Journal of Scientific Research and Studies Vol. 1(6), pp. 95-117, December, 2014
ISSN 2375-8791
Copyright © 2014
Author(s) retain the copyright of this article
http://www.modernrespub.org/jsrs/index.htm
MRP
Full Length Research Paper
A cosmological model based on Type Ia supernova
discoveries at 0.01 < z ≤ 1.55 and dark energy evolution
Ümit Deniz Göker
Department of Physics, Boğaziçi University, Bebek, 34342, Istanbul, Turkey. E-mail: [email protected], Tel: +90212-3596604, Fax: +90-212-2872466.
Accepted 18 December, 2014
In this work, 257 Type Ia supernovae (SNe Ia) which have been discovered from different space
telescopes in the redshift range .
< ≤ .
were investigated to realize the history of cosmic
expansion over the last 13.5 Gyr. The Hubble parameter, ( ) was measured from plotted data. The
observational and the new theoretical luminosity distances of these 257 SNe Ia were determined from
the relations between light-curve shape and luminosity, and Hubble velocities, respectively. Using a
model of the expansion history for a flat universe, the transition between the two epochs was
constrained to be at z = 0.39±0.007 and the Hubble parameters were measured as ΩM (=Ω1) = 0.14±0.06
and ΩΛ (=Ω2) = 0.86±0.02. The radiation parameter did not take into account and accepted that the
universe had started to expand from a spherical distribution at the beginning. The Minkowskian metric
was used, nevertheless some assumptions were presented with the help of Neo-Newtonian cosmology
which shows similar features to Minkowskian metric and enables to apply the Newtonian laws, to make
the Minkowskian metric applicable to the shell model. In this paper, the astrophysical applications for
the Minkowskian metric were mainly presented rather than relativistic calculations.
Key words: Astrophysics, cosmological model, dark energy, supernovae Type Ia, Minkowskian metric, NeoNewtonian cosmology.
INTRODUCTION
In modern cosmology, there exist a number of unsolved
problems at the moment (the origin of dark matter, the
direct calculation of the Hubble constant, the definition of
gravitational energy, etc.). In the late nineties, Riess et al.
(1998) and Perlmutter and Schmidt (2003) measured the
redshift-distance relation of Type Ia supernovae (SNe Ia)
and found that the expansion of the universe is
accelerating rather than decelerating as was previously
thought. This led cosmologists to postulate some
unknown medium named dark energy, which penetrates
the universe and drives the accelerated expansion of
space. The dark energy with high density corresponds to
a low temperature universe (Saadat, 2012) and tends to
increase the rate of expansion of the universe (Riess et
al., 2004).
The expansion of the universe at the present time
appears to be accelerating from the observations of SNe
Ia at redshift < 1 (Riess et al., 2004) and the nearby
ones exhibit the most accurate measurements of the
present expansion rate (Riess et al., 2011), but at
redshift ≥ 1, it can easily be seen that the expansion of
the universe appears to be decelerating (Riess et al.,
2004). The cosmological principle says that our universe
is homogeneous and isotropic on the cosmic scale.
Indeed, the assumption of homogeneity and isotropy is
consistent with data coming from the Cosmic Microwave
Background (CMB) radiation, especially from the
Wilkinson Microwave Anisotropy Probe (WMAP), the
statistics of galaxies and the halo power spectrum, etc.
The main goal of this paper is finding the past and
predict future behaviors of the universe. For this purpose,
the study improved the shell model for the universe (this
shell structure was announced first by Tolga Yarman and
Metin Arik) and compared the theoretical model with the
astrophysical data. In this model, the study did not
consider the earlier stages of the universe formation; that
is, those just after the presumed Big-Bang but that it had
an initial radius before it started its expansion. The initial
96
J. Sci. Res. Stud.
expansion velocity was zero while a characteristic
acceleration was about 4.5 × 10
. The acceleration
decreases as inverse distance squared. The actual
acceleration of the universe, outward, becomes the
residual effects of the initial acceleration, leading to a
velocity near c, for the farthest locations. It is actually
around 10
for
= . The velocity cannot
increase indefinitely and it has to attain at most, the
speed-of-light ceiling.
The shell model was an applicable model but it needed
some additional assumptions to be an acceptable model.
The universe was formed inside a spherical shell with
expanding radius in a flat universe and there is no energy
exchange between inertial frames. For this reason, the
study included relativistic version of Newtonian
cosmological theory (in other words Neo-Newtonian
which the study preferred the name to use in this paper
or Post-Newtonian cosmology as indicated by some
cosmologists (Bain, 2004a; Wuthrich, 2007; Flender and
Schwarz, 2012) inside the Minkowskian metric with some
assumptions to test our understanding of structure
formation. But the question is how reliable are these
methods, since they are Newtonian rather than relativistic
equations and how reliable is Newtonian cosmology on
large scales? The study had to find a correspondence
between relativistic cosmological perturbation theory and
Neo-Newtonian cosmology. Furthermore, redshift space
distortions are due to the peculiar motion of galaxies in
the classical Newtonian cosmology but for the relativistic
formulation of Newtonian cosmology, the study would
receive contributions from the peculiar motion and from
space-time metric perturbations, most importantly the
local variation of the cosmic expansion rate (Flender and
Schwarz, 2012). This clearly demonstrates that the NeoNewtonian cosmology for different SNe Ia can be applied
in different redshifts and mix the quantities defined on
these different spatial hypersurfaces and in the largest
scales Minkowski space assumptions can be used. In this
paper, the author clarified how these different spacetimes
could be applied to each other.
In the cosmological principle, for spatially flat models,
positive cosmological constant (i.e., ΩΛ > 0) and a current
acceleration of the expansion (i.e., < 0) occurs (Riess
et al., 1998). In these models, the matter density
decreases with time in an expanding universe while the
vacuum energy density remains constant (Riess et al.,
1998) and the radiation was dominant for the redshift
> 1.25.
