Eight steps to the understanding the CEH

[marcus]

I realized after posting that this is not really an attempt to start a glossary of terms. I'm trying to explain, very briefly, the Cosmic Event Horizon (abbr. CEH). Comments, additions, suggested changes are welcome. We start with the most basic idea and build step by step.

CMB rest: There is an FAQ for this. An observer at rest relative to the CMB sees approximately the same temperature (of the ancient light) in all directions. There is no Doppler hotspot which would indicate that he or she was moving in that direction. It's like being at rest with respect to the ancient matter when it was more uniformly spread out, or with respect to the expansion process itself.

Universe time: Time as clocked by observers at CMB rest.

Proper distance at a particular time t: What you would measure by any conventional means (radar, tape measure...) if you could stop expansion at some given moment of universe time. Stopping expansion gives you time to measure---the distance won't change while you are sending the radar pulse, for example.

Scale factor a(t): This curve plots the expansion of distance as an increasing function of time. It is normalized to equal 1 at the present time. a(now) = 1. Back when distances between stationary observers were only half what they are today a(then) = 0.5. The slope has not been constant so it's convenient to have the curve as a record of expansion history.

Fractional rate of increase of a(t): A good handle on the rate distances are increasing is the fractional or percentage increase over time. Currently the scalefactor increases by about 1/140 of one percent per million years. So any largescale distance (e.g. between galaxies free of each other's gravity and each approximately at CMB rest,) will increase at that rate. (More precisely using the latest data 1/139 of one percent per million years.) The math expression for this rate, at any time t, is a'(t)/a(t). This is the absolute increase at that time, divided by the current size at that time, IOW a fractional or percentage increase rate.

Hubble rate H(t): By definition H(t) = a'(t)/a(t), just another name for the fractional rate of expansion. The current value of the Hubble rate is denoted H_o. Or you could say H(now), or a'(now)/a(now). It would all mean the same thing. Mathematically it is a fractional rate of increase the current value of which is 1/140 of one percent per million years. (Or 1/139 using the latest data)
That's the rate that distances (between observers at CMB rest) grow, at present. Using proper distance and the universe standard timescale.
In common astronomy units it is 70.4 km/s per Mpc. 70.4 km/s is the rate that a distance of one Mpc is growing.
The Hubble rate is slated to decline in future to sqrt 0.728*H_o

Hubble radius c/H(t): This is the (proper) radius within which distances grow at rates less than c. If a photon is trying to get to us and it can manage to get within this radius then it will begin to approach. The photon is coming faster than the remaining distance is increasing.
Try using the google calculator to find the current Hubble radius in lightyears. Put this in the search window:
1/70.4 km/s per Mpc
When you press return the calculator will say 13.9 billion years.
Multiply by c and you obviously get 13.9 billion lightyears.
This is the current Hubble radius.
Photons within that radius are going to make it.

Cosmic Event Horizon ≈ c/(sqrt 0.728*H_o) ≈ 16 billion lightyears.
Photons heading for us can still make it even if they are OUTSIDE the current Hubble radius as long as the radius is increasing fast enough and reaches out and takes them in.
What would make c/H(t) increase? The denominator H(t) decreasing would. The Hubble expansion rate has decreased sharply in the past which is why we can see such a lot of stuff that we know is receding faster than light.
But according to the standard cosmic model H(t) though still declining is not expected to go below sqrt(0.728) of its current value.
It is expected to level out at sqrt(0.728)*70.4 km/s per Mpc
So what will the Hubble radius be then?
Try putting this in the google window
c/(sqrt(0.728)*70.4 km/s per Mpc) in lightyears
You will get the longterm value of the Cosmic Event Horizon (abbreviated CEH)

 

[marcus]

Here's a second draft of post #1. I corrected the title and added some material.
The existence of the Cosmic Event Horizon (CEH) depends on the fact that the scalefactor curve a(t) has a slope which (although gradually decreasing for roughly the first half of the expansion age) is now slowly increasing. The scalefactor curve is very gradually getting steeper, and is expected to continue doing so.

Picture: http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
the dark solid curve labeled (.27, .73).

This has the interesting consequence that the most distant galaxy which we could, today, send a message to and expect it to arrive is only about 16 billion lightyears away. We currently see galaxies much farther away than that, and if it were not for this gentle acceleration effect we could in principle flash messages which would eventually reach them. But because of the slight acceleration they are actually "beyond our event horizon". And it works both ways: they, as of today, could not send information to us. If a star exploded today in one of those galaxies, we would never see it, no matter how long we waited.

