There’s one coming soon. These past two years have been tough on me, on my health, and with a few rejected papers. But I have plenty of material for the next newsletters including a list compiled by Mike Helland on galaxy formation problems…
“The ACG Newsletter highlights observational results that are anomalous in terms of the Big Bang paradigm.”
Suggestions are welcome. Post them here in this conversation!
@MJoseph315 I made the mistake to submit my last three papers to astronomy journals. STz redshift is now a non-relativistic fully quantized calculation, so it predicts many astrophysical phenomena but is not advanced enough to accurately describe the CMB (a relativistic version of STz is needed for that). My papers were rejected because reviewers expect a fully completed cosmology, “STz is obviously wrong since it can’t predict the CMB.”
Pneumonia & stress triggered an hemolytic anemia. I’ve been improving slowly since then, but a brain with more oxygen works faster! :wink:
@RedshiftDrift writes:
>not advanced enough to accurately describe the CMB (a relativistic version of STz is needed for that).
How can people be so sure that relativity is a panacea for anything?
And by the way, I have always thought that Quantum ElectroDynamics IS already - by definition - relativistically invariant.
My papers were rejected because reviewers expect a fully completed cosmology,
Take the advice by Goethe: In die Beschränkung zeigt sich erst der Meister.
With other words: showing some restraint (and cutting into pieces) would help.
a brain with more oxygen works faster!
Let them drill a few holes in your skull, like they did to me after I fell, though not in the bathtub.🤕
I was just wondering where to post this and still not sure if this is the right discussion, but I grappled with a GR vs ΛCDM paradox this morning.
It is difficult to refute the claim that cosmic expansion cannot overcome atomic and molecular forces. The expansion is too weak and too slow. So even if one suspects this argument is flawed, it cannot be easily disproven. The same ‘slow’ can be argued to not rip apart a gravitational bound system.
But the ‘slow expansion of space’ is also changing the properties of space, upon which GR is built.
The same defense fails for gravitationally bound systems. A star, a galaxy, or even a galaxy cluster is held together by gravity, which, in General Relativity, is the spacetime metric. If space expands, the metric must expand, and gravity must change accordingly. This is not a question of forces overcoming expansion, it is a question of the very fabric of spacetime evolving. Since space and therefore spacetime itself permeates matter (stars, galaxies…) it (spacetime) must also expand. The ‘too weak’ argument then fails over a cosmic time period.
Reminder that ΛCDM does not say that ‘space does not expand within matter’; it merely states that expansion is too weak to pull local forces apart, thereby acknowledging that all space expands. For ΛCDM to deny that space expands within systems, would be tantamount to saying ‘gravity is keeping space together’; a logical fallacy.
Over billions of years, space has expanded by a factor of $$(1+z)$$ since recombination. A primordial cubic meter of spacetime has grown to a cube of $$(1+z)^3$$ cubic meters today. This expansion must have altered the local properties of spacetime — its curvature, its energy density, and its coupling to matter, even within stars, galaxies, and clusters. Yet no such changes are observed. Gravity today appears to behave identically to gravity billions of years ago. The ΛCDM response — that expansion does not affect bound systems — is a logical convenience, not a physical mechanism. If a cubic meter of spacetime has grown, but the matter within it has not expanded at all, the metric must show signs of this tension.
The FLRW metric:
$$ds^2=−c^2dt^2+a(t)^2(dr^2+r^2dΩ^2)$$
In flat space c=s/t must remain constant. In vicinity of matter it requires that the local metric evolve with a(t). An expanding ‘flat space’ requires s to change to give redshift, but then t must also change for c to remain constant. All clocks would be affected. ΛCDM would refute this by saying all clocks are within bound systems, and we are back where we started.
However, if a(t) has also changed by a factor of $$10^3$$ since recombination (or insert your own favorite value for z into $$(1+z)^3$$), then local spacetime properties (within bound systems) must have changed by a corresponding amount. Since no such changes are observed in the gravitational behavior of stars, galaxies, or clusters, the hypothesis of expanding space must be refuted — or at least recognized as a logical paradox that conflicts GR with ΛCDM. Since ΛCDM is built on GR, this paradox cannot remain unresolved.
The claim that bound systems are exempt from expansion is logically circular: it asserts that the metric does not apply inside bound systems, which contradicts the very definition of spacetime. Since it is agreed that bound systems do not expand, it must be concluded the metric does not expand. The only coherent resolution is that redshift is not a metric effect.
This should probably be in its own thread, as it is a complex topic and a fundamental one. It’s not true that expansion of gravitational systems is not observed. We see it in a number of them: the AU, the Moon’s orbit around the Earth, Titan’s orbit around Saturn. The radii all increase in proportion to the Universe’s own expansion. I touched on this in my paper on the origin of the Moon:
a secular increase in the orbital radius of moons about planets and of planets around the Sun (Zhang et al. 2010; Křížek et al. 2022; Vavryčuk 2023).
It also relates to the increase of supermassive black holes over time. While these observations seem to suggest that the Universe is expanding like they say, I argue that it is does not necessarily. On local scales, the force of gravity overpowers the Hubble luminosity completely. The expansion of radii that we see is just the residual outward push when you cancel gravitational infall with centrifugal force outwards. But for the universe as a whole the rotational state is uncertain. Suppose it is not rotating. At the radius corresponding to the Hubble radius c/H, the Hubble luminosity completely balances the gravitational force. (It’s a bit more complicated than that, as the CMB is heavily involved as well, being replenished by the Hubble luminosity.) At the cosmic scale, and in black holes, the Hubble luminosity is the same as the Planck power c^5/4G.
@Francois-Zinserling writes:
>The claim that bound systems are exempt from expansion is logically circular: it asserts that the metric does not apply inside bound systems, which contradicts the very definition of spacetime. Since it is agreed that bound systems do not expand, it must be concluded the metric does not expand. The only coherent resolution is that redshift is not a metric effect.
The problem is completely solved by the Variable Mass (VM) theory. Remember that - according to a the 1993 publication by Jayant Narlikar and Halton Arp, VM is essentially General Relativity in disguise: Finally, the Euclidean, flat spacetime becomes a natural, primary reference frame in which cosmological processes are most simply described. The Variable Mass Theory is compatible with a static Euclidean universe, which is eternal in Atomic time and has moments of creation (i.e. one or more beginnings) in Orbital time.
The above is a quote from Atomic Time, Orbital Time and the Variable Mass Theory. Section 4 of the accompanying PDF is named Orbital ticks. Theory is developed in that section with the explicit condition that space within the - gravitationally bound - orbit is constant and Orbital time is the only variable: the condition (r = constant) is a necessary one.
Theory about Atomic ticks (section 3) is somewhat more complicated and requires knowledge about the technique of Nondimensionalization and the Schrödinger equation for general atoms. Contrary to orbital distances, it turns out that atomic distances are variable with Variable Mass, but the effect of this on energy levels - i.e. the intrinsic redshift - is neutralized within the dimensionless formulation.
