Assignment 9: Big Bang Cosmology

For submission in Canvas
Canvas question may have different formats.

1. Concerning the largest scale, observed features of the universe:
(a) How is matter distributed?
(b) How does it move?
(c) How do we know it moves this way?

We can use Hubble's law to arrive at a crude estimate of the age of the universe. That is, we will calculate how long ago all the galaxies were crammed into our neighborhood of space. This time will be our estimate of how long ago the big bang happened. We will assume that each galaxy has moved at a constant speed for all time, although this speed will vary from galaxy to galaxy.

We will use the value of 20 km/sec per 1,000,000 light-years for Hubble's constant.

2. (a) If a galaxy is 1,000,000 light-years away from us now, according to Hubble's law, how fast is it receding from us?

(b) A galaxy traveling at 1 km/sec will travel one light-year in 300,000 years. How long does the galaxy of (a) require to travel a light-year?

3. Repeat the calculation of 2. for a galaxy now 2,000,000 light years distant from us.

4. Repeat the calculation of 2. for a galaxy now 3,000,000 light-years distant from us.

The final result of 2., 3., and 4. should be the same. At the time calculated, all the matter of universe would have been compressed into our neighborhood. This is our estimate of the age of the universe, often called the "Hubble age."

5. What is the origin of the 2.7K cosmic microwave background radiation?

For discussion in the recitation.

A. It may seem that Hubble's law conflicts with the basic supposition of Friedman Robertson Walker cosmology that the universe is homogeneous and isotropic in space. For Hubble's law tells us that everything is rushing away uniformly from our particular galaxy. Does not that make our galaxy some sort of special center of galactic motion, different from every other galaxy? The following calculations show that the galactic motions of Hubble's law look the same from every galaxy.

Consider (0) our galaxy and galaxies (I) 1,000,000 and (II) 2,000,000, and (III) 3,000,000 and (IV) 4,000,000 light years distant from us, all in the same direction. Compute the velocities of recession of the galaxies (I)-(IV) from us.

Now imagine that you are an observer located on galaxy (I). Recompute the velocities of recession of the other galaxies. Find that Hubble's law still holds. That means that the expansion looks the same to an observer on galaxy 1 as it does from our galaxy. It is not hard to see that the same result will hold for all observers, no matter which galaxy is their home.

(In computing these velocities, use the ordinary Newtonian rule for composing velocities.)

B. The cosmic microwave background consists of radiation so feeble that we have to cool our instruments with liquid helium before we are able even to detect it. How much can we learn about the cosmos from a source so feeble?

C. If the universe turns out to have an open geometry so that space is infinite, then all of our observations are showing us only the tiniest part of space. It is a finite fragment of an infinite expanse. Given that tiny sample, are we justified in asserting that the universe is spatially homogeneous--the same in every place? Or is this fundamental hypothesis of cosmology mere supposition?