New formulas for the Hubble constant in a Euclidean static universe

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New formulas for the Hubble constant in a Euclidean static universe

Lorenzo Zaninetti^a兲

Dipartimento di Fisica Generale, via P. Giuria 1, I-10125 Turin, Italy

共Received 11 January 2010; accepted 9 March 2010; published online 20 April 2010兲

Abstract: It is shown that the Hubble constant can be derived from the standard luminosity function of galaxies as well as from a new luminosity function as deduced from the mass-luminosity relationship for galaxies. An analytical expression for the Hubble constant can be found from the maximum number of galaxies共in a given solid angle and flux兲 as a function of the redshift. A second analytical definition of the Hubble constant can be found from the redshift averaged over a given solid angle and flux. The analysis of two luminosity functions for galaxies brings four new definitions of the Hubble constant. The equation that regulates the Malmquist bias for galaxies is derived and as a consequence it is possible to extract a complete sample. The application of these new formulas to the data of the two-degree field galaxy redshift survey provides a Hubble constant of 共65.26⫾8.22兲 km s⁻¹Mpc⁻¹ for a redshift lower than 0.042. All the results are deduced in a Euclidean universe because the concept of space-time curvature is not necessary as well as in a static universe because two mechanisms for the redshift of galaxies alternative to the Doppler effect are invoked. © 2010 Physics Essays Publication.关DOI: 10.4006/1.3386219兴

Re´sume´: Il est montré que la constante de Hubble peut être dérivé de la fonction de luminosité standard pour les galaxies, ainsi que d’une fonction de luminosité nouvelle déduite de la relation masse-luminosité pour les galaxies. Une expression analytique de la constante de Hubble peut être trouvée par rapport au maximum dans le nombre de galaxies共dans un angle solide donné et flux兲 en fonction du décalage vers le rouge. Une deuxième définition analytique peut être trouvé par la moyenne de décalage vers le rouge d’un angle solide et le flux. Ces deux définitions sont doublées par l’utilisation d’une fonction de luminosité de nouvelles galaxies. L’équation qui régit le biais Malmquist pour les galaxies est dérivé et avec comme conséquence est possible d’extraire un échantillon complet. L’application de ces nouvelles formules pour les données des deux degrés Field Galaxy Redshift Survey fournit une constante de Hubble共65.26⫾8.22兲 km s⁻¹Mpc⁻¹ pour décalage vers le rouge inférieur à 0.042. Tous les résultats sont déduits dans un univers Euclidien parce que le concept de la courbure de l’espace-temps n’est pas nécessaire, ainsi que dans un univers statique car deux mécanismes pour le décalage vers le rouge de galaxies alternative à l’effet Doppler sont appelés.

Key words: Distances; Redshifts; Radial Velocities; Observational Cosmology.

I. INTRODUCTION

The Hubble constant, in the following H₀, is defined as H₀= v

D关km s⁻¹Mpc⁻¹兴, 共1兲

where v = cz is the recession velocity, D is the distance in Mpc, c is the velocity of light, and z is the redshift defined as

z =␭obs−␭em

␭em

, 共2兲

with ␭obs and ␭em denoting, respectively, the wavelengths of the observed and emitted lines as determined from the laboratory source. The first numerical values of the Hubble constant were H₀= 625 km s⁻¹Mpc⁻¹as deduced by Lemaitre,¹H₀= 460 km s⁻¹Mpc⁻¹as deduced by Robertson,² H₀= 500 km s⁻¹Mpc⁻¹ as deduced by Hubble,³ and H₀

= 290 km s⁻¹Mpc⁻¹ as deduced by Oort.⁴ Figure 1 reports

the decrease in the numerical value of the Hubble constant from 1927 to 1980.

