Abstract
We have studied the time dependence of for an expanding universe in the generalised BD theory and have obtained its explicit dependence on the nature of matter contained in the universe in different era.Lastly, we discuss how the observed accelerated expansion of the present universe can be accomodated in the formalism.
TIME DEPENDENCE OF BRANSDICKE PARAMETER
FOR AN EXPANDING UNIVERSE
B.K.Sahoo and L.P.Singh
Department of Physics,
Utkal University,Bhubaneswar751004,India
Key words : Generalised BransDicke theory, timedependent ,
accelerated expansion of the Universe.
PACS NO: 98.80.k,98.80.cq
1 Introduction
The BransDicke theory is defined by a constant coupling function and a scalar field .The relative importance is determined by the arbitary coupling function [1].The generalised BD theory involves an extension of original theory to the case of a time dependent coupling function . Generalised BD model is special for more than one reason . It appears naturally in super gravity theory, KaluzaKlein theories and in all the known effective string actions .It is perhaps the most natural extension of General Relativity ,which may explain its obiquitous appearance in fundamental theories.In the generalised BD theory,sometimes referred to as gravitondilaton theory, is an arbitary function of the scalar field (dilaton) .Hence it includes a number of models, one for every function .GR is obtained when the field = constant and .
BransDicke theory with a constant is a successful theory because it explains almost all the important features of the evolution of the Universe. Some of the problems like inflation , early and late time behaviour of Universe , cosmic acceleration and structure formation , cosmic acceleration,quintessence and coincidence problem , self interacting potential and cosmic acceleration can be explained in the BD formalism . For large BD theory gives the correct amount of inflation and early and late time behaviour, and for small negative it correctly explains cosmic acceleration, structure formation and coincidence problem .
BransDicke theory with a time dependent is also an interesting theory in its own right . Not only it gets a strong support from string and KaluzaKlein theories, a few attempts have also been taken using this formalism to study the dynamics of the Universe . However, all these attempts address the problems of evolution of universe , cosmic accleration and quintessence etc. in a qualitative way without exibiting the explicit time dependence of . Also,Alimi and Serna have shown that this time varing theory includes a number of models; one each for every time dependence of . So it is of natural interest to seek exact derivation of time dependence of from the dynamical equations of the Universe .
Our aim, in this paper therefore, is to derive explicit time dependence of for simple expanding solutions of field and wave equations in the BransDicke theory for all era which can give some information regarding the early and late time behaviour of the universe .This will also provide the oppertunity to examine if the derived timedependence of can be used to explain atleast some of the presently observed properties of the universe like cosmic accleration, structure formation etc.
2 the General
For a Universe filled with perfect fluid and described by FriedmannRobertson Walker spacetime with scale factor and spatial curvature index , the gravitational field equations in BD theory with time dependent , are
(1) 
(2) 
where and P are respectively the energy density and pressure of the fluid distribution . The equation of state of the fluid is given by. Some of the values of for typical cases are 1 (vacuum), 0 (dust), 1/3 (radiation), 1(massless scalar field) .
The wave equation for BransDicke scalar field when is a function of time is
(3) 
Energy conservation equation which can be obtaioned from eqs.,, and is,
(4) 
One important property of BransDicke theory is that it gives simple expanding solutions for field and scale factor which are compatible with solar system experiments . To derive the time dependence of which satisfies both field and wave equations, we assume the time dependence of the scale factor and scalar field in the following form which provides simple expanding solutions,
(5) 
(6) 
and are the present values of and .
From eq.(1) and taking to be consistent with the paradigm of
inflation, we get,
.
This immediatly leads to
=
.
This equation can alternatively,be written in the form
(7) 
where .
Solving eq. one gets
(8) 
Using eq. and in we get
From the above equation, one immediatly obtains,
(9) 
Since =2 +3 ,we are led to the time dependence of of the form,
(10) 
We now proceed to check the consistency of above time dependence of
with eq. which is the wave equation for the scalar field
.
Using eqs. and , eq. reduces to
We neglect in the denominator as we are interested in time dependence only.
