Anisotropic Bianchi Type-I Magnetized String Cosmological Models with Decaying Vacuum Energy Density Λ(t)

Commun. Theor. Phys. 55 011 931 941 Vol. 55, No. 5, May 15, 011 Anisotropic Bianchi Type-I Magnetized String Cosmological Models with Decaying Vacuum Energy Density Λt Anirudh Pradhan Department of Mathematics, Hindu Post-graduate College, Zamania-3 331, Ghazipur, India Received November 9, 010; revised manuscript received December 8, 010 Abstract The present study deals with a spatially homogeneous and anisotropic Bianchi-I cosmological models representing massive strings with magnetic field and decaying vacuum energy density Λ. The energy-momentum tensor, as formulated by Letelier 1983, has been used to construct massive string cosmological models for which we assume the expansion scalar in the models is proportional to one of the components of shear tensor. The Einstein s field equations have been solved by applying a variation law for generalized Hubble s parameter in Bianchi-I space-time. The variation law for Hubble s parameter generates two types of solutions for the average scale factor, one is of power-law type and other is of the exponential form. Using these two forms, Einstein s field equations are solved separately that correspond to expanding singular and non-singular models of the universe respectively. We have made a comparative study of accelerating and decelerating models in the presence of string scenario. The study reveals that massive strings dominate in the decelerating universe whereas strings dominate in the accelerating universe. The strings eventually disappear from the universe for sufficiently large times, which is in agreement with current astronomical observations. The cosmological constant Λ is found to be a positive decreasing function of time which is corroborated by results from recent supernovae Ia observations. The physical and geometric properties of the models have been also discussed in detail. PACS numbers: 98.80.Cq, 98.80.Es, 04.0.-q, 04.0.Jb Key words: cosmic strings, Bianchi-I universe, variable cosmological constant, Hubble s parameter 1 Introduction Since the observed universe is almost homogeneous and isotropic, space-time is usually described by a Friedman- Lemaitre-Robertson-Walker FLRW cosmology. But it is also believed that in the early universe the FLRW model does not give a correct matter description. The anomalies found in the cosmic microwave background CMB and the large structure observations stimulated a growing interest in anisotropic cosmological model of the universe. Observations by the Differential Radiometers on NACA s Cosmic Background Explorer registered anisotropy in various angle scales. It is conjectured, that these anisotropies hide in their hearts the entire history of the cosmic evolution down to recombination, and they are considered to be indicative of the universe geometry and the matter composing the universe. It is expected, that much more will be known about anisotropy of cosmic microwave s background after the investigations of the microwave s anisotropy probe. There is a general agreement among cosmologists that cosmic microwave s background anisotropy in the small angle scale holds the key to the formation of the discrete structure. The theoretical argument 1] and the modern experimental data support the existence of an anisotropic phase, which turns into an isotropic one. In recent years, there has been considerable interest in string cosmology. Cosmic strings are topologically stable objects, which might be found during a phase transition in the early universe. ] Cosmic strings play an important role in the study of the early universe. These arise during the phase transition after the big bang explosion as the temperature goes down below some critical temperature as predicted by grand unified theories Zel dovich et al.; 3] Kibble;,4] Everett; 5] Vilenkin 67]. It is believed that cosmic strings give rise to density perturbations, which lead to the formation of galaxies. 8] Massive closed loops of strings serve as seeds for the formation of large structures like galaxies and cluster of galaxies. While matter is accreted onto loops, they oscillate violently and lose their energy by gravitational radiation and therefore they shrink and disappear. These cosmic strings have stressenergy and couple to the gravitational field. Therefore it is interesting to study the gravitational effects that arise from strings. The pioneering work in the formulation of the energy-momentum tensor for classical massive strings was done by Letelier 9] who considered the massive strings to be formed by geometric strings with particle attached along its extension. Letelier 10] first used this idea in obtaining cosmological solutions in Bianchi-I and Kantowski Sachs space-times. Stachel 11] has also studied massive string. The cosmological constant Λ was introduced by Einstein in 1917 as the universal repulsion to make the Universe static in accordance with generally accepted picture Supported in part by the Council of Science and Technology, Uttar Pradesh, India E-mail: pradhan@iucaa.ernet.in, acpradhan@yahoo.com c 011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

