The CMB Temperature Is Simply the Geometric Mean: T c m b = T m i n T m a x of the Minimum and Maximum Temperature in the Hubble Sphere

In the Hubble sphere, we assume that the wavelength of pure energy spreads out in all directions. The maximum wavelength in the Hubble sphere is then the circumference of the Hubble sphere. We assume the minimum wavelength occurs in a Planck mass black hole, which is given by $4\pi R_{s,p}=8\pi l_p$ . Here, we build further on the geometric mean CMB approach by Haug and Tatum and based on new analysis given in this paper ¹ we conclude that the CMB temperature is simply given as: $T_{cmb}=\sqrt{T_{min}{T}_{max}}$ , which is the geometric mean of the minimum and maximum physically possible temperatures in the Hubble sphere. This again means the CMB temperature simply is the geometric mean of the Hawking temperature of the Hubble sphere (in black hole cosmology) and the Hawking temperature of the Planck mass black hole, se we have also $T_{cmb}=\sqrt{T_{Haw,H}{T}_{Haw,p}}$ .

• CMB Temperature
• Geometric Mean Temperature
• Minimum Temperature
• Maximum Temperature
• Hubble Sphere
• Cosmology

1. Introduction

The Λ-CDM model, despite its success in many areas, is not able to predict the CMB temperature today. See, for example, Narlikar and Padmanabhan [2] , which states:

“The textit The present theory is, however, unable to predict the value of $T$ at $t=t_0$. It is therefore a free parameter in SC (Standard Cosmology).”

The CMB temperature is likely the most precisely measured cosmological parameter [3] - [5] , but it is clearly not fully understood within Λ-CDM cosmology. In recent years, however, there has been a breakthrough in understanding the CMB temperature and its connection to the Hubble parameter, which we will soon revisit.

We will be operating within a black hole $R_{H_t}=ct$ cosmology. Although black hole cosmology is much less well-known than Λ-CDM, it is not new; it dates back at least to 1972 with a paper by Pathria (1972) [6] . The topic continues to be actively discussed by various researchers to this day [7] - [16] .

There are also multiple variations of $R_{H_t}=ct$ cosmologies, all of which share the common feature that the universe has expanded—or is at least related to—he speed of light; see [17] - [22] . The Melia $R_H=ct$ model is the best known among these, and he has done a tremendous job demonstrating that $R_{H_t}=ct$ cosmology can perform at least as well as, and often better than, the Λ-CDM model. However, in this work, we will focus specifically on black hole $R_{H_t}=ct$ cosmology, as described by Haug and Tatum [23] , which is a subcategory within $R_{H_t}=ct$ cosmologies.

2. The CMB Temperature as a Geometric Mean of the Minimum and Maximum Temperature in the Hubble Sphere

The geometric mean plays an important role in thermodynamics and in other areas of physics [24] - [27] . For example, the optimal reheating pressure is given as the geometric mean of the maximum and minimum pressure: $P_{reheating}=\sqrt{P_{max}{P}_{min}}$, and the optimal intercooling in an ideal two-stage compressor is also given by the geometric mean pressure, $P_{intercooling}=\sqrt{P_{max}{P}_{min}}$, see [28] . The geometric mean temperature: $T_{gm}=\sqrt{T_{hot}{T}_{cold}}$ play a central role in Carnot engines where it defines a type of equilibrium, see [29] - [31] . That geometric means could also potentially play an important role in the thermodynamics of cosmic black hole universe temperatures should not come as a surprise.

Haug and Tatum [32] have recently shown that the CMB temperature, at a deeper physical level, is likely linked to the geometric mean of the shortest and longest possible wavelengths in the Hubble sphere. They presented their formula as:

$T_{cmb}=\hslash \frac{c}{\sqrt{{\bar{\lambda }}_{min}{\bar{\lambda }}_{max}}}\frac{1}{4\pi k_b}=\hslash \frac{c}{{\bar{\lambda }}_{gm}}\frac{1}{4\pi k_b}$ (1)

