Classical Cosmology I. Anomalous Redshift for Galaxies in NED-D

Lorenzo Zaninetti

Physics Department, University of Turin, Turin, Italy.

DOI: 10.4236/jhepgc.2024.101002                

Abstract

Three mechanisms for an alternative to the Doppler effect as an explanation for the redshift are reviewed. A fourth mechanism is the attenuation of the light as given by the Beer-Lambert law. The average value of the Hubble constant is therefore derived by processing the galaxies of the NED-D catalog in which the distances are independent of the redshift. The observed anisotropy of the Hubble constant is reproduced by adopting a rim model, a chord model, and both 2D and 3D Voronoi diagrams.

Keywords

Galaxy Groups, Clusters, Superclusters, Large Scale Structure of the Universe Cosmology

1. Introduction

In the recent literature, the Hubble constant has been oscillating between a low value, as determined by the Planck collaboration [1] , $H_0=\left(67.4\pm 0.5\right)\text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}$ , and an high value, $H_0=\left(74.03\pm 1.42\right)\text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}$ , as measured on 70 long-period Cepheids in the Large Magellanic Cloud (LMC) [2] . This difference is referred to as the Hubble constant tension and the weighted mean is $H_0=\left(70.53\pm 4.94\right)\text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}$ . The source of the Hubble constant tension is a current field of research. We report some approaches among others. A model-independent principal component analysis (PCA) was carried out using an eigenmode decomposition of the varying constant during recombination [3] , a color correction in the calibration sample for observations of Type Ia supernovae (SNe) and Cepheids [4] , and a geometric mismatch in the comparison of the measurements which is equal to the temporal diameter of the surface of last scattering [5] . Another interesting topic is the anisotropy in the Hubble constant, which was predicted starting in 1975 [6] [7] [8] . A detailed analysis of the anisotropy in the Hubble constant has been done by a comparison of the constant in the North and South Galactic hemispheres [9] and by contour maps of the constant in Galactic coordinates [10] . These analyses introduce some doubts on the current status of the standard cosmology, see [11] for the status of the tired light cosmology in the period 1929-1939. Therefore the following questions can be posed:

1) Is the Hubble constant an universal constant or is it a measure of the absorption of light in the intergalactic medium?

2) What are the known physical mechanisms which produce the attenuation of the light?

3) Can we build some models for the anisotropy of the anomalous redshift?

In order to answer these questions, Section 2 reviews the statistics of a line crossing the origin, analyses the attenuation of the light as given by the Beer-Lambert law, reviews three models for the anomalous redshift, and introduces the NED catalog for galaxies. Section 3 reviews the luminosity Schechter function for galaxies and the photometric maximum in the number of galaxies versus redshift with an application to the NED catalog. In order to explain the anisotropy of the Hubble constant, three simple models are introduced in Section 4 and a fourth model uses the Voronoi diagrams, see Section 5.

2. Preliminaries

This section reviews the least square fit of a line crossing the origin, the attenuation of light according to the Beer-Lambert law, some physical mechanisms for the anomalous redshift, and the NED catalog for galaxies.

2.1. A Line Crossing the Origin

A sequence of data $x_i\mathrm{,}{y}_i$ and uncertainty ${\sigma }_i$ in the ith variable in the interval $\left[\mathrm{1,}n\right]$ which presents a linear behaviour is processed assuming a straight line of the form

$y\left(x\mathrm{<mo>\; ;\; </mo>}a\mathrm{,}b\right)=a+bx\mathrm{,}$ (1)

where a and b are deduced by minimizing ${\chi }^2$ defined as

${\chi }^2\left(a,b\right)={\displaystyle \underset{i}{\overset{n}{\sum }}}{\left(\frac{y_i-a-b{x}_i}{{\sigma }_i}\right)}^2,$ (2)

see Section 15.2 in [12] . In some cases, we are interested in a model without intercept, i.e. $a=0$ :

$y\left(x\mathrm{<mo>\; ;\; </mo>}b\right)=bx\mathrm{.}$ (3)

