Astrophys Space SciDOI 10.1007/s10509-013-1390-9

O R I G I NA L A RT I C L E

Oscillatory Universe, dark energy and general relativity

Partha Pratim Ghosh · Saibal Ray · A.A. Usmani ·Utpal Mukhopadhyay

Received: 24 October 2012 / Accepted: 12 February 2013© Springer Science+Business Media Dordrecht 2013

Abstract The concept of oscillatory Universe appears to berealistic and buried in the dynamic dark energy equation ofstate. We explore its evolutionary history under the frame-work of general relativity. We observe that oscillations donot go unnoticed with such an equation of state and that theireffects persist later on in cosmic evolution. The ‘classical’general relativity seems to retain the past history of oscilla-tory Universe in the form of increasing scale factor as theclassical thermodynamics retains this history in the form ofincreasing cosmological entropy.

Keywords Oscillatory Universe · Dark Energy · GeneralRelativity

1 Introduction

Although, the belief in Cyclic model related to OscillatoryUniverse dates back to the ancient times (Kanekar et al.

P.P. GhoshDepartment of Physics, A.J.C. Bose Polytechnic, Berachampa,North 24 Parganas, Devalaya 743 424, West Bengal, Indiae-mail: parthapapai@gmail.com

S. Ray (�)Department of Physics, Government College of Engineering &Ceramic Technology, Kolkata 700 010, West Bengal, Indiae-mail: saibal@iucaa.ernet.in

A.A. UsmaniDepartment of Physics, Aligarh Muslim University,Aligarh 202002, Uttar Pradesh, Indiae-mail: anisul@iucaa.ernet.in

U. MukhopadhyaySatyabharati Vidyapith, North 24 Parganas, Kolkata 700 126,West Bengal, Indiae-mail: utpalsbv@gmail.com

2001 and references cited therein), a scientific model for itcould only be proposed during the first half of the twen-tieth century (Friedman 1922). At a stage, cosmologicalmodel of oscillatory Universe that combines both the BigBang and the Big Crunch as part of a cyclical event, wasconsidered as one of the main possibilities of cosmic evo-lution (Dicke et al. 1965). However, Durrer and Lauken-mann (1996) showed it as a viable alternative to inflation.Throughout the past century, the idea has been of scientificimportance rather than of mere belief, which has been a fo-cus of many theoretical investigations till date as summa-rized in the Sect. 2.

Various observations (Riess et al. 1998; Perlmutter et al.1998; Spergel et al. 2003; Tegmark et al. 2004) suggest thatour Universe has been accelerating from 7 Gyr ago. Thereason behind this phenomenon is believed to be a myste-rious dark energy, a term coined recently in 1999 but itshistory traces back as late as Newton’s time (Calder andLahav 2008). The cosmological constant Λ, first adoptedand then abandoned by Einstein for his static model ofthe Universe (Einstein 1917, 1918, 1931), is considered asone of the candidates for the dark energy. It is, however,speculated that Λ is a dynamical term rather than a con-stant. The other candidate is a scalar field often referred toas quintessence (Watterich 1988; Ratra and Peebles 1988;Caldwell et al. 1998).

The dark energy may be described as a perfect fluidthrough the equation of state (EOS) relating the fluid pres-sure and matter density of the physical system through therelation

p = ωρ, (1)

where ω ≡ ω(t) is the barotropic dark energy EOS parame-ter. It plays a significant role in the cosmological evolution.This is in general a function of time and may be a func-tion of scale factor or redshift (Chervon and Zhuravlev 2000;

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Zhuravlev 2001; Peebles and Ratra 2003). Recently, Usmaniet al. (2008) have proposed a time dependent ω for the studyof cosmic evolution with the EOS parameter

ω(t) = ω0 + ω1(tH /H). (2)

Here, H ≡ H(t) is the time varying Hubble parameter. Thedynamics of both expanding and contracting epoch of theUniverse is buried in the sign of H in this simple expression.The H = 0 represents a linearly expanding Universe withω(t) = ω0.

