Once again, this problem set may take you some time to complete, so please do not wait until the last day to start it.
1. Cosmological Numbers
The purpose of this question is to allow you to figure out for yourself the values of some basic numbers in cosmology. You should find all the necessary formulae in the notes; part of the question is to figure out what formula to use.
Assume where necessary that the Hubble constant is H0 = 65 km s-1 Mpc-1.
The ASTR 3740 homepage has links to pages where you can look up physical constants. Possibly helpful numbers: c = 299,792,458 m s-1, = 1.054 572 66 × 10-34 J s, G = 6.672 59 × 10-11 m3 kg-1 s-2. In 1976, the IAU (International Astronomical Union) defined the AU (Astronomical Unit), the mean Sun-Earth distance, to be 1 AU = 149,597,870 km. The pc (parsec) is defined to be the distance at which 1 AU subtends one second of arc, whence 1 pc = 648,000/pi AU = 3.085 677 57 × 1013 km. 1 yr = 31,556,930 s. The prefix M on a unit, as in Mpc (megaparsecs) denotes mega = 106. The prefix G on a unit, as in Gyr (gigayears) denotes giga = 109.
(a) Age of the Universe
What is the age of the Universe (i) for a wide open Universe, , for which t0 = 1 / H0, and (ii) for a flat matter-dominated Universe, , for which t0 = 2 / (3 H0)? Compare your answer to the age t0 = 11.5±1.3 Gyr of globular cluster stars determined post-Hipparcos by Chaboyer (1977).
(b) Critical density
What is the critical density of the Universe? Express your answer in kg m-3. [Just for fun (no credit): To bring this number down to Earth, estimate (order of magnitude) how much volume your body would occupy if it were spread out to the critical density; compare that volume to something familiar.]
(c) Omega in the CMB
Figure out today. The mass-energy density of photons in thermodynamic equilibrium at temperature T is = a T4 / c2, where a = pi2 k4 / (15 c3 hbar3) = 7.56591 × 10-16 J m-3 K-4 is the radiation density constant.
(d) Redshift of matter-radiation equality
At what redshift were the densities of CMB radiation and matter equal, if Omega in matter today is ?
(e) Omega in matter at Recombination
What was in matter at Recombination? Assume that the Universe has been matter-dominated since Recombination, with in matter today, and that Recombination occurred at a redshift of 1 + zR = 1300. [Hint: use the formula you derived in Problem Set 3, with . Recall that 1 + z = a0 / a.]
2. Horizon at Recombination
What angle does the horizon at Recombination subtend on the CMB today?
Assume a flat, matter-dominated Universe.
Express your answer first in terms of the redshift factor
1 + zR of Recombination,
and then translate your answer into degrees for the case
1 + zR = 1300.
In a flat, matter-dominated Universe, the comoving distance to the
horizon at a time when the cosmic scale factor is a
and the Hubble parameter is H is
3. Horizon Problem
(a) Expansion factor
The temperature of the CMB today is T0 = 3 K approximately. By what factor has the Universe expanded (i.e. what is a0 / a) since the temperature was the Planck temperature T = 1032K approximately? [Hint: how does temperature T go with cosmic scale factor a?]
(b) Hubble distance
By what factor has the Hubble distance c / H increased during the expansion of part (a)? Assume that the Universe has been mainly radiation-dominated during this period, and that the Universe is flat. [Hint: For a flat Universe , and for radiation-dominated Universe .]
(c) Comoving Hubble distance
Hence determine by what factor the comoving Hubble distance xH = c / (aH) has increased during the expansion of part (a).
(d) Comoving Hubble distance during inflation
During inflation the Hubble distance c / H remained constant, while the cosmic scale factor a expanded exponentially. What is the relation between the comoving Hubble distance xH = c / (aH) and cosmic scale factor a during inflation? [You should obtain an answer of the form .]
(e) e-foldings to solve the Horizon Problem
By how many e-foldings must the Universe have inflated in order to solve the Horizon Problem? Assume again, as in part (a), that the Universe has been mainly radiation-dominated during expansion from the Planck temperature to the current temperature, and that this radiation-dominated epoch was immediately preceded by a period of inflation. [Hint: Inflation solves the Horizon Problem if the currently observable Universe was within the Hubble distance at the beginning of inflation, i.e. if the comoving xH,0 now is less than the comoving Hubble distance xH,i at the beginning of inflation. The `number of e-foldings' is ln(af / ai) where ln is the natural logarithm, and ai and af are the cosmic scale factors at the beginning (i for initial) and end (f for final) of inflation.]
4. Relation between Horizon and Flatness Problems
Show that Friedmann's equation can be written in the form
(compare Problem Set 3, Question 3a)