This paper presents both an alternative point of view and a physical framework to the discussion on the acceleration of space expansion. That is, we will argue why discoveries and observations lead us to approach another scientific mindset without ending up in an accelerating Universe driven by dark energy. The observation of supernovae Ia (SN-Ia) is an important study in cosmology as we can infer the behaviour of the Universe. The dimming of these astronomical objects provides information about the physical mechanism occurring in space, allowing us to discuss the physics and mathematics of the radiation as it travels through the intergalactic medium (IGM). The first and most important inquiries were conducted analyzing a small set of SN-Ia [1] [2] and extended to several others in further inquiries [3] - [6] which interpret the measurements in terms of expanding space and, what counts in this cosmological framework, an accelerating one. The latter has a strong impact, not only on the current physics of the SN-Ia but also on the mindset in academic research to the extent that cosmologists discard other potential alternative explanations to the SN-Ia dimming. Accordingly, an accelerating Universe is indispensable from dark energy [7] - [10] and its previously-introduced cosmological constant [11] - [13] . As we will see, it is possible to predict the distance modulus-redshift curve simply by analyzing all the physical components in space including the effect of dust on the line of sight as well as the multiple interactions between photons and electrons in the IGM. The study of the radiation from SN-Ia involves important parameters and procedural analysis such as a dimming calibration [14] [15] , light curves, and the application of proper cosmological filters [16] - [18] along with the correlation between absolute magnitude and colour [19] . Starting from this decisive analysis and corrections, it is possible to derive equations that will lead eventually to the distance modulus formula for SN-Ia.

The origin of the cosmological constant Λ starts with Albert Einstein’s General Theory of Relativity (GR) [11] . In this theory, gravitational forces are explained in terms of curvature of space. GR was proven right, for instance, by the precession of the orbit of Mercury, as well as by the light “bending” around the Sun from Eddington’s experiment [20] and so the time came to apply GR to the Universe as a whole. The theory said that the Universe was allowed to expand or contract but could not be static. Einstein’s problem was that all the data from observations using the Doppler effect to measure the velocities of stars that could be seen and measured were going very slowly and in all sorts of random directions. There was no observational data to show an expansion or contraction: it looked static. So, for no other reason than to make the theory “fit” the observations, he included a cosmological constant, Λ, which was a repulsive constant to balance the gravitational effects of collapse and make the Universe static if only for a short time. Furthermore, Vesto Slipher [21] measured the redshifts and hence the recession “velocities” of certain nebula and found these to have huge redshifts. He noticed that “the dimmer the nebulae the greater the redshift”. Previously, Henrietta Leavitt [22] calibrated Cepheid variables so that they could be used to determine distances. Thereafter, Edwin P. Hubble, working with the greatest telescope in the world at the time, measured the distance to the nebulae and showed that they were outside the Milky Way. He measured their redshift to calculate their recessional velocities, or what he called “apparent velocities”, and found that the recessional velocities of distant galaxies are directly proportional to their distance from us [23] . That is, observational evidence implied that the universe was expanding as per GR. From then on cosmology focused on finding an accurate value for the Hubble constant H₀ in this physical and mathematical framework [24] [25] . Allan Sandage [26] put forward a method to determine the eventual outcome of the universe by observing the amount of deceleration as a result of the effects of gravity. This was to construct a Hubble diagram of velocity (or redshift) against distance and hence determine the rate of expansion now, H₀, and compare it to what it was in the past, i.e. determine the gradient. Several attempts were made using galaxies found by Cepheid variables but these distances were too short to show any deviation in the gradient from a straight line. But then, a confirmation came that a special type of SN, the SN-Ia, could be used as standard candles, and these were bright enough to be seen far into the distance and hence far into the past. Presently, the Big Bang theory (BB) parametrized in the ΛCDM (Lambda Cold Dark Matter) model with its expanding Universe is a theory on how the Universe began, and as such, the Hubble constant leads us to the age of the Universe. The Hubble constant is a measure of the rate at which the Universe is expanding and so in principle, we can reverse the expansion to find the point in time when “it all began”. The Supernova Cosmology Project [27] and the High-z Supernova Search Team [28] used high-z SN to explore the expansion rate. In effect, they were drawing a Hubble diagram for galaxies nearby and comparing it with that for galaxies in the distance and hence back in time. What they found was a surprise to everyone: SN-Ia are observed much dimmer than expected meaning that when a Hubble diagram is drawn for distant SN-Ia, the gradient, H, is less than one drawn for local galaxies meaning that the Universe is expanding at a greater rate now than it did in the past. According to a current researcher in the field of Cosmology, no single theory explains the difference between the high redshift diagram and that from nearby [29] . This is how the ΛCDM stood and an accelerating Universe driven by dark energy was proposed. However, there is no evidence for acceleration or dark energy other than it makes the theory fit the observed data. Since evidence in support of the ΛCDM predictions is not so strong, an important question arises concerning the reliability of the concordance model and its expanding space. The New Tired Light theory (NTL) predicts the observed SN-Ia values from first principles relying on the physical properties of the electrons and the photons that are interacting in the IGM. We will see in detail each term of the distance modulus when we introduce the calculation methodology of SN-Ia dimming due to this process. To do that, we have first to introduce the most current development of the New Tired Light (NTL) theory.