Then we arrive at Variable Mass Theory and Observable Universe. The relevant section in there is 2: Length Contraction. It is shown that all distances in the (sub)atomic domain are varying inversely proportional to (elementary particle rest) mass. It affects in this way how all of our measuring devices work. And in the end it is logically concluded that Length Contraction [of bound systems] with VM is observationally equivalent with the Expansion of Empty Space.
Yes, Francois, Empty Space. What do you think? Is your Aether an Empty Space?
@ExpEarth writes:
>the AU, the Moon’s orbit around the Earth, Titan’s orbit around Saturn. The radii all increase in proportion to the Universe’s own expansion.
That’s in agreement with the VM point of view: : the condition (r = constant) is a necessary one. The constant r is thus belonging to Empty Space and “it seems” to expand with the shrinking of our measuring instruments. (What is the meaning of “it seems” in this context)
>At the cosmic scale, and in black holes, the Hubble luminosity is the same as the Planck power c^5/4G.
Have you elaborate this somewhere, Matt, together with your views upon the CMB? (Because as for myself I am stuck with the latter)
@HanDeBruijn no ‘my’ aether is not empty space. I will read your links and hope to respond soon.
The Planck power $$c^5/4G$$ relates to the units of my aether model, with $$P_P=E_P/t_P$$ as one of the basic foundations. (not sure about the $$1/4$$ ratio from @ExpEarth. Along with the other Planck units, $$L_P$$ (length), $$p_P$$ (momentum), but in my model it permeates all of space - not just at the cosmic horizon. It was not assumed but concluded from the gravitational derivation.
I talked about the luminosity c^5/4G (and also the CMB) previously in my shell universe paper #256 . That power is connected to the inner, contained constant luminosity in all black holes and in the black hole universe. The quantity c^5/G is what they call the Planck power or Planck luminosity, which I think is just the Planck unit for power similar to the Planck mass, etc. The extra factor of 1/4 is considered the ‘maximum power’ that one can actually see in nature’s discussed by Schiller and others. I derived this max power from separate principles connected to what I have called the Hubble luminosity. This is the energy released from gravitating systems according to L = - U H. U is conventionally negative and so it becomes positive. On the cosmic scale, and in black holes generally, this becomes the max power c^5/4G.
So Francois, you and I are on the same page again it seems. I agree there is a Planck luminosity filling all space. In my models, there is a constant and equal drain of energy from space to power gravity. Most people using GR just think of gravity as this steady force acting with no energy input at all, just the geometry induced by mass. Totally wrong. I’m curious how you use the Planck power in your model. I’ve made great progress in my model, but the whole question of what is charge and magnetism is still on the to do list.
On the Planck units, I had been highly skeptical of them having any particular significance. Years ago Harry Dart showed that you get a totally different set of values if you derive the values using units of acceleration for G instead of G’s normal units. It could be an interesting exercise for people to try this and see what units they find for mass, length and time. Using units of acceleration takes us into MOND territory. Harry and I submitted a paper to Science about Harry’s new units, but when Harry found out I was using MOND he freaked out and told me to withdraw the paper. He had apparently only agreed to submit it al all because he had fallen in the bath tub and hit his head! But Science rejected the paper before I could withdraw it. @RedshiftDrift might remember this tale. Now the Planck power is different, as it involves only two of the three constants. Yet it seems correct.
@ExpEarth the luminosity $c^5/(4G) = 8.5\times 10^{52}$ Watts. Just to satisfy my curiosity, could you give us the value of the energy density in space away from galaxies, in Joules per cubic meter, that this value implies?
(Such numbers could make me fall in the bath tub! :rofl: )
Okay @RedshiftDrift , here it goes. Better to read outside the bathroom!
We have to go back to how the black hole universe formed and look at the mass density. Since there was enough mass to form a black hole, the mass density has to be the same as the critical density. Let’s say the widely dispersed baryonic mass that formed the shell universe has mass M0 and energy M0c^2. When it collapses to form a black hole, the gravitational energy released is trapped inside. This becomes the CMB eventually.
But when a black hole forms in GR, there is a mass defect which is referred to as the binding energy. In GR they don’t really say where that energy is or what happens to it. You’ll recall that I talked a lot about photon filaments connecting baryons in my earlier papers. This energy was also identified with U. ( @Francois-Zinserling, you’ll remember saying this would break the computer of the universe if it were true.) There’s a simpler way. The photon filaments are just the standing wave nodes connecting the shell baryons. Think of the shell as a cavity resonator. In GR we would say this lattice energy is negative and its pressure negative. Now the energy released and trapped as CMB photons equals the quantity U of the product shell. So what we have is M0c^2 = 1/2M0c^2 (baryon) + 1/4M0c^2 (CMB) + 1/4M0c^2 (lattice).
Because the lattice energy and pressure are negative, they offset the positive CMB energy and pressure. That is key, since radiation pressure and energy add to the active gravitational mass in GR. I totally missed that in my shell paper. That leaves the outward CMB pressure to hold the shell in place.
So getting back to your question, the CMB energy density is 1/4 the critical density, amounting to about 10^-9 erg/cc. If it were spread evenly throughout the shell universe, its temperature would be 27 K. But because of the shell’s gravity gradient, it peaks at 29 K near the shell and falls linearly to 2.7 K at our position.
@ExpEarth How is the average energy density $\bar E = 10^{-9}$ erg/cm^3 related to the luminosity? It is through the “age” and “volume” of the universe?
Question # 2: why does the temperature fall linearly from 29 K to 2.7 K? Shouldn’t the energy density fall linearly instead?
If it’s redshift that causes the linear drop from 29 K to 2.7 K, does that mean the maximum observable redshift is z = 29/2.7 = 10.7 ?
@RedshiftDrift That is correct. If we have the Planck power pumping energy into the universe at a fixed rate, multiply by the recycling time of the universe (not the ‘age’) and divide by the volume, then we will have the CMB energy density you have on the right. But the initial CMB energy comes from gravitational energy released during the shell formation.
In my model the CMB and the gravitational energy U are in balance. So the same equation can be reversed to get the reverse flow of energy. The CMB energy is being Hubble redshifted and that energy goes into gravity. In that case you take the average energy density of the CMB, which matches a 27 K CMB, and multiply by the Hubble constant (which we can write as roughly 10^-18 sec^-1). That just takes the time from your numerator on the left and puts it into the denominator on the right. That yields the Planck power per unit volume on the left side.
Question for @Francois-Zinserling. What happens to the accumulated Planck power energy in your model?
On Question 2, no, it’s the temperature. The temperature of the CMB tracks the gravitational gradient. Wherever radiation comes into thermal equilibrium with a gravity gradient, it will form blackbody radiation having a gradient in T which matches the redshift. This also fits with the supernova time dilation. This is due to the Tolman-Ehrenfest relation.
, even if I paste these direct into a browser. Is anyone else having this problem?