A closed system like an oscillatory Universe may becreated without violating any known conservation law ofPhysics as its total mass (energy) is zero, which is indeedpossible (Tryon 1973). A Universe may enter into an epochof expansion followed by an epoch of contraction after ev-ery bounce. For an expanding Universe that was createdat t = 0 with H(t) = 0, we may arrive at two situations:(i) linearly expanding Universe represented by H = tH , and(ii) Universe that has undergone an adiabatic expansion toa finite Hi followed by a linear expansion (for example,our Universe) that may be represented by another condi-tion H � tH provided Hi � tH . Having the same theo-retical framework these two cases may be represented by asingle set of equations (Usmani et al. 2008) with a differ-ence that inflation leads to a scaling in ω. Thus, it is theEOS that distinguishes between the two. After achieving itsmaximum radius at time t = t0 (τ = 0) with H = 0 andH(t0) = Hmax , the Universe starts collapsing due to its ownmass and enters into an epoch of contraction with a negativesign of H . Thus at a later time t = t0 +τ , where H is definedas H = Hmax − τH , which would continue to decrease tillUniverse achieves its minimum radius. At preceding timescloser to it, a ‘condition of oscillation’, H(t) � |tH |, wouldhold. This condition would still at a time when Universe setsto expand again.

The above EOS enables us to handle this third possibil-ity at least qualitatively when employed in the frameworkof general relativity. The history of oscillations does not gounnoticed by the classical thermodynamics since cosmolog-ical entropy (unidirectional thermodynamical scale of time)keeps on growing after every cycle (Tolman 1934). Thegeneralized second law of thermodynamics relates the darkenergy EOS with cosmological entropy (Bakenstein 1973;Gibbons and Hawking 1977a, 1977b; Unruh and Wald 1982;Davies 1988). Thus the EOS, which translates itself into en-tropy, would manifest similar properties despite the fact thatrelativity breaks down at singularities, whose details are notrequired in the framework. An observer outside the Uni-verse, who measures a growing time (or growing entropy)from point of creation, would distinguish a state in an os-cillation with an identical state in the next oscillation witha decrease in H . This would result in different solutions offield equations for identical states in two different oscilla-tions. The state of expansion, contraction and oscillation of

the Universe is buried in H . We find no reason to believe,why field equations of general relativity would not yield avalid solution for any value of H , no matter how many timesthe Universe has hit the singularity in the middle and hascome up to a finite size again and again for ‘unknown’ rea-sons.

2 A brief overview

The building up of cosmological entropy in each cycle ofoscillation as shown by Tolman (1934) is fated to a ther-modynamical end. This suggests that the Universe had abeginning at a finite time, and thus had undergone a finitenumber of cycles (Zeldovich and Novikov 1983), opposedto the idea of steady-state Universe. When treated classi-cally, it reaches a point of singularity at the end of everycycle resulting in a total breakdown for the general rel-ativity (GR). However, there are models with contractingepoch preceding a bounce in M-theories (Khoury et al. 2001;Steinhardt and Turok 2002), braneworlds (Kanekar et al.2001; Shtanov and Sahni 2003; Hovdebo and Myers 2003;Foffa 2003; Burgess et al. 2004) and loop quantum cosmolo-gies (Lidsey et al. 2004; Ashtekar et al. 2006). The quantumeffects built into these studies aim to provide a non-singularframework for the bounces that avoid a kind of singularityhit by GR. Besides, they all admit a cosmological evolu-tion that has undergone an early oscillatory phase for a fi-nite time, and has finally led to inflation as we see it now.Saha and Boyadjiev (2004) obtained the oscillatory modeof expansion of the Universe from a self-consistent sys-tem of spinor, scalar and gravitational fields in presence ofa perfect fluid and cosmological term Λ (Zeldovich 1967;Weinberg 1989). Evolution of the expanding Universe hasalso been studied by using scalar, vector and tensor cos-mological perturbation theories (Mukhanov et al. 1992;Noh and Hwang 2004; Bartolo et al. 2004; Nakamura 2007).Scalar field models that involve contracting phases are pro-posed as alternatives to the inflationary scenario. Thesemodels have also been explored to distinguish such phasesfrom purely expanding cosmologies. Vector perturbationsuptill the second order have recently been considered as thescalar perturbations may not be able to distinguish it at firstorder (Mena et al. 2007). Oscillatory model arising fromlinearized R2 theory of gravity (Corda 2008) has also beenstudied, which is found in reasonable agreement with someobservational results like the cosmological red shift and theHubble law.

