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In Cosmology Equations Part 4 we looked at the famous Friedmann equation, for which some applications will be discussed in this installment. First the equation again:
where a is universal expansion factor, a-dot = da/dt the expansion rate, H0-bar the normalized Hubble constant, Ω and its Ωi components are the universal energy density parameters, referring to: total energy density, mass density, radiation density and vacuum energy density.
The first exercise is to plot the expansion factor against time. For this we must integrate the expansion rate a-dot against time. Sadly , this is not possible analytically. Happily , we can do it numerically! Engineers do this a lot when they simulate complex dynamic systems. So, without wasting more time, let's plot the expansion curve (a link to the spreadsheet responsible for this will be given later).
The values used are: Ωm = 0.27, Ωv = 0.73, Ωr = 0, H0 = 70 Km/s/Mpc. The vertical blue line at 13.7 Gy represents "now", with values to the left of it representing the past and values to the right of it the future (which is only shown so that the increasing expansion rate becomes more visible).
The horizontal red line at redshift 1.6 illustrates one of the useful applications of this graph: the length of the line, between "now" and the redshift curve, represents the look-back time to objects observed at redshift 1.6, which in this case is 9.7 Gy. It means that the light from a galaxy that we today observed at redshift 1.6, took 9.7 billion years to reach us.
Looking more closely at the blue expansion curve, note that during the first 5 Gy or so, the expansion rate slowed down under the dominance of matter density (Ωm). Radiation density is ignored, because it was only the first thousand years or so that it played a role and that's insignificant on this giga-year scale.
Then there was a period where matter density and vacuum energy density (Ωv) more or less balanced each other - the cosmic 'middle ages' - where the curve is almost straight. From about 10 Gy onwards, it is clear that the expansion rate is increasing under the dominant vacuum energy. This is what we observe today.
The look-back redshift (z) curve simply represents the inverse of the expansion factor (actually z+1 = 1/a). The curve cuts the time axis at t = 13.7 Gy (now) and stops there, because there is no point in considering redshift into our future! To the left, the curve rises to approach infinity at the time of the Big Bang, but is cut off at z = 2 for no other reason than to get good resolution on the graph at the lower values of a and z.
If you want to, you can download the EXCEL spreadsheet used to plot the curve and play around with the values (best is to right-click the link and select "save target as"). The numeric integration shown is of t against a, i.e., t = [integral f(a) da] for a from the 10-49 to 2, because it works better for the near-infinite slope of the expansion curve when t -› 0 (the time after inflation is about 10-49 Gy).
Note that this graph ignores the inflationary epoch, because it's utterly negligible and invisible on a linear scale graph over the times scales considered. According to our best theory, it lasted from about 10-34 to 10-32 seconds after the Big Bang. The next part of this mini-series will deal with the cosmic inflation epoch - and even earlier!
More about the Friedmann equation and its applications are available on the website Relativity 4 Engineers.
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Re: Cosmology Equations Part 5