@ExpEarth in my model the vacuum energy remains constant but it changes mode when interacting with matter. In short, the dark pairs of Annila et. al interact with a fundamental particle of mass (and charge) and the particle changes the mode of the pair to bright. Bright pairs still have $$E=E_P$$ but no longer dark so new rules apply. From this mode change follows a heuristic derivation of gravity and electrostatics. New paper coming soon,
Question # 2: why does the temperature fall linearly from 29 K to 2.7 K? Shouldn’t the energy density fall linearly instead?
If it’s redshift that causes the linear drop from 29 K to 2.7 K, does that mean the maximum observable redshift is z = 29/2.7 = 10.7 ?
On the last part of your question, this had been puzzling me but I think I might have it now. At the link below you can see the layout of the shell model using Ni’s approach. Note that the gravitational gradient, which determines the Hubble constant, actually drops to 0 at the midpoint of the shell. The Ni approach works because matter is drawn from both sides of the shell midpoint towards the midpoint. Now I think the 29 K CMB could apply as an average temperature for the whole shell. This would have z = 10 as you said. But at the midpoint the temperature of the CMB could rise further. If the midpoint corresponds to z = 14, then the temperature would be about 41 K. It could go even higher. There is an odd anomaly for galaxies on the far side of the midpoint. These will seem unusually faint for their redshift.
I notice something else on that graph; the gradient is very large near the centre, as if the Hubble constant becomes much larger than 75 km/s/Mpc at $0.05 R_U$. How can that be?
From what I can make out, we should be able to see beyond the shell. But the redshift will be misleading, because there will be an intervening blueshift as photons move from the exterior space down the gradient towards the midpoint of the shell before they are redshifted again on their way towards our position.
The graph was really just an attempt to project the Ni-Neslusan structure onto a shell universe. I initially had the 75 value representing the extra gradient arising at the centre in their calculations, but superimposed onto the KBC Void, which I was speculating to be at the very centre. Now I have shifted the emphasis off the KBC Void being dead centre, as we actually need it to be off centre to account for the quasar dipole. What I would say now is that the CMB dipole is caused by the outward motion of galaxies in the KBC Void towards the walls of the void and the quasar/cosmic dipole to be die to our being closer to one cosmic wall than the other. The bottom line is that things get a bit confused inside voids due to the generally outward flow of matter in voids.
@ExpEarth writes in this reply :
>Let’s say the widely dispersed baryonic mass that formed the shell universe has mass $M_O$ and energy $M_Oc^2$ [: notation of our own]
It’s all about the “luminosity” $c^5/(4G)$. Let’s see how that fits into the Variable Mass theory. Or not.
Thus energy of mass times the Hubble parameter equals Planck power, apart from a numerical factor of about $2W \approx 7$. I think (?) that’s in concordance with Matt’s Hubble Luminosity in this reply and anywhere.
The following is a copy from a previous posting that is repeated and completed at this place for the sake of convenience.
The gravitational field of a point mass and the electric field of a point charge are structurally similar. Each may be represented by a vector field in which the vectors are directed along radial lines emanating from the point, and the vector magnitudes decrease as the inverse square of the [radial] distance from the point.
The electric field has a scalar energy density field associated with it. When the field vector at a point is $\vec{E}$, then the energy density at the same point is
where $\epsilon_0$ is the permittivity of free space. Energy density is a measure of the energy stored in the field per unit volume of space.
Question: Is there an energy density associated with a Gravitational Field?
Yes, there is! Let us use the similarity between the gravitational and electric fields to construct a gravitational energy density term. Start by noticing how the permittivity of free space enters into the expression for $E$ :
$$
E = kQ/r^2 \quad \mbox{where} \quad k = 1/(4\pi\epsilon_0)
$$
is a universal constant, $Q$ is the source charge, and $r$ is radial distance from the source.
The gravitational field is given by an analogous expression
$$
g = GM/r^2
$$
where $G$ is a universal constant, $M$ is the source mass.
The electrical field energy density may be written in terms of $k$ as
@HanDeBruijn It totally follows that the cosmic mass times the Hubble constant gives the Planck power (Hubble luminosity), as the CMB is 1/4 or 1/2 of it, depending on how you define M0. But that’s an energy emission. For your variable mass model, don’t you need an energy absorption, to increase the mass M0 ? Still, if the Planck power comes from anywhere, then I suppose it could power a mass increase. It’s interesting that there are so many ways of getting the Planck power. Somebody has to be right!
I’m interested your derivation of the gravitational energy density and will look at it more.
Reminder that ΛCDM does not say that ‘space does not expand within matter’; it merely states that expansion is too weak to pull local forces apart, thereby acknowledging that all space expands.
Can you explain this a little more?
My understanding is that in ΛCDM there are no individual bodies, the universe is an “ideal/perfect fluid”, everything everywhere is the same as everything everywhere else.
It’s the cosmological principle, but expressed in math.
Second, the matter content of the universe on large scales is modeled as a perfect fluid. This “perfect fluid assumption” is incredibly important in the case of the Friedmann equations.
This is because a universe modeled as a perfect fluid implies that all matter (again, on large scales) is distributed uniformly – the same everywhere. This is nothing other than homogeneity from earlier!
@mikehelland the way I understand is that yes, space is the perfect fluid, with matter suspended in the fluid. Space expands (according to ΛCDM) but the matter (planets, stars, chair) does not expand.
It’s kind of required that they claim this, I think, because if ‘everything’ expands, then it is as if nothing expands.
But the ‘mass doesn’t expand’ leads to the GR paradox I mention above. We should see differences in cosmic scale gravity.
This is just my layman interpretation. I don’t claim to know much about ΛCDM. Thanks for the link.
@ExpEarth and @HanDeBruijn have different opinions on this.
This should be in a different discussion, but read that again:
Second, the matter content of the universe on large scales is modeled as a perfect fluid. This “perfect fluid assumption” is incredibly important in the case of the Friedmann equations.
Matter isn’t in the perfect fluid. Matter is the perfect fluid, at least in the mathematics of FLRW, which is what LCDM is based on.
@mikehelland you’re right we’ve thoroughly hijacked Louis’ discussion.
From you link:
Second, the matter content of the universe on large scales is modeled as a perfect fluid. This “perfect fluid assumption” is incredibly important in the case of the Friedmann equations.
This is because a universe modeled as a perfect fluid implies that all matter (again, on large scales) is distributed uniformly – the same everywhere.
I agree that on large scale the entirety of space approaches a perfect fluid. The granularity of matter might disappear in large enough samples but that still doesn’t make matter a fluid.
I don’t know enough about FLRW if this was one of their original postulates, or if it is an interpretation by the author. Either way, seems dodgy to me. To say that matter is the fluid, and not space, is an impossible leap for me.
This could be another assumption from the premise that ‘space is nothing, zero, nada, and its properties are borne from nothing’ which makes matter the molecules of the perfect fluid. This is another conundrum M&M and GR have left us with.
where $U=$ total energy, $M_O=$ mass of Observable universe, $c=$ speed of light in empty space, $G=$ gravitational constant, $H=$ Hubble parameter, $W=$ dimensionless constant = 3.383634283, at the place where the Observability function in VM is a maximum.
