Understanding the Expansion and Age of the Universe
by David Sims
YOU GET THE right number for the age of the universe if the Hubble constant is 71 km/sec/Mpc. If that’s the case, then whatever we see in the universe that is observed to have a distance greater than 7.81054e+9 light years has moved, in the time that the light was enroute to us, to a point beyond the cosmological horizon.
A megaparsec (Mpc) is a unit of distance equal to one million parsecs. One parsec is a unit of distance equal to 3.26156377717 light years. So one megaparsec is equal to 3.26156377717e+6 light years.
The cosmological horizon is a spherical boundary that contains the entire observable universe. It is 13.77 billion light years away, or 4222 Mpc.
Just to provide a sense of scale:
Andromeda galaxy, distance = 0.675 Mpc
Virgo cluster of galaxies, distance = 18.4 Mpc
Width of the Laniakea Supercluster = 160 Mpc
Length of the CfA2 Great Wall = 230 Mpc
(The CfA2 Great Wall is composed of three galaxy superclusters that happen to be arranged end-to-end like a long chain.)
The longest structure in the universe that can be recognized as an assemblage of matter is less than 1/18th as long as the distance from an observer to his cosmological horizon.
The speed at which any object in the expanding universe recedes from an observer is proportional to how far away it is. The speed at which distant objects move away from an observer, due purely to the expansion of space, increases by 71 kilometers per second for each Mpc of additional distance.
That is, the Hubble constant is 71 km/sec/Mpc. When you convert the distance units to the same thing and cancel them out, the Hubble constant can be expressed as reciprocal time:
The age of the universe is:
1/H = 4.3460248e+17 sec = 1.3771721e+10 years
When you observe a distant galaxy, you are looking at where it was, not where it is now. The amount of time that the galaxy has moved since it was where you saw it is, not coincidentally, equal in years to the observed distance in light years.
But figuring out that galaxy’s present distance isn’t simple because all the while the galaxy was moving away, it was picking up speed due to the fact that the Hubble recession speed grows as its distance does. The galaxy wasn’t moving away at a constant speed, in other words. Rather, it was accelerating away as it moved away.
So what astronomers do, when they calculate the “now” distance for an observed distant galaxy, is walk time forward a little bit (say, 1,000 years), assuming that the recession speed was constant during that little bit of time, and find a new distance — and then a new recession speed. Then they repeat this calculation for the next unit of 1,000 years, and so on, until they have accumulated as much time as the light had spent traveling from where they observed the galaxy to their eyes. This is a chore that would be much too tedious to do without a computer.
This is a form of numerical integration. It lets you find the “now” distance and the “now” recession speed of an observed galaxy from its “observed” distance and speed, and the Hubble constant.
If you were to observe a galaxy exactly on the cosmological horizon, 13.77 billion light years away, and moving away from us at the speed of light, then its “now” distance would be 37.43 billion light years, and its “now” recession speed would be 2.718 c.
All the matter that appears to be within a spherical volume having a radius of 13.77 billion light years is, in reality, when you correct for the speed-of-light time lag, “now” spread out over a larger spherical volume having a radius of 37.43 billion light years.
You couldn’t actually see anything on the cosmological horizon because the light from it would be infinitely red-shifted. But I was making a thought experiment, not a practical one.
However, you can see galaxies that are 7.81054 billion light years (2,395 Mpc) from Earth. By observing their red-shifts, we can tell that those galaxies were moving away from us at a speed of 0.567145 c. In the time that it took for the light from those galaxies to reach us, they moved to a “now” distance of 13.77 billion light years, i.e., to the cosmological horizon, and their “now” recession speed is 1.000000 c.
So, really, anything that looks to be 7.8 billion light years (2,391 Mpc) is forever out of our reach, even if we make spaceships that can travel at almost the speed of light. That’s a shame, because it means that a lot of the matter that we can see is already lost to us.
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