93 Communications in Theoretical Physics Vol. 55 of that time. In absence of matter described by the stress energy tensor T ij, Λ must be constant, since the Bianchi identities guarantee vanishing covariant divergence of the Einstein tensor, G ij ;j = 0, while gij ;j = 0 by definition. If Hubble parameter and age of the universe as measured from high red-shift would be found to satisfy the bound H 0 t 0 > 1 index zero labels values today, it would require a term in the expansion rate equation that acts as a cosmological constant. Therefore the definitive measurement of H 0 t 0 > 1 and wide rage of observations would necessitate a non-zero cosmological constant today or the abandonment of the standard big bang cosmology. 1] However, a constant Λ, as it was originally introduced by Einstein in 1917, cannot explain why the calculated value of vacuum energy density at Plank epoch following quantum field theory is 13 orders of magnitude larger than its value as observed or as predicted by standard cosmology at the present epoch. 13] In attempt to solve this problem, variable Λ was introduced such that Λ was larger in the early universe and then decayed with the evolution. 14] The idea that Λ might be variable has been studied for more than two decades see Refs. 15 16] and references therein. Linde 17] has suggested that Λ is a function of temperature and is related to the process of broken symmetries. Therefore, it could be a function of time in a spatially homogeneous, expanding universe. 16] In a paper on Λ-variability, Overduin and Cooperstock 18] suggested that Λg ij is shifted onto the right-hand side of the Einstein field equation and treated as part of the matter content. In general relativity, Λ can be regarded as a measure of the energy density of the vacuum and can in principle lead to the avoidance of the big bang singularity that is characterized of other FRW models. However, the rather simplistic properties of the vacuum that follows from the usual form of Einstein equations can be made more realistic if that theory is extended, which in general leads to a variable Λ. Recently, Overduin 190] has given an account of variable Λ-models that have a non-singular origin. Liu and Wesson 1] have studied universe models with variable cosmological constant. Podariu and Ratra ] have examined the consequences of also incorporating constraints from recent measurements of the Hubble parameter and the age of the universe in the constant and time-variable cosmological constant models. A dynamic cosmological term Λt remains a focal point of interest in modern cosmological theories as it solves the cosmological constant problem in a natural way. There are significant observational evidence for the detection of Einstein s cosmological constant, Λ or a component of material content of the universe that varies slowly with time to act like Λ. In the context of quantum field theory, a cosmological term corresponds to the energy density of vacuum. The birth of the universe has been attributed to an excited vacuum fluctuation triggering off an inflationary expansion followed by the super-cooling. The release of locked up vacuum energy results in subsequent reheating. The cosmological term, which is measure of the energy of empty space, provides a repulsive force opposing the gravitational pull between the galaxies. If the cosmological term exists, the energy it represents counts as mass because mass and energy are equivalent. If the cosmological term is large enough, its energy plus the matter in the universe could lead to inflation. Unlike standard inflation, a universe with a cosmological term would expand faster with time because of the push from the cosmological term Croswell 3]. In the absence of any interaction with matter or radiation, the cosmological constant remains a constant. However, in the presence of interactions with matter or radiation, a solution of Einstein equations and the assumed equation of covariant conservation of stressenergy with a time-varying Λ can be found. This entails that energy has to be conserved by a decrease in the energy density of the vacuum component followed by a corresponding increase in the energy density of matter or radiation see also Weinberg, 4] Carroll et al., 5] Peebles, 6] Sahni and Starobinsky, 7] Padmanabhan, 89] Singh et al., 30] Pradhan and Pandey, 313] Pradhan and Singh, 33] Pradhan et al. 3435] The discovery in 1998 that the Universe is actually speeding up its expansion was a total shock to astronomers. In the 1990 s two team of astronomers, the Supernova Cosmology Project Lawrence Berkeley National Laboratory and the High-Z Supernova Search international, were looking for distant Type Ia supernovae in order to measure the expansion rate of the universe with time. They presented evidence that the expansion of the universe is accelerating Garnavich et al., 3637] Perlmutter et al., 3840] Riess et al., 414] Schmidt et al. 43]. These teams have measured the distances to cosmological supernovae by using the fact that the intrinsic luminosity of Type Ia supernovae is closely correlated to their decline rate from maximum brightness, which can be independently measured. These