At the time of writing, two excellent reviews have been written, see Ref. 5 关H0=共63.2⫾1.3共random兲

⫾5.3共systematic兲兲 km s⁻¹Mpc⁻¹兴 and Ref. 6 共H0

⬃70–73 km s⁻¹Mpc⁻¹兲. We now report the methods that use the global properties of galaxies as indicators of distance, as follows:

共1兲 Luminosity classes of spiral galaxies: H₀

=共55⫾3兲 km s⁻¹Mpc⁻¹.⁷

共2兲 21 cm line widths: H0=共59.1⫾2.5兲 km s⁻¹Mpc⁻¹.⁸ 共3兲 Brightest cluster galaxies: H0=共54.2⫾5.4兲 km s⁻¹

Mpc⁻¹.⁹

共4兲 The Dn-␴ ^or fundamental plane method: H₀

=共57⫾4兲 km s⁻¹Mpc⁻¹.⁸

共5兲 Surface brightness fluctuations: H0= 71. 8 km s⁻¹ Mpc⁻¹.⁵

共6兲 Gravitational lens: H0=共72⫾12兲 km s⁻¹Mpc⁻¹.¹⁰

a兲zaninetti@ph.unito.it

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共7兲 The Sunyaev–Zel’dovich effect: H0=共67⫾18兲 km s⁻¹ Mpc⁻¹.¹¹

共8兲 Ks-band Tully–Fisher relation: H0=共84⫾6兲 km s⁻¹ Mpc⁻¹,¹²where the Hubble constant was named Hubble parameter.

At the time of writing, the first important evaluation of the Hubble constant is through Cepheids共key programs with Hubble space telescope兲 and type Ia Supernovae¹³

H₀=共62.3 ⫾ 5兲 km s⁻¹Mpc⁻¹. 共3兲 A second important evaluation comes from the 3 years of observations with the Wilkinson microwave anisotropy probe, see Table II of Ref.14;

H₀=共73.2 ⫾ 3.2兲 km s⁻¹Mpc⁻¹. 共4兲 In the following, we will process galaxies having redshifts as given by the catalog of galaxies. The forthcoming analysis is based on two key assumptions:共i兲 the flux of radiation from galaxies in a given wavelength decreases with the square of the distance and共ii兲 the redshift is assumed to have a linear relationship with distance in Mpc. These two hypotheses allow some new physical mechanisms to be accepted which produce a linear relationship between redshift and distance, for redshifts lower than 1. In this framework, we can speak of a Euclidean universe because the distances are deduced from the Pythagorean theorem and a static universe because it is not expanding. The already listed approaches leave a series of questions unanswered or partially answered:

• Can the Hubble constant be deduced from the Schechter luminosity function of galaxies?

• Can the Hubble constant be deduced from a new luminosity of galaxies alternative to the Schechter function?

• Can the equation that regulates the Malmquist bias be derived in order to deal with a complete sample in apparent magnitude? Can the reference magnitude of the sun be deduced from the luminosity function of galaxies?

In order to answer these questions, Sec. II contains three introductory paragraphs on sample moments, the weighted mean and the determination of the so-called ”exact value” of the Hubble constant. Section III reviews the basic system of magnitudes, a review of two alternative mechanisms for the redshift of galaxies, two analytical definitions of the Hubble constant in terms of the Schechter luminosity function of galaxies, and two other definitions that can be found by adopting a new luminosity function for galaxies. Section IV contains a numerical evaluation of the four new formulas for the Hubble constant as deduced from the data of the two- degree field galaxy redshift survey 共2dFGRS兲. Section V contains a numerical evaluation of the reference magnitude of the sun for a given catalog.

II. PRELIMINARIES

This section reviews the evaluation of the first moment about zero and of the second moment about the mean of a sample of data, the evaluation of the mean and variance when each piece of data of a sample has differing errors, the evaluation of the uncertainty, and the evaluation of H₀from a list of published data.