Standard cosmology is based on the assumption of space expansion, or rather that photons are stretched out during their journey through the vast cosmological distances they travel, leading toward ad hoc cosmological solutions in the EFE (Einstein Field Equations). If we change both our perspective and the assumptions at the beginning of the calculation, we obtain a different result. It turns out that space might be not expanding and that another process can be applied to photons in space. Based on this reasoning, if we not only assume a different process ongoing in space responsible for the cosmological redshift but also provide scientific evidence for the most reliable physics, then the circle closes. The New Tired Light theory (NTL), is a development of the Tired Light (TL) theory of the last century and is based on existing physics. The NTL theory has two basic pillars:

1) Photons are absorbed and re-emitted by the electrons in the IGM which recoil at each event, transferring some of the photon energy to the recoiling electron;

2) Rejection of the old idea of the IGM consisting of a “hot” neutral plasma which is known to be wrong, and replacing it with a neutral plasma where the excess protons are held firm on dust particles whilst the excess electrons can fill the vast spaces in between, arranging themselves on a Wigner crystal lattice. This allows the energy of the incoming photon to be stored as vibrational energy of the electron during the delay between absorption and re-emission whilst allowing recoil to take along the line of sight and thus not blurring the image.

Added to this is an alternative dimming process of the radiation from SN-Ia strictly related to the redshift mechanism and dust interactions and thus making a predicted Distance modulus-redshift relationship that fully agrees with observational data.

NTL is the theory that is the most reliable antagonist of the ΛCDM model and explains the SN-Ia dimming without resorting to an accelerating expansion of space. NTL is a theory with a known mechanism and produces predictions that can be tested. They have good agreement with observations [30] - [33] . NTL is a photon-electron interaction whereby the photon transfers energy to the electrons in the IGM. The energy of the photon is reduced, the frequency is reduced, and the wavelength increased. It has been redshifted. The photons are repeatedly absorbed and re-emitted by the electrons in the IGM which recoil on absorption and re-emission. Some of the energy is transferred to the recoiling electron and it is this that accounts for the redshift.

The distance modulus is a parameter that expresses the distance of any emitting source as measured from Earth, and also in the SN-Ia context, refers to the apparent magnitude determined according to a logarithmic logic based on an approach still in force nowadays. In this sense, the core of this inquiry is to compare the NTL approach with the ΛCDM one for determining the distance modulus μ. SN-Ia are thermonuclear explosions and have almost a constant absolute magnitude M = −19.3 mag as all the white dwarfs, from which they originate, explode when they gain mass from a neighboring star and overcome the Chandrasekhar limit $M>1.4M_{Sun}$. SN-Ia show no H and He signature in the spectra but only a strong ionized Si line which leads to a thermonuclear reaction. SN-Ia are extremely bright objects and allow cosmologists to extend the Hubble diagram to far greater distances than those determined by Cepheids. What was found was that the rate of expansion in the past was less than it is now i.e. the expansion was accelerating and Dark Energy was proposed as the mechanism behind it. However, as we will see in this paper, the same measurements can be explained by another approach in astrophysics and cosmology leading to a totally different interpretation of the Hubble constant with important implications on the belief of an expanding Universe. In support of this statement, it is possible to point out the measurements of the JWST (James Webb Space Telescope) where unexpected galaxies at a higher evolutionary stage have been found at a redshift associated to an epoch where we should see a primordial stage of galaxies [51] . We can also follow the calculation of the angular size which deviates from the ΛCDM expectations [52] .

The perfect fit of the NTL predictions, in a non-expanding space, for the distance modulus of SN-Ia to observational data, is very remarkable and deserves the utmost attention in the scientific community. It is no more a matter of chance or speculation but the physics undergone in this inquiry leads toward a different process not only in the explanation of the dimming of the SN-Ia but also in the understanding of the redshift and its mechanism in the IGM beyond.

