 (
𝑡
𝑡0
)
−𝜇
(30)
Taking Ω 𝐷 = Ω 𝐷0 at the present time, i.e., at 𝑡 = 𝑡0, in equation (30), we get,
𝐴 =
1
Ω 𝐷0
(31)
Using equation (31) in (30) we get,
Ω 𝐷 = Ω 𝐷0 (
𝑡
𝑡0
)
−𝜇
(32)
Using equation (32), the density parameter for matter is given by,
Ω 𝑚 =
𝜌 𝑚
𝜌
=
𝜌−𝜌 𝐷
𝜌
= 1 − Ω 𝐷 = 1 − Ω 𝐷0 (
𝑡
𝑡0
)
−𝜇
(33)
In deriving equation (33), we have used the fact that the present universe is composed
mainly of matter and dark energy, considering all other forms of energy to be negligibly small
[27, 28].
Using equations (32) and (33), the expressions for densities of dark energy and matter
can be respectively written as,
𝜌 𝐷 = 𝜌Ω 𝐷 = 𝜌Ω 𝐷0 (
𝑡
𝑡0
)
−𝜇
(34)
𝜌 𝑚 = 𝜌Ω 𝑚 = 𝜌 [1 − Ω 𝐷0 (
𝑡
𝑡0
)
−𝜇
] (35)
Using equations (29) and (31) in (28), the equation-of-state parameter for dark energy
(𝛾 𝐷) becomes,
𝛾 𝐷 =
𝛾
Ω 𝐷
=
𝛾
Ω 𝐷0
(
𝑡
𝑡0
)
𝜇
(36)
Using equation (25) in (32) and (33), the expressions of Ω 𝐷 and Ω 𝑚 can be written in
terms of redshift (z) as,
Ω 𝐷 = Ω 𝐷0 (1 +
1
𝛼
𝑙𝑛
1
𝑧+1
)
−𝜇/𝛽
(37)
Ω 𝑚 = 1 − Ω 𝐷0 (1 +
1
𝛼
𝑙𝑛
1
𝑧+1
)
−𝜇/𝛽
(38)
The present value of Ω 𝐷0 is close to 0.7 according to several astrophysical observations
[27, 28]. In the evolution of density parameters of the universe, there was a time in the recent
past when Ω 𝐷 = Ω 𝑚 = 0.5 and, the corresponding z value was lying somewhere in the range
of 0 < 𝑧 < 1, as obtained from some recent studies in this regard [19, 20]. That phase of the
universe might be same as (or close to) the one when there was a transition from a decelerated
expansion to an accelerated expansion of the universe because it is the dark energy that is
known to be responsible for the accelerated expansion. According to a recent study, the
transition of the universe from a phase of decelerated expansion to its present state of
accelerated expansion took place in the past at 𝑧 = 0.6818 which was around 7.2371 × 109
years ago [28]. Choosing 𝑧 𝑐 to denote the value of the redshift parameter at which the universe
had Ω 𝐷 = Ω 𝑚, we get the following expression of 𝜇 from equations (37) and (38).
𝜇 =
𝛽 𝑙𝑛(2Ω 𝐷0)
𝑙𝑛[1+
1
𝛼
𝑙𝑛(
1
𝑧 𝑐+1
)]
(39)
Using equations (10), (20), (32) and (33), Ω 𝐷 and Ω 𝑚 can be expressed as functions of
the scalar field (𝜙) by the following two equations.
Ω 𝐷 = Ω 𝐷0 (1 +
1
𝑛𝛼
𝑙𝑛
𝜙
𝜙0
)
−𝜇/𝛽
(40)
7. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 50
Ω 𝑚 = 1 − Ω 𝐷0 (1 +
1
𝑛𝛼
𝑙𝑛
𝜙
𝜙0
)
−𝜇/𝛽
(41)
4.2 Model – 2
We propose the following form of 𝑓(𝑡) for model-2 of density parameter calculation.
𝑓(𝑡) = 𝐴 𝐸𝑥𝑝 (𝜆
𝑡
𝑡0
) (42)
Where, A and 𝜆 are dimensionless constants.
Combining equations (27), (28) and (42) we get,
Ω 𝐷 =
1
𝑓(𝑡)
=
1
𝐴
𝐸𝑥𝑝 (−𝜆
𝑡
𝑡0
) (43)
Using the fact that Ω 𝐷 = Ω 𝐷0 at 𝑡 = 𝑡0 in (43) we get,
𝐴 =
𝐸𝑥𝑝(−𝜆)
Ω 𝐷0
(44)
Substituting for A in equation (43) from equation (44) we get,
Ω 𝐷 = Ω 𝐷0 𝐸𝑥𝑝 {𝜆 (1 −
𝑡
𝑡0
)} (45)
Here Ω 𝐷0 is approximately equal to 0.7, according to various studies on dark energy
[27, 28].