In the shell model, the universe was accepted as
overall electrically neutral, the gravitational interaction
dominates at larger scales and the radiation was
neglected (Yarman and Kholmetskii, 2013). After infinite
time, the Hubble expansion parameter behaves as a
constant (Saadat, 2012). At the end, the study
determined the maximum radius of the universe to be
= 6 × 10 " before it started to collapse. The set of
257 Sne Ia (including high-redshift Sne Ia) data was
used to find the following cosmological parameters:
Hubble constant (# ), the cosmological constant, (ΩΛ),
the vacuum energy density (ΩM), the deceleration
parameter ( ) and the dynamical age of the universe
($ ).
ASTROPHYSICAL APPLICATIONS
Galactic coordinates
The galaxies have been separated uniformly in the
universe but when the expansion rate started to increase,
the galaxies began to separate in an inhomogeneous and
anisotropic way with respect to each other. In the furthest
and largest scales, the universe will become isotropic and
homogeneous again where it is spherically symmetric at
any point (Mitra, 2011). SNe Ia are the best known data
for cosmic analysis because they are standard candles to
measure the distance. The visual magnitude of the
supernovae depends primarily on the distance. The study
calculated the motions of 257 SNe Ia with respect to us.
Similar work had been done by Kalus (2010) but there,
the
Friedmann-Robertson-Lemaitre-Walker
(FRLM)
Metric was used.
The origin in this model, is the galactic center near
Sagittarius A in the Northern part of the galaxy. In the
southern part of the galaxy, there is more dust and it is
not easy to find a reference star. As a result of the orbital
motion of Earth around the Sun, their synchronized
motion with the solar system around the center of the
Milky Way and all their rotations around the center of
Local Group of Galaxies (LGG), there are preferred
directions determined by orbital motions (Borissova,
2006). So, the real universe is anisotropic (inequivalence
of directions) and there are billions of centers of
gravitational attraction, and the universe is also
inhomogeneous (inequivalence of events at same cosmic
time). The homogeneous isotropic metric spaces can
only be possible if there are no stars, galaxies or other
space bodies existing in the universe (Borissova, 2006).
The Galactic coordinates are called Galactic longitude (l)
and Galactic latitude (b). The galactic latitude of the
object measures the angular distance of an object
perpendicular to the galactic equator, positive to the
north, negative to the south (-90°≤ b ≤ +90°) and the
galactic longitude of the object measures the angular
distance of an object eastward along the galactic equator
from the galactic center (0° ≤ l ≤ 360°) (Kalus, 2010).
The study used the results of ground-based telescopes
and space-based searches to study the early expansion
of the Universe. Data were collected for 257 SNe Ia at
0.01 < ≤ 1.55, among them there are 14 SNe Ia data
from Hubble Space Telescope (HST) and ground-based
telescopes such as the Keck telescopes, the Multiple
Mirror Telescope (MMT) and European Southern
Observatory 3.6 m (ESO 3.6-m) (Riess et al., 1998); 22
Göker
SNe Ia data from HST and the ground-based telescopes
as Keck telescopes, Very Large Telescope (VLT) and
Magellan Telescope (Riess et al., 2004); 198 SNe Ia data
from 4-m Blanco Telescope at the Cerro Tololo InterAmerican Observatory (CTIO) (Wood-Vasey et al., 2007)
and 23 SNe Ia data from HST for > 1 (Riess et al.,
2007). The redshift ranges in my work are classified as
low-redshift for < 0.1; intermediate-redshift for 0.1 < ≤
1.01; high-redshift for 1.23 ≤ ≤ 1.39 and highest-redshift
for
> 1.39. The snapshot distance modulus for
supernova 1995ao, 1995ap, 1996R and 1996T were
taken from Riess et al. (1998).
The relation of the proper distance of astronomical
objects to their redshifts encodes information about the
geometry and expansion history of the universe (Riess et
al., 1998); but it is not possible to determine the proper
distance by observations while the radial distance of an
object can easily be calculated from their lights, which
takes a certain time to reach us. The study indicated the
redshift, galactic coordinates, distance modulus, proper
and heliocentric radial velocity values for J2000
coordinates in Table 1.
Distance modulus
The observational distance modulus of supernova can
easily be derived from well-known distance formula. The
distance estimates from SN Ia light curves is given by:
'( = )
(
*+,
-
/
(1)
where L and F are the intrinsic luminosity and observed
flux of the supernovae (SNe) within a given passband,
respectively. In Equation 1, SNe have roughly a common
luminosity which is well calibrated by their light curves
and it is independent of their redshifts. The luminosity
distance '( (in units of megaparsecs) is a distance which
infers based on flux received and it is different from the
proper distance '/ which is a function of redshift and
Hubble parameter. The extinction-corrected distance
modulus (Riess et al., 2004) is shown to be:
0 =
− 2 = 5345 )
67
8/9
- + 25 + ;< + =
(2)
In Equation 2, ;< is the extinction parameter, K is the
correction factor, m is the visual (apparent) magnitude
and M is the absolute magnitude. The extinction
parameter ;< which describes the extinction is due to
dust of the host and Galaxy. The dust between galaxies
made the distant supernovae fainter by absorbing some
of their light (Riess et al., 1998). ;< has been calculated
explicitly as a function of SNe Ia age from accurate
spectrophotometry of SNe Ia. The mean Galactic
reddening-law parameter < = 3.1 is used to determine
the extinction in extragalactic Sne (Riess et al., 2007).
97
The K-correction factor on the right-hand side of
Equation 2 depends on the observed wavelength, the
emitted spectrum and the redshift (Kalus, 2010). Since
astronomical observations are normally made in fixed
band passes on Earth, corrections need to be applied to
account for the differences caused by the spectrum
shifting within these band passes (Perlmutter and
Schmidt, 2003). K-corrections are used to account for the
SNe redshift and provide a transformation between an
observed-frame magnitude and a rest-frame magnitude.
K-corrections were calculated with Vega-normalized
cross-bands (Riess et al., 2004).
