ASTR 3740 Relativity & Cosmology Spring 1998. Problem Set 4.
Due Tue 28 Apr

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 = 65kms-1Mpc-1.

The ASTR 3740 homepage has links to pages where you can look up physical constants. Possibly helpful numbers: c=299,792,458ms-1, hbar=1.05457266נ10-34Js, G=6.67259נ10-11 m3kg-1s-2. In 1976, the IAU (International Astronomical Union) defined the AU (Astronomical Unit), the mean Sun-Earth distance, to be 1AU=149,597,870km. The pc (parsec) is defined to be the distance at which 1AU subtends one second of arc, whence 1pc=648,000/piAU = 3.08567757נ1013km. 1yr=31,556,930s. 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, Omega = 0, for which t0=1/H0, and (ii) for a flat matter-dominated Universe, Omega = Omega_M = 1, for which t0=2/(3H0)? Compare your answer to the age t0=11.51.3Gyr of globular cluster stars determined post-Hipparcos by Chaboyer (1977).

(b) Critical density

What is the critical density $\rho_c$ of the Universe? Express your answer in kgm-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 Omega_CMB today. The mass-energy density of photons in thermodynamic equilibrium at temperature T is rho =aT4/c2, where a=pi2k4/(15c3hbar3) = 7.56591נ10-16Jm-3K-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 Omega_M = 0.3?

(e) Omega in matter at Recombination

What was Omega in matter at Recombination? Assume that the Universe has been matter-dominated since Recombination, with Omega_0 = 0.3 in matter today, and that Recombination occurred at a redshift of 1 + zR=1300. [Hint: use the formula (Omega - 1) / \Omega = 3 kappa c^2 / (8 pi G rho a^2) you derived in Problem Set 3, with rho proportional to a^{-3}. 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. [Hint: 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
x = 2 c /(a H) (2.1)
The angle, in radians, subtended by the horizon at Recombination is the ratio of the comoving horizon size xR at Recombination to the comoving horizon size x0 now:
Angle = x_R / x_0 (2.2)
(the formula is this simple because the geometry is flat). Recall that H = \dot a/a, and from Problem Set 3 that in a flat, matter-dominated Universe
a proportional to t^{2/3} . (2.3)
The redshift at recombination is
1 + z_R = a_0 / a_R . (2.4)
Remember that there are 180 degrees in pi radians.]

3. Horizon Problem

(a) Expansion factor

The temperature of the CMB today is T0=3K 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 H^2 proportional to rho, and for radiation-dominated Universe rho proportional to a^{-4}.]

(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 x_H proportional to a^?.]

(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)
Omega - 1 = kappa (x_H)^2 (3.1)
where x=c/(aH) is the comoving Hubble distance. Use this equation to argue in your own words how the horizon and flatness problems are related. [The main part of this question is not the math but the explanation. You should convince the grader that you understand physically what is going on.]

Andrew Hamilton