Content uploaded by Espen Gaarder Haug
Author content
All content in this area was uploaded by Espen Gaarder Haug on Mar 18, 2024
Content may be subject to copyright.
Planck length from Cosmological Redshifts Solves the
Hubble Tension
Espen Gaarder Haug⇤and Eugene Terry Tatum⇤⇤
⇤Norwegian University of Life Sciences,
Christian Magnus Falsensvei 18, Aas, Norway
˚
As, Norway; espenhaug@mac.com
https://orcid.org/0000-0001-5712-6091
⇤⇤Independent Researcher, Bowling Green, Kentucky, USA
https://orcid.org/0000-0002-1544-7505
March 18, 2024
Abstract
Haug and Tatum have developed a cosmological model that links the CMB temperature,
Hubble constant, redshift, and the Planck length in a manner fully consistent with general
relativity. This means that one can easily extract the Planck length from observed cosmo-
logical redshifts. We demonstrate this by extracting the Planck length from the Union2
supernova database using the observed redshifts from all 580 type Ia supernovae. Our new
approach places constraints on the Hubble constant extracted from supernovae in a manner
that appears to resolve the Hubble tension. Additionally, our model provides a nearly perfect
match to observed redshifts without necessitating the expansion of space beyond the speed of
light cor the introduction of an accelerating dark energy. This approach also strongly favors
black hole Rh=ct cosmological models over the ⇤-CDM model. However, at this point, it
cannot be excluded that the ⇤-CDM model could also be modified to achieve similar results,
considering the model’s history of adaptation based on observational findings.
Keywords: Hubble tension, Planck length, Hubble constant, CMB, Cosmological red-
shift, General relativity, Upsilon equation.
1 Introduction and background
Haug and Tatum [1] have recently presented a cosmological model that is consistent with both
general relativity theory and a newly-quantized version of general relativity theory, which we
will discuss shortly. Haug and Tatum’s cosmological model represents the culmination of
many years of work by several researchers, wherein each piece of the puzzle has gradually
fallen into place, leading to the sudden convergence of various elements into a single model
that appears to be a very simple and powerful cosmological framework. Here, we will outline
some of these components. In 2015, Tatum et al [2] heuristically arrived at the following
predictive formula for the CMB temperature:
Tcmb =~c3
kb8⇡GpMhmp
=~c
kb4⇡pRh2lp
=~c
kb4⇡qc
H02lp
(1)
1
2
where kbis the Boltzmann constant, Mcis the critical mass (mass equivalent) in the Hubble
sphere, Rhis the Hubble radius, lpis the Planck length, and mpis the Planck mass. Firstly,
the formula is very similar to the Hawking radiation formula except that the Min the
denominator is replaced by pMhmp. The formula is extremely accurate at predicting today’s
CMB temperature if one inputs Mh=Mc,whereMcis the mass equivalent of the critical
Friedmann [3] universe Mc=c2RH
2G, in accordance with Flat Space Cosmology.
Relatively few in the astrophysical community have taken notice of this formula until
recently. There could be many reasons for this; two of the reasons are likely that it was pub-
lished in a low-distribution journal and also that the authors presented no detailed derivation
of their formula. However, recently, Haug and Wojnow [4,5] have derived the same formula
from the Stefan-Boltzmann law. Furthermore, as Haug and Tatum [6] have derived the same
formula from general geometrical principles of the Hubble sphere, this suggests that the for-
mula could be valid within multiple black hole cosmological models. In addition, Haug [7] has
also shown how one can derive the formula if one assumes light bending (space-time bending)
is quantized and linked to the Planck scale.
Equation (1) essentially directly binds together the CMB temperature and the Hubble
constant, something that has not been done in other models. It also links the Planck length
with the CMB temperature and the Hubble constant, as elaborated further in great detail
by Haug [4]. We will soon revisit how this is also consistent with a Planck-scale quantized
version of general relativity theory, which quantizes general relativity theory without altering
any output predictions. Insight into general relativity from a deeper perspective indeed seems
to link gravity with the Planck scale, something that was predicted by Eddington [8] as early
as 1918 and has only recently come to pass.
Haug and Tatum [1] recently have also provided mathematical proof that equation (1)is
consistent with a cosmological redshift of:
z=Rh
Rt1 (2)
which Haug and Tatum demonstrate can only be consistent with Tt=T0(1 + z)1
2. Alterna-
tively, equation (1) is also consistent with a cosmological redshift of the form:
z=rRh
Rt1 (3)
which they demonstrate is consistent with the well-known CMB temperature and z relation
Tt=T0(1+z). Observations seem to favor the latter CMB and z relation, as seen in [9], so we
will stick to the latter alternative [i.e., equation (3)], even if the first cannot be fully excluded.
Both redshift models are consistent with the well-known cosmological redshift formula:
z⇡H0d
c(4)
However, equation (4) is only valid when z⌧1, so it is only a good approximation for
short cosmological distances. For long cosmological distances, our model does not give the
same redshift-vs-distance prediction as ⇤-CDM. In particular, at such long distances, for a
given observed redshift, we will claim that the ⇤-CDM model likely predicts a significantly
shorter-than-actual distance, with the actual distance being that of our redshift model, as we
will mathematically demonstrate in the remainder of this section.
From the Haug and Tatum model we also must have:
th=D(1 + z)2
2cz +cz2(5)
3
and
Rh=D(1 + z)2
2z+z2(6)
and
H0=2cz +cz2
D(1 + z)2(7)
applying the first term of the Taylor series expansion, which is valid for z⌧1, gives
H0⇡2cz
D=cz
d(8)
where dnow is a fictional incorrect distance that is half of the real distance to the object of
interest d=1
2D. Furthermore, for any zwe have
D=2cz +cz2
H0(1 + z)2(9)
At long cosmological distances, this result is di↵erent than the distance given by the ⇤-CDM
model, as the first term of its Taylor series expansion alone is given by
D⇡2cz
H0
(10)
which is exactly twice the distance predicted by the ⇤-CDM model. Nevertheless, when
redshift z⌧1, an observer using either our model or the ⇤-CDM model has d⇡cz
H0. So,
either redshift model, ours or ⇤-CDM, makes the exact same redshift-vs-distance observations
at short distances as defined herein. Tension between the two redshift models only becomes
apparent at longer distances, as demonstrated in [1].
It is important to note here that we will be referring to a particular subclass of Rh=ct
cosmology models. Rh=ct cosmology is an actively-explored cosmology to this day. See, for
example, [10–13]. There are multiple sub-classes of Rh=ct cosmology models. Herein, we
will be working under the assumption that the Hubble sphere acts as a growing black hole.
The idea that the Hubble sphere could be treated as a black hole was pointed out as early
as 1972 by Pathria [14]. Even if black hole cosmology models are much less well-known than
the ⇤-CDM model, they are seriously discussed to this day. See, for example, [15–23].
2 Planck-quantized general relativity theory
Max Planck [24,25] assumed in 1899 that there were three universal constants: the speed
of light c, the gravitational constant G, and the Planck constant ~. Combining these with
dimensional analysis, he arrived at a unique length: lp=qG~
c3,time: tp=qG~
c3, mass:
mp=q~c
G, and temperature: Tp=1
kbq~c5
G. These are known today as the Planck units or
natural units. However, in Planck’s day, it was far from clear whether these Planck units had
a real physical significance, as they emerged solely from dimensional analysis. For example,
Bridgman [26], who received the 1946 Nobel Prize in physics, considered them to be pure
mathematical artifacts unrelated to physical reality.
Quantum gravity has been an unsolved challenge for more than 100 years. Already in
1916, Einstein [27] pointed out that the next step in gravity was quantum gravity, or in his
own words:
4
“While, according to the intra-atomic electron movement, atoms might emit not
only electromagnetic but also gravitational energy, albeit in a minuscule amount.