Thus above equation reduces to
(11) 
Eq.(11) has obvious solution
or .
Thus, eq.(9) will satisfy both field and wave equations when or,
.It may be noted in passing that for these two cases in particular the neglect of the term as noted earlier is evidently justified. We now consider these two cases one by one.
CASE  1: .
For eq.(10) gives , eq.(7) gives constant and .
Thus, for Brans Dicke model goes over to General Relativity
.Here, is not related to and its value can be obtained by solving
equations of General relativity .
CASE  2: .
Eq. in this case reduces to,
(12) 
For completeness we also write,
(13) 
(14) 
Thus, to work within the BD model with a timedependent we have to take leading to the above solutions. Clearly, for a given (nonnegative and nonzero )which can be obtained from the observational data the time dependence of , and are fixed through the above equations.
3 for Different Era
It is clear from eq. that, the parameter which takes different values in different era, controls the time dependece of in the respective era as detailed below.
3.1 Vacuum dominated era:
Here,
(15) 
Since for an expanding universe , the time dependence of will always be governed by power of time greater than . Hence, decreases faster than with time.
3.2 Radiation dominated era:
.
Here,
(16) 
.
Here, again is a decreasing function of time as the Universe undergoes a deccelerated expansion in this era with .
3.3 Matter dominated era :
Here,
(17) 
The implications of this time dependence is dicussed in the next section.
3.4 Massless scalar field dominated era :
Here,
(18) 
4 for Present Universe.
The present observable universe contains cold matter of negligible pressure (dust). We,therefore, take the time dependence of as given by eq. i.e,
(19) 
For present time,taking , one gets,
(20) 
where represents present value of . ,, are positive nonzero constants. However, they can all be set equal to with no loss of generality if time t is measured in units of .Therefore,setting ,we get,
(21) 
For the presently observed acceleration to be accomodated
needs to be greater than 1.In that case as given by
eq. has the minimum value of .This result is in
agreement with the observation made by Banerjee and Pavon that
value must be greater than for Newtonian
constant of gravitation G and the scalar field energy density to
remain positive .
We now recall that the present observational data for decceleration parameter is
(22) 
Since by defination,
(23) 
Using eq. we get,
(24) 
We see from eq. and eq. that in order to obtain
between 1 and 0 , need only be greater than .
Putting , where but small, the present
day solutions of a varying BD theory can be
taken as
(25)  
(26)  
(27) 
Thus, we find that time dependence of obtained through consistent solutions of BD field equations and the wave equation leads to a negative constant at the present epoch. This result is consistent with conclusions arrived at by Bertolami and Martins , N.Banerjee and D.Pavon ,Sen and Seshadri that should possess a low negative value for a satisfactory explanation of structure formation,cosmic acceleration, coincidence problem, and to avoid the problems of quintessence etc within the formalism of BD theory.
5 Discussions and Conclusions
.
In this work,we wish to emphasize that we have, for the first time derived the explicit time dependence
of the BransDicke parameter by solving gravitational
field and wave equations of generalised BD theory consistently, assuming power law
behaviour for the scale factor
and scalar field .Interestingly,we find two consistent
solutions of the field and wave equations.One solution leads to
General Relativity with the implication that BD theory is a more
general formulation than GR. The other solution, which is of greater
interest to us,leads to a timedependent whose time
dependence is governed by the EOS parameter . Consequently,
exhibits different temporal behaviour in different epochs of
the evolving Universe characterised by its dominant matter/radiation
component. This,we believe,is an important result which can be used to study various characteristics of an evolving Universe within the generalised BD formalism. In particular,for an accelerated expanding universe,the present
value of comes out to be negative with a minimum value of
. This result, once again, nicely agrees with the conclusions of the earlier works carried within the
formalism of constant BD theory that
needs to be negative for a successful explanations of the
various observed characteristic of the evolving universe.
Acknowledgements
The authors are grateful to DST,Govt.of India for providing financial support.The authors also thank Institute of Physics,Bhubaneswar,India,for providing facility of the computer centre.
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