The Friedmann Equation R = GM R 2. R(t) R R = GM R GM R. d dt. = d dt 1 2 R 2 = GM R + K. Kinetic + potential energy per unit mass = constant

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1 The Friedmann Equation R = GM R R R = GM R R R(t) d dt 1 R = d dt GM R M 1 R = GM R + K Kinetic + potential energy per unit mass = constant

2 The Friedmann Equation 1 R = GM R + K M = ρ 4 3 π R3 1 R = 4πGρR 3 + K R R = 8πGρ + K 3 R a( t) = R( t) ( ) = R R 0 R t = 0 a a = 8πGρ + K 3 R 0 a!a =! R R 0

3 The Friedmann Equation a a = 8πG 3 ρ( t) + K R 0 1 Newtonian form of a( t) Friedman equation General Relativity: Replace density with energy density ( ) ε ( t) ρ t c Constant of integration is curvature of spacetime K R 0 kc R 0 k = E = ( m c 4 + p c ) 1 1 negative curvature 0 flat space +1 positive curvature

4 The Friedmann Equation k=0 k=+1 k=-1

5 The Friedmann Equation a a = 8πG 3c ε ( t) kc R 0 1 a a( t) a H ( t) H ( t) = 8πG 3c ε ( t) If k=0 ε c ( t) = 3c 8πG H ( t) Critical density ε c,0 c = h M Mpc -3

6 The Friedmann Equation Ω t ( ) = ε ( t) ( t) ε c H ( t) = H ( t) Ω( t) kc R 0 1 Ω( t) = kc R 0 Energy density in units of critical density 1 H ( t) a( t) 1 a( t) Ω( t) < 1 k = 1 = 1 k = 0 > 1 k = +1 Sign of 1-Ω does not change as universe expands.

7 The Friedmann Equation At the present epoch: H ( 0 1 Ω ) 0 = kc R 0 Replace curvature constant in Friedman equation: 1 Ω t ( ) ( ) = H 0 1 Ω 0 H ( t) a( t) H ( t) H 0 1 Ω( t) ( ) = 1 Ω 0 a t ( )

8 The Fluid Equation dq = de + PdV 1 st law of thermodynamics E + P V = 0 εv + ε V + P V = 0 V ε + 3 a a ε + P ( ) ε + 3 a a ( ε + P) = 0 = 0 E( t) = ε ( t)v ( t) V ( t) = 4 3 π R( t)3 V = 4 3 π 3R R V = V 3 R R = V 3 a a

9 The Acceleration Equation a a = 8πG 3c ε ( t) kc R 0 1 a( t) a = 8πG 3c ε ( t)a( t) kc R 0 a a = 8πG 3c a a = 4πG 3c ( εa + εa a ) ε a a + ε ε + 3 a a ( ε + P) = 0 ε a a = 3( ε + P)

10 The Acceleration Equation a a = 4πG 3c a a = 4πG 3c ( 3( ε + P) + ε) ( ε + 3P) The energy density is always positive If Pressure is positive then the universe must decelerate. (e.g., baryonic gas, photons, dark matter) If Pressure is negative, the universe can accelerate. (e.g., dark energy)

11 The Equations of Motion of the Universe a a = 8πG 3c ε ( t) kc R 0 1 a( t) Friedmann Equation a a = 4πG 3c ( ε + 3P) Acceleration Equation Two equations and three unknowns: a( t), ε ( t), P( t) Need a third equation: Equation of state P = P( ε) P = wε

12 The Equation of State P = wε Non-relativistic particles: matter (Ideal gas law) P = ρ kt kt = µ µc ε w = kt µc 0 Relativistic particles: radiation P = 1 3 ε w = 1 3 Mildly relativistic particles: 0 < w < 1 3

13 Cosmological Constant / Dark Energy 1915 Einstein s GR equations predict a dynamic universe But Einstein thought the Universe was static, so he introduced the Cosmological Constant, Λ, to his equations of motion. 199 When Hubble discovered the expansion of the universe, Einstein called Λ his greatest blunder SN results show that the universe is accelerating in its expansion so scientists revive Λ The cosmological constant or Dark Energy is thought to be the energy of a vacuum, predicted by quantum mechanics.