Besides the above works on oscillating Universe modelswe would like to have a special mention of the followingtwo models:

(i) The Steinhardt-Turok model which gives a descriptionof two parallel M-branes that collide periodically in a

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higher dimensional space (Steinhardt et al. 2005). Heredark energy corresponds to a force between the branes.

(ii) In the Baum-Frampton model EOS parameter assumesthe form ω < −1, which is known as phantom energycondition in contrast to the Steinhardt-Turok assump-tion where ω is never less than −1 (Baum and Frampton2007).

These two models will provide motivation for choice ofthe form of EOS parameter in connection to our model onoscillatory Universe.

3 The Einstein field equations and their solutions

The Einstein field equations of GR can be given by

Rij − 1

2Rgij = −8πG

[T ij − Λ

8πGgij

], (3)

where Λ is time-dependent Cosmological constant with ve-locity of light in vacua being unity (in relativistic units).The justification for introducing this time varying Λ canbe found directly from the observations on supernova dueto the High-z Supernova Search Team (HZT) and the Su-pernova Cosmology Project (SCP) (Riess et al. 1998; Perl-mutter et al. 1998) as discussed in the Introduction. To un-derstand the physical nature of this exotic energy that tendsto increase the speed of expansion of the space-time of theUniverse several models have been proposed by the investi-gators as can be found in the literature (Overduin and Coop-erstock 1998; Sahni and Starobinsky 2000).

For the spherically symmetric Friedmann-Lemaître-Robertson-Walker metric, the Einstein field equations (3)yield Friedmann and Raychaudhuri equations, respectively,as follows

3H 2 + 3k

a2= 8πGρ + Λ, (4)

3H 2 + 3H = −4πG(ρ + 3p) + Λ. (5)

Here, a = a(t) is the cosmic scale factor, k is the curva-ture constant, H = a/a is the Hubble parameter and G, ρ,p are the gravitational constant, matter-energy density andfluid pressure, respectively. The G is taken to be a constantquantity along with a variable Λ where the generalized en-ergy conservation law may be derived (Shapiro et al. 2005).Using the EOS (1) in Eq. (5) and comparing it with Eq. (4)for flat Universe (k = 0), we arrive at

H = −4πGρ(1 + ω). (6)

In the light of the studies by Ray et al. (2007), we use theansatz Λ = AH 3, where A is a constant of proportionality.As argued by Usmani et al. (2008) this ansatz may find re-alization in the framework of self consistent inflation model(Dymnikova and Khlopov 2000) in which time-dependent

Λ is determined by the rate of Bose condensate evaporationwith A ∼ (mB/mP )2, where mB is the mass of bosons andmP is the Planck mass.

The EOS along with Eqs. (5), (6) and above ansatz leadsto

dH

dH+ 3(1 + ω)H = A(1 + ω)H 3

2H, (7)

with dH/dt = H .The condition H(t) � |tH | with ±H simplifies it further

as

dH

dH± 3ω1τH = 0. (8)

A change of sign for H would amount to the change ofsign for ω1. Thus, we have same solution set for ±H with±ω1 as follows:

a(t) = C exp(B[lnT − 1]), (9)

H(t) = B

tlnT , (10)

ω(t) = ω0 + ω1

(1

lnT

), (11)

ρ(t) = −(

1

4πG

)[B

t2(1 + ω(t))

], (12)

p(t) = ω(t)ρ(t), (13)

Λ(t) = AB3

t2

[(lnT )3 − 3(lnT )2 + 6 lnT − 6

], (14)

with B = t/Eτ , T = DEτt and E = 3ω1, whereas C andD are two constants of integration.

4 Physical features of the oscillatory model

The term lnT demands a positive definite value for T . ThusE, D and ω1 must have identical sign. Therefore, secondterm of Eq. (11) i.e. ω1/ lnT is positive for T > 1 and neg-ative in the range 0 < T < 1. We find ω1 = 0 for a linearlyexpanding Universe and a range −2/3 < ω1 < −0.46 for aninflationary Universe (Usmani et al. 2008). Thus, it is alwaysnegative. For a contracting Universe H is negative, whichmakes the second term of Eq. (2) positive. This, when com-pared with Eq. (11), suggests that ω1 is positive for T > 1and negative for T < 1. Thus, for D = 1, we obtain a condi-tion, E = 3ω1 > 1/τ t . Here t → ∞ means ω1 → 0, whichrepresents a linearly expanding Universe with ω(t) = ω0 =−1/3. For an observer sitting outside the Universe, the rateH would appear to be smaller and smaller for the increasingnumber of oscillations as t would appear to be larger andlarger. It is obvious from Eq. (10) that H is positive definitethat means it grows from zero to a maximum value duringthe epoch of expansion and then approaches to zero duringthe epoch of contraction. We also observe that Eq. (12) for