I want to try to explain where this figure of 16 billion lightyears comes from. This is a first attempt and comments are welcome. It goes in 8 steps starting from the most basic concept. For some readers much of this will be review:

CMB rest: There is an FAQ entry for this. An observer at rest relative to the CMB sees approximately the same temperature (of the ancient light) in all directions. There is no Doppler hotspot which would indicate that he or she was moving in that direction. It's like being at rest with respect to the ancient matter when it was more uniformly spread out, or with respect to the expansion process itself.

Universe time: Time as clocked by observers at CMB rest.

Proper distance at a particular time t: What you would measure by any conventional means (radar, tape measure...) if you could stop expansion at some given moment of universe time. Stopping expansion gives you time to measure---the distance won't change while you are sending the radar pulse, for example.

Scale factor a(t): This curve plots the expansion of distance as an increasing function of time. It is normalized to equal 1 at the present time. a(now) = 1. Back when distances between stationary observers were only half what they are today a(then) = 0.5. The slope a'(t) has not been constant so it's convenient to have the curve as a record of expansion history. Picture: http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid curve labeled (.27, .73) is the one to focus on.

Fractional rate of increase of a(t): A good handle on the rate distances are increasing is the fractional or percentage increase over time. Currently the scalefactor increases by about 1/140 of one percent per million years. So any largescale distance (e.g. between galaxies free of each other's gravity and each approximately at CMB rest,) will increase at that rate. (More precisely using the latest data 1/139 of one percent per million years.) The math expression for this rate, at any time t, is a'(t)/a(t). This is the absolute increase at that time, divided by the current size at that time, IOW a fractional or percentage increase rate.

Hubble rate H(t): By definition H(t) = a'(t)/a(t), just another name for the fractional rate of expansion. The current value of the Hubble rate is denoted H_o. Or you could say H(now), or a'(now)/a(now). It would all mean the same thing. Mathematically it is a fractional rate of increase the current value of which is 1/140 of one percent per million years. (Or 1/139 using the latest data)
That's the rate that distances (between observers at CMB rest) grow, at present. Using proper distance and the universe standard timescale.
In common astronomy units it is 70.4 km/s per Mpc. 70.4 km/s is the speed a distance of one Mpc is growing.
The Hubble rate is slated to decline in future to sqrt 0.728*H_o ≈ 60 km/s per Mpc.

Hubble radius c/H(t): This is the radius within which proper distances increase at speeds less than c. If a photon is trying to get to us and can manage to get within this radius then it will begin to approach. The photon's own speed is then faster than the remaining distance is increasing, so it can make progress towards us and narrow the gap.
The google search window doubles as a calculator. Try using it to find the current Hubble radius in lightyears. I invite you to copy this into the search window:
1/70.4 km/s per Mpc
When you press return, the calculator will say 13.9 billion years.
Multiply by c and you obviously get 13.9 billion lightyears.
This is the current Hubble radius.
Photons within that radius are going to make it.

Cosmic Event Horizon ≈ c/(sqrt 0.728*H_o) ≈ 16 billion lightyears.
Photons heading for us can still make it even if they are OUTSIDE the current Hubble radius as long as the radius itself is increasing fast enough and reaches out and takes them in.
What would make c/H(t) increase? The denominator H(t) decreasing would. The Hubble expansion rate has decreased sharply in the past which is why we can see such a lot of stuff that we know is receding faster than light.
But according to the standard cosmic model H(t) though still declining is not expected to go below sqrt(0.728) of its current value.
It is expected to level out at (sqrt 0.728)*70.4 km/s per Mpc
So what will the Hubble radius be then?
Try putting this in the google window
c/(sqrt 0.728 *70.4 km/s per Mpc) in lightyears
You will get the longterm value of the Cosmic Event Horizon (abbreviated CEH)

====================
The number 0.728 is technical and hard to explain, so I've had to put it in *ad hoc*. It represents a constant VACUUM CURVATURE contributing to the near flatness of space, which would otherwise be negatively curved (e.g. triangles adding to less than 180 degrees). Without such an inherent constant curvature bias, (or cosmological constant) the current density of matter/energy would only be 0.272 (or about 27%) of what was needed for the observed degree of flatness. So (although in my opinion it's a bit confusing to think this way) the number 0.728 could be imagined as a fictitious energy contribution making up the rest of what would be needed without a cosmological constant.
The square root of 0.728 gets into the picture for technical reasons when we want to talk about the longterm value of the Hubble rate, the level below which it is not expected to decline (because of the acceleration in the scalefactor.)