where $k_b$ is the Boltzmann constant and $\hslash =\frac{h}{2\pi }$ is the reduced Planck constant (the Dirac constant). They assumed the shortest wavelength ${\bar{\lambda }}_{min}=l_p=\sqrt{\frac{G\hslash }{c^3}}$ was the Planck [33] [34] length and the maximum wavelength was the diameter of the Hubble sphere ${\bar{\lambda }}_{max}=2R_H$, they mention also the circumference could be the limiting factor. Their geometric mean wavelength is ${\bar{\lambda }}_{gm}=\sqrt{{\bar{\lambda }}_{min}{\bar{\lambda }}_{max}}$. This, again, they demonstrate to be consistent with the CMB formula heuristically first suggested by Tatum, Seshavatharam and Lakshminarayana [35] . Haug and Wojnow [36] [37] have further demonstrated the CMB formula fully consistent with this can be derived from the Stefan-Boltzmann law. The Stefan-Boltzmann [38] [39] law holds for a perfect black body and the CMB is the closest we likely get to a perfect black body in the real world as stated by for example Muller et al. [40] :

“Observations with the COBE satellite have demonstrated that the CMB corresponds to a nearly perfect black body characterized by a temperature $T_0$ at $z=0$, which is measured with very high accuracy, $T_0=2.72548\pm 0.00057\text{K}$.”

Haug [41] has recently expanded on the geometric mean approach of Haug and Tatum and shown that the CMB formula can even be written directly in the form:

$T_{cmb}=\frac{\sqrt{T_{max}{T}_{min}}}{4\pi }=\frac{\sqrt{T_p{T}_{min}}}{4\pi }$ (2)

where he suggested $T_{max}=T_p=\frac{1}{k_b}=\sqrt{\frac{\hslash c^5}{G}}=\frac{m_p{c}^2}{k_b}=\frac{E_p}{k_b}$, which is the Planck [33] [34] temperature, and $T_{min}=\hslash \frac{c}{2R_H}\frac{1}{k_b}$. Furthermore $k_b$ is the Boltzmann constant.

The question is: why does the geometric mean temperature have to be multiplied by the constant $\frac{1}{4\pi }$? We now think we have an answer to that. Electromagnetic waves (such as CMB radiation), as well as matter waves, tend to spread out in all directions—like throwing a stone into a lake, where the waves propagate in all directions. If a Planck-mass Schwarzschild black hole is the smallest black hole and the Hubble sphere is a cosmic black hole, then the maximum wavelength is the circumference of the Hubble sphere, ${\bar{\lambda }}_{max}=4\pi R_H$, and the minimum wavelength is the circumference of a Schwarzschild Planck-mass black hole, which is ${\bar{\lambda }}_{min}=4\pi R_{s,p}=8\pi l_p$.

The minimum and maximum energies and temperatures in the Hubble sphere are then based on the Planck—Einstein relation:

$E_{min}=\hslash \frac{c}{4\pi R_{H_t}},T_{min}=\frac{E_{min}}{k_b}$ (3)

and the maximum energy and maximum temperature is then

$E_{max}=\hslash \frac{c}{4\pi R_{s,p}}=\hslash \frac{c}{8\pi l_p},T_{max}=\frac{E_{max}}{k_b}$ (4)

The CMB temperature now is then given by:

$T_{cmb}=\sqrt{T_{max}{T}_{min}}\approx 2.725\text{K}$ (5)

While the maximum temperature is always constant, $T_{max}=\hslash \frac{c}{8\pi l_p}\frac{1}{k_b}$, the minimum temperature varies as we travel along the cosmic epoch as we assume $R_{H_t}=ct$. It is worth mention that the minimum temperature now always is equal to the Hawking [42] temperature $T_{min}=\frac{\hslash c}{4\pi R_H}=T_{Haw}=\frac{\hslash c}{4\pi R_s}$ when the Hubble radius is equal to the Schwarzschild radius $R_{H_t}=R_{s,t}=ct$ as it will be in a black hole Hubble universe where the equivalent mass is the critical Friedmann mass: $M_{BH}=M_{cr}=\frac{c^2{R}_{H_t}}{2G}$.

