FUGE and Redshift

I have written before about the OE Curve and redshift, but it was blended together with consideration of someone else’s theory about the universe so it’s not as clear as I would have liked.  So, I’ll have another go at it.

---

There are (at least) two ways to think about redshift.

A photon which is emitted by a source that is moving away at a sufficiently high speed can be observed to be redshifted.  That’s not to say there isn’t redshift for slow (non-relativistic) light sources, it’s just that the redshift in that case is so small as to be unobservable.  This form of redshift is effectively the same as Doppler shift, but requires some extra thought due to relativistic effects (see relativistic Doppler shift).

A photon that is emitted by a source that is stationary (at rest relative to the Hubble flow) or “comoving” relative to the observer, if sufficiently distant from that observer, can also be observed to be redshifted.  Again, the redshift is there, irrespective of the distance under those conditions, but at relatively short distances it’s too small to be observed.

(There is a third source of redshift, which is due to gravity.  It’s actually similar to the redshift due to cosmological expansion in a way, but at a different scale, since both involve deformation of space along the path of the photon.  Redshift due to gravity occurs when the photon is moving out from a gravitational field.)

I want to address redshift due to cosmological expansion.  First think about a photon moving through space in a given direction.  It has a wavelength determined by the speed of light, since a photon is a wavelet oscillating with a frequency determined by its energy (which is related to its colour and associated temperature).  That wave traces a path through space like this:

Say we stretch the original space illustrated above by a factor of two (looking at only the first full wavelength), we now get this:

The speed of light remains constant, so what we have here is a photon with twice the wavelength and half the frequency.  It’s an extreme case of redshifting.

To determine the redshift, z, we take the second wavelength, λ_now, subtract the original wavelength, λ_then, and divide by the first wavelength:

z=(λ_now-λ_then)/λ_then=(λ_now/λ_then)-1

Or,

z+1=λ_now/λ_then

Note that this equation is the same as standard redshift, where now=obsv and then=emit.

In a FUGE universe, the radius of the universe at a given time t will be ct.  The wavelength of an arbitrary photon at that time will be some fraction of that radius, so we can consider a photon of wavelength λ_then=ct_then/B.  After a period of expansion that same photon (now) will have the wavelength λ_now=ct_now/B.  Substituting this into the above and we get:

z+1=t_now/t_then

Alternatively, we can think about scale factor over time.  In a FUGE universe, a(t)=ct/ct₀ where is a reference time (usually now, but it doesn’t have to be in the case).  Note that cosmological redshift is given by

z+1=a_now/a_then

So using the scale factor equation above:

z+1=(ct_now/ct₀)/(ct_then/ct₀)

z+1=t_now/t_then

Alternatively, we can think about the OE curve, for which the equation is x'=(ct₀-x).x/ct₀.  Note that this could be confusing, because in all the OE curve articles in which I clarify that x=ct, I use t to refer to the time elapsed between emission time of the photon (or when it was in a particular location) and the observer.  The t used above (and in the redshift literature more generally) is a reference to the age of the universe (for which I use æ).  Also, in the OE curve equation, rather than being just a reference time, t₀ is specifically the current period of time since the beginning of the universe, or perhaps less confusingly t₀ is the current age of the universe, so t_now=t₀.

I do note, in Mathematics for Taking Another Look at the Universe, that in the OE curve equation it would be more accurate to say that it would be more accurate to use Δt and Δx, so Δx=c.Δt and thus

x'=(ct₀-Δx).Δx/ct₀=(ct₀-cΔt).cΔt/ct₀

I’ll use this notation for as much clarity as is possible under the circumstances, noting that Δt is referring to the delta between now and then, so Δt=t_now-t_then=t₀-t_then, and Δx is the distance a photon in static space would travel in that time.

Consider, hypothetically, that redshift might be due to the difference between the actual separation crossed and the distance that the photon needed to travel to cross that separation, or:

z=Δx/x'

Note that what this is effectively doing is comparing the distance between two comoving locations (emitter and observer) at two times (relative to the observer), time of emission and time of observation.

Substituting in the OE curve equation:

z=Δx/((ct₀-Δx).Δx/ct₀)

z=ct₀/(ct₀-Δx)

Noting that Δx=c.Δt:

z=t₀/(t₀-Δt)

And then noting that Δt=t₀-t_then and t₀=t_now:

z=t_now/t_then

This isn’t precisely the same as above, but for sufficiently high values of z, z+1≈z, so it’s a good approximation.