Since this should not be true in nature, it seems that the quantum theory must
modify not only Maxwell’s electrodynamics but also the new theory of gravitation.”
– Albert Einstein, 1916
Einstein devoted much of the remainder of his life to this endeavor, but with little suc-
cess. However, Eddington [8] had already provided an important hint in 1918, suggesting
that quantum gravity likely had to be dependent is some way on the Planck length. Today,
most researchers, particularly those working on developing quantum gravity theory, acknowl-
edge the significance of Planck units [28–30]. The two best-known attempts to develop a
reasonable quantum gravity theory have been string theory and loop quantum gravity theory
(LQG). However, despite the initial enthusiasm, these theories have not achieved any con-
sensus among experts in the field. Despite the considerable talents of researchers involved in
the development of string theory and LQG, development of a useful quantum gravity theory
has eluded them.
Since there has been no lack of talent in this pursuit, and yet quantum gravity theory
remains unsolved even after 100 years, it could be that, at some point in the history of physics,
a wrong turn was taken. If such was the case, we must first backtrack to the point where the
wrong turn was possibly taken and proceed from there. This is what Haug [31] has recently
tried to do, leading to a new and simple way to Planck-quantize general relativity theory.
While this is not the main focus of our paper, a brief historical context will provide the
reader with an interesting background for what we will soon demonstrate: a close connection
between the quantum scale and the cosmic scale, and how the quantum scale, in the form
of the Planck length value, constrains the H0value when considering cosmological redshift.
The result, we propose, will be our newly-quantized version of general relativity theory.
This brief history begins with Newton [32] in 1686. Newton’s original gravity force formula
was simply F=Mnmn
r2. There was no gravitational constant in Newton’s formula, which he
only expressed in words in his Principia. Despite this, Newton provided a series of accurate
gravitational e↵ect predictions. See Cohen [33]. For hundreds of years, Newton’s gravity
force formula was used to find planetary orbital velocities, the masses of planets, as well as
their gravitational accelerations. However, Newton’s mass definition was quite di↵erent from
today’s. Maxwell [34] used Newton’s original framework as late as 1873, describing gravita-
tional acceleration simply as g=Mn
r2(in contrast to today’s formula of g=GM
r2), meaning
that the Newtonian mass of the Earth was Mn=gr2. Since gravitational acceleration has
dimensions of L·T2, this implies that Newtonian mass dimensions were L3·T2. So, ob-
viously, this Newtonian unit was very di↵erent from today’s kilogram mass unit. Maxwell
actually took note of this history. It was known as “astronomical mass” and, for many years,
had been understood in astrophysics in relation to Newton’s theory. However, for earthly
macroscopic objects and even microscopic ones, the kilogram had become the standard in
France and the pound in Great Britain, as Maxwell also mentioned.
There had been discussions for some years on whether it would be favorable to use the same
mass standard for astronomical objects as for everyday macroscopic and microscopic ones,
preferably across countries. The kilogram was ultimately implemented as the standard in all
areas of physics. This meant that the kilogram mass had to be introduced into Newton’s
formula. However, the original Newton formula no longer worked if one simply replaced
Mnand mnwith their kilogram counterparts Mand m. Something was now missing from
the formula, which could be fixed by introducing a constant that soon would be known as
Newton’s gravitational constant, despite Newton never attempting to invent it or even search
for it. Furthermore, Cavendish [35] in 1798 also did not attempt to measure the gravitational
constant, nor did he mention a gravitational constant, as incorrectly claimed in multiple
5
papers and books, including by Feynman. See [36,37]. What is known today as Newton’s
gravitational constant was actually first introduced in 1873 by Cornu and Baille [38] to make
the Newtonian formula still work after replacing the Newtonian mass with the kilogram mass
definition. Th¨uring [39] pointed out that this gravitational constant was introduced without
a deep understanding of its physical significance. We realize that there is nothing inherently
wrong with the gravitational constant; it is indeed a constant and it is clearly needed when
working with the kilogram definition o↵mass. The key question, however, is: “What exactly
does the gravitational constant represent, from a deeper perspective?”
Hossenfelder [40], in her otherwise excellent book, claims: “Newton’s constant (G) quan-
tifies the strength of gravity.” However, this does not seem to be the case, since one could
clearly predict the same Newtonian gravity phenomena with Newton’s original formula, which
had no explicit gravitational constant. As early as 1984, Cahill [41,42] solved the Planck
mass formula for Gand obtained G=~c
m2
p. He suggested that the Planck mass might be more
fundamental than Gand that the gravitational constant could be expressed in this way as
a composite constant. However, in 1987, Cohen correctly pointed out that since no one had
demonstrated how to derive the Planck units without first knowing G, Cahill’s approach led
to a circular problem in reasoning. Such views persisted at least until 2016, as mentioned in
a paper by McCulloch [43], wherein the circular reasoning problem is also addressed.
Nevertheless, in 2017, Haug [44] demonstrated that one could find the Planck length from
even small macroscopic objects using a Cavendish apparatus without knowledge of G, and
later he showed that one can find the Planck length without relying on Gor ~at all [45,46].
Furthermore, in 2021, Haug demonstrated how one could find the Planck length from
cosmological redshift without knowing G, but by assuming z⇡H0c
d, although this formula is
only an approximation valid for low z. Herein, we extend this history by developing a method
that can be used to find the Planck length even from higher zcosmological measurements.
The main focus of our paper, however, is to demonstrate that the Planck length imposes
constraints on the acceptable values of the Hubble constant, even when extracting it from
observed cosmological redshifts.
It is also highly significant that we can find the Planck length using two entirely di↵erent
methods. The first method follows Max Planck’s approach based on dimensional analysis,
yielding:
lp=rG~
c3(11)
Since the Planck constant and the speed of light today are defined as exact (by NIST CODATA
2018) with no uncertainty, the only uncertainty in the Planck length, using this definition,
must come from uncertainty in the gravitational constant. The gravitational constant is one
of the least precise physical constants, and enormous work has been done to measure it more
precisely, see for example [47–50]. Today, the NIST CODATA 2018 standard gives it a value
of G=6.67430 ⇥1011 ±0.00015 ⇥1011m3·kg1·s2. This means we must have
lp=rG~
c3=r6.67430 ⇥1011 ±0.00015 ⇥1011 ⇥1.054571817 ⇥1034
2997924583
=1.616255 ⇥1035 ±0.000018 ⇥1035m
Here, one might well ask how the above historical context may guide us in quantizing
general relativity theory. We propose that if one can find the Planck length without knowledge
of G, it is indeed possible to express Gin terms of Planck units. If so, then we can use G=l2
pc3
~
and now substitute this identity into Einstein’s field equation, yielding (see [51,52]):
6
Rµ⌫1
2Rgµ⌫=8⇡G
c4Tµ⌫
Rµ⌫1
2Rgµ⌫=8⇡l2
p
c~Tµ⌫(12)
The Planck length now becomes a part of Einstein’s field equation. However, the benefit of
doing this is not entirely clear until we solve the equation for certain boundary conditions and
examine the metric solution, such as the Schwarzschild solution. Before doing this, however,
we will first take advantage of one more very simple but, in our view, very important way to
consider a kilogram mass. In 1923, Compton [53] described the Compton wavelength of an
electron as ¯
=~
mc and measured it through Compton scattering. If we solve the Compton
wavelength formula for the mass, we obtain:
m=~
¯
1
c(13)
In the spirit of complementarity, we will assert that any kilogram mass can be represented
in this way, not only the mass of an electron. The idea that protons could also have a
Compton wavelength has been discussed by multiple authors [54,55]. In actuality, it is
likely that only fundamental particles have a ’physical’ Compton wavelength. The Compton
wavelength of a composite mass can be seen as the aggregate of the Compton wavelengths of
all of the constituent elementary particles, including even photons, as the rest mass energy
of the photon can be expressed through the Compton wavelength, see [56]. To make a long
story short, this means, for example, that the Schwarzschild metric can be expressed as (by
replacing Gwith G=l2
pc3
~and the kilogram mass Mwith M=~
¯
1
c):
ds2=✓12GM
c2r◆c2dt2+✓12GM
rc2◆1
dr2r2d⌦2
ds2=✓12lp
r
lp
¯
M◆c2dt2+✓12lp
r
lp
¯
M◆1
dr2r2d⌦2(14)
The term lp
¯
Mrepresents the reduced Compton frequency in the gravitational mass of
interest. This is natural, since we have the reduced Compton frequency per second as f=c
¯
and the reduced Compton frequency per Planck time is then f=c
¯
tp=lp
¯
M, which, in our
view, achieves the quantization of matter and gravity. Interestingly, multiple recent research
studies do indeed indicate that matter ticks at the Compton frequency, see [57,58].
