The Hubble Sphere Radiation Pressure Law is Consistent With the Cosmological Redshift z = Sqrt( R_H/R_t) − 1

1. Introduction

Haug and Wojnow [1] have demonstrated a law similar to the ideal gass law is consistent with the Hubble sphere in certain subgroups of $R_H=ct$ cosmology, their Hubble pressure law is given by:

$$P_{c,t}{V}_{H_t}=n_{H_t}R{T}_t=N_{H_t}{k}_b{T}_t$$

(1)

where R is the ideal gas constant, $P_{c,t}={\displaystyle \frac{3H_t^2{c}^2}{8\pi G}}$ is Hubble radiation pressure. $V_{H_t}={\displaystyle \frac{4}{3}}\pi R_{H,t}^3={\displaystyle \frac{4}{3}}\pi {\displaystyle \frac{c^3}{H_t^3}}$ is the Hubble sphere volume. Further, $N_{H,t}={\displaystyle \frac{E_c}{E_{cmb}}}$, where $E_c={\displaystyle \frac{c^5}{2G{H}_t}}$ is the critical Friedmann energy and $E_{cmb,t}=k_b{T}_t$ is the energy of the CMB temperature at time t. Also, we follow the $R_{H,t}=ct$ principle, where then $H_t={\displaystyle \frac{1}{t}}$. Further, $n_{H,t}={\displaystyle \frac{N_{H,t}}{N_A}}$, where $N_A$ is Avogadro’s constant.

Further, Haug and Wojnow have demonstrated that in $R_{H,t}=ct$ black hole cosmology, we obtain consistency with the CMB temperature formula: $T_0={\displaystyle \frac{T_p}{8\pi }}\sqrt{{\displaystyle \frac{2l_p}{R_{H,t}}}}$, where $T_p={\displaystyle \frac{1}{k_b}}\sqrt{{\displaystyle \frac{\hslash c^5}{G}}}={\displaystyle \frac{m_p{c}^2}{k_b}}$ is the Planck [2,3] temperature, and $l_p=\sqrt{{\displaystyle \frac{G\hslash }{c^3}}}$ is the Planck length (see [4,5,6,7] for the CMB temperature formula itself). This CMB temperature formula can predict the CMB temperature at present, something the $\mathrm{\Lambda }$-CDM model cannot do. This leads to the following relation to the ideal gas constant:

$$R={\displaystyle \frac{mass\times lengt{h}^2}{amount\times temperature\times tim{e}^2}}={\displaystyle \frac{M_{c,t}\times R_t^2}{n_H\times T_t\times t_t^2}}$$

(2)

where $M_{c,t}={\displaystyle \frac{c^3}{2G{H}_t}}$ is the critical mass, $R_{H,t}={\displaystyle \frac{c}{H_t}}$ is the Hubble radius and $t_{H,t}={\displaystyle \frac{1}{H_t}}$ is the Hubble time, and $n_H={\displaystyle \frac{M_{c,t}{c}^2}{E_{cmb}}}={\displaystyle \frac{M_{c,t}}{m_{cmb}}}$, where $E_{cmb}=k_b{T}_t$, this all in a universe where we have $R_{H,t}=ct$, so we get:

$$\begin{array}{ccc}\hfill R& =& {\displaystyle \frac{M_{c,t}\times R_{H,t}^2}{n_H\times T_t\times t_{H,t}^2}}\hfill \\ \hfill R& =& {\displaystyle \frac{\frac{c^3}{2G{H}_t}\times \frac{c^2}{H_t^2}}{\frac{N_H}{N_A}\times T_t\times \frac{1}{H_t^2}}}\hfill \\ \hfill 1& =& {\displaystyle \frac{\frac{c^5}{2G{H}_t}}{N_H\frac{R}{N_A}{T}_t}}\hfill \\ \hfill 1& =& {\displaystyle \frac{E_t}{\frac{E_t}{E_{cmb,t}}{k}_b{T}_t}}\hfill \\ \hfill 1& =& {\displaystyle \frac{E_{cmb,t}}{k_b{T}_t}}\hfill \end{array}$$

(3)

Next we take advantage of the observed relation that we have $T_t=T_0(1+z)$ see [8,9,10]. So replacing $T_t$ with $T_0(1+z)$ we get:

$$\begin{array}{ccc}\hfill 1& =& {\displaystyle \frac{E_{cmb,t}}{k_b{T}_0(1+z)}}\hfill \\ \hfill z& =& {\displaystyle \frac{E_{cmb,t}}{E_{cmb,0}}}-1\hfill \end{array}$$

(4)

Further $E_{cmb,0}=k_b{T}_0={\displaystyle \frac{E_p}{8\pi }}\sqrt{{\displaystyle \frac{2l_p}{R_{H,0}}}}$ and $E_{cmb,t}=k_b{T}_t={\displaystyle \frac{E_p}{8\pi }}\sqrt{{\displaystyle \frac{2l_p}{R_{H,t}}}}$, where $E_p=\sqrt{{\displaystyle \frac{\hslash c^5}{G}}}$ is the Planck energy, this leads to:

$$\begin{array}{ccc}\hfill z& =& {\displaystyle \frac{\frac{E_p}{8\pi }\sqrt{\frac{2l_p}{R_{H,t}}}}{\frac{E_p}{8\pi }\sqrt{\frac{2l_p}{R_{H,0}}}}}-1\hfill \\ \hfill z& =& \sqrt{{\displaystyle \frac{R_H}{R_{H,t}}}}-1\hfill \end{array}$$

(5)

Which is the cosmological redshift first presented by Haug and Tatum [11], which they have demonstrated can be used to match the full distance ladder of SN Ia with a single value of $H_0$. In other words, it has been used to resolve the Hubble tension. This paper also demonstrates that their redshift is fully consistent with the Hubble sphere radiation pressure law presented by Haug and Wojnow.

2. Conclusions

We have demonstrated that the Hubble sphere radiation pressure law, which is linked to the ideal gas law, is consistent with the cosmological redshift of Haug and Tatum: $z=\sqrt{{\displaystyle \frac{R_H}{R_{H,t}}}}-1$, and that this is yet another important piece falling into place in the development of $R_{H,t}=ct$ cosmology. Black hole $R_{H,t}=ct$ cosmology seems robust in addition to outperforming the $\mathrm{\Lambda }$-CDM model on a series of points (see [12]).