14 Cosmological Constant / Dark Energy a a = 8πG 3c ε ( t) kc R 0 1 a( t) + Λ 3 Friedman Equation a a = 4πG 3c ( ε + 3P) + Λ 3 Acceleration Equation ε + 3 a a ( ε + P) = 0 Fluid Equation: If the energy density of Dark Energy is constant with time ε = 3 a ( a ε + P ) = 0 P = ε w = 1

15 Cosmological Constant / Dark Energy a a = 8πG 3c ε ( t) kc R 0 1 a( t) + Λ 3 a a = 8πG 3c ε ( t) + 8πG 3c c Λ 8πG kc R 0 1 a( t) a a = 8πG 3c ( ε m + ε ) Λ kc R 0 1 a( t) H ( t) ( 1 Ω m Ω ) Λ = kc R 0 1 a( t)

16 Cosmological Parameters Cosmological parameter #4: the matter density Ω m Ω m 0.8 ± 0.01

17 Cosmological Parameters Cosmological parameter #5: the baryon density Ω b Ω b ± 0.001

18 Cosmological Parameters Cosmological parameter #6: the dark energy density Ω Λ Ω Λ 0.7 ± 0.01

19 Cosmological Parameters Cosmological parameter #7: the radiation density Ω γ Ω ν Ω γ Ω ν

20 Cosmological Parameters

21 The Evolution of Energy Density ε + 3 a a ( ε + P) = 0 P = wε ε + 3 a a ( 1 + w)ε = 0 dε dt = da dt 3 a 1 + w ( )ε dε ε = da a w ( ) If w is constant with a: lnε = lnε 0 3( 1+ w)ln a 3 1+w ε = ε 0 a ( )

22 The Evolution of Energy Density ε = ε a 3 ( 1+w ) 0 Non-relativistic particles (baryons, dark matter) w = 0 ε = ε 0 a 3 = ε 0 ( 1+ z) 3 Relativistic particles (photons, neutrinos) w = 1 3 ε = ε 0 a 4 = ε 0 ( 1+ z) 4 Dark energy w = 1 ε = ε 0

23 The Friedmann Equation H ( t) H 0 = H ( t) H 0 Ω t ( ) ( ) + 1 Ω 0 a( t) ( ) H H = ε + 1 Ω 0 0 ε c,0 a H H 0 ε c = 3c 8πG H ε = ε c ε = ε ε c ε c,0 ε c ε c,0 ε = ε m + ε r + ε Λ = ε m,0 a 3 + ε r,0 a 4 + ε Λ,0

24 The Friedmann Equation ( ) H H = 1 ε m,0 + ε r,0 0 ε c,0 a 3 a + ε 4 Λ,0 + 1 Ω 0 a H H = Ω m,0 + Ω r,0 + Ω 0 a 3 a 4 Λ,0 + 1 Ω 0 a Ω 0 = Ω m,0 + Ω r,0 + Ω Λ,0 1 Ω 0 = Ω k,0 Ω m,0 + Ω r,0 + Ω Λ,0 + Ω k,0 = 1

25 Cosmological Parameters Cosmological parameter #8: the spatial curvature Ω k Ω k = 0.00 ± 0.004

26 Cosmological Parameters Cosmological parameter #9: the equation of state of dark energy w w 1.04 ± 0.07

27 The Friedmann Equation H ( ) H = Ω r,0 + Ω m,0 + Ω k,0 + Ω Λ,0 0 a 4 a 3 a a 3 1+w Ω m Ω r ( z) = ε m = ε m,0 = ε m,0 ε c,0 = Ω m,0 ε c a 3 ε c a 3 ε c,0 ε c ( z) = Ω r,0 H 0 a 4 H H 0 a 3 H Ω Λ ( z) = Ω Λ,0 H 0 ( ) a 3 1+w H

28 The Friedmann Equation

29 The Friedmann Equation H ( ) H = Ω r,0 + Ω m,0 + Ω k,0 + Ω Λ,0 0 a 4 a 3 a a 3 1+w Solving this equation gives expansion history a(t) (which also specifies the age of the universe t 0 )

30 Solving the Friedmann Equation Special case #1: empty universe Ω k,0 = 1 H ( ) H = Ω r,0 + Ω m,0 + Ω k,0 + Ω Λ,0 0 a 4 a 3 a a 3 1+w H H 0 = 1 a 1 H = H 0 a a = H d a 0 = H 0 d t a 0 t 0 a( t) = H t 0 t 0 = 1 H 0

