Lecture 5: Building a model - dynamics 1. A Newtonian derivation of the Friedmann equation 2. Newtonian to General Relativity (conceptually) 3. The Friedmann Acceleration equation 4. Einstein’s Blunder? • Adding in the Cosmological constant (Λ) • Equation of State of Λ (w=-1) • What is Λ? 5. The Hubble constant revisited 6. Endpoints of the Universe Course Text: Chapter 4 & 6 Wikipedia: Friedmann Equations, Einstein The Robertson-Walker metric Space-time interval/metric Normal spatial polar coords 2 2 2 2 2 % dr 2 2 2 2 ( ds = c dt " R (t) ' 2 + r (d# + sin #d$ )* & 1" k r ) Light travel distance Curvature Term (open, closed, flat) Expansion factor ! This metric is forced by the Cosmological Principle (universal expansion) Spacetime interval for a photon = 0 (from special relativity) If we consider photons only on radial paths in a flat Universe (k=0): c 2dt 2 = R2(t)dr2 ! The Dynamics of the Universe • We currently have a geometric description of the Universe but we lack a description of the dynamics R(t)=? • Notation: R0 = R(t = T), scale-factor today at time T r = R(t), distance dR(t) r! = , velocity dt d 2 R(t) !r!= , acceleration dt 2 d 3R(t) !r!!= , rate of change of acceleration dt3 • GR tells us dynamics depend on contents (interlude) Einstein’s General Relativity • Spacetime geometry tells matter how to move – gravity = effect of curved spacetime – free particles follow geodessic trajectories – ds2 > 0 v < c time-like massive particles – ds2 = 0 v = c null massless particles (photons) – ds2 < 0 v > c space-like tachyons (not observed) • Matter (+energy) tells spacetime how to curve – Einstein field equations – nonlinear – second-order derivatives of metric with respect to space/time coordinates Spacetime Einstein’s Theory of Relativity • The laws of Physics are the same to all observers, i.e., there is no fundamental reference frame. • The speed-of-light represents a fundamental limit and the velocity of a photon is measured to be the same to all observers • Counter-intuitive but has been tested exhaustively! The Equivalence Principle This states that you cannot distinguish between being at rest in a gravitational field and acceleration. i.e., gravity and inertia are indistinguishable GR equates inertial mass with gravitational mass and explains why they are identical Einstein Field Equations GEOMETRY=CONTENTS Gµ" = 8# G Tµ" + $ gµ" 2 µ " gµ" = spacetime metric ( ds = gµ" dx dx ) Gµ" = Einstein tensor (spacetime curvature) G = Netwon's gravitational constant Tµ" = energy - momentum tensor $ = cosmological constant ! The Friedmann eqns: • Adopt standard GR G = 8#GT + $g • Assume µ" µ" µ" – Homogeneity and Isotropy, i.e., uniform fluid – Adiabatic expansion (lossless!) "E = #p"V ! Hubble constant or H(t) > Friedmann eqns: 2 ! • $ R 8!G" kc2 ( Λ = Cosmological constant # & = ' 2 + = vacuum energy ! # R & 3 R 3 = dark energy " % = “Greatest blunder” •• = Inflation R 4!G ! 3p$ ( = ' #" + &+ = Useful Fudge factor? R 3 " c2 % 3 Dynamics of the Universe • What contents do we know about? -3 – Matter, ρm~r -4 – Radiation, ρr~r – Intrinsic curvature/quantum froth/cosmological constant, ρv~ k – Anything else? • Adopting CP allows us to describe the Universe as a smooth fluid comprised of radiation and matter and any intrinsic space-time curvature • Adoption of adiabatic (lossless expansion): dE=-pdV – Fluid equation: • • r p "+ 3 (" + ) = 0 r c 2 • So we have an expression involving change in densities with velocity of the scale factor so we only need an expression for the acceleration/deceleration! (how this changes with time). From Lec 1: Adiabatic Expansion • If U self-contained it must expand without losing energy: (1st law of thermodynamics) dE = "pdV • Can use E=mc2 and rewrite with m= ρ(4πr3/3) where r is some physical radius for expanding region of density ρ. dE d(!4" r3#c 2 3) 4 d# dr dr = = "r3c 2 + #4"r2c 2 = $4"pr2 = $pdV dt dt 3 dt dt dt [Uses: Chain rule + d(x3)=3x2d ! dx • 4 • • • • Use dot notation: i.e., = x "r3c 2 #+ 4"r2c 2# r = $4"r2 pr 3 dt • • Rearrange to get the Fluid Equation: • r p "+ 3 (" + ) = 0 r c 2 ! ! ! Newtonian Analogy Consider, a location and a test particle: CP tells us: • Any point can be the centre • Any location the particle m Particle only feels forces from inside sphere as d gravity from shells outside R acting on m cancel (Newton’s Theorem) " Total energy given by U=T+V (potential+kinetic) 2 1 2 GMm 1 2 4!G"d m U = md! ! = md! ! 