This means the CMB temperature also can be seen as simply the geometric mean of the Hawking temperature of the Hubble sphere and a Planck mass black hole. The Hawking temperature of the Hubble sphere is given as:

$T_{min}=T_{Haw,H}=\frac{\hslash c^3}{k_{b}8\pi G{M}_{BH}}=\hslash \frac{c}{4\pi R_s}\frac{1}{k_b}=\hslash \frac{c}{4\pi R_H}\frac{1}{k_b}$ (6)

We here assume that the relevant mass of the Hubble sphere in relation to the CMB temperature is the critical Friedmann [43] mass, $M_{BH}=M_{cr}=\frac{c^2{R}_H}{2G}$. If we solve for the Hubble radius in terms of the critical Friedmann mass, we get $R_H=\frac{2G{M}_{cr}}{c^2}$, and we can see that it must be identical to the Schwarzschild radius of a black hole with mass equal to the critical Friedmann mass: $R_s=\frac{2G{M}_{BH}}{c^2}$. This is not a new result, but it is important for understanding why we can apply the Hawking temperature to a black hole Hubble sphere universe.

In addition, we have the Hawking—Planck temperature, which is the Hawking temperature of a Schwarzschild black hole, given by:

$T_{max}=T_{Haw,p}=\hslash \frac{c}{4\pi R_{s,p}}\frac{1}{k_b}=\hslash \frac{c}{4\pi 2l_p}\frac{1}{k_b}=\hslash \frac{c}{8\pi l_p}\frac{1}{k_b}$ (7)

This is the maximum possible temperature in the black hole, related to a Planck mass black hole that again likely could be a Planck mass particle linked to quantum gravity.

The CMB temperature is then given by:

$T_{cmb}=\sqrt{T_{max}{T}_{min}}=\sqrt{T_{Haw,H}{T}_{Haw,p}}\approx 2.725\text{K}$ (8)

Based on a $H_0\approx 66.9$ km/s/Mpc as reported by Haug and Tatum [22] . We further assume the Hubble sphere and the CMB temperature follows the $R_{H_t}=ct$ cosmology, where the circumference of the black hole Hubble sphere was smaller in the past.

Alternatively we can express the CMB temperature from energies:

$T_{cmb}=\sqrt{E_{max}{E}_{min}}\frac{1}{k_b}$ (9)

where $E_{max}=k_b{T}_{max}$ and $E_{min}=k_b{T}_{min}$.

The maximum energy is naturally much smaller than the energy in the Hubble sphere. This can be seen as the maximum possible energy from a single particle or photon (or perhaps even a graviton), which we conjecture is linked to the Planck scale and is actually a Schwarzschild Planck mass black hole (Planck mass particle). It is common for researchers working on quantum gravity to assume that the Planck scale will play an important role; see, for example [44] - [47] . Therefore, it should not be a big surprise that the Planck scale also plays an important role in the CMB temperature.

It is naturally remarkable that, based on recent years of research on the CMB temperature, we can now accurately predict the CMB temperature today—something the Λ-CDM model has not been able to do and still cannot, as it is not compatible with $R_{H_t}=ct$ black hole cosmology. Even more important than predicting the CMB temperature is the fact that the approach developed in recent years has found the exact mathematical relationship between the CMB temperature and the Hubble parameter. Tatum, Haug and Wojnow [48] , as well as Haug and Tatum [22] , have recently demonstrated that one can predict the Hubble parameter much more precisely than with other methods. This is possible because the CMB temperature can be used to determine the Hubble constant due to these new exact mathematical relationships.

Based on the geometric mean approach above we get the following formula:

$H_0=\frac{T_{cmb,0}^2}{T_{max}}\frac{k_{b}4\pi }{\hslash }=\frac{T_{cmb,0}^2}{T_{Haw,p}}\frac{k_{b}4\pi }{\hslash }=66.8943\pm 0.0287\text{km}/\text{s}/\text{Mpc}$ (10)

we have used the Fixsen [49] measured CMB temperature now (at $z=0$): $T_0=2.72548\pm 0.00057\text{K}$. This is as expected in line with the research just mentioned above.

The most plausible reason the CMB temperature is simply the geometric mean of the lowest and highest possible Hawking temperatures in the Hubble sphere is likely that the Hubble sphere operates as a Carnot [50] engine. The idea that the universe could be a Carnot engine is not new and has been suggested for the Friedmann-Robertson-Walker (FRW) metric-based universe by Debnath [51] [52] . Despite Debnath’s highly interesting paper, he did not derive a CMB prediction formula for the current temperature and is it even possible inside the FRW model? To our knowledge, this is the very first paper to predict that $T_{cmb}=\sqrt{T_{max}{T}_{min}}=\sqrt{T_{Haw,H}{T}_{Haw,p}}$, which is likely the most elegant way to express the CMB temperature both mathematically and intuitively. Anyone deeply knowledgeable in Carnot theory (heat theory) will recognize it simply as the geometric mean temperature required for an ideal Carnot engine.