As a second ingredient it is assumed that the CMB temperature, denoted as Θ is proportional with cosmological redshift (1+z). The redshift obeys the logarithmic formula $r=(c/H)\ln(1+z)$ - in TL as well as VM - and so
where the subscript (0) denotes the CMB temperature at our place and $r=$ distance.
The Stefan-Boltzmann law expressed as a formula for radiation energy density is:
$$
u_{CMB} = \frac{4}{c}\sigma \Theta^4
$$
where the constant of proportionality, $\sigma$, is called the Stefan–Boltzmann constant.
The total energy of the CMB is thus expressed by the following integral over a sphere with radius $R_O=W(c/H)$ of the Observable (VM) universe, taking $x=(H/c)r$.
Oooff! Keep going @HanDeBruijn. I agree with the exponenial but still working on my own version of it.
Do you have a fundamental equation for VM that is based on distance (or time) and not include $$H_0$$?
My own version (work in progress)
$$1+z=e^τ$$ , where $$τ=∫κρ(x)dx$$ from 0-d
$$κρ(x)dx$$ is the cumulative optical depth. For nearby sources $$(τ≪1)$$ , a Taylor simplification reduces to the Hubble law $$z≈κρ_0d$$ , with $$H_0=κρ_0c$$. The integral can be avoided if homegeneity is certain.
For distant sources, the full exponential form must be used. This implies there is an eventual optic horison. (Opposite of Olber’s paradox, or a solution thereto). Same would apply for your exponential, I think.
τ (Tau – Optical Depth)
The cumulative attenuation of the flux along a path, defined by the integral of the opacity over density and distance.
κ (Kappa – Opacity)
The effective cross-section per unit mass for the interaction of photons with matter or the aether.
ρ (Rho – Density)
The local mass density of matter or aether constituents that contributes to the attenuation of the flux.
An interesting quote from Estimations of Total Mass and Energy of the Observable Universe follows.
>The recent astronomical observations indicate that the expanding universe is homogeneous, isotropic and asymptotically flat. The Euclidean geometry of the universe enables to determine the total gravitational and kinetic energy of the universe by Newtonian gravity in a flat space. By dimensional analysis, we have found the mass of the universe close to the Hoyle-Carvalho formula $M\sim c^3/(GH)$. This value is independent from the cosmological model and infers a size (radius) of the universe close to [or two times the] Hubble distance. [ .. ] Both, the total gravitational and kinetic energies of the universe have been determined in relation to an observer at an arbitrary location. The relativistic calculations for total kinetic energy have been made and the dark energy has been excluded from calculations. The total mechanical energy of the universe has been found close to zero, which is a remarkable result. This result supports the conjecture that the gravitational energy of the universe is approximately balanced with its kinetic energy [ .. ]
According to (our version of) the Variable Mass theory the Hoyle-Carvalho formula is recovered if we replace our beloved constant $W=3.383634283$ by a much simpler value, namely
Then a miracle happens (silently employing the Hubble tension).
W := 2;
H := 67; # CMB
# ..
# Rest as before
# ..
Theta_0 := evalf(Theta_0);
And the outcome is $\large \Theta_0 \approx 2.9098 \mbox{ K} \approx 2.7255 \mbox{ K}$ for the CMB temperature here and now. But there is more ..
x = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } ; y = { 0.0 , 0.1 , 0.2 , 0.3 , 0.4 , 0.5 } .
It is seen in the picture that $x\approx 2$ corresponds with the ΛCDM value for the radius of the Observable universe. Analytically:
where $f(x)$ is the Observability function as defined by formula (8) in #377.
The quantity $R_O=x.c/H\approx (2/H).c = A.c$ corresponds with the “age” of the VM universe times light speed.
Coincidence, or conspiracy, or neither, or both? Anyway the above seems closer to the CMB model by @ExpEarth than ever.
For me, here and now, the interesting part in discussion #127 is given in the text as saved in notification #389 :
>In my model the temperature of the CMB varies from a maximum value of 29 K near the rim of the visible universe to a value of 2.7 K at our position.
Our own calculations “near the rim” follow. Make a choice between once or twice the Hubble distance.
where $R_O=$ radius of Observable (visible) universe, $c=$ speed of light in empty space, $H=$ Hubble parameter (with tension eventually), $z=$ cosmological redshift, $\Theta=$ CMB temperature, $\Theta_0=$ same here and now.
Update
It turns out that the outcomes are extremely sensitive to the precise value of W. If we take W=2.096433787 (for ΛCDM) instead of W=2 then the outcome is $\Theta_0 \approx 2.6041 \mbox{ K}$ instead of $\approx 2.9098 \mbox{ K}$ for the CMB temperature here and now. And the CMB temperature near the rim becomes $\approx 22\mbox{ K}$ instead of $\approx 20\mbox{ K}$.
But if we employ the Hubble tension and use H=73.4 km/s/Mpc (: Riess) together with W=2.096433787 (: ΛCDM ⊂ VM) then the outcome is Θ0 = 2.725 K , which accurately equals the empirical value !!
There’s one coming soon. These past two years have been tough on me, on my health, and with a few rejected papers. But I have plenty of material for the next newsletters including a list compiled by Mike Helland on galaxy formation problems…
Suggestions are welcome. Post them here in this conversation!
I’m very sorry to hear that Louis. If you don’t mind me asking, what happened? Why did your papers get rejected and what went wrong with your health?
@MJoseph315 I made the mistake to submit my last three papers to astronomy journals. STz redshift is now a non-relativistic fully quantized calculation, so it predicts many astrophysical phenomena but is not advanced enough to accurately describe the CMB (a relativistic version of STz is needed for that). My papers were rejected because reviewers expect a fully completed cosmology, “STz is obviously wrong since it can’t predict the CMB.”
Pneumonia & stress triggered an hemolytic anemia. I’ve been improving slowly since then, but a brain with more oxygen works faster! :wink:
@RedshiftDrift writes: >not advanced enough to accurately describe the CMB (a relativistic version of STz is needed for that).
How can people be so sure that relativity is a panacea for anything? And by the way, I have always thought that Quantum ElectroDynamics IS already - by definition - relativistically invariant.
Take the advice by Goethe: In die Beschränkung zeigt sich erst der Meister. With other words: showing some restraint (and cutting into pieces) would help.
Let them drill a few holes in your skull, like they did to me after I fell, though not in the bathtub.🤕
I was just wondering where to post this and still not sure if this is the right discussion, but I grappled with a GR vs ΛCDM paradox this morning.
It is difficult to refute the claim that cosmic expansion cannot overcome atomic and molecular forces. The expansion is too weak and too slow. So even if one suspects this argument is flawed, it cannot be easily disproven. The same ‘slow’ can be argued to not rip apart a gravitational bound system.
But the ‘slow expansion of space’ is also changing the properties of space, upon which GR is built.
The same defense fails for gravitationally bound systems. A star, a galaxy, or even a galaxy cluster is held together by gravity, which, in General Relativity, is the spacetime metric. If space expands, the metric must expand, and gravity must change accordingly. This is not a question of forces overcoming expansion, it is a question of the very fabric of spacetime evolving. Since space and therefore spacetime itself permeates matter (stars, galaxies…) it (spacetime) must also expand. The ‘too weak’ argument then fails over a cosmic time period.
