Consider the epoch of radiation domination where the energy density is dominated by photons with ρ = aT4 (where a is the radiation constant and T is their temperature) and the equation of state is P = ρ/3.
Justify any assumptions you need to make.
- a) How is the scale factor related to T?
- b) How does the scale factor R(t) vary with time during this epoch?
- c) Write the equivalent expression for T(t).
Note: depending on how you approach the problem, it may be easier to reverse the order of (b) and (c). - d) Verify the consistency of the Friedmann equation and the expression for the second time derivative of R.
- a) Assume the early universe is filled with stuff ρ which obeys
the equation of state p = -ρ.
What is ρ(t)?
[Hint: use only the equation of state and conservation of mass-energy.] - b) Derive R(t).
Write H in terms of ρ and Λ. (Note: do not equate these two.)
What do you have to assume to do this? How might you justify this assumption? - c) Why does Inflation predict Ωk = 0?
(or equivalently, Ωm + ΩΛ = 1.)
- a) Write the Friedmann equation in terms of Ωs and the redshift z and its time derivative. Ignore the curvature and cosmological constant terms to obtain an expression for t(z), the age of the universe at redshift z. [This is sometimes called the Einsten-de Sitter limit.]
- b) The leading term of the expression from (a)
involving H0 and Ωm0
can be thought of as the "age" of the universe: t(z=0).
Is this a valid approximation, given the assumption made in (a)?
If not, at what redshift might it become valid?
[Hint: consider curvature and Λ terms as separate cases.] - c) It is fairly straightforward to age-date young stellar populations.
Suppose a galaxy observed at z=3 has spectral characteristics which indicate it has an age of 2 Gyr. What age does this imply for the universe now?
Professor Fink returns excited from an observing run because he has discovered a nova in a galaxy in the Virgo cluster, a key step in the distance scale. He knows that novae can be used as standard candles since they obey a luminosity--fade-time relation
MVpeak = -10.7 + 2.3 log(t2)
where t2 is the time in days that it takes for a nova to fade 2 magnitudes from its peak. (That's a base ten logarithm, as is conventional in astronomy.) Professor Fink hands you the data in the graph, and asks you to determine The Answer.
- a) What is the fade time t2? (Eyeball it.)
- b) What is the distance to Virgo from this?
- c) Supposing that Virgo has a recession velocity of 1400 km/s, what is H0?
- d) Fink assumes there is no extinction. How does the distance change if there is a little extinction, AV = 0.3? How does H0 change?
- e) Assume for the moment that AV = 0. What is the uncertainty in the distance determination?
- f) Not all novae are the same; there is some intrinsic dispersion in the luminosity--fade-time relation, about 0.5 mag. Galactic extinction maps suggest that the extinction is AV = 0.3 +/- 0.02. What now is the distance and its uncertainty? and H0?
(Keep in mind that the modern goal is to measure H0 to better than 10%.)
Light curve of nova in the Virgo cluster.
Use the data of Verheijen 2001, ApJ, 563, 694 to determine the distance to the UMa cluster of galaxies using the I-band and K-band calibration of the HST Key Project (Sakai et al. 2000, ApJ, 529, 698). Be sure to include a plot of the data with calibration lines overlaid.
Assuming the recession velocity of the Ursa Major cluster is 1088 km/s, what Hubble constant(s) do you find? How does this compare to the value given by Sakai et al.?
* Note that Verheijen gives absolute magnitudes for a presumed distance of 15.5 Mpc. Also note that ascii versions of the tables, convenient for plotting, can be downloaded from the journal website.
** For the K'-band data, use the H-band calibrations assuming H-K = 0.25.