Using equation (45), the density of dark energy is given by,
𝜌 𝐷 = 𝜌Ω 𝐷 = 𝜌Ω 𝐷0 𝐸𝑥𝑝 {𝜆 (1 −
𝑡
𝑡0
)} (46)
Using equation (45), the density parameter for matter (dark + baryonic) can be
expressed as,
Ω 𝑚 =
𝜌 𝑚
𝜌
=
𝜌−𝜌 𝐷
𝜌
= 1 − Ω 𝐷 = 1 − Ω 𝐷0 𝐸𝑥𝑝 {𝜆 (1 −
𝑡
𝑡0
)} (47)
Using equation (47), the density of matter can be expressed as,
𝜌 𝑚 = 𝜌Ω 𝑚 = 𝜌 [1 − Ω 𝐷0 𝐸𝑥𝑝 {𝜆 (1 −
𝑡
𝑡0
)}] (48)
Using equations (28), (42) and (44), the equation-of-state parameter for the dark
energy (𝛾 𝐷) can be expressed as,
𝛾 𝐷 =
𝛾
Ω 𝐷0
𝐸𝑥𝑝 {−𝜆 (1 −
𝑡
𝑡0
)} (49)
Combining equations (25), (45) and (47), the expressions of Ω 𝐷 and Ω 𝑚 can be written
in terms of redshift (z) as,
Ω 𝐷 = Ω 𝐷0 𝐸𝑥𝑝 [−𝜆 {[1 −
𝑙𝑛(𝑧+1)
𝛼
]
1/𝛽
− 1}] (50)
Ω 𝑚 = 1 − Ω 𝐷0 𝐸𝑥𝑝 [−𝜆 {[1 −
𝑙𝑛(𝑧+1)
𝛼
] − 1}] (51)
Taking Ω 𝐷 = Ω 𝑚 at 𝑧 = 𝑧 𝑐, one gets the following expression of 𝜆 from equations (50)
and (51).
𝜆 =
𝑙𝑛(
1
2Ω 𝐷0
)
[1−{1−
𝑙𝑛(𝑧 𝑐+1)
𝛼
}
1/𝛽
]
(52)
Using equations (10) and (20) in (45) and (47), Ω 𝐷 and Ω 𝑚 can be expressed as functions
of the scalar field (𝜙) by the following two equations.
Ω 𝐷 = Ω 𝐷0 𝐸𝑥𝑝 {−𝜆 ([
𝑙𝑛(𝜙/𝜙0)
𝑛𝛼
+ 1]
1/𝛽
− 1)} (53)
Ω 𝑚 = 1 − Ω 𝐷0 𝐸𝑥𝑝 {−𝜆 ([
𝑙𝑛(𝜙/𝜙0)
𝑛𝛼
+ 1]
1/𝛽
− 1)} (54)
8. Roy et al., (2019)
51 Volume 06, Issue 04, Version I, Apr’ 2019
4.3 A Comparison of Two Models for Density Parameter Calculation
For a positive value of 𝑧 𝑐, both 𝜇 and 𝜆 are negative, as obtained from model-1 (eqn.
39) and model-2 (eqn. 52) respectively, indicating an increase of Ω 𝐷 ( and a corresponding
decrease of Ω 𝑚) with time. Qualitatively, these two models show the same nature of time
dependence for Ω 𝐷 and Ω 𝑚, with Ω 𝐷 = Ω 𝐷0 and Ω 𝑚 = Ω 𝑚0 at 𝑧 = 0 (present time). The time
derivatives of them, for two models, are respectively given by equations (55) and (56).
Ω̇ 𝐷 =
−𝜇
𝑡0
Ω 𝐷0 (
𝑡
𝑡0
)
−𝜇−1
(55)
Ω̇ 𝐷 =
−𝜆
𝑡0
Ω 𝐷0 𝐸𝑥𝑝 {𝜆 (1 −
𝑡
𝑡0
)} (56)
Since 𝜇 and 𝜆 are negative, both expressions of Ω̇ 𝐷 are positive, being proportional to
Ω 𝐷0
𝑡0
at the present time. Near the present time, they have almost the same behaviour (fig. 5).
V. CALCULATION OF AN EFFECTIVE INTERACTION TERM
The energy conservation equation is expressed as [7, 29],
𝜌̇ + 3𝐻(𝜌 + 𝑃) = 0 (57)
Taking the pressure 𝑃 = 𝛾𝜌 = 𝛾 𝐷 𝜌 𝐷 (eqn. 26) and 𝜌 = 𝜌 𝑚 + 𝜌 𝐷 in equation (57) one gets,
𝜌̇ 𝑚 + 𝜌̇ 𝐷 + 3𝐻[𝜌 𝑚 + 𝜌 𝐷(1 + 𝛾 𝐷)] = 0 (58)
If it is assumed that the two entities, matter and dark energy the universe, have been
interacting with each other, causing the generation of one of them at the cost of the other,
one may define a parameter representing their interaction, on the basis of equation (58). The
interaction term (𝑄) can be represented by the following equations [19, 21, 25, 29].
𝜌̇ 𝑚 + 3𝐻𝜌 𝑚 = 𝑄 (59)
𝜌̇ 𝐷 + 3𝐻𝜌 𝐷(1 + 𝛾 𝐷) = −𝑄 (60)
A negative value of 𝑄 represents a transfer of energy from the matter field to the field
of dark energy and a positive value of 𝑄 represents conversion of dark energy into matter. It
is evident from equations (59) and (60) that 𝑄 has a dependence upon time and, its values,
obtained from these two equations, would not be the same. Let us denote the values of 𝑄,
obtained from these two equations, by 𝑄1 and 𝑄2 respectively.
Using the relations 𝜌 = 𝜌 𝑚 + 𝜌 𝐷 and 𝜌 𝐷 = 𝜌Ω 𝐷 in (59) we get,
𝑄1 = (1 − Ω 𝐷)𝜌̇ − 𝜌Ω̇ 𝐷 + 3𝐻𝜌(1 − Ω 𝐷) (61)
Using the relations 𝜌 𝐷 = 𝜌Ω 𝐷 and 𝛾 𝐷 =
𝛾
Ω 𝐷
(eqn. 36) in equation (60) we get,
𝑄2 = −Ω 𝐷 𝜌̇ − 𝜌Ω̇ 𝐷 − 3𝐻𝜌(Ω 𝐷 + 𝛾) (62)
To estimate the difference (Δ𝑄) between 𝑄1and 𝑄2, we write the following expression.
