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## APPENDIX 1: The Origin of the Zero Point Energy and the Redshift/Distance Equation
The standard form of the recombination equation is *dN/dt = q - rN2*, where *N *is the number of ion pairs per unit volume available for recombination, *r *is the recombination coefficient, and *q *is the number of ion pairs created per unit volume per unit time by any given process, such as ionization [144]. Equation (57) can be reproduced by rationalizing the recombination equation. Alternatively, it is possible to reproduce the above equation by working backwards from Equation (57). We therefore start in (57) without any constants of proportionality since our derivation will produce other constants. Equation (57) then reads
1/U ~ [1 + T] **/ [√ (**1 – T 2)] (60)
Note that *T = (1 – t) *where *t *is dynamical time increasing from the origin of the universe. This means that at the origin time *T = 1 *and *t = 0*, while *T = 0 *at the present epoch when the quantity *t = 1*. We also designate the original number of Planck Particle Pairs (PPP) *= N**1* and make the number of PPP present at time *T = N*. Now the strength of the ZPE per unit volume is proportional to the number of PPP that have combined which is given by *(N**1** - N)*. From the discussion around (53), the quantity *(1 + z) *is inversely proportional to the strength of the ZPE per unit volume. Thus, if we ignore constants of proportionality, we can write
1/U ~ (1 + z ) ~ [1 + T ] **/ [√ (**1 – T 2)] = 1**/** (N1 – N) (61)
If we now make the substitution
N1 - N = M = **[√**(1-T2)] / (1+T)
we then have
d(N1 - N) dT = dM/dT
which means that
-dN/dT = dM/dT
Now
M = [√(1-T2)] **/** (1+T) = (1-T2)1**/**2 / (1+T) (62)
Therefore, the following mathematical operations can be performed on equation (62):
dM/dT = [-2T(1-T2)-1/2 (1+T) -1(1-T2)1/2] / (1+T)2
which can be written
dM/dT = [(1-T2)1/2 / (1+T)] {[-2T(1-T2)-1 (1+T) -1]/(1+T}= M{[-2T/1-T2)] - [1/(1+T)]}
Remembering that (*1 - T**2* )* = *(*1 – T*)(*1 + T*) we can then write
dM/dT = M{[ -2T – (1 – T)] / [(1 – T)(1 + T)]
= M{(-2T-1+T)/[(1+T)(1-T)]}
= -M{(1+T)/[(1+T)(1-T)]}
= -M/(1-T)
Therefore we can write
-dN/dT = dM/dT = -M/(1-T) (63)
We now need to find an expression for *T* in terms of *M*. In order to do this we start with equation (62) which reads
M = [ √(1-T2)] / (1+T)
If we now square both sides of this equation we find that
M2(1+T)2 = (1-T2)
Expanding this allows us to manipulate the equation as follows:
M2(1+T)(1+T) = (1-T)(1+T)
M2(1+T) = (1-T)
M2 + TM2 = 1-T
T(M2+1) = 1-M2
therefore: T = (1-M2) / (1+M2) (64)
Substituting this expression for T from equation (64) back into (63) then gives us
dM/dT = -M / {1-[(1-M2) / (1+M2)]}
= -M(1+M2) / [(1+M2) - (1-M2)]
= -M(1+M2) / 2M2
= - {[1 / (2M)] + (M/2)}
Therefore, if we now insert a constant of proportionality, *k*, which is required to be negative, we have the result that
dM/dT = -dN/dT = dN/dt = k{[1/(2M)] + M/2} (65)
Substituting *(N**1**-N)* for *M* in (65) we obtain the result that
dN/dt = k{1/ [2(N1-N)] + (N1-N)/2} (66)
Now N1 is a constant, which, to avoid confusion with N, we shall call A. Therefore,
dN/dt = k{1/[2(A-N)] + (A-N)/2} (67)
dN/dt = k{1/[2(A-N)] + A/2 - N/2} (68)
We now multiply the last term in (68) by *N/N* to give us the form
dN/dt = k{1/[2(A-N)] + A/2 - N2/(2N)
Now, *A/2* is also a constant, which we shall call *B*. We then proceed as follows:
dN/dt = k{1/[2(A-N)] + B - N2/(2N)}
dN/dt = k{1/[2(A-N)] + B - [1/(2N)][N2/1]}
dN/dt = k{1/(2M) + B] - [1/(2N)][N2/1]} (69)
It can therefore be seen that (69) has the form
dN/dt = k(q-rN2) (70)
which is the standard equation for recombination phenomena where *N *is the number of ion-pairs per unit volume available for recombination, *r *is the recombination coefficient, and *q *is the number of ion pairs created per unit volume per unit time by any given process, such as ionization. In the interpretation here, *N *is the number of Planck Particle Pairs (PPP) available for recombination per unit volume at any given time. The recombination coefficient, *r = 1/(Nt)*, where *t *is the recombination time [145]. As such, the recombination coefficient bears the units of *cm3/(ion-seconds) *as pointed out by Zwaska et al. [146]. Since *N *is the number of PPP available for recombination per unit volume at any given unit of time, the fact that r = 1/(2N) intrinsically has units of *cm3/(PPP-seconds) *is to be expected.
