arXiv:0801.4529v2 [gr-qc] 11 Mar 2008 .A Usmani A. A. state of equation energy dark The o.Nt .Ato.Soc. Astron. R. Not. Mon. cetd.Rcie noiia form original in ; Received . Accepted eurce ntefrgon fcsooy nfact, In cosmology. entity of physical foreground new the a in years resurrected recent etc in parameters, density these matter-energy with paramete parameter, physical Hubble various factor, of evolution scale well) as in (and space time cases with concerned mainly is research Cosmological INTRODUCTION 1 ae oad naclrtn nvre ti eivdb mo by parameter believed cosmological the is that It physicists 19 Universe. the al. accelerating som et an Perlmutter after towards 1998; Einstein cated al. of et equations (Riess field results the servational of part essential an ue oevr u osm n-uigpolm(nw scos as (known problem problem), fine-tuning constant some mological to negat exert due can Moreover, it sure. because acceleration present the driving nw hsclqatte including quantities physical known constant. a than rather eainhpbtencsi pressure cosmic between relationship a al paig n aibeqatt a eedo h prod the construct may on one So, depend quantities. variable can other quantity two of variable one speaking, cally sity hrve 01 ebe ar 03.I at ausof values fact, In Zhuravlev 2003). & Ratra (Chervon & factor Peebles scale 2001; Zhuravlev or red-shift time, of tion ie sa ntne o h rsn rsuels Univers pressure-less present the evolve for of may value instance, it an that As suggest evolution time. cosmic of stages ferent al aito oiae nvre oee,i sconveni is it However, consider Universe. dominated radiation early † ⋆ equatio constant a and varying a between distinguish hardly

