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, TH= ħc3/8πGMkb. All they had done, effectively, was replace the term M with √(MmP), where mP 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)=(mP/2).(t/tP) and the age of the universe (t)
would be such that ꬱ(t)=t/tP:
TH(t)=ħc3/8πGM(t)kb=(mP2.c2/kb)/(8π.(mP/2).(t/tP))
TH(t)=(mP.c2/kb)/(π.ꬱ(t))=TP/(4π.ꬱ(t))
For t=t0:
TH(t0)=TP/(4π.ꬱ(t0))=(TP/(4π.√(2ꬱ(t0))).√(2/ꬱ(t0))
And noting (as shown
in Observable Events Curve - Shifting About
Redshift) that T0=TP/(4π.√(2ꬱ)):
TH(t0)=T0√(2/ꬱ(t0))
Meaning that:
T02=(ꬱ(t0)/2).TH(t0)2
Or:
T02=(m(t0)/mP).TH(t0)2
Just to confirm
this, the Hawking temperature for a Schwarzschild black hole with a radius of
13.8 billion light years is TH(t0)=1.4×10-30K, ꬱ(t0)=t0/tP=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).(TP/(4π.ꬱ(t0)))2=(1/2).(TP/4π)2/ꬱ(t0)
ꬱ(t0)=(TP/T0)2/32π2
So, noting from Return to Constants that Resolve to Unity
that TP=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).TH(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 TH
be at some point in the past, say during recombination at ꬱ=380,000 years?
TH(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=mP, then √(mP.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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