Assume that the expansion velocity as a function of time (measured in billions of years) has the form e^-n/3, so that after 1, 2, etc. billion years the velocity relative to that of light is .72, . 51, .37, .26, .19, .135, and the radius is approximately .86, 1.48, 1.92, 2.24, 2.47, and 2.66. So at this time, the diameter is about 5.32 billion light years and the circumference 16+ billion light years - expressed in megaparsecs, it is about 5000 mpc.
Figure 2: The size of the universe as a function of time (using v = e^-n/3)
We can only observe stars out to about 3 or 4 mpc, but if for simplicity we use 50 mpc as a recent or nearby event, this would only be a separation of 3.6 degrees on the sphere with diameter 5.32. Such a star would lie approximately on the same size sphere as the solar system now occupies (i.e. 5.32 billion light years); and it would have an expansion velocity of about 40,000 km/sec.
The recession velocity of this star would be given by this number, 40,000 km/sec, multiplied by the sine of 3.6/2 degrees (which gives us the value along the line of sight to the star) or .03, which yields a recession velocity for the star of about 1270 km/sec.
From this we can easily find the present value of the Hubble constant, Ho, by dividing by the distance in mpc, that is 50, which yields a value of 25. However, there is an equal contribution to the recession velocity from the projection of our own expansion velocity on the line of sight to the star. So the actual value is Ho = 50 - a surprisingly good agreement with current estimates.
With this model we can also explain the clustering of the supernovae data. Figure 2 shows the growth of the universe as a function of time. It also shows selected points corresponding to supernovae which explode at an age of between 2 and 4 billion years.
The location of the stars we can now see, on the corresponding circles, is determined by the requirement that the light reaches us at the present time (an age of 6 billion years), that is, after 3.5 billion years if the explosion occurred at the age of 2.5 billion years, to use just one example. These points when rotated out of the plane of the diagram each represent circles on a three dimensional sphere. Only the cross section determined by the expansion velocities of the observer and the supernova at time of explosion is shown.
Also shown is the line of sight connecting earth to the star, along with the projections on this line of sight of the expansion velocity of earth at this time, as well as the star, at its time of explosion. The sum of these projections yields the recession velocity of the star relative to an observer here as calculated from the model.
For example for the star which explodes at the age of 2.5 billion years, the expansion velocity was about .44, and the present expansion velocity here on earth, at the age of 6 billion years, is about .135 times the velocity of light.
Calculating the projections, the value of z is then obtained from the formula, (see ref.4, p.266)
For this case we obtain
The above formula is based on relativity theory, and is used by astrophysicists to calculate the recession velocity v(r) from the observed red shift z, which in turn is derived from the relative wavelength shift. For consistency with experimental data the same formula is used in connection with the model above. In this case we work backwards to first obtain the recession velocity from the projections of the expansion velocity onto the line of sight to the star, and then connect this velocity to z, through the same formula used by others.
We need also to remember that only relatively few of the explosions which occur at the age of 2.5 billion years can be seen. Only if they are at a distance in light years away from us that would cause the explosion to be visible at this moment will they appear to us now (that is they must be about 3.5 billion light years away). The curve which looks like a new moon represent those points which exploded within the age range of 2 to 3 billion years that would be visible today. The points to the right of this curve, within this ring, are visible at a later epoch. The points to the left were visible up to a billion or more years ago. This also makes clear why the data are clustered.
Any large stars that exploded in or shortly after the first billion years would have been seen on earth at a much earlier time. They are no longer visible today. Any stars that exploded when the universe was, say, 4 billion years old have to be two billion light years away at the time of explosion. Such stars are most probably from A later generation, since any supernova of the type 1A almost certainly has a life time of at most 3 billion years. Thus, it is the type 1A supernovae with life times of 2 to 3 billion years, and consequently with z values between .35 and 1.0 that we would expect to see here and now - and that is precisely what the data show. No such transparent results are possible without appeal to Euclidian geometry.
In another 6 billion years all the type 1A supernovae will not only have exploded, and been detected, but are past the possibility of detection; the diameter of the universe will still be under 6 billion light years; Ho will be about 6, and observers will wonder what the original mechanism of star formation could possibly have been. Perhaps a few billion years after that, it will probably be noted that this universe is actually collapsing.
Under the scenario suggested here, it appears that the sun was created along with the first, or second generation stars. The material for the planets, with all of the heavier elements, would have been formed later from the residue of the first, or second, generation supernovae.
The original event, which we call 'creation', could have been itself a supernova from a star with a mass of over 10^17 solar masses. This would account for all the mass in the universe including dark matter. This large a star, formed perhaps from an earlier collapse, would have a minuscule life time.
It is also interesting to speculate what would be the contribution to the presently detected background radiation from the black body radiation emitted by 10^14 to 10^15 exploding first generation, and second generation, stars over a period of one to three billion years. Perhaps this is the background radiation that we now attribute to the origin of the universe. Under the model above that original radiation would have passed us about three billion years ago, if it originated from the center six billion years ago.
The theory that supernovae leave behind neutron stars and X ray sources may have to be revised if it turns out that, for any one supernova, the origin in time was the same for all types of emissions, and only the rate of travel is responsible for the observed delay. A rate differential of .001% could result in a delay of as much as 3 thousand years of the slower signal, for stars that are a billion light years away - making it seem as though a supernova remnant was left behind.
Since what can be observed, of the living, actual stars, apart from the early type 1A supernovae is only a minute fraction of one degree in our neighborhood - at best 3 to 4 megaparsecs out of the circumference of 5000 - we should be glad that it is early enough in time that we can still get some hint of the early supernovae as driving force in the development of the cosmos as we know it, and by nucleosynthesis the source of the heavy elements in the planets (but not the sun).