No, Espen Haug, You Cannot Just Magic God Time out of Empty Space

I recently wrote about Eugene Tatum’s assertion that the CMB is related to the Hawking Temperature of the universe.  The coauthor of that particular paper was Espen Gaarder Haug.  I didn’t mention it before, but Tatum is a Doctor, of Anatomic and Clinical Pathology.  Haug is also a Doctor, in the area of finance, specifically quantitative finance.  He got his doctorate in something relevant to options and trading (his doctoral thesis was on that anyway) but I can’t find out what he did his undergraduate in (presumably economics, but not necessarily).  He’s currently a finance professor at the Norwegian University of Life Sciences.

Haug and Tatum are coauthors on a range of papers about the flat cosmological model (which they sometimes refer to as the Haug-Tatum cosmology [HTC] model), while Haug has a quite a few papers of his own about the quantisation of gravity.

During research for No Eugene Tatum, Hawking Temperature is not Related to CMB, I stumbled upon one of Haug’s paper on the quantisation of gravity: God Time = Planck Time: Finally Detected! And Its Relation to Hubble Time.  Note that, like most of Haug’s papers, this one was published at scirp.org, which is a predatory publisher.  That said, being published in an odd place does not necessarily make the content wrong.  It’s just easier to get such publishers to publish bad and low-quality science – and even easier to publish at a blog under a pseudonym.  When you find a paper published by a pay-to-play or predatory publisher, you just need to be more careful and engage your critical thinking more intently than you might otherwise, such as when the source is more reputable.

The very first thing drew my attention was the title.  “God time”?  I wondered if I was looking at the work of someone like Luke Barnes, a person dabbling in physics to support a theological world view.  On closer inspection, and in the context of Haug’s other papers, including two others that mention “god”, it appears that he is just using a hook to draw attention rather than making any serious claim about the existence of an actual god.  Haug does refer to the “the god particle” in at least two of his papers without mentioning that the Higgs boson was know as the “god-damned particle” due to its reticence to reveal itself, which is unfortunate but, in the conclusion to the God Time paper, he indicates that historical references to indivisibility at the smallest scales, including apparent biblical references, are not important.

The basic claim of the God Time paper, and the one that I have a problem with, is that knowledge of G, ħ, and c is unnecessary to establish the value of t_P, the Planck time.  (In another paper, and another, Haug seems most intent on removing the use of G entirely, which is odd.  There are problems in those papers too, but I will try to restrict my efforts here to addressing Haug’s efforts to eliminate fundamental constants.)

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The standard definition of Planck time is:

Given that the defined value consists of G, ħ, and c, and only G, ħ, and c, establishing Planck time without the use of those fundamental constants should be quite a challenge.  Haug does it by using the equation:

Which he later generalises (in a special case) to:

In his numerical example for the first equation, Haug sets object 1 to the Sun and object 2 to the Earth.  The term δ is used to refer to the effect of  gravitational lensing by object 1 (the Sun), expressed in radians.  (Note that in the link provided, the symbol θ is used.) The term g refers to the magnitude of gravitational acceleration at object 2 (the Earth).  Note that the equation at the link has a direction that is implied by the minus sign, it’s not intended to refer to a negative value.  The term λ is a reference to the Compton wavelength of the mass of the relevant object, but in this case it is the reduced variant, in the same way that the reduced Planck constant is given by ħ=h/2π.  So, for clarity, Haug provides the numerical result for this equation (using subscripts E for Earth and S for Sun):

where (per the links above)

If we substitute these into Haug’s equation, we get:

which resolves to

In his generalised version, Haug is talking about a single mass, so no subscripts are required, and:

which also resolves to

So, yes, you can get Planck time using Haug’s equation which does not explicitly use G, ħ and c.  However, there is a problem with his assertion that we can therefore reach a value for Planck time without using any of the fundamental constants because, in order to use his method, we must establish the values for at least gravitational lensing deflection and the reduced Compton wavelength.

The value for gravitational lensing deflection due to the Sun has been measured, first by Eddington in 1919, so while not so easy, this is entirely possible – with some caveats.  The apparent radius r in the equation (per Wikipedia), θ=4GM/c²r, is not necessarily the radius of the body with the mass M, but rather the distance between the centre of the mass and the radiation being deflected.  The value measured by Eddington was for light that grazed the surface of the sun, hence the need to observe a total solar eclipse.  The related deflection is, therefore, at the photosphere, which is considered be the surface of the sun, and r=R_s.

The Compton wavelength, on the other hand, is not something that is measured for, and probably doesn’t even apply to, bodies at the scale of the Earth and the Sun (because it’s a quantum mechanical property).  We can calculate it, sure, but to do so we need to use both ħ and c.  (This could explain Haug’s fixation on eliminating G in his other papers, if he has noted that he can’t get around using ħ and c.  He says it explicitly in section 6 of another paper.)

