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훹 **(3.8.1) The cosmological constant problems**: The cosmological term () in Einstein GR can be considered as standing for vacuum energy:  (3.8.1) Where GR takes the form: (3.8.2) Experimentally observations indicate that vacuum energy negligible small at presen...

훹 **(3.8.1) The cosmological constant problems**: The cosmological term () in Einstein GR can be considered as standing for vacuum energy:  (3.8.1) Where GR takes the form: (3.8.2) Experimentally observations indicate that vacuum energy negligible small at present. On the hand, large vacuum energy at the early universe can produce inflation where,  (3.8.3) This inflation can solve some of cosmological problems. The diversity between the present small value of the vacuum energy and the need of large vacuum energy at the early universe is known as cosmological constant problems. **(3.8.2) The singularity problem:** The **SBB** model stated that the universe started with initial singularity. The cosmic scale factor is zero with infinite density, i.e. (3.8.4) The infinity density is not physically acceptable, since physical quantities should be finite. The infinity density is called singularity problem **(3.8.3) The horizon problem**: The horizon problem result from the fact that the radius of the light sphere  is much less than the radius of the universe.where the horizon radius is given by:  (3.8.4) The radius of the universe is given by: (3.8.5) But for cosmic microwave background:  (3.8.6) Where T is absolute temperature. This means that at blank time  (3.8.7) This means that the initial universe consist of at least Isolated light sphere. Since the maximum propagation of fields and physical interactions are that of light speed, one expects these light sphere to be not interacting with each other. Thus these large number of spheres are co usually disconnected. This means that they have different temperatures and densities as far as the matter and energy have no enough speed to retard themselves to be uniform. This means that the universe is no homogeneous or isotropic at early times. This is in conflict with observed homogeneous and isotropic universe. This problem is called horizon problem. **(3.8.4) The flatness problem**: The flatness problem is related to the fact that at plank time, in the very early universe, the critical density  and the density of the universe are related according to the relation:  (3.8.9) For k=1 If the universe density and plank time was slightly greater than, i.e.  The universe would be e lost and it would collapsed millions of ears age, which is in conflict with the present expansion of the universe but if: (3.8.10) The universe would be open and the present energy density would be very small and much less than the critical density. This is in conflict with the fact that at present the density is nearly equal to the critical density where  (3.8.11) Thus the only way to be in agreement with observation is to set: (3.8.12). I.e. the universe is flat. Which again contradict the fact that the universe is curved, since the light path is no longer straight line when travelling from stars towards the earth. **(3.8.5) the energy problem:** The energy of the universe is given by:  Thus: (3.8.12) According to energy conservation:  (3.8.13) Thus the entropy is constant. This contradicts the fact that the universe entropy increases. **(3.9) The experimental conformation of general relativity within the field of cosmology** The modern history of experimental relativity can be divided roughly into four periods, Genesis, Hibernation, a Golden Era, and the Quest for Strong Gravity. The Genesis (1887--1919) comprises the period of the two great experiments which were the foundation of GR physics. The deflection of light and the perihelion advance of Mercury. Following this was a period of Hibernation (1920--1960) during which Astronomical discoveries (quasars, pulsars, cosmic background radiation) and new experiments pushed GR to the forefront. Experimental gravitation experienced a Golden Era (1960--1980). The period began with an experiment to confirm the gravitational frequency shift of light (1960) and ended with the reported decrease in the orbital period of the binary pulsar at a rate consistent with the general relativity prediction of gravity-wave energy loss (1979). The results all supported GR, and most alternative theories of gravity fell by the wayside. Many of the remaining interesting weak-field predictions of the theory are extremely small and difficult to check, in some cases requiring further technological development to bring them into detectable range. Examples include the use of laser-cooled atom and ion traps to perform ultra precise tests of special relativity; the proposal of a "fifth" force, which led to a host of new tests of the weak equivalence principle; and recent ideas of large extra dimensions, which have motived new tests of the inverse square law of gravity at sub-millimeter scales. Several major ongoing efforts also continue, principally the Stanford Gyroscope experiment, known as Gravity Probe-B. Instead, much of the focus has shifted to experiments which can probe the effects of strong gravitational fields. The principal figure of merit that distinguishes strong from weak gravity is the quantity ǫ ∼ GM/Rc2, where G is the Newtonian gravitational constant, M is the characteristic mass scale of the phenomenon, R is the characteristic distance scale, and c is the speed of light. Near the event horizon of a non-rotating black hole, or for the expanding observable universe, ǫ ∼ 0.5; for neutron stars, ǫ ∼ 0.2. These are the regimes of strong gravity. For the solar system ǫ \< 10−5; this is the regime of weak gravity. At one extreme are the strong gravitational fields associated with Planck-scale physics. Will unification of the forces, or quantization of gravity at this scale leave observable effects accessible by experiment? Dramatically improved tests of the equivalence principle or of the inverse square law are being designed, to search for or bound the imprinted effects of Planck-scale phenomena. At the other extreme are the strong fields associated with compact objects such as black holes or neutron stars. Astrophysical observations and gravitational-wave detectors are being planned to explore and test GR in the strong-field, highly-dynamical regime associated with the formation and dynamics of these objects. **(3.10) Break down of GR within the field of cosmology:** The universe has beginning intimae called big bang associated with initial singularity of friedmann equations. Singular state possibility marks the edge of applicability of general relative attempts to Resolve this problem within GR has no success. The prediction of singularities represents a break down of general relativity in that its classical description of gravitation and matter connot be expected near a space time singularity thus GR break down at scale characteristic by the plank length equal L~p~ =(ћG ∕ c^3^ )^γ^..