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).