Δ𝑄 = 𝑄1 − 𝑄2 = 𝜌̇ + 3𝐻𝜌(1 + 𝛾) (63)
To have an estimate of an effective interaction between matter and dark energy, we
define a parameter 𝑄 𝑒𝑓𝑓 as the average of 𝑄1and 𝑄2. Thus, we have,
𝑄 𝑒𝑓𝑓 =
𝑄1 + 𝑄2
2
= (
1
2
− Ω 𝐷) 𝜌̇ − 𝜌Ω̇ 𝐷 +
3
2
𝐻𝜌(1 − 2Ω 𝐷 − 𝛾) (64)
To judge the plausibility of using 𝑄 𝑒𝑓𝑓 as a measure of interaction between matter and
dark energy, we have calculated the following expression of Δ𝑄/𝑄 𝑒𝑓𝑓, using equations (63),
(64).
9. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 52
Δ𝑄
𝑄 𝑒𝑓𝑓
=
𝜌̇ +3𝐻𝜌(1+𝛾)
(
1
2
−Ω 𝐷)𝜌̇ −𝜌Ω̇ 𝐷+
3
2
𝐻𝜌(1−2Ω 𝐷−𝛾)
(65)
Here we have used model-1 (section 4.1) of density parameter calculation, because two
models predict almost the same nature of time dependence of density parameters near the
present time (fig. 5). To determine the time dependence of the interaction term (𝑄), we have
calculated 𝜌̇ and Ω̇ 𝐷, based on the expressions of 𝜌 and Ω 𝐷 in equations (15) and (32)
respectively.
𝜌̇ = 𝜌𝐻 [𝑛 − 2(1 + 𝑞) −
𝜔𝑚𝑛3
6+6𝑛−𝜔𝑛2] (66)
Ω̇ 𝐷 =
−𝜇Ω 𝐷
𝑡
=
−𝜇Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
(67)
Equations (10) and (11) have been used, along with the definitions of 𝐻 and 𝑞, to
calculate 𝜌̇ (eqn. 66) from (15). Equation (21) has been used to calculate Ω̇ 𝐷 (eqn. 67) from
(32).
Using equations (66) and (67), in (61), (62), (64) and (65) one obtains the following
relations.
𝑄1 = (1 − Ω 𝐷)𝜌𝐻 [𝑛 − 2(1 + 𝑞) −
𝜔𝑚𝑛3
6+6𝑛−𝜔𝑛2] +
𝜇𝜌Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
+ 3𝐻𝜌(1 − Ω 𝐷) (68)
𝑄2 = −Ω 𝐷 𝜌𝐻 [𝑛 − 2(1 + 𝑞) −
𝜔𝑚𝑛3
6+6𝑛−𝜔𝑛2] +
𝜇𝜌Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
− 3𝐻𝜌(Ω 𝐷 + 𝛾) (69)
𝑄 𝑒𝑓𝑓 = (
1
2
− Ω 𝐷) 𝜌𝐻 [𝑛 − 2(1 + 𝑞) −
𝜔𝑚𝑛3
6+6𝑛−𝜔𝑛2] +
𝜇𝜌Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
+
3
2
𝐻𝜌(1 − 2Ω 𝐷 − 𝛾) (70)
Δ𝑄
𝑄 𝑒𝑓𝑓
=
𝜌𝐻[𝑛−2(1+𝑞)−
𝜔𝑚𝑛3
6+6𝑛−𝜔𝑛2]+3𝐻𝜌(1+𝛾)
(
1
2
−Ω 𝐷)𝜌𝐻[𝑛−2(1+𝑞)−
𝜔𝑚𝑛3
6+6𝑛−𝜔𝑛2]−𝜌
−𝜇Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
+
3
2
𝐻𝜌(1−2Ω 𝐷−𝛾)
(71)
If the value of
Δ𝑄
𝑄 𝑒𝑓𝑓
is found to be sufficiently small, it would be quite reasonable to
believe that 𝑄 𝑒𝑓𝑓, defined as an average of 𝑄1and 𝑄2, will truly represent the interaction
between matter and dark energy. Table-1 shows the values of (𝑄 𝑒𝑓𝑓)
𝑡=𝑡0
and (Δ𝑄/𝑄 𝑒𝑓𝑓)
𝑡=𝑡0
for several values of the parameter n. Here, the present value of 𝑄 𝑒𝑓𝑓 is numerically of the
order of 10−44
and it is negative. The present value of Δ𝑄/𝑄 𝑒𝑓𝑓 is so small (~10−15
) that 𝑄 𝑒𝑓𝑓,
as defined by eqn. 64, is likely to describe the interaction with sufficient accuracy. Its negative
value indicates decay of matter into dark energy at the present time. As n becomes more
negative, (𝑄 𝑒𝑓𝑓)
𝑡=𝑡0
becomes more negative. It implies that a greater rate of change of the
scalar field (𝜙) has a connection to a larger interaction between mater and dark energy of the
universe, causing a larger rate of generation of dark energy at the expense of matter.
