Friedman Robertson Walker Cosmological models: A study

Transcription

1 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: Friedman obertson Walker Cosmological models: A study 1 Namrata I. Jain. Shyamsunder S. Bhoga 1 Department of Physics, M.D. College, Parel, Mumbai India -41 Department of Physics, TM Nagpur University, Nagpur India Abstract: Friedman obertson-walker model is the standard model which has been studied thoroughly. The FW model is successful in explaining expansion of the Universe but it has certain limitations which are also discussed here. FW model with and without Cosmological model have been compared which leads to the conclusion that Cosmological Constant plays an important role in the study of the Universe. It is also observed that Cosmological Constant is not really constant but it is varying. Although FW models are inadequate in the study of the Universe but its study will be a milestone for the future research. Keywords: Cosmology, Cosmological Model, Einstein Field Equations. I. INTODUCTION Cosmology is the study of the Universe. The Universe, its origin, evolution and its fate are still a curiosity for mankind and is intensively studied by the scientific community since ancient times. Observing heavenly objects and dynamics by sky watching has been a natural trend involved in the study of the Universe and this study is called Astronomy. It is the astronomical observations of planets, stars and galaxies that have created inquisitiveness to look beyond our Milky Way Galaxy to distant objects from which light may take billion years to reach us. Nowadays, modern techniques of observing the sky to study stars and galaxies have enabled us to understand dynamical and physical behavior of billions of galaxies spread across vast distances. Studying the extragalactic world, large scale structure of the Universe has gained popularity in the nineteenth century, when A. Einstein s Theory of elativity emerged. In the late twenties, tensor calculus revolutionized the geometry of space by involving it in the study. This led to the theoretical study of the Universe and thus a cosmological model could be set up using tensor calculus and metric algebra. The Cosmological Model of the Universe is a Mathematical model which is developed by solving mathematical equations with the help of tensor calculus. The application of tensor calculus with iemannian geometry brought out the concept of Space-Time curvature and geodesics in General elativity. A Geodesic is understood as the minimum stationary distance of anybody on the space time curve which is similar to a curve on a 3D sphere. In tensor calculus, Christofflel tensor and icci tensor are the main tensors to derive equations of Geodesics. With the help of Tensor calculus, A. Einstein published equations in General elativity which related space and geometry in With these equations K. Schwarzchild found internal and external solutions for space time geometry of the spherical distribution of matter thereby developing t he Schwarzchild metric. [1] The Schwarzchid metric has been successfully implemented in the study of the Universe by A. Einstein, W. de Sitter and many others. It has also been successfully applied to explain experimental measurements of the solar system which involve Advance Perihelion of a planet, Bending of Light, Gravitational ed Shift, adar Echo Delay. Einstein and de Sitter set up Cosmological models by line element as Schwarzchild metric. Both models could explain the Universe successfully but at some issues they were contradictory to each other. Einstein s Universe was a Steady State Model and filled with matter without motion; while de Sitter s Universe was an Expanding Model with empty space. Both Page 7

2 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: models were inadequate to explain the expansion of the Universe which was observed experimentally by E. Hubble and M. Humason in 1931 [, 3]. E. Hubble in 199 found that the Universe is expanding by actually calculating moving galaxies with finite velocity which is proportional to its distance from Earth. The proportionality constant is called the Hubble s Constant H.[] Expansion of the Universe was a puzzle until the work carried out by A. Friedmann in 19 and A. Lamaitre, in 197 independently which have been recognized later due to the discovery by E. Hubble in 199 by science community which unfortunately had gone unnoticed. The expanding model could be explained by generalizing the line element so that the non static line element could be used to study various aspects of the Universe including Hubble s Law. The non static line element was derived rigorously by H.P. obertson and A.G. Walker (independently) in the 193s which is called the obertson Walker line element [4,5]. The model thus developed is called the Friedmann Lamaitre- obertson- Walker model which has been worked upon and explored to understand the Universe since its evolution. Since two decades, the FW model of the Universe has become an area of research for many. Despite the successful explanation of expansion and age of the Universe, FW model suffers from some infirmities. These are related to Horizon, Flatness etc. of the Universe. The Cosmological Constant () which is an important parameter in Cosmology, plays a key role in resolving these infirmities. Several researchers have pointed out the importance of () by setting up models. They succeeded in overcoming some of the problems faced by their models without. Among these researchers, some explained dust model with p= while others considered the relation between p and.[see 6 and references therein] While the problems related to age, horizon, flatness are solved to a certain extent, FW models with constant - which is four dimensional model - is still unable to explain certain observational facts. With reference to accelerated expansion, Existence of Dark Energy, Dark Matter and Cosmological Constant Problem (CCP) there is a mismatch between observed and calculated value, which are yet to be resolved. One solution for CCP is to consider the model with time dependent Lambda which has been quite successful in explaining accelerated expansion and other features of the Universe, However CCP is not yet completely solved. Besides CCP, the mystery of Dark Energy and Dark Matter, there is a curiosity among Cosmologists about the early Universe phenomena. ecent researches attempting to overcome these shortcomings and unfolding the early Universe have suggested that one must look for higher dimension cosmology. Kaluza in 191 [7] and Klein in 196 [8] lay the foundation of higher dimensional cosmology which is actually the consequence of their attempt to unify all four types of forces i.e. gravitation with electro-magnetism and gravitation with particle interaction. Higher dimension plays an important role in the analysis and the dynamics of the early Universe. This paper has been organized as: The detailed study of FW model will be taken up in the next section followed by Kinematics and neoclassical study. The shortcomings of FW model will be studied thereafter. Kaluza Klein Cosmology have been analyzed further to enable us to develop a ground for the present work. Final section is the concluding remarks on the present work. II. FIEDMAN OBETSON WALKE COSMOLOGICAL MODEL In the 193s the Universe was observed to be non static by E. Hubble and M. Humason which had to be explained theoretically. Friedman -obertson Walker (FW) model is the model of the Universe representing Friedman universe in obertson Walker metric. The model which is going to be discussed in the present section, is a four dimensional model. In order to study FW model it is necessary to note the assumptions that are accepted to generalize the model for non static conditions..1 The assumptions for the model A. Weyl s postulate It states that in a space-time diagram, world lines representing history of galaxies never intersect each other and form a funnel like structure, increase steadily and these three bundles of nonintersecting geodesics are orthogonal to a series of Page 71

