Curvature of the Universe: Analysis

Curvature of the Universe AnalysisIntroduction1.1 Reviewing of customary Relativity1.1.1 Metric TensorThe compare which describes the relationship between dickens habituated points is c eithered metric and is effrontery byWhere interval of space- clip continuum between two neighboring points, connects these two points and ar the components of contra variant vector. with the function, any break between two points is dependent on the position of them in arrange corpse.The displacement between two points in rectangular coordinates system is free-lance of their components collectible to homogeneousness, so metric is given byWhere are the space- fourth dimension coordinates, is speed of light and is metric for this eccentric and is given byThrough the coordinates transformation from rectangular coordinates,, to curved coordinates system the components ofin a curved coordinates system drop be found . For constructing rectangular coordinates system in a curved coordinates if space- condemnation is locally savorless so it is possible to that locally. From rectangular coordinates system find outd locally in a point of a curved space- metre to a curved coordinates system undersurface be written asSo in this way we can suffer local values of metric tensorThree important properties of metric tensor are is symmetric so we earnmetric tensors are used to clayey or raising indices1.1.2 Riemann Tensor, Ricci Tensor, Ricci ScalarThe tool which plays an important role in identifying the geometric properties of spacetime is Riemann (Curvature) tensor. In shapes 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 also be written asthus useful symmetries of the Riemann Tenser areso due to above symmetries, the Riemann tensor in four dimensional spacetime has only 20 autonomous components. instantly simply spotting the Riemann Tensor over two of the indices we decease Ricci Tensor asabove equation is symmetric so it has at most 10 independent components. flat contracting over remaining two indices we enamour scalar k nowadaysn as Ricci Scalar.Another important symmetry of Riemann Tensor is Bianchi identitiesThis subsequently contracting leads to1.1.3 Einstein equivalenceThe Einstein equation is the equation of question for the metric in general theory of relativity is given byWhere is stress qualification momentum tensor and is Newtons constant of Gravitation. and so the left hand side of this equation measures the curvature of spacetime while the rectify hand side measures the energy and momentum contained in it.Taking trace of both(prenominal) sides of above equation we obtain using 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 gener al relativity two types of momentum-energy tensor,are normally used dust and perfective tense fluent.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 approach,of the innovation at later times when ray is minimal1.4.2 Perfect fluid If there is no heat conduction and viscosity then such type of fluid is perfect fluid and parameterized by its mass engrossment and pressure and is given byIt is an approximation of the human beings at earlier times when radiation dominates so conservation equations for energy momentum tensor are given byIn Minkowski metric it becomes1.1.5 phylogenesis 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 pro portion so in the first term and in the second term soFor a flat, homogeneous and isotropous spacetime which is expanding in its spatial coordinates by a subdue factor, the metric tensor is obtained from Minkowski metric is given byThe Christoffel symbol by interpretation Because Because the only non-zero is so from eq. () conservation law in expanding universe becomes aft(prenominal) resolving above equations we getabove equation is used to find out for both matter and radiation cuticle with expansion. In case of dust approximation we cook soSo energy-density of matter scale varies as .Now the total amount of matter is conserved but quite a little 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, th e cosmological rule is a suspicion, or living up to expectations theory, about the expansive scale organise 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 isotropous at all focuses, is likewise homogeneous. Copernicus rule and this result about homogeneity makes the cosmologic rule (CP) which states that, at a one-time, universe is homogeneous and isotropic. normal 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 excite with or a flat space. In literature shorthanded notation is adequateTo isolate time-dependent term, make the following situationThenwhereIf we introduce conformal time (arc parameter measure of time) asthen we can express the 4D telephone circuit 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 fluidfinally we obtain the components of Ricci tenserThe components areand components areTo get a closed(a) system of equations, we need 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 score of Friedmann equations1.2.3 Solutions of Friedmann EquationsWe are going to comprehend Friedmann equation for the matter dominated and radiation dominated universe and get the manifestation 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 combined 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 in conclusionRadiation-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 condition we finally getNow we can calculate the get along with of universe, which corresponds to the Hubble rate and scale factor to beTaking and we getlong timeRadiation-dominatedThe First Friedmann equation becomesAt the big bang and .Also we have disagreeable 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 colony 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