If we forgive this, then we still have some problems to address.  The accuracy of the value of t_P that can be determined using Haug’s first equation depends on the accuracy of the measurement of:

Deflection due to gravity as light grazes the surface of the sun – 3%

Gravitational acceleration at some point on the Earth with a distance from the centre of R_E (not the defined standard gravity value which is nominal and limited by caveats) – about 0.7% assuming it’s somewhere on the surface

The distance to the centre of the Earth used above (about ±30mm in 6378km) – 0.0000005%

The radius of the photosphere – 0.02%

The mass of the Sun (for calculating the reduced Compton wavelength) – 0.005%

The mass of the Earth (ditto)– 0.01%

We don’t need to worry about the accuracy of h and c, because these values are both defined.  The only inaccuracy in ħ will be due to the approximation to π that is used, but this can be set arbitrarily low by using π to many significant numbers.  So this means the resultant inaccuracy in the measurement of t_P would be, approximately:

3%+(0.7%/2)+0.02%+0.005%+0.0000005%+(0.01%/2)=3.38%

Presumably, from a value of t_P determined this way, we could calculate a value for G using the standard definition of t_P and the defined values of ħ and c.  This value would have an accuracy of approximately 6.76%.

Compare this with the NIST values which have accuracies of 0.0011% for t_P and 0.0022% for G.

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As indicated above, Haug seems to realise that he can’t get around using ħ and c, and is therefore laser focussed on eliminating the use of G, even in the process of determining the value of t_P.

This, to me, is madness.  If expressed in Planck units, all the four values listed above, ħ, c, G and t_P, resolve to unity.  (In addition, m_P, l_P, q_P, v_P, i_P, T_P, E_P, k_B, k_e, m_P, reduced μ₀ and raised ε₀ also resolve to unity.  Even the unitless gravitational and electromagnetic coupling constants resolve to unity when Planck values for mass and charge are used as the reference rather than arbitrary values like a proton or electron mass or the elementary charge.)

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In other papers, like Quantum Gravitational Energy Simplifies Gravitational Physics and Gives a New Einstein Inspired Quantum Field Equation without G and Not Relying on the Newton Gravitational Constant Gives More Accurate Gravitational Predictions, Haug suggests that he’s hit upon some evidence of the quantisation of gravity – all which seem to relate to the notions of “collision length” and “collision time” (which is expanded on in Collision-space-time: Unified quantum gravity – where he assigns photons with a mass (an extremely tiny mass, admittedly)).

In the first two papers above, Haug mentions the “factor”:

Which he refers to as “reduced Compton frequency per Planck time”.  This is an odd way of putting it, for more than one reason.  First, as a simile, we could say that Gm_P²/ħ is “Planck length per Planck time”.  This not completely untrue, because Gm_P²/ħ=c=l_P/t_P, but it’s an odd way of putting it.  The second reason may not be immediately clear, but we can look at Haug’s own words, from Quantum Gravitational Energy Simplifies Gravitational Physics and Gives a New Einstein Inspired Quantum Field Equation without G:

It's not anything “per Planck time”, it’s a value that is multiplied by Planck time, not divided.  If anything, it’s equivalent to “Planck time per reduced Compton period”, noting that there’s another error buried in there.

Haug states that the reduced Compton frequency is the speed of light divided by the reduced Compton wavelength.  Presumably, the vanilla Compton frequency (f_c) is the speed of light (c) divided by the vanilla Compton wavelength (λ_c):

If we implement the reduced Compton wavelength, which is the Compton wavelength divided by 2π, we have (using the bar as an indication of some sort of modification, not necessarily reduction):

So it’s not a reduced Compton frequency, it’s a raised Compton frequency.

He could, however, express things somewhat less awkwardly.  Using the definitions of reduced Compton wavelength and Planck length, we see that:

And, in flat universe, we know that*:

So, Haug’s “factor” is, in fact, simply an expression for the age of the universe (at time t) divided by Planck time (or, to put it another way, the magnitude of the age of the universe when expressed in Planck time).  Or, as I have used frequently elsewhere, ꬱ (first introduced in Avoiding a Contravention of the Extended Consistency Principles).  It’s a useful term in that context, but not so much when mixed up with the notion of a confused (reduced? raised?) Compton frequency.

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I did reach out to Haug to discuss the above but have not, yet, heard back from him.

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* There's a little oversimplification here.  Keen-eyed readers will note that M has not been explained here.  It's not quite the mass of the universe, because in a flat universe, the mass is m_P.ꬱ/2.  To understand what is going on a review of The Conservatory - Notes on the Universe might help.  In brief though, if we think of a standing wave between two nodes (null points), then the minimum full wavelength is twice the minimum distance possible between nodes - because there is a node in the middle of a full wavelength.  The consequence of this, if we have granularity (as in a FUGE Universe), is that when the universe expands by one Planck length, there is one additional node added, allowing for one more half wavelength, or λ(t)=c.t/2.  Each half wavelength corresponds to an extra half a unit of Planck mass/energy, so the M value above (if it were the mass of the universe) would be M(t)=m_P/2.t/t_P.

