A sequence of data x i , y i and uncertainty σ i in the ith variable in the interval [ 1, n ] which presents a linear behaviour is processed assuming a straight line of the form

y ( x ; a , b ) = a + b x , (1)

where a and b are deduced by minimizing χ 2 defined as

χ 2 ( a , b ) = ∑ i n ( y i − a − b x i σ i ) 2 , (2)

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

y ( x ; b ) = b x . (3)

In the absence of uncertainties, the values of b and its uncertainty, σ b , are found by minimizing χ 2 ,

χ 2 ( b ) = ∑ i n ( y i − b x i ) 2 , (4)

and are given by

b = ∑ i = 1 n x i y i ∑ i = 1 n x i 2 , (5a)

σ b = ∑ i = 1 n ( y i − b x i ) 2 ( n − 1 ) ( ∑ i = 1 n x i 2 ) . (5b)

In the presence of uncertainties, the parameter b and its standard error σ b can be determined by minimizing

χ 2 ( b ) = ∑ i n ( y i − b x i σ i ) 2 , (6)

yielding the values

b = ∑ i = 1 n x i y i σ i 2 ∑ i = 1 n x i 2 σ i 2 , (7a)

σ b = 1 ∑ i = 1 n x i 2 σ i 2 . (7b)

A nonlinear regression of the type

ln ( 1 + z ) = b x , (8)

can be made linear with the transformation

z ′ = ln ( 1 + z ) , (9)

which now produces a linear regression

z ′ = b x . (10)

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

d d x ν ( x ) = − k n e ν ( x ) , (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 ν ( 0 ) = n u 0

ν ( x ) = ν 0 e − k n e x . (12)

We now define the redshift as a function of the wavelength λ and the initial wavelength λ 0 by

z = λ − λ 0 λ 0 . (13)

Making use of

ν λ = c , (14)

where c is the speed of light, we obtain

z = e k n e x − 1, (15)

or

ln ( z + 1 ) = k n e x . (16)

Taking a Taylor expansion around z = 0 gives

z = k n e x , (17)

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

v = k n e x c . (18)

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

v = H 0 x km Mpc , (19)

which gives

H 0 = k n e c . (20)

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

k n e = 7.567 × 10 − 27 ( H 0 70 ) 1 m . (21)

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:

ln   ( 1 + z ) = 3.326 × 10 − 25 ∫ 0 R n e d x + γ i − γ 0 ξ ω , (22)

where γ i is the initial photon width, γ 0 is the final photon width, ω is the photon frequency, ξ 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),

ln   ( 1 + z ) = 3.326 × 10 − 25 ∫ 0 R n e d x . (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, 〈 n e 〉 , as a function of the Hubble constant is

〈 n e 〉 = 2.27492 × 10 − 4 ( H 0 70 ) 1 cm 3 , (24)

and the Hubble constant

H 0 = 307701.8 〈 n e 〉 km ⋅ s − 1 ⋅ Mpc − 1 , (25)

where 〈 n e 〉 is expressed in 1/cm³.

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

H 0 = 2 n e h r e m e , (26a)

z = exp H 0 ∗ 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

〈 n e 〉 = 0.55337 ( H 0 70 ) 1 m 3 , (27)

and the Hubble constant

H 0 = 126.49 〈 n e 〉 km ⋅ s − 1 ⋅ Mpc − 1 , (28)

with 〈 n e 〉 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 ∫ 0 D n e d s , (29)

with

w = 3 q e 4 μ 0 512 π m e 2 = 2.33 × 10 − 30   m 2 , (30)

where q e is the charge of the electron, μ 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

〈 n e 〉 = H 0 c w , (31)

or, in numbers,

〈 n e 〉 = 3235.5 ( H 0 70 ) 1 m 3 , (32)

and the Hubble constant is

H 0 = 0.021634 〈 n e 〉 km ⋅ s − 1 ⋅ Mpc − 1 , (33)

with 〈 n e 〉 expressed in 1/m³.

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, 〈 n e 〉 and k n e . The cosmological results are shown in Table 1 for a straight line with and without uncertainty.

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.

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

Φ ( L ) d L = ( Φ * L * ) ( L L * ) α exp ( − L L * ) d L , (34)

where α sets the slope for low values of L, L * is the characteristic luminosity, and Φ * is the normalization. The equivalent distribution in absolute magnitude is

Φ ( M ) d M = 0.921 Φ * 10 0.4 ( α + 1 ) ( M * − M ) exp ( − 10 0.4 ( M * − M ) ) d M , (35)

where M * is the characteristic magnitude as determined from the data. We introduce H 0 = 100 h   km ⋅ s − 1 ⋅ Mpc − 1 , with h = 1 when h is not specified and therefore the scaling with h is M * − 5 log 10 h and Φ * h 3 [ Mpc − 3 ] .

