(Acknowledgements to the participants of the Bad Astronomy - Against the mainstream forum, for their astute observations, corrections & comments)
Before we begin it is useful to be aware of the true nature of the Hubble Constant in SI units.
Hubble rate of recession 73 km s-1 Mpc-1. This breaks down to:
73000m s-1/ 30.857 x 1021m =
2.3675 x 10-18s-1
The distance to the edge of the observable universe without invoking relativity would be the speed of light (c) divided by this value.
c/2.3675 x 10-18 = 1.266 x 1026m = the edge of the observable universe.
It is generally accepted that electromagnetic radiation can travel through the eternity of the vacuum of space without suffering any effects apart from the tug of gravitational fields. It is also accepted that the Heisenberg Uncertainty Principle applies as a direct consequence of observing and interfering with a system. However, if the Uncertainty Principle was more fundamental to the Universe as some are speculating, then there would be further consequences. For example, a small uncertainty could be introduced at every interaction between particles including virtual positron and electron pairs in the vacuum of space. In other words all electromagnetic radiation travelling in every direction in every part of the Universe has a small probability of interacting and losing a tiny quantity of energy, but no more than allowed for by the Uncertainty Principle. This then becomes the background radiation.
Uncertainty is brought into a system when dealing with certain pairs of values where the level of values is between zero and h/4Pi. For example, in measuring the momentum and position particle. We can only know the product of both the momentum and the position as long as we are dealing with values above h/4Pi. At values below that level uncertainty is introduced. Now an interesting result occurs if we allow our particle to be a photon of electromagnetic radiation.
Dp.Dx. > h/(4Pi)
Dh/lambda . Dx > h/(4Pi) (substituting for the momentum of a photon)
Dh.nu/c . Dx > h/(4Pi) (rewriting the wavelength in terms of frequency and light speed)
DE/c . Dx > h/(4Pi)
DE.Dx > hc/(4Pi) (multiplying through by the speed of light) Two alternative derivations of this result
If we assume that it is the distance travelled by the photon that introduces the uncertainty, then this would result in an exponential decay of energy or an exponential change in the observed wavelength or Z (red shift) but no more than the tiny quantity hc/(4Pi) per metre of travel. In fact, if these energy loss was due to encounters with virtual electron positron pairs for example, then in any one interaction the uncertainty or loss could be anywhere between 0 and hc/(4Pi) per metre of travel but no higher. Therefore, to get an average probability we have to divide by two. This gives hc/(8Pi). Its numerical value is 7.9 x 10-27 joule metres. So, if you opt to measure the energy, there will be a fractional loss of information in distance/wavelength. If you opt to measure distance/wavelength there will be a fractional loss in energy. For photons however, the energy is inversely proportional to the wavelength so each is subject to the full value of the uncertainty probability.
Now lets consider a simple model. If you had a bag of coins with a hole in it and you lost 1/100th of your money for every metre you travelled. How far would it be before you lost it all? The naive answer is the inverse 100m and it does look that way in the early part of the journey. However, mathematicians will know that it is really an exponential loss in that the rate of loss depends on what you have at the time.
Now back to the universe. if you naively ask how far you travel before you have lost all the information then the answer comes out at the inverse of the above number, that is 1.265x1026m. 1/1.265x1026m is the coefficient of either energy loss or change in wavelength for every metre of travel and is the apparent size of the observable universe in the early part of the graph below. Think of it as similar to the coefficient of radioactive decay which is time dependent. In this case we have a decay of information which is distance dependent. It is this value that leads to an interesting result when extrapolated across the Universe as an exponential effect. We will first apply it to see how our energy measurements are affected and then the red shift. For energy it leads to an exponential redistribution, which due to the nature of the exponential decay curve looks linear in our region of the Universe.
E = E(0) e(-x./1.265x10^26); where x is the distance travelled in metres.
Here the gradient points to the apparent size of the observable universe on the x axis, which at the beginning points to around 1.265 x 1026 m which is equivalent to a recession speed of 73 km s-1 Mpc-1. At greater distances the gradient points to an apparent larger distance which equates with slower recession speeds. Since it is all illusory there is no need to consider relativistic effects or dark energy.
A fiddle? A coincidence? or something going on for which there is no experimental test? A free travelling photon would have to travel way beyond the solar system before it lost its first bit of energy. With it being an exponential, the relationship looks linear in our own region of the Universe but at greater distances the curve of the exponential kicks in, leading to the fact that the further we looked for our evidence of expansion the greater the radius of the observational universe would appear to become. This also leads to a Universe which gives the illusion of apparently expanding at a greater rate in our own locality than at greater cosmic distances which is in line with current supernova research. There is no need to wonder about dark energy and what is driving the expansion because it does not exist. The Universe only appears to be expanding faster in our own locality as a consequence of the model.
Now we use the same function in a positive way as a Z (red shift) predictor. We assume that this quantity affects our wavelength measurements. Z = e (x /1.265 x 10^26)-1; where x is the distance travelled in metres. The green line is the traditional Hubble line with an apparent recessional velocity of 73 km s-1 Mpc-1
Here we see that the function follows the Hubble line until around Z = 0.2 when it starts to leave the line curving up and giving a higher red shift than the Hubble line for the same distance. This is what has been found with supernova. At greater distances they are showing a higher red shift than that predicted by their brightness. With this model of the Universe it is just a natural feature of the way energy is lost exponentially due to quantum effects. There is no acceleration and no need for dark energy.
Question: How can this be true? The Universe has to be expanding, because the time for supernova to dim takes longer with increasing distance proving they are in a state of recession.
Answer: Could well be. However, following the above scenario, there is also an uncertainty in measuring energy and time DE.Dt > h/4Pi. Therefore there would be a gradual loss of time information with distance leading to longer times and the illusion that time stands still at the edge of the observable Universe. Think of it as a very gradual loss of all information to do with energy, distance and time measurements that increases the further into the Universe that you observe. There is also a selection effect that may have influenced the results called the Malmquist bias. When allowance was made for this it appeared that all evidence for time dilation vanished. So, this is still up in the air at the moment.
Now whether you have followed this, or if I have not explained it adequately, then try and take away one fact. For red shifts below 0.2 you can easily work them out by taking the distance to the galaxy in metres and dividing by 1.265 x 1026m. For example, a galaxy 1025 metres away/1.265 x 1026m has a red shift of 0.079. Try it yourself with other values but remember to go to the exponential form for large red shifts. This is not a number we have derived from observation but from the centre of the boundaries that define the Heisenberg Uncertainty Principle. It should make you think!
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