3 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: space-like hyper surfaces. This was postulated by a mathematician H. Weyl in 193 to understand the regularities in the world lines which are space-time curve for the locations of galaxies.[9] A particle moving in space can be represented by curve in a Space time plane. Thus, for particles moving in a Space-time plane can be shown by several curves as shown in Fig. 1 (i) and (ii). Fig.1 (i) shows the curves intersecting each other while the other figure represents the non-intersecting curves in a space-time plane. In the following figures, lines intersecting each other denote collision of particles. If this would have been situation in the Universe then Einstein Field Equations would be difficult to solve. The real Universe does not appear so messy; therefore the world lines representing galaxies are not intersecting and have a funnellike structure as shown in Fig. 1 (ii). The physical significance of Weyl s Postulate can be explained by considering co-ordinates and metrics of space time. For a typical world line in (1+3) dimensions there are three space co-ordinates and an one-time co-ordinate. For spacelike hyper surfaces the co-ordinates represented by x μ are considered to be constant. the galaxies lying on different lines are assumed to be like smooth fluid and so x μ represents both time as well as space co-ordinates together(, 1,,3). The metric tensor for such system is given by g ij. From the orthogonality condition g μ = where suffix is for time co-ordinate if μ is other than zero. The geodesic equations is given by i k i d x i dx dx kl (1) dx ds ds The above equation is satisfied for constant x μ.we also have i=1,, 3, Christoffel Tensor of second kind is defined as i 1 ij gjl gkj gkl kl g k l j x x x Г μ = for μ=1,, 3.This will give The line element therefore becomes g x. Without loss of generality we take g = 1 i kl which is ds dx gdx dx c dt gdx dx (3) The time co-ordinate in the above equation is called the Cosmic time which is an important parameter for studying dynamics of the Universe. () Page 7