Normally, *q* represents the ionization rate. In our case, *q *is given by *[1/(2M) + B]* in equation (69). This is equivalent to the number of PPP created per unit volume in a given time by the decaying turbulence after the original expansion. This quantity, *q*, also is the basis of the first term in (67) above. In (67), when PPP numbers are high near the origin of the cosmos, the term *(A – N) *which is equal to *M*, is small, so that *1/[2(A-N)] = 1/(2M)* will dominate the equation. This means *q *is dominant in determining ZPE behavior, and hence the shape of the graph, near the origin of the universe. Therefore the term *1/(2M) *attracts our attention since it is the major component of *q *when equation (69) is compared with (70). For the PPP system being considered here, *q *is directly related to the decay in turbulence, *L*. It is generally agreed that the decay in turbulence follows a power law such that for time, *t*, [147]:
L = t-n (71)
Since the number of PPP forming is dependent upon the turbulence, this means that *q *is also proportional to *1/(t**n*). The relationship between *1/(2M) *and *1/(t*n*) *now needs to be established. Since we have already made the identification
dN/dt = k[1/(2M) + M/2]
we can therefore proceed as follows:
t = 2M2 / (1+M2)
t(1+M2) = 2M2
t + tM2 = 2M2
M2(2-t) = t
M2 = t/(2-t) so that M = t1/2(2-t)-1/2 (72)
Focusing for a moment on the term *(2-t)-1/2* and using the binomial expression, we get
(2-t)-1/2 = [2(1-t/2)]-1/2 = (1/**√**2)(1-t/2)-1/2
= (1/**√**2){1+ (-1/2)(-t/2) + [(-1/2)(-3/2)(-t/2)2] / 2! + [(-1/2)(-3/2)(-5/2)(-t/2)3] /3! + ...}
= (1/**√**2){1+t/4 + (3t2/8)/2 + (15t3/16)/6 +...} (73)
Substituting this result from (73) back into (72), we find that the quantity *M* is given by
M = [t1/2/**√**2] + [t3/2/(4**√**2)] + [3t5/2/(8**√**2)]/2 + [15t7/2/(16**√**2)]/6 + ...
Near the origin of the universe, when *t* is small, higher order terms can be ignored so we get
M = t1/2/**√**2 (74)
It thus follows that the quantity
1/(2M) = 1/[**√**2(t1/2)]
so that the expression for *q* becomes
q = [1/(2M) + B] = 1/[**√**2(t1/2)] + B (75)
Since this has the form of *1/(tn*), this means that the quantity *[1/(2M) + B]* is consistent with turbulence. In (71), the value of *n* is equal to one half. Now the value of *n *is different for various systems. For incompressible systems the value of *n *can be much lower than unity. Also, for spatially free turbulence, *n *is lower than for turbulence in a confined system. In confined systems, the value of *n *in incompressible turbulence can be as low as *n = 0.66 *[147]. For spatially free turbulence, it is not unreasonable to expect *n* to drop to a value of *n = 0.5* or lower. Furthermore, the term designated *q* in (70) and (75) allows for a range of values for *n* in (71). Consequently, it is now be possible, at least in principle, to obtain a better fit to the data for redshifts from about *z = 0.8* or greater. Further data are needed before the precise form of the curve, and hence the value of *n*, is obtained. This would then allow a resolution of the redshift/distance discrepancies at the frontiers of the universe, attested to by the brilliance of Type Ia supernovae in distant galaxies, without needing the action of a cosmological constant.
So the formula describing the behavior of a universal PPP system is independent of the expansion of space-time or galaxy motion. Even though it is related to equation (55), all this means is that distance, *x*, in one is related to time, *T*, in the other. Since it derives from the physics of turbulence in the early cosmos, the production and recombination of PPP, and hence the behavior of the ZPE, it has nothing to do with velocities or the relativistic Doppler formula. Indeed, that formula requires a precise square-root, whereas actual observation and the turbulence analysis here indicate that there is an important deviation from that precision.
**References**
[144] See under “recombination coefficient” in American Meteorological Society *Glossary of Meteorology*, available online at: http://amsglossary.allenpress.com/glossary/browse?s=r&p=30 (2003).
[145] See, for example, Astronomy 871 – Physics of the ISM, page III-3, available at: www-astronomy.mps.ohio-state.edu/~pogge/Ast871/Notes/Ionized.pdf (2003)(as of March 7, 2006).
[146] R. M. Zwaska et al., “Beam Tests of Ionization Chambers for the NuMI Neutrino Beam,” IEEE Transactions on Nuclear Scoence, vol. 50, No.4 (August 2003) 1129
[147] M-M. MacLow, R. S. Klessen & A. Burket, *Physical Review Letters ***80:13**, (1998) 27,54.
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