A. Sample moments

Consider a random sample ␹^{= x}1, x₂, . . . , xn and let x_共1兲 艌x共2兲艌 ¯ 艌x共n兲 denote their order statistics so that x_共1兲

= max共x,x, ... ,xn兲,x_共n兲= min共x1, x₂, . . . , xn兲. The sample mean, x_iis

¯ =x 1

n兺^xⁱ^, ^共5兲

and the standard deviation of the sample,␴, is according to Press et al.,¹⁵

␴⁼冑n − 1¹ 兺^共xⁱ^{− x}^¯^兲²^. ^共6兲

B. The weighted mean

The probability, N共x;␮^,␴兲, of a Gaussian 共normal兲 distribution is

N共x;␮,␴兲 = 1

␴共2␲兲^1/2exp −共x −␮兲²

2␴² ^, ^共7兲

where ␮ is the mean and ␴² is the variance. Consider a random sample␹^{= x}1, x₂, . . . , xnwhere each value is from a Gaussian distribution having the same mean but a different standard deviation ␴i. By the maximum likelihood estimate 共MLE兲, in the following MLE,^16,17 an estimate of the weighted mean␮ ^is

␮=

兺_␴^xⁱ

i 2

兺_␴¹

i 2

, 共8兲

and an estimate of the error of the weighted mean,␴共␮兲,

FIG. 1. Logarithmic values of the Hubble constant H₀from 1927 to 1980.

The error bar is evaluated according to the file in Ref.39.

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␴共␮兲 =

冑

^兺¹_␴¹_i²^, ^共9兲

see Ref. 18for a detailed demonstration.

C. Error evaluation

When a numerical value of a constant is derived from a theoretical formula, the uncertainty is found from the error propagation equation 共often called law of errors of Gauss兲 when the covariant terms are neglected 关see Eq. 共3.14兲 in Ref. 17兴. In the presence of more than one evaluation of a constant with different uncertainties, the weighted mean and the error of the weighted mean are found by formulas共8兲and 共9兲. In the following, in each diagram we will specify the technique by which the error bars on the derived quantities are derived.

D. A first statistical application

The determination of the numerical value of the Hubble constant is an active field of research and the file in Ref.19 contains a list of 355 published values during the period 1996–2008. Figure 2 reports the frequencies of such values with the superposition of a Gaussian distribution.

TableIreports the statistics of this sample as well as the minimum, H_0,minand maximum H_0,max.

III. USEFUL FORMULAS

This section reviews three different mechanisms for the redshifts of galaxies: the system of magnitudes, the standard luminosity function共LF兲 in the following LF of galaxies, and a new LF of galaxies as given by the mass-luminosity relationship.

A. The nature of the redshift

In the following, we will present two theories for the redshift of galaxies alternative to the Doppler effect which are based on basic axioms of physics. In these two alterna- tive mechanisms, the distance, r, in a Cartesian coordinate system, x , y , z, is given by the usual Pythagorean theorem r =冑x²+ y²+ z². These two alternative theories do not require any expansion of the universe even though local velocities of the order of ⬇100 km/s are not excluded. These random velocities of galaxies can explain the bending of radiogalaxies.²⁰

Starting from Hubble,³ the suggested correlation between the expansion velocity and distance in the framework of the Doppler effect is

V = H₀D = c z, 共10兲

where H₀ is the Hubble constant H₀= 100h km s⁻¹Mpc⁻¹, with h = 1 when h is not specified, D is the distance in Mpc, c is the velocity of light, and z the redshift. The quantity cz, a velocity, or z, a number, characterizes the catalog of galax- ies. The Doppler effect produces a linear relationship between distance and redshift. The analysis of mechanisms which predict a direct relationship between distance and redshift started with Marmet²¹ and a current list of the various mechanisms can be found in Ref. 22. Here, we select two mechanisms among others. The presence of a hot plasma with low density, such as in the intergalactic medium, produces a relationship of the type

