Imagine the cosmos some 200 million years after time zero. If the cosmos is presently in its prime (like a 30-something years old person), then in human terms the cosmos was about 6 months old at that stage. Very young, but probably a very violent place, with supernovae, not too far apart, going off everywhere. The stars of that time were apparently very hot and heavy and hence very short lived; also, their deaths were mostly violent supernovae. On the same human scale as above, few of them lived longer than 2 weeks - extreme infant mortality!
During those supernovae, lots of matter were thrown out, anything from elementary particles, up to atoms of some metals. Most of the matter were probably caught up in later generations of the star- and planet-formation, but some particles may have survived. For the purpose of this exercise, let's assume a few atoms made it out of that distant region and all the way to our region of the cosmos.[1] Let's also suppose that they were ejected with a velocity of 0.999c relative to the overall cosmic frame of reference,[2] the CMB - like high speed 'bullets' from afar.
What would the velocity of such bullets be when they eventually reach our neighborhood? And how long would they have taken to get here? If the cosmos was only 200 million years old and we can observe light from that era today, the light will be red-shifted to z = 19. Calculations show that when the light was emitted, the source had to be 1.8 billion light years from our region.[3] Let's take this as the original proper distance of the bullets from our neighborhood.
At that time, the cosmos was expanding extremely rapidly, but slowing down under the dominant gravitational force of all the matter in the cosmos. At first, those bullets, traveling at 0.999c, could not make any headway towards us; the expansion took them farther from us for quite some time. However, after the expansion rate diminished somewhat, the bullets did indeed start to come towards us. So, how long would they have taken and at what speed would they arrive?
If there was no cosmic expansion, it would have been rather simple: 1.8 billion light years at a speed of 0.999c would have taken 1.802 billion years and they would have arrived at the original 0.999c. However, with the observed expansion profile of the cosmos, it becomes quite tricky, for two reasons: (i) the bullets were taken away at first and (ii), their speed relative to the CMB would have dropped (or decayed) to a lower value.[4] In the end, the bullets would take almost 18 billion years to reach our neighborhood (~4 billion years from now) and they would arrive at the "sedate speed" of 0.64c.
 Enough of these ramblings - let the graph on the right rather 'speak for itself'. The bullets were ejected at a proper distance of 1.8 Gly (the
start of the dark blue D_proper curve) from where our Galaxy would form shortly thereafter (somewhere on the horizontal axis, D=0).
You can clearly see how the bullets first
moved farther away from our quarters and then later started to get closer to us. They will take almost 18 billion years to reach us - where the D_proper graph intersects the horizontal axis, some 4 billion years into our future.
The green V_peculiar curve shows a bullet's momentum decay. V_peculiar starts at -0.999c and then gradually decreases in magnitude, until it passes us at around -0.64c. The reason for this decay is discussed in[4], but in a nutshell, as the cosmic balloon expands (R increases), any free particle's angular momentum[5] around the hyper-center remains constant; hence, it must lose some surface momentum. The light blue curve represents the Hubble velocity (V_Hubble = H x D_proper), which is the recession speed for the bullet's (then)
proper distance. The red curve shows the proper velocity (V_proper = d(D_proper)/dt) of the bullet over time.
'Sexy' as these curves may appear, they represent some deep truths about the standard cosmological
model. If you understand them, you could well be reckoned as an amateur 'cosmic ballistics detective'. If you don't, here's your chance to learn something more about that interesting trade - just ask...
Notes:
[1] Earth (and our galaxy) could not have been around yet, but that's a long story...
[2] It is known as the local rest frame w.r.t. the Cosmic Microwave Background (CMB), meaning an inertial frame in which the CMB has the same average temperature in all directions.
[3] These values are available from any good cosmological calculator. I have an up-to-date (2009) cosmo-calculator on my website Relativity-4-Engineers.
[4] See Blog entry Particle momentum decay. The equations can be found in my Blog entry on Tethered Galaxies.
[5] This angular momentum is equivalent to the so-called 4-momentum of the particle, which is constant during expansion.
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Re: Cosmic Ballistics Anyone?