Reminder that ΛCDM does not say that ‘space does not expand within matter’; it merely states that expansion is too weak to pull local forces apart, thereby acknowledging that all space expands. For ΛCDM to deny that space expands within systems, would be tantamount to saying ‘gravity is keeping space together’; a logical fallacy.
Over billions of years, space has expanded by a factor of $$(1+z)$$ since recombination. A primordial cubic meter of spacetime has grown to a cube of $$(1+z)^3$$ cubic meters today. This expansion must have altered the local properties of spacetime — its curvature, its energy density, and its coupling to matter, even within stars, galaxies, and clusters. Yet no such changes are observed. Gravity today appears to behave identically to gravity billions of years ago. The ΛCDM response — that expansion does not affect bound systems — is a logical convenience, not a physical mechanism. If a cubic meter of spacetime has grown, but the matter within it has not expanded at all, the metric must show signs of this tension.
The FLRW metric: $$ds^2=−c^2dt^2+a(t)^2(dr^2+r^2dΩ^2)$$
In flat space c=s/t must remain constant. In vicinity of matter it requires that the local metric evolve with a(t). An expanding ‘flat space’ requires s to change to give redshift, but then t must also change for c to remain constant. All clocks would be affected. ΛCDM would refute this by saying all clocks are within bound systems, and we are back where we started.
However, if a(t) has also changed by a factor of $$10^3$$ since recombination (or insert your own favorite value for z into $$(1+z)^3$$), then local spacetime properties (within bound systems) must have changed by a corresponding amount. Since no such changes are observed in the gravitational behavior of stars, galaxies, or clusters, the hypothesis of expanding space must be refuted — or at least recognized as a logical paradox that conflicts GR with ΛCDM. Since ΛCDM is built on GR, this paradox cannot remain unresolved.
The claim that bound systems are exempt from expansion is logically circular: it asserts that the metric does not apply inside bound systems, which contradicts the very definition of spacetime. Since it is agreed that bound systems do not expand, it must be concluded the metric does not expand. The only coherent resolution is that redshift is not a metric effect.
@Francois-Zinserling
This should probably be in its own thread, as it is a complex topic and a fundamental one. It’s not true that expansion of gravitational systems is not observed. We see it in a number of them: the AU, the Moon’s orbit around the Earth, Titan’s orbit around Saturn. The radii all increase in proportion to the Universe’s own expansion. I touched on this in my paper on the origin of the Moon:
It also relates to the increase of supermassive black holes over time. While these observations seem to suggest that the Universe is expanding like they say, I argue that it is does not necessarily. On local scales, the force of gravity overpowers the Hubble luminosity completely. The expansion of radii that we see is just the residual outward push when you cancel gravitational infall with centrifugal force outwards. But for the universe as a whole the rotational state is uncertain. Suppose it is not rotating. At the radius corresponding to the Hubble radius c/H, the Hubble luminosity completely balances the gravitational force. (It’s a bit more complicated than that, as the CMB is heavily involved as well, being replenished by the Hubble luminosity.) At the cosmic scale, and in black holes, the Hubble luminosity is the same as the Planck power c^5/4G.
@Francois-Zinserling writes: >The claim that bound systems are exempt from expansion is logically circular: it asserts that the metric does not apply inside bound systems, which contradicts the very definition of spacetime. Since it is agreed that bound systems do not expand, it must be concluded the metric does not expand. The only coherent resolution is that redshift is not a metric effect.
The problem is completely solved by the Variable Mass (VM) theory. Remember that - according to a the 1993 publication by Jayant Narlikar and Halton Arp, VM is essentially General Relativity in disguise: Finally, the Euclidean, flat spacetime becomes a natural, primary reference frame in which cosmological processes are most simply described. The Variable Mass Theory is compatible with a static Euclidean universe, which is eternal in Atomic time and has moments of creation (i.e. one or more beginnings) in Orbital time.
The above is a quote from Atomic Time, Orbital Time and the Variable Mass Theory. Section 4 of the accompanying PDF is named Orbital ticks. Theory is developed in that section with the explicit condition that space within the - gravitationally bound - orbit is constant and Orbital time is the only variable: the condition (r = constant) is a necessary one.
Theory about Atomic ticks (section 3) is somewhat more complicated and requires knowledge about the technique of Nondimensionalization and the Schrödinger equation for general atoms. Contrary to orbital distances, it turns out that atomic distances are variable with Variable Mass, but the effect of this on energy levels - i.e. the intrinsic redshift - is neutralized within the dimensionless formulation.
Then we arrive at Variable Mass Theory and Observable Universe. The relevant section in there is 2: Length Contraction. It is shown that all distances in the (sub)atomic domain are varying inversely proportional to (elementary particle rest) mass. It affects in this way how all of our measuring devices work. And in the end it is logically concluded that Length Contraction [of bound systems] with VM is observationally equivalent with the Expansion of Empty Space.
Yes, Francois, Empty Space. What do you think? Is your Aether an Empty Space?
@ExpEarth writes: >the AU, the Moon’s orbit around the Earth, Titan’s orbit around Saturn. The radii all increase in proportion to the Universe’s own expansion.
That’s in agreement with the VM point of view: : the condition (r = constant) is a necessary one. The constant r is thus belonging to Empty Space and “it seems” to expand with the shrinking of our measuring instruments. (What is the meaning of “it seems” in this context) >At the cosmic scale, and in black holes, the Hubble luminosity is the same as the Planck power c^5/4G.
Have you elaborate this somewhere, Matt, together with your views upon the CMB? (Because as for myself I am stuck with the latter)
@HanDeBruijn no ‘my’ aether is not empty space. I will read your links and hope to respond soon.
The Planck power $$c^5/4G$$ relates to the units of my aether model, with $$P_P=E_P/t_P$$ as one of the basic foundations. (not sure about the $$1/4$$ ratio from @ExpEarth. Along with the other Planck units, $$L_P$$ (length), $$p_P$$ (momentum), but in my model it permeates all of space - not just at the cosmic horizon. It was not assumed but concluded from the gravitational derivation.
@HanDeBruijn @Francois-Zinserling
I talked about the luminosity c^5/4G (and also the CMB) previously in my shell universe paper #256 . That power is connected to the inner, contained constant luminosity in all black holes and in the black hole universe. The quantity c^5/G is what they call the Planck power or Planck luminosity, which I think is just the Planck unit for power similar to the Planck mass, etc. The extra factor of 1/4 is considered the ‘maximum power’ that one can actually see in nature’s discussed by Schiller and others. I derived this max power from separate principles connected to what I have called the Hubble luminosity. This is the energy released from gravitating systems according to L = - U H. U is conventionally negative and so it becomes positive. On the cosmic scale, and in black holes generally, this becomes the max power c^5/4G.