4 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: B. The Cosmological Principle. Another very important assumption that is adopted while developing a cosmological model is the Cosmological Principle. It states that at any given cosmic time, the Universe is homogeneous and isotropic. This gives the surface of hyper sphere of Einstein s Universe to be smooth and surfaces with constant t exhibit the properties of Einstein Universe. This principle is the result observations made by Hubble s Telescope. From Cosmic microwave background (CMB) observations it is inferred that the Universe appears to be isotropic. In other words, the principle can be stated that at large scale the Universe appears to be the same from everywhere and there is neither a centre nor a boundary. Weyl s postulate itself explains homogeneity and an isotropic condition of the Universe- as we have seen earlier that the lines are not intersecting but parallel to each other and space-like hyper-surfaces are orthogonal to time lines. With this principle, it is also possible to transform the space-time co-ordinates into Cosmic standard co-ordinates representing density, pressure and temperature.. The line element With the above assumptions it is possible to have three types of surfaces - the Universe can have a flat surface, or aclosed surface or an open surface. While a plain surface has zero curvature, an open surface has negative curvature and a closed surface has a positive curvature. These three types of curvatures have some constant value which ensures the properties of homogeneity and isotropy. If the curvature of space differs, than it is possible to get other homogeneous and isotropic spaces by appropriate transformations. To get this, we consider space co-ordinates which are given as x μ where μ = 1,,3,4. Here four instead of the usual three dimensions are used to meet the criteria of the Cosmological Principle [1] Let us consider the surface with negative curvature, x1 x x3 x4 where S is a constant. The substitution x 1 = sinhχ cosθ, x = sinhχ sinθ cos, x 3 = sinhχ sinθ sin, x 4 = coshχ gives (4) dx1 dx dx3 dx4 [ d sinh ( d sin d )] (5) The negative sign for x 4 shows that the surface is embedded in four dimensional pseudo- Euclidean spaces instead of normal 3-d Euclidean space. If we consider r= sinhχ in above equation, then Eq. (5) becomes: dr dσ = +r (dθ +sin θdφ ) (6) 1-kr Where k= for flat surface, k=-1 for open surface (negative curvature), k=1 for closed surface and right hand side of above Eq. is simply Euclidean line element scaled by the factor for flat surface. The scale factor can depend upon cosmic time. The most general line element satisfying Weyl s postulate and the Cosmological Principle is given by dr ds =c dt - (t) +r dθ +sin θdφ (7) 1-kr Where (t) is the expansion factor or scale factor. Eq. (7) is known as the obertson Walker line element or metric which was obtained by H.P. obertson and A.G. Walker in the 193s independently. In the next subsection we will obtain the model using -W metric which is often called the FW model. III. THE FW MODEL WITHOUT LAMBDA A. Friedman developed a cosmological model in 19 with the help of relativistic cosmology which is very similar to that obtained by Newtonian cosmology. Page 73

5 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: The model derived using obertson- Walker metric by A. Friedman and it is called as Friedman-obertson-Walker Cosmological model. Here in this section let us glance at the FW model which will enable us to set up a new model with different metric. Consider ds c dt t r d d 1 kr dr ( ) sin Let x =ct, x 1 =r, x = θ, x 3 =, In a curved space time, the line element is given as ds i j gijdx dx where g ij is a 44 metric tensor. Therefore metric tensor for above element is given by g =1, g 11 = - (t)/(1-kr ), g = - (t)r, g = - (t)sin θ. k is a curvature constant, k= for flat Universe, k=1 for closed Universe, k=-1 for open Universe. To derive the model it is necessary to obtain Einstein Field Equations. Einstein Field Equations can be obtained from the following equation: i 1 i i i j- δ j+λδ j=-8πgt j (9) Where i j = g il g lj, g il is metric Tensor, i j icci Tensor, i j Kronecker delta tensor, -icci scalar, T i j-energymomentum tensor, -Cosmological constant, G- gravitational constant. icci Tensor is written as i j=g il lj icci tensor can be obtained as follows: a a ij ia b a b a ij a j ijba iabj x x where Γ is Christoffel Tensor of second kind and is given by a 1 ab g bj g g ib ij ij g i j b, (11) x x x Generally ћ =c=1 is taken at the cosmic scale , (1) a log g ia i x g = g ij. Field Equations can also be written as i 8πG i i G j=- T j +Λgj c and T ij is the energy momentum tensor for perfect fluid which is given as p T ij= +ρ u iu j-gijp c i dxi u= dt is the 4- vector velocity component such that u i u j =-1 for i=,1,,3 (space -time coordinates) p and ρ are pressure and density of matter distribution of the Universe respectively. Thus, from the above equation Energy momentum tensor T i j = (ρ, -p, -p,-p). This assumption also shows that the energy momentum tensor of the Universe takes the same form as that of perfect fluid. According to the Cosmological Principle, the Universe is homogeneous and isotropic, so it is possible to assume ћ =c=1 for deriving field equations. At present we consider (the Cosmological Constant) to be zero. The Cosmological Constant was first suggested by A. Einstein to show a relation between geometry and matter but was later discarded by him as the Einstein model was inadequate to explain expansion of the Universe. Deriving field equations from (9-15) for FW model without lambda, we get the following equations: (8) (1) (13) (14) (15) Page 74