measurements, combined with red-shift data for the supernovae, led to the prediction of an accelerating universe. Both team obtained Ω M 0.3, Ω Λ 0.7, and strongly ruled out the traditional Ω M, Ω Λ = 1, 0 universe. This value of the density parameter Ω Λ corresponds to a cosmological constant that is small, nevertheless, nonzero and positive, that is, Λ 10 5 m 10 35 s. An intense search is going on, in both theory and observations, to unveil the true nature of this acceleration. It is commonly believed by the cosmological community that a kind of repulsive force, which acts as anti-gravity is responsible for gearing up the Universe some 7 billion years ago. This hitherto unknown exotic physical entity is termed as dark energy. The simplest Dark Energy DE candidate is the cosmological constant Λ, but it needs to be extremely fine-tuned to satisfy the current value of the DE. In recent past several authors 4471] have studied cosmic strings in Bianchi type space-times in different context. The simplest of anisotropic models, which, nevertheless, rather completely describe the anisotropic effects, are Bianchi type-i BI homogeneous models whose spatial sections are flat but the expansion or contraction rate is directional dependent. The advantages of these anisotropic

No. 5 Communications in Theoretical Physics 933 models are that they have a significant role in the description of the evolution of the early phase of the universe and they help in finding more general cosmological models than the isotropic FRW models. The isotropy of the present-day universe makes the BI model a prime candidate for studying the possible effects of an anisotropy in the early universe on modern-day data observations. Recently, Saha and Visinescu 7] and Saha et al. 73] have studied BI models with cosmic string in presence of magnetic flux. Motivated by the above discussions, in this paper, the Einstein s field equations have been solved for massive string in presence of magnetic field with a decaying vacuum energy Λ by applying a variation law for generalized Hubble s parameter in BI space-time. The present paper generalizes the recent work of Pradhan and Chouhan. 74] The paper has the following structure. The metric and the field equations are presented in Sec.. In Sec. 3, we deal with an exact solution of the field equations with cloud of strings. Subsecs 3.1 and 3. deal with power-law and exponential-law solutions and their physical and geometric aspects respectively. Finally, in Sec. 4, we summarize the results. Metric and Field Equations We consider totally anisotropic Bianchi type-i line element, given by ds = dt + A dx + B dy + C dz, 1 where the metric potentials A, B, and C are functions of t alone. This ensures that the model is spatially homogeneous. The energy-momentum tensor for a cloud of massive string with perfect fluid and electromagnetic field is taken as T j i = ρ + pv iv j + pg j i λx ix j + E j i, where p is the isotropic pressure; ρ is the proper energy density for a cloud string with particles attached to them; λ is the string tension density; v i = 0, 0, 0, 1 is the four-velocity of the particles, and x i is a unit spacelike vector representing the direction of strings so that x 1 = 0 = x = x 4 and x 3 0. The vectors v i and x i satisfy the conditions In Eq., E j i Lichnerowicz 75] v i v i = x i x i = 1, v i x i = 0. 3 is the electromagnetic field given by E j i = µ h l h l u i u j + 1 gj i h i h j], 4 where µ is a constant characteristic of the medium and called the magnetic permeability and h i the magnetic flux vector defined by h i = 1 µ F ji u j, 5 where the dual electromagnetic field tensor F ij is defined by Synge 76] g F ij = ǫ ijklf kl. 6 Here F ij is the electromagnetic field tensor and ǫ ijkl is the Levi Civita tensor density. Choosing x i parallel to / x, we have x i = A 1, 0, 0, 0. 7 If the particle density of the configuration is denoted by ρ p, then ρ = ρ p + λ. 8 The incident magnetic field is taken along x-axis so that h 1 0, h = 0 = h 3 = h 4. 9 The first set of Maxwell s equations lead to F ij;k + F jk,i + F ki;j = 0, 10 F 3 = constant = Hsay. 11 The semicolon represents a covariant differentiation. Here F 1 = F 4 = F 34 = 0 due to assumption of infinite electromagnetic conductivity. The only non-vanishing component of F ij is F 3. Hence h 1 = AH µbc. 1 Since h = h l l = h1 1 = g11 h 1, so that h = H µ B C. 13 From Eqs. and 4, we have H E1 1 = µ B C = E = E3 3 = E4 4, 14 T 1 1 = p λ H µ B C, 15 T = T 3 3 = p + H µ B C, 16 T4 ρ 4 = H + µ B C. 17 The Einstein s field equations in gravitational units c = 1, 8πG = 1 R j i 1 Rgj i + Λgj i = T j i, 18 where R j i is the Ricci tensor; R = g ij R ij is the Ricci scalar and Λ is the so-called cosmological constant, assumed here to be time-dependent, viz., Λ = Λt. In a co-moving co-ordinate system, the Einstein s field equation 18 with Eq. for the metric 1 subsequently lead to the following system of equations: B B + C C + ḂĊ BC = p λ H µb C + Λ, 19 C C + Ä A + ĊȦ CA = p + H µb C + Λ, 0 Ä A + B B + ȦḂ AB = p + H µb C + Λ, 1 ȦḂ AB + ḂĊ BC + ĊȦ CA = ρ + H µb C Λ. Here, and also in what follows, a dot indicates ordinary differentiation with respect to t.