In the absence of uncertainties, the values of b and its uncertainty, ${\sigma }_b$ , are found by minimizing ${\chi }^2$ ,

${\chi }^2\left(b\right)={\displaystyle \underset{i}{\overset{n}{\sum }}}{\left(y_i-b{x}_i\right)}^2,$ (4)

and are given by

$b=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i{y}_i}}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i^2}},$ (5a)

${\sigma }_b=\sqrt{\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(y_i-b{x}_i\right)}^2}}{\left(n-1\right)\left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i^2}\right)}}.$ (5b)

In the presence of uncertainties, the parameter b and its standard error ${\sigma }_b$ can be determined by minimizing

${\chi }^2\left(b\right)={\displaystyle \underset{i}{\overset{n}{\sum }}}{\left(\frac{y_i-b{x}_i}{{\sigma }_i}\right)}^2,$ (6)

yielding the values

$b=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{x_i{y}_i}{{\sigma }_i^2}}}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{x_i^2}{{\sigma }_i^2}}},$ (7a)

${\sigma }_b=\sqrt{\frac{1}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{x_i^2}{{\sigma }_i^2}}}}.$ (7b)

A nonlinear regression of the type

$\ln \left(1+z\right)=bx,$ (8)

can be made linear with the transformation

$z^{\prime }=\ln \left(1+z\right)\mathrm{,}$ (9)

which now produces a linear regression

$z^{\prime }=bx\mathrm{.}$ (10)

2.2. The Attenuation of Light

We assume that the frequency $\nu$ of a photon which travels through intergalactic space decreases according to the following ODE, called the Beer-Lambert law, after [11] [13] [14] ,

$\frac{\text{d}}{\text{d}x}\nu \left(x\right)=-k{n}_e\nu \left(x\right)\mathrm{,}$ (11)

where $n_e$ is the number density of matter in 1/m³ and k the attenuation coefficient in m². This ODE is solved while assuming the initial condition $\nu \left(0\right)=n{u}_0$

$\nu \left(x\right)={\nu }_0{\text{e}}^{-k{n}_{e}x}\mathrm{.}$ (12)

We now define the redshift as a function of the wavelength $\lambda$ and the initial wavelength ${\lambda }_0$ by

$z=\frac{\lambda -{\lambda }_0}{{\lambda }_0}\mathrm{.}$ (13)

Making use of

$\nu \lambda =c\mathrm{,}$ (14)

where c is the speed of light, we obtain

$z={\text{e}}^{k{n}_{e}x}-\mathrm{1,}$ (15)

or

$\ln \left(z+1\right)=k{n}_{e}x\mathrm{.}$ (16)

Taking a Taylor expansion around $z=0$ gives

$z=k{n}_{e}x\mathrm{,}$ (17)

and the expansion velocity, v, in the radial Doppler framework can be obtained by multiplying both sides by c,

$v=k{n}_{e}xc\mathrm{.}$ (18)

This equation can be equated to Hubble’s Law [15]

$v=H_{0}x\frac{\text{km}}{\text{Mpc}}\mathrm{,}$ (19)

which gives

$H_0=k{n}_{e}c\mathrm{.}$ (20)

This equation allows determining the product of the attenuation coefficient and the number density

$k{n}_e=7.567\times {10}^{-27}\left(\frac{H_0}{70}\right)\frac{\text{1}}{\text{m}}\mathrm{.}$ (21)

2.3. The Anomalous Redshift

The first case to be analysed is a plasma effect. In the framework of photons which penetrate a hot, sparse electron plasma, it is possible to derive a formula for the redshift:

$\text{ln}\left(1+z\right)=3.326\times {10}^{-25}{\displaystyle \underset{0}{\overset{R}{\int }}{n}_e\text{d}x}+\frac{{\gamma }_i-{\gamma }_0}{\xi \omega }\mathrm{,}$ (22)

where ${\gamma }_i$ is the initial photon width, ${\gamma }_0$ is the final photon width, $\omega$ is the photon frequency, $\xi$ is an adjustment factor and $n_e$ is the number of plasma electrons per cm³, see Equation (20) of [16] or Equation (29) of [17] . The second term of Equation (22) is a small correction to the first term and therefore we have a simple expression for the plasma redshift (in the CGS system),

