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Curved Universe & Observational Cosmology

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Curved Universe & Observational Cosmology Cosmology, lect. 5 Curved Cosmos & Observational Cosmology Einstein Field Equation 18πG R− Rg +Λ g =− T µν 2 µν µν c4 µν 18πG R− Rg =− T −Λ g µν 2 µν c4 µν µν Cosmological Principle A crucial aspect of any particular configuration is the geometry of spacetime: because Einstein’s General Relativity is a metric theory, knowledge of the geometry is essential. Einstein Field Equations are notoriously complex, essentially 10 equations. Solving them for general situations is almost impossible. However, there are some special circumstances that do allow a full solution. The simplest one is also the one that describes our Universe. It is encapsulated in the Cosmological Principle On the basis of this principle, we can constrain the geometry of the Universe and hence find its dynamical evolution. “God is an infinite sphere whose centre is everywhere and its circumference nowhere” Empedocles, 5th cent BC Cosmological Principle: Describes the symmetries in global appearance of the Universe: The Universe is the same everywhere: ● Homogeneous - physical quantities (density, T,p,…) ● Isotropic The Universe looks the same in every direction ● Universality Physical Laws same everywhere The Universe “grows” with same rate in ● Uniformly Expanding - every direction - at every location ”all places in the Universe are alike’’ Einstein, 1931 Fundamental Tenet of (Non-Euclidian = Riemannian) Geometry There exist no more than THREE uniform spaces: 1) Euclidian (flat) Geometry Euclides 2) Hyperbolic Geometry Gauß, Lobachevski, Bolyai 3) Spherical Geometry Riemann uniform= homogeneous & isotropic (cosmological principle) Curvature of the Universe: Robertson-Walker Metric Spherical Surface Distances dφ r Rdsinθφ ds2= dr 2 + R 22sin θφ d 2 rR= θ 2 2 22r 2 ds= dr + Rsin dφ R dr= R dθ Spherical Surface Distances: alternative – geodetic distance x ds= Rsinθφ d = x d φ rr dφ x≡ Rsin ⇒=dxcos dr RR r 2 22r ⇒=−dx 1 sin dr R Rdsinθφ 1 dx2 κ = ⇒=dr 2 Rx221−κ ds2= dr 2 + R 22sin θφ d 2 2 2 22r 2 ds= dr + Rsin dφ R ⇓ dx2 ds2= + x22 dφ 1−κ x2 Spherical Space Distances 2 2 22r 2 2 2 ds= dr++ Rsin { dθsin θφ d } R Spherical surface is a 2-D section through an isotropic curved 3D space: dφ generalization to 3D solid angle (θ,φ) r Rdsinθφ ds2= dr 2 + R 22sin θφ d 2 rR= θ 2 2 22r 2 ds= dr + Rsin dφ R dr= R dθ Spherical Space Distances alternative: geodetic distance 2 2 22r 2 2 2 ds= dr++ Rsin { dθsin θφ d } R dφ r Rdsinθφ ds2= dr 2 + R 22sin θφ d 2 dx2 ds2=++x22( dθ sin 2 θφ d 2 ) 1−κ x2 Minkowski Metric spherically isotropic 3D space 2 22 2 2 2r 2 2 2 ds= c dt −+ dr Rsin { dθ+sin θφ d } R dφ r Rdsinθφ ds2= dr 2 + R 22sin θφ d 2 dx2 2=−++ 22 2θ 2 2 θφ 2 ds c dt 2 x( d sin d ) 1−κ x Distances in a uniformly curved spacetime is specified in terms of the Robertson-Walker metric. The spacetime distance of a point at coordinate (r,θ,φ) is: r 2=−+ 22 2 2 22 θ2 + 2 θφ 2 ds c dt a() t dr Rck S dsin d Rc r sin k = +1 Rc where the function Sk(r/Rc) specifies the effect of curvature rr on the distances between Skk = = 0 points in spacetime RRcc r sinh k = −1 Rc Conformal Time Proper time τ Robertson-Walker metric 2 2 2 22 2 2 22r 2 c dτψ==−+ ds c dt a() t dr Rck S d Rc ddψ2= θ 2 + sin 22 θφ d t 2 cd2τ 2 cdt 22 r r c dt ηψ2==−+22 dτ 22d Sdk η = RR Rcc R ()t ∫ 0 Rt() 2′′ 22 2 =−+dηψ{ dr Sk ( r) d } Conformal Time η(t) Observational Cosmology in FRW Universe Redshift Cosmic Time Dilation In an (expanding) space with Robertson-Walker metric, r 2=−+ 22 2 2 22 θ2 + 2 θφ 2 ds c dt a() t dr Rck S dsin d Rc In a RW metric, light travels with ds =0: c22 dt −= a( t ) 2 dr 2 0 ⇓ dt cdt= a() t dr : = cst. at() ⇓ ∆ Cosmic Time Dilation: te ∆=tobs at()e ∆te ∆=tobs at()e In Evidence Cosmic Time Dilation: light curves supernovae (exploding stars): characteristic time interval over which the supernova rises and then dims: systematic shift with redshift (depth) Hubble Expansion • Einstein, de Sitter, Friedmann and Lemaitre all realized that in General Relativity, there cannot be a stable and static Universe: • The Universe either expands, or it contracts … • Expansion Universe encapsulated in a GLOBAL expansion factor a(t) • All distances/dimensions of objects uniformly increase by a(t): at time t, the distance between two objects i and j has increased to rij−= r at()( r i,0 − r j ,0 ) • Note: by definition we chose a(t0)=1, i.e. the present-day expansion