Similarly, this approach can be applied to other metric solutions, such as the Kerr [59] or
Kerr-Newman [60,61] solution, which are often used to describe black holes. The extent to
which this quantized general relativity can be unified with quantum mechanics is beyond the
scope of this paper, but will be addressed in the near future.
For our current purpose, we now have a straightforward formulation of general relativity
that includes the Planck length, yet does not alter any predictions from general relativity.
The potential benefit of equations (12) and (14) is that they likely allow for a new and deeper
insight into the phenomenon of quantum gravity. Modern cosmology theory has its origin in
general relativity theory; and Rh=ct cosmology clearly has a general relativistic framework.
As one will see from our particular sub-class of Rh=ct cosmology models, it is now possible
to extract the Planck length directly from cosmological redshift, by using the entire Union2
supernova redshift database. In view of our new approach to general relativity theory, this
result is fully consistent. What is most important in this paper is that the Planck length and
its mathematical relationship to the Hubble constant appears to impose a constraint on the
7
Hubble constant that one can extract from the supernova database. We maintain that this
discovery appears to solve the Hubble tension. Nevertheless, extraordinary claims require
extraordinary proofs; so we will attempt to demonstrate this carefully in the next sections.
3 How finding the Planck length from the Union2
supernova redshift database leads to solving the Hub-
ble tension by putting constraints on H0
Here, we will demonstrate that one can find the Planck length from the Union2 supernova
database without first relying on a knowledge of the value of G. Let us first revisit the CMB
prediction redshift formula of Haug and Tatum [1]. In its most general form, it is given by:
Tcmb,t =~c
kb4⇡pRt2lp
=T0(1 + z) (15)
Solving for Rtwe get
Rt=✓~c
T0(1 + zobs)kb4⇡◆21
2lp
(16)
Next, we input this for Rtin our cosmological redshift formula:
zpre =rRh
Rt1=v
u
u
u
t
c
H0
⇣~c
T0(1+zobs)kb4⇡⌘21
2lp
1 (17)
To use this redshift prediction formula to predict or, more precisely, attempt to match, the
redshift from the Union2 supernova database, we need to know the value of the Planck
constant. It is defined exactly today based on the NIST CODATA 2018 value of ~=
1.054571817 ⇥1034 J·s. The speed of light is also exactly defined as c= 299792458 m/s,
and the Boltzmann constant, which is also exactly defined as kb=1.380649 ⇥1023 J·K1.
Therefore, in c,~, and kb, there is no uncertainty. In addition, we need the CMB temperature,
the Hubble constant, and the Planck length to predict the cosmological redhsift. For T0,we
require the CMB temperature at present. The CMB temperature in the current cosmic epoch
is measured very accurately in a series of recent studies, as seen in [62–65]. We will use the
CMB temperature published by Dhal et al. [65] of 2.725007 ±0.000024K.
For the Hubble constant, we will initially use the value from one of the most recent
H0studies involving supernovae by one of the leading research teams. It is important to
note here that there is little or no disagreement regarding the CMB temperature at present.
For example, the 2009 Fixen [64] study of the current CMB temperature reports the value
2.72548 ±0.00057K. However, for the Hubble constant, the standard uncertainties are much
larger. Additionally, measurements of the Hubble constant from supernovae have yielded
considerably di↵erent values compared to those obtained from the CMB. This phenomenon
is known as the Hubble tension.
In our model, remarkably, we also need the Planck length. It’s important to note that,
in the standard view of Newtonian physics and general relativity theory, there hasn’t been
a successful attempt to connect gravity theory with the Planck length. As pointed out in
the last section, Haug has recently claimed to have developed a quantized version of general
relativity theory where, from a deeper perspective, we see the Planck length and the Compton
wavelength as playing an important role. Theory is one thing, but herein we will use real
observations in comparison to theoretical predictions, in much of the remainder of this paper.
8
Here, we will assume that we do not know the exact value of the Planck length except for
very rough estimates and qualified guesses. We will start by guessing the Planck length as
lp=5⇥1035 meters. Then, we input this value together with the CMB temperature from the
Dhal study and the Hubble constant from Murakami et al. [66] of 73.01±0.85 km/s/Mpc into
our redshift prediction formula for each type Ia supernova, and plot our findings relative to the
real observations. This is illustrated in Figure 1. The predicted redshifts based on our Planck
length guess of lp=5⇥1035 are represented by the red line, which is far above the observed
redshifts represented by the blue line. This indicates that our guess for the Planck length is
too high. Therefore, we make another qualified, but still wild, guess of the Planck length being
lp=0.5⇥1035. Based on this Planck length guess, we obtain predicted redshifts represented
by the green line. As we can see, the green line gives way too low predicted redshifts compared
to the observed ones. Hence, we deduce that the Planck length must lie between 0.5⇥1035 m
and 5⇥1035 m. We can continue with trial-and-error like this to minimize the error between
the observed and predicted redshifts: min P580
i=1 zobs(i)zpred (i). The number 580 is used in
our calculation because we take into account every single supernova in the Union2 database;
there are 580 observed type Ia supernova redshifts in the database. A simple “manual” trial-
and-error method will work, or we can use a more efficient and ”intelligent” trial-and-error
method like the bisection method or the Newton-Raphson method. Both algorithms are much
faster. One can even use the nearly instantaneous Goal Seek function in Excel, which likely
employs a bisection method.
Figure 1: This figure shows the predicted supernova redshifts with an assumed H0=
73.01 km/s/Mpc and wild guesses of the Planck length of 5 ⇥1035 mand 0.5⇥1035 m, as well as
the observed redshifts.
Our trial-and-error method that minimizes the di↵erence between the predicted and ob-
served redshifts yields an estimated Planck length of lp⇡1.7646 ⇥1035 m, as shown in
Figure 2. It is important to note that this estimated Planck length carries additional uncer-
tainty due to the uncertainties in the Cosmic Microwave Background (CMB) temperature and
the Hubble constant (H0) that we used. The uncertainty in CMB observations is significantly
9
smaller compared to that for the Hubble constant. Specifically, considering the Hubble ten-
sion, the uncertainty in H0becomes considerably large. Not surprisingly, a Planck length of
1.7646⇥1035msignificantly deviates from the Planck length estimated through dimensional
analysis. Returning to Max Planck’s formula, the Planck length is defined as lp=qG~
c3,
where the uncertainty in the Planck length estimate then primarily arises from the uncer-
tainty in the best estimates of G. According to NIST CODATA 2018, the reported value of lp
is 1.616255 ⇥1035 m, with a standard uncertainty of 0.000018 ⇥1035m. The Planck length
estimate derived from the supernova database, using the Hubble constant from Murakami et
al. of 73.01 ±0.85 km/s/Mpc, is 8241outside the NIST CODATA estimate (based solely
on dimensional analysis). We can slightly adjust the CMB temperature in our input based
on its standard deviation, but we still remain far from the recognized Planck length based on
dimensional analysis.
Figure 2: This figure shows that if we assume the Hubble constant H0= 73.01 ±0.81 km/s/Mpc
then we find the Planck length must be 1.7646 ±0.0020 ⇥1035 m.