So Francois, you and I are on the same page again it seems. I agree there is a Planck luminosity filling all space. In my models, there is a constant and equal drain of energy from space to power gravity. Most people using GR just think of gravity as this steady force acting with no energy input at all, just the geometry induced by mass. Totally wrong. I’m curious how you use the Planck power in your model. I’ve made great progress in my model, but the whole question of what is charge and magnetism is still on the to do list.
On the Planck units, I had been highly skeptical of them having any particular significance. Years ago Harry Dart showed that you get a totally different set of values if you derive the values using units of acceleration for G instead of G’s normal units. It could be an interesting exercise for people to try this and see what units they find for mass, length and time. Using units of acceleration takes us into MOND territory. Harry and I submitted a paper to Science about Harry’s new units, but when Harry found out I was using MOND he freaked out and told me to withdraw the paper. He had apparently only agreed to submit it al all because he had fallen in the bath tub and hit his head! But Science rejected the paper before I could withdraw it. @RedshiftDrift might remember this tale. Now the Planck power is different, as it involves only two of the three constants. Yet it seems correct.
@ExpEarth the luminosity $c^5/(4G) = 8.5\times 10^{52}$ Watts. Just to satisfy my curiosity, could you give us the value of the energy density in space away from galaxies, in Joules per cubic meter, that this value implies?
(Such numbers could make me fall in the bath tub! :rofl: )
Okay @RedshiftDrift , here it goes. Better to read outside the bathroom!
We have to go back to how the black hole universe formed and look at the mass density. Since there was enough mass to form a black hole, the mass density has to be the same as the critical density. Let’s say the widely dispersed baryonic mass that formed the shell universe has mass M0 and energy M0c^2. When it collapses to form a black hole, the gravitational energy released is trapped inside. This becomes the CMB eventually.
But when a black hole forms in GR, there is a mass defect which is referred to as the binding energy. In GR they don’t really say where that energy is or what happens to it. You’ll recall that I talked a lot about photon filaments connecting baryons in my earlier papers. This energy was also identified with U. ( @Francois-Zinserling, you’ll remember saying this would break the computer of the universe if it were true.) There’s a simpler way. The photon filaments are just the standing wave nodes connecting the shell baryons. Think of the shell as a cavity resonator. In GR we would say this lattice energy is negative and its pressure negative. Now the energy released and trapped as CMB photons equals the quantity U of the product shell. So what we have is M0c^2 = 1/2M0c^2 (baryon) + 1/4M0c^2 (CMB) + 1/4M0c^2 (lattice).
Because the lattice energy and pressure are negative, they offset the positive CMB energy and pressure. That is key, since radiation pressure and energy add to the active gravitational mass in GR. I totally missed that in my shell paper. That leaves the outward CMB pressure to hold the shell in place.
So getting back to your question, the CMB energy density is 1/4 the critical density, amounting to about 10^-9 erg/cc. If it were spread evenly throughout the shell universe, its temperature would be 27 K. But because of the shell’s gravity gradient, it peaks at 29 K near the shell and falls linearly to 2.7 K at our position.
@ExpEarth How is the average energy density $\bar E = 10^{-9}$ erg/cm^3 related to the luminosity? It is through the “age” and “volume” of the universe?
$$ \frac{c^5/(4G) \times 10^{18} \mbox{ s}}{(10^{27} \mbox{ m})^3} \approx 10^{-10} \mbox{ J/m}^3 (= 10^{-9} \mbox{erg/cm}^3) $$
Question # 2: why does the temperature fall linearly from 29 K to 2.7 K? Shouldn’t the energy density fall linearly instead? If it’s redshift that causes the linear drop from 29 K to 2.7 K, does that mean the maximum observable redshift is z = 29/2.7 = 10.7 ?
@RedshiftDrift That is correct. If we have the Planck power pumping energy into the universe at a fixed rate, multiply by the recycling time of the universe (not the ‘age’) and divide by the volume, then we will have the CMB energy density you have on the right. But the initial CMB energy comes from gravitational energy released during the shell formation.
In my model the CMB and the gravitational energy U are in balance. So the same equation can be reversed to get the reverse flow of energy. The CMB energy is being Hubble redshifted and that energy goes into gravity. In that case you take the average energy density of the CMB, which matches a 27 K CMB, and multiply by the Hubble constant (which we can write as roughly 10^-18 sec^-1). That just takes the time from your numerator on the left and puts it into the denominator on the right. That yields the Planck power per unit volume on the left side.
Question for @Francois-Zinserling. What happens to the accumulated Planck power energy in your model?
On Question 2, no, it’s the temperature. The temperature of the CMB tracks the gravitational gradient. Wherever radiation comes into thermal equilibrium with a gravity gradient, it will form blackbody radiation having a gradient in T which matches the redshift. This also fits with the supernova time dilation. This is due to the Tolman-Ehrenfest relation.
@HanDeBruijn I am unable to acces two of your links: https://github.com/orgs/a-cosmology-group/discussions/377 https://github.com/orgs/a-cosmology-group/discussions/344
, even if I paste these direct into a browser. Is anyone else having this problem?
@ExpEarth in my model the vacuum energy remains constant but it changes mode when interacting with matter. In short, the dark pairs of Annila et. al interact with a fundamental particle of mass (and charge) and the particle changes the mode of the pair to bright. Bright pairs still have $$E=E_P$$ but no longer dark so new rules apply. From this mode change follows a heuristic derivation of gravity and electrostatics. New paper coming soon,
@HanDeBruijn has no problem and doesn’t know what to do about it. Maybe a maintainer (like @RedshiftDrift ?) can tell us what’s going on.
@Francois-Zinserling Try again, github.com was down earlier today $\rightarrow$ https://www.theverge.com/tech/980978/github-is-down
It’s working today 🥳
@RedshiftDrift
On the last part of your question, this had been puzzling me but I think I might have it now. At the link below you can see the layout of the shell model using Ni’s approach. Note that the gravitational gradient, which determines the Hubble constant, actually drops to 0 at the midpoint of the shell. The Ni approach works because matter is drawn from both sides of the shell midpoint towards the midpoint. Now I think the 29 K CMB could apply as an average temperature for the whole shell. This would have z = 10 as you said. But at the midpoint the temperature of the CMB could rise further. If the midpoint corresponds to z = 14, then the temperature would be about 41 K. It could go even higher. There is an odd anomaly for galaxies on the far side of the midpoint. These will seem unusually faint for their redshift.
https://www.mdpi.com/astronomy/astronomy-03-00014/article_deploy/html/images/astronomy-03-00014-g001.png
@Francois-Zinserling Your paper sounds interesting. I like the Annila part. Good luck!
@ExpEarth Can we see beyond the shell?
I notice something else on that graph; the gradient is very large near the centre, as if the Hubble constant becomes much larger than 75 km/s/Mpc at $0.05 R_U$. How can that be?
From what I can make out, we should be able to see beyond the shell. But the redshift will be misleading, because there will be an intervening blueshift as photons move from the exterior space down the gradient towards the midpoint of the shell before they are redshifted again on their way towards our position.
