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PHYSICS ESSAYS 23, 2 共2010兲
New formulas for the Hubble constant in a Euclidean static universe
Lorenzo Zaninettia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−1 Mpc−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
Résumé: 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−1 Mpc−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 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−1 Mpc−1兴 and Ref. 6 共H0
⬃ 70– 73 km s−1 Mpc−1兲. We now report the methods that
use the global properties of galaxies as indicators of distance,
as follows:
The Hubble constant, in the following H0, is defined as
H0 =
v
关km s−1 Mpc−1兴,
D
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
共1兲 Luminosity
classes
of
spiral
galaxies:
H0
= 共55⫾ 3兲 km s−1 Mpc−1.7
共2兲 21 cm line widths: H0 = 共59.1⫾ 2 . 5兲 km s−1 Mpc−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−1 Mpc−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−1 Mpc−1.10
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−1 Mpc−1 as deduced by
Lemaitre,1 H0 = 460 km s−1 Mpc−1 as deduced by Robertson,2
H0 = 500 km s−1 Mpc−1 as deduced by Hubble,3 and H0
= 290 km s−1 Mpc−1 as deduced by Oort.4 Figure 1 reports
a兲
[email protected]
0836-1398/2010/23共2兲/298/8/$25.00
298
© 2010 Physics Essays Publication
Phys. Essays 23, 2 共2010兲
299
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 twodegree field galaxy redshift survey 共2dFGRS兲. Section V
contains a numerical evaluation of the reference magnitude
of the sun for a given catalog.
FIG. 1. Logarithmic values of the Hubble constant H0 from 1927 to 1980.
The error bar is evaluated according to the file in Ref. 39.
II. PRELIMINARIES
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−1,12 where the Hubble constant was named Hubble
parameter.
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 H0 from
a list of published data.
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−1 Mpc−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−1 Mpc−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?
A. Sample moments
Consider a random sample ␹ = x1 , 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, xi is
x̄ =
1
n
兺 xi ,
共5兲
and the standard deviation of the sample, ␴, is according to
Press et al.,15
␴=
冑
1
n−1
兺 共xi − x̄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x − ␮兲2
exp
−
,
␴共2␲兲1/2
2␴2
共7兲
where ␮ is the mean and ␴2 is the variance. Consider a
random sample ␹ = x1 , x2 , . . . , xn where 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
x
␮=
兺 ␴i2
i
1
兺 ␴2
,
共8兲
i
and an estimate of the error of the weighted mean, ␴共␮兲,
Phys. Essays 23, 2 共2010兲
300
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
FIG. 2. Histogram of frequencies of 355 published values of H0 during 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.
␴共␮兲 =
冑
1
1
兺 ␴2
共9兲
,
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.
Table I reports the statistics of this sample as well as the
minimum, H0,min and maximum H0,max.
TABLE I. The Hubble constant from a list of published values during the
period 1996–2008.
n
x̄
␴
H0 , max
H0 , min
␮
␴共␮兲
共10兲
V = H0D = c z,
i
see Ref. 18 for a detailed demonstration.
Entity
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 alternative mechanisms, the distance, r, in a Cartesian coordinate
system, x , y , z, is given by the usual Pythagorean theorem
r = 冑x2 + y 2 + 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
Definition
Value
No of samples
Average
Standard deviation
Maximum
Minimum
Weighted mean
Error of the weighted mean
355
65.85 km s−1 Mpc−1
10 km s−1 Mpc−1
98 km s−1 Mpc−1
30 km s−1 Mpc−1
66.04 km s−1 Mpc−1
0.25 km s−1 Mpc−1
where H0 is the Hubble constant H0 = 100h km s−1 Mpc−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 galaxies. 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
ln共1 + z兲 cm,
共Ne兲av
共11兲
where the averaged density of electrons, 共Ne兲av, is
共Ne兲av =
冉 冊
H0
−4 H0
cm−3 ,
5 ⬇ 2.42 ⫻ 10
3.076 ⫻ 10
74.5
共12兲
see Eqs. 共48兲 and 共49兲 in Ref. 23 or 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 冊
␲bc ␦␭2
D,
4
␭3
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
Phys. Essays 23, 2 共2010兲
301
冉 冊
M = m − 5 log
cz
− 25.
H0
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␲DL2
共15兲
where DL represents the distance of the galaxy expressed in
Mpc and
cz
.
DL =
H0
共16兲
The relationship connecting the absolute magnitude, M, of a
galaxy to its luminosity is
L
= 100.4共M 䉺−M兲 ,
L䉺
共17兲
2
zcrit
=
H20Lⴱ
.