934 Communications in Theoretical Physics Vol. 55 The energy conservation equation T ij ;j = 0, leads to the following expression: Ȧ Λ + ρ + ρ + p A + Ḃ B + Ċ λȧ C A = 0, 3 which is a consequence of the field equations 19. 3 Solutions of Field Equations Equations 19 are four equations in seven unknowns A, B, C, p, ρ, λ, and Λ. Three additional constraints relating these parameters are required to obtain explicit solutions of the system. We first assume that the component σ1 1 of the shear tensor σj i is proportional to the expansion scalar θ, i.e., σ1 1 θ. This condition leads to the following relation between the metric potentials: A = BC m, 4 where m is a positive constant. The motivation behind assuming this condition is explained with sources suggest reference to Thorne, 77] the observations of the velocityred-shift relation for extragalactic that Hubble expansion of the universe is isotropic today within 30 per cent Kantowski and Sachs; 78] Kristian and Sachs 79]. To put more precisely, red-shift studies place the limit σ/h 0.3, on the ratio of shear σ to Hubble constant H in the neighbourhood of our Galaxy today. Collins et al. 80] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous expansion satisfies that the condition σ/θ is constant. Considering ABC 1/3 as the average scale factor of the anisotropic Bianchi-I space-time, the Hubble parameter may be written as H = 1 Ȧ 3 A + Ḃ B + Ċ. 5 C Secondly, we utilize the special law of variation for the Hubble parameter given by Berman 1983, which yields a constant value of deceleration parameter. Here, the law reads as H = labc n/3, 6 where l > 0 and n 0 are constants. Such type of relations have firstly been considered by Berman 81] and Berman & Gomide 8] for solving FRW models. Latter on many authors see, Saha and Rikhvitsky; 83] Saha; 84] Singh et al.; 8587] Singh and Chaubey; 8889] Zeyauddin and Ram; 90] Singh and Baghel; 91] Pradhan and Jotania 9] have studied flat FRW and Bianchi type models by using this law to find their solutions. From Eqs. 5 and 6, we get 1 Ȧ 3 A + Ḃ B + Ċ = labc n/3. 7 C Integration of Eq. 7 gives ABC = nlt + c 1 3/n, n 0, 8 ABC = c 3 e3lt, n = 0, 9 where c 1 and c are constants of integration. Thus, the law 6 provides two types of the expansion in the universe i.e., i Power-law 8 and ii Exponential-law 9. The value of deceleration parameter q, is then found to be q = n 1, 30 which is a constant. The sign of q indicates whether the model inflates or not. A positive sign of q, i.e., n > 1 corresponds to the standard decelerating model whereas the negative sign of q, i.e., 0 n < 1 indicates inflation. It is remarkable to mention here that though the current observations of SNe Ia Perlmutter et al.; 3840] Riess et al. 414] and CMBR favour accelerating models, but both do not altogether rule out the decelerating ones, which are also consistent with these observations see, Vishwakarma 93]. Subtracting Eq. 0 from Eq. 1, and taking integral of the resulting equation two times, we get B ] C = c 3 exp ABC 1 dt, 31 where c 3 and are constants of integration. In the following subsections, we discuss the string cosmology using the power-law 8 and exponential-law 9 of expansion of the universe. 3.1 String Cosmology with Power-Law Solving Eqs. 4, 8, and 31, we obtain the metric functions as At = nlt + c 1 3m/nm+1, 3 Bt = c 3 nlt + c 1 3/nm+1 exp ln 3 nlt + c 1 n3/n], 33 Ct = 1 c3 nlt + c 1 3/nm+1 exp ln 3 nlt + c 1 n3/n], 34 provided n 3. Hence the model 1 is reduced to ds = dt + nlt + c 1 6m/nm+1 dx + c 3 nlt + c 1 3/nm+1 exp ln 3 nlt + c 1 n3/n] dy + 1 nlt + c 1 3/nm+1 c 3 exp ln 3 nlt + c 1 n3/n] dz. 35 By using the transformation x = X, y = Y, z = Z, nlt + c 1 = T, the space-time 35 is reduced to ds = dt nl + T 6m/nm+1 dx + c 3 T 3 nm+1 exp + 1 c 3 T 3/nm+1 exp ln 3 T n3/n] dy ln 3 T n3/n] dz.36 The expressions for the isotropic pressure p, the proper energy density ρ, the string tension λ and the

No. 5 Communications in Theoretical Physics 935 particle density ρ p for the model 36 are given by p = 3l 4m n 3 + 6mn 1 + n 3] 4 T 4 T 6/n KT 6/nm+1 Λ, 37 ρ = 9l 4 4 T 4 T 6/n KT 6/nm+1 +Λ, 38 λ = 3l m 1n 3 T KT 6/nm+1 39 ρ p = 3l 34 m 1n 3] 4 T, 4 T 6/n + KT 6/nm+1 + Λ, 40 where K = H / µ. For the specification of Λt, we assume that the fluid obeys an equation of state of the form p = γρ, 41 where γ 0 γ 1 is a constant. Using Eq. 41 in Eqs. 37 and 38, we obtain 1 + γρ = 3l 4m n 3 + 6mn + 1 + n] 4 T T 6/n KT 6/nm+1. 