Fall 2000 ASTR 1120-001 General Astronomy Problem Set 6 Fall 2000 ASTR 1120-001 General Astronomy: Stars & Galaxies.
Problem Set 6. Due Th 14 Dec

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Please show your working on this sheet, and write your answers on this sheet. Attach extra sheets if you need them.

1. Accelerating Universe

In January 1998 two competing teams rocked the cosmological world by reporting that the Hubble diagram of high redshift supernovae indicates that the Universe is accelerating, not decelerating as might be expected if gravity is attractive.

Suppose that the Universe contains only matter (M) energy and vacuum energy (a cosmological constant L), and that it is geometrically flat

WM + WL = 1
(1.1)
where WM = rM/rc and WL = rL/rc are the mass-energy densities of the Universe in matter and vacuum, relative to the critical density rc. How big must WL be for the Universe to be accelerating? [Hint: According to General Relativity, the condition for the Universe to be accelerating is that
r + 3 p < 0
 
where r = rM + rL is the total energy density, and p = pM + pL is the total pressure (momentum density). Ordinary matter has mass-energy density rM but essentially no pressure, pM = 0, while vacuum has negative pressure equal to its mass-energy density, pL = - rL.]

For the Universe to accelerate, WL should be less/greater (pick one) than ____________________ .

2. Critical density

(a) Critical density

The critical density rc of the Universe, the density which makes it geometrically flat, is

rc = 3  H02
8 pG
 
where H0 is the Hubble constant, and G is Newton's gravitational constant. What is the critical density if H0 = 65 km s-1 Mpc-1? [Hint: You may quote the answer to Problem Set 5, Question 3b, if you wish. 1 pc = 3.1 × 1013 km; G = 6.673 × 10-11 m3 kg-1 s-2.]

The critical density of the Universe is ____________________ kg/m3.

(b) How big is that?

Estimate approximately how much volume you personally would fill if you were spread out to critical density. If this volume were a cube, roughly how long would be the side of the cube? Compare this distance to that of some familiar object.

I would occupy a volume of about ____________________ m3.

This corresponds to a cube of side about ____________________ m.

This is about ____________________ times as big as ____________________  .

3. Recombination

The Cosmic Microwave Background is thought to have come to us from the epoch of Recombination, when hydrogen and helium in the Universe changed from being ionized to neutral atoms.

(a) Size of the Universe at Recombination

Wien's law states that the peak wavelength lpeak varies inversely with the temperature, lpeak µ 1/T. General relativity shows that the wavelength stretches in proportion to the cosmic scale factor, lpeak µ a. If the temperature of the Cosmic Background was 3,000 K at Recombination, and is now 2.73 K, by what factor has the Universe expanded since Recombination?

Since Recombination, the Universe has expanded by a factor of ____________________ .

(b) Age of the Universe at Recombination

The cosmic scale factor a varies with time t approximately as a µ t2/3. If the current age of the Universe is about 14 Gyr, approximately how old was the Universe at the time of Recombination? [Hint: You will need to use your answer from part (a). Note that a µ t2/3 is strictly true only for a matter-dominated Universe; the actual relation depends on what the Universe is made of.]

At Recombination, the Universe was ____________________ years old.

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On 28 Nov 2000, 02:15.