Table – 1: Values of 𝑄 𝑒𝑓𝑓 and |𝛥𝑄/𝑄 𝑒𝑓𝑓| at the present time for several values of the parameter
n which determines the time variation of the scalar field (𝜙)
𝑛 (𝑄 𝑒𝑓𝑓)
𝑡=𝑡0
|(
Δ𝑄
𝑄 𝑒𝑓𝑓
)
𝑡=𝑡0
|
-1.60 −1.153 × 10−44
1.296 × 10−14
-1.62 −1.606 × 10−44
3.719 × 10−15
-1.64 −2.046 × 10−44
2.433 × 10−15
10. Roy et al., (2019)
53 Volume 06, Issue 04, Version I, Apr’ 2019
-1.66 −2.473 × 10−44
3.020 × 10−15
-1.68 −2.886 × 10−44
1.021 × 10−15
-1.70 −3.285 × 10−44
9.549 × 10−15
-1.72 −3.670 × 10−44
4.883 × 10−15
-1.74 −4.042 × 10−44
4.188 × 10−15
-1.76 −4.400 × 10−44
2.602 × 10−15
-1.78 −4.744 × 10−44
2.001 × 10−15
-1.80 −5.075 × 10−44
1.014 × 10−15
-1.82 −5.392 × 10−44
6.463 × 10−15
-1.84 −5.696 × 10−44
2.797 × 10−15
-1.86 −5.986 × 10−44
1.663 × 10−15
Time dependence of 𝜌̇ 𝑚 and 𝜌̇ 𝐷 can be calculated from equations (59) and (60) by taking
𝑄 = 𝑄 𝑒𝑓𝑓. Use of these two equations for this purpose, instead of (48) and (46), ensures
consistency with the energy conservation equation given by (57). Thus, we can write,
𝜌̇ 𝑚 = 𝑄 𝑒𝑓𝑓 − 3𝐻𝜌 𝑚 (72)
𝜌̇ 𝐷 = −𝑄 𝑒𝑓𝑓 − 3𝐻𝜌 𝐷(1 + 𝛾 𝐷) (73)
Combining equation (70) with (72) and (73), one gets the following expressions (eqns.
74 and 75) for the time derivatives of matter and dark energy (𝜌̇ 𝑚 and 𝜌̇ 𝐷).
𝜌̇ 𝑚 = (
1
2
− Ω 𝐷) 𝜌𝐻 [𝑛 − 2(1 + 𝑞) −
𝜔𝑚𝑛3
6 + 6𝑛 − 𝜔𝑛2
] +
𝜇𝜌Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
+
3
2
𝐻𝜌(1 − 2Ω 𝐷 − 𝛾) − 3𝐻𝜌(1 − Ω 𝐷) (74)
𝜌̇ 𝐷 = − (
1
2
− Ω 𝐷) 𝜌𝐻 [𝑛 − 2(1 + 𝑞) −
𝜔𝑚𝑛3
6 + 6𝑛 − 𝜔𝑛2
] −
𝜇𝜌Ω 𝐷
𝑡0
(
𝐻𝑡0
𝛼𝛽
)
1
1−𝛽
−
3
2
𝐻𝜌(1 − 2Ω 𝐷 − 𝛾) − 3𝐻𝜌Ω 𝐷(1 + 𝛾 𝐷) (75)
Using these relations, we have studied the time dependence of 𝜌̇ 𝑚 and 𝜌̇ 𝐷 graphically
(figs. 13-16). It is evident here that the nature of their time variation must depend upon the
parameters 𝑛 and 𝜇, where 𝜇 is connected to the value of 𝑧 𝑐, as given by equation (39). A
negative value of 𝜌̇ 𝑚 and a simultaneous positive value of 𝜌̇ 𝐷 imply a conversion of matter
into dark energy.
VI. CHOICE OF VALUES FOR THE PARAMETER n
The parameter n controls the time evolution of the scalar field (𝜙). Time variations of
𝜔, 𝜌 and 𝛾 depend on n, as we know that both m and 𝜔0 are functions of n. The scalar field
(ϕ) is the reciprocal of the gravitational constant (𝐺) [12, 23, 24]. Hence, using equations (10) and
(20), we can write,
𝐺 =
1
𝜙
=
1
𝜙0
(
𝑎
𝑎0
)
−𝑛
=
1
𝜙0
𝐸𝑥𝑝[−𝑛𝛼{(𝑡/𝑡0) 𝛽
− 1}] (76)
According to some studies, the gravitational constant increases with time [30-32]. In the
expression of 𝐺 in equation (76), the scale factor (𝑎) is an increasing function of time. For
negative values of the parameter 𝑛, 𝐺 would be an increasing function of time. Therefore, in
the present study we have used only negative values of the parameter n.
11. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
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Volume 06, Issue 04, Version I, Apr’ 2019 54
Using equations (21) and (76), we get,
𝐺̇
𝐺
= −𝑛
𝑎̇
𝑎
= −𝑛𝐻 = −𝑛
𝛼𝛽
𝑡0
(
𝑡
𝑡0
)
𝛽−1
(77)
Writing equation (77) for the present time, i.e., 𝑡 = 𝑡0, one obtains,
𝑛 = −
𝑡0
𝛼𝛽
(
𝐺̇
𝐺
)
𝑡=𝑡0
(78)
Using equation (78), the parameter n can be estimated from the experimental
observations regarding [𝐺̇/𝐺] 𝑡=𝑡0
, as reported in articles on time varying gravitational
constant [33]. According to a study by S. Weinberg, the largest possible value of |
𝐺̇
𝐺
|
𝑡=𝑡0
is
4 × 10−10
per year [34]. Using equation (78), one gets 𝑛~ − 5.5 for this upper limit.
