Estimating the Cosmological Constant

I was talking to an old friend on the phone the other day when he asked a question about the Cosmological Constant.  I said that you can estimate the value of the Cosmological Constant if you know the size of the Observable  Universe. This was an idea which had occurred to me while watching one of Prof. Leonard Susskind's awesome youtube lectures on Cosmology. So I was then determined to show exactly how that works out.

Start with the Friedmann Equations.

Now assume that matter and radiation are negligible and only the Cosmological term remains. Further assume that space is flat.

Now the Friedmann Equations read as follows.

The two equations are consistent with each other and the first order equation is like a "first integral" of the second order one.

Recalling that

gives a simple expression for the Cosmological Constant in terms of the Hubble parameter.

or

If we are now truly at a point in time where the cosmological term dominates, as appears to be the conventional wisdom, then we can get an estimate for

(Sincere apologies for the large lambda since we all know it is supposed to be very small)

by plugging in a value for H something like

, which is more or less the current value of the Hubble Parameter.

This gives the following.

Converting to Planck units gives

Cool. This is the usually quoted  value for the Cosmological Constant in Planck units.

So what about the size of the Observable Universe? Well, we are not exactly in a case of Cosmological Term only. But humor me for a moment and assume that we are near to that case, with Cosmological Term dominating. Consider our Hubble Sphere, the radius of which is the distance at which galaxies recede from us at the speed of light. (In the case of Cosmological Term Only our Hubble Sphere is our true Cosmological Event Horizon and so I will use the terms interchangeably. )

If R denotes the radius of our Hubble Sphere then

And so, as I told my friend, the Cosmological Constant is small because our Observable Universe is big.

There is a bit more. Under the assumption of flat space, the surface area of our Hubble Sphere is

And so the Cosmological Constant is inversely proportional to the surface area of our Hubble Sphere or Cosmological Event Horizon.

But also the surface area of our Cosmological Horizon SHOULD BE proportional to it's entropy.

Under this assumption the Cosmological Constant, can be related to the entropy (S), or information content (S/k), of our Cosmological Horizon.

Also, as usual, the Cosmological Constant can be expressed in the form of an energy density, the density of "dark energy".

So

The density of dark energy would then also be inversely proportional to the entropy of the horizon.

In this calculation I have taken the density of dark energy to be measured in kg/m3 and followed units carefully at each stage in order to see that they work out correctly.

Just for hahas let's plug in some numbers and estimate the density of dark energy.

so

The accepted value of the density is

So my simple calculation is off by a factor of 5. Hey, that's the best I can come up with having such limited resources as only Purina Cat Chow, dry erase markers, and some whiteboard. In my defense I will point out that it's off by a factor of 5 in 27 orders of magnitude.

Well I am not much for error analysis but this seems like more than just sloppy arithmetic. Possibly the assumption that only the Cosmological Term remains in Friedmann's Equations is a bit too stringent.

Thank you for your kind consideration of my blitherings. In as much as this post has been written by a large orange cat and is not peer reviewed, I ask that you challenge these ideas if you find them lacking. As usual, please use the comment section to let me know if you think I am all wet here.

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