D =3.0064⫻ 10²⁴ 共Ne兲av

ln共1 + z兲 cm, 共11兲

where the averaged density of electrons,共Ne兲av, is 共Ne兲av= H₀

3.076⫻ 10⁵⬇ 2.42 ⫻ 10⁻⁴冉74.5^H⁰ 冊^cm⁻³^, ^共12兲

see Eqs.共48兲 and 共49兲 in Ref. 23or Eq. 共27兲 in Ref.24. A second explanation for the redshift is the dispersive extinction theory 共DET兲 in which the redshift is caused by the dispersive extinction of star light by the intergalactic medium. In this theory

z =冉^␲4^bc冊^␦␭^␭³²D, 共13兲 where␦␭ is the natural linewidth and b is a parameter that characterizes the linearity of the extinction, see formula共17兲 in Ref.25.

B. System of magnitudes

The absolute magnitude of a galaxy, M, is connected to the apparent magnitude m through the relationship

TABLE I. The Hubble constant from a list of published values during the period 1996–2008.

Entity Definition Value

n No of samples 355

¯x Average 65.85 km s⁻¹Mpc⁻¹

␴ Standard deviation 10 km s⁻¹Mpc⁻¹

H₀, max Maximum 98 km s⁻¹Mpc⁻¹

H₀, min Minimum 30 km s⁻¹Mpc⁻¹

␮ Weighted mean 66.04 km s⁻¹Mpc⁻¹

␴共␮兲 Error of the weighted mean 0.25 km s⁻¹Mpc⁻¹ FIG. 2. Histogram of frequencies of 355 published values of H₀during the period 1996–2008 with error bars computed as the square root of the frequencies. The continuous line fit represents a Gaussian distribution with mean from Eq.共8兲and standard deviation from Eq.共9兲.

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M = m − 5 log冉^H^cz0冊^{− 25.} ^共14兲

In a Euclidean, nonrelativistic and homogeneous universe, the flux of radiation, f, expressed in L_䉺/Mpc² units, where L_䉺represents the luminosity of the sun, is

f = L 4␲^DL

2, 共15兲

where DLrepresents the distance of the galaxy expressed in Mpc and

DL= cz

H₀. 共16兲

The relationship connecting the absolute magnitude, M, of a galaxy to its luminosity is

L

L_䉺= 10^0.4^共M^䉺^−M^兲, 共17兲

where M_䉺is the reference magnitude of the sun in the band- pass under consideration.

The flux expressed in L_䉺/Mpc²units as a function of the apparent magnitude is

f = 7.957⫻ 10⁸e^0.921M^䉺^−0.921m L_䉺

Mpc², 共18兲

and the inverse relationship is

m = M_䉺− 1.0857 ln共0.1256 ⫻ 10⁻⁸f兲. 共19兲

C. The Schechter function

The Schechter function, introduced by Schechter,²⁶provides a useful fit for the luminosity of galaxies,

⌽共L兲dL =⌽^ⴱ

L^ⴱ冉L^L^ⴱ冊^␣^exp冉⁻ L^L^ⴱ冊^dL. ^共20兲

Here,␣sets the slope for low values of L, L^ⴱis the characteristic luminosity, and⌽^ⴱis the normalization. The equiva- lent distribution in absolute magnitude is

⌽共M兲dM = 共0.4 ln 10兲⌽^ⴱ10^0.4^{共␣+1兲共M}^ⴱ^−M^兲

⫻ exp共− 10^0.4共M^ⴱ^−M兲兲dM, 共21兲 where M^ⴱis the characteristic magnitude as derived from the data. The joint distribution in z and f for galaxies, see for- mula共1.104兲 in Ref.27or formula共1.117兲 in Ref.28, is

dN

d⍀dzdf= 4␲冉^H^c0冊⁵^z⁴^⌽冉^z^zcrit²

2 冊^, ^共22兲

where d⍀, dz, and df represent the differential of the solid angle, redshift, and flux, respectively. This relationship has been derived assuming z⬇V/c⬇H0r/c and using Eq. 共15兲.