Note that because, in the FUGE conception, increments are in half wavelengths, Haug’s “factor” would become (noting the caveats above):

No, Eugene Tatum, Hawking Temperature is not Related to CMB

When I first wrote about redshift (in the context of the OE Curve), about six months ago, I took a pretty hard turn (away from) and ended up mostly covering the ideas of Eugene Tatum, who has quite a few papers out with various colleagues, mostly self-published and with no discernible peer review. I have now broken out the redshift discussion into a separate article and am using this article to focus properly on some key equations from Tatum’s “flat space cosmology”.

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The first thing to note is that Tatum’s flat space cosmology (FSC) is a model of the universe that is:

flat (“cosmic radius R and total mass M follow the Schwarzschild formula … at all times”),

expands such that the “cosmic event horizon” expands at R=ct (“the cosmic event horizon translates at speed of light c with respect to its geometric center”), and

has a Hubble parameter such that H=1/t, where t is the age of the universe (“(the) cosmic Hubble parameter H can be expressed as c/R and Hubble time (universal age) can be expressed as R/c for any stage of cosmic expansion”).

These are all consistent with a FUGE universe, so it was very exciting to see someone else working on a similar idea with a large number of papers published and a raft of supporting equations.

Tatum goes beyond a FUGE universe however when he introduces the notion of “(the) cosmic linear velocity of rotation” which he calls an angular velocity (without initially introducing a 2π term) and goes on to say, after mentioning Hawking’s black hole temperature formula, that “at any radius R the cosmic temperature T is inversely proportional to the geometric mean of cosmic total mass M and Planck mass”.

Note that, while some close reading is required, it clear in context that, by “cosmic temperature”, Tatum means the CMB radiation temperature, which today is determined to be 2.725K.

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To understand why there are problems with Tatum’s equation(s), we must take a look at Hawking (radiation) temperature.  For any black hole of mass M, there is an associated temperature given by:

For both a FUGE universe and a universe with FSC, the mass of the universe of radius R=ct is given by M=m_P/2*R/ct_P=m_P/2*t/t_P, where m_P is the Planck mass, and t_P is the Planck time.

That value, assuming a universe that is t₀=t_now=13.8 billion years old, is M_now=8.79×10⁵²kg.  The corresponding Hawking temperature, for a black hole with a radius R_S=R_now=ct_now, would be T_{H_now}=1.40x10^-30K, which is substantially less than the CMB radiation temperature today (2.725K).

Tatum does not claim however that the CMB temperature is equal to the Hawking temperature of a black hole with “cosmic radius” of the universe, merely that there is a relationship, which he establishes (in Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database) as:

First note that I have changed all of his subscripts for clarity, since he used T_CMB,0, k_b, M_H (referring to the radius when H(t)=H₀ rather than Hawking, as it means in T_H) and m_p.  Tatum’s use of subscripts is quite inconsistent, with Planck units sometimes getting _pl or _Pl, current values getting ₀, _H, _r and _R.

Second note that Tatum points out that his equation is “quite similar” to the Hawking radiation temperature formula.  I would put it differently, what he is effectively claiming is that:

Noting that M_now=m_P/2*t_now/t_P:

We calculated above that T_{H_now}=1.40x10^-30K, and t_now/t_P=8.08×10⁶⁰ (if t=13.8 billion years), so:

This is about right, so that is indeed intriguing.  However, if we look at the standard time for when the CMB was formed – the decoupling/recombination event at t_d=380,000 years, when t_d/t_P=2.22×10⁵⁷ and the T_{H_d}=5.07×10^-26K – we get:

So what we have here is a deviation from the Standard Model and what we understand about decoupling/recombination (hence use of the symbol “≠”), specifically that the temperature at that time had to be ~3000K.

In a FUGE universe, and indeed with FSC, the universe does not expand at a rate of H(t)=2/3t after decoupling/recombination (and in fact it doesn’t in the Standard Model either, or at least not for the entire period, but that’s another issue). 

Instead, with a flat universe, H(t)=1/t so the timing of decoupling/recombination is not the same as for the Standard Model.  In a flat universe, t_d=12.5 million years, when t_d/t_P=7.32×10⁵⁷ and the T_{H_d}=1.54×10^-27 K, meaning that:

This is effectively a proof that Tatum’s equation is invalid.  By coincidence, it works for current values (with an error of about 3%), but fails for past and future values.

