Thermodynamic Cosmology from the CMB Temperature and the Planck Temperature and The Relation to Quantum Cosmology

1. CMB Temperature from the Stefan-Boltzman Law

Haug and Wojnow [1,2] have demonstrated that, from the Stefan-Boltzmann [3,4] law, when assuming $R_h=ct$ cosmology, one can derive the following equation for the CMB temperature:

$$T_0={\displaystyle \frac{T_p}{8\pi }}\sqrt{{\displaystyle \frac{2l_p}{R_h}}}$$

(1)

where $R_h={\displaystyle \frac{c}{H_0}}$ is the Hubble radius. This can also be written as $T_0={\displaystyle \frac{T_p}{8\pi }}\sqrt{{\displaystyle \frac{2l_p{H}_0}{c}}}={\displaystyle \frac{T_p}{8\pi }}\sqrt{2t_p{H}_0}$, where $l_p$ is the Planck [5,6] length, $t_p$ is the Planck time, and $T_p={\displaystyle \frac{1}{k_b}}\sqrt{{\displaystyle \frac{\hslash c^5}{G}}}={\displaystyle \frac{m_p{c}^2}{k_b}}={\displaystyle \frac{E_p}{k_b}}$ is the Planck temperature, with $k_b$ being the Boltzmann constant.

The importance of deriving the CMB temperature from the Stefan-Boltzmann law becomes clear when one understands that the CMB is an almost perfect black body and that the Stefan-Boltzmann law is valid only for systems close to perfect black bodies. Muller et al. [7] point out:

“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, $T_0=2.72548\pm 0.00057K$."

The equation derived by Haug and Wojnow from the Stefan-Boltzmann law has also been shown to be fully consistent with a CMB formula suggested heuristically by Tatum et al. [8]. Additionally, the same CMB formula has recently been demonstrated to be derivable using a geometric mean approach by Haug and Tatum [9], as well as an approach involving quantized bending of light [10].

Equation (1) is valid for any previous cosmic epoch in an $R_h=ct$ universe and is expressed in its more general form as:

$$T_0={\displaystyle \frac{T_p}{8\pi }}\sqrt{{\displaystyle \frac{2l_p}{R_t}}}={\displaystyle \frac{T_p}{8\pi }}\sqrt{2t_p{H}_t}$$

(2)

where the Hubble radius $R_H=R_t={\displaystyle \frac{c}{H_t}}$ or the time dependent Hubble parameter: $H_t$ is the only time-dependent parameter that follows the $R_t=ct$ principle (see [11,12,13,14,15,16]). This implies that the universe either expanded only at the speed of light or that we could be living in a steady-state black hole universe, where light signals inside the sphere relative to us adhere to the $R_t=ct$ principle. Simply put, photons cannot travel faster than the well-known speed of light, c. The idea that the Hubble sphere could be a gigantic black hole is not new and have been suggested at least since 1972 by Pathria [17] and the idea of a black hole universe is actively discussed to this day (see [18,19,20,21,22]). The hypothesis of black hole universe is clearly much less popular than the $\Lambda$-CDM model, but it in our view deserved further investigation. It is first when the different hypothesis of the universe is extensively explored one can truly compare the different models in great detail.

2. Thermodynamic Cosmology from the CMB Temperature and the Planck Temperature

The Friedmann [23] equation is given by:

$$H_0^2={\displaystyle \frac{8\pi G{\rho }_c+\Lambda c^2}{3}}-{\displaystyle \frac{k^2{c}^2}{3}}$$

(3)

In the case of the critical Friedmann equation under flat space ($k=0$), we have:

$$H_0^2={\displaystyle \frac{8\pi G{\rho }_c}{3}}$$

(4)

where ${\rho }_c={\displaystyle \frac{M_c}{\frac{4}{3}\pi R_H^3}}={\displaystyle \frac{3H_0}{8\pi G}}$ is the critical Friedmann density.