In the calculations, the magnitude differences (0 ) were
found by using different references (for 0.0141 ≤ z ≤
0.061 (Riess et al., 2004); 0.16 ≤ z ≤ 0.97 (Riess et al.,
1998); 0.0154 ≤ z ≤ 1.01 (Wood-Vasey et al., 2007);
0.839 ≤ z ≤ 1.39 (Kalus, 2010) and 0.216 ≤ z ≤ 1.551
(Riess et al., 2007) from the literature. In Figure 1, the
Hubble diagram of distance modulus (observational)
versus redshift for 257 SNe Ia is shown and these SNe Ia
span a wide range of redshift (0.01< z ≤ 1.55). The
observational luminosity distances are plotted by using a
classical cosmology model (ΩM = 0.29 and ΩΛ = 0.71) for
the flat universe. The best fit value was found as > =
0.9906 and it is in a good correlation with the distance
modulus of Riess et al. (2004).
Heliocentric radial velocity and peculiar velocities
Heliocentric radial velocity indicates the radial velocity of
an object from us by using the position of the Sun and
rotation around the center of the Milky-Way in today's
coordinates and it is given in Table 1. The orbit of the
-1
Sun around the center of the galaxy is about 220 kms
and its orbital period is about Porb=240 million years
(Borissova, 2006).
The effects of peculiar velocities on cosmological
parameters include our own peculiar motion, SNe's
motion and coherent bulk motion. The distance from the
sun to the center of galaxy is very small, at about 0.002%
of the nearest SNe Ia from the sun. So, calculations were
done from the center of galaxy to make the model easier
and to decrease the effect of the perturbations originating
from the rotation.
In Figure 2, the heliocentric radial velocities and
logarithmic ?/ values relate with the Hubble constant and
proper distances are shown. Different authors have used
different data (e.g., Cepheids, supernova, etc.) and found
several Hubble constant values. All of them were plotted
and found the best fit value for # = 71 km/s/Mpc. As wee
see from the logarithmic ?/ values in Figure 2b, the path
of SNe Ia data show the similar increase with the
observational luminosity distance, 0 . The best fit value is
0.9995. The Hubble flow is given by:
/
=
≈ # ?/
(3)
98
J. Sci. Res. Stud.
Table 1. The Redshift, Galactic Coordinates, Distance Modulus, Proper Velocity, Heliocentric Radial Velocity and Relativistic Radial Velocity values for J2000 Coordinates.
SN Name
a
SN 1991ag
c
SN 2001cn
c
SN 2001V
c
SN 2001cz
c
SN 1996bo
c
SN 2000dk
c
SN 1997Y
c
SN 1998ef
c
SN 1996bv
c
SN 1994S
c
SN 1998V
c
SN 1999ek
c
SN 1992bo
c
SN 1992bc
c
SN 2000fa
c
SN 1995ak
c
SN 2000cn
c
SN 1998eg
c
SN 1994M
c
SN 2000ca
c
SN 1993H
c
SN 1992ag
c
SN 1992P
c
SN 1999gp
c
SN 1996C
c
SN 1998ab
c
SN 1997dg
c
SN 2001ba
c
SN 1990O
c
SN 1999cc
a
SN 1991U
c
SN 1996bl
c
SN 1994T
c
SN 2000cf
c
SN 1999aw
Redshift
(z)
a
0.0141
c
0.0154
c
0.0162
c
0.0163
c
0.0163
c
0.0164
c
0.0166
c
0.0167
c
0.0167
c
0.0160
c
0.0172
c
0.0176
c
0.0181
c
0.0198
c
0.0218
c
0.0220
c
0.0232
c
0.0235
c
0.0243
c
0.0245
c
0.0248
c
0.0259
c
0.0263
c
0.0260
c
0.0275
c
0.0279
c
0.0297
c
0.0305
c
0.0306
c
0.0315
a
0.0331
c
0.0348
c
0.0357
c
0.0365
c
0.0392
Distance modulus
(µ
µ0)
a
34.130
c
34.058
c
34.142
c
34.275
c
33.983
c
34.370
c
34.530
c
34.164
c
34.171
c
34.350
c
34.360
c
34.279
c
34.734
c
34.838
c
34.901
c
34.702
c
35.118
c
35.318
c
35.242
c
35.245
c
35.098
c
35.138
c
35.597
c
35.624
c
35.940
c
35.175
c
36.149
c
35.876
c
35.805
c
35.822
a
35.540
c
36.090
c
36.024
c
36.363
c
36.539
Galactic Longitude
(l)
f
342.56
f
329.64
f
218.93
f
302.11
f
144.46
f
126.83
f
124.77
f
125.88
f
157.34
f
187.38
f
43.940
f
189.40
f
261.99
f
245.69
f
194.17
f
169.66
f
53.440
f
76.460
f
291.69
f
313.20
f
318.22
f
312.49
f
295.61
f
143.25
f
99.620
f
124.86
f
103.61
f
285.39
f
37.650
f
59.667
f
311.82
f
116.99
f
318.01
f
99.880
f
260.24
Galactic Latitude
(b)
f
-31.64
f
-24.05
f
+77.73
f
+23.28
f
-48.96
f
-30.34
f
+62.37
f
-30.56
f
+17.97
f
+85.14
f
+13.35
f
-08.23
f
-80.35
f
-59.64
f
+15.48
f
-48.98
f
+23.32
f
-42.06
f
+63.03
f
+27.83
f
+30.34
f
+38.38
f
+73.11
f
-19.50
f
+65.03
f
+75.19
f
-33.98
f
+28.03
f
+28.36
f
+48.74
f
+36.21
f
-51.30
f
+59.84
f
+42.16
f
+47.45
Proper Velocity
(km/s)
f
4197
f
4497
f
4857
f
5096
f
5096
f
4917
f
4977
f
5396
f
5096
f
4497
f
5096
f
5276
f
5336
f
5576
f
6535
f
6895
f
6985
f
7015
f
6895
f
7495
f
7195
f
7855
f
7945
f
7795
f
8994
f
8334
f
8904
f
9294
f
9204
f
9473
f
9593
f
10793
f
10493
f
10793
f
11992
Heliocentric Radial
Velocity (km/s)
f
4168
f
4463
f
4817
f
5053
f
5053
f
4876
f
4935
f
5348
f
5053
f
4463
f
5053
f
5230
f
5289
f
5524
f
6464
f
6816
f
6904
f
6933
f
6816
f
7401
f
7109
f
7752
f
7839
f
7693
f
8859
f
8218
f
8772
f
9150
f
9062
f
9324
f
9440
f
10598
f
10309
f
10598
f
11752
Relativistic Radial
Velocity (km/s)
*
29.0508
*
34.0683
*
40.0938
*
43.1110
*
43.1110
*
41.0685
*
42.1034
*
48.1389
*
43.1110
*
34.0684
*
43.1110
*
46.1273
*
47.1330
*
52.1606
*
71.3012
*
79.3730
*
81.3925
*
82.4007
*
79.3730
*
94.5243
*
86.4421
*
103.6310
*
106.6643
*
102.6153
*
136.0843
*
116.7999
*
133.0391
*
145.2351
*
143.1943
*
150.3278
*
154.3982
*
197.2563
*
186.0121
*
197.2563
*
243.4306
Göker
99
Table 1. Contd.