2 d 2 3 ! d = R(t)r1 = rr1 2 2 2 2 2 1 "d(rr )% 4!G"r r m 1 2 2 4!G"r r m U = 1 ! 1 = r! r ! 1 2#$ dt &' 3 2 1 3 Rearrange : 2 ( r!+ 8!G" 2U * - = + 2 2 ) r , 3 mr1 r Newton à GR From analogy to GR form: 2 2U Set: kc = ! 2 mr1 2 " r!% 8!G" kc2 $ ' = ! # r & 3 r2 !2U [k = 2 2 = curvature term in RW metric=0 if Universe flat] mc r1 Now differentiate this expression using the Fluid Eqn: 2 1 d r! d( 2 ) 8!G!! 2 ( r ) = ! kc r dt 3 dt LHS: 2 1 1 d(r! 2 ) 2 d( 2 ) 2 2 1 d(r! ) 2 2r!!r! 2r! r! r! r!r!! r! r = + r! r = ! = 2 [ ] dt r2 dt dt r2 r3 r r2 1 2 d( 2 ) r! r!r!! r! 8!G!! 2 2 [ ] = ! kc r r r2 3 dt Sub for ! from fluid equation: r! r!r!! r!2 8!G r!" p % 2kc2r! 2 [ ] = ! 3 $! + '+ r r2 3 r # c2 & r3 !r! r!2 " p % kc2 ! = !4!G$" + '+ r r2 # c2 & r2 Sub for: (r! )2 and rearrange to get: r !r! p 8!G" = !4!G" ! 4!G + r c2 3 !r! 4!G 3p = ! (" + ) r 3 c2 Deceleration equation Basic Friedmann Equations Curvature term k=0=FLAT 2 2 k=-1=OPEN Velocity of ! r!$ 8!G" kc k=+1=CLOSED Scale factor # & = ' " r % 3 r2 !r! 4!G 3p Deceleration of = ' (" + ) Scale factor r 3 c2 Contents acting via density and pressure Problem: Unless universe is empty it will decelerate, i.e., non-static We measure both matter and radiation to be positive. Einstein wanted a static (infinite) Universe Einstein’s Blunder? • Einstein introduced a constant density, ρv • How does constant density behave? dE = !pdV d(!R3 ) = !p / c2d(R3 ) If !v = constant p ! = ! v c2 Exerts -ve pressure!!?! 2 (NB : in equation of state: w!vc = p, so w=-1) • Physical motivations for Λ? – Quantum froth – particles flicking in and out of existence – Intrinsic curvature – intrinsic curvature of empty spacetime ? – Dark Energy – hypothetical energy which pervades spacetime Λ 2 ! r!$ 8!G(" + " + " ) kc2 # & = M R v ' " r % 3 r2 2 ! r!$ 8!G" kc2 8!G" # & = ' + v " r % 3 r2 3 Define: ( = 8!G"v 2 2 ! r!$ 8!G" kc ( The Friedmann # & = ' + Equation " r % 3 r2 3 Λ introduced to create a static universe by providing an acceleration to counter-balance the deceleration implied by matter and radiation. Λ !r! 4!G 3(p + p + p ) = ! (" + " + " + M R v ) r 3 M R v c2 2 But : pv = !"vc => !r! 4!G 3p = ! (" + ! 2" ) r 3 c2 v !r! 4!G 3p 8!G = ! (" + )+ " r 3 c2 3 v !r! 4!G 3p " = ! (" + )+ Friedmann Acceleration r 3 c2 3 Equation Λ introduced to create a static universe by providing an acceleration to counter-balance the deceleration implied by matter and radiation. Notation • Todays values carry a suffix, i.e., R(t) and R0 – i.e., ρ à ρo etc The Hubble Constant v ! r!$ ! r!$ • Hubble constant is: H0 = = # & , H(t)=# & d " r % " r % 0 • Hubble constant is only constant in space not time • Can rewrite Friedmann equation now with H: 2 2 8!G" kc " H = ! + 3 r2 3 Question? • Show for a flat Universe that the critical density can be given by: 3H 2 ! = 0 crit 8"G • In a flat Universe k=0, assuming Λ=0 2 ! r!$ 8!G" kc2 ( # & = ' + " r % 3 r2 3 2 8!G" H = crit o 3 3H 2 " = o ) 10'26 kg / m3 crit 8!G If "o > "crit k is +ve Universe is closed (will recollapse) If "o < "crit k is -ve Universe is open (will expand forever) " kc2 [i.e., 1' o = ' ] H 2r2 "crit 0 Expand forever or recollapse? Empty Open Flat 1/H0 Closed Size of Universeof Size Now Time With Λ Did we actually need GR? 2 ! r!$ 8!G" 2U ' # & = + 2 2 + Newtonian " r % 3 mr1 r 3 2 ! r!$ 8!G" kc2 ' # & = ( + GR " r % 3 r2 3 For a static Universe U=0, i.e., potential and kinetic energies are balanced. For a static flat Universe Newton = GR For any non-static Universe the extra term manifests as an energy imbalance in the Newtonian Universe but a curvature in the GR Universe which are fundamentally different. GR’s success at explaining precession of Mercury, occultation times, and gravitational lensing favours it but Newtonian models are still being explored. Conclusion: GR is required if either motion or curvature exist. =>Expansion demands GR or Newton interpretation. .