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While working through this, I noticed some other errors.  For example, in The Basics of Flat Space Cosmology (where he lays out FSC), Tatum starts the equation with:

He doesn’t derive this equation but it’s not the relationship that I have issues with, but the use of the equivalence symbol.  Here’s the derivation, for a flat universe (both FUGE and FSC), noting that M_R is the mass of the universe when it has a cosmic radius of R=ct, where t is the age of the universe:

So noting that R=ct, we get the equation above.  But it’s not approximately equal to, it’s precisely equal to, so “=” should be used, not “≅”.

Saying that the Hubble parameter is angular velocity just seems to be nonsense (where is the 2π term?), and the notion of a Planck mass angular velocity is nonsense on top of nonsense – as is the later discussion of “galactic revolving speed” (in which, again the obligatory 2π term is forgotten).  The Hubble parameter in any flat universe model is merely the inverse age of the universe, nothing more.  For that reason, in the discussion below, I will either ignore any term in which Tatum uses any form of ω, or I will replace it with 1/t, which it clearly equates to given that he states that ω≅c/R≅H, again with erroneous use of “≅”.

That all, I want to look more closely at the third equation.  Making all the following subscripts consistent as discussed above:

I have put the “≠” in there to highlight that there is an issue, one that follows through from an error made in the second equation in which Tatum writes (again with error highlighted):

In the final term, they divided by two in the wrong spot.  It should have been within the square root, so:

This means the third equation should have been:

This is probably not news to Tatum though, because in Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, the full first equation as presented is:

It still doesn’t work for any value other than those that apply right now, but he has fixed at least one problem (albeit without fixing the reference [The Basics of Flat Space Cosmology] used as support for quoting this equation).

A Relationship Between Hawking Temperature and CMB Radiation Temperature?

In Observable Events Curve - Shifting About Redshift (Including an Alternate FUGE Universe), I looked at a paper by Espen Haug and Eugene Tatum (Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database).  I noted that they had a rather awkward “composite constant” that they label upsilon (Ʊ), which is used in one equation, namely H₀=ƱT₀², where T₀ is the (current) temperature of the CMB, and that there was a more simple expression of upsilon than the one they had used.  I also noted that, in a different paper by Tatum, Seshavatharam and Lakshminarayana (The Basics of Flat Space Cosmology), there was an equation of T₀ which was related to the equation for Hawking temperature, T_H= ħc³/8πGMk_b.  All they had done, effectively, was replace the term M with √(Mm_P), where m_P is the Planck mass.

It is certainly interesting that there might be a relationship between the Hawking temperature of Schwarzschild black hole with today’s Hubble radius and the current CMB temperature, T₀.  But I would suggest going about it a different way, in the context of a FUGE universe.

In a FUGE universe, M(t)=(m_P/2).(t/t_P) and the age of the universe (t) would be such that ꬱ(t)=t/t_P:

T_H(t)=ħc³/8πGM(t)k_b=(m_P².c²/k_b)/(8π.(m_P/2).(t/t_P))

T_H(t)=(m_P.c²/k_b)/(π.ꬱ(t))=T_P/(4π.ꬱ(t))

For t=t₀:

T_H(t₀)=T_P/(4π.ꬱ(t₀))=(T_P/(4π.√(2ꬱ(t₀))).√(2/ꬱ(t₀))

And noting (as shown in Observable Events Curve - Shifting About Redshift) that T₀=T_P/(4π.√(2ꬱ)):

T_H(t₀)=T₀√(2/ꬱ(t₀))

Meaning that:

T₀²=(ꬱ(t₀)/2).T_H(t₀)²

Or:

T₀²=(m(t₀)/m_P).T_H(t₀)²

Just to confirm this, the Hawking temperature for a Schwarzschild black hole with a radius of 13.8 billion light years is T_H(t₀)=1.4×10^-30K, ꬱ(t₀)=t₀/t_P=8×10⁶⁰, and

T₀²=((8×10⁶⁰)/2).(1.4×10^-30)²=4.(1.4)²=(2.8)²

So,

T₀=2.8K

As Rhodri Evans put it, simples.

Why there is such a scaling, well … not quite so simples.  While it does seem to be there, without a clear underlying rationale for the relationship, it could be a coincidence.

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Note that there is some roughness above.  I used 8×10⁶⁰ as the value of ꬱ(t₀), which corresponds to an age of the universe of 13.66 billion years.  Also the CMB temperature is 2.72548±0.00057K (from 2009).

We can arrange the equations above to get:

T₀²=(ꬱ(t₀)/2).(T_P/(4π.ꬱ(t₀)))²=(1/2).(T_P/4π)²/ꬱ(t₀)

ꬱ(t₀)=(T_P/T₀)²/32π²

So, noting from Return to Constants that Resolve to Unity that T_P=1.416×10³²K:

ꬱ(t₀)=(1.416×10³²/2.72548)²/32π²=8.5465×10⁶⁰

which corresponds to a universe that is 14.60 billion years old, with a Hubble parameter of 66.99km/s/Mpc (as arrived at in Observable Events Curve - Shifting About Redshift).

