Espen Gaarder Haug¹, Eugene Terry Tatum^2
1Tempus Gravitational Laboratory, Aas, Norway.
²Independent Researcher, Bowling Green, Kentucky, USA.
DOI: 10.4236/jmp.2026.176031   PDF    HTML   XML   21 Downloads   154 Views   Citations

Abstract

Based on the Haug-Tatum $R_{H_t}=ct$ cosmology model redshift formula, $z=\sqrt{\frac{R_{H_0}}{R_{H_t}}}-1=\sqrt{\frac{H_t}{H_0}}-1$ , we demonstrate that the time-dependent Lambda parameter as a function of $z$ must be ${\Lambda }_t={\Lambda }_0{\left(1+z\right)}^4$ . This equation is also consistent with the Haug and Tatum formula, ${\Lambda }_t=3\frac{H_t^2}{c^2}=3{\left(\frac{\mho T_t^2}{c}\right)}^2$ and the observation-based relation, $T_t=T_0\left(1+z\right)$ . We also show how $H_0$ predicted by the Haug-Tatum model is most consistent with the DESI-based $H_0$ estimates from the ${\omega }_0{\omega }_a$ CDM model that allows for dynamic dark energy.

Keywords

Cosmological Constant, DESI, Hubble Parameter, Time-Dependent Lambda, Model, Dynamic Dark Energy, Cosmological Redshift, Cosmological Entropy, Black Hole Cosmology

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1. Introduction and Background

The Haug and Tatum cosmology model is rooted in a series of discoveries that fit together nicely in a consistent theory. Tatum et al. [1] suggested in 2015 the following formula for the Cosmic Microwave Background (CMB) temperature:

$T_{cmb}=\frac{\hslash c}{k_{b}4\pi \sqrt{R_{H}2l_p}}$ (1)

where $\hslash =\frac{h}{2\pi }$ is the reduced Planck constant, $k_b$ is the Boltzmann constant, $R_H$ is the Hubble radius, and $l_p$ is the Planck length. Furthermore, one assumes an $R_H=ct$ cosmology. Haug and Wojnow [2] [3] subsequently proved that this formula could be derived from the Stefan-Boltzmann law. It is important to note here that observations have already confirmed that the CMB spectrum is that of a nearly perfect black body; see, for example, Muller et al. [4], who, based on COBE satellite observations, stated:

“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 and Tatum [5] [6] have discussed and mathematically demonstrated how the CMB temperature can be derived by simply assuming that it is the geometric mean temperature of the maximum and minimum possible temperatures inthe Hubble sphere. Specifically, $T_{cmb}=\sqrt{T_{\max }{T}_{\min }}=\frac{\hslash c}{k_{b}4\pi \sqrt{R_{H}2l_p}}$ where $T_{\min }=\frac{\hslash c}{k_{b}4\pi R_H}$ is the minimum possible Hawking temperature anywhere in the Hubble sphere, and $T_{\max }=\frac{\hslash c}{k_{b}8\pi l_p}$ is the maximum possible Hawking

temperature anywhere in the Hubble sphere. It is unlikely to be a coincidence that both the geometric mean temperature calculated in this way and the Stefan-Boltzmann law lead to exactly the same CMB temperature prediction. The Stefan-Boltzmann law is rooted in fundamental thermodynamics, and the geometric mean temperature is linked to Carnot theory. The CMB temperature Equation (1) can also be solved for $H_0$ . Since the CMB temperature is the most precisely measured cosmic parameter, this leads to a much more precise prediction of $H_0$ than any other known method; see, for example, Tatum et al. [7] for further details. In addition, Haug and Tatum [8] demonstrated that a thermodynamic Friedmann equation, which links the CMB temperature with the density of the universe, can also be derived from Equation (1). They have shown how this leads to a much more precise estimate of the density of the universe than other competing theories can provide. Again, the reason for this is the direct link Equation (1) allows for between the CMB temperature and other cosmic parameters.