4 Eliminating ρt between Eqs. 38 and 4, we obtain 1 + γλ = 3l 4m n 3 + 6mn 1 + n 3 3γ4] 4 T + γ 1KT 6/nm+1 + γ 1 T 6/n. 43 4 Using above solutions, it can be easily seen that the energy conservation equation 3 is identically satisfied, as expected. It is evident that the energy conditions ρ 0 and ρ p 0 are satisfied under the appropriate choice of constants. We observe that all the parameters diverge at T = 0. Therefore, the model has a singularity at T = 0. This singularity is of Point Type MacCallum 94] since all the scale factors diverge at T = 0. The cosmological evolution of B-I space-time is expansionary, with all the three scale factors monotonically increasing function of time. So, the universe starts expanding with a big bang singularity in the derived model. The parameters p, ρ, ρ p, λ, and Λ start off with extremely large values, which continue to decrease with the expansion of the universe provided m < 1. In particular, the large values of ρ p and λ in the beginning suggest that strings dominate the early universe. For sufficiently large times, the ρ p and λ become negligible. Therefore, the strings disappear from the universe for large times. That is why, the strings are not observable in the present universe. For n < 1, the model is accelerating whereas for n > 1 it goes to decelerating phase. In what follows, we compare the two modes of evolution through graphical analysis of various parameters. We have chosen n =, i.e. q = 1 to describe the decelerating phase while the accelerating mode has been accounted by choosing n = 0.4, i.e. q = 0.6. The other constants are chosen as l =, K = 1, = 1, m = 0.3. Figure 1 depicts the variation of pressure versus time in the two modes of evolution of the universe. We observe that the pressure is positive in the decelerating universe, which decreases with the evolution of the universe. But in the accelerating phase, the pressure rises suddenly in the early phase and attains its maximum and then it decreases with time. The pressure in both phases is always positive and becomes negligible at late time. Fig. 1 Plots of isotropic pressure p vs. time T in powerlaw string cosmology for q > 0 and q < 0. From Eq. 4, it is observed that the rest energy density ρ is a decreasing function of time and ρ > 0 always. The rest energy density has been graphed versus time in Fig.. It is evident that the rest energy density remains positive in both modes of evolution. However, it decreases more sharply with the cosmic time in the decelerating universe compare to accelerating universe. From Eq. 39, it is observed that the tension density λ is a decreasing function of time and λ > 0 always. Figure 3 shows the plots of string tension density verses time in both decelerating and accelerating mode of the universe. It is evident that the λ remains positive in both modes of evolution. However, it decreases more sharply with the cosmic time in the accelerating universe compare to decelerating universe. In the early phase of universe, the string tension density of both mode will dominate the dynamics and later time it approaches to zero. It is worth mentioning that string tension density is less in decelerating phase compare to accelerating phase and due to this in

936 Communications in Theoretical Physics Vol. 55 decelerating phase the massive strings disappear from the evolution phase of the universe at later stage i.e. present epoch. was Einstein s original motivation for introducing the cosmological constant. Fig. Plots of rest energy density ρ vs. time T in powerlaw string cosmology for q > 0 and q < 0. Fig. 3 Plots of string tension density λ vs. time T in power-law string cosmology for q > 0 and q < 0. From Eq. 40, it is evident that the particle density ρ p is a decreasing function of time and ρ p > 0 for all time. Figure 4 shows the plots of particle density verses time in both decelerating and accelerating mode of the universe. Here it is to be noted that ρ p in the decelerating phase is more than the accelerating phase through the evolution of the universe. In the accelerating phase ρ p increases rapidly in initial stage, it attains maximum value at some epoch closer to the early phase of the universe. In the later stage, it decreases from its maximum value with time and approaches to small positive value at late time. The behaviour of the universe in this model will be determined by the cosmological term Λ, this term has the same effect as a uniform mass density ρ eff = Λ, which is constant in time. A positive value of Λ corresponds to a negative effective mass density repulsion. Hence, we expect that in the universe with a positive value of Λ the expansion will tend to accelerate whereas in the universe with negative value of Λ the expansion will slow down, stop and reverse. In a universe with both matter and vacuum energy, there is a competition between the tendency of Λ to cause acceleration and the tendency of matter to cause deceleration with the ultimate fate of the universe depending on the precise amounts of each component. This continues to be true in the presence of spatial curvature, and with a nonzero cosmological constant it is