The critical value of z , z_critis

z_crit² = H₀²L^ⴱ

4␲^fc²^. ^共23兲

The number of galaxies in z and f as given by formula共22兲 has a maximum at z = z_pos-max, where

z_pos-max= z_crit冑_␣+ 2, 共24兲

which can be re-expressed as

z_pos-max=冑2 +␣冑10^0.4M^䉺^−0.4M^ⴱ冑2 +␣H₀

2冑_␲冑fc . 共25兲

From the previous formula, it is possible to derive a first Hubble constant adopting for the velocity of light c

= 299 792.458 km/s, Mohr and Taylor,²⁹ H₀^I= N^I

D^I km s⁻¹MPc⁻¹,

N^I= 2.997⫻ 10¹⁰z_pos-max冑e^0.921M^䉺^−0.921m,

D^I=冑2 +␣冑10^0.4M^䉺^−0.4M^ⴱ. 共26兲

The mean redshift of galaxies with a flux f, see formula 共1.105兲 in Ref.27or formula共1.119兲 in Ref.28is

具z典 = zcrit

⌫共3 +␣兲

⌫共5/2 +␣兲. 共27兲

A second Hubble constant can be derived from the observed averaged redshift for a given magnitude,

H₀^II=N^II

D^IIkm s⁻¹Mpc⁻¹,

N^II= 1.691⫻ 10¹⁰具z典obs⫻冑_␲冑e^0.921M^䉺^−0.921m⌫共5/2 +␣兲, D^II=⌫共3 +␣兲冑10^0.4M^䉺^−0.4M^ⴱ, 共28兲 where 具z典obsis the averaged redshift as evaluated from the considered catalog.

From formula共27兲, it is also possible to derive the ref- erence magnitude of the sun M_䉺for the given catalog

M_䉺= M^ⴱ+ 1.085 ln冉^1.129^{⫻ 10}¹²^具z典^obs²^H0^{f共⌫共2.5 +}^␣^兲兲² 2共⌫共3 +␣兲兲² 冊^.

共29兲 In this case, M_䉺is the unknown and H₀is an input parameter.

D. The mass-luminosity relationship

A new LF of galaxies as derived in Ref.30is

⌿共L兲dL =冉a⌫共c¹ f兲冊冉^⌿^L^ⴱ^ⴱ冊冉^L^L^ⴱ冊^共c^f^−a兲/a

⫻ exp冉⁻冉^L^L^ⴱ冊^1/a冊^dL, ^共30兲

where ⌿^ⴱ is a normalization factor that defines the overall density of galaxies, a number per cubic MPc, 1/a is an

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exponent that connects the mass to the luminosity, and cf

is connected with the dimensionality of the fragmentation, cf= 2d, where d represents the dimensionality of the space being considered: 1, 2, and 3. The distribution in absolute magnitude is

⌿共M兲dM =冉^{0.4 ln 10}a⌫共c¹ f兲冊^⌿^ⴱ¹⁰^0.4共c^f^/a兲共M^ⴱ^−M兲

⫻ exp共− 10^0.4^共M^ⴱ^−M^兲共1/a兲兲dM. 共31兲 This function contains the parameters M^ⴱ, a, cf, and ⌿^ⴱ, which are derived from the operation of fitting the experi- mental data. The joint distribution in z and f, in the presence of theM-L luminosity 关Eq.共30兲兴, is

dN

d⍀dzdf= 4␲冉^H^c0冊⁵^z⁴^⌿冉^z^zcrit²

2 冊^. ^共32兲

The number of galaxies, N_M-L共zfmin, f_max兲, comprised be- tween f_minand f_max, can be computed through the following integral:

N_M-L共z兲 =冕f_min f_max

4␲冉^H^c0冊⁵^z⁴^⌿冉^z^zcrit²

2 冊^{df ,} ^共33兲

and also in this case a numerical integration must be per- formed.