We plan to follow up on our findings with an in-depth analysis of why the Hubble sphere is likely a black hole Carnot engine, probably governed by the extremal solution of Reissner-Nordström [53] [54] , Kerr [55] , or the Haug-Spavieri [56] metric rather than the FRW metric. However, we consider the findings in this paper so important that making them publicly available at this stage could benefit the research community, despite the deeper mathematical foundation will first be laid out in future papers. For considerably deeper mathematical evidence linked to Carnot engine theory, see our follow-up working paper [57] .

3. The Link to Cosmological Redshift

It is well known from observations that the CMB temperature in relation to cosmological redshift is given by $T_{cmb,t}=T_{cmb,0}\left(1+z\right)$, where $z$ is the cosmological redshift, see [58] - [60] . This means to be consistent with this we must have:

$T_{cmb,t}=\sqrt{T_{max}{T}_{min,t}}=\sqrt{T_{max}{T}_{min,0}}\left(1+z\right)$ (11)

and since $T_{max}=T_{Haw,max}=\frac{\hslash c}{k_{b}8\pi l_p}$ and $T_{min,t}=T_{Haw,min,t}=\frac{\hslash c}{k_{b}4\pi R_{H_t}}$ so we can also write this as:

$T_{cmb.t}=\sqrt{T_{Haw,max}{T}_{Haw,min,t}}=\sqrt{T_{Haw,max}{T}_{Haw,min,0}}\left(1+z\right)$ (12)

and this means we also must have:

$z=\frac{T_{cmb,t}}{\sqrt{T_{max}{T}_{min}}}-1=\frac{T_{cmb,t}}{\sqrt{T_{Haw,max}{T}_{Haw,min,0}}}-1=\sqrt{\frac{R_{H_0}}{R_{H_t}}}-1$ (13)

which is fully consistent with the cosmological redshift findings of Haug and Tatum [61] . Interestingly, this also leads to the conclusion that the Hawking temperature of the whole Hubble sphere in relation to cosmological redshift is found by solving the following equation for $T_{Haw,min,0}$, which is the Hawking temperature for the whole Hubble sphere:

$T_{Haw,H_0}=T_{Haw,min,0}=\frac{T_{cmb.t}^2}{T_{Haw,max,0}{\left(1+z\right)}^2}$ (14)

where, again, $T_{Haw,max,0}=\frac{\hslash c}{k_{b}8\pi l_p}$. Since we know from observations that $T_{cmb,t}=T_{cmb,0}\left(1+z\right)$, we must have:

$T_{Haw,H_0}=\frac{T_{cmb,0}^2{\left(1+z\right)}^2}{T_{Haw,max,0}{\left(1+z\right)}^2}=\frac{T_{cmb,0}^2}{T_{Haw,max,0}}$ (15)

which is naturally consistent with our main findings that $T_{cmb}=\sqrt{T_{Haw,max}{T}_{Haw,min}}$. Equation (15) also demonstrates that the Hawking temperature now for the hole Hubble sphere is naturally independent of cosmological redshift.

4. Comparison to the Λ-CDM model

The Λ-CDM model cannot predict the CMB temperature now, despite the CMB temperature being the most precisely measured cosmological parameter. However, using the geometric mean approach, we can predict the CMB temperature now. Furthermore, this also link the CMB temperature to the Hubble constant in a veruy powerful way. If we know the CMB temperature we know the Hubble parameter now, or vice versa.

Haug [62] has recently also demonstrated that the CMB radiation density parameter consistent with this way to express the CMB temperature is given by ${\Omega }_{\gamma }=\frac{{\rho }_{\gamma }}{{\rho }_{cr}}=\frac{1}{5760\pi }\approx 5.52621330180192\times {10}^{-5}$, which is well within the 95% confidence interval of the Particle Data Group (PDG)²: 5.08 × 10⁻⁵ to 5.68 × 10⁻⁵. Our model predicts an exact CMB temperature density parameter.

