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The Photon Energy Density Parameter Ωγ of the Universe Exactly Derived

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Espen Haug^*

This version is not peer-reviewed

Submitted:

26 March 2024

Posted:

27 March 2024

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Abstract

We will demonstrate that the photon energy density parameter Ωγ of the universe can be derived exactly in Rh = ct cosmology. We find that it must be Ωγ = 1/5760π ≈ 5.52621330180192× 10^(−5). This indicates that there is no uncertainty in the radiation density within at least somesub-classes of Rh = ct cosmology.

Keywords:

Subject: Physical Sciences - Astronomy and Astrophysics

Deriving the Photon Energy Density Parameter $Ω_{γ}$

The photon energy density:

ρ_{γ}

is normally found by doing the following integral:

ρ_{γ, 0} c^{2} = \int_{0}^{\infty} h v n (v) d v = a_{b} T_{0}^{4}

(1)

which mean we have:

ρ_{γ, 0} = \frac{a_{b} T_{0}^{4}}{c^{2}}

(2)

where the radiation density constant:

a_{b}

is given by:

a_{b} = \frac{8 π^{5} k_{b}^{4}}{15 c^{3} h^{3}} = \frac{4}{c} σ

(3)

In 1978, Emslie and Green [1] presented the photon energy density relative to the critical Friedmann energy density as:

\begin{matrix} Ω_{γ, 0} & = & \frac{ρ_{γ}}{ρ_{c}} \\ Ω_{γ, 0} & = & \frac{\frac{a T_{0}^{4}}{c^{2}}}{\frac{3 H_{0}^{2}}{8 π G}} \end{matrix}

(4)

Haug and Tatum [2] have recently shown that the Friedmann [3] equation can be expressed in thermodynamic form, indicating that the critical density must be given by:

ρ_{c, 0} = \frac{3 H_{0}}{8 π G} = T_{0}^{4} \frac{23040 π}{c^{3}} σ

(5)

By replacing this into equation (4) one get:

\begin{matrix} Ω_{γ, 0} & = & \frac{\frac{a T^{4}}{c^{2}}}{T^{4} \frac{23040 π}{c^{3}} σ} \\ Ω_{γ, 0} & = & \frac{4 σ}{23040 π σ} \\ Ω_{γ, 0}, & = & \frac{1}{5760 π} \approx 5.52621330180192 \times 10^{- 5} \end{matrix}

(6)

This implies that the photon density parameter remains exact for all cosmic time epochs. This consistency aligns with at least two potential types of

R_{h} = c t

cosmological models. It is conceivable within growing black hole models, as exemplified in [4], and also within steady-state black hole cosmology, where the metric from general relativity suggests that density inside the black hole varies as one approaches the center, as discussed in [5,6]. Additionally, this holds true in the extremal universe scenario, where the density inside the black hole’s Hubble sphere varies, as indicated in [7]. The idea of black hole cosmology goes at least back to 1972 by Pathria [8] and is actively discussed to this day [9,10,11,12,13,14,15,16,17], even if the

Λ

-CDM model currently takes most headlines.

When observing the black hole (Hubble sphere), it resembles a growing black hole model due to the density change along the radius in the steady-state black hole.

R_{h} = c t

black hole cosmology in earlier epochs of the universe, we must have:

ρ_{γ, t} = \frac{a_{b} T_{0}^{4} {(1 + z)}^{4}}{c^{2}} = \frac{a_{b} T_{t}^{4}}{c^{2}}

(7)

Furthermore, we must have a critical density, as demonstrated by Haug and Tatum [2,18], of:

ρ_{c, t} = \frac{3 H_{t}}{8 π G} = T_{0}^{4} {(1 + z)}^{4} \frac{23040 π}{c^{3}} σ = T_{t} \frac{23040 π}{c^{3}} σ

(8)

By substituting these into the equation below, we obtain:

\begin{matrix} Ω_{γ, t} & = & \frac{ρ_{γ}}{ρ_{c, t}} \\ Ω_{γ, t} & = & \frac{\frac{a T_{t}^{4}}{c^{2}}}{T_{t}^{4} \frac{23040 π}{c^{3}} σ} \\ Ω_{γ, t} & = & \frac{4 σ}{23040 π σ} \\ Ω_{γ, t} & = & \frac{1}{5760 π} \approx 5.52621330180192 \times 10^{- 5} \end{matrix}

(9)

In a black hole growing

R_{h} = c t

cosmology during the entire cosmic epoch, the photon radiation density ratio remains constant and exact. This is likely inconsistent with the predictions of the

Λ

-CDM model in earlier times, but it should be correct for the current time as well.

Data Availability Statement

No data was used for this study.

Conflicts of Interest

The author declare no conflict of interest.

References

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