The number of galaxies as given by formula共32兲has a maximum at z_pos-maxwhere

z_pos-max= z_crit共cf+ a兲^a^/2, 共34兲

which can be re-expressed as

z_pos-max=共a + cf兲^1/2a冑10^0.4M^䉺^−0.4M^ⴱH₀

2冑␲冑fc . 共35兲

A third Hubble constant as deduced from the maximum in the number of galaxies as a function of z is

H₀^III= N^III

D^III km s⁻¹Mpc⁻¹, 共36兲

N^III= 2.997⫻ 10¹⁰z_pos-max冑e^0.921M^䉺^−0.921m,

D^III=共cf+ a兲^0.5a冑10.0^0.4M^䉺^−0.4M^ⴱ. 共37兲 The mean redshift connected with theM-L LF is

具z典 = zcrit

2 4⁻^共2a+c^f^兲/a⌫共2a + cf兲2^共2c^f^+3a^兲/a

⌫共cf+ 3/2a兲 , 共38兲

and the fourth Hubble constant is H₀^IV=N^IV

D^IVkm s⁻¹Mpc⁻¹

N^IV= 8.457⫻ 10⁹具z典obs冑_␲冑e^0.921M^䉺^−0.921m⌫共cf+ 3/2a兲, D^IV= 4⁻^共2a+c^f^兲/a冑10^0.4M^䉺^−0.4M^ⴱ⌫共2a + cf兲2^共2c^f^+3a^兲/a. 共39兲

IV. NUMERICAL VALUE OF THE HUBBLE CONSTANT The formulas previously derived are now tested on the catalog from the 2dFGRS, available at the website in Ref.31. In particular, we added together the file parent.ngp.

txt, which contains 145 652 entries for NGP strip sources and the file parent.sgp.txt, which contains 204 490 entries for SGP strip sources. Once the heliocentric redshift was selected, we processed 219 107 galaxies with 0 . 01艋z艋0.3 and two strips of the 2dFGRS are shown in Fig.3. From the previous figure the nonhomogeneous structure of the universe is clear and this concept can be clarified by counting the number of galaxies in one of the two slices as a function of the redshift when a sector with a central angle of 1° is considered, see Fig. 4.

Conversely, when the two slices are considered together the behavior of the number of galaxies as a function of the redshift is more continuous, see Fig.5. In this quasihomogeneous universe, some statistical properties such as the theoretical position of the maximum in the number of galaxies

FIG. 3. Cone-diagram of all the galaxies in the 2dFGRS. This plot contains 203 249 galaxies.

FIG. 4. Histogram共step-diagram兲 of the number of galaxies as a function of the redshift in the slice to the right of Fig.3, the number of bins is 50. The circular sector has a central angle of 1°.

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agree with the observations and Fig. 6 reports the observed maximum in the 2dFGRS as well as the theoretical curve as a function of the magnitude. Before reducing the data, we should discuss the Malmquist bias, see Refs. 32 and 33, which was originally applied to the stars and was then applied to the galaxies by Behr.³⁴ We therefore introduce the concept of limiting apparent magnitude and the corresponding completeness in absolute magnitude of the considered catalog as a function of the redshift. The observable absolute magnitude as a function of the limiting apparent magnitude, mL, is

M_L= m_L− 5 log₁₀冉^H^cz0冊^{− 25.} ^共40兲

The previous formula predicts, from a theoretical point of view, an upper limit on the absolute maximum magnitude that can be observed in a catalog of galaxies characterized by a given limiting magnitude and Fig. 7 reports such a curve

and the galaxies of the 2dFGRS.