no longer true that the negatively curved open universes expand indefinitely while positively curved closed universes will necessarily re-collapse-each of the four combinations of negative or positive curvature and eternal expansion or eventual re-collapse become possible for appropriate values of the parameters. There may even be a delicate balance, in which the competition between matter and vacuum energy is needed drawn and the universe is static non expanding. The search for such a solution Fig. 4 Plots of particle density ρ p vs. time T in powerlaw string cosmology for q > 0 and q < 0. Figure 5 is the plots of cosmological term Λ versus time in decelerating and accelerating modes of the universes. It is observed that in the decelerating phase, the cosmological parameter is negative in early time and it increases very rapidly in very short time period approaching to maximum positive and then decreasing function of time and follows normal evolution. But it is observed that in the accelerating phase the cosmological term is negative and increasing function of time. In accelerating phase, though Λ is negative at initial stage yet it is an increasing function of time. This results in the accelerating mode of expansion because as we proceed we get a less positive pressure than the previous one. This is the physics behind it. Models with negative cosmological constant have been investigated by Saha and Boyadjiev, 95] Pedram et al., 96] Biswas and Mazumdar, 97] Yadav. 98] Really at present the estimation of Λ is not only complicated but it is uncertain and indirect too. Recent cosmological observations Garnavich et al.; 3637] Perlmutter et al.; 3840]

No. 5 Communications in Theoretical Physics 937 Riess et al.; 414] Schmidt et al. 43] suggested the existence of a positive cosmological constant Λ with the magnitude ΛG /c 3 10 13. These observations on magnitude and red-shift of type Ia supernova suggest that our universe may be an accelerating one with induced cosmological density through the cosmological Λ-term. But this does not rule out the decelerating ones, which are also consistent with these observations Vishwakarma 93]. Thus the nature of Λ in our derived models are supported by recent observations. et al., 69] strings dominate the universe evolving with acceleration. If this is so, we should have some signature of massive string at present epoch of the observations. However, it is not been seen so far. Fig. 7 Plots of ρ p and λ vs. time T in power-law string cosmology for q < 0. Fig. 5 Plots of cosmological term Λ vs. time T in powerlaw string cosmology for q > 0 and q < 0. Figure 6 shows the comparative behaviour of particle energy density and string tension versus time in the decelerating mode. It is observed that ρ p > λ, i.e., particle energy density remains larger than the string tension density during the cosmic expansion see, Refs. ] and 67], especially in early universe. This shows that massive strings dominate the early universe evolving with decelerating and in later phase it will disappear, which is in agreement with current astronomical observations. Fig. 6 Plots of ρ p and λ vs. time T in power-law string cosmology for q > 0. Figure 7 demonstrates the variation of ρ p and λ versus the cosmic time for q = 0.6. In this case, we observe that ρ p < λ. Therefore, according to Kibble ] and Krori It follows that the dynamics of the strings depends on the value of n or q. Further it is observed that for sufficiently large times, the ρ p and λ tend to zero. Therefore, the strings disappear from the universe at late time i.e. present epoch. The same is predicted by the current observations. The rate of expansion H i in the direction of x, y, and z read as H x = Ȧ A = 3ml T 1, 44 H y = Ḃ B = 3l T 1 + 1 T 3/n, 45 H z = Ċ C = 3l T 1 1 T 3/n. 46 The Hubble parameter, expansion scalar and shear of the model are, respectively given by H = lt 1, 47 θ = 3lT 1, 48 σ = 3l m 1 T + 1 4 T 6/n. 49 The spatial volume V and anisotropy parameter Ā are found to be V = T 3/n, 50 m 1 Ā = + 6l Tn3/n. 51 From the above results, it can be seen that the spatial volume is zero at T = 0, and it increases with the cosmic time. The parameters H i, H, θ, and σ diverge at the initial singularity. The mean anisotropic parameter is an increasing function of time for n > 3 whereas for n < 3 it decreases with time. Thus, the dynamics of the mean anisotropy parameter depends on the value of n. Since σ /θ = constant from early to late time, the model does not approach isotropy through the whole evolution of the

