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There is some evidence that Planck’s constant has changed by something like one part in a million since the early universe. And cosmological inflation theory talks about another constant changing, I think. Although even after reading about it I can’t tell which one(s).

  1. Can inflation theory be characterized as a fundamental constant(s) changing? If so which one(s) and how do they relate to expansion?

  2. Do constants have to change relative to each other for it to matter? At first I thought that doubling all the constants tomorrow would be undetectable, or even meaningless by construction. But then I thought since some equations are nonlinear, changing all the constants relative to the past would matter. Which of these is right?

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    $\begingroup$ Which quantities do you want to double in $\alpha=\dfrac{ke^2}{c\hbar},\,k=\dfrac{1}{4\pi\varepsilon_0}=\dfrac{\mu_0c^2}{4\pi}$? $\endgroup$
    – J.G.
    Jul 30, 2021 at 14:14
  • $\begingroup$ I always thought there was a clear definition of the list of “fundamental constants”? Like constants for “strong nuclear force, weak one, electrostatic force, speed of light in vacuum, charge of an electron, mass of a proton (or maybe a quark?)” stuff like that. I dont know much about it, but does that answer? Looking at that I would guess probably not talking about k or alpha $\endgroup$
    – Al Brown
    Jul 30, 2021 at 14:19
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    $\begingroup$ It's hard enough deciding whether $\alpha$ or $\sqrt{\alpha}$ should be shortlisted. $\endgroup$
    – J.G.
    Jul 30, 2021 at 14:22
  • $\begingroup$ I think i see. So maybe the answer is “in theory there may be a long list of constants that could all be doubled and not matter or even have any meaning” but if so we dont confidently know that list and it would be somewhere around X constants long. What is X? What about the first question, #1 above, if willing? $\endgroup$
    – Al Brown
    Jul 30, 2021 at 14:25
  • $\begingroup$ @Al Brown If all constants changed in a very specific way, i.e. proportional to their length dimensions, then it wouldn't be noticeable physics.stackexchange.com/questions/620794/… whether it is meaningful is another question... $\endgroup$ Jul 30, 2021 at 16:01

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It is not possible for all constants to change proportionally, due to the various relationships between constants. A good example is the fine structure constant which can be written as: $$\alpha =\frac{\mu_0 e^2 c}{2h}$$ So if we double all of the terms on the right then the fine structure constant goes up by a factor of 8.

To answer your broader question the only type of physical constant changes that would produce physically measurable results would be those that change the dimensionless constants like the fine structure constant. So, for example, if $e$ doubled and $h$ quadrupled, with everything else on the right staying the same, then we would not detect any measurable difference.

Sometimes there are various theories where some dimensionful constant changes, but the actual physical change is due to the resulting change in the dimensionless constants.

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  • $\begingroup$ If all the terms were to go up by a factor $2^n$ where $n$ is the number of length dimensions in each quantity, then $\alpha$ would be unchanged. $\endgroup$ Jul 30, 2021 at 16:03
  • $\begingroup$ Sure, that would just be changing the SI definition of a meter. You could do the same with any units. That is precisely why such changes don’t change the physics $\endgroup$
    – Dale
    Jul 30, 2021 at 16:05
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    $\begingroup$ If it were proportional to time like $h = h_0 e^{2kt}$, (the 2 is for the length dimension of h), then no change would be noticed locally, but distant stars would have atoms that are smaller than atoms here, so it's tricky to decide whether it could be happening like that... $\endgroup$ Jul 30, 2021 at 16:11
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    $\begingroup$ @AlBrown the core concept is right, but it is not a list of constants so much as combinations of constants. The definition/criteria is very clear. Any combination of constants that leave the dimensionless constants unchanged could be scaled without effect. $\endgroup$
    – Dale
    Jul 30, 2021 at 17:14
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    $\begingroup$ @AlBrown yes, any combination of units in any system of units could be scaled arbitrarily with no change to the dimensionless constants. $\endgroup$
    – Dale
    Jul 30, 2021 at 17:49
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Yes, many constants could be changing and we wouldn't know...

It's possible that an expansion of all length scales can be happening, as in the cartoon below.

enter image description here

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...

It would be hard to rule out such an expansion and changing of constants, especially if the redshift of light from distant stars is regarded as due to it. 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.

It also 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.

enter image description here and enter image description here

The diagrams show the distance modulus predicited by this type of expansion, 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 this type of expansion, 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.

So the answer to your question is that the fundamental constants could be changing in proportion - and as such a changing constants situation actually matches all observations, it is very difficult to rule it out.

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