Latest revision 26-06-2023

index

Hubble Parameter

$ \def \hieruit {\quad \Longrightarrow \quad} \def \slechts {\quad \Longleftrightarrow \quad} \def \SP {\quad ; \quad} \def \half {\frac{1}{2}} $ Narlikar's Law is repeated once again: $$ \frac{m}{m_0} =\left(\frac{T-A}{T_0-A}\right)^2 $$ With $T=$ orbital time, $m=$ elementary particle rest mass, $m_0=$ reference mass (here and now), $T_0=$ gravitational reference time, which is the "nowadays" timestamp, $A=$ gravitational timestamp corresponding with a beginning (Alpha), at the time when mass is created out of nothing. Up to now, $A$ is the only unknown quantity in Narlikar's Law. Our aim is to find it's value.

On a large scale there is Hubble's Law, which is essentially a manifestation of the Doppler effect. We opt for the non-relativistic approach. Then, according to standard knowledge we have: $$ v_{rs} \equiv c_0\,z \SP z = \frac{\lambda_o}{\lambda_e} - 1 = \frac{v_{rs}}{c_0} = \frac{H_0\,D}{c_0} $$ Here $v_{rs}=$ recessional velocity, $c_0=$ (standard) lightspeed, $z=$redshift, $\lambda=$ wavelength of light, $o=$ observed, $e=$ emitted, $H_0$ = Hubble constant, $D=$ distance.
According to Halton Arp's Theory, however, Intrinsic Redshift is not caused by the Doppler effect. Formulas have been found in the latter reference. Consistency reveals that $\lambda = \lambda_o$, $\lambda_0 = \lambda_e$. So we have, with $t=$ classical/cosmic time: $$ 1+z = \frac{\lambda_o}{\lambda_e} = 1 + \frac{H_0\,D}{c_0} = 1 + H_0\,t = \frac{\lambda}{\lambda_0} $$ On the other hand we have Intrinsic Redshift $\,z\,$ and Narlikar's Law: $$ \frac{\lambda}{\lambda_0} = \frac{m_0}{m} = \left(\frac{T_0-A}{T-A}\right)^2 $$ In order to establish a correspondence with the linear classical result, part of the Taylor series expansion around $T=T_0$ is considered: $$ \left.\frac{d}{dT}\frac{m_0}{m}\right|_{T=T_0} = -2\frac{(T_0-A)^2}{(T_0-A)^3} \hieruit 1+z = \frac{m_0}{m} \approx 1 + 2\frac{T_0-T}{T_0-A} $$ Where $\,T_0-T\,$ corresponds with $\,t\,$ in $\,1+z=1 + H_0\,t\,$. It is thus seen that, as a first (linear) approximation: $$ \frac{2}{T_0-A} = H_0 \hieruit A-T_0 = 2(-1/H_0) \hieruit \\ \mbox{Creation time} = 2 \times \mbox{Hubble time} \SP \mbox{Hubble time} = \mbox{Creation time}\;/\;2 $$ Here $(T_0-A)$ is the creation time Alpha (apart from a minor "nowadays" correction $T_0$) and $-1/H_0$ is known as the Hubble time $\approx -14.4 \text{ billion years}$. Thus, according to intrinsic redshift theory, Alpha must be $-28,8$ billion years. The latter result could have more impact than one might think.
The age of the universe according to nowadays (2022) standards is 13.8 billion years. With UAC the (gravitational) age is twice as large; there is plenty of time to form anything. Are we still in the need of cosmic Inflation? In the book by [Fahr 2016] at page 215 we read: [ .. ] Aus den chemischen Isotopenanomalien in meteoritischen Gesteinen, die auf die Erde gefallen sind, erschließt man ein Meteoritenalter von 15 bis 18 Milliarden Jahren, und die ältesten Sterne in den Kugelsternhaufen unserer Galaxie datiert man auf ein Alter von 18 bis 20 Milliarden Jahren. Es kann aber nicht angehen, dass solche Dinge wie Meteoriten oder Sterne älter als der Urknall sind [..]. With Google translate: Chemical isotope anomalies in meteoritic rocks that have fallen to Earth suggest meteorite ages of 15 to 18 billion years, and the oldest stars in our galaxy's globular clusters are dated to 18 to 20 billion years. But it cannot be that things like meteorites or stars are older than the Big Bang [ .. ] Precisely ! But these ages do fit quite well within the (orbital) timespan as predicted by UAC, which is $\ge -28,8$ billion years, that is $\,2/H_0\,$ instead of $\,\approx 1/H_0\,$. At Doug Marett's webpage The 100 Year Wrong Turn in Cosmology, two other objects are mentioned that may be older than the Big Bang: a galaxy and a star.

