Ch-17 Homework

1 The galaxy RD1 has a redshift z=5.34. The light from this galaxy includes the Lyman-alpha spectral line of hydrogen, with an unshifted wavelength of 121.6 nm. Calculate the wavelength at which we detect the Lyman-alpha photons from RD1. In what part of the electromagnetic spectrum does this wavelength lie?

2 Calculate the mass density of radiation in each of the following situations, and explain whether each situation is matter-dominated or radiation-dominated:
(a) the photosphere of the Sun (T=5800 K, ρ_m = 3x10^-4 kg/m³);
(b) the center of the Sun (T=1.55x10⁷ K, ρ_m = 1.6x10⁵ kg/m³);
(c) the solar corona (T=2x10⁶ K, ρ_m = 5x10^-13 kg/m³).

3
Consider the galaxy RD1 again, which has z=5.34.
(a) Suppose that in the present-day universe, two clusters of galaxies are 500 Mpc apart. At the time that the light was emitted from RD1 to produce an image tonight, how far apart were those two clusters?
(b) What was the average density of matter (ρ_m) at that time? Assume that in today's universe, ρ_m =2x10^-27 kg/m³.
(c) What were the temperatures of the cosmic background radiation and the mass density of radiation ( ρ_rad) at that time?
(d) At this time in the remote past, was the universe matter-dominated, radiation-dominated, or dark-energy-dominated? Explain.

4 Suppose we lived in a flat universe that had q₀=0. What would be the value of the dark energy density parameter in such a universe? Which would be dominant in such a universe, matter or dark energy? Explain.

5 In general, the deceleration parameter is not constant but varies with time. For a flat universe, the deceleration parameter at a redshift z is given by the formula
q_z = (1/2) - (3/2)Ω_Λ / [Ω_Λ + (1-Ω_Λ)(1+z)³]
where Ω_Λ is the dark energy density parameter. Using Ω_Λ = 0.7, find the value of q_z for
(a) z=0.5 and
(b) z=1.0.
(c) Explain how your results show that the expansion of the universe was actually decelerating at z=1.0, but changed from deceleration to acceleration between z=1.0 and z=0.5.

6 The dark energy density parameter Ω_Λ is related to the value of the cosmological constant Λ by the formula
Ω_Λ = (Λc²)/(3H₀²)
where c=3x10⁸m/s is the speed of light. Determine the value of Λ if Ω_Λ =0.7 and the Hubble constant is H₀=70 km/s/Mpc. (Hint: You will need to convert units to eliminate kilometers and megaparsecs.)

1	The galaxy RD1 has a redshift z=5.34. The light from this galaxy includes the Lyman-alpha spectral line of hydrogen, with an unshifted wavelength of 121.6 nm. Calculate the wavelength at which we detect the Lyman-alpha photons from RD1. In what part of the electromagnetic spectrum does this wavelength lie?
2	Calculate the mass density of radiation in each of the following situations, and explain whether each situation is matter-dominated or radiation-dominated: (a) the photosphere of the Sun (T=5800 K, ρ_m = 3x10^-4 kg/m³); (b) the center of the Sun (T=1.55x10⁷ K, ρ_m = 1.6x10⁵ kg/m³); (c) the solar corona (T=2x10⁶ K, ρ_m = 5x10^-13 kg/m³).
3	Consider the galaxy RD1 again, which has z=5.34. (a) Suppose that in the present-day universe, two clusters of galaxies are 500 Mpc apart. At the time that the light was emitted from RD1 to produce an image tonight, how far apart were those two clusters? (b) What was the average density of matter (ρ_m) at that time? Assume that in today's universe, ρ_m =2x10^-27 kg/m³. (c) What were the temperatures of the cosmic background radiation and the mass density of radiation ( ρ_rad) at that time? (d) At this time in the remote past, was the universe matter-dominated, radiation-dominated, or dark-energy-dominated? Explain.
4	Suppose we lived in a flat universe that had q₀=0. What would be the value of the dark energy density parameter in such a universe? Which would be dominant in such a universe, matter or dark energy? Explain.
5	In general, the deceleration parameter is not constant but varies with time. For a flat universe, the deceleration parameter at a redshift z is given by the formula q_z = (1/2) - (3/2)Ω_Λ / [Ω_Λ + (1-Ω_Λ)(1+z)³] where Ω_Λ is the dark energy density parameter. Using Ω_Λ = 0.7, find the value of q_z for (a) z=0.5 and (b) z=1.0. (c) Explain how your results show that the expansion of the universe was actually decelerating at z=1.0, but changed from deceleration to acceleration between z=1.0 and z=0.5.
6	The dark energy density parameter Ω_Λ is related to the value of the cosmological constant Λ by the formula Ω_Λ = (Λc²)/(3H₀²) where c=3x10⁸m/s is the speed of light. Determine the value of Λ if Ω_Λ =0.7 and the Hubble constant is H₀=70 km/s/Mpc. (Hint: You will need to convert units to eliminate kilometers and megaparsecs.)