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.(adsbygoogle = window.adsbygoogle || []).push({});

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 distanceat 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 rateH(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 radiusc/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)

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# Eight steps to the understanding the CEH

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