For the purpose of explaining the dimming of SN-Ia for increasing redshifts, we introduced the multiple interactions between photons and crystallized electrons in the IGM as well as the interaction between photons and dust grains which dim the incoming radiation interaction after interaction.

The main non-expanding mindset is based on existing physics and it does not contain any adjustable parameters. We demonstrate how by making a comparison between raw distance modulus measurements without dust corrections (from the Pantheon+ analysis) and a raw distance modulus analysis in a non-expanding space (through a NTL process) which includes the dust effect on the radiation, the two curves coincide.

The two approaches undertaken allow us to compare the cosmological models and to draw conclusions in an objective manner. If we did not remove the dust correction from the observational data, the comparison would be falsified. But it is not the case. This means that, in general, a tired light process, and specifically the NTL theory, can explain several main aspects of modern cosmology, from the dimming of SN-Ia to the cosmological redshift. The latter is strictly related to the SN-Ia dimming as shown in this article and all its appendices.

The number electron density in the IGM together with the presence of dust is a prerogative for obtaining the right distance modulus. It implies backwards that the interaction between photons and crystallized electrons in the IGM is very reliable and cosmology should focus on it for explaining the dimming of supernovae Ia. Laboratory tests should be planned for this new mindset and new physics departments should embark on an alternative theory in parallel to standard cosmology. Only in this way, the ΛCDM model can be confirmed or debunked. In absence of antagonist working projects in cosmology, science might commit irreversible mistakes.

The expanding Universe of the Big Bang Theory, currently parametrized in the ΛCDM model, has found the Hubble diagram extended up to large redshifts difficult, if not impossible to explain. There is not one relationship that will explain both near and far distance modulus-redshift relationships and so it has been necessary to resort to new physics such as space acceleration and the accompanying dark energy said to drive the space acceleration. However, no one has any clue as to what the dark energy consists of. It is an attempt to make the theory fit the data.

The ΛCDM model remains just a theory for which several points must be explained: the Hubble diagram associated to the apparent recession velocities of the galaxies, the Hubble Tension giving different Hubble constants depending on the adopted method and direction in the sky and, last but not least, the dark energy driving the space expansion.

The ΛCDM model relies on the prior assumption that the Universe is expanding in order to describe any cosmological phenomenon including the cosmological redshift and the SN-Ia dimming described by the analysis of distance modulus. On the contrary, New Tired Light is based on present day Physics and gives quantitative predictions on redshift and distance. Based on the distance to the supernovae-Ia and knowing the absolute magnitude of these standard candles it can predict a value for the distance modulus of each object.

In NTL, the photons lose energy as they travel, interacting with crystallized electrons in the IGM and so a correction must be added to the distance modulus for this. Since each photon has less energy on arrival than when it set off the apparent magnitude of the supernovae will be more and the supernovae appear farther away. Dust also has an effect on the distance modulus and a predicted adjustment is made. When the final quantitative value of the distance modulus is found by taking the distance modulus as predicted by NTL along with corrections for photon energy loss and dust, the predictions match the data perfectly with no adjustable parameters or any new required Physics. Moreover, in this new framework, the Hubble constant is no more related to an expanding and accelerating space, but rather it has just the meaning of photon energy loss, due to the interaction with Wigner electron crystals, as a function of the electron properties.

For the first time, for the first time overall, we derive exactly how the distance modulus in a NTL process is provided by the sum of four contributions as for Equation (49) and Equation (52): the contribution due to calculated distance in a non-expanding Universe, the contribution due to energy loss in the IGM as photons transfer energy to the recoiling electrons, the contribution related to the unity of measurement due to the conversion of the distances from parsecs to megaparsecs for accounting for the cosmological scales and, last but not least, the contribution caused by dust extinction dimming of the SN-Ia, the latter included in the dust parameter. The sum of these contributions provides exactly the same observational data trend determined by the ΛCDM cosmology by removing the dust corrections.

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

The authors confirm being the sole contributor of this work and have approved it for publication.

