The Photon Energy Density Parameter Ωγ of the Universe Exactly Derived

Espen Haug

doi:10.20944/preprints202403.1663.v1

Submitted:

26 March 2024

Posted:

27 March 2024

You are already at the latest version

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:

Photon energy density parameter

;

Friedmann equation

;

critical density

;

Thermodynamical Friedmann equation

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.

In

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

A. G. Emslie and R. M. Green. Can the Einstein-de sitter model adequately describe the universe at any epoch. Astrophysics and Space Science 1978, 58, 181. [CrossRef]
E. G. Haug and E. T. Tatum. Planck length from cosmological redshifts solves the Hubble tension. ResearchGate.org, 2024a. URL https://www.preprints.org/manuscript/202403.1241/v1.
A. Friedmann. Über die krüng des raumes. Zeitschrift für Physik 1922, 10, 377. [CrossRef]
E. T. Tatum, U. V. S. Seshavatharam, and S. Lakshminarayana. The basics of flat space cosmology. International Journal of Astronomy and Astrophysics 2015, 5, 116. [CrossRef]
E. G. Haug and G. Spavieri. Mass-charge metric in curved spacetime. International Journal of Theoretical Physics 2023, 62, 248. [CrossRef]
E. G. Haug and G. Spavieri. New exact solution to Einstein’s field equation gives a new cosmological model. Researchgate.org Pre-print 2023. [CrossRef]
E. G. Haug. The extremal universe exact solution from Einstein’s field equation gives the cosmological constant directly. Journal of High Energy Physics, Gravitation and Cosmology 2024, 10, 386. [CrossRef]
R. K. Pathria. The universe as a black hole. Nature 1972, 240, 298. [CrossRef]
P. Christillin. The Machian origin of linear inertial forces from our gravitationally radiating black hole universe. The European Physical Journal Plus 2014, 129, 286. [CrossRef]
T. X. Zhang and C. Frederick. Acceleration of black hole universe. Astrophysics and Space Science 2014, 349, 567. [CrossRef]
N. Popławski. The universe in a black hole in Einstein–Cartan gravity. The Astrophysical Journal 2016, 832, 96. [CrossRef]
T. X. Zhang. The principles and laws of black hole universe. Journal of Modern Physics 2018, 9, 1838. [CrossRef]
D. A. Easson and R. H. Brandenberger. Universe generation from black hole interiors. Journal of High Energy Physics 2001, 2001. [CrossRef]
E. Gaztanaga. The black hole universe, part i. Symmetry 2022, 14, 1849. [CrossRef]
Z. Roupas. Detectable universes inside regular black holes. The European Physical Journal C 2022, 82, 255. [CrossRef]
E. Siegel. Are we living in a baby universe that looks like a black hole to outsiders? Hard Science, Big Think, January 27, 2022. https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/.
C. H. Lineweaver and V. M. Patel. All objects and some questions. American Journal of Physics 2023, 91.
E. G. Haug and E. T. Tatum. Planck length from cosmological redshifts solves the Hubble tension. ResearchGate.org 2024. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

The Photon Energy Density Parameter Ωγ of the Universe Exactly Derived

Abstract

Keywords:

Subject:

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

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe

The Photon Energy Density Parameter Ωγ of the Universe Exactly Derived

Abstract

Keywords:

Subject:

Deriving the Photon Energy Density Parameter Ω γ

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe

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