Table – 2: Values of 𝜔0, 𝑚, 𝛾0, 𝛾 𝐷0, (
𝐺̇
𝐺
)
𝑡=𝑡0
for Several Values of the Parameter
𝑛 which determines the Time Variation of the Scalar Field (𝜙)
𝑛 𝜔0 𝑚 𝛾0
𝛾 𝐷0
( 𝑧 𝑐 = 0.7)
(
𝐺̇
𝐺
)
𝑡=𝑡0
(𝑌𝑟−1)
-1.63 -1.514 -0.183 -1.796 -2.566 1.198× 10−10
-1.64 -1.518 -0.168 -1.691 -2.416 1.205× 10−10
-1.65 -1.522 -0.154 -1.588 -2.268 1.212× 10−10
-1.66 -1.525 -0.140 -1.486 -2.123 1.220× 10−10
-1.67 -1.529 -0.127 -1.386 -1.979 1.227× 10−10
-1.68 -1.532 -0.115 -1.287 -1.839 1.234× 10−10
-1.69 -1.535 -0.104 -1.190 -1.700 1.242× 10−10
-1.70 -1.538 -0.093 -1.095 -1.564 1.249× 10−10
-1.71 -1.540 -0.083 -1.001 -1.430 1.256× 10−10
-1.72 -1.543 -0.073 -0.909 -1.299 1.264× 10−10
-1.73 -1.545 -0.064 -0.819 -1.170 1.271× 10−10
-1.74 -1.547 -0.055 -0.730 -1.043 1.279× 10−10
-1.75 -1.549 -0.046 -0.643 -0.918 1.286× 10−10
-1.76 -1.551 -0.038 -0.558 -0.796 1.293× 10−10
-1.77 -1.553 -0.031 -0.474 -0.677 1.301× 10−10
-1.78 -1.554 -0.024 -0.392 -0.559 1.308× 10−10
-1.79 -1.555 -0.017 -0.311 -0.444 1.315× 10−10
-1.80 -1.557 -0.011 -0.232 -0.332 1.323× 10−10
-1.81 -1.558 -0.005 -0.155 -0.221 1.330× 10−10
-1.82 -1.559 0.001 -0.079 -0.113 1.337× 10−10
-1.83 -1.560 0.006 -0.005 -0.008 1.345× 10−10
-1.84 -1.561 0.012 0.067 0.095 1.352× 10−10
-1.85 -1.561 0.016 0.137 0.196 1.359× 10−10
12. Roy et al., (2019)
55 Volume 06, Issue 04, Version I, Apr’ 2019
Table-2 shows the values of 𝜔0, 𝑚, 𝛾0, 𝛾 𝐷0, (
𝐺̇
𝐺
)
𝑡=𝑡0
for different values of 𝑛 which
governs the time variation of the scalar field (𝜙), as per equation (10). For the values of n over
the range of −1.76 < 𝑛 < −1.67, the values of the present EoS parameter (𝛾0) in this table are
consistent with the range of values (approximately −1.7 < 𝛾0 < −0.6) obtained from
astrophysical observations, as reported in some recent articles on the time variation of EoS
parameter, based on an anisotropic space-time [17,18]. Over this range, the values of m are
negative, indicating that 𝜔 would be less negative as 𝜙 increases, as evident from equation
(11), where 𝜔0 is negative. For 𝑛 < −1.81 in this table we have 𝑚 > 0, implying that 𝜔 would
be more negative as 𝜙 increases. Both of these natures of variations of 𝜔(𝜙) are reflected in
the plots of some other studies on Brans-Dicke theory [35, 36]. But the second type of variation
is not physically valid, since it corresponds to those values of EoS parameter which are not in
conformity with the astrophysical observations in this regard. The values of (
𝐺̇
𝐺
)
𝑡=𝑡0
in this
table are absolutely within the range of permissible values shown by S. Weinberg [34].
VII. GRAPHICAL INTERPRETATION OF THEORETICAL FINDINGS
Fig. 1 shows the variation of the Brans-Dicke parameter (𝜔) as a function of time, for
different values of the parameter n. Here, 𝜔 has a negative value and it becomes more
negative with time. For more negative values of n, it decreases less rapidly. This nature of 𝜔,
which is gradually becoming more negative with time, is also obtained from other studies on
Brans-Dicke theory [35, 36]. The time dependence of 𝜔, obtained from the present study, is based
on a scale factor which leads to a time dependent deceleration parameter (that shows a
signature flip with time from positive to negative), unlike some recent studies that are based
on time independent deceleration parameters [7, 14, 15]. The value of 𝜔0 (close to −1.55) and the
gradual decrease of 𝜔 with time are consistent with the results described in the section of
‘conclusions’ of a recent article based on Kantowski-Sachs space-time [15].
Fig. 2 shows the time variation of energy density (𝜌) with time, for different values of
the parameter n. Here, 𝜌 rises steeply to a peak and then decreases at a slower rate. More
negative values of n cause quicker attainment of the peak and also a larger peak value of
energy density. It falls more steeply for more negative values of n. This behaviour is
consistent with studies based on General Relativity [17].
Fig. 3 shows the time dependence of the equation of state (EoS) parameter (𝛾) for three
different values of the parameter n. These curves show a steep rise initially and then they
gradually become asymptotic to a small negative value. Saturation is reached faster for less
negative values of n. The value of 𝛾 at the present time (𝑡 = 𝑡0) is −0.73. The ranges of 𝛾 from
observational results based on SN Ia data and SN Ia data collaborated with galaxy clustering
statistics and CMBR anisotropy are −1.67 < 𝛾 < −0.62 and −1.33 < 𝛾 < −0.79 respectively [37,
38]. These curves are similar to those obtained from dark energy models based on LRS Bianchi
type-V metric in the framework of Einstein’s general theory of relativity [18].
Fig. 4 shows a single plot EoS parameter, for 𝑛 = −1.74, on a longer time scale
compared to figure 3. It shows a pattern that is very much similar to the one depicted in one
of the earlier studies based on dark energy models in anisotropic Bianchi type-I space-time,
in the framework of Einstein’s general theory of relativity [17].
13. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 56
The plots in the Figs. 2-4 show very clearly that the time evolution of energy density
and EoS parameter, obtained here for FRW space-time in the framework of Brans-Dicke
theory, are quite in agreement with the results obtained from anisotropic space-time in the
framework of general relativity [17, 18].