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${\mu |}_{P+,\Lambda CDM}=-2.5{\log }_{10}\left(\frac{L_o}{4\pi d_{L,\Lambda CDM}^2}\frac{4\pi d_{L,10pc}^2}{L_o}\right)+K_{\nu },$(A1)

${\mu |}_{P+,\Lambda CDM}=-2.5{\log }_{10}{\left(\frac{d_{L,10pc}}{d_{L,\Lambda CDM}}\right)}^2+K_{\nu },$(A2)

${\mu |}_{P+,\Lambda CDM}=-2\cdot 2.5{\log }_{10}\left(\frac{d_{L,10pc}}{d_{L,\Lambda CDM}}\right)+K_{\nu },$(A3)

${\mu |}_{P+,\Lambda CDM}=-5{\log }_{10}\left(\frac{d_{L,10pc}}{d_{L,\Lambda CDM}}\right)+K_{\nu },$(A4)

Moreover, due to Equation (24), we can write that

${\mu |}_{P+,\Lambda CDM}=-5{\log }_{10}\left(\frac{10}{d_{L,\Lambda CDM}}\right)+K_{\nu },$(A5)

so that

${\mu |}_{P+,\Lambda CDM}=-5{\log }_{10}\left(10\right)-\left[-5{\log }_{10}\left(d_{L,\Lambda CDM}\right)\right]+K_{\nu },$(A6)

${\mu |}_{P+,\Lambda CDM}=-5+5{\log }_{10}\left(d_{L,\Lambda CDM}\right)+K_{\nu }.$(A7)

The observe luminosity is

$L_o=\frac{E_o}{T_0}=\frac{h{\nu }_o}{T_0}=\frac{hc}{T_0{\lambda }_o}.$(B1)

where $E_o$ is the observed photon energy given by the scala product of the Planck constant h multiplied by the observed photon frequency ${\nu }_o$. $T_0$ is the observed time, c is the speed of light and ${\lambda }_o$ is the observed photon wavelength. From the definition of redshift

$z=\frac{{\lambda }_o-{\lambda }_e}{{\lambda }_e}=\frac{{\lambda }_o}{{\lambda }_e}-1,$(B2)

which implies

$\frac{{\lambda }_o}{{\lambda }_e}=1+z=\frac{c{\nu }_o}{c{\nu }_e}=\frac{T_o}{T_e}=\frac{R_o}{R_e}=\frac{1}{R},$(B3)

in which we can apply the scale factor in our “today epoch”

$R_o=1.$(B4)

It is important to stress how the ΛCDM cosmology in Equation (B3) conceives the existence of time dilation as the photon rate at emission and reception may be different. Indeed, in the same equation we are interested in the expression of the following parameter

$T_o=\frac{T_e}{R}.$(B5)

Thus, substituting Equation (B5) into Equation (B1), we obtain

$L_o=\frac{hc}{T_e}\frac{R}{{\lambda }_o}.$(B6)

Moreover, in terms of emitted physical quantities, it is possible to write

$L_e=\frac{E_e}{T_e}=\frac{h{\nu }_e}{T_e}=\frac{hc}{T_e{\lambda }_e},$(B7)

where $E_e$ is the emitted photon energy given by the scala product of the Planck constant h multiplied by the emitted photon frequency ${\nu }_e$. $T_e$ is the emitted time and ${\lambda }_e$ is the emitted photon wavelength. It implies that

$\frac{hc}{T_e}={\lambda }_e{L}_e.$(B8)

Considering the result of Equation (B8), Equation (B6) becomes

$L_o={\lambda }_e{L}_e\frac{R}{{\lambda }_o}=L_e{R}^2=\frac{L_e}{{\left(1+z\right)}^2}.$(B9)

From Equation (29), we can write that

$F_{bol}=\frac{L_{bol}}{4\pi d_{L.\text{Λ}CDM}^2}.$(C1)

This equation recalls Equation (28) and therefore, we can infer that the following parameters are equal, both

$L_{bol}\equiv L_e,$(C2)

and

$d_{L,\text{Λ}CDM}^2\equiv d_{\text{Λ}CDM}^2{\left(1+z\right)}^2.$(C3)

or rather

$d_{L,\Lambda CDM}\equiv d_{\Lambda CDM}\left(1+z\right).$(C4)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left[d_{\Lambda CDM}\left(1+z\right)\right]-5+K_{\nu },$(D1)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(d_{\Lambda CDM}\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu }.$(D2)

As the distance refers to the parsec scale. Equation (D2) can be better expressed as

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(\frac{d_{\Lambda CDM}}{\text{pc}}\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu },$(D3)

and since we are dealing with large cosmological distances, cosmologists express them in megaparsecs instead of parsecs, therefore, we have to convert the overall distance scale as follows

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(\frac{d_{\Lambda CDM}}{\text{pc}}\frac{{10}^6\text{pc}}{\text{Mpc}}\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu },$(D4)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(\frac{d_{\Lambda CDM}\cdot {10}^6}{\text{Mpc}}\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu }.$(D5)

Going back to our previous terminology, removing the unit of measurements in the equations, we can write that

