Indian J Phys (August 2012) 86(8):755–761 DOI 10.1007/s12648-012-0123-1 ORIGINAL PAPER Interacting and non-interacting two-fluid cosmological models in a Bianchi type VI0 space-times S Oli* B C T Kumaon Engineering College, Almora 263 653, Uttarakhand, India Received: 10 June 2011 / Accepted: 13 February 2012 / Published online: 27 June 2012 Abstract: In this paper we have presented anisotropic, homogeneous two-fluid cosmological models in a Bianchi VI0 space-time. We have obtained classes of cosmological models where the two fluids are in interactive phase and another when the two fluids are non-interacting. These models have one or two distinct real singularities depending upon the sign of the constant of integration. The behavior of associated fluid parameters and kinematical parameters has been discussed in detail. Keywords: Two fluid; Bianchi VI0 space-time; Interacting and non-interacting cosmological models PACS Nos.: 04.20.-q; 98.80.-k 1. Introduction The detection of 2.73°K isotropic cosmic microwave background radiation (CMBR) by Penzias and Wilson has been a notable observational pillar on which stands the entire speculative edifice of the theoretical cosmology. The discovery of CMBR encouraged many cosmologists to investigate Friedmann–Robertson–Walker (FRW) cosmological models, which contain both matter and radiation [1–3]. In the two-fluid model, one fluid represents the matter content of the universe and another fluid is chosen to model the CMBR. In the last decade space investigations have detected anisotropies in CMBR at various angular scales. In 1992, Cosmic Background Explorer (COBE) discovered temperature variations in the CMB level of 1 part in 100,000 [4]. Proponents of inflation believe that there anisotropies encode immense wealth of information about various parameters of the universe. More about CMBR anisotropy is expected to be uncovered by the investigation of Microwave Anisotropy Probe (MAP) and Cosmic Background Radiation Anisotropy Satellite–Satellite for Measurement of Background Anisotropies (COBRAS– *Corresponding author, E-mail: sanjayoli@rediffmail.com SAMBA), known as Planck Surveyor satellite. From different instruments such as, Cosmic Anisotropy Telescope (CAT), Bolometeres and High Electron Mobility Transistor (HEMT), we are able to detect and measure anisotropy more precisely. The observed anisotropies of CMB at various angular scales make a point of fresh look in the investigation of two-fluid models, especially those spacetimes shows anisotropy. Bianchi space-times provide spatially homogeneous and anisotropic models of universe as against the homogeneous and isotropic FRW models. Billyard and Coley [5] have investigated spatially flat isotropic cosmological models which contain a scalar field with an exponential potential and a perfect fluid with a linear equation of state. Recently, Kalita et al. [6] have compared matter and radiation densities with a constant vacuum energy density of positive cosmological constant, from a few seconds of the universe till the present epoch. Two-fluid cosmological models in Bianchi I, II, V, and VI0 space-times with co-moving twofluid source and different geometries have been investigated by a number of authors [7–12] for the description of the large-scale behavior of the actual universe. Saha [13] studied the behavior of matter distribution in the framework of Bianchi type II space-time. In this paper we have presented general relativistic, new two-fluid cosmological