factor • Cosmic Expansion manifests itself in the in a recession velocity which linearly increases with distance • this is the same for any galaxy within the Universe ! • There is no centre of the Universe: would be in conflict with the Cosmological Principle Cosmic Expansion is a uniform expansion of space Objects do not move themselves: they are like beacons tied to a uniformly expanding sheet: rt()= atx () a Ht()= a r() t= a () t x = ax = H () t r a a Cosmic Expansion is a uniform expansion of space Objects do not move themselves: they are like beacons tied to a uniformly expandingHubbleComoving Parameter:sheet: Position Comoving Position Hubble “constant”: H0H(t=t0) rt()= atx () a Ht()= a r() t= a () t x = ax = H () t r a a For a long time, the correct value of the Hubble constant H0 was a major unsettled issue: -1 -1 -1 -1 H0 = 50 km s Mpc H0 = 100 km s Mpc This meant distances and timescales in the Universe had to deal with uncertainties of a factor 2 !!! Following major programs, such as Hubble Key Project, the Supernova key projects and the WMAP CMB measurements, +2.6 −− 1 1 H0 = 71.9−2.7 km s Mpc Hubble Expansion Edwin Hubble (1889-1953) v = H r Hubble Expansion For a long time, the correct value of the Hubble constant H0 was a major unsettled issue: -1 -1 -1 -1 H0 = 50 km s Mpc H0 = 100 km s Mpc This meant distances and timescales in the Universe had to deal with uncertainties of a factor 2 !!! Following major programs, such as Hubble Key Project, the Supernova key projects and the WMAP CMB measurements, +2.6 −− 1 1 H0 = 71.9−2.7 km s Mpc Hubble Expansion Space expands: v= Hr displacement - distance Hubble law: velocity - distance Deformation NonlinearCosmic Volume Descriptions Element The evolution of a fluid element on its path through space may be specified by its velocity gradient: in which Hv=1 ∇⋅ 3 θ: velocity divergence contraction/expansion σ: velocity shear deformation ω: vorticity rotation of element Deformation NonlinearCosmic Volume Descriptions Element Global Anisotropic expansion/contraction Anisotropic Relativistic Universe Models: Bianchi I-IX Universe models • expand anisotropically • have to be characterized by at least 3 Hubble parameters (expansion rate different in different directions • Only marginal claims indicate the possibility on the basis of CMB anisotropies Deformation NonlinearCosmic Volume Descriptions Element Local Anisotropic Flows: “fatal” attractions • In our local neighbourhood the cosmic flow field has a significant shear • This shear is a manifestation of - infall of our Local Group into the Local Supercluster - motion towards the Great Attractor - possibly motion towards even larger mass entities: Shapley concentration Horologium supercluster Deformation NonlinearCosmic Volume Descriptions Element Global Hubble Expansion Observations over large regions of the sky, out to large cosmic depth: • the Hubble expansion offers a very good description of the actual Universe • the Hubble expansion is the same in whatever direction you look: isotropic Hubble flow: 1 Hv=3 ∇⋅ Pure expansion/contraction Cosmic Distances In an (expanding) space with Robertson-Walker metric, r 2=−+ 22 2 2 22 θ2 + 2 θφ 2 ds c dt a() t dr Rck S dsin d Rc ds =0: c22 dt −= a( t ) 2 dr 2 0 radial comoving distance r travelled by radiation ⇓ in a RW space: cdt= a() t dr t 0 cdt r=−= rt() rt () 0 e ∫ at() te Distance Measure In an (expanding) space with Robertson-Walker metric, r 2=−+ 22 2 2 22 θ2 + 2 θφ 2 ds c dt a() t dr Rck S dsin d Rc there are several definitions for distance, dependent on how you measure it. They all involve the central distance function, the RW Distance Measure, r Dr()= RSck Rc Light propagation in a RW metric (curved space): ds2 = 0 ⇒=−c dt R() t dr c ψ 2 = d 0 R0 dr= dz Note: - light propagation is along radial lines Hz() - the “-” sign is an expression for the fact that the light ray propagating towards you moves in opposite direction of radial coordinate r After some simplification and reordering, we find R Rt()= atR () = 0 0 + dR dR dR 1 z c dt= c = c = c dR R HR ⇓ dR 1 dt = − dz Rz1+ Observing in a FRW Universe, we locate galaxies in terms of their redshift z. To connect this to their true physical distance, we need to know what the coordinate distance r of an object with redshift z, c R dr= dz 0 Hz() In a FRW Universe, the dependence of the Hubble expansion rate H(z) at any redshift z depends on the content of matter, dark energy and radiation, as well ss its curvature.
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