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 = z c H 0 . (36)

The flux of radiation, f, is introduced as

f = L 4 π r 2 , (37)

and is here expressed in L ⊙ Mpc 2 . The joint distribution in distance, r, and flux, f, for the number of galaxies is

d N d Ω d r d f = 1 4 π ∫ 0 ∞     4 π r 2 Φ ( L L * ) δ ( f − L 4 π r 2 ) d r , (38)

were the factor ( 1 4 π ) 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

d N d Ω d z d f = 4 π ( c H 0 ) 5 z 4 Φ ( z 2 z c r i t 2 ) , (39)

where d Ω , d z , and d f represent the differential of the solid angle, the red shift, and the flux, respectively, and

z c r i t 2 = H 0 2 L * 4 π f c 2 , (40)

where c denotes the speed of light; CODATA recommends c = 299792.458   km ⋅ s − 1 . The mean red shift of galaxies with a flux f, see Formula (1.105) in [21] , is

〈 z ( f ) 〉 = z c r i t Γ ( 3 + α ) Γ ( 5 / 2 + α ) . (41)

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

z max ( f ) = z c r i t α + 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 〈 z 〉 and z max , have been derived for a fixed value of f, which appears in the definition of z c r i t ; 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

〈 z ( f min , f max ) 〉 = − 0.2115 e 0.4605 M b o l , ⊙ − 0.4605 M * H 0 Γ ( α + 3 ) ( f max 2 − f min 2 ) Γ ( α + 2.5 ) ( f max f min 2 − f min f max 2 ) c , (43)

where M b o l , ⊙ is the bolometric luminosity of reference, which is different for each catalog. In this formula, we have used

L * = 10 0.4 M b o l , ⊙ − 0.4 M * . (44)

The number of galaxies with flux lying between f min and f max is

N ( z ; f min , f max ) = 1 α + 1 ( F * z 2 + α H 0 − 3 − α c 3 + α π α 2 ( 10 0.4 M b o l , ⊙ − 0.4 M * ) − α 2 2 α         × ( f max α 2 e − 2 π f max z 2 c 2 H 0 2 10 0.4 M b o l , ⊙ − 0.4 M * M α 2 , α 2 + 1 2 ( 4 π f max z 2 c 2 H 0 2 10 0.4 M b o l   ⊙   − 0.4 M * )   − f min α 2 e − 2 π f min z 2 c 2 H 0 2 10 0.4 M b o l , ⊙ − 0.4 M * M α 2 , α 2 + 1 2 ( 4 π f min z 2 c 2 H 0 2 10 0.4 M b o l , ⊙ − 0.4 M * ) ) ) , (45)

where M κ , μ ( z ) 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 〈 z ( f min , f max ) 〉 and 〈 z ( f min , f max ) 〉 .

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.

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

R V ¯ = 18.23 h Mpc , (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

n e ( r ) = n 0     if     0 ≤ r < a n e ( r ) = n 1     if     a ≤ r < b n e ( r ) = n 0     if     r ≥ b (49)

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 = 2 b and split into two parts,

Part I, three pieces if 0 ≤ y < a

Part II, two pieces if a ≤ 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

〈 n e 〉 = n 0 b − n 0 b 2 − y 2 + n 1 b 2 − y 2 − n 1 a 2 − y 2 + n 0 a 2 − y 2 b       if       0 ≤ y < a 〈 n e 〉 = n 0 b − n 0 b 2 − y 2 + n 1 b 2 − y 2 b       if     a ≤ y < b (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.

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

G ( x ) d x = π 4 x 2 F ( x ) d x ∫ 0 ∞ π 4 x 2 F ( x ) d x = x 2 F ( x ) d x 〈 x 2 〉 . (51)

Given a line which intersects a sphere of diameter x, the probability that the distance from the centre lies in the range r , r + d r is

p ( r ) = 2 π r d r π 4 x 2 , (52)

and the length of the chord is

l = x 2 − 4 r 2 , (53)

see Figure 6.

The probability that spheres in the range ( x , x + d x ) are intersected to produce chords with lengths in the range ( l , l + d l ) is

G ( x ) d x 2 l d l x 2 = 2 l d l 〈 x 2 〉 F ( x ) d x . (54)

The probability of having a chord with length between ( l , l + d l ) is

g ( l ) = 2 l 〈 x 2 〉 ∫ l ∞     F ( x ) d x . (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 ( x ) = δ ( x − 2 R ) . (56)

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

g ( l ) = 1 2 l R 2 , (57)

which has an average value

〈 l 〉 = 4 3 R , (58)

and a variance

σ 2 = 2 R 2 9 . (59)

The distribution function, DF, for equal diameter spheres is

G ( l ) = l 2 4 R 2 , (60)

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

l = 2 U R , (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.