6 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: k G 3 3 8G (16) 1 k G1 8 Gp (17) 1 3 G1 G G3 Additionally, the conservation of energy requires T i j;j = which implies : ( p)3 (18) In equivalent form d 3 d 3 ( ) p ( ) dt dt (19) The above relation is similar to the energy conservation equation given by de + pdv= in thermodynamics. It should be noted that the field equations satisfy the energy conservation equation. The above field equations are solved for (i) dust Universe for which p= and can be studied for open, closed and flat curvature of the Universe (ii) Equation Of State (EOS) for perfect fluid given by p= where is constant of EOS. determines the relation between pressure and density. If = then the Universe is dust filled or matter-dominated and if = 1/3 then it is in a radiation-dominated phase. Including EOS in field equations, Eq. (17) and (18) are now rewritten as : + k 8πGρ = 3 We also have k + + =-8πGωρ Solving the above equations, a general solution is obtained as, () (1) -(1+3ω) =A -k () A is a constant of integration, which can be determined from initial conditions. From the above equation it is possible to obtain the solution for a matter-dominated and radiation-dominated phase of the Universe However, the solution is analyzed for matter-dominated phase for simplicity as the present Universe is matterdominated. A radiation-dominated phase will be analyzed for the model with lambda in the later section. 3.1 Matter Dominated Phase If the Universe is matter-dominated then EOS for it is given by taking =, Thus Eq. () takes the form as -1 =A -k (3) Scale factors for flat, closed and open Universe can be determined by substituting k=, k=1 and k=-1 respectively in Eq. (3) which are as follows. 1. Flat Universe, k=, 3 = At+B 3 Where B is constant of integration and is assumed to be zero, so t /3. Close Universe, k=1 A = (1-cos ) (5) (4) Page 75

7 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: A t= ( -sin ) (5a) To arrive at above Eq. initial conditions t=o, = are assumed. 3. Open Universe, k=-1 A (cosh 1) (6) A t (sinh ) (6a) Here also we have = at t=. Plot of all three models is shown in Fig. below. Fig. Types of Universe open, flat and closed models of the Universe. P, Q and correspond to present epochs. From Fig. 1., it is observed that open and flat models represent an expanding Universe, while a closed model has expansion for some time and thereafter it may contract. ecent observations lead to the conclusion that the Universe is flat. But FW models suffer from flatness problem as it can be seen from Equations for scale factor that constant A has not been determined from initial conditions. The problems faced by FW models in absence of cosmological constant will be discussed in next section. Let us have an overview of the cosmological parameters in the following section. IV. COSMOLOGICAL PAAMETES To understand the geometry of the Universe, several important parameters are defined; which play a vital role in observing and understanding the Universe i.e. observational cosmology. These parameters are i) Hubble parameter ii) Deceleration parameter iii) Cosmological density parameter. A short introduction of these parameters is given below. A. Hubble Parameter Hubble parameter is derived from Hubble s Law which is defined for velocity distance relation. In 199, Hubble established linearity between the velocity of a galaxy and a red shift of the light received on Earth by which the distance of the galaxy from the Earth could be determined. Hubble s Law states that the velocity of a galaxy is directly proportional to the distance between it and the earth. It is given by v = H D. Page 76

8 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: V-velocity of galaxy, D- distance between galaxy and earth, H - Hubble s constant at present epoch. H has dimensions of km/s/mpc, mpc (mega parsec ) is the unit for distance measurements on the cosmic scale, hence the Hubble Constant has a dimension inverse of time. Hence, it can be used to determine the age of the Universe. This (t) constant is now called the Hubble Parameter which, for present Universe, is written as H =, [(t) is a scale (t) t factor. A dot over scale factor (t) is the derivative of (t) with respect to time]. The Hubble parameter not only determines age of the Universe but is also an important parameter in the measure of observable size of the Universe. The inverse of H is called Hubble s Time. B. Deceleration Parameter Deceleration Parameter q is required for the explanation of expansion of the Universe and is useful for the expression of density of the Universe. q is formulated as q =-. t q is the deceleration parameter at present time. According to recent observations by COBE, WMAP, BAO and several other experiments, the expansion of the Universe is accelerating. Hence the deceleration parameter is defined to measure the accelerated expansion. The value of q should be negative for the acceleration of the Universe. For the matter-dominated FW closed model, it is found to be greater than 1/, so the model is decelerating; hence it is thought that the present Universe is not a closed one. For open and flat models, since q is negative the Universe is accelerating forever. For flat Universe q -1 shows that the Universe is accelerating at a constant rate. From field equations, the acceleration equation is derived from Eqs. (16) and (17) which is given by, 3 =-4πG(ρ+3p)=-4πGρ(1+3ω) In the above EOS, p= has been substituted. With Eq. (7), q is obtained as 1 k q= (1+3ω) 1+ H Where q=- (9) at any given time. For a flat universe >-1/3 i.e. universe filled with any fluid accelerates forever. q can also be represented in terms of the Hubble Parameter as, (7) (8) H =-(1+q) H The above equation can also be applied for closed or open models. (3) C. Density Parameter To define the expansion of the Universe, a Density Parameter is formulated in terms of proper density and Hubble parameter, which is written as: 8πGρ Ω= (31) 3H For present epoch we have 8πρ Ω = 3H when G= 1 is assumed. Page 77