Furthermore, Haug and Tatum [9]-[11] have also derived their cosmological redshift formula as:

$z=\sqrt{\frac{R_{H_0}}{R_{H_t}}}-1=\sqrt{\frac{H_t}{H_0}}-1$ (2)

if one wants to be consistent with the observation-based [12]-[14] $T_t=T_0\left(1+z\right)$ relation. The Haug-Tatum model is also, to our knowledge, the only model wherein the $T_t=T_0\left(1+z\right)$ relation is derived directly from model assumptions. Key model assumptions and derivations with respect to Equation (2) are clearly given in [9]. In the standard Λ-CDM model, $T_t=T_0\left(1+z\right)$ comes only from observations. In the Haug-Tatum model, by contrast, it comes from derivations that have subsequently been supported by observations. Equation (2), therefore, also means:

$H_t=H_0{\left(1+z\right)}^2$ (3)

Most current researchers in cosmology work on the Λ-CDM model. Our model is within the class of $R_H=ct$ cosmologies. One of the best-known models in this class is the Melia et al. [15]-[18] $R_H=ct$ model. Melia [19] [20] has compared his model to the Λ-CDM model and demonstrated that it outperforms Λ-CDM when compared across many types of observational studies. However, the Melia model is likely not compatible with Equation (1) and is also not compatible with the Haug-Tatum redshift. The Haug-Tatum model gives far more precise estimates of cosmological parameters, such as $H_0$ and $R_H$ , than both the Λ-CDM model and the Melia model. We have compared our model to the Λ-CDM model and demonstrated that it performs better in a long series of observational studies. Naturally, extraordinary claims require extraordinary evidence, so another recent direct comparison of the Haug-Tatum model and the Λ-CDM model can also be found here [21]. That said, there are naturally outstanding research questions concerning all of these models. For example, one of the multiple outstanding questions is whether Lambda should be a constant over time or evolve over time. This paper clearly demonstrates that, within the Haug-Tatum model, Lambda must evolve over time. Furthermore, this appears to be consistent with the recent DESI observations.

2. Time-Dependent Lambda

While deriving their new thermodynamic type Friedmann equation mentioned above, Haug and Tatum pointed out that the Lambda term in their $R_{H_t}=ct$ cosmology must be a time-dependent parameter according to:

${\Lambda }_t=3\frac{H_t^2}{c^2}=\frac{3}{R_{H_t}^2}$ (4)

Interestingly, Einstein [22], in 1917, suggested a “cosmological constant” $\Lambda =\frac{1}{r^2}$ . However, without a time-varying $r$ , he actually introduced his Lambda in order to preserve a steady state universe. This was in keeping with his simplifying assumption, at the time, of a universe which neither contracted nor expanded. Lambda in the standard Λ-CDM model is still assumed to be a constant of the form $\Lambda =3\frac{H_0^2}{c^2}{\Omega }_{\Lambda }$ , thus constant throughout the lifetime of the universe. Unfortunately, the recent DESI observations appear to contradict this assumption.

Furthermore, based on the time-dependent Lambda in the Haug-Tatum cosmology model (HTC), we must naturally also have:

${\Lambda }_t=3\frac{1}{t_t^2{c}^2}$ (5)

where $t_t=\frac{1}{H_t}$ is the Hubble time of $R_{H_t}=ct$ cosmology.

Substituting the $H_t$ expression from Equation (3) gives:¹

${\Lambda }_t=3\frac{{\left(H_0{\left(1+z\right)}^2\right)}^2}{c^2}$ (6)

and also:

${\Lambda }_t=3\frac{{\left(1+z\right)}^4}{t_0^2{c}^2}$ (7)

which means we must have:

${\Lambda }_t={\Lambda }_0{\left(1+z\right)}^4$ (8)

Haug and Tatum [8] have additionally linked their time-dependent Lambda to the current CMB temperature by:

${\Lambda }_t=3{\left(\frac{\mho T_t^2}{c}\right)}^2=3\frac{{\mho }^2{T}_t^4}{c^2}=3{\left(\frac{\mho T_0^2{\left(1+z\right)}^2}{c}\right)}^2$ (9)

where our composite constant $\mho =\frac{k_b^{2}32{\pi }^2{G}^{1/2}}{c^{5/2}{\hslash }^{3/2}}$ . Since the current CMB temperature has been measured much more precisely than the Hubble parameter (see [24]-[27]), this also means that they have constrained the current value of Lambda much more narrowly than the Λ-CDM model, as also shown in [8]. The entire purpose of our paper is to provide our Figure 1 and Table 1 so that the derivations made from our model can be compared to current and future observational data.