c
SN 1992bl
c
SN 1992bh
a
SN 1992J
c
SN 1995ac
a
SN 1990T
c
SN 1990af
c
SN 1993O
c
SN 1998dx
a
SN 1991S
c
SN 1993ag
a
SN 1992au
c
SN 1992bs
c
SN 1993B
c
SN 1992ae
c
SN 1992bp
c
SN 1992br
c
SN 1992aq
c
m075
c
SN 1996ab
c
m032
c
e020
b
SN 1996R
c
k429
c
m057
c
d086
c
d099
c
h363
a
SN 2002kc
c
n404
c
g005
c
e132
c
m039
b
SN 1995ao
b
SN 1996T
c
m022
c
SNLS 04D3ez
c
m043
c
0.0429
c
0.0451
a
0.046
c
0.0488
a
0.040
c
0.0502
c
0.0519
c
0.0537
a
0.0560
c
0.0500
a
0.0610
c
0.0634
c
0.0707
c
0.0748
c
0.0789
c
0.0878
c
0.1009
c
0.1020
c
0.1242
c
0.155
c
0.159
b
0.160
c
0.181
c
0.184
c
0.205
c
0.211
c
0.213
a
0.216
c
0.216
c
0.218
c
0.239
c
0.249
b
0.240
b
0.240
c
0.240
c
0.263
c
0.266
c
36.487
c
36.906
a
36.350
c
36.566
a
36.380
c
36.691
c
37.123
c
36.917
a
37.310
c
37.070
a
37.300
c
37.642
c
37.783
c
37.723
c
37.782
c
37.764
c
38.799
c
40.785
c
38.899
c
39.954
c
39.786
b
39.080
c
39.891
c
41.327
c
40.075
c
40.422
c
40.333
e
40.330
c
40.590
c
40.371
c
40.424
c
40.799
b
40.330
b
40.680
c
41.634
c
40.765
c
40.819
f
344.13
f
267.85
f
263.54
f
58.690
f
341.50
f
330.82
f
312.41
f
77.670
f
214.06
f
268.43
f
319.11
f
240.03
f
273.32
f
332.70
f
208.83
f
288.01
f
1.7758
f
71.150
f
43.160
f
70.675
f
69.862
f
259.07
f
177.087
f
166.34
f
70.610
f
163.93
f
164.71
f
223.34
f
180.043
f
72.391
f
164.62
f
177.10
f
178.19
f
247.59
f
73.104
d
96.130
f
72.436
f
-63.92
f
-37.33
f
+23.54
f
-55.05
f
-31.52
f
-42.23
f
+28.92
f
+26.66
f
+57.42
f
+15.93
f
-65.88
f
-55.34
f
+20.46
f
-41.99
f
-51.10
f
-59.43
f
-65.32
f
-62.28
f
+56.93
f
-64.19
f
-63.17
f
+54.36
f
-59.98
f
-60.44
f
-63.33
f
-60.11
f
-61.27
f
-54.39
f
-60.16
f
-63.02
f
-60.01
f
-59.98
f
-50.52
f
+36.99
f
-63.26
d
+59.38
f
-63.50
f
12891
f
13491
f
13491
f
14990
f
11992
f
14990
f
15589
f
16129
f
16489
f
14990
f
18287
f
19187
f
21285
f
22484
f
23684
f
26382
f
30279
f
30579
f
36874
f
46468
f
49166
f
45269
f
51265
f
55162
f
60858
f
62956
f
63256
f
64755
f
63256
f
65355
f
73149
f
74648
f
89938
f
72250
f
71950
f
77946
f
79745
f
12614
f
13187
f
13187
f
14615
f
11752
f
14615
f
15184
f
15696
f
16036
f
14615
f
17731
f
18574
f
20531
f
21643
f
22751
f
25225
f
28757
f
29027
f
34622
f
42904
f
45180
f
41884
f
46935
f
50158
f
54785
f
56464
f
56703
f
57893
f
56703
f
58367
f
64434
f
65580
f
76898
f
63743
f
63512
f
68078
f
69427
*
281.5787
*
309.5063
*
309.5063
*
383.3994
*
243.4306
*
383.3995
*
414.8189
*
444.2466
*
465.3034
*
383.3995
*
574.6323
*
635.6534
*
788.4557
*
884.0182
*
986.1364
*
1239.4860
*
1667.1656
*
1703.1768
*
2581.5775
*
4451.4739
*
5124.2626
*
4176.9738
*
5701.9621
*
6918.1148
*
9110.5553
*
10072.5303
*
10217.9958
*
10977.1999
*
10217.9958
*
11297.5266
*
16484.7096
*
17764.4245
*
42283.4671
*
17569.5660
*
15538.7032
*
21024.4085
*
23116.1612
100
J. Sci. Res. Stud.
Table 1. Contd.