9 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: The Density Parameter also determines the geometry of the Universe. For a flat Universe, the density of the universe is same as critical density. Critical density is defined as a watershed point between expansion and contraction of the Universe []. Its expression is derived from the field equations of the Friedman model for flat Universe, and is given as 3H ρ C =.Its value is equal to h g/cm 3. So the present cosmological density parameter is expressed as 8πG ρ Ω=.If the Universe is flat, then =1. If the Universe is closed and has stopped expanding, then >1, If the ρ C Universe is open and expands forever, then <1 then it. ecent observations by WMAP have concluded that is nearly equal to one which implies that the present Universe is flat. When the Cosmological Constant is introduced then a more general expression for c where relates to and k can be expressed i.e. and k respectively. represents vacuum density parameter defined as, represents vacuum energy i.e. (/3H ) [ role of in cosmology will be discussed later in detail in this chapter] and k is expressed as ( k/ H ) which corresponds to density of curvature. The total density of the Universe is now given as + + k which should be equal to 1. At early stages of evolution of the Universe, visible matter was dominated by radiation, that is to say that it was a radiation-dominated stage. If the visible matter in the Universe is contained in the galaxies the density is approximately equal to 1-31 gm/cm 3 over the largest scales. This is equivalent to one proton per cubic meter. In this stage the Universe also contains neutrinos, gravitational waves which are called as primordial radiations. Apart from visible matter nowadays researchers are looking for dark matter too. Hence, the total density parameter is not only related to visible matter and dark energy but also includes dark matter density which is yet to be cracked. Dark matter and Dark energy were not parts of the model during the time of Friedman, Hence his field equations do not have any expression related to these. Although the FW model has become the standard model in cosmology; some observations inferred the inadequacies of the model without the Cosmological constant. Thus developing a FW model with the Cosmological constant is being undertaken by many in cosmology. In the next section a FW model with the Cosmological Constant will be explained. V. FW MODEL WITH COSMOLOGICAL CONSTANT The FW model without the Cosmological constant suffered from some drawbacks. Basically these related to the Horizon and Flatness. A. The Horizon Drawback In Cosmology, two kinds of Horizons are often discussed. These are the Particle Horizon - which relates to the communication in past and the Event Horizon which relates to limits on communication in the future. The Horizon drawback deals with the problem of the largest distance traveled by light to reach an observer since the time of the Big-Bang. The largest radial distance for such a case is written as t cdt' r max = (t'). From r max we get information that has been obtained from those particles which could be detected and the three surfaces in space-time having this radius is called as particle horizon. Physical distance to the Horizon at the time of t is D horizon (t) = (t) r max (t). D horizon (t) is found to be equal to ct for radiation-dominated phase for which p = ρ/3 and D horizon (t) = 3ct for matter dominated phase for which p=. Page 78

10 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: The different value of D horizon (t) for different phases is called the Horizon Problem. This problem raises the question of homogeneity of the Universe, as cosmic microwave radiation data clearly indicates that we live in a nearly homogeneous Universe. B. Flatness Drawback The Flatness Drawback is due to total density calculated for open, flat and closed Universe, where we define total density ρ given by Ω= where ρ c is the critical density of Universe. Ω at t is calculated for present epoch. ρ c According to the FW model, Ω should deviate further and further from unity for k = ±1. However, even when deviating continuously from unity, it reaches unity for k = ±1 (even though retaining unity for k=). From present day observations, Ω has values between.1 and 1. The discrepancy between Theoretical and Observed values is the Flatness Problem. An inflationary model can provide solutions for the two basic problems. But it can only give information about the early Universe. It cannot explain the present Universe. There have been several unsuccessful attempts by researchers to solve these problems, leading to the conclusion that the finite Cosmological Constant existed and it has some physical significance in the study of the Universe. The importance of the Cosmological Constant will be discussed subsequently in the present thesis. Since we know that the FW model is a basic, standard model the FW model with Cosmological Constant will be discussed here. VI. THE FIELD EQUATIONS AND THE MODEL Field equations with Λ are obtained from FW metric and these are given below:- k 1 8πGρ G = + - Λ= k G 1= + + -Λ=-8πGp 1 3 G 1=G =G3 (3) (33) The above equations can be solved with the help of EOS where p=. Solving the above equations, the first order differential Equation is given by -(1+3ω) Λ k =A ω For dust model Universe, we have =, Thus the equation (34) will now be given as: -1 Λ =A + -k 3 The real solution of the above equation can be obtained if A=. In such a case we have: (34) (35) For flat model Λ t =Ce 3, For open model i.e. k=-1, =Ccosh For closed model k=1 =Csinh Λ t 3 Λ t 3, Where C is a constant. The above solutions are quite different from the results obtained for =. The models for positive lambda represent steady state models which resemble Einstein s static models. A Model with < will not give a real solution for closed or open models. Page 79