$\text{ln}\left(1+z\right)=3.326\times {10}^{-25}{\displaystyle \underset{0}{\overset{R}{\int }}{n}_e\text{d}x}\mathrm{.}$ (23)

The above logarithmic dependence is similar to that of the attenuation of light as represented by Formula (16). The average electron plasma number density, $\langle n_e\rangle$ , as a function of the Hubble constant is

$\langle n_e\rangle =2.27492\times {10}^{-4}\left(\frac{H_0}{70}\right)\frac{1}{{\text{cm}}^3}\mathrm{,}$ (24)

and the Hubble constant

$H_0=307701.8\langle n_e\rangle \text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}\mathrm{,}$ (25)

where $\langle n_e\rangle$ is expressed in 1/cm³.

The second case to be analysed is the New Tired Light (NTL) hypothesis which has equations

$H_0=\frac{2n_{e}h{r}_e}{m_e}\mathrm{,}$ (26a)

$z=\exp \frac{H_0\ast d}{c}-1$ (26b)

where $n_e$ is the number density of matter, h is Planck’s constant, $r_e$ is the classical radius of the electron, $m_e$ is the mass of the electron, c is the speed of light, d is the distance and H₀ is the Hubble constant; see Equations (13) and (14) in [18] . Also in this case we have a logarithmic dependence similar to that of the attenuation of light as represented by Formula (16). The average number density of matter in NTL is

$\langle n_e\rangle =0.55337\left(\frac{H_0}{70}\right)\frac{1}{{\text{m}}^3}\mathrm{,}$ (27)

and the Hubble constant

$H_0=126.49\langle n_e\rangle \text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}\mathrm{,}$ (28)

with $\langle n_e\rangle$ expressed in 1/m³. The third case is represented by the interaction of a low density electromagnetic wave with an electron, see equation at page 3 in [19] , which in the SI system produces the following redshift

$z=w{\displaystyle {\int }_0^D{n}_e\text{d}s}\mathrm{,}$ (29)

with

$w=\frac{3q_e^4{\mu }_0}{512\pi m_e^2}=2.33\times {10}^{-30}{\text{m}}^2\mathrm{,}$ (30)

where $q_e$ is the charge of the electron, ${\mu }_0$ is the magnetic permeability of the vacuum, and $m_e$ is the mass of the electron. The average electron number density in the interaction of a low density electromagnetic wave with an electron is

$\langle n_e\rangle =\frac{H_0}{cw}\mathrm{,}$ (31)

or, in numbers,

$\langle n_e\rangle =3235.5\left(\frac{H_0}{70}\right)\frac{1}{{\text{m}}^3}\mathrm{,}$ (32)

and the Hubble constant is

$H_0=0.021634\langle n_e\rangle \text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}\mathrm{,}$ (33)

with $\langle n_e\rangle$ expressed in 1/m³.

2.4. The NED-D Catalog

The NASA/IPAC Extragalactic Database of Distances (NED-D) contains an estimate of the redshift versus an independent distance for 11,699 galaxies, see Figure 1. Figure 2 reports the redshift against distance for the above catalog and the connecting line with uncertainty which allows the determination of H₀, $\langle n_e\rangle$ and $k{n}_e$ . The cosmological results are shown in Table 1 for a straight line with and without uncertainty.

Figure 1. Mercator projection in galactic coordinates of the NED’s galaxies.

Figure 2. Redshift versus distance for galaxies, green points, and fitted red line with uncertainty.

Table 1. Numerical values of the cosmological parameters for the NED-D catalog for different models in the data analysis.

The above nonlinear regression can also be processed with the Levenberg-Marquardt method [12] , which modifies H₀ with uncertainties from 67.76 km∙s⁻¹∙Mpc⁻¹ to 67.80 km∙s⁻¹∙Mpc⁻¹. In the NED’s data, see Figure 2, we note the presence of outliers. The main problem with outlier removal is that it may also introduce a bias in the estimation of H₀ and therefore we processed all the data provided by the catalog.