c
n326
c
k396
c
p455
c
SNLS 03D4ag
c
m027
c
SNLS 03D3ba
c
g055
c
n278
c
d117
b
SN 1995ap
b
SN 1996J
c
m062
c
b016
c
SNLS 03D1fc
c
e029
c
d083
c
SNLS 04D3kr
c
SNLS 04D3nh
c
m193
c
d149
c
h364
c
SNLS 03D1bp
c
h359
c
d087
c
g097
c
e136
c
SNLS 04D3fk
c
SNLS 04D2fs
e
HST04Kur
c
d093
c
n263
c
SNLS 04D2cf
c
SNLS 03D3ay
c
g052
b
SN 1996K
c
f308
c
g142
c
0.268
c
0.271
c
0.284
c
0.285
c
0.286
c
0.291
c
0.302
c
0.309
c
0.309
b
0.300
b
0.300
c
0.317
c
0.329
c
0.331
c
0.332
c
0.333
c
0.337
c
0.340
c
0.341
c
0.342
c
0.344
c
0.346
c
0.348
c
0.340
c
0.340
c
0.352
c
0.358
c
0.357
e
0.359
c
0.363
c
0.368
c
0.369
c
0.371
c
0.383
b
0.380
c
0.394
c
0.399
c
40.813
c
40.289
c
41.102
c
40.977
c
41.532
c
40.565
c
41.391
c
41.163
c
41.424
b
40.740
b
40.990
c
41.279
c
41.349
c
41.242
c
41.505
c
40.709
c
41.456
c
41.635
c
41.291
c
41.626
c
41.323
c
41.497
c
41.888
c
41.320
c
41.559
c
41.618
c
41.414
c
41.645
e
41.230
c
41.726
c
41.445
c
41.807
c
41.800
c
41.563
b
42.210
c
42.429
c
41.960
f
72.574
f
71.807
f
165.96
f
39.149
f
132.58
d
95.740
f
135.00
f
71.123
f
179.67
f
179.36
f
253.22
f
132.76
f
71.545
d
172.02
f
132.60
f
179.51
d
95.910
d
94.410
f
177.35
f
166.29
f
179.16
d
173.22
f
166.27
f
133.53
f
70.538
f
165.59
d
95.560
f
237.28
f
223.66
f
178.93
f
164.61
f
237.47
d
95.600
f
70.234
f
224.38
f
178.89
f
72.250
f
-63.48
f
-62.80
f
-60.20
f
-52.76
f
-62.40
d
+60.18
f
-62.51
f
-63.54
f
-60.13
f
-46.15
f
+34.30
f
-61.91
f
-64.01
d
-57.24
f
-61.52
f
-60.64
d
+60.08
d
+59.58
f
-59.83
f
-60.43
f
-60.37
d
-57.60
f
-61.33
f
-62.24
f
-63.57
f
-59.89
d
+59.85
f
+41.84
f
-54.42
f
-60.22
f
-61.69
f
+42.23
d
+59.94
f
-63.47
f
+20.46
f
-60.36
f
-63.15
f
80344
f
89938
f
85141
f
85441
f
85741
f
86940
f
90537
f
92636
f
88739
f
68952
f
89938
f
94734
f
97433
f
99531
f
100430
f
99831
f
101929
f
101929
f
100730
f
101630
f
103129
f
104028
f
104328
f
101929
f
101929
f
107925
f
107925
f
107026
f
107625
f
108825
f
110324
f
110623
f
111193
f
115120
f
113921
f
118118
f
119317
f
69875
f
76898
f
73419
f
73639
f
73858
f
74731
f
77329
f
78827
f
76035
f
61190
f
76898
f
80313
f
82207
f
83665
f
84287
f
83873
f
85318
f
85318
f
84494
f
85112
f
86138
f
86751
f
86955
f
85318
f
85318
f
89382
f
89382
f
88779
f
89181
f
89984
f
90981
f
91180
f
91557
f
94135
f
93352
f
96076
f
96846
*
23869.3304
*
42283.4607
*
31266.7299
*
31826.6748
*
32402.2208
*
34854.1089
*
44043.3396
*
51186.4927
*
39056.4217
*
13427.1443
*
42283.4670
*
60297.8325
*
76500.0549
*
94850.4285
*
105162.2468
*
98089.9242
*
127421.9473
*
127421.9473
*
109045.528
*
122349.4435
*
152157.1827
*
177115.1715
*
187177.0299
*
127421.9473
*
127421.9473
*
524632.1486
*
524632.1486
*
366156.5581
*
458925.9816
*
908038.5096
5173606.694
2245016.866
1087882.098
250770.4155
324047.2086
163479.3353
144473.2738
Göker
Table 1. Contd.
