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ASTR 3740 Relativity & Cosmology Spring 1999. Problem Set 5. Due Tue 27 Apr

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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^-1Mpc^-1.

Here are links to values of physical constants. Possibly helpful numbers:
c = 299,792,458  m  s^-1,
(^h/_2p) = 1.054 572 66 ×10^-34 J s,
G = 6.672 59 ×10^-11 m³ kg^-1s^-2.
In 1976, the IAU (International Astronomical Union) defined the AU (Astronomical Unit), the mean Sun-Earth distance, to be 1 AU ș 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 ș 648,000/p 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, W = 0, for which t₀ = 1/H₀, and (ii) for a flat matter-dominated Universe, W = W_M = 1, 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 et al. (1998, Astrophysical Journal, 494, 96).

(b) Critical density

What is the critical density r_c 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 W_CMB today. The mass-energy density of photons in thermodynamic equilibrium at temperature T is r = a T⁴/c², where a = p² k⁴/(15 c³ (^h/_2p)³) = 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 W_M = 0.3?

(e) Omega in matter at Recombination

What was W in matter at Recombination? Assume that the Universe has been matter-dominated since Recombination, with W₀ = 0.3 in matter today, and that Recombination occurred at a redshift of 1 + z_R = 1300. [Hint: use the formula (W - 1) / W = 3 k c² / (8 p G r a²) you derived in Problem Set 4, with r ” 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 + z_R of Recombination, and then translate your answer into degrees for the case 1 + z_R » 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

3. Horizon Problem

(a) Expansion factor

The temperature of the CMB today is T₀ » 3  K. By what factor has the Universe expanded (i.e. what is a₀/a) since the temperature was the Planck temperature T » 10³²  K? [Hint: you already did this in Problem Set 4, Question 3c.]

(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 H² ” r, and for radiation-dominated Universe r ” a^-4.]

(c) Comoving Hubble distance

Hence determine by what factor the comoving Hubble distance x_H = 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 x_H = c/(aH) and cosmic scale factor a during inflation? [You should obtain an answer of the form x_H ” 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 x_H,0 now is less than the comoving Hubble distance x_H,i at the beginning of inflation. The `number of e-foldings' is ln(a_f / a_i), where ln is the natural logarithm, and a_i and a_f 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 4, Question 3a)