3. The Photometric Maximum

This section reviews the Schechter’s luminosity function for galaxies and the photometric maximum in the number of galaxies versus redshift.

3.1. The Schechter Function

The Schechter function, introduced by [20] , provides a useful fit for the LF of galaxies

$\Phi \left(L\right)\text{d}L=\left(\frac{{\Phi }^{\mathrm{<mo>\; *\; </mo>}}}{L^{\mathrm{<mo>\; *\; </mo>}}}\right){\left(\frac{L}{L^{\mathrm{<mo>\; *\; </mo>}}}\right)}^{\alpha }\exp \left(-\frac{L}{L^{\mathrm{<mo>\; *\; </mo>}}}\right)\text{d}L\mathrm{,}$ (34)

where $\alpha$ sets the slope for low values of L, $L^{\mathrm{<mo>\; *\; </mo>}}$ is the characteristic luminosity, and ${\Phi }^{\mathrm{<mo>\; *\; </mo>}}$ is the normalization. The equivalent distribution in absolute magnitude is

$\Phi \left(M\right)\text{d}M=0.921{\Phi }^{\mathrm{<mo>\; *\; </mo>}}{10}^{0.4\left(\alpha +1\right)\left(M^{\mathrm{<mo>\; *\; </mo>}}-M\right)}\exp \left(-{10}^{0.4\left(M^{\mathrm{<mo>\; *\; </mo>}}-M\right)}\right)\text{d}M\mathrm{,}$ (35)

where $M^{\mathrm{<mo>\; *\; </mo>}}$ is the characteristic magnitude as determined from the data. We introduce $H_0=100h\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}$ , with $h=1$ when h is not specified and therefore the scaling with h is $M^{\mathrm{<mo>\; *\; </mo>}}-5{\log }_{10}h$ and ${\Phi }^{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mi>\; </mi>}{h}^3\left[{\text{Mpc}}^{-3}\right]$ .

3.2. The Behaviour of the Schechter Function

Under the hypothesis of spherical symmetry, we use the symbol r for the distance d. The radius r in the pseudo-Euclidean cosmology has the following dependence:

$r=\frac{zc}{H_0}\mathrm{.}$ (36)

The flux of radiation, f, is introduced as

$f=\frac{L}{4\pi r^2}\mathrm{,}$ (37)

and is here expressed in $\frac{L_{\odot }}{{\text{Mpc}}^2}$ . The joint distribution in distance, r, and flux, f, for the number of galaxies is

$\frac{\text{d}N}{\text{d}\Omega \text{d}r\text{d}f}=\frac{1}{4\pi }{\displaystyle {\int }_0^{\infty }}4\pi r^2\Phi \left(\frac{L}{L^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(f-\frac{L}{4\pi r^2}\right)\text{d}r\mathrm{,}$ (38)

were the factor ( $\frac{1}{4\pi }$ ) converts the number density into the density for a solid angle and the Dirac delta function selects the required flux. In the case of the Schechter LF, we have

$\frac{\text{d}N}{\text{d}\Omega \text{d}z\text{d}f}=4\pi {\left(\frac{c}{H_0}\right)}^5{z}^4\Phi \left(\frac{z^2}{z_{crit}^2}\right)\mathrm{,}$ (39)

where $\text{d}\Omega$ , $\text{d}z$ , and $\text{d}f$ represent the differential of the solid angle, the red shift, and the flux, respectively, and

$z_{crit}^2=\frac{H_0^2{L}^{\mathrm{<mo>\; *\; </mo>}}}{4\pi f{c}^2}\mathrm{,}$ (40)

where c denotes the speed of light; CODATA recommends $c=299792.458\text{km}\cdot {\text{s}}^{-1}$ . The mean red shift of galaxies with a flux f, see Formula (1.105) in [21] , is