31 Solving the Friedmann Equation Special case #: radiation dominated universe Ω r,0 = 1 H ( ) H = Ω r,0 + Ω m,0 + Ω k,0 + Ω Λ,0 0 a 4 a 3 a a 3 1+w H H 0 = 1 a H = H 0 a = H a 0 a a d a a = H 0 d t 0 t 0 a( t) = ( H t) 1 0 t 0 = 1 H 0

32 Solving the Friedmann Equation Special case #3: matter dominated universe Ω m,0 = 1 H ( ) H = Ω r,0 + Ω m,0 + Ω k,0 + Ω Λ,0 0 a 4 a 3 a a 3 1+w H H 0 = 1 a 3 H = H 0 1 a 3 a = H 0 a 1 a a d a = H 0 d t 0 t 0 a( t) = 3 H 0 t 3 t 0 = 3H 0

33 Solving the Friedmann Equation Special case #4: dark energy dominated universe Ω Λ,0 = 1 H H 0 = Ω r,0 a 4 + Ω m,0 a 3 + Ω k,0 a + Ω Λ,0 H H 0 = H = H a = H 0 0a d a = H 0 d t a 1 a t 0 t a( t) = e H 0 ( t t 0 ) t 0 =

34 Solving the Friedmann Equation

35 Solving the Friedmann Equation Knop et al. (003)

36 Distance Measures in Cosmology Hubble time: Time it took the universe to reach its present size if expansion rate has always been the same. t H 1 H 0 = 9.78h 1 Gyr Hubble distance: Distance light travels in a Hubble time. D H c H 0 = 3000h 1 Mpc Proper distance: Distance measured in rulers between two points. D P

37 Distance Measures in Cosmology Proper distance between point at a and a+da: dd P = c da a = c da a a a = c da ah = c da H 0 a H H 0 1 dz = D H ( 1 + z)e( z) d z Integrating: D P = D H ( 1 + z )E z z 0 ( )

38 The Friedmann Equation H ( ) H = Ω r,0 + Ω m,0 + Ω k,0 + Ω Λ,0 0 a 4 a 3 a a 3 1+w 0 H = Ω ( m,0 1+ z) 3 + Ω ( k,0 1+ z) + Ω ( Λ,0 1 + z) 3( 1+w) = E( z) H 0

39 Distance Measures in Cosmology Comoving distance: Distance between points that remains constant if both points move with the hubble flow. D C D P a = D P z 0 d z D C = D H E z ( ) ( 1 + z)

40 Distance Measures in Cosmology Comoving distance (line-of-sight) D C z 0 d z D C = D H E z ( ) Comoving distance (transverse) D M = 1 D H sinh( Ω k D C D ) H for Ω k > 0 Ω k D C for Ω k = 0 1 D H sin( Ω k D C D ) H for Ω k < 0 Ω k D M

41 Distance Measures in Cosmology Angular diameter distance: Ratio of object s physical size to angular size. D A = D M 1+ z ( ) θ R D A = R θ D A

42 Distance Measures in Cosmology Luminosity distance: Distance that defines the relationship between luminosity and flux. D L = L 4πF D L = ( 1 + z) D M = ( 1 + z) D A

43 Distance Measures in Cosmology Comoving volume: Volume in which densities of non-evolving objects are constant with redshift. dv C = D H ( 1 + z) D A E( z) dωdz V C = 4π 3 D 3 M for Ω k = 0

44 Distance Measures in Cosmology Lookback time: Difference between age of universe now and age of universe at the time photons were emitted from object. z 0 d z t L = t H ( 1+ z )E z ( ) Age of the universe at redshift z: t 0 t ( L z) t L t = 0 t ( z) t 0

45 Constraining Cosmological Parameters Dunkley et al. (009)

46 Constraining Cosmological Parameters Ω k > 0 Ω k = 0 Ω k < 0

47 Constraining Cosmological Parameters D A = λ θ = l π λ Measurement Physics Cosmological parameters D A = H 0 c ( 1+ z) z 0 d z E ( z ) E( z) = Ω ( m,0 1 + z) 3 + Ω ( k,0 1 + z) + Ω ( Λ,0 1 + z) 3( 1+w)

48 Constraining Cosmological Parameters Dunkley et al. (009)

49 Constraining Cosmological Parameters Komatsu et al. (009)

50 Constraining Cosmological Parameters w = w 0 + w a a Hinshaw et al. (013)

51 Constraining Cosmological Parameters WMAP SPT ACT Hinshaw et al. (013)