---

The problem, of course, as identified in an early article about redshift, is that the value for CMB redshift calculated this way is in the order of z≈36,000 (assuming t_d=380,000 years).  The accepted value is z≈1100.

However, this value is based on the assumption that the universe expanded such that H(t)=2/3t since decoupling/recombination (at t=t_d) which in turn would mean that z=(t_d/t₀)^⅔.  Note that 36,000^⅔≈1100.

I think that, if the FUGE universe model were to be correct, the redshift value associated with the CMB could still be correct, but the timing would be different, by a factor of ~32, so decoupling/recombination would have happened at t_d≈12.5 million years (which would have little impact on the oldest star – Methuselah – which is calculated to be 14.46 billion years old).

From what I can establish from looking at widely available information on the topic, the key event of decoupling/recombination is that the universe was sufficiently cool for the electrons to be captured by atomic nuclei.  This is 3000K, and the remnant radiation (in the form of the CMB) is currently 2.7K, so this suggests a redshift value of z=1100.  Then they have worked backwards, assuming the Standard Model (which is entirely reasonable) to say that the universe was at this temperature when it was 380,000 years old.

In the FUGE model, however, decoupling/recombination would have happened at 2.7/3000*13.8×10⁹=12.5 million years.

---

There’s another aspect of redshift, which I haven’t really mentioned above, which is the shift in colour.  The colour of a photon is related to its wavelength – red photons have a wavelength in the region 625-740 nm.  Blue photons have a wavelength in the region 450-495nm.  You can see, therefore, that as a photon’s wavelength is increased, its colour is changed in the red direction.

If the colour of a photon is pushed even further, it leaves the visible spectrum into infrared and eventually into the microwave spectrum (pushing it further you could even get radio waves).

We can think of the CMB as being a blackbody radiation spectrum associated 2.75K and indeed the correspondence is nigh on perfect with that (source):

The find the peak of this spectrum, we can use Wein’s displacement law, λ_max=b/T, where b(=2.897×10^-3 m.K) is the constant of proportionality and T is the temperature in kelvin.

Given the current temperature and the temperature at decoupling/recombination (t_d=3000K), we can work out the wavelengths:

λ_{max_CMB}(t₀)=2.897×10^-3/2.75=1.05×10^-3m

λ_{max_CMB}(t_d)=2.897×10^-3/3000=966×10^-9m

Note that this latter wavelength actually corresponds with infrared and the first, unsurprisingly, is in the microwave spectrum.

---

There is another way to arrive at a value of redshift in a FUGE universe.  This is via equations in notes from Oxford university (3.c. (Solution)), where it is stated that H(t_d)=H₀.1000√1000 and that (t_d/t₀)^⅔=1/1000.  If so, then it follows that (H(t_d)/H₀)^⅔=1000=(t₀/t_d)^⅔, and thus H(t_d)/H₀=t₀/t_d.  Since t₀=t_now and, in this instance, t_d=t_then, we have:

z≈t_now/t_then=H(t_then)/H(t_now)

Again, this should come as no surprise as, in a FUGE universe, H(t)=1/t.  However, if there were any direct evidence that, at decoupling/recombination, the value of the Hubble parameter was in fact H(t_d)≈ 7700, then we’d have another way to work out that z_CMB≈1100.

FUGE Clarifications

Out of interest, I put the link to Flat Uniform Granular Expansion into Google’s NotebookLM tool and listened to the two person “deep dive” that it generated (transcript here).  Generally, I was happy with it – even if it was very heavy on me being “out there”.  In a few instances however, the AI appears to have hallucinated, reaching conclusions that aren’t in the text and ascribing them to me.  However, it’s possible that I managed (unintentionally) to imply certain aspects that can be extracted by the human reader as well so here are some clarifications.

---

Origin of the FUGE model

The logic that leads to the FUGE model begins with three observations: the universe is expanding, the universe is flat and the age of the universe is approximately 13.8 billion years.

A flat universe has critical density.  Critical density is proportional to the square of the Hubble parameter (with all other components being constants).  Also, critical density is equal to the density of a Schwarzschild black hole with a radius equal to the Hubble length (which is merely the speed of light divided by the Hubble parameter).  The value of the Hubble parameter is equivalent to the inverse age of the universe.  (Note that the preceding link uses a Hubble parameter value of 67.8 km/s/Mpc which was determined by the Planck collaboration in 2015.  There is however a tension given that increasing accurate values for the Hubble parameter value are zeroing in on different values. Data from the JWST, SH0ES and more recent data from DESI indicate a value closer to 74 km/s/Mpc – which corresponds to a Hubble time of 13.3 billion years.  The Hubble parameter value which precisely corresponds with 13.8 billion years is 70.8 km/s/Mpc – which is clearly within the margin of error given the results that have been achieved.)