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(d_{\Lambda CDM}\cdot {10}^6\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu },$(D6)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(d_{\Lambda CDM}\right)+5{\log }_{10}\left({10}^6\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu },$(D7)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(d_{\Lambda CDM}\right)+6\cdot 5{\log }_{10}\left(10\right)+5{\log }_{10}\left(1+z\right)-5+K_{\nu },$(D8)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(d_{\Lambda CDM}\right)+30+5{\log }_{10}\left(1+z\right)-5+K_{\nu },$(D9)

${\mu |}_{P+,\Lambda CDM}=5{\log }_{10}\left(d_{\Lambda CDM}\right)+5{\log }_{10}\left(1+z\right)+25+K_{\nu }.$(D10)

Based on generic Equation (23), we can write that

$F_o=\frac{L_o}{4\pi d_{L,NTL}^2},$(E1)

The definition of emitted intensity as power per unit area A allows us to write

$I_o=\frac{L_o}{A},$(E2)

from which

$L_o=A{I}_o.$(E3)

Due to this, the observed flux of Equation (E1) becomes

$F_o=\frac{A{I}_o}{4\pi d_{L,NTL}^2}.$(E4)

Our target is now to determine the expression for the observed intensity of radiation $I_o$ as function of the emitted intensity as we want to consider the reduction of the intensity itself due to the presence of the dust in the IGM. The radiative transfer equation allows us to describe the absorption of the light along its travel as follows

$\frac{\text{d}I}{\text{d}r}=-{\alpha }_{\nu }I,$(E5)

where r represents the distance parameter and ${\alpha }_{\nu }$ the dust absorption coefficient. Reordering the factors, we can obtain that

$\frac{\text{d}I}{I}=-{\alpha }_{\nu }\text{d}r,$(E6)

and integrating the variables between our earthly position and the generic distance of an emitting source x

${\displaystyle {\int }_0^{d_x}\frac{\text{d}I}{I}}={\displaystyle {\int }_0^{d_x}-{\alpha }_{\nu }\text{d}r},$(E7)

we obtain that

$\ln \left[\frac{I\left(d_x\right)}{I\left(0\right)}\right]=-{\alpha }_{\nu }{d}_x,$ (E8)

or namely

$\frac{I\left(d_x\right)}{I\left(0\right)}={\text{e}}^{-{\alpha }_{\nu }{d}_x}.$(E9)

The intensity after the distance travelled $d_x$ coincides with the observed intensity $I_o$ so that $I\left(d_x\right)\equiv I_o$ and that the intensity emitted at the source coincides with the emitted intensity $I_e$ expressed by $I\left(0\right)\equiv I_e$. Moreover, for coherence with the terminology, in this framework, we have to set

$d_x\equiv d_{NTL},$(E10)

where in our inquiry $x\equiv SNIa$ as we are dealing with supernovae Ia. Due to the previous considerations, Equation (E9) can be written as

$I_o=\frac{I_e}{{\text{e}}^{{\alpha }_{\nu }{d}_{NTL}}}.$(E11)

According to this, Equation (E4) becomes

$F_o=\frac{A}{4\pi d_{L,NTL}^2}\frac{I_e}{{\text{e}}^{{\alpha }_{\nu }{d}_{NTL}}}$(E12)

The distance of Equation (E11) in a non-expanding Universe can be obtained from NTL redshift distance shown in the right-hand side term of Equation (1) as follows

$d_{NTL}=\frac{c}{H_{NTL}}\ln \left(1+z\right).$(E13)

Due to that, Equation (E12) evolves in

$F_o=\frac{A}{4\pi d_{L,NTL}^2}\frac{I_e}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}\ln \left(1+z\right)}}.$(E14)

which can be also written as

$F_o=\frac{A{I}_e}{4\pi d_{L,NTL}^2}\frac{1}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}\left(1+z\right)}.$(E15)

In a non-expanding space, the observed luminosity is

$L_o=\frac{E_o}{T_0}=\frac{h{\nu }_o}{T_0}=\frac{hc}{T{\lambda }_o},$(F1)

and there is no time dilation as there is no change of photon rate between emission and reception. It is expressed by

$T_e=T_0=T.$(F2)

From the generic definition of redshift written in Equation (B2), it implies

$\frac{{\lambda }_o}{{\lambda }_e}=1+z.$(F3)

Therefore, by defining and developing the expression of the emitted luminosity as for Equation (B7) but now including the previous considerations in a non-expanding space, we can write that