  1. Now Even Further: Ancient Galaxy is Latest Candidate for Most Distant - Universe Today
  2. The oldest known star in the universe, HD140283, also known as the Methuselah star
Needless to say that the ages of these objects too fit seamingless into the UAC timespan.

In the book Dark Matter, Missing Planets & New Comets by Tom van Flandern we find in the footnote at page 95 the following.

Now I know that the logarithmic relationship was not invented solely by Tom van Flandern. This has been investigated by Louis Marmet via A Cosmology Group (18 dec. 2022 04:23): Out of the references I have compiled, most models (72 out of 92) that derive the redshift from a physical process give a logarithmic relationship D = (c/Ho) ln(1 + z).
Let us employ the result that has been called Milne's Formula: $$ t-t_0 = (T_0-A)\ln\left(\frac{T-A}{T_0-A}\right) $$ On the other hand we have, using Intrinsic Redshift (called $\,z_i\,$ in the sequel) and Narlikar's Law: $$ \frac{m}{m_0} = \left(\frac{T-A}{T_0-A}\right)^2 \hieruit 1+z_i = \frac{m_0}{m} = \left(\frac{T_0-A}{T-A}\right)^2 \\ \hieruit \ln(1+z_i) = \ln\left[\left(\frac{T_0-A}{T-A}\right)^2\right] = -2\ln\left(\frac{T-A}{T_0-A}\right) $$ Now let $D=$ distance in deep space, $c_0=$ standard lightspeed in empty space, $H_i=$ Hubble parameter belonging to intrinsic redshift. With $\;D = c_0(t_0-t)\;$ we are looking into the past: $t < t_0\;$. Then the above can be rewritten as: $$ D = c_0(A-T_0)\left[-\frac{1}{2}\ln(1+z_i)\right] = c_0\cdot 2(-1/H_i)[-\ln(1+z_i)/2] \hieruit \\ \large \boxed{D=\ln(1+z_i)\,c_0/H_i} $$ Which is exactly Tom van Flandern's formula, apart from notation issues: distance $d\to D$ , lightspeed $c\to c_0$ , Hubble's law constant $H\to H_i$. Inverse function is the Intrinsic Redshift: $$ D = \ln(1+z_i)c_0/H_i \hieruit z_i = \exp\left(\frac{H_i D}{c_0}\right)-1 $$ Which means that the Redshift $z_i$ is rapidly increasing with cosmic Distance, beyond a certain distance. On the other hand we have: $$ 1+z_i = \frac{m_0}{m} \hieruit \frac{m}{m_0} = \exp\left(-\frac{H_i D}{c_0}\right) = e^{-H_i(t_0-t)} = e^{+H_i(t-t_0)} $$ So elementary particle rest mass is exponentially decreasing with cosmic Distance / atomic time reversed. But it never becomes absolutely zero. This is consistent with an infinite universe and the absence of a Big Bang. And elementary particle rest mass is exponentially increasing with future atomic (proper) time. Consistency is granted: $$ \frac{m}{m_0} = e^{+H_i(t-t_0)} = \left(\frac{T-A}{T_0-A}\right)^{H_i(T_0-A)} = \left(\frac{T-A}{T_0-A}\right)^2 \slechts \large \boxed{\,H_i(T_0-A)=2\,} $$ Here-and-now time $\,T_0\,$, of course, is continuously increasing. Which means that the the Hubble parameter is actually thought to be decreasing with time, as quoted from the Wikipedia webpage about Hubble's law. The same relationship between the Hubble parameter and age is found in the article by Jayant Narlikar and Halton Arp: FLAT SPACETIME COSMOLOGY: A UNIFIED FRAMEWORK FOR EXTRAGALACTIC REDSHIFTS. It is formula (13) in the article.