models in Bianchi VI0 space-time and the physical behavior of the model for interacting and non-interacting case has been studied in detail. Ó 2012 IACS 756 S Oli 2. Einstein’s field equations for Bianchi VI0 space-time G44 The metric for Bianchi type VI0 space-time is ds2 ¼ c2 dt2  X2 ðtÞdx2  !2 ðtÞe2x dy2  Z2 ðtÞe2x dz2 : Einstein’s field equation for a two-fluid source is i 8pG h iðmÞ iðrÞ Gij ¼  4 Tj þ Tj ; c ð1Þ ð2Þ where Gij  Rij  12 dij R þ kc2 is the Einstein tensor, the iðmÞ energy momentum tensor for matter field Tj iðrÞ radiation field Tj and that for are given by [12]: m Timj ¼ ðpm þ qm c2 Þ um i uj  pm gi j ð3Þ Tirj ¼ ð4=3Þ qr c2 uri urj  1=3 qr c2 gi j ð4Þ G14  X_ Y_ Y_ Z_ X_ Z_ 1 þ þ  2  XY YZ XZ X  8pG 4 2 2 r 2 ¼  2 ðpm þ qm c2 Þðum 4 Þ þ qr c ðu4 Þ c 3 i q ðpm þ r c2 Þ þ kc2 ð12Þ 3   8pG 4 Z_ Y_ m 2 r r q u þ c u u :   ¼  2 ðpm þ qm c2 Þum 1 4 c 3 r 1 4 Z Y  ð13Þ Thus, we have seven equations in ten unknowns X, Y, Z, m r r qm, pm qr, um 1 , u4 , u1 and u4. In view of Eq. (13), we note that assuming the field variables Y and Z are equal the four-velocity becomes co-moving and its vice versa. In the next section, we have obtained the cosmological models with commoving matter and radiation. r Four-velocity um i and ui satisfy m ij r r gi j um i uj ¼ 1; g ui uj ¼ 1: ð5Þ Consequently, from the off diagonal of Eqs. (2–4) together with energy conditions pm þ qm c2 [ 0, qm, qr [ 0 imply that ðmÞ u2 ðmÞ ¼ u3 ðrÞ ðrÞ ¼ u2 ¼ u3 ¼ 0: ð6Þ In view of Eqs. (5) and (6), the non-vanishing components of the four-velocity of the matter field ui(m) is 2 ðum 4Þ  2 ðum 1Þ ¼1 2 X ð7Þ The non-vanishing components of the four-velocity of the radiation field ui(r) is ður4 Þ2 ður Þ2  12 ¼ 1: X ð8Þ From Eqs. (1), (6–8), Einstein’s field Eq. (2) are   € Y€ Y_ Z_ Z þ þ G11  1 þ X 2 Z Y XZ      2 4 8pG 2 r 2 ¼ 2 p m þ qm c 2 u m 1 þ qr c u1 c 3  qr 2 i 2 2 2 þ X pm þ c ð9Þ þ kc X 3   € Z € X_ Z_ X 1 þ þ  2 G22  e2x Y 2 X Z  XZ X  q r c2 2 2x 2 8pG p þ ¼ e !  kc ð10Þ m c2 3 " # € X_ !_ € ! 1 2x 2 X þ þ  G33  e Z X ! X! X 2    8pG q c2 ¼ e2x Z2 2 pm þ r ð11Þ  kc2 c 3 3. Models with co-moving matter and radiation Here we assume that the matter and radiation both are comoving: r m r um 4 ¼ u4 ¼ 1 and u1 ¼ u1 ¼ 0: ð14Þ Consequently, from Eq. (13), we obtain YðtÞ ¼ ZðtÞ: ð15Þ From Eqs. (14) and (15), field Eqs. (9–13) are reduced to three independent equations in five unknowns X, Y, Z, qr, qm and pm: € X_ !_ !_ 2 € ! X 2  þ þ ¼0  X ! X! !2 X 2 8pGqr ¼  ð16Þ € 3X_ !_ € 3! 24pGpm 3X 3   þ þ 3kc2  X ! X! X 2 c2 ð17Þ 8pGqm ¼ € € 3! 24pGpm 3X 3X_ !_ 4 Y_ 2 þ þ 5  þ þ  4kc2 X ! X! X 2 Y 2 c2 ð18Þ Thus to get a solution, two additional relations are needed among the field variables X, Y and the fluid variables, qm, qr and pm. One of the additional relations will be provided by the equation of the state of the matter field, for which we consider the c - law of equation of state. Another relation in X and Y will be needed to solve the Eq. (16). Assuming field variables X and Y in power law form with c - law of equation of state for the matter field, Coley and Dunn [12] have obtained the solution of field Eqs. (9–13). Two-fluid cosmological models 757 4. A class of physically meaningful models with c-law 3 t [ t ¼ ðDÞ4ð1nÞ : ð25Þ By the transformation X ¼ aðtÞY; ð19Þ Eq. (16) yields " Z # 2 4 dt _ 3 Y 2 ¼ ðaÞ 1 þ D ; 3 aðaÞ _ 3 ð20Þ provided a_ 6¼ 0. By a proper choice of a(t), we obtain the field variables X and Y from Eqs. (19) and (20). Consequently, we find