Calculating age of the universe using redshift?

[zeion]

Hello. This is one of my coursework questions I was wondering if I could get some insight here.. here is the question:

 

The size of the Universe if conveniently parameterized by a scale factor, a(t), which simply describes how big the Universe is at other times relative to its present size (ie. at the present we say that a is 1, and at some time in the past when the Universe was half as big as it was today, then a was 0.5). A matter-dominated Universe grows with time as [math]a \propto t^{\frac{2}{3}} [/math]. Assuming the Universe is 13.5 billion years old at present, how old is the Universe at redshifts, z, of z = 0.5 ... etc, z= 100? Assume that we presently live in a matter-dominated Universe, and that the Universe is matter-dominated out to redshifts of at least 100.

 

 

The formula for redshift relative to scale factor is [math] 1 + z = \frac{a_{now}}{a_{then}} [/math] Then, since [math]a \propto t^{\frac{2}{3}}[/math] then [math] 1 + z = t^{\frac{2}{3}}[/math]

Then I plug in z and solve for t, then divide the current age by t?

[modest]

The formula for redshift relative to scale factor is [math] 1 + z = \frac{a_{now}}{a_{then}} [/math] Then, since [math]a \propto t^{\frac{2}{3}}[/math] then [math] 1 + z = t^{\frac{2}{3}}[/math]

Then I plug in z and solve for t, then divide the current age by t?

 

More or less, yes. Consider t a ratio.

 

You know:

[math]\frac{a_0}{a} = \left( \frac{t_0}{t} \right)^{2/3}[/math]

you also know [math]z+1 = a_0/a[/math], sub:

[math]z+1 = \left( \frac{t_0}{t} \right)^{2/3}[/math]

solve for t:

[math]t = \frac{t_0}{(z+1)^{3/2}}[/math]

 

Check your results with Ned Wright's cosmology calculator where Lambda is set to zero:

Ned Wright's Javascript Cosmology Calculator

 

~modest

[zeion]

thx modest you've been an enormous help

[modest]

No trouble, mate.

 

Good luck with your course. You seem to be taking it seriously and doing well

 

~modest