Fig. 5 shows the variations of density parameters, for dark energy and matter, as
functions of the redshift parameter, using models-1, 2 (Sec. 4.1, 4.2). Here, the value of 𝜇 has
been taken to be −0.557, causing 𝑧 𝑐 = 0.7, implying that the dark energy took over at around
𝑧 = 0.7. These plots are quite similar in nature to the plots obtained from some recent studies
[19, 20].
Fig. 6 shows the plots of density parameters, for dark energy and matter, as functions
of time (Sec. 4.1), for two different values of 𝑧 𝑐. For higher values of 𝑧 𝑐 (implying less negative
values of 𝜇, as per eqn. 39), |Ω̇ | gradually decreases with time.
Fig. 7 shows the time variation of the EoS parameter for dark energy (𝛾 𝐷) for three
different values of the parameter n. These plots are very much similar to the ones obtained
from studies based on anisotropic space-time in the framework of general relativity [17, 18].
Plots with less negative values of n take longer time to attain saturation.
Fig. 8 shows the time variations of EoS parameters for total energy and dark energy.
Here 𝛾 𝐷 has a more negative value compared with 𝛾. Their difference decreases with time, as
the proportion of dark energy increases in the universe. For 𝑛 = −1.74 the values of 𝛾0 and
𝛾 𝐷0 are respectively −0.73 and −1.04 respectively. These values are quite consistent with the
ranges of values obtained from several astrophysical observations and described in some
studies regarding dark energy and time varying EoS parameter [17-19].
The parameter n, in this model, plays an important role in governing the time
evolution of various cosmological quantities. Both 𝜔0 and m, which controls the value of 𝜔,
are dependent upon n (eqns. 16, 17). Larger negative value of n means faster change of 𝜙
(≡ 1/𝐺) with time. Taking 𝑛 = −1.74 we get, (
𝐺̇
𝐺
)
𝑡=𝑡0
= 1.28 × 10−10
𝑌𝑟−1
from equation (77).
This is well within its upper limit, i.e. 4 × 10−10
𝑌𝑟−1
, predicted by S. Weinberg [34].
Fig. 9 shows the variation of the effective interaction term 𝑄 𝑒𝑓𝑓 as a function of time
for three different values of 𝑧 𝑐. It is found to decrease with time from an initial positive value,
gradually becoming negative at a later time. This behaviour of signature flip is consistent
with the results of a recent study on holographic dark energy [21]. For smaller values of 𝑧 𝑐
(which correspond to more negative values of 𝜇, as per eqn. 39), 𝑄 𝑒𝑓𝑓 attains its negative phase
faster. Its positive phase indicates that dark energy initially decayed into matter and, in the
later stage, where 𝑄 𝑒𝑓𝑓 < 0, matter decays into dark energy.
Fig. 10 shows the variation of 𝑄 𝑒𝑓𝑓 as a function of time for three different values of
the parameter n. Two of these curves decrease with time and change their signs from positive
to negative. These two curves show that the transfer of energy, between matter and dark
energy, reverses its direction, producing dark energy at the expense of matter at the later
stage. The third curve, with 𝑛 = −1.8, is entirely in the negative domain, becoming gradually
closer to zero with time. It clearly shows that, there exists a value of the parameter n, below
which we have an unidirectional energy transfer, from the matter field to the field of dark
energy, at a gradually decreasing rate.
14. Roy et al., (2019)
57 Volume 06, Issue 04, Version I, Apr’ 2019
Fig. 11 depicts the change of 𝑄 𝑒𝑓𝑓 as a function of z, for three different values of 𝑧 𝑐. It
is found that, for smaller values of 𝑧 𝑐, 𝑄 𝑒𝑓𝑓 attains its negative phase faster.
Fig. 12 shows the variation of 𝑄 𝑒𝑓𝑓 as a function of the redshift parameter (z) for three
different values of the parameter n. Like figure 10, one of these three plots also indicate the
existence of a threshold value for n, below which the direction of energy transfer is always
from the matter field to the field of dark energy with 𝑄 𝑒𝑓𝑓 becoming less negative with time.
Above this threshold of n, the interaction term shows a signature flip from positive to negative
as time progresses. The curve for 𝑛 = −1.8 in this figure is similar to the behaviour obtained
from a study on the interaction term between dark matter and dark energy [22]. Since around
80% of the matter content is dark matter, the interactions of dark energy with matter consist
mainly of the interactions of dark energy with dark matter.
Fig. 13 shows the variation of 𝜌̇ 𝑚 as a function of time for three different values of 𝑧 𝑐.
Each graph shows a decrease with time, becoming positive to negative prior to the present
time (𝑡 = 𝑡0). Smaller values of 𝑧 𝑐 cause faster transition from positive to negative values. In
an interacting model of matter and dark energy, a negative value of 𝜌̇ 𝑚 means a transfer of
energy from the sector of matter to that of dark energy.
Fig. 14 depicts the variation of 𝜌̇ 𝐷 as a function of time for three different values of 𝑧 𝑐.
Each plot shows a decrease of 𝜌̇ 𝐷 with time. For the larger values of 𝑧 𝑐, we find a faster fall of
these graphs. For a conversion matter into dark energy, at the present time, it is necessary
that 𝜌̇ 𝐷 is positive when 𝜌̇ 𝑚 is negative. Figures 13, 14 show that it is possible to find a value
of 𝑧 𝑐 for which this condition is satisfied during a span close to the present time (𝑡 = 𝑡0).
Fig. 15: shows the variation of 𝜌̇ 𝑚 as a function of time for three different values of
the parameter 𝑛. For less negative values of this parameter, these curves show a decrease
with time, from positive to the negative region. For more negative values of n, 𝜌̇ 𝑚 remains
negative throughout, with a gradually decreasing slope. For less negative values of n, 𝜌̇ 𝑚
becomes negative, from an initial positive value, prior to the present time (𝑡 = 𝑡0). It means
that the matter content started decaying into dark energy somewhere in the recent past,
causing an accelerated expansion of the universe.