expressions for the fluid parameters from Eqs. (17) and (18). Here we assume a power law from for a: a ¼ tn : Xm ¼ 5. Cosmological parameters The cosmological parameters [14] for the solutions of Eqs. (22–24) are given by " #1 " #1 4ð1nÞ 2 4ð1nÞ 2 ð1þ2nÞ D þ t 3 1 D þ t 3 l ¼ t3 ; l1 ¼ t 3 ; nð1  nÞ nð1  nÞ ð26Þ " #1 4ð1nÞ 2 ð1nÞ D þ t 3 l2 ¼ l3 ¼ t 3 nð1  nÞ ð21Þ ðn  1Þf6ð4n  3Þt 8ð1nÞ 3   4ð1nÞ 4ð1nÞ 2 þ 2ð13n  8Þt 3 D þ 2ðn þ 1ÞD2 g  12kc2 t2 D þ t 3 h i 4ð1nÞ 2 ð4  3cÞ D þ ð3  2nÞt 3 ð27Þ 8ð1nÞ Xr ¼ ½18ð1  nÞ þ 9cð1  nÞð2n  3Þt 3 þ 3ðn2  1Þðc  2ÞD2   4ð1nÞ 2 4ð1nÞ 9kcc2 t2 D þ t 3 þ½ð22n2 þ 14n þ 8Þ þ cð3n2 þ 21n  18ÞDt 3 þ   4ð1nÞ 2 ð4  3cÞ D þ ð3  2nÞt 3 The resulting solution is given by " # " # 4ð1nÞ 4ð1nÞ ð1þ2nÞ D þ t 3 ð1nÞ D þ t 3 1 1 2; ! ¼ t 3 2 X¼t 3 nð1  nÞ nð1  nÞ ð22Þ provided 0 \ n \ 1. Consequently, we obtain 8pGqm ¼ 4ð1nÞ 3 þ 2ðn2  1ÞD 4kc2 h i  4ð1nÞ ð4  3cÞ 3ð4  3cÞt2 D þ t 3 6ðn  1Þð4n  3Þt ð23Þ 8pGqr ¼ 9ð1  nÞf2 þ cð2n  3Þgt þ þ 8ð1nÞ 3 þ 3ðn2  1Þðc  2ÞD2 ½ð22n2 þ 14n þ 8Þ þ cð3n2 þ 21n  18ÞDt 3kcc2 ð4  3cÞ 3ð4  3cÞt2 ðD þ t 4ð1nÞ 3 4ð1nÞ 3 Þ2 ð24Þ where D is an arbitrary constant, which can have positive as well as negative value in the range 0 \ n \ 1. From Eq. (22), we note that, in the case D \ 0, the validity of the solution keeps a lower bound on t, q¼ h 2nð3  2nÞt h ð28Þ 8ð1nÞ 3  2ð4n2  7n þ 1ÞDt h i 4ð1nÞ 2 D þ ð3  2nÞt 3 i ð3  2nÞt þD h i h ¼ 3H ¼ ; 4ð1nÞ t Dþt 3 4ð1nÞ 3 r2 ¼ 4ð1nÞ 3 n2 3t2 þ 2D2 i ð29Þ ð30Þ For t = 0, the spatial volume is zero and it increases with the increase of t. This shows that the universe starts evolving with zero volume at t = 0. In view of Eq. (30) we note that r2 is non-negative for all values of t (0 \ t \ ?) and drops to zero at infinite time (t ? ?). Thus the model is anisotropic for 0 \ t \ ?. q is also non-negative for all t and is independent of the sign of D in the case pffiffiffi ð33 2Þ  n\1: For 0\n\ 34, Xm [ 0 for t [ t1 with 4 Xm(t1) = 0, whereas for 34  n\1, Xm is negative for all t and independent of the sign of D. The total energy density (Xm ? Xr) lies between 0 to 1 and its value decreases as n increases for the large values of D and t. We also find that 758 S Oli the ratio r/h does not tend to zero as t ? ?, which indicates that the shear does not tend to zero faster than expansion scalar. This indicates that the model does not tend to isotropy for large value of t and therefore it remains 8pGqr ¼ 8ð1nÞ 3 4ð1nÞ 6ðn  1Þð4n  3Þt 3 þ 2ðn2  1ÞD h i 8pGqm ¼ 4ð1nÞ 3t2 D þ t 3 4ð1nÞ þ ð1  nÞð25n  10ÞDt 3  3ðn2  1ÞD2 h i 4ð1nÞ 2 3t2 D þ t 3 h ih i 4ð1nÞ 4ð1nÞ 6ðn  1Þð4n  3Þt 3 þ 2ðn2  1ÞD D þ t 3 9ð1  2nÞðn  1Þt qm ¼ 8ð1nÞ 4ð1nÞ qr 9ð1  2nÞðn  1Þt 3 þ ð1  nÞð25n  10ÞDt 3  3ðn2  1ÞD2 anisotropic throughout the evolution. Here, we get the following two cases depending upon the sign of D: Case (i) D C 0. In this case, the solution has point type singularity at t = 0. For c = 1 and k = 0, Xr [ 0 for 0 B t \ t2 in the case 0\n\ 12 with Xr(t2) = 0. For pffiffiffi ð33 2Þ 1 ,q is negative for all t. For 0\n\  n\1, X r 4 2 is negative in the interval t3 \ t \ t4 where q(t3, t4) = 0. h is negative for all t. Case (ii) D \ 0. In this case, the solution have two point singularity, viz., at t = 0 and t = t*. For c = 1 and k = 0, Xr [ 0 for 0 B t \ t5 in the case 0\n  12 with D¼ ð31Þ t0 ¼ 5  1017 s; In this section we have presented a detailed study of the solutions of Eqs. (22–24) with k = 0 and pm = 0(c = 1). The expressions for qr, qm and qm =qr are ð34Þ which is estimated from the