c
d085
c
k448
c
f096
c
SNLS 04D2fp
c
k485
c
f076
c
h342
c
g133
c
f235
c
b020
c
b013
c
e148
c
d097
c
d089
b
SN 1996E
b
SN 1996U
c
SNLS 04D2gb
c
SNLS 03D3aw
b
SN 1997ce
c
SNLS 04D3gt
c
p425
c
SNLS 03D3cd
c
SNLS 03D3cc
c
m158
c
e108
e
HST04Yow
e
SN 2002dc
c
SNLS 04D3df
b
SN 1995K
c
g160
c
h319
c
SNLS 03D1ax
c
e149
e
HST04Haw
c
h283
c
SNLS 03D1au
c
p524
c
0.401
c
0.401
c
0.412
c
0.415
c
0.416
c
0.410
c
0.421
c
0.421
c
0.422
c
0.425
c
0.426
c
0.429
c
0.436
c
0.436
b
0.430
b
0.430
c
0.430
c
0.449
b
0.440
c
0.451
c
0.453
c
0.461
c
0.463
c
0.463
c
0.469
e
0.460
e
0.475
c
0.470
b
0.480
c
0.493
c
0.495
c
0.496
c
0.497
e
0.490
c
0.502
c
0.504
c
0.508
c
41.956
c
42.342
c
41.613
c
42.062
c
42.163
c
41.473
c
42.179
c
42.216
c
41.777
c
41.766
c
41.976
c
42.249
c
42.097
c
42.048
b
42.030
b
42.340
c
41.809
c
42.073
b
42.260
c
41.350
c
42.267
c
42.125
c
42.254
c
42.580
c
42.275
e
42.230
e
42.250
c
42.027
b
42.490
c
42.385
c
42.395
c
42.329
c
42.230
e
42.540
c
42.495
c
42.553
c
42.428
f
71.680
f
132.38
f
70.619
f
236.50
f
165.49
f
132.64
f
179.88
f
165.55
f
134.06
f
164.64
f
179.51
f
179.51
f
179.68
f
177.98
f
253.12
f
259.36
f
237.54
d
95.120
f
69.240
d
94.820
f
73.041
d
95.620
d
95.280
f
69.507
f
179.83
d
125.53
d
125.49
d
95.280
f
259.95
f
179.83
f
165.45
f
171.72
f
179.59
d
125.72
f
164.19
d
171.37
f
179.54
f
-63.18
f
-62.54
f
-62.43
f
+41.98
f
-60.46
f
-63.28
f
-59.93
f
-60.39
f
-62.09
f
+61.95
f
-59.98
f
-59.98
f
-59.87
f
-60.21
f
34.31
f
+68.00
d
+42.70
d
+59.54
f
+36.62
d
+59.34
f
-63.25
d
+59.77
d
+59.71
f
-62.80
f
-60.52
d
+55.14
d
+55.12
d
+60.07
f
+43.33
f
-60.13
f
-60.75
f
-58.49
f
-59.95
d
+55.14
f
-61.75
d
-57.42
f
-60.38
f
121416
f
120217
f
122315
f
124414
f
124714
f
123215
f
126213
f
125913
f
125013
f
127412
f
128311
f
128011
f
130110
f
128611
f
127412
f
128911
f
128911
f
134607
f
131909
f
134907
f
135806
f
131909
f
140902
f
138804
f
141502
f
137005
f
142401
f
140902
f
143301
f
147798
f
143301
f
149896
f
148997
f
146898
f
148397
f
149896
f
153194
f
98185
f
97421
f
98756
f
100079
f
100267
f
99324
f
101204
f
101017
f
100455
f
101950
f
102508
f
102322
f
103616
f
102693
f
101950
f
102878
f
102878
f
106353
f
104717
f
106534
f
107075
f
104717
f
110104
f
108864
f
110457
f
107793
f
110984
f
110104
f
111509
f
114108
f
111509
f
115305
f
114793
f
113592
f
114451
f
115305
f
117166
120917.3315
133180.6273
113300.3665
99276.7421
97604.5566
106741.7072
90179.0525
91551.1713
95997.9017
85167.0632
81838.8384
82912.0988
76099.8686
80805.6389
85167.0632
79804.2771
79804.2771
65465.5418
71311.7799
64894.3658
63264.2497
71311.7799
55855.4676
58582.3006
55143.2084
61264.4661
54127.7413
55855.4676
53169.6482
49075.4887
53169.6482
47490.593
48148.7925
49811.2776
48604.0959
47490.593
45323.2092
101
102
J. Sci. Res. Stud.
Table 1. Contd.
b
SN 1997cj
c
d084
c
g120
c
SNLS 04D2gc
e
HST05Zwi
c
n258
a
SN 2002hr
c
SNLS 04D1ak
c
n285
c
d033
c
SNLS 03D3af
c
f011
c
SNLS 03D1gt
c
f244
c
SNLS 04D3hn
c
SNLS 04D1ag
c
SNLS 04D4bq
c
f041
c
m034
c
SNLS 03D4gl
b
SN 1996I
c
SNLS 03D1aw
c
SNLS 03D4gf
c
m138
c
k430
c
d058
c
b010
c
SNLS 03D4gg
c
f216
c
h323
c
SNLS 03D4dy
c
e138
c
SNLS 04D4an
c
f231
c
SNLS 04D3do
c
SNLS 03D4dh
b
SN 1996H
b
0.500
c
0.519
c
0.510
c
0.521
e
0.521
c
0.522
a
0.526
c
0.526
c
0.528
c
0.531
c
0.532
c
0.539
c
0.548
c
0.540
c
0.552
c
0.557
c
0.550
c
0.561
c
0.562
c
0.571
b
0.570
c
0.582
c
0.581
c
0.582
c
0.582
c
0.583
c
0.591
c
0.592
c
0.599
c
0.603
c
0.604
c
0.612
c
0.613
c
0.619
c
0.610
c
0.627
b
0.620
b
42.700
c
42.948
c
42.304
c
42.436
e
42.050
c
42.740
e
43.080
c
42.494
c
42.631
c
42.960
c
42.844
c
42.661
c
42.417
c
42.721
c
42.279
c
42.598
c
42.749
c
42.718
c
42.799
c
42.445
b
42.830
c
43.119
c
42.846
c
42.815
c
43.311
c
43.103
c
42.984
c
42.869
c
43.339
c
43.009
c
42.790
c
42.990
c
43.076
c
43.046
c
42.816
c
42.938
b
43.010
f
125.80
f
179.54
f
134.65
f
237.41
f
223.41
f
164.52
f
223.38
d
172.87
f
70.979
f
69.938
f
94.950
f
166.76
d
171.73
f
176.79
d
94.340
d
171.86
d
39.840
f
131.89
f
177.42
d
40.300
f
276.85
d
172.00
d
39.890
f
132.25
f
134.67
f
131.92
f
178.67
f
38.768
f
180.16
f
178.62
f
38.827
f
180.29
d
40.220
f
73.212
d
95.360
f
39.725
f
290.75
f
+54.61
f
+60.82
f
-61.78
d
+42.55
f
-54.36
f
-61.02
f
-54.44
d
-57.06
f
-62.06
f
-63.07
d
+59.56
f
-60.96
d
-57.36
f
-59.93
d
+59.69
d
-57.52
d
-53.31
f
-62.13
f
-60.21
d
-53.17
f
+60.01
d
-57.42
d
-53.18
f
-61.89
f
-61.75
f
-62.53
f
-59.84
d
-53.84
f
-58.85
f
-60.09
f
-52.86
f
-59.53
d
-53.50
f
-63.35
d
+60.12
d
-53.82
f
+62.24
f
149896
f
156492
f
151695
f
156192
f
156192
f
156492
f
157691
f
157691
f
158290
f
157091
f
159490
f
161588
f
167884
f
163087
f
164886
f
166984
f
164886
f
167284
f
168483
f
170882
f
170882
f
173880
f
173880
f
174479
f
172680
f
174779
f
175978
f
177477
f
179576
f
180775
f
182873
f
183173
f
183773
f
185871
f
182873
f
187970
f
185871
f
115305
f
119003
f
116323
f
118837
f
118837
f
119003
f
119665
f
119665
f
119995
f
119335
f
120653
f
121796
f
125170
f
122607
f
123574
f
124693
f
123574
f
124852
f
125487
f
126747
f
126747
f
128306
f
128306
f
128616
f
127685
f
128770
f
129387
f
130153
f
131218
f
131823
f
132875
f
133024
f
133323
f
134362
f
132875
f
135393
f
134362
47490.593
43475.5004
46264.4123
43631.9402
43631.9402
43475.5004
42869.8975
42869.8975
42578.3451
43167.2057
42017.9842
41104.1597
38781.4089
40497.7874
39816.4059
39079.6813
39816.4059
38979.4284
38587.6461
37855.5876
37855.5876
37022.3927
37022.3927
36865.0191
37344.4458
36788.6826
36486.8285
36127.4964
35652.7672
35394.7419
34965.4705
34907.1319
34790.3018
34398.9598
34965.4705
34030.8123
34398.9598
Göker
Table 1. Contd.
