Some Thoughts About Cosmological Horizons and Thermodynamics

Well, I was looking at the super simple model described in my last post. This model has flat space with a positive Cosmological Constant and no matter or energy. So basically I'm thinking about de Sitter space.

Just to recap a bit about that post, 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. The only difference is that the first equation requires a positive Cosmological Constant, while the second one does not. To satisfy both equations, in this simplest of models, the Cosmological Constant must be positive.

So we get a solution for a(t) which looks like

We can have an exponentially expanding solution, an exponentially contracting solution, or a linear combination of the two like a hyperbolic cosine solution.

Let's look at the metric in the x-t plane, for the expanding case. For simplicity, I will take units for which c=1.

Here I am calling x the Co-Moving coordinate, even though there is nothing to move in this model which is devoid of Matter or Regular Energy. Proper distance is then defined by

In order to investigate the Horizon, I need to work out the trajectories of light rays. Using the metric above things proceed in the usual way. To find the Horizon, I am looking for a light ray which reaches x=0 at t=infinity. The calculation goes as follows.

If I choose the other sign the solution is just flipped across the t-axis. Here is a graphical representation.

It is clear that I will never receive a signal from any events which occur in the shaded region, even if I wait till the end of time. So this is my true Cosmological Horizon. In terms of proper distance, this becomes...

So, in this case, my Horizon remains fixed at constant proper distance L=1/H = c/H. In other words, according to the Hubble Law, the Horizon is at exactly that proper distance at which Galaxies recede at the speed of light. So my Hubble Sphere and my Horizon, are the same thing, at least in this simple model.

This is not the case when boring ordinary Matter and or Energy are included in the mix. You can learn a lot about a much more complicated LCDM model in the Doctoral Dissertation of Prof. Tamara Davis, which was also published, in part, as an article in Scientific American.

In Prof. Davis's dissertation, she uses several different coordinate systems to make certain points clear. In particular, she uses Co-Moving coordinate vs so-called Conformal Time and in this coordinate system light rays are rendered as straight lines. Conformal Time is defined as follows...

So in the simple case under consideration here, this becomes

Notice that the Conformal Time is defined on the finite interval [0,1/H], as opposed to Hubble Time, t, which is on the interval [0, infinity).

That is

(I have chosen to begin Hubble Time at t=0. I could have started at
t = - infinity, but the results I am looking for would be essentially the same. So let's go with this.)

And so, my entire existence in this veil of tears must sadly end at the event T=1/H, x=0, which is the end of Conformal Time at the origin of Co-Moving coordinates.

Light rays travel in straight lines with slope equal to +-1, so I can make a diagram like this.

It is a simple matter to check that the line T+x=1/H is, in fact, the same as the Horizon/Hubble Sphere previously mentioned.

A few observations can be made from looking at the diagram.

1) Events outside the backwards light cone can never signal us. This makes evident our true Cosmological Horizon.

2) Events occurring near the Horizon, but inside the light cone, can be seen only near the end of Conformal Time.

3) Galaxies will cross the Horizon in finite Conformal Time, but we will not see this until the bitter end of days

4) We can reach, and indeed cross, the Horizon in our primitive, clunker, no warp drive, spaceship and do it in finite Conformal Time. But again, home base will neither see this nor get our report until the end of the world.

In general, no matter what the form of the scale factor a, light rays travel on straight lines at 45-degree slope when viewed in Conformal Time vs Co-Moving Coordinates. The calculation is as follows.

I mentioned above that for this model, with an exponential scale factor, the Conformal Time coordinate lives on a finite interval. This is not the case where the Cosmological Constant is zero and either Matter or Regular Energy dominate. In those cases, the scale factor, a, goes like a power of t and not an exponential.

 In the Radiation-Dominated case, for example, Conformal Time looks like this.

So there is no end to Conformal Time and the backward light cone of an observer at x=0 fills all Spacetime as T approaches infinity. The Spacetime diagram looks like this.

The Matter-Dominated case goes the same way. There is no Horizon when the Cosmological Term is absent, only an ever expanding Hubble Sphere.

Let's calculate the acceleration of a typical galaxy/test particle caught up in the flow of cosmic expansion. Assume the Galaxy is originally at Co-Moving Coordinate x0. At Hubble Time t, the Galaxy is still at Co-Moving Coordinate x0 but it's proper distance, velocity and acceleration are given by the following.

I want to get the acceleration of a Galaxy which is near the Horizon.

Notice how stuff accelerates towards Horizon like it is being pulled by "gravity". I guess this is the anti-gravity of the mysterious dark energy. I had a hard time with this because Newtonian Gravity says if you have a spherical hollow in a uniform material, say spherical shell or infinite medium, the force on a particle inside the hollow cancels and nobody accelerates.  But that's not what's going on here. It is the flow of cosmic expansion which gives us this acceleration or "force".

So far this has been pretty tame. Warning: The following is a very weak heuristic argument if even that. Take with a grain of salt.

Let's calculate the temperature of the horizon. If a photon is added to the Horizon an amount of Entropy is also added.

Let's choose the photon wavelength so that the smallest possible Entropy is added. That would be one e bit of information or one k (Boltzmann's Constant) of Entropy. The photon should have the smallest possible Energy, in other words, the longest possible wavelength. I choose the wavelength to be the size of the observable universe! (Cool huh? I can just do that because it's my blog. Who knows? Let's go with this for a bit.)

And that gives the temperature of my Horizon. OK. That agrees within a geometric factor with the usual result for the de Sitter temperature.

Actually, this is disingenuous, If I assume hc/lmax = kT, where lmax is the "largest possible wavelength available", whatever that might mean, then assuming lmax= R is already equivalent to stating the temperature goes like 1/R. So the argument is circular. I disavow this line of reasoning completely. I refute my own argument!

But hey. Let's go with this just a bit further. If I substitute the above result for the acceleration at the Horizon, I can express the temperature in terms of that acceleration, in a manner similar to the Unruh-Davies temperature experienced by an accelerated observer.

The usual Unruh-Davies result is

It is my hope that this heuristic "argument", as messed up as it is, can somehow shed some light on this subject matter for someone. All this is similar to my other post about Black Hole temperature, the basic idea being cribbed from Susskind's book, "The Black Hole Wars". The argument is a crazy one, which, none the less, gives more or less the right answer.

As usual, this post has not been subject to peer review. And so, dear reader, I ask you to please educate me concerning any flaws or failings in this post. I did reach out to someone who is specialized in this area, asking for comments. But to date, I have received no reply.

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