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FLRW Models: Kinematics, Dynamics, Geometry Cosmology Block Course 2014

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FLRW Models: Kinematics, Dynamics, Geometry Cosmology Block Course 2014 FLRW models: Kinematics, dynamics, geometry Cosmology Block Course 2014 Simon Glover & Markus Possel¨ Institut fur¨ Theoretische Astrophysik/Haus der Astronomie 29-30 July 2014 FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Contents 1 FLRW expansion: Kinematics and basic properties 2 The dynamics of FLRW universes 3 Examples for FLRW universes 4 Light propagation / distances in FLRW spacetimes 5 What does cosmic expansion mean? Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Cosmic time and FLRW metric Friedmann-Robertson-Walker-Metric: Substrate of “galaxy dust” moving with the cosmic flow (scale factor expansion). Distances between substrate galaxies change proportionally to the cosmic scale factor a(t). Time coordinate t (cosmic time) is proper time of substrate particles. Different kinds of possible geometry parametrized by K = −1; 0; +1 (hyperbolic, flat, spherical): " # dr2 ds2 = −c2dt2 + a(t)2 + r2dΩ = −c2dτ2: 1 − Kr2 with dΩ ≡ dθ2 + sin2 θ dφ2: Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Taylor expansion of the scale factor Generic Taylor expansion: 1 a(t) = a(t ) + a˙(t )(t − t ) + a¨ · (t − t )2 + ::: 0 0 0 2 0 Re-define the expansion parameters by introducing two functions a˙(t) a¨(t)a(t) H(t) ≡ and q(t) ≡ − a(t) a˙(t)2 and corresponding constants H0 ≡ H(t0) and q0 ≡ q(t0) " # 1 a(t) = a 1 + (t − t )H − q H2(t − t )2 + ::: 0 0 0 2 0 0 0 Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Some nomenclature and values 1/2 t0 is the standard symbol for the present time. If coordinates are chosen so cosmic time t = 0 denotes the time of the big bang (phase), then t0 is the age of the universe. Sometimes, the age of the universe is denoted by τ. H(t) is the Hubble parameter (sometimes misleadingly Hubble constant) H0 ≡ H(t0) is the Hubble constant. Current values are around km=s H = 70 ≈ 10−10=a 0 Mpc Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Some nomenclature and values 2/2 Sometimes, the Hubble constant is written as km=s H = h · 100 0 Mpc to keep one’s options open with h the dimensionless Hubble constant. The inverse of the Hubble constant is the Hubble time (cf. the linear case and the models later on). 1 ≈ h−1 · 1010 a: km=s h · 100 Mpc Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Light in an FRW universe For light, often easier to use ds2 = 0 instead of the geodetic equation. Also: use symmetries! Move origin of your coordinate system wherever convenient. Look only at radial movement. " # dr2 ds2 = −c2dt2 + a(t)2 + r2dΩ 1 − Kr2 becomes a(t) dr=c dt = ± p : 1 − Kr2 Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Light in an FRW universe Integrate to obtain Z t1 dt Z r1 dr c = ± p : 2 t2 a(t) r2 1 − Kr Plus/minus: light moving towards us or away from us. The key to astronomical observations in an FRW universe: Z t0 dt Z r1 dr c = p : 2 t1 a(t) 0 1 − Kr where, by convention, t0 is present time, t1 < t0 emission time of particle, r1 (constant) coordinate value for distant source. Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Light signals chasing each other 1/2 Imagine two signals leaving a distant galaxy at r = r1 at consecutive times t1 and t1 + δt1, arriving at t0 and δt0. Then t0 r1 Z c dt Z dr = p : a(t) 1 − Kr2 t1 0 and t0+δt0 r1 Z c dt Z dr = p a(t) 1 − Kr2 t1+δt1 0 t0+δt0 t1+δt1 Z dt Z dt − = 0: a(t) a(t) t0 t1 Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Light signals chasing each other 2/2 For small δt, Z¯t+δt f (t) dt ≈ f (¯t) · δt; ¯t so in our case δt δt 0 = 1 a(t0) a(t1) Signals could be anything — in particular: consecutive crests (or troughs) of elementary light waves of frequency f / 1/δt: f a(t ) λ a(t ) 0 = 1 ; wavelengths change as 0 = 0 : f1 a(t0) λ1 a(t1) Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Frequency shift by expansion Frequency shift z (commonly redshift) defined as λ − λ a(t ) z = 0 1 = 0 − 1 λ1 a(t1) or a(t ) 1 + z = 0 a(t1) For co-moving galaxies: z is directly related to r1. For monotonous a(t): distance measure. Relation depends on dynamics ) later! Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Cosmological redshift Wavelength scaling with scale factor: wavelength Redshift for a(t0) > a(t1); blueshift for a(t0) < a(t1) Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion For “nearby” galaxies. 2 . use the Taylor expansion a(t) = a(t0)[1 + H0(t − t0) + O((t − t0) )]: 1 a(t1) 1 − z ≈ = ≈ 1 + H0(t1 − t0) 1 + z a(t0) or z ≈ H0(t0 − t1) ≈ H0d=c for small z, small t0 − t1, d the distance of the galaxy from us This is Hubble’s law. Originally discovered by Alexander Friedmann (cf. Stigler’s law). Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Pedestrian derivation of Hubble’s law and redshift For scale factor expansion, d(t) = a(t)=a(t0) · d(t0): “Instantaneous speed” of a galaxy a˙(t) v(t) = d(t) = H(t) d(t) ≈ H d(t): a(t) 0 Classical (moving-source) Doppler effect: cz = v in other words: cz = H0 d: “Speed of recession” – we’ll come back later to the question whether or not that is a real speed. Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Solving Einstein’s equations for FRW 00 component of Einstein’s eq.: a˙2 + Kc2 3 = 8πG ρ a2 i0 components vanish. ij components give a¨ a˙2 + Kc2 8πG 2 + = − p a a2 c2 These are the Friedmann equations. Their solutions are the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universes. Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Re-casting the Friedmann equations Take what we will call the first-order Friedmann equation a˙2 + Kc2 8πG ρ = (1) a2 3 and for a˙ , 0, differentiating the above and inserting the ij-equation, derive a˙ ρ˙ = −3 (ρ + p=c2) = −3H(t)(ρ + p=c2): a — this amounts to energy conservation (as you’ve seen in Monday’s exercise). Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion The physics behind the Friedmann equations Multiply a˙ ρ˙ = −3 (ρ + p=c2) a by a3c2 and integrate: d da3 (ρc2 a3) + p = 0: dt dt The volume of a small ball 0 ≤ r ≤ r1 is p 3 V = grrgθθgφφ dr dθ dφ = a v(r1): Using this, rewrite$ d dV (ρc2V) + p = 0: dt dt Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion The physics behind the Friedmann equations d dV (ρc2V) + p = 0: dt dt but ρc2 is energy density — ρV = U is the system’s energy! ) dE = −p dV — change in energy is the “expansion work”. If p = 0 (dust universe), dE = 0, so energy/mass is conserved! Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Physics behind the Friedmann: deceleration Recombine Friedmann equations to give equation for a¨ (which we’ll call the second-order Friedmann equation): a¨ 4πG = − (ρ + 3p=c2) (2) a 3 Almost Newtonian — but in general relativity, pressure is a source of gravity, as well! (E.g. stellar collapse.) This leads to an expression for the deceleration parameter: 4πG q = (ρ + 3p =c2) 0 3 0 0 (with ρ0 and p0 the present density/pressure). Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Newtonian analogy Using purely Newtonian reasoning, one can derive the Friedmann equations for dust for K = 0 or, with a slight modification for the source terms of Newtonian gravity, also for matter with pressure. Details ) You’ve done this in Monday’s exercise Simon Glover & Markus Possel¨ FLRW models: Kinematics, dynamics, geometry FLRW spacetime basics FLRW dynamics FLRW examples Light propagation / distances Meaning of Expansion Different equations of state Now, assume equation of state p = wρc2.
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