Remarkably, the Hubble tension also seems to be resolved when applying this CMB temperature relation in addition to a cosmological redshift of $z=\sqrt{\frac{R_{H_0}}{R_{H_t}}}-1$, as demonstrated by Haug and Tatum [22] [63] . They use a more complicated formula for the CMB that at a deeper level is the same as the one presented in this paper. They demonstrate that one obtains a basically perfect match between predicted and observed supernova data using the full distance ladder of observed SN Ia (the PhantonPlusSH0ES database) with only one $H_0$ value ( $H_0\approx 66.9$ km/s/Mpc).

It is important to understand that all of this is based on $R_{H_t}=ct$ cosmology. Melia [21] (see also the Melia references in section 1) has demonstrated that $R_{H_t}=ct$ cosmology is outperforming the Λ-CDM model on a long list of testable points. That said, the CMB temperature presented here is consistent with only a subclass of $R_{H_t}=ct$ models, namely black hole $R_{H_t}=ct$ cosmological models, so clearly, much more research is needed here, for example, also in relation to comparison studies with the interesting Melia $R_{H_t}=ct$ model.

The most plausible reason the CMB temperature is simply the geometric mean of the lowest and highest possible Hawking temperatures in the Hubble sphere is that the Hubble sphere likely operates as a Carnot [50] engine. The idea that the universe could be a Carnot engine is intriguing. We plan to follow up on our findings with an in-depth analysis of why the Hubble sphere is likely a black hole Carnot engine, possibly governed by the extremal solution of the Reissner-Nordström, Kerr, or Haug-Spavieri metric rather than the FRW metric. However, we believe that the findings in this paper—that the CMB temperature is simply the geometric mean of the maximum and minimum temperatures in the Hubble sphere given by the very elegant formula first presented in this paper: $T_{cmb}=\sqrt{T_{max}{T}_{min}}\approx 2.725\text{K}$—are so significant that making this result publicly available now could greatly benefit the black hole and cosmological research community. For a considerably deeper mathematical analysis linked to Carnot engine theory and black hole cosmology, see our follow-up working paper [57] .

5. Conclusions

Based on years of research on the CMB temperature by several authors, we can now conclude that the CMB temperature, in its simplest and most understandable form, is simply the geometric mean of the minimum and maximum temperatures possible in the Hubble sphere. The CMB temperature is given by $T_{cmb}=\sqrt{T_{max}{T}_{min}}$. This also means that the CMB temperature is the geometric mean temperature of the Hawking Hubble temperature and the Hawking Planck temperature: $T_{cmb}=\sqrt{T_{max}{T}_{min}}=\sqrt{T_{Haw,H}{T}_{Haw,p}}\approx 2.725\text{K}$.

This has important implications, as it provides a precise mathematical relationship between the CMB temperature and the Hubble parameters, as well as a deeper physical understanding of the CMB temperature. Unlike in the Λ-CDM model, we can now accurately predict the CMB temperature in black hole $R_{H_t}=ct$ cosmology. In addition, we can predict the Hubble parameter much more precisely, as recentyl demonstrated by Tatum, Haug and Wojnow [48] and Haug and Tatum [22] .

NOTES

¹This paper is a strongly improved version of our preprint [1] .

²See https://pdg.lbl.gov/2023/reviews/rpp2023-rev-astrophysical-constants.pdf .

References

1. Haug, E.G. (2025) The CMB Temperature Is Simply the Geometric Mean: of the Minimum and Maximum Temperature in the Hubble Sphere. Cambridge University Press, Cambridge Engage, Physics and Astronomy: Preprint Archive. >https://doi.org/10.33774/coe-2025-s7wnb

2. Narlikar, J.V. and Padmanabhan, T. (2001) Standard Cosmology and Alternatives: A Critical Appraisal. Annual Review of Astronomy and Astrophysics, 39, 211-248. >https://doi.org/10.1146/annurev.astro.39.1.211

3. Fixsen, D.J., Kogut, A., Levin, S., Limon, M., Lubin, P., Mirel, P., et al. (2004) The Temperature of the Cosmic Microwave Background at 10 GHz. The Astrophysical Journal, 612, 86-95. >https://doi.org/10.1086/421993

4. Noterdaeme, P., Petitjean, P., Srianand, R., Ledoux, C. and López, S. (2011) The Evolution of the Cosmic Microwave Background Temperature. Astronomy&Astrophysics, 526, Article No. L7. >https://doi.org/10.1051/0004-6361/201016140