$\langle z\left(f\right)\rangle =z_{crit}\frac{\Gamma \left(3+\alpha \right)}{\Gamma \left(5/2+\alpha \right)}\mathrm{.}$ (41)

The number of galaxies in z and f as given by Formula (39) has a maximum at

$z_{\max }\left(f\right)=z_{crit}\sqrt{\alpha +2}.$ (42)

The value of $z_{\max }$ can be determined from the histogram of the observed number of galaxies expressed as a function of z. These two formulae, for $\langle z\rangle$ and $z_{\max }$ , have been derived for a fixed value of f, which appears in the definition of $z_{crit}$ ; we now analyse the case of all the galaxies with $f_{\min }<f<f_{\max }$ . The average value of the number of redshifts with flux lying between $f_{\min }$ and $f_{\max }$ is

$$\langle z\left(f_{\min },f_{\max }\right)\rangle =-\frac{0.2115{\text{e}}^{0.4605M_{bol,\odot }-0.4605M^{*}}{H}_0\Gamma \left(\alpha +3\right)\left(f_{\max }^2-f_{\min }^2\right)}{\Gamma \left(\alpha +2.5\right)\left(\sqrt{f_{\max }}{f}_{\min }^2-\sqrt{f_{\min }}{f}_{\max }^2\right)c}\mathrm{,}$$ (43)

where $M_{bol,\odot }$ is the bolometric luminosity of reference, which is different for each catalog. In this formula, we have used

$L^{\mathrm{<mo>\; *\; </mo>}}={10}^{0.4M_{bol,\odot }-0.4M^{*}}\mathrm{.}$ (44)

The number of galaxies with flux lying between $f_{\min }$ and $f_{\max }$ is

$$\begin{array}{c}N\left(z;f_{\min },f_{\max }\right)=\frac{1}{\alpha +1}(F^{*}{z}^{2+\alpha }{H}_0^{-3-\alpha }{c}^{3+\alpha }{\pi }^{\frac{\alpha }{2}}{\left({10}^{0.4M_{bol,\odot }-0.4M^{*}}\right)}^{-\frac{\alpha }{2}}{2}^{\alpha }\begin{array}{c}\\ \\ \end{array}\\ \times (f_{\max }^{\frac{\alpha }{2}}{\text{e}}^{-\frac{2\pi f_{\max }{z}^2{c}^2}{H_0^2{10}^{0.4M_{bol,\odot }-0.4M^{*}}}}{M}_{\frac{\alpha }{2}\mathrm{,}\frac{\alpha }{2}+\frac{1}{2}}\left(\frac{4\pi f_{\max }{z}^2{c}^2}{H_0^2{10}^{0.4M_{bol\odot }-0.4M^{*}}}\right)\\ -f_{\min }^{\frac{\alpha }{2}}{\text{e}}^{-\frac{2\pi f_{\min }{z}^2{c}^2}{H_0^2{10}^{0.4M_{bol,\odot }-0.4M^{*}}}}{M}_{\frac{\alpha }{2}\mathrm{,}\frac{\alpha }{2}+\frac{1}{2}}\left(\frac{4\pi f_{\min }{z}^2{c}^2}{H_0^2{10}^{0.4M_{bol,\odot }-0.4M^{*}}}\right)))\mathrm{,}\end{array}$$ (45)

where $M_{\kappa \mathrm{,}\mu }\left(z\right)$ is the Whittaker function, see [22] . Table 2 reports the typical parameters of different catalogs.

Figure 3 treats all the observed galaxies of the NED and the theoretical curve with data as in Table 2. Table 3 presents a comparison between theory and observation of $\langle z\left(f_{\min },f_{\max }\right)\rangle$ and $\langle z\left(f_{\min },f_{\max }\right)\rangle$ .

Figure 3. All the galaxies of NED, organized in frequencies versus heliocentric redshift, (empty circles); the error bar is given by the square root of the frequency. The maximum frequency of observed galaxies is at $z=0.0148$ . The full line is the theoretical curve generated by N as given by the application of the Schechter LF which is Equation (45).