In Figure 3, we propose that the Planck length must fall within its standard deviation
(STD) as defined by the NIST CODATA 2018. This assumption leads to an estimated Hub-
ble constant from the Union2 redshift database of 66.8711+0.0019
0.0019,km/s/Mpc. This estimation
potentially resolves the Hubble tension, because it utilizes the entire supernova database, with
observed redshifts ranging from z=0.015 to z=1.414, to find the matching Hubble con-
stant. In other words, to maintain the Planck length within its uncertainty range, especially
when considering the small standard deviation in the Cosmic Microwave Background (CMB)
temperature, this is the matching value necessary for the Hubble constant.
10
Figure 3: This figure shows the predicted redshift and Planck length when using a Hubble constant
of H0= 66.8711 km/s/Mpc. We take into account the uncertainty in the current CMB temperature
from the Dhal et.al study and find that there is a match with the NIST CODATA 2018 value of the
Planck length as a constraint on the Planck length. We find that, to be inside the acceptable Planck
length uncertainty, we must match the Union2 supernova redshift database with a Hubble constant
value of H0= 66.8711 ±0.0019 km/s/Mpc. To put it another way, if we want the Hubble constant
value to be outside this value in relation to the observed supernova redshifts, then we must accept a
Planck length outside of the one STD uncertainty given by NIST CODATA for the Planck length.
As we have seen from the previous figure, a Hubble constant value of around 72 to 73 km/s/Mpc is
totally unacceptable in our model, as it leads to unacceptable Planck length tension. We conclude
that neither Hubble tension nor Planck length tension is necessary, if one uses our model.
4 Supernova team H0determinations point to in-
correct CMB temperature predictions from Union2
supernova redshifts
We can also find the CMB temperature from cosmological redshift based on equation (17)
that we repeat here for convinence:
zpre =rRh
Rt1=v
u
u
u
t
c
H0
⇣~c
T0(1+zobs)kb4⇡⌘21
2lp
1 (18)
This time we will assume that the Planck length is known, and we will used the NIST
CODATA 2018 value for it. In addition, we will use the H0value estimated by Riess et
al. [67] at H0= 73.04 ±1.04 km/s/Mpc. Based on these inputs, we can now minimize the
di↵erence between the predicted and observed redshifts by adjusting the T0until we have
minimized the error, according to min P580
Izobs,i zpre,i. This is illustrated in Figure 4.
11
Figure 4: This figure shows that, based on the Riess et.al Hubble constant determination of H0=
73.04 ±1.04 km/s/Mpc, one can match the observed Union2 supernova redshifts in our model only
if one accepts a current CMB best-fit temperature of T0=2.8479+0.0203
0.0203K. This is far outside the
measured current CMB temperature of T0=2.725007 ±0.000024Kby Dhal et. al and indicates that
the Riess et. al H0value is way too high, and not consistent with our new model.
5 Additional arguments in support of a Hubble ten-
sion solution
In addition to the above arguments in support of a Hubble tension solution, one can also
employ a di↵erent approach which reaches the same basic conclusion and is complementary
to the one above. This approach makes use of our newly-derived “Upsilon equation” which
couples H0with the current CMB temperature T0(see [68–70]) by the simple and exact
formula:
H0=
⌦
T2
0(19)
The Latin Capital Upsilon symbol
⌦
is a compound coupling constant with the following
value:
⌦
=2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2. This is the value for this
composite constant based on the NIST CODATA values of its constituent constants. This
composite constant was derived in relation to equation (19) and given first by Tatum et. al
[68]:
⌦
=k2
b32⇡2G1/2
c5/2~3/2=2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2(20)
This is a composite constant composed entirely of already very well-known physical constants.
There is no uncertainty in the Boltzmann constant kb, the reduced Planck constant ~, or the
speed of light c, as these are all exactly defined in today’s most updated S.I. unit system,
the NIST CODATA 2018 standard: kb=1.380649 ⇥1023J·K1;c= 299792458m/s; ~=
1.054571817 ⇥1034J·s. Only the gravitational constant Ghas a residual small uncertainty;
its NIST CODATA value is given as G=6.67430 ⇥1011 ±0.00015 ⇥1011m3·kg1·s2.
12
Rearrangement of equation (19) gives :
T0=✓H0
⌦
◆1/2
(21)
Which can be used to calculate a current T0value for any given H0or vice versa. We can
then compare the coupled H0and T0values from four recent CMB temperature studies (see
Table 1) with the coupled H0and T0values of the most recent high precision SH0ES Team
study reported in 2022 by Riess et al [67].
Table 1: This table presents Hubble constant estimates derived from Equation (19) across several
notable CMB studies. The uncertainties in the predicted H0account for both the uncertainty in the
measured T0and the uncertainty in the Upsilon constant.
CMB Study : Temperature Measurement : H0=
⌦
T2
0:
2004: Fixsen et. al [62]: 2.721 ±0.010K H0= 66.68 ±0.49 km/s/Mpc
2009: Fixsen et. al [64]: 2.72548 ±0.00057K H0= 66.8944 ±0.0287 km/s/Mpc
2011: Noterdaeme et. al [63]: 2.725 ±0.002K H0= 66.8708 ±0.0989 km/s/Mpc
2023: Dhal et. al [65]: 2.725007 ±0.000024K H0= 66.8712 ±0.0019 km/s/Mpc
In Table 1, the values of H0in units of km/s/Mpc (after conversion from their S.I. unit
values) are coupled to a tight T0range of 2.721 ±0.010Kto 2.72548 ±0.00057K. As a result
of these high precision T0measurements, the calculated H0values using equation (19)showa
tight range of 66.68 ±0.49 km/s/Mpc to 66.8712 ±0.00190 km/s/Mpc. This is much higher
H0precision than given by any other method and is fully consistent with our findings from
the last section, wherein we incorporated all 580 type Ia supernova redshifts in the Union2
database, and found a best-fitting H0= 66.8711 ±0.0019 km/s/Mpc.
We can then compare what T0value would, according to equation (21), be coupled to the
SH0ES study H0value of 73.04±1.04km/s/Mpc. Given this H0range of 72.0 to 74.08km/s/Mpc
(once converted to S.I. units), equation (17) indicates that the coupled T0value should be
2.8479 ±0.0203K. This surprisingly high T0value, greater than 0.1K higher than the mea-
sured T0value, is clearly an outlier when analyzing it using our “Upsilon equation.”
Using our CMB redshift prediction formula and method of reference [1], we can also show
how the Union2 database of 580 type Ia supernova redshifts, in combination with the Riess
et al H0value of 73.04 ±1.04km/s/Mpc, is a best match for a T0value of 2.8479 ±0.0203K.
See Figure 4. This is simply yet another way to show the same outlier appearance of local
H0determination, in comparison to H0determinations made from CMB studies. So, we
conclude that the Planck length and Upsilon equation approaches demonstrated herein add
further support to the impression that the Hubble tension is now solved in favor of the
Planck Collaboration result, particularly since, in every case, we have used the local universe
supernova redshift data to do so.
Furthermore, if we solve the Planck length formula lp=qG~
c3for G, we get G=l2
pc3
~(see
[51]), so the Upsilon constant used above can also be expressed as:
⌦
=k2
b32⇡2lp
~2c=2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2(22)
13
or from the Planck time:
⌦
=k2
b32⇡2tp
~2=2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2(23)
Since the Planck energy is given by Ep=~
lpc, we can also re-write the Upsilon constant
as:
⌦
=k2
b32⇡2
~Ep
=2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2(24)
Again, the only uncertainty in the Upsilon constant comes from Gor alterntively lp, as we
have Ep=q~c5
G=~
lpc. The uncertainty now comes from the Planck length NIST CODATA
lp=1.616255⇥1035 ±0.000018 ⇥1035 m. The relative standard uncertainty in the Planck
length: 0.000018⇥1035 m
lp=1.1⇥105is exactly half that of the relative standard uncertainty
in G:0.00015⇥1011 m3·kg1·s2
G⇡2.2⇥105. However, the uncertainty in Upsilon is the same
no matter if calculated from Gor lp. Because the formula relying on Guses p(G) and the
Planck length formula uses G=l2
pc3
~, the uncertainty a↵ecting Upsilon must be the same.