3. Predictions with Respect to Cosmological Redshift and Cosmic Entropy

Figure 1 shows how the time-dependent Lambda varies with observed cosmological redshift in HTC. This result appears to be in line with DESI observations [28] [29], in that Lambda cannot be a constant. Haug and Tatum have already demonstrated that they get a perfect match to the full distance ladder of SNe Ia observations without the need for accelerated expansion of the universe. So, it appears to be likely that the universe is either expanding at a constant velocity according to $R_{H_t}=ct$ , or that it is in a steady state and yet somehow gives the same predictions as $R_{H_t}=ct$ as a consequence of following an extremal universe model; see [30].

Figure 1. HTC time-dependent Lambda as a function of cosmological redshift.

We can also express time-dependent Lambda as a function of cosmic entropy. Haug and Tatum [11] have previously derived:

$z={\left(\frac{S_{BH,0}}{S_{BH,t}}\right)}^{\frac{1}{4}}-1=\sqrt{\frac{H_t}{H_0}}-1$ (10)

Therefore, time-dependent Lambda can also be written as:

${\Lambda }_t={\Lambda }_0{\left(1+z\right)}^4={\Lambda }_0\frac{S_{BH,0}}{S_{BH,t}}$ (11)

where the number of entropic states (defined according to the Bekenstein-Hawking black hole entropy formula [31] [32]) within HTC is given by:

$S_{BH,t}=\frac{4\pi R_{H_t}^2}{4l_p^2}$ (12)

following $R_{H_t}=ct$ , where $l_p=\sqrt{\frac{G\hslash }{c^3}}$ is the Planck length, as defined by Max Planck [33] [34]. See also Tatum and Haug [35], which discusses how a time-dependent entropy in $R_{H_t}=ct$ cosmology could be directly linked to a dynamic dark energy without necessarily requiring acceleration in the expansion of the universe. We have also demonstrated [11] that, within HTC $R_{H_t}=ct$ cosmology, one must have:

$z={\left(\frac{{\rho }_{cr,t}}{{\rho }_{cr,0}}\right)}^{\frac{1}{4}}-1$ (13)

Therefore, we can also express time-dependent Lambda in HTC as:

${\Lambda }_t={\Lambda }_0{\left(1+z\right)}^4={\Lambda }_0\frac{{\rho }_{cr,t}}{{\rho }_{cr,0}}$ (14)

where ${\rho }_{ct,t}=\frac{3c^4}{R_{H_t}^{2}8\pi G}$ .

4. Hubble Constant Predicted by the HTC Model Compared to DESI Study Results

Haug and Tatum have used the full distance ladder of SNe Ia to extract $H_0=66.8943\pm 0.0287$ km/s/Mpc. In addition, Tatum et al. [7] have, using only the CMB temperature, found the same value for $H_0$ . The reason for obtaining the same $H_0$ value from SNe Ia redshift observations and the current CMB temperature is the exact mathematical relationships between the current CMB temperature, $H_0$ , and cosmological redshift in the HTC model.

Table 1 shows our predicted $H_0$ values in comparison to recent DESI-based $H_0$ values. It should be noted that the DESI estimated values for $H_0$ are different for different models. We see, for example, that the pure ΛCDM model gives $H_0$ values slightly above those derived from the Haug-Tatum $R_{H_t}=ct$ model. The ${\omega }_0{\omega }_a$ CDM model, on the other hand, allows for dynamic dark energy which, by definition, changes over cosmic time. One can readily see that the Haug-Tatum values for $H_0$ are, for most of the DESI-related values of $H_0$ , well within the 95% confidence intervals. However, note that the Haug-Tatum model gives much more precise $H_0$ values than the other models, due to its exact mathematical relationships between $H_0$ , CMB and cosmological redshift $z$ . The models with ${\displaystyle \sum m_{\nu }}$ also allow for the neutrino mass parameter to vary, assuming a ${\displaystyle \sum m_{\nu }}>0$ ; see [29].

Table 1. This table shows $H_0$ values from the Haug-Tatum model in comparison to the $H_0$ values from DESI observations.