938 Communications in Theoretical Physics Vol. 55 universe. If we set K = 0 and Λ = 0 in our solutions, we get the solutions obtained by Pradhan and Chouhan. 74] 3. String Cosmology with Exponential-Law Solving Eqs. 4, 9, and 31, we obtain the metric functions as At = c 3m/m+1 3ml exp t, 5 Bt = c 3 c 3/m+1 3l exp t 6lc 3 e 3lt], 53 Ct = c3/m+1 exp c3 3l t + 6lc 3 e 3lt]. 54 Hence the model 1 is reduced to ds = dt + c 6m/m+1 6ml exp t dx + c 3 c 3/m+1 exp e 3lt] dy + c3/m+1 c 3 exp 3l t 3lc 3 3l t + 3lc 3 e 3lt] dz. 55 The pressure p, the energy density ρ, the string tension λ, and the particle density ρ p for the model 55 are given by p = 9l 4m + 4 K c 6/m+1 exp ρ = 9l 4 4 K c 6/m+1 λ = 9l 1 m exp 6lt 4c 6 e 6lt 4c 6 e 6lt Λ, 56 6lt + Λ, 57 K c 6/m+1 exp ρ p = 9l 4m + 6m 1 4 + K c 6/m+1 exp c 6/m+1 4c 6 e 6lt 6lt 6lt, 58 + Λ. 59 For the specification of Λt, we use the equation of state 41 in Eqs. 56 and 57, we obtain 1 + γρ = 9l mm 1 1 c4 e 6lt c 3 K exp 6lt. 60 Eliminating ρt between Eqs. 57 and 60, we obtain 1 + γλ = 9l 4γ + 4m + ] 4 + 1 4 c4 exp γ 1e 6lt + c 6lt γ 1K c 6/m+1. 61 Using above solutions, it can easily be seen that the energy conservation equation 3 is identically satisfied, as expected. Figure 8 depicts the variation of pressure versus time in exponential-law string cosmology for q = 1. We observe that the pressure is negative, it attains negative maximum value and then it is almost constant negative value as aspected. This negative pressure repulsive force can be viewed as a source of the initial inflation. Fig. 8 Plots of isotropic pressure p vs. time t in exponential-law string cosmology for q = 1. From Eq. 60, it is observed that the rest energy density ρ is a decreasing function of time and ρ > 0 always. The rest energy density has been graphed versus time in Fig. 9. One can see this behaviour of ρt in Fig. 9. Fig. 9 Plots of rest energy density ρ vs. time t in exponential-law string cosmology for q = 1. Figure 10 shows the plot of string tension density verses time. It is observed that λ suddenly increases and becomes constant. It is always positive. Figure 11 shows the plots of particle density verses time in accelerating mode of the universe. It is observed that ρ p is decreasing function of time and always positive. It is also seen that the particle density disappears at late

No. 5 Communications in Theoretical Physics 939 time i.e. at present epoch. Fig. 10 Plot of string tension density λ vs. time t in exponential-law string cosmology for q = 1. Fig. 11 Plot of ρ p vs. time t in exponential-law string cosmology for q = 1. Fig. 1 Plot of cosmological term Λ vs. time t in exponential-law string cosmology for q = 1. From Eq. 61, it is observed that the cosmological constant Λ is a decreasing function of time and it approaches a small positive value at late time. Figure 1 is the plot of cosmological term Λ versus time in accelerating mode of the universe, which shows this property of Λt. The expressions for kinematic parameters i.e. the scalar of expansion θ, shear scalar σ, the spatial volume V, the average anisotropy parameter Ā, and deceleration parameter q for the model 55 are given by θ = 3l, 6 σ = 3l m 1 4 4c 6 e 6lt, 63 V = c 3 e 3lt, 64 m 1 Ā = 6l c 6 e 6lt, 65 q = 1. 66 The rate of expansion H i in the direction of x, y, and z and the Hubble parameter are obtained as H x = 3ml, 67 3l H y = + e 3lt, 68 c 3 3l H z = c 3 e 3lt, 69 H = l. 70 From above results, it is observed that the physical and kinematic quantities are all constant at t = 0. The kinematic parameters tend to zero as t. The expansion in the model is uniform throughout the time of evolution. Since σ /θ = constant from early to late time, the model does not approach isotropy at any time. The derived model is non-singular. If we set K = 0 and Λ = 0 in our solutions, we get the solutions obtained by Pradhan and Chouhan. 74] 4 Concluding Remarks In this paper, a spatially homogeneous and anisotropic Bianchi-I models representing massive strings in general relativity has been studied. The proposed law of variation for the Hubble s parameter yields a constant value of deceleration parameter. The law of variation for Hubble s parameter defined by Eq. 6 for Bianchi-I spacetime model gives two types of cosmologies, i First form for n 0 gives the solution for positive value of deceleration parameter for n > 0 and also negative value of deceleration parameter for n < 0 indicating the power law expansion of the universe whereas ii Second one for n = 0 gives the solution for negative value of deceleration parameter, which shows the exponential expansion of the universe. The power law solution represents the singular model where the spatial scale factors and volume vanish at T = 0. All the physical parameters are infinite at this initial epoch and tend to zero as T. There is a Point Type singularity MacCallum 94] at T = 0 in the model 36. The exponential solutions represents singularity free model of the universe. It is observed that