This Hubble parameter value is what WMAP and Planck Collaboration measurements seem to be zeroing in on.  The age of the universe does not match with their estimates, but it should be noted that we are using a different model here, so some difference should be expected.

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As mentioned above, despite the close alignment there is no immediately obvious reason why the CMB temperature (as measured today) should have any relationship to Hawking temperature/radiation whatsoever.  The notion of the Hawking temperature is that a black hole will radiate from its surface, which is equated to the Hubble radius.  However, no photons from the Hubble radius at any time in the past or future would or will ever reach us since that radius is recessing away from us at the speed of light.

That said … the surface defined by the Hubble radius is not a surface per se.  It’s a nominal surface.  It might be more accurate to consider the universe as a whole as being a surface.

Purely as a thought experiment, consider that entire surface to be emitting Hawking radiation such that each location on that surface has the appropriate Hawking temperature.  Effectively, half of the radiation would be emitted in an observer’s direction and the other half would be emitted in the other direction.  That would give us division by a factor of 2.  And then there would be the summation of all the radiation over the entirety of the radius of the universe, which for a FUGE universe, is ꬱ(t₀) fundamental segments.

Of course, each of the packets of radiation would take time to get to the observer, during which time the universe would have expanded, reducing the temperature related to the radiation via redshift.  However, Hawking radiation is inversely proportional to mass (and therefore also radius) – meaning higher temperature in the past.  My suggestion, currently unsupported by mathematics, is that the decrease in temperature due to expansion could be precisely balanced by the higher temperature in the past such that the result of the summation is T₀²=(ꬱ(t₀)/2).T_H(t₀)².

While satisfying, this suggests a totally different interpretation of the CMB and what the universe was like in the very early era.

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For interest’s sake, what would the value of T_H be at some point in the past, say during recombination at ꬱ=380,000 years?

T_H(380,000 years)=5.1×10^-26K

And, using Tatum et al.’s equation, the equivalent of the CMB at that time would have been:

T(380,000 years)=537K

Recall that the standard temperature attributed to recombination is ~3000K, so Tatum et al.’s calculation does not appear to work.  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) - but I am far from convinced that this would work either.

Using the same process for various ages of the universe, we get the following chart:

Note that I had to cut this off because if we include values of 380,000 years or less, we get something like this:

We can resolve this scale problem by using logarithmic scales:

Note that a temperature in the order of 3000K appears at 12,000,000 years.

We can also look at a comparison between background temperature and Hawking temperature:

The lines intersect at the point at which the universe has an age of two units of Planck time (as expected because if M=m_P, then √(m_P.M)=M and these are the terms that differentiate Tatum’s equation [as expressed in The Basics of Flat Space Cosmology] from Hawking’s).

I must stress that, if this works, it has nothing to do with a single photon being emitted at the time of recombination which is observed by us about 13.8 billion years later with some appropriate redshift.  Instead it would effectively be a photon emitted, whenever it was emitted, at the temperature that it had at that time, which would have decreased temperature because of expansion and increased temperature due to having picked up Hawking radiation on the way through, but since that is very small, the overall effect would still be cooling, just at a significantly reduced rate.

Does it work?  It is really difficult to say.  The coincidence, if it is indeed a coincidence that the cosmic microwave background temperature of the universe has an apparently meaningful relationship to the Hawking temperature, is striking.  But the relationship does not quite make sense.  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 – especially given that we already have a very good explanation for the cosmic microwave background in terms of cosmological redshift.

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I did say that this would be a shorter post.  My apologies to anyone who was relying on that.

Observable Events Curve - Shifting About Redshift (Including an Alternate FUGE-like Universe)

I tried calculating redshift in a FUGE universe, and ran into what looked to be a problem.

My logic was as follows: if we are observing a photon from an event that occurred a time t ago, then that photon had a proper distance of x'=x.(ct₀-x)/ct₀ and the event location (at the time of observation) will be x=ct from the observation location.  So for any arbitrary wavelength of the photon:

1+z=λ_observed/λ_emitted=x/x'=ct₀/(ct₀-x)

z=ct₀/(ct₀-x)-1=(ct₀-(ct₀-x))/(ct₀-x)=x/(ct₀-x)

z=t/(t₀-t)