1 2 3 5 4 c eateto hsc,AiahMsi nvriy Aligarh University, Muslim Aligarh Physics, of Department aaBammyeVdaadr arpli hangr743 Shyamnagar Matripalli, Vidyamandir, Brahmamoyee Tara aybaaiVdaih aaal,Nrh2 agns Kol Parganas, 24 North Nabapalli, Vidyapith, Satyabharati eateto hsc,BrstGvrmn olg,Nrh2 North College, Government Barasat Physics, of Department eateto ahmtc,Gvrmn olg fEngineer of College Government Mathematics, of Department -al [email protected] E-mail: -al [email protected] E-mail: 08RAS 2008 ρ o,i re oseiyeattm-eedneo h un- the of time-dependence exact specify to order in Now, novn h qaino tt parameter state of equation the involving ω ω scniee szr,weesisvlewas value its whereas zero, as considered is sacntn uniybcueosrainldt can data observational because quantity constant a as 1 ⋆ 000 .P Ghosh P. P. , , ?? Λ – ?? srgre savral quantity variable a as regarded is 20)Pitd7Nvme 08(NL (MN 2018 November 7 Printed (2008) epromasuyo omceouinwt neuto fsta of equation an with evolution cosmic of study a perform We ABSTRACT e words: Key possi been Higgs-Boson. has and It energy delineated. dark been have other the and pointed ω qain.W oieta h nainlast cln nt in scaling a to leads inflation the ω that inflation notice and We expanding equations. linearly both explains proposition Λ 1 ( n a otk eoreof recourse take to has one , t ( ) t p n ec neuto fsae nti prah n fist its of one approach, this In state. of equation in hence and , H/H ˙ n atreeg den- matter-energy and 2 ta Mukhopadhyay Utpal , Λ ) srsosbefor responsible is yslcigaphenomenological a selecting by ω rvtto omlgclprmtr omlg:theory cosmology: - parameters cosmological - gravitation Λ Mathemati- . aa7016 etBna,India Bengal, West 126, 700 kata ω a become has 0 0,UtrPaeh India Pradesh, Uttar 002, 202 1 agns okt 0 2,Ws egl India Bengal, West 124, 700 Kolkata Parganas, 4 n n ete ehooy okt 0 9,Ws egl In Bengal, West 098, 700 Kolkata Technology, Leather and ing safunc- a as / 9 indi- 99) 2,Nrh2 agns etBna,India Bengal, West Parganas, 24 North 127, v pres- ive ω Along . 3 slike rs tdif- at n to ent Λ ,the e, nthe in 2000; ob- e some tof st with of n has uct - tmyhv o-ierrltosi as relationship non-linear a have may it ( Ko ta.20)weesrfie auswr niae yth cluster by galaxy is indicated and which anisotropy) were statistics CMB values (with data refined SNIa whereas combined 2003) al. et (Knop ω fo form suggests This ple 2003). Linder 2001; Chevallier & (Polarski hc a o nepii iedpnec htdsper wi disappears that dependence condition, time explicit an got has which cl atro omlgclrdhf.I oncint red to linearly, connection depend In redshift. may cosmological it or factor scale tt Kjte l 02 atlane l 05.Hr oeu some Here 2005). al. et on Bartelmann limits 2002; al. et (Kujat state nainr nvre.Frti,aphenomenological a expan this, linearly For viz., Universes. evolution, inflationary cosmic of stages different time-dependent uhpdyy hs,Klpv&Ry20) hri sev- 2007; wherein 2007), phenomenological 2005, eral Ray & Duttachowdhury Khlopov & Ghosh, Ray Mukhopadhyay, Mukhopadhyay, Duttachowdhur & Mukhopadhyay (Ray, 2007; mat works are There motivated equations. field ically Einstein the solve to selected dω/dz ( t = ) A sn bv rpsto,w xlr h hsclfaue o features physical the explore we proposition, above Using ssae above, stated As T E tl l v2.2) file style X ) ω z ω 0 =0 3 t + H sapae rmSI aaare data SNIa from appeared as .C Ray C. P. , ˙ Htrr&Tre 01 elr&Abeh 02 or 2002) Albrecht & Weller 2001; Turner & (Huterer ω = Λ 1 ω ( H t . oe fteform, the of model − H/H ˙ . 1 . 33 ω ) Λ < ω < a aeafntoa eainhpwith relationship functional a have may , ω r nvre ihasnl e of set single a with Universes ary oeshv enivsiae for investigated been have models ( 4 l oso oncinbetween connection a show to ble z ablRay Saibal , = ) eeuto fsaeparameter, state of equation he − oprmtr aebe pin been have parameters wo 0 eparameter te . 79 ω dia ω Tgake l 2004). al. et (Tegmark o Λ ( ˙ z + = ) al Universe. early - ∼ − ω 1 . H ′ ω 67 z o 5 where , 3 ω † + hssimple This . < ω < ( ω t Λ = ) 1 z/ igand ding oe is model 1+ (1 sim- a r hemat- ω − ω hthe th ′ seful 0 shift 0 . ing (1) 62 z + = y e ) f 2 Usmani, Ghosh, Mukhopadhyay, Ray & Ray