5. Dhal, S. and Paul, R.K. (2023) Investigation on CMB Monopole and Dipole Using Blackbody Radiation Inversion. Scientific Reports, 13, Article No. 3316. >https://doi.org/10.1038/s41598-023-30414-4

6. Pathria, R.K. (1972) The Universe as a Black Hole. Nature, 240, 298-299. >https://doi.org/10.1038/240298a0

7. Stuckey, W.M. (1994) The Observable Universe Inside a Black Hole. American Journal of Physics, 62, 788-795. >https://doi.org/10.1119/1.17460

8. Christillin, P. (2014) The Machian Origin of Linear Inertial Forces from Our Gravitationally Radiating Black Hole Universe. The European Physical Journal Plus, 129, Article No. 175. >https://doi.org/10.1140/epjp/i2014-14175-2

9. Zhang, T.X. and Frederick, C. (2014) Acceleration of Black Hole Universe. Astrophysics and Space Science, 349, 567-573. >https://doi.org/10.1007/s10509-013-1644-6

10. Popławski, N. (2016) Universe in a Black Hole in Einstein-Cartan Gravity. The Astrophysical Journal, 832, Article 96. >https://doi.org/10.3847/0004-637x/832/2/96

11. Zhang, T.X. (2018) The Principles and Laws of Black Hole Universe. Journal of Modern Physics, 9, 1838-1865. >https://doi.org/10.4236/jmp.2018.99117

12. Easson, D.A. and Brandenberger, R.H. (2001) Universe Generation from Black Hole Interiors. Journal of High Energy Physics, 2001, Article 24. >https://doi.org/10.1088/1126-6708/2001/06/024

13. Gaztanaga, E. (2022) The Black Hole Universe, Part I. Symmetry, 14, Article 1849. >https://doi.org/10.3390/sym14091849

14. Roupas, Z. (2022) Detectable Universes Inside Regular Black Holes. The European Physical Journal C, 82, Article No. 255. >https://doi.org/10.1140/epjc/s10052-022-10202-6

15. Siegel, E. (2022) Are We Living in a Baby Universe that Looks Like a Black Hole to Outsiders? Hard Science, Big Think. >https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/

16. Lineweaver, C.H. and Patel, V.M. (2023) All Objects and Some Questions. American Journal of Physics, 91, 819-825. >https://doi.org/10.1119/5.0150209

17. Melia, F. (2018) A Comparison of the and ΛCDM Cosmologies Using the Cosmic Distance Duality Relation. Monthly Notices of the Royal Astronomical Society, 481, 4855-4862. >https://doi.org/10.1093/mnras/sty2596

18. John, M.V. (2019) and the Eternal Coasting Cosmological Model. Monthly Notices of the Royal Astronomical Society: Letters, 484, L35-L37. >https://doi.org/10.1093/mnrasl/sly243

19. Melia, F. (2021) Thermodynamics of the Universe: A Simplification of Cosmic Entropy. The European Physical Journal C, 81, Article No. 234. >https://doi.org/10.1140/epjc/s10052-021-09028-5

20. Melia, F. (2023) Model Selection with Baryonic Acoustic Oscillations in the Lyman-Forest. Europhysics Letters, 143, Article 59004. >https://doi.org/10.1209/0295-5075/acf60c

21. Melia, F. (2024) Strong Observational Support for the Timeline in the Early Universe. Physics of the Dark Universe, 46, Article 101587. >https://doi.org/10.1016/j.dark.2024.101587

22. Haug, E.G. and Tatum, E.T. (2025) Solving the Hubble Tension Using the PantheonPlusSH0ES Supernova Database. Journal of Applied Mathematics and Physics, 13, 593-622. >https://doi.org/10.4236/jamp.2025.132033

23. Haug, E. G. and Tatum. E.T. (2024) Friedmann Type Equations in Thermodynamical Form Lead to Much Tighter Constraints on the Energy Density of the Universe. Pre-print. >https://doi.org/10.20944/preprints202403.1241.v1

24. Henderson, R.F. (1965) The Theorem of the Arithmetic and Geometric Means and Its Application to a Problem in Gas Dynamics. Zeitschrift für angewandte Mathematik und Physik ZAMP, 16, 788-803. >https://doi.org/10.1007/bf01614106

25. Zhang, L., Gong, L.Y. and Tong, P.Q. (2011) The Geometric Mean Density of States and Its Application to One-Dimensional Nonuniform Systems. The European Physical Journal B, 80, 485-492. >https://doi.org/10.1140/epjb/e2011-20062-9

