Some useful formulae in ΛCDM cosmology can be expressed in terms of a Padé approximant. The basic parameters are: the Hubble constant, H 0 , expressed in km s⁻¹・Mpc⁻¹, the velocity of light, c, expressed in km・s⁻¹, and the three numbers Ω M , Ω K , and Ω Λ , see [14] for more details. In the case of the Union 2.1 compilation, see [15] , the parameters are H 0 = 69.81   km ⋅   s − 1 ⋅ Mpc − 1 , Ω M = 0.239 and Ω Λ = 0.651 . To have the luminosity distance, D L ( z ; H 0 , c , Ω M , Ω Λ ) , as a function of the redshift only, we apply the minimax rational approximation, which is characterized by the two parameters p and q. We find a simplified expression for the luminosity distance, D L ,6,2 , when p = 6 and q = 2

D L , 6 , 2 = N D 0.284483 + 0.153266 z + 0.0681615 z 2   for   0.001 < z < 4 , (1)

where

N D = − 0.0017 + 1221.80 z + 1592.35 z 2 + 504.386 z 3     + 85.8574 z 4 + 0.41684 z 5 + 0.186189 z 6 , (2)

The inverse of the above function, i.e. the redshift z 6,2 as function of the luminosity distance, is

z 6 , 2 = 3.3754 × 10 − 5 D L − 0.46438 + 2.1625 × 10 − 14     × 2.4363 × 10 18 D L 2 + 3.9538 × 10 23 D L + 4.6114 × 10 26 . (3)

We used the Schechter function, see [16] , as a luminosity function (LF) for galaxies

Φ ( L ) d L = ( Φ * L * ) ( L L * ) α exp ( − L L * ) d L , (4)

here α sets the slope for low values of luminosity, L , L * is the characteristic luminosity and Φ * is the normalisation. 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 , (5)

where M * is the characteristic magnitude as derived from the data. The scaling with h is M * − 5 l o g 10 h and Φ * h 3 [ Mpc − 3 ] .

We processed the SDSS Photometric Catalogue DR 12, see [17] , which contains 10,450,256 galaxies (elliptical + spiral) with redshift. In the following we will use the generic term galaxies without distinction between the two types, elliptical and spiral. The number of galaxies for an area in redshift of 0.025 × 0.025 of the u-band is reported in Figure 1 as a contour plot and in Figure 2 as a cut along a line.

The flux, f, is

f = L 4 π r 2 , (6)

where r is the luminosity distance. The luminosity distance is

r = D L , 6 , 2 , (7)

and the relationship between d r and d z is

d r = N D d z , (8)

where

N = 74813.67 + 10.9263 z 7 + 49.05768 z 6 + 2642.64259 z 5 + 16024.51314 z 4     + 54307.16663 z 3 + 127258.486 z 2 + 195005.8564 z , (9)

and

D = ( z 2 + 2.248575472 z + 4.173664398 ) 2 . (10)

The joint distribution in z and f for the number of galaxies is

d N d Ω d z d f = 1 4 π ∫ 0 ∞ 4 π r 2 d r   Φ ( L ; L * , σ ) δ ( z − ( z 6 , 2 ) ) δ ( f − L 4 π r 2 ) , (11)

where δ is the Dirac delta function, Φ ( L ; L * , σ ) has been defined in Equation (4) and z 6,2 has been defined in Equation (3). The explicit version is

d N d Ω d z d f = N N D D , (12)

where

N N = 644.3 ( z + 1.058 ) 4 ( z − 0.000001412 ) 4 ( z 2 + 4.889 z + 13.34 ) 4     × ( z 2 − 3.708 z + 464.6 ) 4 e A A B B 2 L * ( C C B B 2 L * ) α Φ * ( z + 0.5328 )     × ( z 2 + 5.047 z + 7.9141 ) ( z 2 + 0.7 z + 7.011 ) ( z 2 − 1.79 z + 231.58 ) , (13)

C C = f ( z + 1.058 ) 2 ( z − 0.0000014124 ) 2 ( z 2 + 4.889 z + 13.343 ) 2     × ( z 2 − 3.708 z + 464.6 ) 2 (14)

D D = ( z 2 + 2.248 z + 4.173 ) 6 L * (15)

A A = − 93.76 f z 12 − 419.8 f z 11 − 86945.4 f z 10 − 701622.4 f z 9     − 22679411 f z 8 − 239083307 f z 7 − 1430432291 f z 6     − 4912205831 f z 5 + ( 4.540788 α L * − 10191896880 f ) z 4     + ( 20.4206 α L * − 10524532830 f ) z 3 + ( 60.862 α L * − 4037704632 f ) z 2     + ( 85.22 α L * + 11406.2 f ) z + 79.098 α L * − 0.008055 f , (16)

B B = z 2 + 2.2485 z + 4.173. (17)

Figure 3 presents the number of galaxies that are observed in SDSS DR 12 as a function of the redshift for a given window in flux, in addition to the theoretical curve. The theoretical number of galaxies is reported in Figure 4 as a function of the flux and redshift, and is reported in Figure 5 as a function of α and redshift.

The total number of galaxies comprised between a minimum value of flux, f min , and a maximum value of flux f max , for the Schechter LF can be computed through the integral

d N d Ω d z = ∫ f min f max N N D D d f . (18)

This integral has a complicated analytical solution in terms of the Whittaker function M κ , μ ( z ) , see [18] . Figure 6 reports all of the galaxies of SDSS DR12 and also the theoretical curve.

A theoretical surface/contour of the Great Wall is displayed in Figure 7