11 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: For a radiation dominated Universe = 1/3 Therefore Eq. (34) is simplified as, - Λ k =A If A,then, we have the following equation 4 Λ k =A+ - 3 If k=,i.e. Flat Universe, the model is given by, 1 3A Λ = sinh t+c 3 If k=1, Closed Universe, we have, 3A 9 Λ 3 = - sinh t+c + Λ 16Λ 3 4Λ For k=-1, Open Universe, 1 1 3A 9 Λ 3 = - sinh t+c - Λ 16Λ 3 (4) 4Λ From the above, and for simplicity we choose C= which will imply that the behavior of all the models are similar in the radiation-dominated phase. All are expanding in radiation-dominated phase. A Flat model appears to have singularity while others are nonsingular models. It can be observed that the positive cosmological constant can play an important role in the study of the Universe which was initially discarded by Einstein, but since then it has been found that results of FW with lambda models can be reconciled with observational data [3]. There are several reportings which have shown the importance of lambda in the study of the Universe. The following Fig. (4) represents the types of Universes in the presence of lambda. (36) (37) (38) (39) Fig.3 Dynamics of the Universe when k= -1,, 1 and positive and negative lambda [1] Error! No text of specified style in document. From Figure it can be easily understood that when k= and k=-1 with positive lambda represents an expanding model; whereas when k=1 there is a bouncing Universe model. Page 8

12 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: VII. AGE OF THE UNIVESE FO FW MODEL WITH COSMOLOGICAL CONSTANT. In Cosmology, the age of the Universe is estimated by calculating H -1 i.e. Hubble parameter at present time. Since it was known that the present Universe is matter-dominated, hence by age t H -1 it is calculated as 14.4Gyr. According to observational data, Age= Gyrs. is calculated. A small discrepancy in the age of the Universe can be resolved by considering density corresponding to Lambda along with matter density. The relation between Age and density parameter is explained here. Two important Cosmological parameters (Hubble parameter and deceleration parameter) at present time are given by k 1 8πGρ H =, q =-, considering = at t=t Eq.(3) in presence of lambda is rewritten as H + - Λ= t t 3 3 (41) Considering the definition density parameter, we have m + + k =1, where we consider density parameter for 8πGρ matter as Ω m =. Other density parameters for lambda and curvature constant are defined in a similar way. 3H Values of matter density and vacuum density (density corresponding to ) are obtained as m =.3 and =.7 approximately as per observational data. Eq. (33) is rewritten as k 1-q H + -Λ=-8πGωρ The above equation is further simplified to : 1 q = (1+3ω)Ω m-ωλ For a dust-filled Universe ω= Hence we have q = Ωm/- Ω Λ. (4) (43) Current observations estimate Ωm=.3 and Ω Λ =.7 values. So, for a dust-filled Universe, the value of deceleration parameter is determined as q = -.55 which confirms that Universe is accelerating. The Age of the Universe can be determined in the following way: If we assume the present Universe is in a matter-dominated phase i.e. dust-filled, flat Universe, then, Scale factor (t) ~ t /3 and t = /3H - which is approximately calculated as 9Gyr. But the age of the oldest star is found to be 1Gyr. The discrepancy between the age of a star and that estimated through Hubble parameter forces us to accept the existence of a finite, positive Cosmological constant. In the presence of the Cosmological constant, the age of a flat Universe is determined by calculating t. (t) -1 d' t=h ' In terms of red shift, (44) Consider 1+z= /, where z is red-shift magnitude, we can find age as given by formula (1+z) -1-1 d' =H (45) ' Hence from Friedman s equations, it is possible to represent the Hubble parameter as a function of z. Page 81

