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although , the belief in oscillatory universe dates back to the ancient times ( @xcite and references cited therein ) , a scientific model for it could only be proposed during the first half of the twentieth century @xcite . at a stage ,
oscillatory universe was considered as one of the main possibilities of cosmic evolution @xcite .
however , @xcite showed it as a viable alternative to inflation . throughout the past century
, the idea has been of scientific importance rather than of mere belief , which has been a focus of many theoretical investigations till date as summarized below .
the building up of cosmological entropy in each cycle of oscillation as shown by @xcite is fated to a thermodynamical end .
this had suggested that the universe had a beginning at a finite time , and thus had undergone a finite number of cycles @xcite , opposed to the idea of steady - state universe . when treated classically , it reaches a point of singularity at the end of every cycle resulting in a total breakdown for the general relativity ( gr ) . however , there are models with contracting epoch preceding a bounce in m - theories @xcite , braneworlds @xcite and loop quantum cosmologies @xcite .
the quantum effects built into these studies aim to provide a non singular framework for the bounces that avoid a kind of singularity hit by gr . besides , they all admit a cosmological evolution that has undergone an early oscillatory phase for a finite time , and has finally led to inflation as we see it now .
evolution of the expanding universe has also been studied using scalar , vector and tensor cosmological perturbation theories @xcite .
scalar field models that involve contracting phases are proposed as alternatives to the inflationary scenario .
these models have also been explored to distinguish such phases from purely expanding cosmologies .
vector perturbations uptill the second order have recently been considered as the scalar perturbations may not be able to distinguish it at first order @xcite .
oscillatory model arising from linearized @xmath0 theory of gravity @xcite has also been studied , which is found in reasonable agreement with some observational results like the cosmological red shift and the hubble law .
@xcite obtained the oscillatory mode of expansion of the universe from a self - consistent system of spinor , scalar and gravitational fields in presence of a perfect fluid and cosmological term @xmath1 @xcite .
various observations @xcite suggest that our universe is accelerating . and
the reason behind it is believed to be a mysterious _ dark energy _ , a term coined recently in 1999 but its history traces back as late as newton s time @xcite .
the cosmological constant @xmath1 , first adopted and then abandoned by einstein for his static model of the universe @xcite , is considered as one of the candidates for the dark energy .
the other candidate is a scalar field often referred to as quintessence @xcite .
it is , however , speculated that @xmath1 is a dynamical term rather than a constant .
the dark energy may be described as a perfect fluid through the equation of state @xmath2 relating the fluid pressure and matter density of the physical system through the relation @xmath3 , where @xmath4 is the barotropic dark energy equation of state parameter .
it plays a significant role in the cosmological evolution .
this is in general a function of time and may be a function of scale factor or redshift @xcite .
recently , @xcite have proposed a time dependent @xmath5 for the study of cosmic evolution with the equation of state parameter @xmath6 here , @xmath7 is the time varying hubble parameter .
the dynamics of both expanding and contracting epoch of the universe is buried in the sign of @xmath8 in this simple expression .
the @xmath8=0 represents a linearly expanding universe with @xmath9 .
a close system like an oscillatory universe may be created without violating any known conservation law of physics as its total mass ( energy ) is zero , which is indeed possible @xcite .
a universe may enter into an epoch of expansion followed by an epoch of contraction after every bounce . for an expanding universe that was created at @xmath10 with @xmath11
, we may arrive at two situations ( i ) linearly expanding universe represented by @xmath12 ( ii ) and universe that has undergone an adiabatic expansion to a finite @xmath13 followed by a linear expansion ( for example , our universe ) that may be represented by another condition @xmath14 provided @xmath15 .
having the same theoretical framework these two cases may be represented by a single set of equations @xcite with a difference that inflation leads to a scaling in @xmath16 .
thus , it is the equation of state that distinguishes between the two . after achieving its maximum radius at time @xmath17 ( @xmath18 ) with @xmath19 and @xmath20 ,
the universe starts collapsing due to its own mass and enters into an epoch of contraction with a negative sign of @xmath8 .
thus at a later time @xmath21 , @xmath22 is defined as @xmath23 , which would continue to decrease till universe achieves its minimum radius . at preceding times closer to it , a ` condition of oscillation ' , @xmath24 , would hold .
this condition would still at a time when universe sets to expand again .
the above equation of state enables us to handle this third possibility at least qualitatively when employed in the framework of general relativity .
the history of oscillations does not go unnoticed by the classical thermodynamics since cosmological entropy ( unidirectional thermodynamical scale of time ) keeps on growing after every cycle @xcite .
the generalized second law ( gsl ) of thermodynamics relates the dark energy equation of state with cosmological entropy @xcite .
thus the equation of state , which translates itself into entropy , would manifest similar properties despite the fact that relativity breaks down at singularities , whose details are not required in the framework .
an observer outside the universe , who measures a growing time ( or growing entropy ) from point of creation , would distinguish a state in an oscillation with an identical state in the next oscillation with a decrease in @xmath8 .
this would result in different solutions of field equations for identical states in two different oscillations .
the state of expansion , contraction and oscillation of the universe is buried in @xmath8 .
we find no reason to believe , why field equations of general relativity would not yield a valid solution for any value of @xmath8 no matter how many times the universe has hit the singularity in the middle and has come up to a finite size again and again for ` unknown ' reasons .
the einstein field equations of general relativity are @xmath25,\ ] ] where @xmath1 is time - dependent cosmological term with vacuum velocity of light being unity in relativistic units . for the spherically symmetric friedmann - lematre - robertson - walker metric , einstein field equations ( [ eq3 ] ) yield friedmann and raychaudhuri equations , respectively ,    @xmath26    here , @xmath27 is the cosmic scale factor , @xmath28 is the curvature constant , @xmath29 is the hubble parameter and @xmath30 , @xmath31 , @xmath32 are the gravitational constant , matter - energy density and fluid pressure , respectively .
the @xmath30 is taken to be a constant quantity along with a variable @xmath1 in which case the generalized energy conservation law may be derived @xcite . using equation of state ( @xmath33 ) in eq .