The problem is that using this equation, redshift for the CMB does not come out to be what is generally attributed to it (z_CMB=1100).  Recall that t₀ here is the age of the universe and t is the transit time for an observed photon (see most recent posts, particularly Mathematics for Taking Another Look at the Universe), so for a photon emitted during recombination, the event (also confusingly known as decoupling) that led to the CMB when the universe was 380,000 years old, t_CMB=(13.8×10⁹-380,000) years and:

z_CMB=t_CMB/(t₀-t_CMB)

z_CMB=(13.8×10⁹-380,000)/(13.8×10⁹-(13.8×10⁹-380,000))=36,300

This is 33 times higher than the standard answer.  We can reorganise the equation above to work out how old the universe must have been for photons from the CMB to have a redshift of z=1100.  Using ꬱ_CMB=t₀-t_CMB (and thus t_CMB=t₀-ꬱ_CMB):

z_CMB=(t₀-ꬱ_CMB)/ꬱ_CMB=t₀/ꬱ_CMB-1

t₀/ꬱ_CMB=z_CMB+1

ꬱ_CMB=t₀/z_CMB+1

For z_CMB=1100, we get an ꬱ_CMB=1.25×10⁶ years.  Hm, it appears that something is not right.

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The question that immediately arises, at least for me, is how do we know that the redshift for photons from recombination is 1100 and how do we know that recombination happened when the universe was 380,000 years old?  And what are the error bars associated with these values?

The redshift value is a little rubbery, but is usually quoted as simply 1100, although it’s probably a bit lower.  Rhodri Evans (astrophysicist and author of The Cosmic Microwave Background - How It Changed Our Understanding of the Universe) has a blog post on the CMB redshift which gives a bit of the history, indicating that the value comes from a comparison between the temperature of the CMB radiation today and that at the time of recombination, so:

z_CMB=T_recomb/T_now=3000/2.725≈1100

This equation is a slight approximation, since it should be z+1=T_recomb/T_now and I will be using the non-approximation from now on.

Note also that the equation does not explicitly rely on the timing of the event.  The recombination is thought to have happened when the universe got sufficiently cool, so the timing isn’t actually key.  While Evans does write that “as the Universe expands, the temperature (..) decreases in inverse proportion to its size. Double the size of the Universe, and the temperature will halve”, to know when the temperature of the universe was 3000K, we would have to know what the temperature was at some other time, what that time was and what the relevant expansion rate was.

Note that in a FUGE universe, the radius of the universe is directly proportional to its age (so currently r₀=ct₀).  So, noting the inverse relationship, we could simply replace T_recomb and T_now with 1/ꬱ_recomb=1/380,000 years and 1/ꬱ_now=1/t₀=1/13.8×10⁹ years.  Which gives us … z≈36,300.

So, I still have questions about timing and redshift and now also temperature.

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When I expanded my search to find out how the relevant temperatures are calculated, I stumbled on what looks to be a variant of the FUGE universe, described in papers that have been published in what appear to be legitimate journals.  This model is referred to more frequently as Flat Space Cosmology, but there are also references to “r_H=ct models” (presumably those similar to Melia's, where t is the age of the universe) and “growing black hole models” which seems to describe something akin to the FUGE universe (I disagree with the terminology but that may just be a matter of perspective).

Espen Haug and Eugene Tatum (and others) have, in the past half a year, published a number of papers, for the most part on open archive sites but sometimes in journals (for example the International Journal of Theoretical Physics).  The paper that most attracted my attention provides a method for calculating temperature, Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database (available from HAL open science, which is an open archive).

I’m not going to get into whether or not they have actually solved the Hubble Tension, instead I am going to look at equations in that paper that I have issues with.

The first appears late in the paper:

The complexity of this equation does not appear justified, since it resolves to:

Ʊ=2(4π/T_P)²/t_P

where T_P is Planck temperature and t_P is Planck time.  This follows from T_P=E_P/k_b where E_P=m_Pc² is Planck energy and k_b is the Boltzmann constant, noting that m_P=√(ħc/G), so that:

Ʊ=k_b²32π²G^1/2/c^5/2ħ^3/2=((E_P/T_P)².2.(4π)²/c²ħ).√(G/ħc)

Ʊ=((m_Pc²/T_P)².2.(4π)²/c²ħ)/m_P=(m_Pc²/ħ).2.(4π)²/T_P²

Ʊ=(√(ħc/G).c²/ħ).2.(4π/T_P)²=(√(c⁵/ħG).c²/ħ).2.(4π/T_P)²

And since t_P=(√(ħG/c⁵):

Ʊ=2.(4π/T_P)²/t_P=2.923×10^-19K^-2s^-1

This “composite constant” upsilon, which has no other apparent name than the Latinised Greek letter used to denote it, has no other apparent use than in the equation H₀=ƱT₀², where T₀ is the (current) temperature of the CMB.  In the paper in which upsilon is introduced, Upsilon Constants and Their Usefulness in Planck Scale Quantum Cosmology, it is derived purely from this simpler relationship.  (I should provide a warning here that SCIRP is considered to be a predatory publisher, meaning that there is no peer review for articles which are published after payment.  Tatum (who is an anatomic & clinical pathology specialist during the day) indicates that he consulted Dr. Rudolph Schild of Harvard-Smithsonian Center for Astrophysics.  Unfortunately, Schild apparently publishes in a fringe (and allegedly predatory) astronomy journal, Journal of Cosmology, of which he is the editor in chief.  Tatum has published in that journal at least twice.  That all said, if the mathematics is correct, it is correct irrespective of where it has been published, even if the author paid to have it published.)