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 = zpos-max, where
zpos-max = zcrit冑␣ + 2,
共24兲
which can be re-expressed as
冑2 + ␣冑100.4M䉺−0.4Mⴱ冑2 + ␣H0
.
zpos-max =
2冑␲冑 fc
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
HI0 =
NI
km s−1 MPc−1 ,
DI
NI = 2.997 ⫻ 1010zpos-max冑e0.921M 䉺−0.921m ,
DI = 冑2 + ␣冑100.4M 䉺−0.4M .
ⴱ
where M 䉺 is the reference magnitude of the sun in the bandpass under consideration.
The flux expressed in L䉺 / Mpc2 units as a function of the
apparent magnitude is
f = 7.957 ⫻ 10 e
8 0.921M 䉺−0.921m
共26兲
The mean redshift of galaxies with a flux f, see formula
共1.105兲 in Ref. 27 or formula 共1.119兲 in Ref. 28 is
具z典 = zcrit
L䉺
,
Mpc2
共25兲
共18兲
⌫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,
and the inverse relationship is
m = M 䉺 − 1.0857 ln共0.1256 ⫻ 10−8 f兲.
共19兲
HII0 =
NII
km s−1 Mpc−1 ,
DII
NII = 1.691 ⫻ 1010具z典obs ⫻ 冑␲冑e0.921M 䉺−0.921m⌫共5/2 + ␣兲,
C. The Schechter function
⌽共L兲dL =
冉 冊 冉 冊
⌽ⴱ L
Lⴱ Lⴱ
␣
exp −
L
dL.
Lⴱ
共20兲
Here, ␣ sets the slope for low values of L, Lⴱ is the characteristic luminosity, and ⌽ⴱ is the normalization. The equivalent distribution in absolute magnitude is
⌽共M兲dM = 共0.4 ln 10兲⌽ⴱ100.4共␣+1兲共M
0.4共M ⴱ−M兲
⫻ exp共− 10
ⴱ−M兲
兲dM ,
共21兲
where M ⴱ is the characteristic magnitude as derived from the
data. The joint distribution in z and f for galaxies, see formula 共1.104兲 in Ref. 27 or formula 共1.117兲 in Ref. 28, is
冉 冊 冉 冊
c
dN
= 4␲
d⍀dzdf
H0
5
z2
z⌽ 2 ,
zcrit
4
DII = ⌫共3 + ␣兲冑100.4M 䉺−0.4M ,
ⴱ
The Schechter function, introduced by Schechter,26 provides a useful fit for the luminosity of galaxies,
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 , zcrit is
共28兲
where 具z典obs is the averaged redshift as evaluated from the
considered catalog.
From formula 共27兲, it is also possible to derive the reference magnitude of the sun M 䉺 for the given catalog
冉
M 䉺 = M ⴱ + 1.085 ln 1.129 ⫻ 1012
2
具z典obs
f共⌫共2.5 + ␣兲兲2
H20共⌫共3 + ␣兲兲2
冊
.
共29兲
In this case, M 䉺 is the unknown and H0 is an input parameter.
D. The mass-luminosity relationship
A new LF of galaxies as derived in Ref. 30 is
⌿共L兲dL =
冉 冊冉 冊冉 冊
冉冉 冊冊
1
a⌫共c f 兲
⌿ⴱ
Lⴱ
⫻ exp −
L
Lⴱ
L
Lⴱ
共c f −a兲/a
1/a
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
Phys. Essays 23, 2 共2010兲
302
exponent that connects the mass to the luminosity, and c f
is connected with the dimensionality of the fragmentation,
c f = 2d, where d represents the dimensionality of the space
being considered: 1, 2, and 3. The distribution in absolute
magnitude is
冉
⌿共M兲dM = 0.4 ln 10
冊
1
ⴱ
⌿ⴱ100.4共c f /a兲共M −M兲
a⌫共c f 兲
⫻ exp共− 100.4共M
ⴱ−M兲共1/a兲
兲dM .
共31兲
This function contains the parameters M ⴱ, a, c f , and ⌿ⴱ,
which are derived from the operation of fitting the experimental data. The joint distribution in z and f, in the presence
of the M-L luminosity 关Eq. 共30兲兴, is
冉 冊 冉 冊
c
dN
= 4␲
d⍀dzdf
H0
5
z2
.
2
zcrit
z 4⌿
共32兲
The number of galaxies, NM-L共zf min , f max兲, comprised between f min and f max, can be computed through the following
integral:
N M-L共z兲 =
冕
f max
f min
冉 冊 冉 冊
c
4␲
H0
5
z2
z ⌿ 2 df ,
zcrit
4
IV. NUMERICAL VALUE OF THE HUBBLE CONSTANT
共33兲
and also in this case a numerical integration must be performed.