Then I propose a hypothesis – what would it look like if the universe was expanding such that its radius was increasing at the speed of light?  The value of the Hubble parameter would always be the inverse of the age of the universe.  And, the radius of the universe would always be the age of the universe times the speed of light.

But, if the universe were flat, then the density would have to be critical (and according to Sean Carroll, if flat then the universe has always been and will always be flat).  Note however that the critical density is proportional to the square of the Hubble parameter, which is inversely proportional to the radius of the universe – so the critical density is inversely proportional to the square of the radius.  However, the density of a fixed mass in a sphere (such as a Schwarzschild black hole) is inversely proportional to the cube of the radius.

This means that the mass-energy content of an expanding volume that maintains critical density must be increasing.  Then it’s relatively simple mathematics to show that mass-energy in a sphere at critical density, if it increases in radius by one Planck length must increase by the equivalent of half a Planck mass to remain critical.  To be increasing at the speed of light, the increase of one Planck length has to happen in one Planck time.

Another aspect of being flat is that the universe is smooth – as lumpiness at a sufficiently large scale could lead to curvature.  It follows therefore that the rate and distribution of mass-energy entering the universe would be smooth.

If we start with those conclusions, which are based on observations, the critical density and the volume being defined by the age of the universe, we arrive at a total mass-energy in the universe being equal to the baryonic (ordinary) matter attributed to the observable universe.  It is from this that the suggestion in the Standard Model that the total mass-energy of universe is about twenty times that of the baryonic matter appears to be a problem.  The FUGE model does not arise from eliminating Dark Matter and Dark Energy, but rather it is the case that the model doesn’t accommodate either of them.  The suggestion is that the phenomena which those dark substances were postulated to explain would have to be explained otherwise (if indeed they need explaining).

Note again that the critical density is equal to the density of a Schwarzschild black hole with a radius that is equal to the radius of a FUGE universe.  As the universe increases in size, while it does increase in mass, it decreases in density.  The same applies to a black hole (the Schwarzschild black hole, being one that is neither rotating nor charged, is the largest you can have with a given mass).  A universe which is at least 46 light years across, per the Standard Model, with a density of approximately 10^-26 kg/m³, is about one hundred times more dense than a black hole of that size.  (Note that if the universe were infinitely large, then this might not be a problem, it might only be a problem if there is a notional boundary, as there would be if the universe was initially constrained in a relatively small volume from which it expanded, as per the both the Standard Model and the FUGE model.)

 

Age of the universe

In the FUGE model, the universe is in the order of 14 billion years old.  It is not past eternal.  That said, if our universe were being “fed” by a “progenitor universe” that is/was orthogonal to ours, then that universe could be effectively eternal (in its own orthogonal history) with its eternity being entirely (and orthogonally) coincident with the initiation of our universe.

That said, our universe certainly had a beginning in the FUGE model with what is sometimes called an “instanton” – a Planck sphere (a sphere with a radius of a Planck length) containing the mass-energy equivalent of half a Planck mass at time t = one Planck time.

Note that in the FUGE model, you could work out how old the universe is in two ways.  You could look at the Hubble parameter or you could look at the density of the universe.  If the Hubble parameter were approximately 70 km/s/Mpc in a FUGE universe, then the universe would be approximately 14 billion years old.  If the density were approximately 9×10^-26 kg/m³ in a FUGE universe, then the universe would be approximately 14 billion years old.

 

Contents of the Universe (and the Distribution thereof)

The FUGE model does not, in itself, make comment on the nature of the mass-energy in the universe, nor its distribution.  All that the model suggests is that half a Planck mass is added to it every Planck time, and this has been the case for approximately 14 billion years.

The notion that the universe was opaque until about 380,000 years after the instanton is not problematic.

 

Threads?

No idea where this came from.  The expansion of space in the FUGE model is granular, in that individual “grains” of space are added stochastically (or randomly) to the volume.  The grains of space which appear are not linked, so it would be wrong to think of them as threads.  Note that it would result in a simultaneity problem if they were linked or coordinated in any way – with a whole new form of spooky action at a distance.