c
SNLS 04D3co
c
n256
c
e140
c
g050
c
SNLS 03D4at
e
HST05Dic
c
SNLS 04D3cy
c
e147
a
SN 2003be
c
m026
c
m226
c
SNLS 03D1co
a
SN 2003bd
c
g240
c
h300
c
SNLS 03D1fl
c
k441
c
SNLS 04D2iu
c
SNLS 03D4cz
c
p454
c
SNLS 04D2gp
c
SNLS 04D3is
c
SNLS 04D1aj
a
SN 2002kd
c
SNLS 04D3fq
c
SNLS 04D2ja
e
HST04Rak
c
SNLS 04D3ks
c
SNLS 04D3oe
c
h311
c
SNLS 03D4fd
c
SNLS 04D4dm
c
SNLS 04D3nc
c
SNLS 04D3ny
c
SNLS 04D3lu
a
SN 2003eq
e
HST05Spo
c
0.620
c
0.631
c
0.631
c
0.633
c
0.633
e
0.638
c
0.643
c
0.645
a
0.640
c
0.653
c
0.671
c
0.679
a
0.670
c
0.687
c
0.687
c
0.688
c
0.680
c
0.691
c
0.695
c
0.695
c
0.707
c
0.710
c
0.721
a
0.735
c
0.730
c
0.741
e
0.740
c
0.752
c
0.756
c
0.750
c
0.791
c
0.811
c
0.817
c
0.810
c
0.822
a
0.839
e
0.839
c
43.203
c
43.086
c
42.893
c
42.767
c
43.257
e
42.890
c
43.336
c
43.015
e
43.010
c
43.023
c
43.129
c
43.586
e
43.190
c
43.038
c
43.092
c
43.128
c
43.243
c
43.420
c
43.110
c
43.530
c
43.438
c
43.714
c
43.455
e
43.140
c
43.571
c
43.579
e
43.380
c
43.360
c
43.538
c
43.445
c
43.754
c
43.917
c
43.721
c
43.637
c
43.759
e
43.670
d
43.370
d
96.270
f
177.25
f
178.79
f
70.978
d
39.860
f
126.09
d
95.780
f
179.50
f
125.98
f
71.359
f
164.40
f
172.65
f
125.77
f
73.351
f
179.94
d
172.08
f
178.82
f
236.74
d
39.970
f
164.28
f
236.27
d
96.290
f
172.61
f
223.45
d
95.690
f
236.97
f
223.44
d
94.180
d
95.320
f
70.786
d
40.730
d
40.840
d
95.700
d
94.990
d
95.560
f
125.69
d
125.42
d
+59.59
f
-60.03
f
-59.76
f
-64.32
d
-53.20
f
+54.86
d
+59.78
f
-60.08
f
+54.92
f
-63.57
f
-61.35
f
-58.33
f
+54.83
f
-63.37
f
-60.06
d
-57.18
f
-60.03
d
+42.78
d
-53.76
f
-60.09
d
+42.45
d
+59.80
f
-58.44
f
-54.45
d
+60.11
d
+41.91
f
+54.46
d
+59.67
d
+59.71
f
-62.39
d
-53.45
d
-53.21
d
+60.26
d
+60.05
d
+59.25
f
+54.83
d
+55.09
f
185871
f
188869
f
184073
f
189769
f
190068
f
191268
f
191867
f
192167
f
191867
f
196664
f
201161
f
203559
f
200861
f
205957
f
196664
f
205957
f
201161
f
209855
f
208356
f
207157
f
219448
f
212853
f
216150
f
220347
f
218848
f
221846
f
221547
f
224844
f
226643
f
227842
f
237136
f
243132
f
244930
f
242832
f
246369
f
251526
f
251526
f
134362
f
135833
f
133472
f
136270
f
136416
f
136996
f
137286
f
137430
f
137286
f
139576
f
141684
f
142794
f
141545
f
143893
f
139576
f
143893
f
141684
f
145658
f
144982
f
144439
f
149890
f
146997
f
148452
f
150278
f
149630
f
150923
f
150795
f
152202
f
152963
f
153467
f
157295
f
159693
f
160402
f
159575
f
160966
f
162961
f
162961
34398.9598
33878.7671
34732.9391
33732.4096
33683.4095
33495.1332
33401.97
33356.8749
33401.97
32715.9840
32147.4792
31869.1917
32182.9622
31607.6212
32715.9840
31607.6212
32147.4792
31213.5042
31361.0496
31482.4902
30383.8796
30934.6450
30649.3603
30315.4376
30340.5423
30203.5667
30225.3317
29990.4551
29868.6639
29790.2428
29245.4585
28945.2629
28861.6943
28959.0407
28796.8798
28579.5619
28579.5619
103
104
J. Sci. Res. Stud.
Table 1. Contd.