26. Yamagami, S. (2008) Geometric Mean of States and Transition Amplitudes. Letters in Mathematical Physics, 84, 123-137. >https://doi.org/10.1007/s11005-008-0238-7

27. Kocic, M. (2021) Geometric Mean of Bimetric Spacetimes. Classical and Quantum Gravity, 38, Article 075023. >https://doi.org/10.1088/1361-6382/abdf28

28. Lümmen, N. (2020) Termodynamikk, Kort og Godt. Universitetsforlaget.

29. M. Bucher. (1996) Maximum-Power Efficiency of a Carnot Engine. International Journal of Engineering, 12, 47-50. >https://www.ijee.ie/articles/Vol12-1/12-1-08.PDF

30. Johal, R.S. (2017) Pythagorean Means and Carnot Machines. Resonance, 22, 1193-1203. >https://doi.org/10.1007/s12045-017-0581-z

31. Johal, R.S. (2017) Efficiencies of Power Plants, Quasi-Static Models and the Geometric-Mean Temperature. The European Physical Journal Special Topics, 226, 489-498. >https://doi.org/10.1140/epjst/e2016-60265-9

32. Haug, E.G. and Tatum, E.T. (2024) The Hawking Hubble Temperature as the Minimum Temperature, the Planck Temperature as the Maximum Temperature, and the CMB Temperature as Their Geometric Mean Temperature. Journal of Applied Mathematics and Physics, 12, 3328-3348. >https://doi.org/10.4236/jamp.2024.1210198

33. Planck, M. (1899) Natuerliche Masseinheiten. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. >https://www.biodiversitylibrary.org/item/93034#page/7/mode/1up

34. Planck, M. (1906) Vorlesungen über die Theorie der Wärmestrahlung. Legare Street Press.

35. Tatum, E.T., Seshavatharam, U.V.S. and Lakshminarayana, S. (2015) The Basics of Flat Space Cosmology. International Journal of Astronomy and Astrophysics, 5, 116-124. >https://doi.org/10.4236/ijaa.2015.52015

36. Haug, E.G. (2024) CMB, Hawking, Planck, and Hubble Scale Relations Consistent with Recent Quantization of General Relativity Theory. International Journal of Theoretical Physics, 63, Article No. 57. >https://doi.org/10.1007/s10773-024-05570-6

37. Haug, E.G. and Wojnow, S. (2024) How to Predict the Temperature of the CMB Directly Using the Hubble Parameter and the Planck Scale Using the Stefan-Boltzmann Law. Journal of Applied Mathematics and Physics, 12, 3552-3566. >https://doi.org/10.4236/jamp.2024.1210211

38. Stefan, J. (1879) Über die beziehung zwischen der Wärmestrahlung und der temperatur. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien. Mathematisch—Naturwissenschaftliche Klasse. II. Abteilung, 79, 391-428.

39. Boltzmann, L. (1884) Ableitung des Stefan’schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie. Annalen der Physik, 258, 291-294. >https://doi.org/10.1002/andp.18842580616

40. Muller, S., Beelen, A., Black, J.H., Curran, S.J., Horellou, C., Aalto, S., et al. (2013) A Precise and Accurate Determination of the Cosmic Microwave Background Temperature at . Astronomy&Astrophysics, 551, Article No. A109. >https://doi.org/10.1051/0004-6361/201220613

41. Haug, E.G. (2025) Geometric Mean Cosmology Predicts the CMB Temperature Now and in Past Epochs. Cambridge University Press, Preprint. >https://doi.org/10.33774/coe-2025-1dlls

42. Hawking, S.W. (1974) Black Hole Explosions? Nature, 248, 30-31. >https://doi.org/10.1038/248030a0

43. Friedman, A. (1922) Über die krüng des raumes. Zeitschrift für Physik, 10, 377-386. >https://doi.org/10.1007/bf01332580

44. Hossenfelder, S. (2012) Can We Measure Structures to a Precision Better than the Planck Length? Classical and Quantum Gravity, 29, Article 115011. >https://doi.org/10.1088/0264-9381/29/11/115011

45. Hossenfelder, S. (2013) Minimal Length Scale Scenarios for Quantum Gravity. Living Reviews in Relativity, 16, Article No. 2. >https://doi.org/10.12942/lrr-2013-2