Table 2. Catalog, reference and numerical values of $\alpha$ , $M^{\mathrm{<mo>\; *\; </mo>}}$ , $M_{bol,\odot }$ , $H_0$ , $f_{\min }$ and $f_{\max }$ .

Table 3. Theoretical and observed $\langle z\left(f_{\min },f_{\max }\right)\rangle$ and $z_{\max }\left(f_{\min },f_{\max }\right)$ for the NED catalog.

4. Simple Models for the Spatial Anisotropy

The mechanisms for the redshift here considered require the evaluation of the average density of electrons along the line of sight, which can be computed by

$\langle n_e\rangle =\frac{{\displaystyle {\int }_0^D}{n}_e\left(x\right)\text{d}x}{D}\mathrm{,}$ (46)

where D is the considered distance. The discrete version is

$\langle n_e\rangle =\frac{{\displaystyle {\sum }_i{n}_{e,i}{l}_i}}{{\displaystyle {\sum }_i{l}_i}},$ (47)

where $n_{e\mathrm{,}i}$ and $l_i$ denote the number density of electrons and length for different zones. This will be called the fundamental integral or fundamental sum.

4.1. Anisotropy in the Catalogs

The first catalog to be analysed is the HST Key Project, which explored the value of H₀ in all directions of the sky; the data are reported in Table 1 and Table 2 of [10] . Table 4 reports the average, the standard deviation, the weighted mean, the error of the weighted mean, the minimum, $H_{0,\min }$ and the maximum $H_{0,\max }$ computed as in [26] [27] . Asymmetries for H₀ are also found in the CMB dipole direction considering the angular distribution of type Ia supernovae [28] and in quasars and gamma-ray bursts [29] . We then analysed the NED catalog selecting 64 sectors with equal solid angle, and determined H₀ through a straight line without uncertainties, see Table 4.

4.2. The Rim Model

The radius of the cosmic voids as given by the Sloan Digital Sky Survey (SDSS) R7, has been modeled by spheres which have average radius

$\overline{R_V}=\frac{18.23}{h}\text{Mpc}\mathrm{,}$ (48)

see [30] . One model for the asymmetric Hubble constant assumes that the galaxies are situated on the thick surface of spheres having thickness t much smaller than the average radius. We suggest that the density of the free electrons follows the previous trend, and therefore

$\begin{array}{l}{n}_e\left(r\right)=n_0\text{if}0\le r<a\\ n_e\left(r\right)=n_1\text{if}a\le r<b\\ n_e\left(r\right)=n_0\text{if}r\ge b\end{array}$ (49)

Table 4. The Hubble constant of the HST key project.

where r is the distance from the origin of a Cartesian 3D reference system. This means that the electron density increases from $n_0$ at the centre of the sphere to $n_1$ at $r=a$ , remains constant up to $r=b$ (the radius of the void), and then decreases again to $n_0$ outside the sphere. The fundamental integral of the density as given by Equation (46) can be done in the x-direction over the length $D=2b$ and split into two parts,

Part I, three pieces if $0\le y<a$

Part II, two pieces if $a\le y<b$

Corresponding to the lines of sight $s_1$ and $s_2$ in Figure 4. The result of the integral of the average electron density, see (46), is

$\begin{array}{l}\langle n_e\rangle =\frac{n_{0}b-n_0\sqrt{b^2-y^2}+n_1\sqrt{b^2-y^2}-n_1\sqrt{a^2-y^2}+n_0\sqrt{a^2-y^2}}{b}\text{if}0\le y<a\\ \langle n_e\rangle =\frac{n_{0}b-n_0\sqrt{b^2-y^2}+n_1\sqrt{b^2-y^2}}{b}\text{if}a\le y<b\end{array}$ (50)

The Hubble constant can be obtained from the average electron density by multiplying by a numerical factor, see Equation (28) for the photo-absorption process. Figure 5 shows the behaviour of Hubble’s constant as a function of the observer’s position and Table 5, the statistical parameters along the line of sight for this first model.

