Cosmology - an expansion of all length scales From the link Is non-mainstream physics appropriate for this site?
"a question that proposes a new concept or paradigm, but asks for evaluation of that concept within the framework of current (mainstream) physics is OK."
Here is a concept, evaluation within the framework of current (mainstream) physics would be welcome.
Is it possible that an expansion of all length scales can be happening, as in the cartoon below?

It shows all lengths increasing, the size of atoms, people, stars and the distances between all objects.
Each physical quantity and constant varies depending on the number of length dimensions in it.  For example since Planck's constant has a length dimension of 2, so it's change with time is
$h=h_0e^{2Ht}$
where $H$ is an expansion constant and $t$ is time.
\begin{array}{c|c|c}
{quantity} & {length-dimension} & {change}\\
\hline
length & 1 & e^{Ht}\\
mass & 0 & constant\\
time & 0 & constant\\
h & 2 & e^{2Ht}\\
c & 1 & e^{Ht}\\
G & 3 & e^{3Ht}\\
Area & 2 & e^{2Ht}\\
\end{array}
etc...
Can this type of expansion be ruled out A) locally  or B) by distant measurements e.g. of distant stars or galaxies, from within mainstream physics?
The expansion referred to occurs for the whole universe.  It's proposed as there could be another reason for the redshift of light from distant stars.  If the energy of a photon is conserved during flight, but was emitted when Planck's constant was lower, then from $E=hf$, the frequency of the received photon would be lower and the light from a distant star would be redshifted.
A bounty has now been added.  A convincing reason why the above type of expansion cannot be occurring would be welcome.
Here is the work done so far.
It is to determine the apparent matter density that would be concluded in a flat universe, with a matter density of $1.0$ and the type of expansion above.
It leads to the conclusion that the matter density would be measured to be $0.25$ or $0.33$ from galaxy clusters and supernovae data respectively.  A Diagram of supernovae data is below and then more details of the calculations.
  and  
The diagrams show the distance modulus predicited by the type of expansion in the question, top curve.  Concordance cosmology with a matter density of 0.3 and 1.0 are the middle and bottom curve respectively.  The second diagram is an enlargement of the first.
Matter density from Galaxy Clusters etc...
Traditionally the scale factor of the universe at redshift $z$ is
$a=\frac{1}{1+z}\tag{1}$
If the energy of the photon is conserved during flight, from $E=hf$ and $h=h_0e^{2Ht}$
For an emitted wavelength of $\lambda_1$
$z=\frac{\lambda_1e^{2Ht}-\lambda_1}{\lambda_1}$
$1+z = e^{2Ht}=a^{-2}$ ,
($a$ decreases with increasing $z$ in an expanding universe)
so
$a=\frac{1}{\sqrt{1+z}}\tag{2}$
For small distance $d$
$\frac{v}{c} =z= e^{2H\frac{d}{c}}-1=\frac{2Hd}{c}$
$v=2Hd\tag{3}$
i.e. Hubble’s law is still valid but we identify the expansion parameter $H$ with half of Hubble’s constant $H_0$
this leads to the conclusion that the matter density will be measured to be $\frac{1}{4}$ of the true value, as follows.
$\Omega_m = \frac{\rho}{\rho_{crit}}\tag{4}$
$\rho_{crit}=\frac{3H(z)^2}{8\pi G}\tag{5}$
If the value for $ H(z)$ used in $\rho_{crit}$ is twice the true value, then the apparent matter density would be measured as $0.25$ instead of $1$.
Matter Density from Supernovae Data.
In LCDM the Hubble parameter is
$H(z)=H_0\sqrt{\Omega_m {(1+z)}^3+\Omega_k{(1+z)}^2+\Omega_\Lambda}$
The comoving distance is obtained from
$D_M=\int_0^z \frac{c}{H(z)} dz$
Using a flat universe approximation, omitting $\frac{c}{H_0}$ and using $m$ for $\Omega_m$ ,the comoving distance, for small $z$ is
$\int_0^z(m(1+3z+3z^2+\dots )+1-m)^{-\frac{1}{2}}dz$
$=\int_0^z(1+3mz+3mz^2)^{-\frac{1}{2}}dz =\int_0^z(1-\frac{3}{2}mz+\dots)dz$
$=z-\frac{3mz^2}{4}\tag{6}$
For the type of expansion that we hope to rule out,
The co-moving distance is
$D_M=\int_t^0 \frac{c}{a(t)} dt$
$a=\frac{1}{\sqrt{1+z}}$
$\frac{da}{dt}=\frac{da}{dz} \times \frac{dz}{dt} ={-\frac{1}{2}(1+z)^{-\frac{3}{2}}}\times\frac{dz}{dt}$
$H(z)=H=\frac{\dot{a}}{a}=\frac{-1}{2(1+z)}\times\frac{dz}{dt}$
$dt=\frac{-1}{2H(1+z)}dz$
$D_M=\int_0^z \frac{c}{2H}{(1+z)}^{-\frac{1}{2}} dz$
$D_M=\frac{2c}{H_0}(\sqrt{1+z}-1)\tag{7}$
again omitting $\frac{c}{H_0}$ and for small $z$, $(7)$ becomes
$2(1+\frac{1}{2}z-\frac{1}{8}z^2-1)$
$=z-\frac{z^2}{4}\tag{8}$
there is a match between $(6)$ and $(8)$ if $m=\frac{1}{3}$
So we conclude from Galaxy and supernovae data, or combinations of data sets, that the matter density would be measured, with the type of expansion in the question, at between $0.25$ and $0.33$.  As it is measured at this value, it's concluded that the expansion cannot be ruled out this way.  A diagram with supernovae data is above.
Is there a convincing reason why the expansion described should be ruled out?
 A: Mainstream physics/cosmology says that local systems that are held together e.g. by gravity or electromagnetic forces do not take part in the global expansion. Our solar system had the same size billions of years ago (there is certainly no evidence to the contrary) and atoms in galaxies billions of light years (in space and time) away have the same size as those locally (as one can conclude from the spectra of distant objects).
See also this reference https://arxiv.org/abs/gr-qc/0508052
Anyway, if your ruler expands as well (like you have drawn above) there would not be an expansion of the universe in the first place, as you would always measure the same distance to a galaxy.
A: 
the frequency of the received photon would be lower