The interval covered by the LF of galaxies,⌬M, is de- fined by

⌬M = Mmax− M_min, 共41兲

where M_maxand M_minare the maximum and minimum absolute magnitudes of the LF for the considered catalog. The real observable interval in absolute magnitude, ⌬ML, is

⌬ML= ML− M_min. 共42兲

We can therefore introduce the range of observable absolute maximum magnitude expressed in percent,苸s共z兲, as

苸s共z兲 =⌬ML

⌬M ⫻ 100%. 共43兲

This is a number that represents the completeness of the sample and, given the fact that the limiting magnitude of the 2dFGRS is mL= 19.61, it is possible to conclude that the 2dFGRS is complete for z艋0.0442. This efficiency expressed as a percentage can be considered a version of the Malmquist bias. In our case, we have chosen to process the galaxies of the 2dFGRS with z艋0.0442 of which there are 22 071; in other words our sample is complete. Another quantity that should be fixed in order to continue is the ab- solute magnitude of the sun in the bJfilter, M_䉺= 5.33.^35–37

We now outline the algorithm that allows to deduce z_pos-maxand具z典obsfrom a catalog of galaxies.

共1兲 We fix a given flux or magnitude, for example, bJ, and a relative narrow window.

共2兲 We organize the selected galaxies according to frequency versus redshift, see a typical histogram in Fig.8.

共3兲 Once the histogram is made, we compute the astronomi- cal z = z_pos-max, which is inserted in formulas 共26兲 and 共36兲in order to deduce the Hubble constant.

共4兲 The selected sample of galaxies with a given magnitude allows an easy determination of具z典obs.

共5兲 Particular attention should be paid to the completeness

FIG. 5. Histogram共step-diagram兲 of the number of galaxies as a function of the redshift when the two slices of Fig.3are added together. The number of bins is 50.

FIG. 6. Value of z_pos-maxat which the number of galaxies in the 2dFGRS is maximum as a function of the apparent magnitude bJ共stars兲 and theoretical curve of the maximum for the Schechter function as represented by formula 共25兲共full line兲. In this plot, M_䉺= 5.33 and H₀= 65.26 km s⁻¹Mpc⁻¹. The horizontal dotted line represents the boundary between complete and incom- plete samples.

FIG. 7. 共Color online兲 The absolute magnitude M of 202 923 galaxies belonging to the 2dFGRS when M_䉺= 5.33 and H₀= 66.04 km s⁻¹Mpc⁻¹ 共points兲. The upper theoretical curve as represented by Eq.共40兲is reported as the thick line when mL= 19.61.

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of the sample and Fig.9reports the maximum value in redshift z_maxfor each run in magnitude/flux.

TableII reports the four values of the Hubble constant deduced here and Fig.10displays the data corresponding to the constant deduced from Eq. 共28兲.

From a practical point of view,苸, the percentage reli- ability of our results can also be introduced,

苸 =冉^{1 −}^兩共Q^obs^Q^{− Q}obs^num^兲兩冊^{⫻ 100%,} ^共44兲

where Q_obsis the quantity given by the astronomical obser- vations and Q_num is the analogous quantity calculated by us. The value of H₀ as found by us with the weighted mean is, see fifth row in Table II, H₀= 65. 26 km s⁻¹Mpc⁻¹ and the observed value, see the weighted mean in Table I, H₀= 66. 04 km s⁻¹Mpc⁻¹.

V. THE ABSOLUTE MAGNITUDE OF THE SUN

The reference absolute magnitude of the sun 共the unknown variable兲 can be derived from formula共29兲but in this case the value of H₀ 共known variable兲 should be specified.

Perhaps the best choice is the weighted mean reported in TableI, H₀= 66. 04 km s⁻¹Mpc⁻¹. Adopting this value of H₀, the absolute reference magnitude of the sun can be plotted in Fig.11and the averaged value is

M_䉺=共5.50 ⫾ 0.35兲mag. 共45兲

The efficiency in deriving the absolute reference magnitude of the sun is

苸 = 96.63%. 共46兲

VI. CONCLUSIONS

A careful study of the standard LF of galaxies allows the determination of the position of the maximum in the theoretical number of galaxies versus redshift and the theoretical averaged redshift. From the two previous analytical results, it is possible to extract two new formulas for the Hubble constant, Eqs.共26兲and共28兲. The same procedure can be applied by analogy to a new LF as given by the mass-luminosity relationship, see Eqs. 共36兲and 共39兲. The weighted mean of the four values of H₀ as deduced from TableIIgives H₀=共65.26 ⫾ 8.22兲 km s⁻¹Mpc⁻¹ when z艋 0.042. 共47兲

TABLE II. Numerical values of the Hubble constant as deduced from ten different apparent magnitudes.