The relative standard uncertainty in Upsilon based on inputs from NIST CODATA 2018 must
be
0.00003279 ⇥1019 s1·K2
⌦
⇡1.1⇥105(25)
In other words, exactly the same as for the Planck length, as we would expect. So, from
equation (19) we can most readily see that there are key inter-relationships between the
CMB temperature, H0and the Planck length, fully consistent with the previous sections of
this paper. This also means that we naturally have
H0=
⌦
T2
0
H0=k2
b32⇡2lp
~2cT2
0
lp=H0
T2
0
~2c
k2
b32⇡2(26)
and since we have Tt=T0(1 + z) we must also have :
lp=H0(1 + z)2
T2
t
~2c
k2
b32⇡2(27)
Equation (26) was recently presented and discussed by Haug [52], who derived it by simply
rearranging the formula of Tatum et al. [2]. One can readily see that this equation yields the
Planck length from T0and H0. One can do this derivation either by using best current high
precision measurements of the CMB temperature and the Hubble constant, or even more
precisely by incorporating all 580 type Ia supernova redshifts in the Union2 database, as
demonstrated herein. This confirms the consistency of our framework and also supports the
contention that the Planck length is constant through time, as expected by many physicists
and quantum gravity theorists. Furthermore, since ~2c
k2
b32⇡2has an exact value due to ~, and
cand kbbeing exact constants, it implies also that, if lpis constant over cosmic time, there
are compelling reasons to believe that, by Equation (26), lpimposes a constraint on the ratio
14
relation between H0and T0, as clearly seen in our Upsilon equation and other work in this
paper. Given that T0is extremely accurately measured (with very low standard deviation), it
is the uncertainty in H0that we have greatly reduced by understanding this deeper relation.
We also can easily incorporate cosmological redshift into our Upsilon equation. We can
start with:
Tt=T0(1 + z) (28)
and then naturally we must have:
T0=Tt
(1 + z)(29)
We can now replace T0with this in the Upsilon equation and we get
H0=
⌦
T2
t
(1 + z)2(30)
and
Tt=✓H0
⌦
◆1/2
(1 + z) (31)
Furthermore, we can also have
D=cz +cz2
⌦
T2
0(1 + z)2(32)
Table 2 summarizes additional key inter-relationships between the CMB temperature, H0,
and the Planck units. It is important to note here that the only uncertainty in the Planck
units arises from difficulties in measuring lpwith precision. The uncertainty is the same in
every Planck unit since we have tp=lp
c,mp=~
lp
1
c,Ep=mpc2=~c
lp, and ap=c2
lp.Since
cand ~are exact constants, the uncertainty in all of these arises only from lp. The consid-
erable uncertainty in lpand tpshould not be surprising, as it is likely that they respectively
represent the smallest length and time interval possible. They cannot be measured directly,
but indirectly we can measure them, for example, by finding the best fit with high precision
Hubble constant and CMB temperature values, and even with cosmological redshifts (see
also [45]). Despite the high uncertainty in lp, its uncertainty is still very small compared to
that found in H0by traditional studies. We believe that our Upsilon equation (19)isthekey
to minimizing uncertainty in H0, thus representing an important development in quantum
cosmology.
Table 3 gives additional relationships between cosmic parameters, expressed in terms of
the Upsilon constant.
Table 4 summarizes a series of relations between cosmic parameters in past cosmic epochs,
in terms of the Upsilon constant.
15
Table 2: This table illustrates how to determine the Hubble constant from the current CMB tem-
perature and various Planck units as well as the CMB temperature from H0and various Planck units.
Hubble constant H0: CMB temperature T0:
From Upsilon formula : H0=
⌦
T2
0T0=⇣H0
⌦
⌘1/2
From Upsilon formula : H0=
⌦
T2
t
(1+z)2Tt=⇣H0
⌦
⌘1/2(1 + z)
where
⌦
=k2
b32⇡2G1/2
c5/2~3/2is a composite constant
or
⌦
=k2
b32⇡2lp
~2cthe same as above, but re-written.
or
⌦
=k2
b32⇡2tp
~2the same as above, but re-written.
or
⌦
=k2
b32⇡2
~Epthe same as above, but re-written.
Planck unit:
From Planck length : H0=lpT2
0
k2
b32⇡2
~2cT0=qH0c
lp
~
kb⇡p32
From Planck time : H0=tpT2
0
k2
b32⇡2
~2T0=qH0
tp
~
kb⇡p32
From Planck mass : H0=T2
0
mp
k2
b32⇡2
~c2T0=cpH0mp~
kb⇡p32
From Planck energy : H0=T2
0
Ep
k2
b32⇡2
~T0=pH0Ep~
kb⇡p32
From Planck acceleration : H0=T2
0
ap
ck2
b32⇡2
~2T0=qH0ap
c
~
kb⇡p32
From Planck force : H0=T2
0lp
Fp
k2
b32⇡2
~T0=pH0Fp~
kb⇡p32lp
Table 3: This table summarizes a series of relations between cosmic parameters, expressed in terms
of the Upsilon constant.
Entity : Equation :
Upsilon constant
⌦
=k2
b32⇡2G1/2
c5/2~3/2
or
⌦
=k2
b32⇡2tp
~2
value (NIST CODATA 2018) 2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2
Hubble constant : H0=
⌦
T2
0
CMB temperature T0=qH0
⌦
Hubble time : tH=1
⌦
T2
0
CMB temperature T0=q1
⌦
tH
Hubble radius : RH=c
⌦
T2
0
CMB temperature T0=qc
⌦
RH
Cosmological redshift z=Ttq
⌦
H01= Tt
T01=qRh
Rt1
Cosmological redshift z⇡D
⌦
T2
0
2c=d
⌦
T2
0
c=dH0
c
Redshift distance D=2cz+cz2
⌦
T2
0(1+z)2
Redshift distance D=2cz+cz2
⌦
T2
t
Redshift distance D⇡2cz
⌦
T2
0
=2cz
H0,whenz⌧1
Critical mass (Friedmann) Mc=c3
2GT 2
0
⌦
Critical density (Friedmann) ⇢c=3
⌦
2T4
0
8⇡G
CMB temperature T0=qc3
2GMc
⌦
16
Table 4: This table summarizes a series of relations between cosmic parameters in past cosmic
epochs, in terms of the Upsilon constant.
Entity : Equation :
Upsilon constant
⌦
=k2
b32⇡2G1/2
c5/2~3/2
or
⌦
=k2
b32⇡2tp
~2
value (NIST CODATA 2018) 2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2
Hubble constant in the past: Ht=
⌦
T2
0(1 + z)2
Hubble constant in the past: Ht=
⌦
T2
t
Hubble (CMB) temperature in the past : Tt=qH0
⌦
(1 + z)
Hubble time in the past : tt=1
⌦
T2
0(1+z)2
Hubble radius in the past : Rt=c
⌦
T2
0(1+z)2
Critical mass (Friedmann) Mt=c3
2GT 2
0(1+z)2
⌦
Critical density (Friedmann) ⇢c,t =3
⌦
2T4
0(1+z)4
8⇡G=pc(1 + z)4
CMB temperature in the past Tt=qHt
⌦
CMB temperature in the past Tt=(1+z)qc3
2GMc
⌦
CMB temperature in the past Tt=(1+z)qH0
⌦
CMB temperature in the past Tt=(⇢c,t8⇡G)1
4
31
4p
⌦
17
6 Possible underlying reasons why we appear to
have solved the Hubble tension and dramatically in-
creased precision in estimates
We have claimed to have solved the Hubble tension and, in addition, to have dramatically
reduced the uncertainty (standard deviation, STD) in H0estimates. This almost seems too
good to be true. However, we have carefully reviewed our logic and derivations, and find no
obvious errors. The reason our method appears to be much more powerful than the existing
⇤-CDM model is that we have established exact relations between H0,T0,z, and the Planck
length, as we illustrate in Figure 5. These appear to be exact relations between the smallest
and largest scales of the universe. Thus, our model appears to have a solid framework. If we
know any three of these parameters, we can find the remaining one with high precision. It
makes sense to take what is measured most accurately, namely, the CMB temperature, the
Planck length, and the redshifts, and then use these high-precision measurements to determine
the Hubble constant with high precision. In the ⇤-CDM model, such exact relations have not
yet been established between these parameters; and for this reason, the ⇤-CDM model does
not appear to be as good at describing certain aspects of the universe as the model we have
presented; nor can it predict the Hubble constant with the precision that we can achieve.
