a-cosmology-group/acg#188

@RedshiftDrift writes: >while your equation seems to be pulled out of a hat.

Maybe not. It has cost me some search, but ..

Here is a series of slides, simply titled Cosmology Slides. I think it’s a decent overview of the current state in cosmological science(fiction). Talking about distances on slide 60: >As mentioned earlier in the course within the frame of work of Robertson-Walker metric one can define a number of distances. This is due to the fact that distances are measured between objects that have different cosmic times, different expansion rates (along or across the line of sight) etc. It is this mixing of time and proper coordinates in the metric that gives rise to very different possibilities of defining distances in the Universe.

On slide 47, indeed, two formulas as mentioned by @mikehelland are discovered:

It’s based on the speed of light being $v = c - Hd$.

I actually found that you can use $v = c - Hd$ to fully calculate an FLRW model. You can check out this notebook.

https://github.com/mikehelland/hubbles-law/blob/master/other/python/cosmodistance.ipynb

This assumes the - Hd comes from offsetting expanding space $v = Hd$.

In my model, this comes from time running slower. Time is observed to run slower at $z > 0$. Clocks run at $1/(1+z)$, or $a$.

The logic is:

• Light is observed at $c$ for all observers, on their local contemporary clock
• An observer in the past has a clock that runs at $1/(1+z)$ compared to ours
• They still measure light at $c$, on their clock
• Therefore, the light around them is moving at $v = c/(1+z)$ compared to our clock

If you take:

$$1 + z = \frac{\lambda_{obs}}{\lambda_{emit}}$$

Multiply both sides by $f_{obs} / f_{obs}$ you get:

$$1 + z = \frac{c}{v}$$

Where we don’t know what $v$ is, but if $z > 0$ then $c > v$. Since $v < c$, then we can say $v = c - Hd$, where we don’t know what $d$ it, but we know when we multiply it by $H$ we get a velocity, and if we subtract that from $c$, we get another velocity, that if we divide $c$ by it, well, we might get a $z$.

$$c - Hd = \frac{c}{1+z}$$

$$\frac{1}{1+ z} = \frac{c-Hd}{c }$$

$$\frac{1}{1+ z} = 1 - \frac{Hd}{c }$$

$$\frac{z}{1+ z} = \frac{Hd}{c }$$

$$d = \frac{z}{1+ z} \frac{c}{H }$$

Well, that’s a lovely thought and all, but does it work? This code verifies the results:

data = [0]

H = 70
H = H / 3.08e19 * 60 * 60 * 24 * 365 * 1e6
c = 1
z = 0
t = 0
x = 0

while x < 7000:
    t -= 1
    x += c / (1 + z)
    z = c / (c - H * x) - 1

Plot x over z.

Then plot t over z. You’ll see the log() function.

(but I challenge you to convince Eric Lerner to reject this relationship).

If someone is in touch with him, send this page his way.

@mikehelland

Time is observed to run slower at $z\gt 0$

But then you have $1/(1+z)\lt 1$ so Time is observed to run faster instead of slower. This already destroys the rest of your reasoning, unless I’ve made an obvious mistake.

Angular size measurements of galaxies show that we see them at a distance as far as 35 Gly that is much larger than a Hubble distance, yet $\displaystyle d = \frac{c}{H} \frac{z}{1+z}$ implies a maximum distance beyond which we should not see galaxies.

@RedshiftDrift

Angular size measurements of galaxies show that we see them at a distance as far as 35 Gly that is much larger than a Hubble distance, yet d=cHz1+z implies a maximum distance beyond which we should not see galaxies.

Is there a good dataset you recommend you to investigate that further?

Let’s suppose $z=9$. LCDM gives a comoving distance of 30 Gly.

My model says the galaxy is at a distance of $0.9 c/H$.

Accounting for the time dilation experienced by a photon traveling from $z=9$ to $z=0$, is $c/H \log (1+9) = 32$ Gly.

The angular size the galaxy appears as should be based on this value.

Note that at $z>8$ galaxies in my model have a larger luminosity distance than LCDM.

@HanDeBruijn

But then you have $1/(1+z) < 1$ so Time is observed to run faster instead of slower.

Clock rates are $1/(1+z)$, or $a$.

So at $z=1$, the clock (there and then) is running at $1/2$ speed as compared to our clock .

@mikehelland . >Clock rates are $1/(1+z)$ or $a$

I don’t know what you mean by “rate”. Clock periods anyway are proportional to $(1+z)$.

So at $z=1$, the clock (there and then) is running with period $2$, that is indeed $1/2$ speed as compared to our clock .

But how about your purported derivation?

@mikehelland A recent data set, which includes hundreds of galaxies and JWST observations, is given here https://www.mdpi.com/2075-4434/10/6/108.