940 Communications in Theoretical Physics Vol. 55 the physical and kinematic quantities are all constant at t = 0. The kinematic parameters tend to zero as t. The main features of the models are as follows: i The models are based on exact solutions of the Einstein s field equations for the anisotropic Bianchi-I spacetime filled with massive strings in presence of magnetic field with variable Λ-term. The literature has hardly witnessed this sort of exact solutions for the anisotropic Bianchi-I space-time. So the derived models add one more feather to the literature. ii The models present the dynamics of strings in the accelerating and decelerating modes of evolution of the universe. It has been found that massive strings dominate in the decelerating universe whereas strings dominate in the accelerating universe. iii The strings dominate in the early universe and eventually disappear from the universe for sufficiently large times. At early universe, the possible occupation of cosmic strings is not allowed to exceed over 10% due to constraints of latest CMB data. At late time evolution, the strings become negligible even then still play an important role in astronomical experiments. Recent results from the PAMELA 99] and ATIC 100] experiments have indicated an excess power of cosmic ray positron flux compared to what is predicted from astrophysical backgrounds alone. Recently, Brandenberger et al. 101] have studied cosmic ray positron from cosmic strings showing that very few leptonic cosmic strings could decay into leptons and may be applied to explain recently discovered positron anomaly by Pamela data. iv The nature of decaying vacuum energy density Λt in our derived models are supported by recent cosmological observations Garnavich et al.; 3637] Perlmutter et al.; 3840] Riess et al.; 414] Schmidt et al. 43]. These observations on magnitude and red-shift of type Ia supernova suggest that our universe may be an accelerating one with induced cosmological density through the cosmological Λ- term. v If we set K = 0 and Λ = 0 in the present work i.e. in absence of magnetic field and cosmological constant, we derive the results recently obtained by Pradhan and Chouhan. 74] Thus we have generalized the previous solutions in the present work. Acknowledgments The author would like to thank the Inter-University Centre for Astronomy and Astrophysics IUCAA, Pune, India for providing facility and support under the Visiting Associateship Programme where this work was done. The author also thanks the anonymous referee for his constructive suggestions. References 1] C.W. Misner, Astrophys. J. 151 1968 431. ] T.W.B. Kibble, J. Phys. A: Math. Gen. 9 1976 1387. 3] Ya.B. Zel dovich, I.Yu. Kobzarev, and L.B. Okun, Sov. Phys.-JETP 40 1975 1. 4] T.W.B. Kibble, Phys. Rep. 67 1980 183. 5] A.E. Everett, Phys. Rev. 4 1981 858. 6] A. Vilenkin, Phys. Rev. D 4 1981 08. 7] A. Vilenkin, Phys. Rep 11 1985 65. 8] Ya.B. Zel dovich, Mon. Not. R. Astron. Soc. 19 1980 663. 9] P.S. Letelier, Phys. Rev. D 0 1979 194. 10] P.S. Letelier, Phys. Rev. D 8 1983 414. 11] J. Stachel, Phys. Rev. D 1 1980 171. 1] L.M. Krauss and M.S. Turner, Gen. Rel. Grav. 7 1995 1137. 13] S. Weinberg, Rev. Mod. Phys. 61 1989 1. 14] A.D. Dolgov, in The Very Early Universe, eds. by G.W. Gibbons, S.W. Hawking, and S.T.C. Siklos, Cambridge University Press, Cambridge 1983 p. 449. 15] S.L. Adler, Rev. Mod. Phys. 54 198 79. 16] S. Weinberg, Phys. Rev. Lett. 19 1967 164. 17] A.D. Linde, Sov. Phys. Lett. 19 1974 183. 18] J.M. Overduin and F.I. Cooperstock, Phys. Rev. D 58 1998 043506. 19] J.M. Overduin, Ap. J. 517 1999 L1. 0] J.M. Overduin, Phys. Rev. D 6 000 10001. 1] H. Liu and P.S. Wesson, Asrophys. J. 56 001 1. ] S. Podariu and B. Ratra, Asrophys. J. 53 000 109. 3] K. Croswell, New Scientist April 1994 18. 4] S. Weinberge, Gravitation and Cosmology, Wiley, New York 197. 5] S.M. Caroll, W.H. Press, and E.L. Turner, Ann. Rev. Astron. Astrophys. 30 199 499. 6] P.J.E. Peebles, Rev. Mod. Phys. 75 003 559. 7] V. Sahani and A. Starobinsky, Int. J. Mod. Phys. D 9 000 373. 8] T. Padmanabhan, Phys. Rep. 380 003 35. 9] T. Padmanabhan, Gen. Rel. Grav. 40 008 59. 30] C.P. Singh, S. Kumar, and A. Pradhan, Class. Quantum Grav. 4 007 455. 31] A. Pradhan and O.P. Pandey, Int. J. Mod. Phys. D 1 003 199. 3] A. Pradhan and P. Pandey, Astrophys. Space Sci. 301 006 1. 33] A. Pradhan and S. Singh, Int. J. Mod. Phys. D 13 004 503. 34] A. Pradhan, A.K. Singh, and S. Otarod, Roman. J. Phys. 5 007 415. 35] A. Pradhan, K. Jotania, and A. Singh, Braz. J. Phys. 38 008 167. 36] P.M. Garnavich, et al., Astrophys. J. 493 1998 L53. 37] P.M. Garnavich, et al., Astrophys. J. 509 1998 74. 38] S. Perlmutter, et al., Astrophys. J. 483 1997 565. 39] S. Perlmutter, et al., Nature 391 1998 51. 40] S. Perlmutter, et al., Astrophys. J. 517 1999 5.

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