2 FIELD EQUATIONS FOR A STATIC SPHERICALLY conjunction with our proposition (equation 1) encorporate both lin- SYMMETRIC SOURCE early expanding and inflationary Universes. The Einstein field equations are 1 Λ Rij − Rgij = −8πG T ij − gij , (2) 4 LINEARLY EXPANDING UNIVERSE 2 8πG h i We consider a situation in which our Universe started ex- where Λ is the time-dependent cosmological term with vacuum ve- panding linearly (Crane 1979; Azuma & Tomimatsu 1982; locity of light being unity in relativistic units. Calzetta & Castagnino 1983) since its very beginning at a rate From equation (2) and Robertson-Walker metric, we get the H˙ = dH/dt with H(t = 0) = 0 at the point of singularity. Thus Friedmann and Raychaudhuri equations, respectively at a later time t> 0, the observable H(t) would be determined by 2 3k the relation, H(t)= tH˙ . The H˙ is the present value of H divided 3H + = 8πGρ + Λ, (3) a2 by the age of the Universe. In this case, equation (11) reduces to 2 3H + 3H˙ = −4πG(ρ + 3p) + Λ. (4) 3 dH˙ A(1 + W )H +3(1+ W )H = (12) Here, a = a(t) is the scale factor and k is the curvature con- dH 2H˙ − stant which assumes values 1, 0 and +1 for open, flat and closed where W = ω0 + ω1. models of the Universe respectively. Also, H =a/a ˙ is the Hub- Solution set for the differential equation (12) in connection to ble parameter and G, ρ, p are the gravitational constant, mat- different physical parameters is given below, ter energy density and pressure respectively. However, the gener- 1/E alized energy conservation law for variable G and Λ is derived a(t) = C(Et + D) , (13) by Shapiro, Sol`a& Stefan˘ci´c(2005)˘ using Renormalization Group 1 H(t) = , (14) Theory and also by Vereschagin & Yegorian (2006) using a for- Et + D mula of Gurzadyan & Xue (2003). For variable Λ and constant G, 1 0 1 the generalized conservation law reduces to the form ω(t) = ω + ω D , (15)  1+ Et  ρ˙ + 3(p + ρ)H = −Λ˙ /(8πG). (5) E ρ(t) = , (16) 4πG(Et + D)2(1 + ω(t)) p(t) = ω(t)ρ(t), (17) 3 COSMOLOGICAL MODELS FOR VARIABLE A Λ(t) = − . (18) EQUATION OF STATE PARAMETER 2E(Et + D)2 The barotropic equation of state which relates the pressure and den- Here, C and D are integration constants and E reads as sity of the physical system is given by E = 3(1 + W )+ 9(1 + W )2 + 4A(1 + W ) /4. (19) p = ωρ. (6) h p i Using this equation with equation (5), we arrive at With the fact that A << W , we may neglect the term involv- ing A in the above equation, which would yield E ≈ 3(1 + 8πGρ˙ + Λ=˙ −24πG(1 + ω)ρH. (7) W )/2. However, this would amount to be neglecting r.h.s term, 3 ˙ For a flat Universe(k = 0), equation (3) yields A(1 + W )H /2H, of equation (12), which suggests that the effect of this term is small. It is also obvious from equation (12) that this − 4πGρ = H/˙ (1 + ω). (8) term matters only at an early stage of the evolution of the Universe where H ∼ A. However, at this regime quantum effects become The equivalence of three phenomenological Λ-models (viz., 2 important and hence are of no relevance in our general relativistic Λ ∼ (a/a ˙ ) , Λ ∼ a/a¨ and Λ ∼ ρ) have been studied in detail approach. by Ray, Mukhopadhyay & Duttachowdhury (2007) for constant ω. With the consideration, H(t) = H˙ , equation (1) does not in- So, it is reasonable to study a variable-Λ model with a variable ω. 3 volve any explicit time dependence. So is equation (15) provided Let us, therefore, use the ansatz Λ˙ ∝ H , so that D = 0. We notice that with E = 1 and integration constants 3 Λ=˙ AH . (9) D = 0 and C = 1, equation (13) becomes a perfect example of a linearly expanding Robertson-Walker Universe, a(t) = t. How- This ansatz may find realization in the framework of self consis- ever, E = 1 suggests a value W = w0 + w1 = ω(t) = −1/3, tent inflation model (Dymnikova & Khlopov 2000, 2001), in which which is well above the minimum limit of ω(t) i.e. −0.79. We time-variation of Λ is determined by the rate of Bose condensate would see it later that inflation scales it to a lower value. From equa- evaporation (Dymnikova & Khlopov 2000) with A ∼ (m /m )2 B P tion (14), deceleration parameter, q, is deduced to be q = E − 1, (where m is the mass of bosons and m is the Planck mass). B P which thus is zero for such a linearly expanding Universe. From equations (4),(6),(8) and (9), we get 2 d2H 6 dH + = A. (10) (1 + ω)H3 dt2 H2 dt 5 INFLATIONARY UNIVERSE With dH/dt = H˙ , equation (10) reduces to We now consider a physical situation in which our Universe ini- 3 tially inflated non-linearly up to a certain value of time t = t0 << dH˙ A(1 + ω)H +3(1+ ω)H = . (11) 1 second (Guth 1981; Linde 1982; Albrecht & Steinhardt 1982). dH 2H˙ Since this time onward the expansion of the Universe is assumed to We would now show, how does these field equations used in be quite linear, which is described by the rate H˙ = dH/dτ. Here