From Fig. 5 (copied below) with $d_A \propto 1/\theta$, comparing the angular size at $z_1=0.1$ and $z_2=10$ gives an observed ratio $\theta_1/\theta_2 = d_A(z_2)/d_A(z_1)$ of about $25$ (marked on the graph).

Comparing that number with theory: $\displaystyle d_A = \frac{c}{H}z \quad$ gives a ratio of $100$; the linear “distance vs z” model proposed by Lerner is falsified. $\displaystyle d_A = \frac{c}{H} \frac{z}{1+z} \quad$ gives a ratio of $10.0$; your model @mikehelland is falsified. $\displaystyle d_A = \frac{c}{H_0}\ln(1+z)$ (labeled ‘TL’ on the figure) gives a ratio of $25.2$ which fits the data very well. $\displaystyle d_A = \frac{c}{H_0} \int_0^z \frac{{\rm d}z’}{\sqrt{\Omega_m(1+z’)^3 + \Omega_\Lambda}} \quad$ gives a ratio of $2.29$; ΛCDM is very falsified.

In the ‘non-expansion paradigm’, there is no currently observed limit to the size of the universe up to a distance of 35 Gly (z = 13.2; spectroscopically confirmed) and up to 40 Gly (z = 20; photometrically confirmed).

@RedshiftDrift

Thanks for the link.

As mentioned in my previous comment, the physical distance ($(c/H) (z/(1+z))$) is not how it appears on the sky in my model. That’s affected by time dilation.

“Accounting for the time dilation experienced by a photon traveling from $z=9$ to $z=0$, is $c/H \log (1+9) = 32$ Gly. The angular size the galaxy appears as should be based on this value. “

@mikehelland . According to that scheme on slide 47 in a previous reply there would have been an opportunity to test your formula $d = z/(1+z).c/H$ in a universe with Radiation alone. But unfortunately that Photon epoch has “occurred” Between 10 seconds and 370,000 years after the Big Bang, which is quite a while ago :-(

@RedshiftDrift . A more or less preliminary version of this posting is found in thread 160. The mathematics below is meant as a heuristics for the ‘TL’ formula $\displaystyle d_A = \frac{c}{H_0}\ln(1+z)$ where we both agree upon. Take a look at the picture, depicting a timeline in cosmos back in time. Let (2) be our position on earth, (1) is the position of a far away civilization like ours, (0) is the position of a galaxy observed by observers on both habitable planets. Let $\lambda=$ wavelength of light. Then it is obvious that:

$$ \frac{\lambda_1}{\lambda_0} \frac{\lambda_2}{\lambda_1} = \frac{\lambda_2}{\lambda_0} $$

And therefore, by definition of $z_{ab}=(\lambda_b-\lambda_a)/\lambda_a$ :

$$ 1+z_{02} = (1+z_{01})(1+z_{12}) $$

It is reasonable to assume that the whole $\Delta t_{02}$ is the sum of the parts $\Delta t_{01}$ and $\Delta t_{12}$ and so

$$ 1+z = F(\Delta t) \quad \mbox{where} \quad F(\Delta t_{02}) = F(\Delta t_{01}).F(\Delta t_{12}) = F(\Delta t_{01} + \Delta t_{12}) $$

This makes the function $F$ a special one, because it is seen now that the following property must hold for all $(t,\tau)$.

$$ \ln(F(t+\tau)) = \ln(F(t)) + \ln(F(\tau)) $$

We need another preliminary: Additive functions and measure theory. From that reference we infer that

$$ \ln(F(t)) = \kappa.t \quad \Longrightarrow \quad F(t) = 1+z = e^{\kappa.t} $$

where $\kappa$ is a constant that has to be determined from physics. Employing $1+z = e^{\kappa.t} \approx 1+\kappa.t$ and Hubble’s law $z = - H_0.t = \kappa.t$ for (relatively) small cosmic lightyears we get that $\kappa=-H_0$ . And $d_A = -c.t$ then leads to the conclusion:

$$ 1+z = e^{-H_0.t} \quad \Longrightarrow \quad d_A = \frac{c}{H_0}\ln(1+z) $$

@HanDeBruijn

The equation:

$$d = \frac{z}{1+z} \frac{c}{H_0}$$

appears as the angular diameter distance in de Sitter, $\Omega_\Lambda=1, \Omega_m=0$, and as the light travel time ins the Milne Model, $\Omega_\Lambda=0, \Omega_m=0$.

This is mostly a result of how the Friedmann equation works. Just because they have the same equation, doesn’t mean they’re the same. Here they represent two different things.

Your example from a “radiation dominated” universe is like it. Sure, it’s the same equation. It means something different in that context though.

My model, unlike FLRW, doesn’t have radiation, matter, and dark energy epochs. That’s a different model entirely.