( [ eq6 ] ) and differentiating eq .
( [ eq5 ] ) with time @xmath34 for flat universe ( @xmath35 , we arrive at    @xmath36    in the light of the studies by @xcite , we use the _ ansatz _ , @xmath37 .
as argued by @xcite this _ ansatz _ may find realization in the framework of self consistent inflation model @xcite in which time - dependent @xmath1 is determined by the rate of bose condensate evaporation with @xmath38 ( where @xmath39 is the mass of bosons and @xmath40 is the planck mass ) .
the equation of state along with eqs .
( [ eq6 ] ) ,  ( [ eq10 ] ) and above _ ansatz _ leads to    @xmath41    the condition @xmath24 with @xmath42 simplifies it further    @xmath43    a change of sign for @xmath8 would amount to the change of sign for @xmath44 .
thus , we have same solution set for @xmath42 with @xmath45 as    @xmath46),\end{aligned}\ ] ]    @xmath47    @xmath48    @xmath49,\end{aligned}\ ] ]    @xmath50    @xmath51,\end{aligned}\ ] ]    where , @xmath52 , @xmath53 and @xmath54 and @xmath55 are integration constants .
the logarithm @xmath56 demands a positive definite value for @xmath57 .
thus @xmath58 , @xmath55 and @xmath44 must have identical sign .
therefore , second term of eq .
( [ eq17 ] ) i.e. @xmath59 is positive for @xmath60 and negative in the range @xmath61 .
we find @xmath62 for a linearly expanding universe and a range @xmath63 for an inflationary universe @xcite .
thus , it is always negative . for a contracting universe @xmath8 is negative , which makes the second term of eq .
( [ eq2 ] ) positive .
this , when compared with eq .
( [ eq17 ] ) , suggests that @xmath44 is positive for @xmath64 and negative for @xmath65 .
thus , for @xmath66 , we obtain a condition , @xmath67 . here
@xmath68 means @xmath69 , which represents a linearly expanding universe with @xmath70 . for an observer sitting outside the universe , the rate @xmath8 would appear to be smaller and smaller for the increasing number of oscillations as @xmath34 would appear to be larger and larger .
it is obvious from eq .
( [ eq16 ] ) that @xmath22 is positive definite that means it grows from zero to a maximum value during the epoch of expansion and and then approaches to zero during the epoch of contraction .
we also observe that eq .
( [ eq18 ] ) for the density is singular at @xmath71 , so is the pressure ( @xmath72 ) , which turns out to be positive for @xmath73 .
the same restriction is found for the expanding universe using same equation of state @xcite , which is in agreement with the results of gsl of thermodynamics that too restricts the equation of state , @xmath74 , in an expanding universe . beyond this limit , for example @xmath75 , pressure is negative .
let us now try to extract the two phases of expanding and contracting universe .
as mentioned earlier , we had the condition @xmath76 @xcite which we must satisfy here also and thereby take @xmath77 and @xmath78
. with this our @xmath31 will always be positive either universe is expanding or contracting . in our solution
@xmath57 depends upon the constants @xmath55 , @xmath58 , @xmath79 and also @xmath34 .
@xmath58 , via the relation @xmath80 , is a negative quantity
. thus @xmath55 must be negative to make @xmath57 positive and between @xmath81 and @xmath82 .
hence pressure will be positive and universe will contract .
it also depends upon constant @xmath55 , so if @xmath83 multiplied by @xmath79 is less than @xmath82 pressure would be positive . in other words
if @xmath58 is very very small , even for a large @xmath79 , pressure will be positive and universe will contract . but with increasing @xmath79 , @xmath84 will eventually be greater than equal to one and pressure will again be negative and the universe will start expanding .
we , therefore , construct the story for a universe like ours , which had started with an adiabatic expansion with @xmath85 , then it had been slowed down to a linearly expanding universe with @xmath86 , which was then followed by an epoch of contraction with a positive value of @xmath59 . for smaller values of @xmath57 or @xmath34 ,
@xmath59 may reach the condition @xmath75 , which will make the pressure positive and thus would receive a bounce .
this process would continue repeating without violating any known conservation law .
however , @xmath87 for @xmath88 .
thus , the universe would appear to be a linearly expanding universe with @xmath70 and would never reach the condition for contraction or bounce i.e. @xmath75 .
thus , it recommends for initial oscillations only .
one may suggest improvements in the equation of state , however we have shown a way to incorporate oscillatory universe in general relativity through it .
for the successive oscillations having equal epoch of expansion and contraction .
the solid , dotted , dashed , long - dashed and chain curves represent @xmath89 during first , second , third , fourth , fifth and sixth oscillations , respectively .
the oscillation period for the universe is assumed to be ( @xmath90 units of h ) . ]
authors ( sr and aau ) are thankful to the authority of inter - university centre for astronomy and astrophysics , pune , india for providing associateship programme under which a part of this work was carried out .
the concept of oscillatory universe appears to be realistic and buried in the dynamic dark energy equation of state . 
 we explore its evolutionary history under the frame work of general relativity . 
 we observe that oscillations do not go unnoticed with such an equation of state and that their effects persist later on in cosmic evolution . 
 the ` classical ' general relativity seems to retain the past history of oscillatory universe in the form of increasing scale factor as the classical thermodynamics retains this history in the form of increasing cosmological entropy .    
 gravitation - cosmological parameters - cosmology : theory - early universe .