Fig. 16 shows the time variation of 𝜌̇ 𝐷 for three different values of the parameter 𝑛.
For 𝑧 𝑐 = 0.1, 𝜌̇ 𝐷 is positive for all n values here (implying a creation of dark energy) and it
decreases with time. The curves with more negative values of n are lying below the ones with
less negative values of the same parameter. It is evident that the more negative values of n
cause slower rate of production of dark energy.
VIII. CONCLUDING REMARKS
The present study is based on a spatially flat, homogeneous and isotropic space-time,
in the framework of Brans-Dicke theory. Here, the time evolution of Brans-Dicke parameter
(𝜔), energy density (𝜌) and EoS parameter (𝛾) have been determined from the field equations.
The choice of the scale factor, that ensures a signature flip of the deceleration parameter with
time, is likely to give more credibility to the results of this model. Without involving any self-
interaction potential (𝑉𝜙) and cosmological constant (Λ), the time evolutions of density
parameters for matter and dark energy have been determined and these results are
sufficiently consistent with those obtained through different models and methods [19, 20]. The
parameters 𝜇 and 𝜆, in model-1 and model-2 (sec. 4.1, 4.2), govern the time dependence of Ω 𝐷,
15. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 58
Ω 𝑚 and 𝛾 𝐷. It has been shown that 𝜇 and 𝜆 can be determined if one gets information from
astrophysical observations regarding the time or redshift (z) at which we had Ω 𝐷 = Ω 𝑚 in the
recent past of the expanding universe. Assuming an interaction to be taking place between
matter and dark energy, an effective interaction term (𝑄 𝑒𝑓𝑓) has been theoretically proposed
and its value has been determined on the basis of the energy conservation equation. For some
combinations of n and 𝑧 𝑐 (or n and 𝜇), this term changes its sign from positive to negative,
showing a change of direction of energy transfer between the sectors of matter and dark
energy. Its negative value at the present time implies a decay of matter into dark energy
which is responsible for the accelerated expansion of the universe. The behaviour of this
interaction term is consistent with the results of some recent studies carried out in a
completely different way [21, 22]. Using the expression of 𝑄 𝑒𝑓𝑓 we have determined the time
variations of 𝜌̇ 𝑚 and 𝜌̇ 𝐷. The conditions under which they respectively have negative and
positive signs during a span of cosmic time, matter may be regarded as decaying into dark
energy in that phase of expansion of the universe. The parameter n, which actually controls
the rate of change of gravitational constant with time (as per eqn. 76), determines the nature
of time variation of some cosmological quantities (𝜔, 𝜌 and 𝛾) discussed above. It implies that
the dependence of the scalar field (𝜙 ≡ 1/𝐺) upon the scale factor (𝑎) (eqn. 10) plays a very
important role in cosmic expansion. The dependence of 𝜔 upon 𝜑 is controlled by 𝜔0 and 𝑚
which are all functions of n. Positive values of n leads to large negative values of 𝛾 𝐷0, contrary
to its range of values (close to −1) obtained from recent astrophysical observations [19]. This
result is in favour of using negative values of n, implying an increase of gravitational constant
with time, as per equation (76). This behaviour of gravitational constant is also obtained from
other studies [30, 31]. Keeping in view all these facts, one may think of improving this model by
choosing a relation, between 𝜙 and 𝑎, which is different from the ansatz of equation (10), in
order to have more than one tunable parameter like n controlling the time variation of the
scalar field (𝜙). As a future project, one may think of determining the time dependence of 𝜌 𝑚
and 𝜌 𝐷 from equations (59) and (60) by assuming 𝑄 to have a form like 𝑄 = 𝜅𝐻 where 𝜅 is a
constant and H is the Hubble parameter. In the second equation, 𝛾 𝐷 can be replaced by 𝛾 𝐷0,
considering a very slow time variation of 𝛾 𝐷, as evident from figure 8, over a span of time close
to 𝑡 = 𝑡0. The time dependence of H can be taken from equation (21). Using the solutions of
these differential equations, (59) and (60), one can then calculate Ω 𝐷 and Ω 𝑚, taking the
energy density (𝜌) from equation (15). It would be possible to make an estimate of 𝜅 by using
the known values of Ω 𝐷 and Ω 𝑚 at 𝑡 = 𝑡0 in the expressions of the density parameters for
matter and dark energy. Then, using equations (10) and (20), one might make an attempt to
express the density parameters Ω 𝐷 and Ω 𝑚 in terms of the scalar field (𝜙).
IX. ACKNOWLEDGEMENTS
The authors of this article express their thanks and gratitude, very heartily, to the
researchers and academicians whose works have enriched them immensely and inspired
them to undertake the present study. S. Roy would like to thank all his colleagues in the
Department of Physics, St. Xavier’s College, Kolkata, India for creating and maintaining an
ambience of erudition and an atmosphere of cooperation and support. The authors also thank
the anonymous reviewers of this article for their valuable comments and useful suggestions.
16. Roy et al., (2019)
59 Volume 06, Issue 04, Version I, Apr’ 2019
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XI. FIGURES
Figure – 1: Plots of Brans-Dicke parameter versus time for three values of the parameter n.
