Jorrie's calculator (Lightcone) makes cosmic history tables which tell you among other things the(adsbygoogle = window.adsbygoogle || []).push({}); Hubble timesin past years. For convenience let's temporarily use greek Theta Θ to stand for T_{Hubble}so we don't have to write so much.

Basic facts (definitions actually) are that Θ = 1/H and the Hubble radius R = cΘ.

Someone asked me this recently: if they want to work out critical density of mass they use:

ρ = 3/(8πG Θ^2)

And replace Θ with Hubble time.

What mass unit would this be in? And would they have to convert it at all?

I think that's a good simple exercise in quantitative cosmology so I want to reply to the question in open thread. I use google calculator:

If you open Lightcone it tells you the current Hubble radius R = 14.4 billion lightyears. So that means Θ = 14.4 billion years.

So you can type this into the google window:

3/(8 pi G (14.4e9 years)^2)

If you paste that into google window, you get:

8.66146267 × 10^{-27}kg / m^{3}

So that is the answer to the question "what mass unit would it be in?" Google tends to give you answers expressed in standard metric units, likekilogram. If you want the answer in grams per cubic meter then you should type this in:

3/(8 pi G (14.4e9 years)^2) in g per m^3

If you paste that in, it will realize you want the answer in terms of g per m^3 and it will say:

8.66146267 × 10^{-24}g per (m^3)

Or you can paste in:

3/(8 pi G (14.4e9 years)^2) in g per cubic kilometer

and then it will tell you:

8.66146267 × 10^{-15}g per (cubic kilometer)

Basically you get the answer in whatever units you want and specify to your calculator.

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# Calculating critical density of mass, from Hubble time, for exercise

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