From:	Hubble Parameter:	Study
CMB	$H_0=67.4\pm 0.5$ km/s/Mpc	Planck Collobration [36]
CMB	$H_0=66.8943\pm 0.0287$ km/s/Mpc	Tatum et al [7]
SNe Ia Pantheon+	$H_0=66.8943\pm 0.0287$ km/s/Mpc	Haug and Tatum [9]
SNe Ia Union2	$H_0=66.8711\pm 0.0019$ km/s/Mpc	Haug and Tatum [37]
ΛCDM		See TABLE V. [29]
DESI + BBN	$H_0=68.51\pm 0.58$ km/s/Mpc
DESI + CMB	$H_0=68.17\pm 0.28$ km/s/Mpc
ωCDM		See TABLE V. [29]
DESI + CMB	$H_0=69.51\pm 0.92$ km/s/Mpc
DESI + CMB + Pantheon+	$H_0=66.75\pm 0.56$ km/s/Mpc
DESI + CMB + Union3	$H_0=68.01\pm 0.68$ km/s/Mpc
DESI + CMB + DESY5	$H_0=67.34\pm 0.54$ km/s/Mpc
${\omega }_0{\omega }_a$ CDM
DESI BAO + CMB + Pantheon+	$H_0=67.51\pm 0.59$ km/s/Mpc	See TABLE V. [29]
DESI BAO + CMB + Union3	$H_0=65.91\pm 0.84$ km/s/Mpc
DESI BAO + CMB + DESY5	$H_0=66.74\pm 0.56$ km/s/Mpc
ΛCDM + ${\displaystyle \sum m_{\nu }}$ :		See TABLE VII. [29]
DESI BAO + CMB [Camspec]	$H_0=68.36\pm 0.29$ km/s/Mpc
DESI BAO + CMB [L-H]	$H_0=68.48\pm 0.30$ km/s/Mpc
DESI BAO + CMB [Plik]	$H_0=68.56\pm 0.31$ km/s/Mpc
ωCDM + ${\displaystyle \sum m_{\nu }}$ :
DESI BAO + CMB	$H_0=69.28\pm 0.92$ km/s/Mpc	See TABLE VII. [29]
DESI BAO + CMB + Pantheon+	$H_0=67.94\pm 0.58$ km/s/Mpc
DESI BAO + CMB + Union3	$H_0=67.93\pm \pm 0.69$ km/s/Mpc
DESI BAO + CMB + DESY5	$H_0=67.34\pm 0.53$ km/s/Mpc
${\omega }_0{\omega }_a$ CDM + ${\displaystyle \sum m_{\nu }}$
DESI BAO + CMB	$H_0=69.28\pm 0.92$ km/s/Mpc	See TABLE VII. [29]
DESI BAO + CMB + Pantheon+	$H_0=67.54\pm 0.59$ km/s/Mpc
DESI BAO + CMB + Union3	$H_0=65.96\pm 0.84$ km/s/Mpc
DESI BAO + CMB + DESY5	$H_0=66.75\pm 0.56$ km/s/Mpc

The DESI comparison in Table 1 can be judged by the simple rule of overlap of confidence intervals. All papers referenced in this table explain how the estimates and their confidence intervals were calculated. Forthcoming papers of ours will delve further into statistical comparisons with various DESI data sets, including Baryonic Acoustic Oscillations data. The current paper does not perform a new fit to DESI data and instead compares HTC predictions with published parameter estimates from other analyses.

5. Conclusion

We have shown that we must have a time-dependent Lambda according to ${\Lambda }_t={\Lambda }_0{\left(1+z\right)}^4$ in our HTC variant of $R_{H_t}=ct$ black hole cosmology. This is not only consistent with the observation-based relation $T_t=T_0\left(1+z\right)$ , but also consistent with a non-accelerating expansion of the universe following the $R_{H_t}=ct$ principle. We eagerly look forward to seeing further dynamic dark energy observational evidence along the lines shown in our Figure 1.

NOTES

¹Wojnow [23] in a recent working paper suggests: ${\Lambda }_t={\Lambda }_0{\left(1+z\right)}^2$ . However, this is incompatible with the observation-based CMB temperature relation $T_t=T_0\left(1+z\right)$ inside $R_{H_t}=ct$ cosmology, as demonstrated by Haug and Tatum. Instead, the relations presented in this paper are fully consistent with this observation-based CMB temperature versus redshift relation.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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