That said, the ⇤-CDM model has evolved from work over time; it has been adjusted over
time. We do not exclude the possibility that our findings can be incorporated into that model
as well. Both models should be carefully investigated and compared by multiple researchers
over time.
Figure 5: This figure illustrates that we have established exact relations between the Hubble constant
H0, the current CMB temperature T0, and CMB temperatures from the past Tt, the cosmological
redshift z, and even the Planck length lp. It is these newly-established exact inter-relationships that
appear to have allowed us to solve the Hubble tension. In addition, they allow for dramatically-
improved Hubble constant prediction. Here, there appear to be foundational relationships between
the microcosmos and the macrocosmos.
18
Figure 6 shows key equations for the current universal parameters which incorporate the
Upsilon constant.
Figure 6: This figure shows how the Upsilon constant can be used to calculate current universal
parameters plus the cosmological redshift equation at the bottom gives the link between past and
present.
Figure 7 illustrates the di↵erent global parameters of the universe in past cosmic epochs,
and shows how they are inter-related in terms of the Upsilon constant; this scenario is based
on expansion of the universe in so-called Rh=ct growing black hole cosmology.
Figure 7: This figure illustrates the di↵erent global parameters of the universe in past cosmic epochs
in terms of the Upsilon constant and cosmological redshift.
Figures 6 and 7 bring us back to Isaac Newton. In his Principia, he actually mentioned
an absolute minimum time interval, and further stated that all of his philosophy was based
on minimum units (see [71]). Herein, we have demonstrated that the Planck scale plays an
important role in cosmology. The Planck length and the Planck time impose constraints on
the values acceptable for such parameters as the Hubble constant. They even appear to solve
the Hubble tension in favor of the Planck Collaboration Hubble constant value. Of course,
our claims should not be taken for granted. Our work should be studied and scrutinized over
time by multiple researchers. After all, time is the best referee.
19
7 Conclusion
We have demonstrated that it is possible to extract the Planck length from the 580 type Ia
supernova redshifts in the Union2 database by using the current temperature of the Cosmic
Microwave Background (CMB) and the Hubble constant. However, this Planck length ex-
traction approach imposes significant constraints on the Hubble constant value, which must
be 66.8711+0.0019
0.0019,km/s/Mpc to match the observed redshifts in the Union2 database, so long
as we accept the NIST CODATA value for the Planck length. Alternatively, we would have
to introduce the idea that the Planck length has changed since the begining of the universe,
something that seems much less likely. For the local universe Hubble constant determina-
tions by Riess and others to be compatible with the Union2 supernova redshift database, the
best-fitting Planck length would have to deviate by more than 8241from the NIST CO-
DATA value. This deviation appears to be unacceptable, as the Planck length almost surely
must remain constant. Although the uncertainty in the Planck length may be higher than in
most constants, it nevertheless imposes significant constraints on the Hubble constant value
that best-fits the redshift database. To our knowledge, ours is likely the first cosmological
model to establish a clear connection between the CMB temperature, the Hubble constant
(H0), cosmological redshift (z), and the Planck length. We believe that the ⇤-CDM model
does not provide for a method for estimating the Planck length by imposing constraints on
the Hubble constant. Our method of extracting a Hubble constant tightly constrained by
the uncertainty in the Planck length, suggests a solution to the Hubble tension in favor of
the Planck Collaboration CMB Hubble constant determination. We invite others to study
our model and to evaluate its potential usefulness in the context of Planck-scale quantum
cosmology.
References
[1] E. G. Haug and E. T. Tatum. Solving the Hubble tension by extracting current CMB
temperature from the Union2 supernova database. Hal archive, 2024. URL https:
//hal.science/hal-04368837.
[2] E. T. Tatum, U. V. S. Seshavatharam, and S. Lakshminarayana. The basics of flat space
cosmology. International Journal of Astronomy and Astrophysics, 5:116, 2015. URL
http://dx.doi.org/10.4236/ijaa.2015.52015.
[3] A. Friedmann. ¨
Uber die kr¨ung des raumes. Zeitschrift f¨ur Physik, 10:377, 1922. URL
https://doi.org/10.1007/BF01332580.
[4] E. G. Haug. CMB, Hawking, Planck, and Hubble scale relations consistent with recent
quantization of general relativity theory. International Journal of Theoretical Physics,
63(57), 2024. URL https://doi.org/10.1007/s10773-024-05570-6.
[5] E. G. Haug and S. Wojnow. How to predict the temperature of the CMB directly using
the Hubble parameter and the Planck scale using the Stefan-Boltzman law. Research
Square, Pre-print, under consideration by journal, 2023. URL https://doi.org/10.
21203/rs.3.rs-3576675/v1.
[6] E. G. Haug and E. T. Tatum. The Hawking Hubble temperature as a minimum
temperature, the Planck temperature as a maximum temperature and the CMB tem-
perature as their geometric mean temperature. Hal archive, 2023. URL https:
//hal.science/hal-04308132.
20
[7] E. G. Haug. Planck quantized bending of light leads to the CMB temperature. Hal
archive, 2023. URL https://hal.science/hal-04357053.
[8] A. S. Eddington. Report On The Relativity Theory Of Gravitation. The Physical Society
Of London, Fleetway Press, London, 1918.
[9] D.A. Riechers, A. Weiss, and F. et al. Walter. Microwave background temperature at a
redshift of 6.34 from h2oabsorption. Nature, 602:58, 2022. URL https://doi.org/10.
1038/s41586-021-04294-5.
[10] M. V. John. rh=ct and the eternal coasting cosmological model. Monthly Notices of
the Royal Astronomical Society, 484, 2019. URL https://doi.org/10.1093/mnrasl/
sly243.
[11] F. Melia and Shevchuk A. S. H. The rh=ct universe. Monthly Notices of the Royal
Astronomical Society, 419:2579, 2012. URL hhttps://doi.org/10.1111/j.1365-2966.
2011.19906.x.
[12] F. Melia. Thermodynamics of the rh=ct universe: A simplification of cosmic entropye.
European Journal of Physics C, 81:234, 2021. URL https://doi.org/10.1140/epjc/
s10052-021-09028-5.
[13] F. Melia. A resolution of the monopole problem in the rh=ct universe. The Dark
Universe, 42:101329, 2023. URL https://doi.org/10.1016/j.dark.2023.101329.
[14] R. K. Pathria. The universe as a black hole. Nature, 240:298, 1972. URL https:
//doi.org/10.1038/240298a0.
[15] P. Christillin. The Machian origin of linear inertial forces from our gravitationally ra-
diating black hole universe. The European Physical Journal Plus, 129:175, 2014. URL
https://doi.org/10.1140/epjp/i2014-14175-2.
[16] T. X. Zhang and C. Frederick. Acceleration of black hole universe. Astrophysics and
Space Science, 349:567, 2014. URL https://doi.org/10.1007/s10509-013-1644-6.