The graph was really just an attempt to project the Ni-Neslusan structure onto a shell universe. I initially had the 75 value representing the extra gradient arising at the centre in their calculations, but superimposed onto the KBC Void, which I was speculating to be at the very centre. Now I have shifted the emphasis off the KBC Void being dead centre, as we actually need it to be off centre to account for the quasar dipole. What I would say now is that the CMB dipole is caused by the outward motion of galaxies in the KBC Void towards the walls of the void and the quasar/cosmic dipole to be die to our being closer to one cosmic wall than the other. The bottom line is that things get a bit confused inside voids due to the generally outward flow of matter in voids.
@ExpEarth writes in this reply : >Let’s say the widely dispersed baryonic mass that formed the shell universe has mass $M_O$ and energy $M_Oc^2$ [: notation of our own]
It’s all about the “luminosity” $c^5/(4G)$. Let’s see how that fits into the Variable Mass theory. Or not.
In section 8 of the paper in Variable Mass Theory and Observable Universe formula (13) is derived for the mass of our Black Hole Bubble / the VM Observable universe:
$$ M_O = \frac{c^3W}{2GH} \quad \Longrightarrow \quad M_Oc^2.H = 2W\frac{c^5}{4G} = U.H = L \quad (?) $$
Thus energy of mass times the Hubble parameter equals Planck power, apart from a numerical factor of about $2W \approx 7$. I think (?) that’s in concordance with Matt’s Hubble Luminosity in this reply and anywhere.
The following is a copy from a previous posting that is repeated and completed at this place for the sake of convenience.
Essential reading: A Possible Scalar Term Describing Energy Density in the Gravitational Field .
The gravitational field of a point mass and the electric field of a point charge are structurally similar. Each may be represented by a vector field in which the vectors are directed along radial lines emanating from the point, and the vector magnitudes decrease as the inverse square of the [radial] distance from the point. The electric field has a scalar energy density field associated with it. When the field vector at a point is $\vec{E}$, then the energy density at the same point is
$$ u_E = \frac{1}{2}\epsilon_0(\vec{E}\cdot\vec{E}) = \frac{1}{2}\epsilon_0 E^2 $$
where $\epsilon_0$ is the permittivity of free space. Energy density is a measure of the energy stored in the field per unit volume of space. Question: Is there an energy density associated with a Gravitational Field? Yes, there is! Let us use the similarity between the gravitational and electric fields to construct a gravitational energy density term. Start by noticing how the permittivity of free space enters into the expression for $E$ :
$$ E = kQ/r^2 \quad \mbox{where} \quad k = 1/(4\pi\epsilon_0) $$
is a universal constant, $Q$ is the source charge, and $r$ is radial distance from the source. The gravitational field is given by an analogous expression
$$ g = GM/r^2 $$
where $G$ is a universal constant, $M$ is the source mass. The electrical field energy density may be written in terms of $k$ as
$$ u_E = \frac{1}{2}\cdot 1/(4\pi k)\cdot E^2 = E^2/(8\pi k) $$
By analogy, a candidate gravitational energy density term may now be constructed:
$$ u_G = -g^2/(8\pi G) $$
We will use Newton’s law of gravitation without modification, and incorporate the energy density term described above.
$$ g = GM/r^2 \quad \mbox{so that} \quad u_G = - GM^2/(8\pi r^4) $$
Integrate the above to find the gravitational energy of the entire Observable universe.
$$ U_G = \int_{R_O}^\infty - \frac{GM_O^2}{8\pi r^4}4\pi r^2 dr = - \frac{GM_O^2}{2}\int_{R_O}^\infty\frac{dr}{r^2} = - \frac{GM_O^2}{2R_O} $$
Substitute the VM values from our paper:
$$ M_O = \frac{c^3W}{2GH} \quad \mbox{and} \quad R_O = W\frac{c}{H} $$
And the questionable factor 1/4 appears spontaneously:
$$ U_G = - \frac{G}{2}\left(\frac{c^3W}{2GH}\right)^2\frac{H}{W.c} = - \frac{c^5.W}{8GH} = - \frac{1}{4}M_Oc^2 $$
@HanDeBruijn It totally follows that the cosmic mass times the Hubble constant gives the Planck power (Hubble luminosity), as the CMB is 1/4 or 1/2 of it, depending on how you define M0. But that’s an energy emission. For your variable mass model, don’t you need an energy absorption, to increase the mass M0 ? Still, if the Planck power comes from anywhere, then I suppose it could power a mass increase. It’s interesting that there are so many ways of getting the Planck power. Somebody has to be right!
I’m interested your derivation of the gravitational energy density and will look at it more.
@Francois-Zinserling
Can you explain this a little more?
My understanding is that in ΛCDM there are no individual bodies, the universe is an “ideal/perfect fluid”, everything everywhere is the same as everything everywhere else.
It’s the cosmological principle, but expressed in math.
https://profoundphysics.com/the-friedmann-equations-explained-a-complete-guide/
@mikehelland the way I understand is that yes, space is the perfect fluid, with matter suspended in the fluid. Space expands (according to ΛCDM) but the matter (planets, stars, chair) does not expand. It’s kind of required that they claim this, I think, because if ‘everything’ expands, then it is as if nothing expands. But the ‘mass doesn’t expand’ leads to the GR paradox I mention above. We should see differences in cosmic scale gravity.
This is just my layman interpretation. I don’t claim to know much about ΛCDM. Thanks for the link. @ExpEarth and @HanDeBruijn have different opinions on this.
@Francois-Zinserling
This should be in a different discussion, but read that again:
Matter isn’t in the perfect fluid. Matter is the perfect fluid, at least in the mathematics of FLRW, which is what LCDM is based on.
@mikehelland you’re right we’ve thoroughly hijacked Louis’ discussion.
From you link:
I agree that on large scale the entirety of space approaches a perfect fluid. The granularity of matter might disappear in large enough samples but that still doesn’t make matter a fluid.
I don’t know enough about FLRW if this was one of their original postulates, or if it is an interpretation by the author. Either way, seems dodgy to me. To say that matter is the fluid, and not space, is an impossible leap for me.
This could be another assumption from the premise that ‘space is nothing, zero, nada, and its properties are borne from nothing’ which makes matter the molecules of the perfect fluid. This is another conundrum M&M and GR have left us with.