13 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: For a matter-dominated Universe H (z) is derived from Eqs.(18) and (3) which is given as 1 m 3 k Λ H(z)= =H Ω (1+z) +Ω (1+z) +Ω Thus we write H(z) = H f(z), now we have / = H f(z)/(1+z). (46) We have further d/ = -dz/(1+z). This implies that Eq. (44) can be rewritten as -1 dz t=h - [(1+z)f(z)] Hence we have (47) t = sinh Ω 1/ -1 Λ ΩΛ 3H Ωm 1/ Substituting the values of m and i.e..3 and.7 respectively, the age of the Universe for the present model is calculated as 1. Gyrs. From experimental observations by WMAP, COBE, the age of the Universe is found to be ±.37 Gyrs. The size of the Universe is also an important parameter which is related to the age of the Universe Volume of the Universe is normally calculated as: π π 3 r drsinθdθdφ V= 1/ (1-kr ) (49) It will be an infinitely large, flat Universe. Although this is a theoretical estimate, we need to find the size of the observable Universe so that the particle horizon can be found which can give past information of the Universe. In Cosmology there are two kinds of horizons - Event Horizon and Particle Horizon. Particle Horizon relates to the communication in the past and Event Horizon relates to limits on communications in future. Since we have limitations for observable Universe but still size of the Universe can be found out for FW model by determining Hubble radius r H defined by ct H. Cosmologically c=1 so r H = t H.It is roughly estimated to be 998h -1 mpc [4] It can be determined more accurately by the following expression: Let the light ray start from r=r 1 at time t 1 and it reaches us at r= at t. Therefore for light received, we have: t dt - dr=r 1= r1 t (t) 1 (5) For a dust-filled Universe r 1 e a(t 1 -t ) - where a depends upon Lambda. The Horizon problem is solved if the Cosmological constant is included in the field equation. (t) represents an inflationary era. In such a situation the particle at one point will still be in casual contact with another point for a very large distance between them. This is possible because the expansion rate is an exponential function of time. In a similar manner, flatness of the Universe is solved [4]. From Eqs. (38-4) time evolution of an accelerating Universe is as shown below. Fig.4 illustrates the area of the Universe in an accelerated phase for which the Universe is flat and has some finite Cosmological Constant, whereas decelerating phase of the Universe may be due to k= and zero Cosmological Constant. It is also observed from the fig.(4) that there is an outward bulging during the accelerating phase so as to have age more as compared to the Universe with zero Cosmological Constant. It should also be noted that the accelerating expansion due to vacuum energy causing negative pressure tends to have flat Universe. Hence the model with finite, positive Cosmological Constant solves the Age problem and the Flatness problem. (48) Page 8

14 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: Fig. 4 Time evolution of the Universe Accelerating and Decelerating Universe Although FW model with finite Cosmological Constant has solved the Horizon problem, Age Problem and Flatness Problem it still t faces some other problems. In next section these problems are going to be discussed. VIII. LIMITATIONS OF FOU DIMENSIONAL FW MODEL WITH COSMOLOGICAL CONSTANT. Finite Cosmological Constant representing vacuum energy density plays an important role in explaining dynamics of the Universe and it can explain accelerated expansion of the Universe. With Cosmological Constant it is also possible to predict the age of the Universe and relate it to observational data. However recent researches in cosmology have pointed out some key problems related to Cosmological constant i.e. Lambda Cosmology. These are given below, A. The Cosmological constant problem 4 The value of the vacuum energy density is given by as ~ħk max which relates to momentum of the zero mode of vacuum oscillations. By quantum field theory, energy density ε related to cosmological constant is found to be ~ (M pl ) 4 ~ (8πG) - which is calculated as ~(1 18 GeV) 4 = 1 11 erg/cm 3. From observational calculations, it is found that (1-1 GeV) 4 ~ 1-1 erg/cm 3. Discrepancy between the two values of total density is around 1-1 which is called the Cosmological problem.[5]. Several proposals have been put up by several scientists which are based on string theory, super symmetry, scalar field theory etc. But the most promising solution of CCP is suggested as Lambda Decay Cosmology indicating time dependent Cosmological Constant which decays with time. B. Cosmological Coincidence Problem. The Cosmological Coincidence Problem is regarding the values of the Cosmological constant and vacuum density. It is found that Vacuum density is comparable to matter density at some cosmic time and reduces drastically with the expansion of the Universe. However the value of lambda does not reduced correspondingly. Seemingly, there may not necessarily be any coincidence between lambda and Vacuum density; although it can be expressed in terms of lambda. To solve this problem, Quintessence Cosmological model [6] with self interaction, Guassian potential etc have been suggested by many researchers. In order to solve both Cosmological constant problems, coincidence problems, time varying lambda becomes the most satisfactory answer. Page 83