ages of low luminosity population II stars in globular clusters [15]. The density of CMB is qr0 ¼ 4:67  1034 gc3 ; ð35Þ which obeys the blackbody radiation law of temperature 2.73°K. Using Eqs. (34, 35) in Eq. (32), we obtain 2ðV0  3  3n2 Þ 6. Investigation of dust models and determination of the constant D ð33Þ In what follows, in order to evaluate arbitrary constants, we shall assume that the present age of the universe is h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 4ð1nÞ f2V0 þ 5ðn  1Þð5n  2Þg  ð1  nÞ 16V0 þ 409n2  608n þ 208 t0 3 Xr(t5) = 0. In the case 12 \n  0:533747 and 0.9528047 B n \ 1, Xr [ 0 for t6 [ t \ t7 with Xr(t6, t7) = 0. For 0.533747 \ n \ 0.9528047, Xr is non-negative for all t. h is negative in the interval t8 \ t \ t* with h(t8) = 0. Thus the model expands in the interval 0 \ t \ t8, contracts in the interval t8 \ t \ t*. Finally, it expands forever for t [ t*. ð32Þ ð36Þ where V0 ¼ 24p G qr0 t02 : ð37Þ In view of Eq. (36), we note that the sign of the constant D depends upon the value of n and V0. D assumes positive as well as negative values in the case 0 \ n \ 0.500065 and 0.9999021 B n \ 0.9999347. Also, D \ 0 for 0.5000652 B n B 0.5338026, 0.9527499 B n \ 0.9999021 and 0.9999347 \ n \ 1. However, for n1 \ n \ n2, D is imaginary, where n1 and n2 are the roots of the equation: 409n2  608 n þ 208 þ 16 V0 ¼ 0: ð38Þ In view of Eqs. (34) and (35), the roots of Eq. (38) are given by n1 ¼ 0:5338026 and n2 ¼ 0:9527499: ð39Þ Two-fluid cosmological models 759 From Eq. (31), we further note that qm is negative for all t in the case 34 \n\1. Thus our model is physically meaningful in the range 0 \ n \ 0.533802 and accordingly we shall discuss the behavior of the parameters in the following. We come across to three types of situations depending upon the constant D. Case (i) D [ 0. In this case, we shall discuss it under two sub-cases depending upon the parameter n: Case (ia). 0 \ n \ 0.5. Here qm for t [ t1, with qm(t1) = 0; for t [ t1, qm is initially zero and increases till it attains its maximum value and decreases monotonically afterwards. qr [ 0 and monotonically decreasing function of t for 0 B t \ t2, with qr(t2) = 0. It however, to be noted that since there is a single conservation law and qm ? qr [ 0 for t [ t2, the validity of the model holds beyond t2. We further note that, for pffiffiffi  0\n\ 3  3 2 =4, the model displays inflation in the interval t3 \ t \ t4, where ðt3 ; t4 Þ " 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #4ð1nÞ ð4n  7n þ 1ÞD  ð1  nÞ 16n2  24n þ 1D ¼ nð2  3nÞ have, D = 3.5003649 9 1011 and qm0 ¼ 6:8618075 1030 gc3 . Case (iiib). 0.5 \ n B n1. The behavior of qm is similar to that in the preceding case. qr [ 0 for t6 \ t \ t7 with qr(t6,t7) = 0. A physically meaningful model exists in the finite interval t* B t B t6. In this case, the behavior of physical parameters qm and qr is similar to that in case (iiia). Also, this corresponds to the range, 5:854231  1030  qm0 \6:8618075  1030 for the matter density at the present epoch. Here, we have obtained classes of models portraying the epoch when two fluids are interacting in phase. The fluid parameters qm, qr and qm/qr behave reasonably. Here, one of the classes of model shows an inflationary epoch in a finite future. For particular values of qr0, t0 and n, we obtain range for matter density for the present epoch which is close to the observation limits. 