46. Adler, R.J. (2010) Six Easy Roads to the Planck Scale. American Journal of Physics, 78, 925-932. >https://doi.org/10.1119/1.3439650

47. Haug, E.G. (2023) Quantized Newton and General Relativity Theory. Qeios. >https://doi.org/10.32388/6ASRSQ

48. Tatum, E.T., Haug, E.G. and Wojnow, S. (2024) Predicting High Precision Hubble Constant Determinations Based on a New Theoretical Relationship between CMB Temperature and . Journal of Modern Physics, 15, 1708-1716. >https://doi.org/10.4236/jmp.2024.1511075

49. Fixsen, D.J. (2009) The Temperature of the Cosmic Microwave Background. The Astrophysical Journal, 707, 916-920. >https://doi.org/10.1088/0004-637x/707/2/916

50. Carnot, S. (1824) Réflexions sur la puissance motrice du feu et sur les machines propres á développer cette puissance. Annales scientifiques de l’É.N.S., 2, 393-457.

51. Debnath, U. (2020) Thermodynamics of FRW Universe: Heat Engine. Physics Letters B, 810, Article 135807. >https://doi.org/10.1016/j.physletb.2020.135807

52. Debnath, U. (2022) General Thermodynamic Properties of FRW Universe and Heat Engine. Universe, 8, Article 400. >https://doi.org/10.3390/universe8080400

53. Reissner, H. (1916) Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie. Annalen der Physik, 355, 106-120. >https://doi.org/10.1002/andp.19163550905

54. Nordström, G. (1918) On the Energy of the Gravitation Field in Einstein’s Theory. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 20, 1238-1245.

55. Kerr, R.P. (1963) Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Physical Review Letters, 11, 237-238. >https://doi.org/10.1103/physrevlett.11.237

56. Haug, E.G. and Spavieri, G. (2023) Mass-Charge Metric in Curved Spacetime. International Journal of Theoretical Physics, 62, Article No. 248. >https://doi.org/10.1007/s10773-023-05503-9

57. Haug, E.G. (2025) The Hubble Sphere as an Extremal Reissner-Nordström Black Hole Carnot Engine Operating at the CMB Temperature of: . Cambridge University Press, Preprint. >https://doi.org/10.33774/coe-2025-w2dxp

58. de Martino, I., Atrio-Barandela, F., da Silva, A., Ebeling, H., Kashlinsky, A., Kocevski, D., et al. (2012) Measuring the Redshift Dependence of the Cosmic Microwave Background Monopole Temperature with Planck Data. The Astrophysical Journal, 757, Article 144. >https://doi.org/10.1088/0004-637x/757/2/144

59. Li, Y.Y., Hincks, A.D., Amodeo, S., Battistelli, E.S., Bond, J.R., Calabrese, E., et al. (2021) Constraining Cosmic Microwave Background Temperature Evolution with Sunyaev-Zel’Dovich Galaxy Clusters from the Atacama Cosmology Telescope. The Astrophysical Journal, 922, Article 136. >https://doi.org/10.3847/1538-4357/ac26b6

60. Riechers, D.A., Weiss, A., Walter, F., Carilli, C.L., Cox, P., Decarli, R., et al. (2022) Microwave Background Temperature at a Redshift of 6.34 from H ₂O Absorption. Nature, 602, 58-62. >https://doi.org/10.1038/s41586-021-04294-5

61. Haug, E.G. and Tatum, E.T. (2025) A Newly-Derived Cosmological Redshift Formula Which Solves the Hubble Tension and Yet Maintains Consistency with , the Principle and the Stefan-Boltzmann Law. European Journal of Applied Physics, 7, 48-50. >https://doi.org/10.24018/ejphysics.2025.7.1.368

62. Haug, E.G. (2025) An Exact CMB Photon Radiation Density of the Universe Derived from Cosmology. Cambridge University Press, Preprint. >https://doi.org/10.33774/coe-2025-jfg4t

63. Haug, E.G. (2024) Closed Form Solution to the Hubble Tension Based on Cosmology for Generalized Cosmological Redshift Scaling of the Form: Tested against the Full Distance Ladder of Observed SN Ia Redshift. Preprint. >https://doi.org/10.20944/preprints202409.1697.v2