Why would it ? Since $c=\lambda f$ and $c$ and $\lambda$ change in the same proportion then $f$ is constant. All you are doing is changing the units in which length is measured. You get exactly the same effect if you measure the wavelength in furlongs instead of metres, and denominate the speed of light in furlongs per second - frequency remains unchanged.
A: I ll just simply write couple of reasons why its not possible.

*

*If you are talking about everything getting bigger in size then the measurement devices (such as rulers, etc) will also get bigger at the same amount. So even that is the point theres no real way of measuring it. So its not reasonable to talk about it. I can also argue that everything is getting smaller ? Can you argue that as well ?


*The force between the two electrons is about $10^{40}$ times larger than the gravitational force. If the expansion of the universe does not have any effect in our solar system (which is governed by gravitational force), then clearly (and logically) the expansion of the universe cannot have any effect on the atomic-scales.


*From a simplest point of view, if just my size increases but my mass stays the same my density must get lower and lower which is not the case for me or for any other object that is around you.


*Occam's razor - Why everything should be expanded at the same amount in the first place ? Whats the point ?
A: If all things expand, atoms become bigger. Bigger atoms have at least different electromagnetic spectra. Hence, if we look at the hydrogen lines from the andromeda galaxy, we see them like they were 2.5 million years ago, and that would mean, emitted by much smaller atoms. The energy levels of the hydrogen atom are proportional to the reciprocal of the Bohr radius $a_0$ and if this was smaller in the past, that means higher energy differences, or higher frequencies.
Of course the wavelength of the received light would have been expanded since then, due to the alleged spatial expansion, but their frequency would have stayed the same, because time is not affected by the expansion, as I understand you.
Hence, we should see different atomic spectra for distant objects, namely blue-shifted ones (higher atomic frequencies in the past), which has not been observed, as to my knowledge.
A: It is hard to write a precise answer because it is not clear to me that the concept being presented is sufficiently worked out to make something one can assess. This may be a failure of understanding on my part, but I hope it will help if I simply give a few reactions so that you can see what a physicist with reasonable general knowledge of this area makes of this (but I am not a cosmologist).
At the start one first suspects that the question is asking about something unobservable but I think it is not doing that. It is proposing observable things. But it is not clear to me that it hangs together. For example, the question assumes that the combination
$h f$ for a photon is conserved as it propagates long distances, and if $E$ is
constant and $h$ varies as $h = h_0 e^{2 Ht}$ then one has
$$
f = f_0 e^{-2 H t}.
$$
Meanwhile the question also proposes $c = c_0 e^{Ht}$ so this gives, for the wavelength,
$$
\lambda = \frac{c}{f} = \frac{h_0 c_0}{E} e^{3 Ht}.
$$
The next equation in the question reads
$$
z \stackrel{?}{=} \frac{\lambda_1 e^{2 Ht} - \lambda_1}{\lambda_1}
$$
where I have put a query in order to signal that I am not sure where this equation came from since I was expecting $e^{3 Ht}$. I suppose what might have happened is that $z$
was being defined as $z \equiv (f_{r}^{-1} - f_{e}^{-1})/f_e^{-1}$, where the subscripts stand for
'received' and 'emitted', so then one has
$$
z \equiv \frac{f_{r}^{-1} - f_{e}^{-1}}{f_e^{-1}} = \frac{f_e}{f_r} - 1 = e^{2 Ht} - 1
$$
and the rest of the question assumes this result.
The above reveals one difficulty I have with the whole approach. It is not clear to
me that it can hang together overall. When we detect the light from distant supernovae, the instruments to measure red shift use, I think, optical methods such as diffraction
gratings and Michelson interferometers, so it is wavelength not frequency that they are
measuring. But when they detect luminosity then it is, I think, more like an energy measurement. Meanwhile, when we do stellar physics calculations to describe the supernovae (or other stars) we adopt the standard methods and mostly think in terms of
energy and frequency. So if frequency and wavelength are scaling differently as cosmological time goes on, we don't have, at the outset, any idea of which of our calculations to trust, or which aspect of them, until a lot more working out has been done. And the suspicion is that that working-out cannot be done because it won't in fact hang together as a logical whole. I am not being so bold as to assert that; I am simply giving a response which signals some of the issues which need to be addressed before the ideas can be taken forward.