7. Non-interacting models 2 ð40Þ it is to be noted that t1 \ t2 \ t3 \ t4. Thus, we have obtain realistic models which an inflationary epoch in a finite future. In view of Eqs. (34–37), the range for n, viz., 0 \ n \ 0.5 corresponds to the range for the matter density at the present epoch 2:9127006  1031 \ qm0 \2:3844505  1030 . Case (ib) 0.5 B n \ 0.5000652.In this case, the behavior of qm is similar to that in preceding case. qr [ 0 for all t and decreases monotonically as t increases. In view of Eqs. (34–37), the range for n, viz., 0.5 B n \ 0.5000652, corresponds to the range 2:3884505  1030 \qm0 \ 2:3900456  1030 for the matter density at the present epoch. For the particular case n = 0.5, we have, D = 2.9593469 9 1008 and qm0 ¼ 2:3844505 1030 gc3 . Case (ii) D = 0. In this case, the solutions Eqs. (31–33) reduce to the power-law solution given in [12]. Case (iii) D \ 0. Here, also we consider two sub-cases depending upon the parameter n. Case (iiia). 0 \ n B 0.5. In this case, qm [ 0 for t [ t* and qr [ 0 for 0 B t \ t5, with qr(t5) = 0. We note that the validity of the model holds in the finite interval t* B t B t5. At t = t*, qm and qr have maximum values, and decreases monotonically afterwards. From Eqs. (34–37), the range for n, corresponds to the range 6:8618075  1030  qm0 \5:8902721  1029 for the matter density at the present epoch. For n = 0.5, we From Eq. (4), the separate conservation of radiation leads to 4 qr ¼ A X Y 2 3 ; ð41Þ where A is constant of integration. Thus, Einstein’s field Eqs. (9–13) for the Bianchi VI0 space-time result into three independent equations in four unknowns X, Y, qm and pm: € X_ !_ !_ 2 € ! X 2  þ  ¼0  X ! X! !2 X 2 8pGpm ¼  € !_ 2 8pGpr 2! 1   þ kc2  ! !2 X 2 3 8pGqm ¼ 8pGqr þ 2X_ !_ 1 Y_ 2  2 þ 2  kc2 : Y X! X ð42Þ ð43Þ ð44Þ To solve these equations, we require an additional relation. Here, we assume a relation between the field variables X and Y, instead of the equation of state for the matter field. Using the transformation in Eq. (19) and the assumption in Eq. (21), the resulting solution is given by " #1 " #1 4ð1nÞ 2 4ð1nÞ 2 ð1þ2nÞ D þ t 3 ð1nÞ D þ t 3 X¼t 3 ; !¼t 3 ð45Þ nð1  nÞ nð1  nÞ provided 0 \ n \ 1. Consequently, expressions for fluid parameters, in the absence of the cosmological constant, are given by 760 S Oli A qr ¼   4ð1nÞ 2 4 t3 D þ t 3 2 ð46Þ 8pGqm h i 8ð1nÞ 4ð1nÞ ð1  nÞ ð9  6nÞt 3 þ ð6  nÞDt 3 þ ð1 þ nÞD2 ¼   4ð1nÞ 2 3t2 D þ t 3  8pGqr ð47Þ 24pGpm c2 h i 8ð1nÞ 4ð1nÞ ð1  nÞ 9ð2n  1Þt 3  5ð5n  2ÞDt 3 þ 3ð1 þ nÞD2 ¼   4ð1nÞ 2 3t2 D þ t 3  8pGqr ð48Þ Here A is the integration constant. From the expressions (45– 48), we note that pm becomes negative for n B 0.5, whereas qm/qr is a decreasing function of t for n C 0.75. Thus the validity of the model is limited to 0.5 \ n \ 0.75. For 0.5 \ n \ 0.75, qm and qr decrease monotonically, whereas pm increases till it attains its maximum value and decreases monotonically afterwards. qm/qr initially decreases till it attain its minimum value and increases monotonically afterwards. We also note that for values of n close to 0.5, qm/ qr increases more rapidly than for values of n close 0.75. At the present epoch, Eq. (47) yields D¼ 24pGAt3 Xr ¼ h i : 4ð1nÞ 2 D þ ð3  2nÞt 3 In this section, we have obtained classes of models describing the cosmic regime when the two fluids go on evolving independently. For the value of n from 0.5 \ n \ 0.75, the fluid parameters qm, qr and qm/qr behave reasonably. For given values of qr0, t0 and n, we can obtain lower bound for matter density for the present epoch from Eq. (51). Here, we