Exactly the same formula as the one above - though with a different meaning at first sight - is encountered in the Wikipedia page about the cosmological Scale factor. Quote from the header of this page: The relative expansion of the universe is parametrized by a dimensionless scale factor $a$. As a Detail it is seen that, in general, the (time dependent) Hubble parameter is defined as: $$ H(t) = \frac{\dot{a}(t)}{a} $$ In concordance with this is our own formula, found in Dark-energy-dominated era, where the Hubble parameter is supposed to be a constant: $$ a(t)\propto \exp(Ht) $$ In our theory, of course, there is no place for a "Dark-energy-dominated era". As we have argued elsewhere, dark matter apart from common matter may not be much of an issue. The same holds for dark energy. Varying elementary particle rest mass can be conceived as dark entities in disguise. Dark energy is not outside but inside common matter as well.
But how then a relative expansion of the universe with exactly the same law as the one for increasing elementary particle rest mass can be explained?
The secret is in the laws of Length Contraction. Any length that is bound to matter, whether that is a (measuring) rod or a light ray cast back and forth to the moon, obeys the following law. The length of a length measuring device is inversely proportional to varying rest mass: $$ \frac{L}{L_0} = \frac{m_0}{m} $$ There do not exist other measuring devices than those bound to matter in this way. But then we must necessarily conclude that any length that is measured by a rod, or by a light ray cast back and forth is observed as being directly proportional to varying rest mass. Consequently, our dimensionless cosmic scale factor $\,a\,$ must be like: $$ \boxed{\large\;\frac{a(t)}{a_0} = \frac{m}{m_0} = e^{H_i(t-t_0)} = \left(\frac{T-A}{T_0-A}\right)^2\;} $$ According to the Big Bang paradigm, Dark energy is supposed to be the cause of an acceleration in the expansion of the universe. Which can be readily accounted for by our theory. One has the choice between two flavours: atomic time or orbital time. $$ a(t) = a_0\,e^{H_i(t-t_0)} \hieruit \dot{a}(t) = a_0\,H_i\,e^{H_i(t-t_0)} \hieruit \ddot{a}(t) = a_0\,H_i^2\,e^{H_i(t-t_0)} \\ a(T) = a_0 \left(\frac{T-A}{T_0-A}\right)^2 \hieruit \dot{a}(T) = a_0\,\frac{2}{T_0-A}\left(\frac{T-A}{T_0-A}\right) \hieruit \ddot{a}(T) = a_0\,\frac{2}{(T_0-A)^2} \\ \frac{\ddot{a}(t_0)}{a_0} = H_i^2 \SP \frac{\ddot{a}(T)}{a_0} = \half H_i^2 $$ This might explain some. Remember, however, that our universe - being infinite in atomic time - is quite different from the ones that are advertised in e.g. Wikipedia:

  1. Our universe is not at all (nearly) empty; it contains all possible matter.
  2. An expansion of the universe has only to do with empty space outside atoms.
  3. This expansion can also be thought as the shrinking of our means for measuring lengths.
There is an atomic time analogue of the second formulation of Narlikar's law, where the last part is valid for small mass and time differences $(\Delta m=m_2-m_1\,,\,\Delta t=t_2-t_1)$: $$ \frac{m_2/m_0}{m_1/m_0} = e^{H_i(t_2-t_0)} / e^{H_i(t_1-t_0)} = e^{H_i(t_2-t_1)} \hieruit \\ \frac{m_2-m_1}{m_1} = e^{H_i(t_2-t_1)} - 1 \hieruit \frac{\Delta m}{m} \approx H_i\Delta t $$