Let’s proceed with the CMB, taking my previous reply as a starting point, everything in concordance with the Variable Mass (VM) theory.
$$ U_{CMB} = -U_G = \frac{1}{4}M_Oc^2 \quad \mbox{with} \quad M_O = \frac{c^3W}{2GH} \qquad (1) $$
where $U=$ total energy, $M_O=$ mass of Observable universe, $c=$ speed of light in empty space, $G=$ gravitational constant, $H=$ Hubble parameter, $W=$ dimensionless constant = 3.383634283, at the place where the Observability function in VM is a maximum. As a second ingredient it is assumed that the CMB temperature, denoted as Θ is proportional with cosmological redshift (1+z). The redshift obeys the logarithmic formula $r=(c/H)\ln(1+z)$ - in TL as well as VM - and so
$$ \Theta = \Theta_0 (1+z) = \Theta_0 e^{(H/c)r} $$
where the subscript (0) denotes the CMB temperature at our place and $r=$ distance. The Stefan-Boltzmann law expressed as a formula for radiation energy density is:
$$ u_{CMB} = \frac{4}{c}\sigma \Theta^4 $$
where the constant of proportionality, $\sigma$, is called the Stefan–Boltzmann constant. The total energy of the CMB is thus expressed by the following integral over a sphere with radius $R_O=W(c/H)$ of the Observable (VM) universe, taking $x=(H/c)r$.
$$ U_{CMB} = \int_0^{R_O} \frac{4}{c}\sigma \left(\Theta_0 e^{(H/c)r}\right)^4 . 4\pi r^2 dr = \frac{4}{c}\sigma \mbox{ }\Theta_0^4 . 4\pi \left(\frac{c}{H}\right)^3 \int_0^W e^{4x}x^2dx \qquad (2) $$
The integral is calculated to be
$$ II = \int_0^W e^{4x}x^2dx = e^{4W}\left(\frac{1}{4}W^2-\frac{1}{8}W+\frac{1}{32}\right)-\frac{1}{32} $$
The end-result is found by combining (1) and (2).
$$ \Theta_0 = \left[\frac{M_Oc^2/4}{(4\sigma/c).4\pi.(c/H)^3.II}\right]^{1/4} $$
which should give the CMB temperature at our place.
Alas: $\Theta_0=0.6529075366\mbox{ K}\ne 2.725\mbox{ K}$. No matching, no stretching; back to the drawing board :-(
Oooff! Keep going @HanDeBruijn. I agree with the exponenial but still working on my own version of it. Do you have a fundamental equation for VM that is based on distance (or time) and not include $$H_0$$?
My own version (work in progress) $$1+z=e^τ$$ , where $$τ=∫κρ(x)dx$$ from 0-d $$κρ(x)dx$$ is the cumulative optical depth. For nearby sources $$(τ≪1)$$ , a Taylor simplification reduces to the Hubble law $$z≈κρ_0d$$ , with $$H_0=κρ_0c$$. The integral can be avoided if homegeneity is certain.
For distant sources, the full exponential form must be used. This implies there is an eventual optic horison. (Opposite of Olber’s paradox, or a solution thereto). Same would apply for your exponential, I think. τ (Tau – Optical Depth) The cumulative attenuation of the flux along a path, defined by the integral of the opacity over density and distance.
κ (Kappa – Opacity) The effective cross-section per unit mass for the interaction of photons with matter or the aether.
ρ (Rho – Density) The local mass density of matter or aether constituents that contributes to the attenuation of the flux.
An interesting quote from Estimations of Total Mass and Energy of the Observable Universe follows. >The recent astronomical observations indicate that the expanding universe is homogeneous, isotropic and asymptotically flat. The Euclidean geometry of the universe enables to determine the total gravitational and kinetic energy of the universe by Newtonian gravity in a flat space. By dimensional analysis, we have found the mass of the universe close to the Hoyle-Carvalho formula $M\sim c^3/(GH)$. This value is independent from the cosmological model and infers a size (radius) of the universe close to [or two times the] Hubble distance. [ .. ] Both, the total gravitational and kinetic energies of the universe have been determined in relation to an observer at an arbitrary location. The relativistic calculations for total kinetic energy have been made and the dark energy has been excluded from calculations. The total mechanical energy of the universe has been found close to zero, which is a remarkable result. This result supports the conjecture that the gravitational energy of the universe is approximately balanced with its kinetic energy [ .. ]
According to (our version of) the Variable Mass theory the Hoyle-Carvalho formula is recovered if we replace our beloved constant $W=3.383634283$ by a much simpler value, namely
$$ M_O = \frac{Wc^3}{2GH} \quad \mbox{with} \quad W=2 $$
Same story for the integral gives
$$ II = \int_0^{W=2} e^{4x}x^2dx = e^{4\times 2}\left(\frac{1}{4}2^2-\frac{1}{8}2+\frac{1}{32}\right)-\frac{1}{32} = \frac{25}{32}e^8 $$
Then a miracle happens (silently employing the Hubble tension).
And the outcome is $\large \Theta_0 \approx 2.9098 \mbox{ K} \approx 2.7255 \mbox{ K}$ for the CMB temperature here and now. But there is more ..
x = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } ; y = { 0.0 , 0.1 , 0.2 , 0.3 , 0.4 , 0.5 } . It is seen in the picture that $x\approx 2$ corresponds with the ΛCDM value for the radius of the Observable universe. Analytically:
$$ f(x) = \frac{2-e^{-x}(x^2+2x+2)}{x} = \frac{1}{3} \quad \Longrightarrow \quad x = 2.096433787 \approx 2 $$
where $f(x)$ is the Observability function as defined by formula (8) in #377. The quantity $R_O=x.c/H\approx (2/H).c = A.c$ corresponds with the “age” of the VM universe times light speed.
Coincidence, or conspiracy, or neither, or both? Anyway the above seems closer to the CMB model by @ExpEarth than ever.
By the way, @ExpEarth , your thread #127 seems to be gone. Where is it?
@HanDeBruijn Thanks for pointing this out. @ExpEarth There seems to be a part of the discussion that is corrupt… See notification #389
For me, here and now, the interesting part in discussion #127 is given in the text as saved in notification #389 : >In my model the temperature of the CMB varies from a maximum value of 29 K near the rim of the visible universe to a value of 2.7 K at our position.
Our own calculations “near the rim” follow. Make a choice between once or twice the Hubble distance.
$$ R_O = \frac{c}{H} \quad \Longrightarrow \quad 1+z = e^{(H/c)(c/H)} = e \quad \Longrightarrow \quad \Theta = \Theta_0 (1+z) = 2.725\times 2.718 = 7.407 \mbox{ K} $$
$$ R_O = 2\frac{c}{H} \quad \Longrightarrow \quad 1+z = e^{(H/c)2(c/H)} = e^2 \quad \Longrightarrow \quad \Theta = \Theta_0 (1+z) = 2.725\times 7.389 = 20.135 \mbox{ K} $$
where $R_O=$ radius of Observable (visible) universe, $c=$ speed of light in empty space, $H=$ Hubble parameter (with tension eventually), $z=$ cosmological redshift, $\Theta=$ CMB temperature, $\Theta_0=$ same here and now.
Update
It turns out that the outcomes are extremely sensitive to the precise value of W. If we take W=2.096433787 (for ΛCDM) instead of W=2 then the outcome is $\Theta_0 \approx 2.6041 \mbox{ K}$ instead of $\approx 2.9098 \mbox{ K}$ for the CMB temperature here and now. And the CMB temperature near the rim becomes $\approx 22\mbox{ K}$ instead of $\approx 20\mbox{ K}$.
But if we employ the Hubble tension and use H=73.4 km/s/Mpc (: Riess) together with W=2.096433787 (: ΛCDM ⊂ VM) then the outcome is Θ0 = 2.725 K , which accurately equals the empirical value !!