≈10^-33^ cm GR possess a singularity at t=0 where the density diverges and the proper distance between any two points tends to zero. This singularity is called the big bang -- it is a consequence of four things: ( І) cosmological principle. ( II) Einstein equation in the absence of a cosmological constant. ( III) The expansion of the universe. \(IV) The assumed from of the equation of state. It clear that the big bang might well just be consequence of extrapolating based on the theory of GR which is invalid, Indeed, Einstein (1950) himself wrote. \"The Theory is based on separation of the concepts of the gravitational field and matter, while this may be a valid approximation for weak field. It may invalid for very high densities , that means there is no singularity\". This leads us to take into a count quantum gravity, maybe we can avoid singularity without appealing to quantum gravitational and remaining inside Einstein theory of gravity. This may indicate that the classical theory must be replaced by a quantum theory of gravitation. Quantum effects may avoid the presence of the singularity leading to a complete regular cosmological model. GR suffers also from the so-called horizon, entropy and faltness problems \[ \]. Such problems were solved by suggesting inflationary scenario. **Chapter 4** **The generalized model for nonsingular solution** **(4.1) Introduction** The incompatibility of general relativity as a strong field model and its prediction of the gravitational collapse, needs new gravitational model with solution of the gravitational equation to be nonsingular. To do that a number of approaches and schemes had been adopted to modify GR with a hope that strong field gravity will be described both physically and mathematically. There are a number of common grounds between the gravitational and electromagnetic phenomena, first there is a certain correspondence between the metric tensor components and the potential of the electromagnetic field, second the inverse square law of Newtonian gravitational and Colombian electrostatic is a common feature between gravitational and electromagnetic interaction in the weak field areas. Mathematically there is a difference that the gravitational equations describe a tensor field are nonlinear and physically the gravity influences its source where as Maxwell\'s equation describe a vector field are nonlinear in the field variables and electric charge is not affected by its field. Due to analogy between gravity and electromagnetism quadratic lagrangians one recommended to be a basis for gravitational equations which may improved the general relativity theory to be quantizable similar to electromagnetic if a quadratic term is included in the lagrangians**.** \| 2 (4.4.6) **Solution of equation of motion with constant scalar curvature:** One of the simplest solution of equation ( ) can be achieved by considering the scalar curvature to be constant i.e. A direct substitution in equation (5.2.28) yields   This requires to be constant so as to make  constant. On of the possible solutions which requires to be constant is to suggest inflation solution of the form  Thus  Inserting equation (5.3.4) in equation (5.2.27) yields  Where takes one of the possible values:  For to be constant, the only possible value for  is to make Thus  In view of equation (5.3.2) and (5.3.8) Inflation requires  to be large \[see eq. (5.3.3)\]. Thus according to equation (5.3.8) is large. According to equation (5.2.22), the quadratic term become dominant, compared to the linear one  This is equivalent to the setting. Thus eq. (5.3.9) reads  But since should be positive for inflation to take place, and as for the term  reflects the matter or vacuum density \[ \], i.e.  As far as is positive, the only way to make the left hand side positive is to make  negative, i.e. Thus according to eq. (5.3.11)  The inflation takes place when one takes the positive sign, i.e. In view of equation (5.3.10)  The minus sign indicates that inflation is driven by repulsive force. This conforms with some inflation models. The solution with constant scalar curvature can also be achieved by suggesting  In view of equation (5.2.27)  Becomes constant, when it vanishes. This requires  i.e.  When Equation (5.3.16) reads  This equation shows that the universe is expanding, which agrees with astronomical observations. This solution is non-singular, since at  If one assumes that radiation dominates at the early universe, such that matter energy-momentum tensor is conserved, one gets if the universe of mass  has special shape with radius It follows that the universe density is  Multiply both sides by , one gets  Comparing (5.3.19) with (5.3.24) yields Another solution can be achieved when  In this case equation (5.3.16) and (5.3.18) reads This means that no expansion takes place. This conforms with the fact that when no matter exist, k vanishes and the universe is empty. Thus no atoms, that can exert a pressure to make the universe expand, exist. However, when k is positive, i.e.  Thus equations (5.3.18) and (5.3.16) gives Thus  is imaginary. This agrees with the fact that at the early universe quantum laws dominate. Imaginary quantities are associated with the wave function and operators **5.4 Dynamical scalar curvature for non-singular expanding during vacuum and radiation area** The new equation can also allow solutions that describe scalar curvature with time. To do this, consider again the simplest solution In case equation (5.2.27) gives  Where Therefore  Where  For vacuum era Thus insertion of (5.4.2,4,6) in (5.2.28) gives  One of the possible solutions is to set  In view of equation (5.2.23) one gets Thus according to equation energy density is given by  Another solution of (5.4.7, 8) requires Which is satisfied when   For c to be real Thus equation (5.4.11) gives  Taking the plus sign requires equation (5.4.1) to be The scalar curvature vanishes since according to equation (5.4.2) and (5.4.11) one gets  Thus the universe expands with vanishing scalar curvature. For minus sign of equation (5.4.14), equation (5.4.1) gives In this case contraction takes place and the radius vanishes, when  When the curvature parameter is positive, i.e. It follows that  Thus the radius parameter in equation (5.4.1) reads This confirms the models which assumes that in the early universe at plank era quantum mechanical laws are suitable for describing the behavior of the universe. Quantum laws are described by complex operators and wave functions. A third alternative can be tried by using equation (5.2.8, 25), (5.4.2, 3, 4, 5) to get  In view of equation (5.4.5) Thus the universe density is given by  It is very interesting to note that this expression consists of two terms. The first term which indicates that the density decreases as the universe expands, which agrees with observations. The second constant term is assumed to stand for vacuum energy. **(5.5) universe evolution in the presence of Gravitational**