Calculating the cosmological constant from scratch

[marcus]

Here is a fun thing to try

The Google calculator knows the values of pi, the speed of light c, Newton's G and stuff like that. So you can easily make it calculate the cosmological constant from a few familiar facts.

You have to know that the usual estimate of the Hubble parameter is 71 km/s/Mpc

(71 kilometers per second per Megaparsec)

from that you can easily find out the critical density and astronomers regularly take that to be the density of the universe, because it looks approximately flat.

Also they estimate that the darkenergy density is about 75 percent of the total density.

So that's all you need. You just go to Google and type in
3((c*71 km/s/Mpc)^2)/(8pi*G)

that will give the critical density, IOW the actual density of the universe, in joules per cubic meter.

Or, since the darkenergy is about 75 percent of that, you can do the whole thing at once by typing in
0.75*3((c*71 km/s/Mpc)^2)/(8pi*G)
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this works fine for me. Let me know if it doesn't work for you.

The only sophisticated part of this is that the usual formula for the critical density is

$$\frac{3c^2 H^2}{8\pi G}$$

and this is translated into a form that the Google calcuator can understand

 

[marcus]

When you type
3((c*71 km/s/Mpc)^2)/(8pi*G)

into the box and press search you then Google does the calculation for you
and comes up with

8.5 x 10^-10 pascal (which is the same as joules per cubic meter)

so it is telling you that the density of the universe is 0.85 joules per cubic kilometer

and then when you type

0.75*3((c*71 km/s/Mpc)^2)/(8pi*G)

into the box and press search, Google does the calculation and comes up with

6.4 x 10^-10 pascal

again, one of the things that pascal can mean is joules per cubic meter, so that means the dark energy density version of the cosmological constant is

0.6 joules per cubic kilometer

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It can be fun using the Google calculator because it puts the values of many of the constants in for you like c, G, hbar, k (Boltzmann constant), electron charge, mass of Earth etc etc

that way you don't have to remember those things and bother typing the numbers in, you just type in the symbol

I was reminded of this by something Robousy was asking.