$L_e=\frac{E_e}{T_e}=\frac{h{\nu }_e}{T_e}=\frac{hc}{T{\lambda }_e}=\frac{hc}{T}\frac{1+z}{{\lambda }_o}=L_o\left(1+z\right),$(F4)

from which the observed luminosity assumes now the mathematical expression

$L_o=\frac{L_e}{1+z}.$(F5)

Therefore, going back to the observed flux of Equation (E1), we can write that

$F_o=\frac{1}{4\pi d_{NTL}^2}\frac{L_e}{1+z}$(F6)

Due to Equation (29), it is possible to write

$F_{bol}=\frac{L_{bol}}{4\pi d_{L,NTL}^2}.$(G1)

We can identify similarity in the form of the formulae between Equation (F1) and Equation (E15). Therefore, we can infer that

$L_{bol}\equiv \frac{L_e}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}\left(1+z\right)},$(G2)

where the emitted luminosity is

$L_e=A{I}_e,$(G3)

as well as

$d_L^2\equiv d_{NTL}^2\left(1+z\right),$(G4)

from which we obtain the important formula in a non-expanding Universe

$d_L\equiv d_{NTL}\sqrt{1+z},$(G5)

Combining the results of Appendix E, Appendix F, Appendix G, respectively through Equations (E15), (F6), (G5), we can write that

$F_o=\frac{L_e}{4\pi d_{NTL}^2\left(1+z\right)}\frac{1}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}\left(1+z\right)},$(H1)

in which, collecting the $1+z$ terms, it evolves into

$F_o=\frac{L_e}{4\pi d_{NTL}^2{\left(1+z\right)}^2}\frac{1}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}}.$(H2)

From Equation (E15)

$F_{10pc}=\frac{A}{4\pi \cdot {10}^2}\frac{I_e}{{\text{e}}^{0\cdot d_{NTL}}},$(I1)

in which, as shown, our luminosity distance is now

$d_{L,NTL}=10\text{pc},$(I2)

and moreover, we assume that in this short interstellar distance (not cosmological) the absorption coefficient is negligible (no absorption due to dust between Earth and this short distance) and therefore

${\alpha }_{10pc}\cong 0.$(I3)

Accordingly, Equation (I1) becomes

$F_{10pc}=\frac{A{I}_e}{4\pi {10}^2}.$(I4)

Due to previous Equation (G3), it yields

$F_{10pc}=\frac{L_e}{4\pi {10}^2}.$(I5)

${{\mu }_{raw}|}_{NTL}=-2.5{\log }_{10}\left[\frac{L_e}{4\pi d_{NTL}^2{\left(1+z\right)}^2}\frac{1}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}}\frac{4\pi {10}^2}{L_e}\right],$(J1)

in which both the emitted luminosity and the solid angle 4π cancel out so that what remains is

${{\mu }_{raw}|}_{NTL}=-2.5{\log }_{10}\left[\frac{{10}^2}{d_{NTL}^2{\left(1+z\right)}^2}\frac{1}{{\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}}\right],$(J2)

${{\mu }_{raw}|}_{NTL}=-2.5{\log }_{10}\left[\frac{{10}^2}{d_{NTL}^2{\left(1+z\right)}^2}\right]-\left[-2.5{\log }_{10}\left({\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}\right)\right],$(J3)

${{\mu }_{raw}|}_{NTL}=-2.5{\log }_{10}\left[\frac{{10}^2}{d_{NTL}^2{\left(1+z\right)}^2}\right]+2.5{\log }_{10}\left({\text{e}}^{{\alpha }_{\nu }\frac{c}{H_{NTL}}}\right),$(J4)

${{\mu }_{raw}|}_{NTL}=-2.5{\log }_{10}\left({10}^2\right)-\left\{-2.5{\log }_{10}\left[d_{NTL}^2{\left(1+z\right)}^2\right]\right\}+2.5{\alpha }_{\nu }\frac{c}{H_{NTL}}{\log }_{10}\left(\text{e}\right),$ (J5)

${{\mu }_{raw}|}_{NTL}=-2\cdot 2.5{\log }_{10}\left(10\right)+2.5{\log }_{10}\left[d_{NTL}^2{\left(1+z\right)}^2\right]+2.5{\alpha }_{\nu }\frac{c}{H_{NTL}}{\log }_{10}\left(\text{e}\right),$ (J6)

${{\mu }_{raw}|}_{NTL}=-5{\log }_{10}\left(10\right)+2.5{\log }_{10}\left(d_{NTL}^2\right)+2.5{\log }_{10}{\left(1+z\right)}^2+2.5{\alpha }_{\nu }\frac{c}{H_{NTL}}\cdot 0.434,$(J7)