4.3. The Cord

The starting point is a probability density function (PDF) for the diameter of the astrophysical region, $F\left(x\right)$ , where x indicates the diameter. The probability, $G\left(x\right)\text{d}x$ , that a sphere having diameter between x and $x+\text{d}x$ intersects a random line is proportional to their cross section

$G\left(x\right)\text{d}x=\frac{\frac{\pi }{4}{x}^{2}F\left(x\right)\text{d}x}{{\displaystyle {\int }_0^{\infty }}\frac{\pi }{4}{x}^{2}F\left(x\right)\text{d}x}=\frac{x^{2}F\left(x\right)\text{d}x}{\langle x^2\rangle }.$ (51)

Given a line which intersects a sphere of diameter x, the probability that the distance from the centre lies in the range $r\mathrm{,}r+\text{d}r$ is

Figure 4. The two circles (sections of spheres) which include the region with an enhancement in electron density are represented by full lines. The observer is situated along the x direction, and two lines of sight are indicated.

Figure 5. Value of H₀ in the rim model for the photo-absorption process as function of the position; $b=18$ Mpc, $a=17$ Mpc, $n_0=4.85\times {10}^{-7}$ particles cm⁻³, and $n_1=8.72\times {10}^{-7}$ particles cm⁻³.

Table 5. The Hubble constant for the photo-absorption process in the rim model.

$p\left(r\right)=\frac{2\pi r\text{d}r}{\frac{\pi }{4}{x}^2}\mathrm{,}$ (52)

and the length of the chord is

$l=\sqrt{x^2-4r^2}\mathrm{,}$ (53)

see Figure 6.

The probability that spheres in the range $\left(x\mathrm{,}x+\text{d}x\right)$ are intersected to produce chords with lengths in the range $\left(l\mathrm{,}l+\text{d}l\right)$ is

$G\left(x\right)\text{d}x\frac{2l\text{d}l}{x^2}=\frac{2l\text{d}l}{\langle x^2\rangle }F\left(x\right)\text{d}x.$ (54)

The probability of having a chord with length between $\left(l\mathrm{,}l+\text{d}l\right)$ is

$g\left(l\right)=\frac{2l}{\langle x^2\rangle }{\displaystyle {\int }_l^{\infty }}F\left(x\right)\text{d}x\mathrm{.}$ (55)

The previous demonstration has been adapted from [31] . A test of the previous integral can be done by inserting as a distribution for the diameters a Dirac delta function

$F\left(x\right)=\delta \left(x-2R\right)\mathrm{.}$ (56)

As a consequence, the following PDF for chords is obtained:

$g\left(l\right)=\frac{1}{2}\frac{l}{R^2}\mathrm{,}$ (57)

which has an average value

$\langle l\rangle =\frac{4}{3}R\mathrm{,}$ (58)

and a variance

${\sigma }^2=\frac{2R^2}{9}\mathrm{.}$ (59)

The distribution function, DF, for equal diameter spheres is

Figure 6. The section having diameter x of the intersected sphere. The chord is drawn with the thicker line and marked with l; the distance between the chord and the centre is r.

$G\left(l\right)=\frac{l^2}{4R^2}\mathrm{,}$ (60)

and the random generation of the variate is obtained from the formula

$l=2\sqrt{U}R\mathrm{,}$ (61)

where U the unit rectangular variate.

Another model for the asymmetric Hubble constant assumes that

1) The light travels a void and the chord is evaluated with Equation (61) where the radius is $R_v$ and the number density $n_0$ .

2) After the void, the light intercepts the halo of a galaxy characterized by a radius $R_h$ and density $n_1$ . The length of the chord is evaluated with Equation (61).

3) The process restarts from (1) and the two chords are different.

The average density is evaluated with Formula (47) and Table 6 presents the results for the asymmetric Hubble constant in this second model.