Earlier in Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, there was something that really attracted my attention, an equation for the current CMP temperature T₀:

Once again however, this can be simplified.  Note that R_H=c/H₀ is the Hubble radius which, in a flat universe, is equal to R_H=ꬱ.l_P, and that l_P=√(ħG/c³) so:

T₀=ħc/(4π.k_b.√(ꬱ.l_P.2.l_P))=ħc/(4π.(m_Pc²/T_P).l_P.√(ꬱ.2))

T₀=ħc/(4π.(√(ħc/G).c²/T_P).√(ħG/c³).√(2ꬱ))

T₀=T_P/(4π.√(2ꬱ))

Meaning that the only equation in which upsilon is used, H₀=ƱT₀², resolves to:

H₀=ƱT₀²=2.(4π/T_P)²/t_P.(T_P/(4π.√(2ꬱ))²

H₀=1/(ꬱ.t_P)

This is precisely what one would expect from a flat universe.

There is a slightly different approach, just using R_H=H_o.c, but it is messy. 
This messiness can be alleviated if we work with T₀², so that:

T₀²=(ħc/(4π.k_b.√((c/H_o).2.l_P)))²=ħ²c²/((4π)².k_b².((c/H_o).2.l_P))

T₀²=ħ²c²/((4π)².(m_Pc²/T_P)².((c/H_o).2.√(ħG/c³)))

T₀²=ħ²c²/((4π)².(m_P²c⁴/T_P²).((c/H_o).2.√(ħG/c³)))

T₀²=ħ²c²/((4π)².((ħc/G).c⁴/T_P²).((c/H_o).2.√(ħG/c³)))

Then rationalising and rearranging

T₀²=√(ħG/c⁵).H_o/(2.(4π)²/T_P²)=t_P.H_o/(2.(4π)²/T_P²)=H_o/Ʊ

So, of course, H₀=ƱT₀².

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A potential critique, if one were to only consider Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, is that all Haug and Tatum have done here is shuffle numbers around in order to hide H₀ in T₀, and then created a new constant (Ʊ) which does nothing more than reveal the previously hidden H₀.  While that may appear to be the case, such critique ignores the fact that T₀ is a measured value, specifically the temperature of the CMB today or 2.72548±0.00057 K (from 2009 but seemingly still most commonly used).  As Tatum showed in Upsilon Constants and Their Usefulness in Planck Scale Quantum Cosmology, upsilon can be used to extract H₀ from the CMB temperature:

ƱT₀²=(2.923×10^-19K^-2s^-1).(2.72548K)²=2.171×10^-18s^-1

Noting the km to Mpc ratio, 3.086×10¹⁹km/Mpc, we find:

ƱT₀²=66.99km/s/Mpc≈H₀

This would correspond to a FUGE universe which is 14.60 billion years old.

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The equation for T₀ was presented in a slightly different format in an earlier paper by Tatum, Seshavatharam and Lakshminarayana The Basics of Flat Space Cosmology:

Why the third term is there is beyond me, given the parallels between the second and the fourth.  The first two terms are a modification to the Hawking radiation temperature equation, as Tatum acknowledges in Upsilon Constants and Their Usefulness in Planck Scale Quantum Cosmology, (noting that k_b is the same thing as k_B).  The standard Hawking radiation temperature equation:

The implication here is that Tatum et al. are conceptually equating the temperature of the CMB to the temperature of a black hole with the same radius, but with a different value for mass – which doesn’t make sense because with a different mass you are no longer talking about a black hole.  Alternatively, they are effectively scaling that temperature (and the expression of mass in their equation as being less than that of an equivalent Schwarzschild black hole is misguided).

This leads to an intriguing notion and the potential for a different approach, in terms of establishing redshift.

Consider the Hawking radiation temperature for a black hole with the mass and radius of the prevailing Hubble sphere, now and at the time of recombination (when the CMB was generated).  Recall that, per Carroll, “a spatially flat universe remains spatially flat forever” and “the corresponding Schwarzschild radius … equals the Hubble length”.  As a consequence, the radius of the universe now is r_H-0=c/H₀ and, at recombination, it would have been r_H-recomb=c/H_recomb, where H_recomb is Hubble parameter for that time. 