Figure – 2: Plots of energy density versus time for three values of the parameter n
0.5 1.0 1.5 2.0
-1.70
-1.65
-1.60
-1.55
-1.50
-1.45
t / t0
n = -1.75
n = -1.74
n = -1.73
0.5 1.0 1.5 2.0
0.0
2.0x10
-27
4.0x10
-27
6.0x10
-27
8.0x10
-27
1.0x10
-26
1.2x10
-26
(Kg/m
3
)
t / t0
n = -1.75
n = -1.74
n = -1.73
19. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
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Volume 06, Issue 04, Version I, Apr’ 2019 62
Figure – 3: Plots of the equation of state (EoS) parameter versus time for three values of the
parameter n
Figure – 4: Plot of the equation of state (EoS) parameter versus time, for 𝑛 = −1.74, on a time
scale longer than Fig. 3
Figure – 5: Plots of 𝛺 𝐷 (solid curves) and 𝛺 𝑚 (dashed curves), from two models, for 𝑧 𝑐 = 0.3,
against the redshift parameter
0.5 1.0 1.5
-12
-8
-4
0
t / t0
n = -1.75
n = -1.74
n = -1.73
0.0 0.5 1.0 1.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
t / t0
n = -1.74
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
zc
= 0.3
Model-1
Model-2
Model-1
Model-2
DensityParameters
z
20. Roy et al., (2019)
63 Volume 06, Issue 04, Version I, Apr’ 2019
Figure – 6: Plots of 𝛺 𝐷 and 𝛺 𝑚 (using model-1) as functions of time, for two values of the
parameter 𝑧 𝑐
Figure – 7: Plots of the equation of state (EoS) parameter for dark energy versus time for
three values of the parameter n
Figure – 8: Plots of the EoS parameters versus time, for total energy (𝛾) and dark energy (𝛾 𝐷).
Here, 𝛾0 = −0.73 and 𝛾 𝐷0 = −1.04 for 𝑛 = −1.74
0.00 0.25 0.50 0.75 1.00
0.0
0.2
0.4
0.6
0.8
1.0
D
m
zc
= 0.7
zc
= 0.3
zc
= 0.7
zc
= 0.3
DensityParameters
t / t0
0.0 0.5 1.0 1.5
-100
-80
-60
-40
-20
0
20
40
60
80
100
D
t / t0
n = -1.75
n = -1.74
n = -1.73
0.6 0.8 1.0 1.2
-5
-4
-3
-2
-1
0
EoSParameters
t / t0
D
n = -1.74
21. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 64
Figure – 9: Plots of the effective interaction term (𝑄 𝑒𝑓𝑓) versus time for three values of the
parameter 𝑧 𝑐
Figure – 10: Plots of the effective interaction term (𝑄 𝑒𝑓𝑓) versus time for three values of the
parameter n
Figure – 11: Plots of the effective interaction term (𝑄 𝑒𝑓𝑓) versus redshift (z) for three values of
the parameter 𝑧 𝑐
0.5 1.0 1.5
0.0
2.0x10
-43
4.0x10
-43
Qeff
t / t0
zc
= 0.1
zc
= 0.4
zc
= 0.7
n = -1.74
0.5 1.0 1.5
0.0
2.0x10
-42
4.0x10
-42
Qeff
t / t0
n = -1.8
n = -1.7
n = -1.6
zc
= 0.1
-0.5 0.0 0.5 1.0
-2.0x10
-43
0.0
2.0x10
-43
4.0x10
-43
6.0x10
-43
8.0x10
-43
Qeff
z
zc
= 0.1
zc
= 0.4
zc
= 0.7
n = -1.74
22. Roy et al., (2019)
65 Volume 06, Issue 04, Version I, Apr’ 2019
Figure – 12: Plots of the effective interaction term (𝑄 𝑒𝑓𝑓) versus redshift (z) for three values of
the parameter n
Figure – 13: Plots of
𝑑𝜌 𝑚
𝑑𝑡
versus time for three values of the parameter 𝑧 𝑐
Figure – 14: Plots of
𝑑𝜌 𝐷
𝑑𝑡
versus time for three values of the parameter 𝑧 𝑐
-0.5 0.0 0.5 1.0 1.5 2.0
0.0
1.0x10
-42
2.0x10
-42
3.0x10
-42
4.0x10
-42
Qeff
z
n = -1.8
n = -1.7
n = -1.6
zc
= 0.1
0.7 0.8 0.9 1.0 1.1 1.2 1.3
-5.00x10
-44
-2.50x10
-44
0.00
2.50x10
-44
5.00x10
-44
dm
/dt
t / t0
zc
= 0.1
zc
= 0.4
zc
= 0.7
n = -1.74
0.7 0.8 0.9 1.0 1.1 1.2 1.3
-2.0x10
-44
0.0
2.0x10
-44
4.0x10
-44
6.0x10
-44
8.0x10
-44
1.0x10
-43
dD
/dt
t / t0
zc
= 0.1
zc
= 0.4
zc
= 0.7
n = -1.74
23. Time Evolution of Density Parameters for Matter and Dark Energy and their Interaction Term in
Brans-Dicke Gravity
Volume 06, Issue 04, Version I, Apr’ 2019 66
Figure – 15: Plots of
𝑑𝜌 𝑚
𝑑𝑡
versus time for three values of the parameter n
Figure – 16: Plots of
𝑑𝜌 𝐷
𝑑𝑡
versus time for three values of the parameter n
0.7 0.8 0.9 1.0 1.1 1.2 1.3
-2.0x10
-43
0.0
2.0x10
-43
4.0x10
-43
6.0x10
-43
dm
/dt
t / t0
n = -1.8
n = -1.7
n = -1.6
zc
= 0.1
0.7 0.8 0.9 1.0 1.1 1.2 1.3
5.0x10
-44
1.0x10
-43
1.5x10
-43
2.0x10
-43
2.5x10
-43
dD
/dt
t / t0
n = -1.8
n = -1.7
n = -1.6
zc
= 0.1