LF Matching z 共km s⁻¹Mpc⁻¹兲

1 Schechter z_pos-max 共58.35⫾30兲

2 Schechter 具z典obs 共71.73⫾12兲

3 M-L z_pos-max 共60.72⫾32兲

4 M-L 具z典obs 共71.20⫾12兲

5 Weighted mean 共65.26⫾8.22兲

6 Sample mean 共62.88⫾6.0兲

FIG. 8. The galaxies of the 2dFGRS, with b_J⬇14.385 or f

⬇189 983L䉺/Mpc², are isolated in order to represent a chosen value of m or f and then organized according to frequency versus heliocentric redshift.

The error bars are computed as the square root of the frequencies. The maximum in the frequency of observed galaxies is at z = 0 . 006 when M_䉺

= 5.33.

FIG. 9. Plot of z_maxas a function of the chosen magnitude共empty stars兲. The error bar in z is computed as the width of the bin. The dashed line represents the lower limit of the complete sample,苸s共z兲=100%, and the dash-dot-dash

line corresponds to苸s共z兲=90%. FIG. 10. The Hubble constant as deduced by the second method, see Eq.

共28兲, as a function of the selected magnitude共empty stars兲.

(8)

This value lies between the value deduced from the Cepheids¹³ and formula 共3兲 and the value deduced from- Wilkinson Microwave Anisotropy Probe¹⁴ and formula共4兲.

The developed framework also enables the deduction of the reference magnitude of the sun, see formula共29兲, and the application to the 2dFGRS gives

M_䉺=共5.5 ⫾ 0.35兲. 共48兲

Assuming that the exact value is M_䉺= 5.33, the efficiency in deriving the reference magnitude of the sun is 苸=96.63%

when H₀= 66. 04 km s⁻¹Mpc⁻¹. We briefly review the basic cosmological assumptions adopted here to derive the Hubble constant.

• The mechanism that produces the redshift, here ex- tracted from the catalog of galaxies, is not specified but we remember that the plasma redshift and DET do not produce a geocentric model for the universe as given by the Doppler shift.³⁸

• The number of galaxies as a function of redshift as well as the averaged redshift is evaluated in a Euclid- ean space or, in other words, the effects of space- curvature are ignored.

• The spatial inhomogeneities present in the catalog of galaxies are partially neutralized by the operation of adding together the data of the south and north galactic pole of the 2dFGRS. The transition from a nonhomogeneous to a quasihomogeneous universe is clear when Figs.5 and4 are carefully analyzed.

• The initial assumptions of 共i兲 natural flux decreasing, as given by Eq. 共15兲, and 共ii兲 the linear relationship between redshift and distance, which are present in the joint distribution in z and f for the number of galaxies, are justified by the acceptable results obtained for the theoretical maximum in the number of galaxies, see Fig. 6. This fact allows us to speak of a Euclidean universe up to z艋0.042.

• The presence of the Malmquist bias does not allow to

extrapolate the concept of a Euclidean, static universe for distances greater than z⬎0.042 when the 2dFGRS catalog is considered.

ACKNOWLEDGMENTS

I would like to thank the Smithsonian Astrophysical Observatory and John Huchra for the public file http://www.

cfa.harvard.edu/huchra/hubble.plot.dat which contains the published values of the Hubble constant.

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FIG. 11. The absolute reference magnitude of the sun, see Eq.共29兲, as a function of the selected magnitude共empty stars兲.