As $H_0=32{\pi }^2{f}_p{\displaystyle \frac{T_0^2}{T_p^2}}$, where $f_p={\displaystyle \frac{c}{l_p}}$ is the Planck frquency, we can replace $H_0$ in the Friedmann equation, resulting in:

$$\begin{array}{ccc}\hfill T_0^4& =& {\displaystyle \frac{G{\rho }_c{t}_p^2{T}_p^4}{384{\pi }^3}}\hfill \\ \hfill T_0^4& =& {\displaystyle \frac{l_p^{4}c{\rho }_c{T}_p^4}{\hslash 384{\pi }^3}}\hfill \\ \hfill T_0^4& =& {\displaystyle \frac{{\rho }_c{T}_p^4}{\frac{m_p}{\frac{4}{3}\pi l_p^3}512{\pi }^4}}\hfill \\ \hfill T_0^4& =& T_p^4{\displaystyle \frac{{\rho }_c}{512{\pi }^4{\rho }_p}}\hfill \end{array}$$

(5)

where ${\rho }_p={\displaystyle \frac{m_p}{\frac{4}{3}{l}_p^3}}$ is the Planck mass density for a Planck sphere. Often the Planck density is for simplicty defined as a Planck cube ${\displaystyle \frac{m_p}{l_p^3}}$ see Unnikrishnan and Gillies [24], but since we are working with black holes and as we will see the spherical Schwarzschild metric sphere density is more relevant.

This provides a way to express the Friedmann equation in thermodynamic form. This approach is rooted in what has been presented by Haug and Tatum [25], yielding the same output, but here the Friedmann equation itself written on a somewhat different form, with an emphasis on expressing it in terms of the Planck temperature in addition to the CMB temperature, thereby adhering more strictly to thermodynamics. Solving for the critical mass, we obtain:

$$\begin{array}{ccc}\hfill {\rho }_c& =& 512{\pi }^4{\rho }_p{\displaystyle \frac{T_0^4}{T_p^4}}\hfill \end{array}$$

(6)

For the case of the critical density in past cosmic epochs, we have:

$$\begin{array}{ccc}\hfill {\rho }_{c,t}& =& 512{\pi }^4{\rho }_p{\displaystyle \frac{T_0^4{(1+z)}^4}{T_p^4}}=512{\pi }^4{\rho }_p{\displaystyle \frac{T_t^4}{T_p^4}}\hfill \end{array}$$

(7)

and further:

$$T_0={\displaystyle \frac{T_p}{4\pi }}{\left({\displaystyle \frac{{\rho }_c}{2{\rho }_p}}\right)}^{1/4}$$

(8)

Table 1 shows a series of cosmological properties described as functions of the CMB temperature and the Planck temperature. In any of these formulas, $T_0$ can be replaced with $T_t$ if one wants predictions for earlier cosmic epochs. This has major implications, as the CMB temperature has been measured much more precisely than $H_0$. Additionally, it provides new insights into how the CMB temperature and Planck temperature appear to be closely connected to the cosmos.

The mass gap is likely new to most readers. This is based on the assumption that the maximum reduced Compton wavelength is limited by the radius or alternatively the circumference of the Hubble sphere. The mass gap is if the Hubble radius is the limitation of this wavelength equal to $m_{g,r}={\displaystyle \frac{m_p^2}{2M_c}}$, where $m_p=\sqrt{{\displaystyle \frac{\hslash c}{G}}}$ is the Planck mass and $M_c={\displaystyle \frac{c^3}{2G{H}_0}}$ is the critical Friedmann mass, for derivation of the mass gap see [38]. Interestingly the mass gap leads to a gravitational gap, or we can call it a minimum acceleration equal to:

$$g_{min}={\displaystyle \frac{G{m}_{g,r}}{l_p^2}}={\displaystyle \frac{G\frac{m_p^2}{2M_c}}{l_p^2}}=a_{p}32{\pi }^2{\displaystyle \frac{T_0^2}{T_p^2}}\approx 6.5\times {10}^{-10}m/s^2$$

