,-1)/2 Question 1 Define an the scale factor at which wavelength k equals the comoving Hubble radius, via an Hajj) = k. Given that keq = (eg H(Qeq) = Ho (2.Am Acq (50) express ay/acq in terms of k and keq. Also, show that in the limit k keq, this expression reduces to keq lim 2k kken den [9 marks) Question 2 Show that olaj) = :-(1 + y)-? (2 a Hy)-1 dy la Hy) where y = a/acq (12 marks] Cosmology Redshin: 1 1 (1) (1 Friedman equation: ? 8 G K (2) H = P- 3 a? (3) Fluid equation: P+ 3 H(p+P) = 0 Acceleration equation: 47 G r - - (+3P)a 3 (4) Acceleration parameter: (5) 90=- (to) 1 alto) H.2 Hubble distance and time: a (6) th=dh H Conformal time: (7) no 1 -dz' = H Dimensionless density: Pio 12= Per (8) Pressures: Pm= 0, P,= - Pr, PA=-PA 3 P. = 52Pa Densities: Pm ca-?, praa-4 PA = constant (10) Critical density: 3 Per = H2 (11) 87 G Comoving distance: dz (12) x= Jo (2) Angular diameter distance: 1 x aflat da = (13) (1 + z) Statistical physics: Distributions (14) 1 where - e(E-)/T + 1 is for "Bose-Einstein" and + is for "Fermi-Dirac" Number Density: (15) ni = Si Se o fica, (27)3 Equilibrium number density S 1 02 TOP nasilens Doson with 0 : 0) (#/30) for medless fermon with 10 = 0) 7 *** r = 4. p? 0,=* SM) (23) DIP 7 Energy momentum equation: 1 = 22.4.p.? (21) Temperature To (22 ) ) General Relativity: ,-1)/2 Question 1 Define an the scale factor at which wavelength k equals the comoving Hubble radius, via an Hajj) = k. Given that keq = (eg H(Qeq) = Ho (2.Am Acq (50) express ay/acq in terms of k and keq. Also, show that in the limit k keq, this expression reduces to keq lim 2k kken den [9 marks) Question 2 Show that olaj) = :-(1 + y)-? (2 a Hy)-1 dy la Hy) where y = a/acq (12 marks] Cosmology Redshin: 1 1 (1) (1 Friedman equation: ? 8 G K (2) H = P- 3 a? (3) Fluid equation: P+ 3 H(p+P) = 0 Acceleration equation: 47 G r - - (+3P)a 3 (4) Acceleration parameter: (5) 90=- (to) 1 alto) H.2 Hubble distance and time: a (6) th=dh H Conformal time: (7) no 1 -dz' = H Dimensionless density: Pio 12= Per (8) Pressures: Pm= 0, P,= - Pr, PA=-PA 3 P. = 52Pa Densities: Pm ca-?, praa-4 PA = constant (10) Critical density: 3 Per = H2 (11) 87 G Comoving distance: dz (12) x= Jo (2) Angular diameter distance: 1 x aflat da = (13) (1 + z) Statistical physics: Distributions (14) 1 where - e(E-)/T + 1 is for "Bose-Einstein" and + is for "Fermi-Dirac" Number Density: (15) ni = Si Se o fica, (27)3 Equilibrium number density S 1 02 TOP nasilens Doson with 0 : 0) (#/30) for medless fermon with 10 = 0) 7 *** r = 4. p? 0,=* SM) (23) DIP 7 Energy momentum equation: 1 = 22.4.p.? (21) Temperature To (22 ) ) General Relativity