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Fig. 1 Growth of scale factor a(t) for the successive oscillationshaving equal epoch of expansion and contraction. The solid, dotted,dashed, long-dashed and chain curves represent a(t) during first, sec-ond, third, fourth and fifth oscillations, respectively. The oscillationperiod for the Universe is assumed to be (t = 1 units of H )

the density is singular at 1 + ω(t) = 0, so is the pressure(p = ωρ), which turns out to be positive for 0 > ω(t) > −1.The same restriction is found for the expanding Universe us-ing same EOS (Usmani et al. 2008), which is in agreementwith the results of GSL of thermodynamics that too restrictsthe EOS, ω(t) > −1, in an expanding Universe. Beyond thislimit, for example ω(t) > 0, pressure is negative.

Let us now try to extract the two phases of expandingand contracting Universe. As mentioned earlier, we had thecondition −1 < ω1 < −0.46 (Usmani et al. 2008) whichwe must satisfy here also and thereby take ω0 = −1/3 and−2/3 < ω1 < −0.46. With this our ρ will always be positiveeither Universe is expanding or contracting. In our solutionT depends upon the constants D, E, τ and also t . E, viathe relation E = 3ω1, is a negative quantity. Thus D mustbe negative to make T positive and between 0 and 1. Hencepressure will be positive and Universe will contract. It alsodepends upon constant D, so if |D| multiplied by τ is lessthan 1 pressure would be positive. In other words if E is verysmall, even for a large τ , pressure will be positive and Uni-verse will contract. But with increasing τ , |D|τ will eventu-ally be greater than equal to one and pressure will again benegative and the Universe will start expanding.

In Fig. 1 we have depicted the feature of expansion andcontraction via the scale factor a(t). It can be observed fromthe figure that growth of scale factor a(t) for the successiveoscillations have the equal epoch of expansion and contrac-tion.

5 Conclusion

We have constructed in the present work the story for a Uni-verse like ours own. The Universe had started with an adia-batic expansion with ω(t) < −1/3, then it had been slowed

down to a linearly expanding Universe with ω(t) = −1/3,which was then followed by an epoch of contraction witha positive value of ω1/ lnT . For smaller values of T ort , ω1/ lnT may reach the condition ω(t) > 0, which willmake the pressure positive and thus would receive a bounce.This process would continue repeating without violating anyknown conservation law. However, ω1/ lnT → 0 for T →∞. Thus, the Universe would appear to be a linearly ex-panding Universe with ω(t) = ω0 = −1/3 and would neverreach the condition for contraction or bounce i.e. ω(t) > 0.Thus, it recommends for initial oscillations only. One maysuggest improvements in the EOS, however we have showna way to incorporate oscillating Universe in the general rel-ativity through it.

Some other suggestions may also be correlated with thepresent work, e.g. pertinent issues on the origin of the ob-served baryon asymmetry and also physical nature of thedark matter. In connection to the evolution of the baryoncharge in the oscillating baryon asymmetric Universe wewould like to mention a few recent works (Buchmülleraet al. 2005; Davoudiasl et al. 2010; Canettia et al. 2012),in particular the review work of Canettia et al. (2012) wherethey discuss a testable minimal particle physics model thatsimultaneously explains the baryon asymmetry of the uni-verse, neutrino oscillations and dark matter. So, followingthe ‘mechanism for generating both the baryon and darkmatter densities of the Universe’ (Davoudiasl et al. 2010)a future project may be undertaken within the purview ofour present investigation.

Acknowledgements SR and AAU are thankful to the authority ofInter-University Centre for Astronomy and Astrophysics, Pune, Indiafor providing Visiting Associateship under which a part of this workwas carried out. We all are also thankful to the referee for raisingsome pertinent issues which helped us to improve the standard of themanuscript.

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