c 2008 RAS, MNRAS 000, ??–?? The dark energy equation of state 3

30 5

20 4

Gp(t) 3 π 10 4

a(t) 2 0 0.0 1.0 2.0 3.0 4.0 5.0 1 −2 −4 0 Gp(t) 0 1 2 3 4 5 π

4 t (Units of H) −6

−8 Figure 3. The scale factors for the curves shown in Figure 2. 0 1 2 3 4 5 3 t (Units of H) dH˙ A(1 + ω0)H +3(1+ ω0)H = . (20) dH 2H˙ Figure 1. The upper panel represents 1 + ω(t) < 0. In this panel, If we substitute W at the place of ω0 in equation (12), we arrive dotted, dashed, long-dashed, and chain curves correspond to 1 ω = at equation (20). The solution set obtained for the linearly expand- −0.7, −0.8, −0.9 and 1.0, respectively. In the lower panel representing ing Universe is still valid for the inflationary Universe provided we 1 + ω(t) > 0, same curves correspond to ω1 = 0.0, −0.1, −0.2 and −0.3, respectively. The solid, thick dashed and thick long-dashed lines rep- substitute ω0 at the place of W in equation (19). This scaling from resent ω1 = −0.4, −0.5 and −0.6, respectively. For all these ω0 is taken W to ω0 in equation (19) may be attributed to the adiabatic expan- to be −1/3. sion of the Universe till time t0. The r.h.s. of equation (20) may be always neglected in this case because H is evolved to a large value compared to the values of A during inflation. 0 With the consideration that A<<ω0, we obtain ω0 = −1/3. Thus, the value ω(t) = ω0 + ω1 = −1/3 as obtained for linearly expanding Universe now corresponds to ω0 = ω(t) − ω1 = −1/3 −2 for an inflationary Universe. Therefore, the values ω0 = −1/3 and ω1 = 0 correspond to previously discussed linearly expand- −4 ing Universe and a nonzero value for ω1 represents inflationary Universe. Thus, we notice a direct correlation between ω(t) and Gp(t)

π −6 the inflation of the Robertson-walker Universe, which is buried

4 in the value of the parameter ω1. With ω0 = −1/3, the range − − −8 of the values 1.0 < ω1 < 0.46 falls in the suggested range −1.33 <ω(t) < −0.79. We may invoke a time dependence in equation (15) through D. −10 0 1 2 3 However, as mentioned earlier, data do not suggest any significant explicit time dependence in ω(t), thus D is set to zero. The non- t (Units of H) linearity in a(t) may be invoked through ω0 in E by choosing a different value for it other than −1/3. Thus for a linear behaviour Figure 2. The dotted, dashed, long-dashed, chain and solid curves represent after inflation this value is fixed to −1/3. The equation (16) for ρ is ω0 = −0.1, −0.2, −0.3, −0.4 and −1/3, respectively. The thick dashed singular at 1+ ω(t) = 0. So is equation (17) for p, which has been and thick long-dashed lines represent ω = −0.5 and -0.6, respectively. For plotted in Figure 1. For the negative pressure, as required by the all these, ω1 is adjusted using ω(t) = ω0 + ω1 = −0.8 dark energy, it applies a constraint on ω(t) such that ω(t) > −1 or ω1 = ω(t) − ω0 > −2/3. We find a range −2/3 < ω1 < −0.46 with ω0 = −1/3. τ is the measure of the time from t = t0. This leads to a translation in H such that H(t = t0 + τ) = H(t0)+ τH˙ . We assume that inflation has led to a condition H(t0) >> τH˙ , which implies that 6 DISCUSSION AND REMARKS H(t)= H(t0 +τ) >> τH˙ . With the consideration that the period of inflation has been very very brief compared to the age of the Uni- We have discussed two Universes: (i) a linearly expanding Universe verse, we may write t ≈ t0 + τ and H˙ = dH/dt ≈ dH/dτ. How- from its very beginning, (ii) and also the Universe like ours, which ever, the value of H˙ would be different from the previous case of has gone through an inflation at its very early stage followed by a linearly expanding Universe. Under these conditions, equation (11) linear expansion later. We notice that these two kind of Universes, reduces to which are direct consequence of our proposition (equation 1), are

