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Physical Cosmology

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.

  1. How does tex2html_wrap_inline235 scale with R(t)? If the current density, i.e. tex2html_wrap_inline237 (today), is denoted tex2html_wrap_inline239 , and the current scale factor is denoted tex2html_wrap_inline241 , write tex2html_wrap_inline235 as a function of tex2html_wrap_inline239 , tex2html_wrap_inline241 , and R(t).
  2. 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).
  3. Use Hubble's law and the result of part 2 to find the age of the Universe, tex2html_wrap_inline249 , in terms of tex2html_wrap_inline221 . Let tex2html_wrap_inline253 . Express tex2html_wrap_inline249 in years.

Now (1) is valid only for a critical E = 0 Universe. A more general calculation, allowing for nonzero energy, would show




  1. Now consider a universe with no matter. Evaluate (2) to find R(t).
  2. What is the age of the Universe, tex2html_wrap_inline249 , as a function of tex2html_wrap_inline221 for this empty model? What is tex2html_wrap_inline249 in years if tex2html_wrap_inline253 ?
  3. 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:


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


  2. Use the defininitions of the acceleration parameter tex2html_wrap_inline265 , the matter density parameter tex2html_wrap_inline267 , and the vacuum energy density parameter tex2html_wrap_inline269 from your notes (you know, the ones that you took in class) with equation (4) to find tex2html_wrap_inline265 as a function of tex2html_wrap_inline267 and tex2html_wrap_inline269 .

There is considerable observational evidence and theoretical research pointing toward the notion that we live in a critical k = 0 Universe.

  1. If this is true, what is tex2html_wrap_inline277 ? Express tex2html_wrap_inline267 in terms of tex2html_wrap_inline269 .
  2. Let's say that some intrepid astronomer measures tex2html_wrap_inline283 . What are the values of tex2html_wrap_inline269 and tex2html_wrap_inline267 ? What are the implications to the expansion and fate of the Universe?

up previous
Up: No Title Previous: The Mass of the

Astronomy 7
Thu Nov 11 17:12:29 PST 1999