${{\mu }_{raw}|}_{NTL}=-5+2\cdot 2.5{\log }_{10}\left(d_{NTL}\right)+2\cdot 2.5{\log }_{10}\left(1+z\right)+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}},$(J8)

${{\mu }_{raw}|}_{NTL}=-5+5{\log }_{10}\left(d_{NTL}\right)+5{\log }_{10}\left(1+z\right)+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}}.$(J9)

Since we are dealing with high cosmological distances, we can express the distance in megaparsecs instead of parsecs, therefore, we have to convert the distance scale as

${{\mu }_{raw}|}_{NTL}=-5+5{\log }_{10}\left(\frac{d_{NTL}}{\text{pc}}\frac{{10}^6\text{pc}}{\text{Mpc}}\right)+5{\log }_{10}\left(1+z\right)+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}},$(J10)

which leads to

${{\mu }_{raw}|}_{NTL}=-5+5{\log }_{10}\left(d_{NTL}\right)+5{\log }_{10}\left({10}^6\right)+5{\log }_{10}\left(1+z\right)+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}},$ (J11)

${{\mu }_{raw}|}_{NTL}=-5+5{\log }_{10}\left(d_{NTL}\right)+6\cdot 5{\log }_{10}\left(10\right)+5{\log }_{10}\left(1+z\right)+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}},$ (J12)

${{\mu }_{raw}|}_{NTL}=-5+5{\log }_{10}\left(d_{NTL}\right)+30+5{\log }_{10}\left(1+z\right)+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}},$(J13)

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left(d_{NTL}\right)+5{\log }_{10}\left(1+z\right)+25+1.086{\alpha }_{\nu }\frac{c}{H_{NTL}}.$ (J14)

Due to Equation (46), it yields

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left(d_{NTL}\right)+5{\log }_{10}\left(1+z\right)+25+1.086{\alpha }_{\nu }\cdot 1.463\times {10}^{26},$(J15)

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left(d_{NTL}\right)+5{\log }_{10}\left(1+z\right)+25+1.588\times {10}^{26}{\alpha }_{\nu }.$(J16)

Substituting Equation (E13) into Equation (47), we can write that

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left[\frac{\frac{c}{H_{NTL}}\ln \left(1+z\right)}{3.086\times {10}^{22}}\right]+5{\log }_{10}\left(1+z\right)+25+1.588\times {10}^{26}{\alpha }_{\nu },$(K1)

where the factor 3.086E+22 is due to the conversion $1\text{m}=3.086\times {10}^{22}\text{Mpc}$. It leads to

$\begin{array}{c}{{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left(\frac{\frac{c}{H_{NTL}}}{3.086\times {10}^{22}}\right)+5{\log }_{10}\left[\ln \left(1+z\right)\right]+5{\log }_{10}\left(1+z\right)\\ +25+1.588\times {10}^{26}{\alpha }_{\nu },\end{array}$(K2)

$\begin{array}{c}{{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left(1.463\times {10}^{26}\right)-5{\log }_{10}\left(3.086\times {10}^{22}\right)+5{\log }_{10}\left[\ln \left(1+z\right)\right]\\ +5{\log }_{10}\left(1+z\right)+25+1.588\times {10}^{26}{\alpha }_{\nu }.\end{array}$(K3)

Our target is now to express the formula just as a function of the redshift z as follows

$\begin{array}{c}{{\mu }_{raw}|}_{NTL}=130.826-112.447+5{\log }_{10}\left[\ln \left(1+z\right)\right]+5{\log }_{10}\left(1+z\right)\\ +25+1.588\times {10}^{26}{\alpha }_{\nu },\end{array}$(K4)

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left[\ln \left(1+z\right)\right]+5{\log }_{10}\left(1+z\right)+43.379+1.588\times {10}^{26}{\alpha }_{\nu },$(K5)

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left[\left(1+z\right)\cdot \ln \left(1+z\right)\right]+43.379+1.588\times {10}^{26}{\alpha }_{\nu },$(K6)

${{\mu }_{raw}|}_{NTL}=5{\log }_{10}\left[\ln {\left(1+z\right)}^{\left(1+z\right)}\right]+43.379+1.588\times {10}^{26}{\alpha }_{\nu }.$(K7)

We can determine the value of ${\alpha }_{\nu }$ analytically as follows: the dust absorption coefficient can be expressed by the scalar product of the dust number density with the dust cross section

${\alpha }_{\nu }=n_{dust}{\sigma }_{dust},$(L1)