5. Voronoi’s Models for the Spatial Anisotropy

The faces of the Voronoi Polyhedra share the same property: they are equally distant from two nuclei or seeds. The intersection between a plane and these faces produces diagrams that are similar to the displacement of the edges in 2D Voronoi diagrams. From the point of view of the observations, it is very useful to study the intersection between a slice which crosses the centre of a box and the faces of the irregular polyhedra where the galaxies presumably reside. According to the nomenclature of [32] , this cut is classified as $V_P\left(\mathrm{2,3}\right)$ . The parameters that will be used in the following are the kind of nuclei, which can be Poissonian or not Poissonian, the number of nuclei, and the side of the box in Mpc. These are used to build the diagrams, see [33] . In order to calibrate the model, the average radius of the voids should be $R_v=18.23h^{-1}$ Mpc. The density of free electrons can be found by: 1) computing the distance d of a 3D grid point from the nearest face, 2) inserting such a distance in the following piecewise function for the number density of electrons

Table 6. The asymmetric Hubble constant for the second model when $R_v=18.23$ Mpc, $R_h=5$ Mpc, $n_0=2.8\times {10}^{-7}$ particles cm⁻³, and $n_1=1.4\times {10}^{-6}$ particles cm⁻³.

$\begin{array}{l}{n}_e\left(d\right)=n_1\text{if}d<c\\ n_e\left(d\right)=n_0\text{if}d\ge c\end{array}$ (62)

In 2D Figure 7 shows the behaviour of the number density of electrons that follows the $V_P\left(\mathrm{2,3}\right)$ behaviour; the parameters of the simulation are presented in Table 7. Figure 8 presents the behaviour of the Hubble constant when the position of the observer varies progressively from left to right. Table 8 presents the statistics for H₀ in the third model.

In 3D the contours of H₀ in Mollweide projection can be drawn when the observer is at the centre of the box, see Figure 9; Table 9 gives the statistics of the Hubble constant for the fourth model.

Figure 7. Behaviour of the number density in 10⁻⁷ particles cm⁻³ for the intersection, $V_P\left(\mathrm{2,3}\right)$ , between a plane and the Voronoi faces.

Figure 8. Behaviour of H₀ along different lines of sight.

Table 7. Numerical values for the parameters of the voronoi diagrams.

Table 8. The Hubble constant for $V_P\left(\mathrm{2,3}\right)$ for different lines of sight, third model.

Table 9. The Hubble constant in 3D for different lines of sight in the fourth model.

Figure 9. False color contour plot of H₀ as given by the Voronoi’s diagram in the Mollweide map in galactic coordinates with the Galactic center in the middle, parameters as in Table 7.

Figure 10. The present tension on H₀ (black line with two arrows) and our result in the case of the NED sample (blue line).

6. Conclusions

Anomalous redshift

We revitalized the Beer-Lambert law which allows deriving an expression for the redshift as a function of the number density of electrons, see Equation (16). We reviewed three physical mechanisms which produce a definition of the Hubble constant as a function of the number density of electrons, namely, a plasma effect, see Equation (25), an NTL approach, see Equation (28) and the interaction of an electromagnetic wave with an electron, see Equation (33).

Values of H₀

The Hubble constant as determined by processing the NED’s galaxies, turns out to be $H_0=\left(64.24\pm 0.14\right)\text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}$ in the case of a fit with a straight line without uncertainties and $H_0=\left(67.76\pm 0.01\right)\text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}$ in the case of a fit with a straight line with uncertainties, see Table 1. The weighted mean with the standard deviation as error is

$\overline{H_0}=\left(67.74\pm 2.49\right)\text{km}\cdot {\text{s}}^{-\text{1}}\cdot {\text{Mpc}}^{-\text{1}}\text{Ned-catalog}\mathrm{.}$ (63)

Figure 10 presents the values of reference for H₀ and the value here derived.

Anisotropy

The problem of the anisotropy in the Hubble constant has been explained by four methods, which require an increase in the number density of relativistic electrons in the intergalactic medium around the clusters of galaxies, see Table 5, Table 6, Table 8 and Table 9. These models agree with the revised version of the cosmological principle suggested in 2012 in [34] : “The galaxies, as well our galaxy, are situated on the faces of a Voronoi polyhedron. The spatial distribution of galaxies will follow approximately the geometrical rules as well the standard photometric rules.”

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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