ASTR 3740 Problem Set 7

ASTR 3740 Spring 2004 Homepage

ASTR 3740 Relativity & Cosmology Spring 2004. Problem Set 7. Due Fri 28 Apr

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1. 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 ť 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

(1.1)

The angle, in radians, subtended by the horizon at Recombination is the ratio of the comoving horizon size x_R at Recombination to the comoving horizon size x₀ now:

Angle =

x_R

x₀

(1.2)

(the formula is this simple because the geometry is flat). Recall that H = (da/dt)/a, and from Problem Set 5 that in a flat, matter-dominated Universe

a ľ t^2/3 .

(1.3)

The redshift at recombination is

1 + z_R =

a₀

a_R

(1.4)

Remember that there are 180 degrees in p radians.]

2. Horizon Problem

(a) Expansion factor

The temperature of the CMB today is T₀ ť 3 K. By what factor has the Universe expanded (i.e. what is a₀/a) since the temperature was the Planck temperature T ť 10³² K? [Hint: you already did this in Problem Set 5, 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 ľ r, and for radiation-dominated Universe r ľ 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 ľ 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.]

3. Relation between Horizon and Flatness Problems

Show that Friedmann's equation can be written in the form (compare Problem Set 6, Question 3a)

W - 1 = k x_H²

(2.1)

where x_H ş c / (a H) 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.]