Curvature of the Universe: Analysis

Curvature of the Universe AnalysisIntroduction1.1 Reviewing of General Relativity1.1.1 Metric TensorThe par which describes the relationship between twain disposed(p) points is called mensurableal and is given byWhere interval of space-time between two neighboring points, connects these two points and be the components of contra variant vector. Through the function, any displacement between two points is dependent on the position of them in coordinate scheme.The displacement between two points in rectangular coordinates system is independent of their components due to homogeneity, so metric is given byWhere are the space-time coordinates, is speed of light and is metric for this case and is given byThrough the coordinates transformation from rectangular coordinates,, to curved coordinates system the components ofin a curved coordinates system post be found . For constructing rectangular coordinates system in a curved coordinates if space-time is locally flat consequently it is possible to that locally. From rectangular coordinates system defined locally in a point of a curved space-time to a curved coordinates system fucking be written asSo in this way we can find local values of metric tensorThree important properties of metric tensor are is symmetric so we abidemetric tensors are used to lowering or raising indices1.1.2 Riemann Tensor, Ricci Tensor, Ricci ScalarThe tool which plays an important role in identifying the geometrical properties of spacetime is Riemann (Curvature) tensor. In circumstances of Christoffel symbols it is defined asWhere .If the Riemann Tensor vanishes everywhere then the spacetime is considered to be flat. In term of spacetime metric Riemann Tensor can in any case be written asthus useful symmetries of the Riemann Tenser areso due to preceding(prenominal) symmetries, the Riemann tensor in four dimensional spacetime has only 20 independent components. Now simply detection the Riemann Tensor everywhere two of the indi ces we affirm Ricci Tensor asabove equation is symmetric so it has at most 10 independent components. Now contracting over remaining two indices we get scalar known as Ricci Scalar.Another important symmetry of Riemann Tensor is Bianchi identitiesThis after contracting leads to1.1.3 star equationThe Einstein equation is the equation of motion for the metric in general theory of relativity is given byWhere is stress vigor momentum tensor and is Newtons regular of Gravitation. Thus the left hand side of this equation measures the curvature of spacetime while the right hand side measures the energy and momentum contained in it.Taking trace of twain sides of above equation we obtainusing this equation in eq. ( ), we getIn vacuum so for this case Einstein equation isWe define the Einstein tensor byTaking divergence of above eq. we get1.1.4 Conservation Equations for Energy momentum TensorIn general relativity two types of momentum-energy tensor,are commonly used stud and perfect bland.1.4.1 Dust It is simplest possible energy-momentum tensor and is given byThe 4-velocity vector for commoving observer is given by, so energy momentum tensor is given byIt is an approximation,of the universe at later times when radiation is negligible1.4.2 Perfect fluid If there is no heat conduction and viscosity then such type of fluid is perfect fluid and parameterized by its mass density and pressure and is given byIt is an approximation of the universe at earlier times when radiation dominates so conservation equations for energy momentum tensor are given byIn Minkowski metric it becomes1.1.5 Evolution of Energy-Momentum Tensor with TimeWe can use eq. () to determine how components pf energy-momentum tensor evolved with time. The mixed energy-momentum tensor is given byand its conservation is given byConsider componentNow all non-diagonal terms of vanish because of isotropy so in the first term and in the second term soFor a flat, homogeneous and isotropic spacetime which is expanding in its spatial coordinates by a descale factor, the metric tensor is obtained from Minkowski metric is given byThe Christoffel symbol by definition Because Because the only non-zero is so from eq. () conservation law in expanding universe becomes after solving above equations we getabove equation is used to find out for both matter and radiation scale with expansion. In case of dust approximation we have soSo energy-density of matter scale varies as .Now the total amount of matter is conserved but volume of the universe goes as so In case of radiation so from eq.() we obtainWhich implies that, science energy density is directly proportional to the energy per particle and inversely proportional to the volume, that is, because so the energy per particle decreases as the universe expands.1.2 CosmologyIn physical cosmology, the cosmological rule is a suspicion, or living up to expectations theory, about the expansive scale structure of the universe. Throughout the time of Copernicus, much data were not accessible for the universe with the exception of Earth, few stars and planets so he expected that the universe energy be same from all different planets likewise as it looked from the Earth. It suggests isotropy of the universe at all focuses. Once more, a space which is isotropic at all focuses, is likewise homogeneous. Copernicus rule and this result about homogeneity makes the cosmogonical rule (CP) which states that, at a one-time, universe is homogeneous and isotropic. General covariance ensures validity of Cosmological Principle at other times also.1.2.1 Cosmological metricThink about a 3D circle inserted in a 4d hyperspacewhere is the radius of the 3D sphere. The distance between two points in 4D space is given bysolving we getnow becomesIn spherical coordinatesFinally we obtainWe could also have a saddle with or a flat space. In literature shorthanded notation is adaptedTo isolate time-dependent term, make the following situation thenwhereIf we introduce conformal time (arc parameter measure of time) asthen we can express the 4D line element in term of FRW metric1.2.2 Friedmann EquationWe can now figure out Einstein field mathematical statement for perfect fluid. All the calculations are carried out in comoving frame whereand energy-momentum tensor is given byRaising the index of the Einstein tensor equationwe getAfter contracting over indices and we getso Einsteins Equation can be written asIt is easily found for perfect fluid last we obtain the components of Ricci tenserThe components areand components areTo get a closed system of equations, we wishing a relationship of equation states which relates and so solvingAt this point when we joined together with equation 62 comparisons in the connection of energy-momentum tensor and the equation of states, we get a closed frame work of Friedmann equations1.2.3 Solutions of Friedmann EquationsWe are going to comprehend Friedmann equation for the matter domin ated and radiation dominated universe and get the formulation of scale factor. From the definition of Hubbles lawMatter Dominated Universe It is showed by dust approximation As both and, for flat universe (), ( an) for . When combine with equation, this yields critical densityCurrently it value is (we used).The quantity provide relationship between the density of the universe and the critical density so it is given by Now the second Friedmann equation for matter dominated Universe becomesso lastlyRadiation-dominated Universe It is showed by perfect fluid approximation with The second Friedmann Equation becomesFlat Universe Matter Dominated Universe (dust approximation) The first Friedmann equation becomesAt the Big bang Using convention and universe flat configuration we finally getNow we can calculate the age of universe, which corresponds to the Hubble rate and scale factor to beTaking and we getYearsRadiation-dominatedThe First Friedmann equation becomesAt the big bang a nd .Also we have Closed Universe Matter-dominatedThe first Equation becomesIn term of conformal time we can rewrite the above integral asAfter substituting and using equationThen but we have so we get.Nowbut we have at sets. So we have now the dependence of scale factor in term of the time parameterized by the conformal time asRadiation-dominated UniverseThe first Friedmann equation becomesIn term of conformal time we can re write the integral as but we have conditions at sets so we getand the requirement at sets , finally we haveOpen Universe Matter-dominated (dust approximation)The first Friedmann equation In term of conformal time we can rewrite the integral asTake