The number of galaxies as given by formula 共32兲 has a
maximum at zpos-max where
zpos-max = zcrit共c f + a兲a/2 ,
共34兲
which can be re-expressed as
共a + c f 兲1/2a冑100.4M 䉺−0.4M H0
ⴱ
zpos-max =
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
HIII
0 =
NIII
km s−1 Mpc−1 ,
DIII
FIG. 3. Cone-diagram of all the galaxies in the 2dFGRS. This plot contains
203 249 galaxies.
共36兲
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
NIII = 2.997 ⫻ 1010zpos-max冑e0.921M 䉺−0.921m ,
DIII = 共c f + a兲0.5a冑10.00.4M 䉺−0.4M .
ⴱ
共37兲
The mean redshift connected with the M-L LF is
具z典 = zcrit
2 4−共2a+c f 兲/a⌫共2a + c f 兲2共2c f +3a兲/a
,
⌫共c f + 3/2a兲
共38兲
and the fourth Hubble constant is
HIV
0 =
NIV
km s−1 Mpc−1
DIV
NIV = 8.457 ⫻ 109具z典obs冑␲冑e0.921M 䉺−0.921m⌫共c f + 3/2a兲,
DIV = 4−共2a+c f 兲/a冑100.4M 䉺−0.4M ⌫共2a + c f 兲2共2c f +3a兲/a . 共39兲
ⴱ
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°.
Phys. Essays 23, 2 共2010兲
303
FIG. 5. Histogram 共step-diagram兲 of the number of galaxies as a function of
the redshift when the two slices of Fig. 3 are added together. The number of
bins is 50.
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 = mL − 5 log10
冉 冊
cz
− 25.
H0
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
FIG. 6. Value of zpos-max at 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−1 Mpc−1. The
horizontal dotted line represents the boundary between complete and incomplete 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−1 Mpc−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.
and the galaxies of the 2dFGRS.
The interval covered by the LF of galaxies, ⌬M, is defined by
⌬M = M max − M min ,
共41兲
where M max and M min are the maximum and minimum absolute magnitudes of the LF for the considered catalog. The
real observable interval in absolute magnitude, ⌬M L, is
⌬M L = M L − 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 =
⌬M L
⫻ 100%.
⌬M
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 absolute magnitude of the sun in the bJ filter, M䉺 = 5.33.35–37
We now outline the algorithm that allows to deduce
zpos-max and 具z典obs from 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 astronomical z = zpos-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
Phys. Essays 23, 2 共2010兲
304
TABLE II. Numerical values of the Hubble constant as deduced from ten
different apparent magnitudes.
1
2
3
4
5
6
LF
Matching z
共km s−1 Mpc−1兲
Schechter
Schechter
M-L
M-L
Weighted mean
Sample mean
zpos-max
具z典obs
zpos-max
具z典obs
共58.35⫾ 30兲
共71.73⫾ 12兲
共60.72⫾ 32兲
共71.20⫾ 12兲
共65.26⫾ 8.22兲
共62.88⫾ 6.0兲
V. THE ABSOLUTE MAGNITUDE OF THE SUN
FIG. 8. The galaxies of the 2dFGRS, with bJ ⬇ 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.
of the sample and Fig. 9 reports the maximum value in
redshift zmax for each run in magnitude/flux.
Table II reports the four values of the Hubble constant
deduced here and Fig. 10 displays the data corresponding to
the constant deduced from Eq. 共28兲.
From a practical point of view, 苸, the percentage reliability of our results can also be introduced,
冉
苸= 1−
冊
兩共Qobs − Qnum兲兩
⫻ 100%,
Qobs
共44兲
where Qobs is the quantity given by the astronomical observations and Qnum 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−1 Mpc−1
and the observed value, see the weighted mean in Table I,
H0 = 66. 04 km s−1 Mpc−1.
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
Table I, H0 = 66. 04 km s−1 Mpc−1. Adopting this value of H0,
the absolute reference magnitude of the sun can be plotted in
Fig. 11 and the averaged value is
M 䉺 = 共5.50 ⫾ 0.35兲mag.
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 Table II gives
H0 = 共65.26 ⫾ 8.22兲 km s−1 Mpc−1 when z 艋 0.042.
FIG. 9. Plot of zmax as 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%.
共45兲
共47兲
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.
Phys. Essays 23, 2 共2010兲
305
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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See http://www.cfa.harvard.edu/huchra/hubble.plot.dat for the evaluated
error bar.
1
2
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.
This value lies between the value deduced from the
Cepheids13 and formula 共3兲 and the value deduced fromWilkinson 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−1 Mpc−1. We briefly review the basic
cosmological assumptions adopted here to derive the Hubble
constant.
•
The mechanism that produces the redshift, here extracted 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 Euclidean space or, in other words, the effects of spacecurvature 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 and 4 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