c 2008 RAS, MNRAS 000, ??–?? 4 Usmani, Ghosh, Mukhopadhyay, Ray & Ray represented by the same set of equations with a translational shift Knop R. A. et al., 2003, Astrophys. J. 598 102. in the equation of state parameter in the latter case compared to the Kujat J. et al., 2002, Astrophys. J. 572 1. former. In both the cases, a(t) = 1 demands E = 1, which applies Linde A., 1982, Phys. Lett. B 108 389. a constraint on the equation of state parameter. For the inflationary Linder E. V., 2003, Phys. Rev. Lett. 90 91301. Universe, we have pin pointed ω0 = −1/3 and have delineated Mukhopadhyay U., Ray S. and Duttachowdhury S. B., 2005, the other parameter with a range −2/3 < ω1 < −0.46. We ob- astro-ph/0510549. serve that former is a special case of the latter with ω0 = −1/3 Mukhopadhyay U., Ray S. and Duttachowdhury S. B., 2007, and ω1=0. Any other value of ω0 would invoke a non-linear be- astro-ph/0708.0680 (to appear in Int. J. Mod. Phys. D). haviour in a(t) through E. The effect of the variation of ω0 on p is Mukhopadhyay U., Ghosh P. P., Khlopov M. and Ray S., 2007, presented in Figure 2 for a constant ω = ω0 + ω1 = −0.80 ob- astro-ph/0711.0686. tained by adjusting ω1 accordingly. The ω1 has nothing to do with Padmanabhan T. and Roychowdhury T., 2003, Mon. Not. R. As- E and hence has nothing to do with a(t). However, its value is a tron. Soc. 344 823. measure of translation in ω due to inflation. The equations for ρ and Peebles P. J. E. and Ratra B., 2003, Rev. Mod. Phys. 75 559. p involve ω and hence would remain unchanged with its constant Perlmutter S. J. et al., 1999, Astrophys. J. 517 565. value. Thus, variations in curves of Figure 2 is purely due to the Polarski D. and Chevallier M., 2001, Int. J. Mod. Phys. D 10 213. variation in ω0. The corresponding variations in a(t) are shown in Ray S., Mukhopadhyay U. and Meng X.-H., 2007, Gravit. Cos- Figure 3. mol. 13 142 [arXiv: astro-ph/0407295]. A negligible value of A is shown to be physically pos- Ray S., Mukhopadhyay U. and Duttachowdhury S. B., 2007, Int. sible from the viewpoint of cosmology and particle physics, J. Mod. Phys. D 16 1791. which means the absence of Λ in the field equations. So, Riess A. G. et al., 1998, Astron. J. 116 1009. both from physical and mathematical point of view the nul- Riess A. G. et al., 2001, Astrophys. J. 560 49. lity of Λ is achieved for the same Λ model. Again, the ex- Shapiro I. L., Sol`aL. and H. Stefan˘ci´cH.,˘ 2005, J. Cosmol. As- pression of q in this case has a striking similarity with that troparticlePhys. 1 012. of Ray, Mukhopadhyay & Duttachowdhury (2007). This work sug- Tegmark M. et al., 2004, Astrophys. J. 606 70. gests that in the late phase of the Universe, where tH˙ = H, Vereschagin G. V. and Yegorian G., 2006, Class. Quatum Grav. 23 the equation of state parameter behaves as a constant. Perhaps for 5049. this reason current data cannot distinguish clearly between a time- Weller J. and Albrecht A., 2002, Phys. Rev. D 65 103512. dependent ω and a constant one as pointed out by some workers Zhuravlev V. M., 2001, Zh. Eksp. Teor. Fiz. 120 1042. (Kujat et al. 2002; Bartelmann et al. 2005). Separating the entire cosmic history into two phases, it has been possible to derive the time-dependent expressions for the scale factor and the other physical parameters of each phase. It has been found that for inflationary phase, the deceleration param- eter q depends on time whereas for the linearly expanding phase it is constant, rather zero. This supports the opinion that q has changed during the course of time (Riess et al. 2001; Amendola 2003; Padmanabhan & Roychowdhury 2003).

ACKNOWLEDGMENTS Authors (AAU and SR) 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.

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c 2008 RAS, MNRAS 000, ??–??