[17] N. Pop lawski. The universe in a black hole in Einstein–Cartan gravity. The Astrophysical
Journal, 832:96, 2016. URL https://doi.org/10.3847/0004-637X/832/2/96.
[18] T. X. Zhang. The principles and laws of black hole universe. Journal of Modern Physics,
9:1838, 2018. URL https://doi.org/10.4236/jmp.2018.99117.
[19] D. A. Easson and R. H. Brandenberger. Universe generation from black hole interi-
ors. Journal of High Energy Physics, 2001, 2001. URL https://doi.org/10.1088/
1126-6708/2001/06/024.
[20] E. Gaztanaga. The black hole universe, part i. Symmetry, 2022:1849, 2022. URL
https://doi.org/10.3390/sym14091849.
[21] Z. Roupas. Detectable universes inside regular black holes. The European Physical
Journal C, 82:255, 2022. URL https://doi.org/10.1140/epjc/s10052-022-10202-6.
[22] E. Siegel. Are we living in a baby universe that looks like a black hole to outsiders? Hard
Science, Big Think, January 27, 2022. URL https://bigthink.com/hard-science/
baby-universes-black-holes-dark-matter/.
21
[23] C. H. Lineweaver and V. M. Patel. All objects and some questions. American Journal
of Physics, 91(819), 2023.
[24] M. Planck. Natuerliche Masseinheiten. Der K¨oniglich Preussischen Akademie Der Wis-
senschaften: Berlin, Germany, 1899. URL https://www.biodiversitylibrary.org/
item/93034#page/7/mode/1up.
[25] M. Planck. Vorlesungen ¨uber die Theorie der W¨armestrahlung. Leipzig: J.A. Barth, p.
163, see also the English translation “The Theory of Radiation” (1959) Dover, 1906.
[26] P. W. Bridgman. Dimensional Analysis. New Haven: Yale University Press, 1922.
[27] A. Einstein. N¨aherungsweise integration der feldgleichungen der gravitation. Sitzungs-
berichte der K¨oniglich Preussischen Akademie der Wissenschaften Berlin, 1916.
[28] S. L. Adler. Six easy roads to the Planck scale. American Journal of Physics, 78:925,
2010. URL https://doi.org/10.1119/1.3439650.
[29] S. Hossenfelder. Can we measure structures to a precision better than the Planck length?
Classical and Quantum Gravity, 29:115011, 2012. URL https://doi.org/10.1088/
0264-9381/29/11/115011.
[30] S. Hossenfelder. Minimal length scale scenarios for quantum gravity. Living Reviews in
Relativity, 16:2, 2013. URL https://doi.org/10.12942/lrr-2013-2.
[31] E. G. Haug. Di↵erent mass definitions and their pluses and minuses related to gravity:
The kilogram mass is likely incomplete. Preprint at Researchgate.net, under review by
journal, 2023. URL https://doi.org/10.13140/RG.2.2.11581.74720.
[32] I Newton. Philosophiae Naturalis Principia Mathematica. London, UK, Jussu Societatis
Regiae ac Typis Josephi Streater, 1686.
[33] I. B. Cohen. Newton’s determination of the masses and densities of the sun, jupiter,
saturn, and the earth. Archive for History of Exact Sciences, 53(1):83, 1998. URL
https://doi.org/10.1007/s004070050022.
[34] C. Maxwell. A Treatise on Electricity and Magnetism. Macmillan and Co., Oxford, UK,
1873.
[35] H. Cavendish. Experiments to determine the density of the earth. Philosophical Trans-
actions of the Royal Society of London, (part II), 88:469, 1798.
[36] B. E. Clotfelter. The Cavendish experiment as cavendish knew it. American Journal of
Physics, 55:210, 1987. URL https://doi.org/10.1119/1.15214.
[37] L. Sean. Henry Cavendish and the density of the earth. The Physics Teacher, 37:34,
1999. URL https://doi.org/10.1119/1.880145.
[38] A. Cornu and J. B. Baille. D´etermination nouvelle de la constante de l’attraction et de
la densit´e moyenne de la terre. C. R. Acad. Sci. Paris, 76, 1873.
[39] B. Th¨uring. The gravitational constant. Ann. Acad. Sci. Fennicae A, page 269, 1961.
[40] S. Hossenfelder. Existential Physics: A Scientist’s Guide to Life’s Biggest Questions.
Viking books, 2022.
22
[41] K. Cahill. The gravitational constant. Lettere al Nuovo Cimento, 39:181, 1984. URL
https://doi.org/10.1007/BF02790586.
[42] K. Cahill. Tetrads, broken symmetries, and the gravitational constant. Zeitschrift F¨ur
Physik C Particles and Fields, 23:353, 1984. URL https://doi:10.1007/bf01572659.
[43] M. E. McCulloch. Quantised inertia from relativity and the uncertainty principle.
Europhysics Letters (EPL), 115(6):69001, 2016. URL https://doi.org/10.1209/
0295-5075/115/69001.
[44] E. G. Haug. Can the Planck length be found independent of big G?Applied Physics
Research, 9(6):58, 2017. URL https://doi.org/10.5539/apr.v9n6p58.
[45] E. G. Haug. Finding the Planck length multiplied by the speed of light without any
knowledge of G,c, or h, using a newton force spring. Journal Physics Communication,
4:075001, 2020. URL https://doi.org/10.1088/2399-6528/ab9dd7.
[46] E. G. Haug. Planck units measured totally independently of big G.Open Journal of
Microphysics, 12:55, 2022. URL https://doi.org/10.4236/ojm.2022.122004.
[47] B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich. Atom interferometer
measurement of the Newtonian constant of gravity. Science, 315, 2007. URL DOI:
10.1126/science.1135459.
[48] Felix Scholkmann. Indications for a diurnal and annual variation in the anisotropy of
di↵usion pattern SNa reanalysis of data presented by J. Dai (2014, nat. sci.). Progress
in Physics, 10, 2014.
[49] J. J. Andrews and J. S. Bobowski. Automation of the Cavendish torsion-balance
experiment to measure G.European Journal of Physics, 40:035001, 2019. URL
https://doi.org/10.1088/1361-6404/ab01a8.
[50] C. Xue and et. al. Precision measurement of the Newtonian gravitational constant.
National Science Review, 7:1803, 2020. URL https://doi.org/10.1093/nsr/nwaa165.
[51] E. G. Haug. Progress in the composite view of the Newton gravitational constant and
its link to the Planck scale. Universe, 8(454), 2022. URL https://doi.org/10.3390/
universe8090454.
[52] E. G. Haug. CMB, Hawking, Planck, and Hubble scale relations consistent with recent
quantization of general relativity theory. International Journal of Theoretical Physics,
Nature-Springer, 63(57), 2024. URL https://doi.org/10.1007/s10773-024-05570-6.
[53] A. H. Compton. A quantum theory of the scattering of x-rays by light elements. Physical
Review, 21(5):483, 1923. URL https://doi.org/10.1103/PhysRev.21.483.
[54] L.S. Levitt. The proton Compton wavelength as the ‘quantum’ of length. Experientia,
14:233, 1958. URL https://doi.org/10.1007/BF02159173.
[55] O. L. Trinhammer and H. G. Bohr. On proton charge radius definition. EPL, 128:21001,
2019. URL https://doi.org/10.1209/0295-5075/128/21001.
[56] E. G. Haug. The Compton wavelength is the true matter wavelength, linked to the photon
wavelength, while the de Broglie wavelength is simply a mathematical derivative. Qeios,
2023. URL https://doi.org/10.32388/OZ0IRU.2.
23
[57] S. Lan, P. Kuan, B. Estey, D. English, J. M. Brown, M. A. Hohensee, and M¨uller. A
clock directly linking time to a particle’s mass. Science, 339:554, 2013. URL https:
//doi.org/10.1126/science.1230767.