 

The Big Bang

When I first heard the artificial podcasters suggest that the FUGE model does away with the Big Bang, I reacted.  I didn’t mention the Big Bang in the Flat Uniform Granular Expansion article at all.  However, thinking about it, it could be that they were right, depending on the definition of the Big Bang.  If, and only if, you include inflation as a necessary component of the Big Bang theory, then the FUGE does not include the entirety of the Big Bang.  If, on the other hand, you consider the Big Bang to be “a physical theory that describes how the universe expanded from an initial state of high density and temperature” (per Wikipedia), then the FUGE model is consistent with the Big Bang.  An instanton is as dense as you can get, and the corresponding temperature for the mass-energy in an instanton is the Planck temperature, which is as hot as you can get at about 1.4×10³² kelvin (remember things cool down as you expand them, although we usually only notice this with gas it applies across the board for any form of matter that is compressible).

I would have to disagree with the podcast on this point, but only to the extent that they should have mentioned inflation, rather than the Big Bang.

Note also that I would have to disagree that the CMB is evidence of the Big Bang in the form that they must be referring to.  That is to say, the CMB is not evidence of inflation.  That is almost precisely backwards – inflation was introduced as an explanation for the statistical isotropy of the CMB.  If there is another mechanism to explain that phenomenon, then we can lose inflation.  However, the CMB is in fact evidence of the version of the Big Bang that the FUGE is consistent with (that the universe used to be a lot smaller and hotter about 13-ish billion years ago).  It is not, as the pseudo podcasters suggested, a “a result of the thermalization of all that energy that's constantly being added to the universe” – that is pure hallucination on the part of the controlling AI.

 

Red shift

Again, I reacted to this – especially as the artificial podcasters make it the central plank of their final part of the podcast.  I didn’t make any comments about redshift that were specific to the FUGE model.  The only mention was in terms of the sorts of observations that Hubble was making back in the 1920s, that all the stars around us are redshifted and the further away they are the more redshifted they are, and therefore the universe is expanding.

That said, there is a link in the article to The Conservatory - Notes on the Universe, which does hint at redshift.  However, this article doesn’t even mention the term and the one reference to stretching involves a god of physics making a circle out of nothing.  The only suggestion I make in that article is that if we consider a “carrier photon” on which the mass-energy of the universe were carried, after having added half a wavelength every Planck time, since the instanton, then that carrier photon would have an energy that is equivalent to the mass of the universe (which I know sounds crazy, it’s just that the mathematics stacks up).

I made no comment about the redshift of a photon emitted at the CMB in either article.  But I did in another article, which I am not going to link here, but you can find if you are interested (just look for the term “redshift”).  For dramatic reasons I got distracted in that article, which it isn’t really about redshift at all, it’s more about another model that I found when looking into the topic.  Although it does eventually curve back into considerations of redshift in a FUGE universe, that article makes no suggestion that redshift works the way that the podcasters mention.  Instead, I arrived at the equation z=t/(t₀-t) using a number of different approaches.

It is possible that yet another of my articles was accessed by the AI.  There is one that is about a claimed relationship between the temperature of the CMB and Hawking temperature, but also mentions redshift.  This could conceivably have been source of the comment:  “So instead of being leftover heat from the Big Bang, the CMB in (a) FUGE (universe) is more like a constant hum from all th(e) new energy being added to the universe.”

In that article there is this sentence: “The currently observed CMB, per Tatum's implied mechanism, is not from recombination per se but is instead a summation of all the redshifted temperatures across the Hubble radius (which is to say across the past).”  This was not something I said on my own part, but rather was musing on an implied mechanism that seems to be buried in Eugene Tatum’s Flat Space Cosmology.  Later in the article, I go on to clarify, in reference to the relationship between CMB temperature and Hawking temperature, that “the relationship does not quite make sense” and that “I cannot, at this time, work out how the temperature contributions would collate in precisely the right way to give the result that Tatum suggests and this is a major stumbling block as far as my accepting the reality of the relationship goes”.

---

I added this article to the AI (along with the Flat Uniform Granular Expansion article), and it still hallucinates - suggesting a past eternal, that the FUGE model has mechanisms to explain the acceleration of universal expansion for which Dark Energy is a proposed explanation and that the phenomenon explained by Dark Matter is accounted for in the FUGE model.  I want to clarify again for any human readers, the FUGE model does not include a past eternal (at least not in our time dimension) and the FUGE model does not allow for Dark Energy or Dark Matter that would require the universe to be 20 times more massive.

The FUGE model, if correct, would require alternate explanations for the related phenomena, perhaps something akin to MoND to replace Dark Matter (other alternative theories may be available) and acceptance that the measured acceleration of the universe is merely an artefact of that measurement - thus negating the need for Dark Energy (see also Dark Energy - Observational Skeptism).  The FUGE model does not, however, provide mechanisms for Dark Energy or Dark Matter.