15 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: An Anthropic solution and existence of Dark Energy are also solutions suggested by cosmologists, but an anthropic solution means disturbing the structure of the Universe. Since the existence of Dark Energy is still a puzzle and the Dark Energy content is still unknown, cosmologists are still working on it and this area is also the current topic of research. Dark Energy and Dark Matter is still a mystery which is yet to be solved and it is the latest research topic. It is also pointed out that the solution to Cosmological Constant problem may draw some definite conclusion on Dark Energy in the Universe which fills almost 7% of the Universe. As discussed previously here it is necessary to review Lambda Decay Cosmology which motivates us to choose time dependent lambda for further research work. IX. LAMBDA DECAY COSMOLOGY Although a cosmological model with constant explains the expansion of the Universe to some extent as seen in preceding sections; a huge discrepancy exists between the observed value and the predicted value by the Standard Model described in Particle Physics. This discrepancy is also called the Cosmological Constant Problem (CCP). To solve this problem, a popular suggestion given by many scientists was that the Cosmological Constant may have a large value, and with the evolution of time its value decreased. Since the Big-Bang the Universe has undergone several phases. The early phase of the Universe is also called the inflationary phase. In this phase, the value of the cosmological constant is assumed to be very high. However, in present phase, value of Λ is calculated to be very small. The Cosmological constant problem was first discussed by H. Bondi in 196 [7]. There are several articles by quantum field theorists and others ( [8], and ref. therein ) in which sources of vacuum energy have been considered as potential sources like scalar fields, tensor fields, non local effects, worm holes etc. These articles concluded that the Cosmological Constant is not really constant but is varying. The relation between scale factor and cosmological constant was also established by [8] in their paper. V. Sahni and others [9] have stated that vacuum energy density as calculated by Zeldovich [1] is given by ε vac. = ρ vac. c = (Gm /λ)/ λ 3 = Gm 6 c 4 / ћ 4 where λ=ћ/mc. When density calculated for pion mass compared with density 1 m π related to Planck mass it is given as ρ Λ = ρ 4 π m pl Here ρ pl is the Planck density. 6 pl =1.45 ρ pl 1-13 gm/cc The above expression shows that there is a huge difference between energy due to Λ and that of Planck energy. Here value of ρ pl is calculated as = c 5 /G ћ gm/cc. With this, during Planck epoch t pl ~ 1-43 sec, it would involve a fine-tuning of one part in ! In some papers this time is calculated as 1-35 sec. This shows that Vacuum effects also play an important role, if the Universe is expanding. The huge difference between the values of Λ for the early universe and present universe has been a key factor in the study of lambda decay cosmology. There are many projects [11, 1, 13,14] in which a Four dimensional model with time varying cosmological constant have been studied extensively. Overduin and Cooperstock explained time variations of Λ as Λ~a -m, Λ~t -t Λ ~ q r,λ~h, Λ~H. In these models, they considered specific values of m, t, r which are constants and discussed several oscillating and non-oscillating models. Berman and others [15] had concluded Λ~t -. Arbab I. Arbab [16] has also reviewed cosmological models with Λ~H, Λ~ q and Λ~ and studied cosmic acceleration for positive cosmological constant and its implications. A list of time varying cosmological constant is available in the articles by Sahni V. [9]. Page 84

16 International Journal of Mathematics and Physical Sciences esearch ISSN (Online) Vol., Issue, pp: (7-86), Month: October 14 March 15, Available at: Table I. List Of Time Varying Cosmological Constant t - [8,14, 15] t - [8] A + B exp(- t) (Beesham & others) [17] a - [Chen &Wu] [8] a - [8] exp(- a) [ajeev S.G.][18] T [Canuto & others][19] H [16, ] Lima and others H + Aa - Carvalho, Waga [1] f (H) g(, H) Hiscock [4] euter [5] Lima and Maia, [] Lima & Trodden [3] The Phenomenological models with variations of Λ as scale factor a (t), cosmic time t, Cosmic temperature T, Hubble constant H as shown in the above table, have been discussed by many researchers[9] and references therein) X. CONCLUSION In this chapter we conclude that the FW model, the so-called standard model of the Universe is quite successful in providing positive answers to the age of the Universe problem, Horizon problem, and expansion of the Universe problem;, but there are several puzzles still to be solved. Lambda decay Cosmology has gained attention recently for solving puzzle of Dark Energy, Dark matter, Large structure of the Universe, the phases of the Early Universe etc. In the meantime, existence of higher dimensions has also become a matter of interest to study unifying all types of forces. Pioneering work, in the study of higher dimensions, by Kaluza T. in 191 and Klein O. in 196 in five dimensional physics is of great importance and has a wider coverage area in cosmology. Five dimensional FW Cosmological model was first discussed by Chodos and Detweilar [6] in 1984 and later on investigated by many authors in different contexts. Five dimensional FW cosmology is alternately called as Kaluza- Klein Cosmology which is consistently gaining response from research community. ecently, using Kaluza Klein cosmology, Paul Wesson developed a new theory, called it Space-Time Matter theory [7] and explained 5D models in detail in his books. ACKNOWLEDGEMENT It is my pleasure to give my sincere gratitude to my Co-Guide Prof G.S. Khadekar for providing his valuable guidance. EFEENCES [1] Narlikar J.V., An Introduction to Cosmology, Chapter 3, 3 rd edition, Cambridge University Press, Cambridge () [] Wikipedia,(The free encyclopedia) Big Bang en.wikipedia.org/wiki/big_bang [3] Eddington A, Space, Time and Gravitation. An Outline of the General elativity Theory Cambridge University Press, Cambridge (19) [4] Banerji S., Banerjee A. General elativity and Cosmology, Chapter 11, eed Elsevier Pvt. ltd., Elsevier (7) [5] Caroll H. Press T. Ann. ev. Astron. Astrophysics. 3, 499 (199) Page 85

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