(9)

where $a_p$ is the gravitational acceleration. Or if alternatively the circumference of the Hubble sphere is the limitation factor for the maximum wavelength we get a minimum acceleration (gravitational acceleration gap) of:

$$g_{min}={\displaystyle \frac{G{m}_{g,c}}{l_p^2}}=a_{p}16\pi {\displaystyle \frac{T_0^2}{T_p^2}}\approx 1.04\times {10}^{-10}m/s^2$$

(10)

This is very close to the minimum acceleration one operates with in the MOND model of Milgrom [26] that is one of the alternative models to explain galaxy rotation curves. The advantage of the minimum acceleration that comes out from the model here is that it also give a simple explanation for the cause of observed minimum gravitational acceleration. it is simply related to that all matter can never have a smaller gravitational acceleration than what is related to the mass gap inside the Hubble sphere. Again this seems to lead to no need for dark matter. That said this should naturally be carefully studied before one makes firm conclusions, but it is a new hypothesis worth investigating, but is the main topic of the paper just referred to and not of this paper which is more an overview paper of how one can write a series of cosmological properties on what we can call thermodynamic forms.

As the CMB temperature [29,30,31] can be much more precisely measured than $H_0$ (see [32,33,34,35]), this leads to a dramatic improvement in predictions of all these cosmological properties inside $R_H=ct$ cosmology; see [36]. This is naturally true even after taking into account uncertainty in the Planck units used in the formulas above. It even leads to a much more precise Hubble constant, as first demonstrated by Tatum et al. [37].

3. The Hawking Temperature Based Friedmann Equation

Haug [1] has demonstrated that:

$$H_0={\displaystyle \frac{1}{2}}{f}_p{\displaystyle \frac{T_{Haw}^2}{T_0^2}}$$

(11)

where $T_{Haw}$ is the Hawking [39] temperature:

$$T_{Haw}={\displaystyle \frac{\hslash c}{k_{b}4\pi R_s}}$$

(12)

Here, $R_s$ is the Schwarzschild radius, which we have shown in the previous section to be identical to the Hubble radius in a critical Friedmann universe. Thus, we have:

$$T_{Haw}={\displaystyle \frac{\hslash c{H}_0}{k_{b}4\pi c}}$$

(13)

Unlike the Planck temperature, determining the Hawking temperature still depends on knowing the Hubble constant. However, this provides valuable insight into how many cosmological parameters of the universe relate to the ratio between the CMB temperature and the Hawking temperature in a black hole universe.

This means we can re-write the critical Friedmann equation simply by replacing $H_0$ with this expression, resulting in:

$$\begin{array}{ccc}\hfill H_0^2& =& {\displaystyle \frac{8\pi G{\rho }_c}{3}}\hfill \\ \hfill {\displaystyle \frac{1}{4}}{f}_p^2{\displaystyle \frac{T_{Haw}^4}{T_0^4}}& =& {\displaystyle \frac{8\pi G{\rho }_c}{3}}\hfill \\ \hfill T_0^4& =& {\displaystyle \frac{T_{Haw}^4}{8}}{\displaystyle \frac{{\rho }_p}{{\rho }_c}}\hfill \\ \hfill T_0& =& T_{Haw}{\left({\displaystyle \frac{{\rho }_p}{8{\rho }_c}}\right)}^{1/4}\hfill \end{array}$$

(14)

Solved for the critical density we get:

$${\rho }_c={\displaystyle \frac{1}{8}}{\rho }_p{\displaystyle \frac{T_{Haw}^4}{T_0^4}}$$

(15)

where ${\rho }_p={\displaystyle \frac{m_p}{\frac{4}{3}\pi l_p^3}}$ is the Planck mass density. Table 2 shows a series of cosmological properties expressed from the Hawking temperature and the CMB temperature consistent with $R_H=ct$ cosmology.