[58] D. Dolce and A. Perali. On the Compton clock and the undulatory nature of particle
mass in graphene systems. The European Physical Journal Plus, 130(41):41, 2015. URL
https://doi.org/10.1140/epjp/i2015-15041-5.
[59] R. P. Kerr. Gravitational field of a spinning mass as an example of algebraically spe-
cial metrics. Physical Review Letters, 11:237, 1963. URL https://doi.org/10.1103/
PhysRevLett.11.237.
[60] E. T. Newman and A. I. Janis. Note on the Kerr spinning-particle metric. Journal of
Mathematical Physics, 6:915, 1965. URL https://doi.org/10.1063/1.1704350.
[61] E. Newman, E. Couch, K. Chinnapared, A.; Exton, A. Prakash, and R. Torrence. Metric
of a rotating, charged mass. Journal of Mathematical Physics, 6:918, 1965. URL https:
//doi.org/10.1063/1.1704351.
[62] D. J. Fixsen and et. al. The temperature of the cosmic microwave background at 10 ghz.
The Astrophysical Journal, 612:86, 2004. URL https://doi.org/10.1086/421993.
[63] P. Noterdaeme, P. Petitjean, R. Srianand, C . Ledoux, and S. L´opez. The evolution of
the cosmic microwave background temperature. Astronomy and Astrophysics, 526, 2011.
URL https://doi.org/10.1051/0004-6361/201016140.
[64] D. J. Fixsen. The temperature of the cosmic microwave background. The Astrophysical
Journal, 707:916, 2009. URL https://doi.org/10.1088/0004-637X/707/2/916.
[65] S. Dhal, S. Singh, K. Konar, and R. K. Paul. Calculation of cosmic microwave back-
ground radiation parameters using cobe/firas dataset. Experimental Astronomy (2023),
612:86, 2023. URL https://doi.org/10.1007/s10686-023-09904-w.
[66] Yukei S. Murakami and A. et. al Y. S., Riess. Leveraging SN Ia spectroscopic similarity
to improve the measurement of h0.arXiv:2306.00070.
[67] A. G. Riess and et. al. A comprehensive measurement of the local value of the Hubble
constant with 1 km s1Mpc1uncertainty from the Hubble space telescope and the
sh0es team. The Astrophysical Journal, 934, 2021. URL https://doi.org/10.3847/
2041-8213/ac5c5b.
[68] E. T. Tatum, E. G. Haug, and S. Wojnow. High precision Hubble constant determina-
tions based upon a new theoretical relationship between CMB temperature and H0.Hal
archive, 2023. URL https://hal.science/hal-04268732.
[69] E. T. Tatum and U. V. S. Seshavatharam. How the flat space cosmology model correlates
the recombination CMB temperature of 3000K with a redshift of 1100. Journal of Modern
Physics, Accepted and forthcoming February, 2024, 2024.
[70] E. T. Tatum. Upsilon constants and their usefulness in Planck scale quantum cosmology.
Journal of Modern Physics, 15:167, 2024. URL https://doi.org/10.4236/jmp.2024.
152007.
[71] E. G. Haug. God time = Planck time. Open Journal of Microphysics, accepted and
forthcoming, 2024. URL hhttps://hal.archives-ouvertes.fr/hal-03769825/.
24
Appendix
In the paper above, we have based the cosmological model on the assumption that Tt=
T0(1 + z)1when = 0, a choice seemingly favored by observational findings. However, it
is important to note that every observational study in cosmology typically relies on certain
assumptions. Haug and Tatum have demonstrated that Tt=T0(1 + z), that is (= 0), is
only consistent in the Rh=ct black hole cosmology framework with a redshift scaling of
z=qRh
Rt1, which is the framework utilized in this paper.
If one desires a redshift scaling of z=Rh
Rt1, then must equal 1
2, meaning Tt=T0(1+z)1
2.
The tables below replicate those in the main text but adjust the a↵ected formulas to be
consistent with =1
2instead of = 0.
While we believe it is unlikely that =1
2represents the true scenario, it cannot be
conclusively ruled out at this juncture. Therefore, we find it necessary to inform the research
community about this possibility as well, so that further investigation may potentially provide
stronger evidence regarding the favored model within the Rh=ct black hole cosmology
framework.
Table 5: This table illustrates how to determine the Hubble constant from the current CMB tem-
perature and various Planck units as well as the CMB temperature from H0and various Planck units.
Hubble constant H0: CMB temperature T0:
From Upsilon formula : H0=
⌦
T2
0T0=⇣H0
⌦
⌘1/2
From Upsilon formula : H0=
⌦
T2
t
(1+z)Tt=⇣H0(1+z)
⌦
⌘1/2
where
⌦
=k2
b32⇡2G1/2
c5/2~3/2is a composite constant
or
⌦
=k2
b32⇡2lp
~2cthe same as above, but re-written.
or
⌦
=k2
b32⇡2tp
~2the same as above, but re-written.
or
⌦
=k2
b32⇡2
~Epthe same as above, but re-written.
Table 5 gives relationships between cosmic parameters, expressed in terms of the Upsilon
constant.
Table 6 summarizes a series of relations between cosmic parameters in past cosmic epochs,
in terms of the Upsilon constant.
25
Table 6: This table summarizes a series of relations between cosmic parameters, expressed in terms
of the Upsilon constant.
Entity : Equation :
Upsilon constant
⌦
=k2
b32⇡2G1/2
c5/2~3/2
or
⌦
=k2
b32⇡2tp
~2
value (NIST CODATA 2018) 2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2
Hubble constant : H0=
⌦
T2
0
CMB temperature T0=qH0
⌦
Hubble time : tH=1
⌦
T2
0
CMB temperature T0=q1
⌦
tH
Hubble radius : RH=c
⌦
T2
0
CMB temperature T0=qc
⌦
RH
Cosmological redshift z=T2
t
⌦
H01=T2
t
T2
01=Rh
Rt1
Cosmological redshift z⇡d
⌦
T2
0
c=dH0
c
Redshift distance D=cz
⌦
T2
0(1+z)
Redshift distance D=cz
⌦
T2
t
Redshift distance D⇡cz
⌦
T2
0
=cz
H0,whenz⌧1
Critical mass (Friedmann) Mc=c3
2GT 2
0
⌦
Critical density (Friedmann) ⇢c=3
⌦
2T4
0
8⇡G
CMB temperature T0=qc3
2GMc
⌦
Table 7: This table summarizes a series of relations between cosmic parameters in past cosmic
epochs, in terms of the Upsilon constant.
Entity : Equation :
Upsilon constant
⌦
=k2
b32⇡2G1/2
c5/2~3/2
or
⌦
=k2
b32⇡2tp
~2
value (NIST CODATA 2018) 2.91845601 ⇥1019 ±0.00003279 ⇥1019 s1·K2
Hubble constant in the past: Ht=
⌦
T2
0(1 + z)
Hubble constant in the past: Ht=
⌦
T2
t
Hubble (CMB) temperature in the past : Tt=qH0(1+z)
⌦
Hubble time in the past : tt=1
⌦
T2
0(1+z)
Hubble radius in the past : Rt=c
⌦
T2
0(1+z)
Critical mass (Friedmann) Mt=c3
2GT 2
0(1+z)
⌦
Critical density (Friedmann) ⇢c,t =3
⌦
2T4
0(1+z)2
8⇡G=pc(1 + z)2
CMB temperature in the past Tt=qHt
⌦
CMB temperature in the past Tt=qc3(1+z)
2GMc
⌦
CMB temperature in the past Tt=qH0(1+z)
⌦
Critical density (Friedmann) Tt=(⇢c,t8⇡G)1
4
31
4p
⌦
26
Declarations
Conflict of interest
The authors declare no conflict of interest.
Data availability statements
The supernova Union-2 database that we have used can be found here:
https://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt.
