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 æ(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.æ(t))=T_P/(4π.æ(t))

For t=t₀:

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

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

T_H(t₀)=T₀√(2/æ(t₀))

Meaning that:

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

æ(t₀)=(T_P/T₀)²/32π²

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

æ(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 æ(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=(æ(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 æ=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.