4. Quantum Cosmology

Table 2 in the last section shows a series of cosmological properties and parameters expressed in terms of the ratio between the Hawking temperature and the CMB temperature. Haug [1] has demonstrated that:

$${\displaystyle \frac{T_0^2}{T_{Haw}^2}}={\displaystyle \frac{l_p}{{\overline{\lambda }}_c}}$$

(16)

where ${\overline{\lambda }}_c$ is the reduced Compton wavelength of the critical mass in the Hubble sphere: ${\overline{\lambda }}_c={\displaystyle \frac{\hslash }{M_{c}c}}$. The reduced Compton frequency over the Planck time is given by $f={\displaystyle \frac{c}{{\overline{\lambda }}_c}}{t}_p={\displaystyle \frac{l_p}{{\overline{\lambda }}_c}}$. Haug [1,40] has discussed how this can be interpreted as the quantization of gravity—specifically, that the reduced Compton frequency over the Planck time can be incorporated into a quantized version of general relativity theory. While quantizing general relativity alone is not sufficient to unify it with quantum mechanics, it is a step toward better understanding gravity.

In Table 1, the ratio between the Planck temperature and the CMB temperature was used. At a deeper level, this ratio is equal to:

$${\displaystyle \frac{T_p^2}{T_0^2}}=32{\pi }^2{\displaystyle \frac{l_p}{{\overline{\lambda }}_c}}$$

(17)

So again, we arrive at the reduced Compton frequency over the Planck time, multiplied by the constant $32{\pi }^2$. When we replace ${\displaystyle \frac{T_0^2}{T_{Haw}^2}}$ with its deeper constituent, ${\displaystyle \frac{l_p}{{\overline{\lambda }}_c}}$, and ${\displaystyle \frac{T_p^2}{T_0^2}}$ with its deeper constituent, $32{\pi }^2{\displaystyle \frac{l_p}{{\overline{\lambda }}_c}}$, we find that both Table 1 and Table 2 converge into Table 3. This essentially represents quantized cosmology, where ${\displaystyle \frac{l_p}{{\overline{\lambda }}_c}}$ again signifies the reduced Compton frequency over the Planck time, offering a new perspective on the quantization of gravity within general relativity theory [1].

This is all consistent with a new approach to expressing the Schwarzschild metric. Specifically [1,42], we have:

$$\begin{array}{c}\hfill d{s}^2=-\left(1-{\displaystyle \frac{2GM}{c^{2}r}}\right)c^{2}d{t}^2+{\left(1-{\displaystyle \frac{2GM}{c^{2}r}}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2\\ \hfill d{s}^2=-\left(1-{\displaystyle \frac{2l_p}{r}}{\displaystyle \frac{l_p}{\overline{\lambda }}}\right)c^{2}d{t}^2+{\left(1-{\displaystyle \frac{2l_p}{r}}{\displaystyle \frac{l_p}{\overline{\lambda }}}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2\end{array}$$

(18)

where $\overline{\lambda }$ is the reduced Compton wavelength of the mass M. So, we see the same reduced Compton frequency over the Planck time here: ${\displaystyle \frac{l_p}{\overline{\lambda }}}$ in the Schwarzschild metric itself, which can again be linked to the ratio of the Planck temperature to the CMB temperature or the Hawking temperature to the CMB temperature, under the assumption of black hole $R_h=ct$ cosmology.

5. Conclusion

All cosmological properties that are typically expressed through the Hubble parameter can just as effectively be expressed through the observed CMB temperature and the Planck temperature. This is fully consistent with a new thermodynamic version of the Friedmann equation. Within the framework of $R_H=ct$ cosmology, this means we are free to make predictions based on either the Hubble parameter or the CMB temperature. Since the CMB temperature has been measured with far greater precision, this leads to a dramatic improvement in prediction accuracy, as demonstrated in a series of recent papers.