According to Andrei Starinets’ General Relativity and Cosmology solution notes, the Hubble parameter at recombination (note that the event is referred to in the notes as “decoupling”) is H(t_d)=H_recomb= H₀.1000√1000.  Noting that Schwarzschild radius is directionally proportional to mass, and Hawking radiation temperature is inversely proportional to radius, we have (per the equation above from Rhodri Evans, without the approximation):

z_CMB+1=T_H-recomb/T_H-0=r_H-0/r_H-recomb=(c/H_now)/(c/ H_recomb)=H_recomb/H₀

z_CMB+1=1000√1000=31,600

Hm, still not right but it is closer to my answer but given that a(t₀)/a(t_d)=1000 should be almost certainly thought of as a(t₀)/a(t_d)=10³, rather than a(t₀)/a(t_d)=1.000×10³.   The tutorial notes go on to state that:

So, if we plug in the values t_d=380,000 years and t₀=13.8×10⁹ years, we get:

Curiously, if a(t₀)/a(t_d)=1100 is used, we get z=36,000.

Note also that the solution notes do something akin to a pea and cup trick (similar to Tatum’s apparent hiding and revealing of H₀ above).  He presents three equations:

a(t_d)=a(t₀)/1000 … H(t_d)= H₀1000√1000 … (t_d/t₀)^2/3= a(t₀)/a(t_d)=1/1000

It is not difficult to see, when these equations are put side by side to see that:

H(t_d)/H₀=1000^3/2 and t₀/t_d=1000^3/2

So:

H(t_d)/H₀=t₀/t_d

It is unclear why this much simpler and, in retrospect, obvious relationship is not used.

We can go further to show that, since H(t_d)=H_recomb, t_d=t₀-t_CMB (where t_CMB as defined above is how long ago the CMB was generated) and z_CMB+1=H_recomb/H₀:

 z_CMB+1=t₀/(t₀-t_CMB)=t₀/(t₀-t_CMB)-(t₀-t_CMB)/(t₀-t_CMB)+1=t_CMB)/(t₀-t_CMB)+1

And generalising:

z=t/(t₀-t)

Which is the result that I arrived at, using a second approach

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Another approach is on the basis of a FUGE universe using Evan’s equation z_CMB=T_H-recomb/T_H-0.

In a FUGE universe, the current mass is M(t₀)=(m_P/2)t₀/t_P and at any time t ago, M(t)=(m_P/2).(t₀-t)/t_P.  Noting that mass is directly proportional to Schwarzschild radius and temperature is inversely proportional to radius, the equation above becomes:

z+1=T_recomb/T_now=M(t₀)/M(t_CMB)=((m_P/2)t₀/t_P)/((m_P/2).(t₀-t)/t_P)

z+1=t₀/(t₀-t)=t₀/(t₀-t)-(t₀-t)/(t₀-t)+1=t/(t₀-t)+1

This is precisely what I arrived from Starinets solution notes, so we have the same result using a third (slightly different) approach.

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In Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, Haug and Tatum get a third value for z.  This is established from use of his (apparently) scaled temperatures.  This ends up with him comparing T₀=T_P/(4π.√(2ꬱ₀)) with T_CMB=T_P/(4π.√(2ꬱ_CMB)) so, noting that the subscript CMB here is equivalent to recomb used above:

z+1=T_CMB/T_now=T_recomb/T_now=(T_P/(4π.√(2ꬱ₀)))/(T_P/(4π.√(2ꬱ_recomb)))

z+1=√(ꬱ₀/ꬱ_recomb)=√(t₀/(t₀-t))=√(36,000)

It should be noted that they state that there was a choice between T_t=T₀√(1+z) and T_t=T₀(1+z).  They chose the latter, but if we choose the former, we get:

z+1=(T_recomb)²/(T_now)²=(T_P/(4π.√(2ꬱ₀)))²/(T_P/(4π.√(2ꬱ_recomb)))²

z+1=ꬱ₀/ꬱ_recomb=t₀/(t₀-t)=t₀/(t₀-t)-(t₀-t)/(t₀-t)+1

z=t/(t₀-t)

So we have a fourth approach to arrive at the same result (albeit via the questionable route of taking Haug and Tatum seriously).

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Finally, returning to Starinets’ solution notes, he notes that in the “matter-dominated era” a(t)=(t/t₀)^2/3.  This is a Standard Model thing.  In a FUGE universe, a(t)=(t/t₀) at all times, which is to say that there has been more stretching of the universe over the period between recombination and today and more redshift, so it’s precisely what we should expect.

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As I have managed to find five* methods for arriving at equations for redshift which indicate that, for an event that occurred a period t ago, z=t/(t₀-t), I am no longer convinced that there is a problem with my calculations.

There is however some confusion on the part of Tatum et al. associated with the introduction of their “composite constant” upsilon.  This will be touched on again in the next (shorter) post.

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* I understand that not all five methods are entirely distinct.  Also questions remain about Tatum's equations, not only with respect to his methods for getting the papers that contain them published but also with respect to whether the implied relationship he identifies via those equations is anything more than a rather startling coincidence.