also note that the value of total energy density (Xm ? Xr) lies between 0 and 1 and behavior is similar to the discussed for interacting case. 8. Conclusions We have obtained two-fluid cosmological models in Bianchi VI0 spatially homogeneous and isotropic spacetime. By transforming the field equation containing only the geometrical variables into a linear equation we have been able to obtain classes of models where the matter and radiation interact and matter field obeying with c - law of equation of state. These models have one or two distinct real singularities depending upon the sign of the constant of integration. One of the classes of models has an inflationary epoch in the finite future and the total density decreases h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 4ð1nÞ ð2U0 þ þn2  7n þ 6Þ  ð1  nÞ 16U0 þ 25n2  24n t0 3 2ðU0  1  n2 Þ With U0 ¼ 24pGðqr0 þ qm0 Þt02 ð50Þ The realistic condition leads to lower bound for qm0: qm0 [ nð24  25nÞ  qr0 : 384pGt02 ð51Þ In view of Eqs. (34) and (35), for a given value of qm0 satisfying Eq. (51), A and D are obtained from Eqs. (46 and 49). The cosmological parameters expect Xm and Xr, have the same expressions as given in Eqs. (26–30). Xm and Xr are given by ð53Þ ð49Þ with t. These models are valid over a finite interval of t. In another type of models, matter density is initially zero and qm, qr and qm/qr decrease monotonically as t increases. These models are valid for t [ t1 with qm(t1) = 0. Apart from this, we have obtained another class of models where qm and qr are infinite in the singularity and decreases monotonically afterwards. qm/qr increases with t and models are valid over a finite interval of time. Each of the above cosmological models gives rise to a narrow range for qm0 which is close to the observation limits. It is also observed that the rate of expansion slows down with time h i 8ð1nÞ 4ð1nÞ 2 ð1  nÞ ð9  6nÞt 3 þ ð6  nÞDt 3 þ ð1 þ nÞD2  24pGAt3 Xm ¼ h i 4ð1nÞ 2 D þ ð3  2nÞt 3 ð52Þ Two-fluid cosmological models and drops to zero as t ? ? and the universe remains anisotropic throughout the evolution. Lastly, we have obtained classes of models where the two fluids are non-interacting. In these model, qm and qr decrease with time, where pm is initially zero and increases till it attains its maximum value and decreases monotonically afterwards. qm/qr decreases initially till it attains its minimum value and increases afterwards. Here, we obtain a lower bound for qm0. 761 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] References [1] W Davidson Mon. Not. R. Astron. Soc. 124 79 (1962) [2] C B G McIntosh Mon. Not. R. Astron. Soc. 140 461 (1968) [13] [14] [15] A A Coley and B O J Tupper J. Math. Phys. 27 406 (1986) E L Wright et al Astrophys. J. 396 L7 (1992) A P Billyard and A A Coley Phys. Rev. D 61 083503 (2000) S Kalita, H L Duorah and K Duorah Indian J. Phys. 84 629 (2010) A Pradhan and K Jotania Indian J. Phys. 85 497 (2011) S Agarwal, R K Pandey and A Pradhan Indian J. Phys. 86 61 (2012) D N Pant and S Oli Astrophys. Space Sci. 281 623 (2002) S Oli Astrophys. Space Sci. 314 95 (2008) K S Adhav, S M Borikar, M S Desale and R B Raut Int. J. Theor. Phys. 50 1846 (2011) S Dey, J P Gewali, A K Jha, L Chhaigte and Y S Jain Indian J. Phys. 85 1309 (2011) B Saha Cent. Eur. J. Phys. 9 939 (2011) J Wainwright and G F R Ellis Dynamical Systems in Cosmology (Cambridge: Cambridge University press) p 1 (1997) E Harrison Cosmology (Cambridge: Cambridge University press) p 392 (2000)