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In class we learned that for a critical Universe (i.e. one with exactly E = 0)
where R(t) is a scale factor indicative of the size of the Universe at time t.

How does scale with R(t)? If the current density, i.e. (today), is denoted , and the current scale factor is denoted , write as a function of , , and R(t).

Plug this result into (1). Rearrange the equation into the form
(function of R)dR = (function of t)dt
and integrate to get R(t).

Use Hubble's law and the result of part 2 to find the age of the Universe, , in terms of . Let . Express in years.
Now (1) is valid only for a critical E = 0 Universe. A more general calculation, allowing for nonzero energy, would show
where

Now consider a universe with no matter. Evaluate (2) to find R(t).

What is the age of the Universe, , as a function of for this empty model? What is in years if ?

Given that the real Universe does contain goats and other matter, what can you say about the real age of the Universe relative to what you found in part 5?
We can make (2) even more general by including a term with Einstein's cosmological constant:

Plug your answer for part 1 into (3). take the derivative with respect to time of both sides of the equation to show

Use the defininitions of the acceleration parameter , the matter density parameter , and the vacuum energy density parameter from your notes (you know, the ones that you took in class) with equation (4) to find as a function of and .
There is considerable observational evidence and theoretical research pointing toward the notion that we live in a critical k = 0 Universe.

If this is true, what is ? Express in terms of .

Let's say that some intrepid astronomer measures . What are the values of and ? What are the implications to the expansion and fate of the Universe?
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Astronomy 7
Thu Nov 11 17:12:29 PST 1999