Reading: Chapter 29 and 30 of Carroll & Ostlie
Section 8.3 of Sparke & Gallagher
The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. [8 pts] A galaxy is observed with a redshift of 2.34. How old was the universe at the time this galaxy emitted the light received today if the universe is closed with qo = 1, Lambda=0? Relative to the present size, what was the size of the universe when the light was emitted?
2. [8 pts] A galaxy is observed with a redshift of 1.078. How old was the universe at the time this galaxy emitted the light received today if the universe is open with qo = 0.25, Lambda=0? Relative to the present size, what was the size of the universe when the light was emitted?
3.
[8 pts]
Some quantities obey an exponential time-behavior of the form f(t) = fo exp(t/tau) where tau is the characteristic time for the system under
consideration.
(a) Show that tau = [(1/f) (df/dt)]-1. This expression can be
used to define a characteristic time for any function, regardless of whether
its behavior is exponential.
(b) Use the scale factor, R(t), to show that the characteristic time for
the expansion of the universe is tauexpan(t) = 1/H(t).
4. [8 pts] Using the Einstein-de Sitter model, estimate the epoch at which the matter and radiation densities were equal. For this calculation, take rhoo = 10-29 g cm-3 and uo = 10-13 erg cm-3 and express your answer as the fraction of the age of the universe.
5.
[9 pts]
Consider a cosmological model with the following properties:
lambda=P=0 (pressure free Friedmann model)
qo = 1.5
Ho = 75 km/sec Mpc-1
(a) Calculate the present density (rhoo) in gm/cm3.
(b) Calculate the present age of the universe (to) in years.
(c) What is the maximum age of this model universe, if any?
6.
[9 pts]
Consider a cosmological model with the following properties:
lambda=P=0 (pressure free Friedmann model)
qo = 0.3
Ho = 75 km/sec Mpc-1
(a) Calculate the present density (rhoo) in g/cm3.
(b) Calculate the present age of the universe (to) in years.
(c) What is the maximum age of this model universe, if any?