${\alpha }_{\nu }=\frac{N_{dust}}{\bar{V}}\cdot \frac{A_{dust}}{N_{dust}}=\frac{A_{dust}}{\bar{V}}=\frac{\frac{\pi D_{dust}^2}{4}}{\bar{V}},$(L2)

where we defined, respectively, the number density of dust per each cubic meter and the geometrical cross-section per dust grain. $\bar{V}$ is the average volume of space. Dimensional analysis

$\left[\frac{\text{1}}{\text{m}}\right]=\left[\frac{\text{dust}}{{\text{m}}^{\text{3}}}\cdot \frac{{\text{m}}^{\text{2}}}{\text{dust}}\right].$

From important scientific research [54] , the diameter of the cosmic dust is

$10<D_{dust}<1000\mu \text{m}.$(L3)

On earth, they have found in Antarctica the following dust grains

$12<D_{dust}^{Earth}<700\mu \text{m}.$(L4)

Moreover, the estimated dust grain density from averaging all its single material components [55] is

${\bar{\rho }}_{dust}\cong 3200\frac{\text{kg}}{{\text{m}}^{\text{3}}}.$(L5)

The average volume of space $\bar{V}$ can be determined as follows in the next rows.

Let’s consider a spherical volume, with diameter L, in which we place two half grains at the extremities as shown in Figure L1 . The sum of the two masses in the volume has to be equal to the average mass in the estimated volume:

${\bar{m}}_V\cong \frac{1}{2}{m}_{dust}+\frac{1}{2}{m}_{dust},$(L6)

${\bar{m}}_V\cong m_{dust},$(L7)

${\bar{\rho }}_{m}V\cong {\bar{\rho }}_{dust}{V}_{dust},$(L8)

${\bar{\rho }}_m\frac{4}{3}\pi L^3\cong {\bar{\rho }}_{dust}\frac{4}{3}\pi D_{dust}^3,$(L9)

${\bar{\rho }}_m{L}^3\cong {\bar{\rho }}_{dust}{D}_{dust}^3,$(L10)

$L^3=\bar{V}\cong \frac{{\bar{\rho }}_{dust}}{{\bar{\rho }}_m}{D}_{dust}^3.$(L11)

Therefore, due to Equation (L2)

${\alpha }_{\nu }=\frac{\pi D_{dust}^2}{4}\frac{{\bar{\rho }}_m}{{\bar{\rho }}_{dust}}\frac{1}{D_{dust}^3},$(L12)

${\alpha }_{\nu }=\frac{\pi }{4}\frac{{\bar{\rho }}_m}{{\bar{\rho }}_{dust}}\frac{1}{D_{dust}}.$(L13)

The value of the average baryonic mass density of matter in the Universe is

${\bar{\rho }}_{m,b}=4.201\times {10}^{-28}\frac{\text{kg}}{{\text{m}}^{\text{3}}}.$(L14)

Indeed, dust belongs only to baryonic matter. Therefore, we can calculate that for ${\bar{\rho }}_{m,b}$ and for an average dust grain diameter

${\bar{D}}_{dust}=500\mu \text{m},$(L15)

we obtain that

${\alpha }_{\nu ,b}=\frac{\pi }{4}\frac{{\bar{\rho }}_{m,b}}{{\bar{\rho }}_{dust}}\frac{1}{{\bar{D}}_{dust}}=2.062\times {10}^{-28}\frac{\text{1}}{\text{m}}.$(L16)

${{\mu }_{raw}|}_{NTL,b}=5{\log }_{10}\left[\ln {\left(1+z\right)}^{\left(1+z\right)}\right]+43.379+1.588\times {10}^{26}{\alpha }_{\nu ,b},$(M1)

${{\mu }_{raw}|}_{NTL,b}=5{\log }_{10}\left[\ln {\left(1+z\right)}^{\left(1+z\right)}\right]+43.379+1.588\times {10}^{26}\times 2.062\times {10}^{-28},$(M2)

${{\mu }_{raw}|}_{NTL,b}=5{\log }_{10}\left[\ln {\left(1+z\right)}^{\left(1+z\right)}\right]+43.379+0.033,$(M3)

${{\mu }_{raw}|}_{NTL,b}=5{\log }_{10}\left[\ln {\left(1+z\right)}^{\left(1+z\right)}\right]+43.412,$(M4)

${{\mu }_{raw}|}_{NTL,b}\cong 5{\log }_{10}\left[\ln {\left(1+z\right)}^{\left(1+z\right)}\right]+\mathrm{43.4.}$ (M5)