The CMB Luminosity Distance and Cosmological Redshifts

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The CMB Luminosity Distance and Cosmological Redshifts

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Espen Haug^*

This version is not peer-reviewed

Submitted:

28 September 2024

Posted:

30 September 2024

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Abstract

We will outline the relationship between luminosity distance and cosmological redshifts, demonstrating that it is consistent with a new cosmological model recently proposed by Haug and Tatum [1], which appears to resolve the Hubble tension within the Rh = ct cosmology.

Keywords:

Subject: Physical Sciences - Astronomy and Astrophysics

1. Luminosity Distance Consistent with the Haug and Tatum Cosmological Model

According to the Stefan-Boltzmann law [2,3], the luminosity of a spherical black body is given by:

L = 4 π R^{2} σ T^{4}

(1)

This must also hold true for a black hole, where

R = R_{s}

, meaning the radius is equal to the Schwarzschild radius. Solving for the radius, one gets:

D_{L} = R = \sqrt{\frac{L}{4 π σ T^{4}}}

(2)

Furthermore, the flux density is given by:

F = \frac{L}{4 π R^{2}}

(3)

This means we also have:

D_{L} = R = \sqrt{\frac{L}{4 π F}}

(4)

This is also called the luminosity distance. Moreover, we must have:

\frac{F_{t}}{F_{H}} = \frac{R_{t}^{2}}{R_{H}^{2}}

(5)

where

F_{t}

is the flux density in a growing black hole in the

R_{h} = c t

cosmological model at time t and at radius

R_{t}

, and

F_{H}

is the flux density.

Haug and Tatum have demonstrated that if

T_{t} = T_{0} (1 + z)

, then in their type of

R_{h} = c t

cosmology, one must have:

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

(6)

Solved for

R_{h}

, this gives:

R_{h} = R_{t} {(1 + z)}^{2}

(7)

where

R_{t}

is the comoving transverse luminosity distance. This also means the luminosity distance must follow accordingly. We can rearrange this to:

\begin{matrix} R_{h} & = & R_{t} {(1 + z)}^{2} \\ R_{h} & = & R_{t} {(1 + z)}^{2} \\ R_{h} & = & R_{t} + 2 R_{t} z + R_{t} z^{2} \\ R_{h} - R_{t} & = & 2 R_{h} z + R_{h} z^{2} \\ R_{h} - R_{t} & = & 2 \frac{c}{H_{0}} z + \frac{c}{H_{0}} z^{2} \\ R_{h} - R_{t} & = & \frac{2 c z + c z^{2}}{H_{0}} \end{matrix}

(8)

We define

R_{h} - R_{t} = D

, which is the distance to the object emitting photons toward us, so we have:

D = \frac{2 c z + c z^{2}}{H_{0}}

(9)

This is the same formula that Haug and Tatum have presented, namely:

D = \frac{c}{H_{0}} (1 + \frac{1}{{(1 + z)}^{2}}) = \frac{2 c z + c z^{2}}{H_{0}}

(10)

The well-known Etherington equation gives the relationship between the luminosity distance of standard candles and the angular diameter distance, which is given by:

D_{L} = {(1 + z)}^{2} D_{A}

(11)

Here,

D_{A}

represents the angular-diameter distance. We can see that this bears close similarities to our equation (7). We have already proven that the luminosity distance in our

R_{h} = c t

cosmology is identical to

R_{h}

(at present), and that before the present time it corresponds to

R_{t}

. This implies that we have essentially demonstrated the angular diameter distance is identical to the luminosity distance in the Haug and Tatum cosmology model, which assumes

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

. However, this should naturally be studied carefully over time. It is also worth mentioning that there is a different variation of the Haug and Tatum cosmology, where they propose

z = \frac{R_{h}}{R_{t}} - 1

, which would yield different results.

2. The Mattig Formula

The Mattig [4] formula (see also [5]) is used to calculate the luminosity distance in terms of redshift and is given by:

R = \frac{c}{H_{0}} \frac{q_{0} z + (q_{0} - 1) (\sqrt{1 + 2 q_{0} z} - 1)}{q_{0}^{2} (1 + z)}

(12)

If we set

q_{0} = 1

, we obtain:

\begin{matrix} d = R = \frac{c z}{H_{0} (1 + z)} \end{matrix}

(13)

This is the same distance as given by the Haug and Tatum model when they assume

z = \frac{R_{h}}{R_{t}} - 1

. The first-term Taylor series expansion of this is the well-known distance formula often used in the

Λ

-CDM model:

\begin{matrix} d \approx \frac{c z}{H_{0}} \end{matrix}

(14)

3. Speculative Equivalent Mattig Formula for $z = \sqrt{\frac{R_{h}}{R_{t}}} - 1$ Scaling

We are suggesting a speculative modification to the Mattig [4] formula in an attempt to make it consistent with

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

. This is the only formula in this paper that is not strictly based on derivation, so it should simply be regarded as an initial guess. The idea is that the Mattig formula may require modification to accommodate the redshift scaling

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

. Based on a qualified assumption, we propose the following:

d = \frac{c}{H_{0}} \frac{2 q_{0} z + z^{2} q_{0}^{2} + (q_{0} - 1) (\sqrt{1 + 2 q_{0} z} - 1)}{q_{0}^{2} {(1 + z)}^{2}}

(15)

When

q_{0} = 1

, we obtain:

d = \frac{2 c z + c z^{2}}{H_{0} {(1 + z)}^{2}}

(16)

This is equivalent to the distance Haug and Tatum [1] presented when they assume

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

. Naturally, someone should derive an equivalent Mattig formula from scratch as our speculative suggestion in this section could be wrong as it not is based on derivations from first principles.

4. Conclusions

The Haug-Tatum cosmological model, in which they propose the redshift scaling

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

, leads to a comoving luminosity distance relationship of

R_{h} = R_{t} {(1 + z)}^{2}

, as was already indirectly suggested in their paper. We have pointed out that the angular diameter distance in their model appears to be identical to this. In our

R_{h} = c t

cosmological model, it seems that many types of cosmological distances converge to the same value. We suspect that some of the many different types of cosmological distances in the

Λ

-CDM model may result from an overly complex framework that does not represent the cosmos as accurately.

Conflicts of Interest

The author declare no conflict of interest.

References

E. G. Haug and E. T. Tatum. Solving the Hubble tension using the Union2 supernova database. Preprints.org, 2024. [CrossRef]
L. Boltzmann. Ableitung des stefanschen gesetzes, betreffend die abhängigkeit der wärmestrahlung von der temperatur aus der electromagnetischen lichttheori. Annalen der Physik und Chemie, 22:291, 1879.
Stefan J. Über die beziehung zwischen der wärmestrahlung und der temperatur. Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften in Wien, 79:391, 1879.
W. Mattig. Über den zusammenhang zwischen rotverschiebung und scheinbarer helligkeit. Astronomische Nachrichten, 284:97, 1957. [CrossRef]
J. Terrell. The luminosity distance in Friedmann cosmology. American Journal of Physics, 45:869, November 1977. [CrossRef]

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The CMB Luminosity Distance and Cosmological Redshifts

Abstract

1. Luminosity Distance Consistent with the Haug and Tatum Cosmological Model

2. The Mattig Formula

3. Speculative Equivalent Mattig Formula for z = R h R t − 1 Scaling

4. Conclusions

Conflicts of Interest

References

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3. Speculative Equivalent Mattig Formula for $z = \sqrt{\frac{R_{h}}{R_{t}}} - 1$ Scaling