SNLS 04D3cpc
SNLS 04D4bkc
HST04Mane
SNLS 03D1ewc
SN 2003eba
SNLS 03D4dic
SNLS 04D3gxc
SNLS 03D4cyc
SNLS 04D3kic
SNLS 03D4cxc
SN 2003esa
HST04Thae
SNLS 04D3mlc
SN 2002dde
SNLS 04D3nrc
HST04Ombe
SN 1997ckb
HST04Pate
SNLS 04D3ddc
HST05Stre
HST04Eage
HST05Fere
HST05Gabe
SN 2002kia
HST04Gree
HST05Rede
HST05Koee
HST05Lane
SN 2003az a
SN 2002hpa
SN 2003aja
SN 2002fwa
SN 2003dy a
HST04Mcge
HST04Sase
SN 2002fxa
SN 2003aka
(a)
0.830c
0.840c
0.854e
0.868c
0.899a
0.905c
0.910c
0.927c
0.930c
0.949c
0.954a
0.954e
0.950c
0.950e
0.960c
0.975e
0.970b
0.970e
1.010c
1.010e
1.020e
1.020e
1.120e
1.141a
1.140e
1.189e
1.230e
1.230e
1.265a
1.305a
1.307a
1.300a
1.340a
1.357e
1.390e
1.400a
1.551a
43.617c
43.880c
43.760d
43.954c
43.640e
43.838c
44.206c
44.153c
44.434c
44.256c
44.300e
43.980d
44.136c
43.970d
44.292c
44.270d
44.300b
44.430d
44.700c
44.500d
44.380d
44.030d
44.570d
44.710e
44.440e
43.640e
44.990d
45.080d
44.640e
44.510e
44.990e
45.060e
44.920e
45.230e
44.930d
45.280e
45.070e
95.520d
39.530d
125.52d
172.15d
125.80f
40.230d
95.524f
39.080f
95.685f
39.353f
125.87f
125.86f
96.700d
125.87f
94.800d
223.47f
57.602f
125.20d
95.620d
125.58d
125.37d
125.56d
125.60d
125.74f
223.52f
125.85f
125.56d
125.46d
125.77f
223.51f
223.77f
223.53f
125.83f
223.60f
125.48d
223.44f
223.77f
+59.44d
-53.46d
+55.08d
-57.86d
+54.82f
-53.04d
+59.85d
-52.50f
+59.91d
-52.26f
+54.83f
+54.83f
+59.61d
+54.83f
+59.33d
-53.95d
+38.45f
+55.04d
+59.96d
+55.16d
+55.12d
+55.08d
+55.13d
+54.71f
-54.46f
+54.84f
+55.03d
+55.13d
+54.74f
-54.44f
-54.39f
-54.40f
+54.86f
-54.51f
+55.20d
-54.50f
-54.38f
248828f
263817f
256023f
260220f
269513f
269513f
272811f
277908f
278807f
284503f
286002f
286002f
284803f
284803f
287801f
292298f
290799f
290799f
302790f
307887f
305489f
305788f
335768f
341763f
341763f
356453f
368745f
370244f
379237f
391229f
391829f
389730f
401722f
406818f
416712f
419709f
464978f
161922f
167562f
164669f
166237f
169623f
169623f
170797f
172582f
172893f
174843f
175349f
175349f
174944f
174944f
175952f
177444f
176949f
176949f
180830f
182428f
181679f
181773f
190666f
192332f
192332f
196268f
199408f
199782f
201984f
204817f
204955f
204469f
207201f
208329f
210463f
211096f
219929f
28690.6283
28140.0434
28406.9324
28258.6016
27968.1679
27968.1679
27876.2570
27745.1731
27723.4357
27592.0318
27559.7873
27559.7873
27585.7299
27585.7299
27522.4674
27433.3722
27462.4598
27462.4598
27251.4903
27174.9105
27210.4292
27205.8166
26861.0203
26812.4423
26812.4423
26715.1688
26654.3564
26647.9753
26614.6420
26580.7884
26579.5965
26584.4605
26560.4091
26553.0065
26543.4333
26541.4686
26560.0800
(b)
Riess et al. (2004): Data have been taken from Hubble Space Telescope (HST) and Ground Based Telescopes such as the Keck Telescope, Very Large Telescope (VLT) and Magellan Telescope; Riess et al. (1998): Data
(c)
have been taken from Hubble Space Telescope (HST) and Ground Based Telescopes such as the Keck Telescope, the Multiple-Mirror (MMT) Telescope and European Southern Observatory (ESO 3.6-m); Wood-Vasey et al.
(d)
(2007): Data have been received from 4-m Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO); Benedit Klaus (2010): The Galactic Longitude and Galactic Latitude values are taken for some Supernovae
Remnants; (e) Riess et al. (2007): Data have been taken for the redshifts z > 1 from Hubble Space Telescope; (f) Proper velocities and Heliocentric Radial Velocities are taken from the SIMBAD Database; (*) The asterisks indicate the
converging relativistic radial velocities to our reference frame (to the center of Milky Way).
Göker
105
Figure 1. The observational distance modulus versus redshift is given.
Figure 2. The Proper distances and the radial velocities versus redshift.
Here, / is the proper velocity, ?/ is the proper distance,
# is the Hubble constant, z is the redshift and c is the
velocity of light. The cz values have been taken from
SIMBAD (see. SIMBAD Database).
SNe Ia which were taken from (Riess et al., 1998,
2004; Wood-Vasey, 2007; Riess et al., 2007) are shown
with red, blue, green and purple colors, respectively. The
best fit value is found as > = 0.9906. The observational
luminosity distances are plotted by using a classical
cosmology model ΩM = 0.29 and ΩΛ = 0.71) for the flat
Universe.
On the left side, The proper distances was used for
possible Hubble constants from different data and fit the
best value for # = 71 km/s/Mpc. As it is seen from the
logarithmic ?/ values in Figure 2b, the path of SNe Ia data
show the similar increase with the observational
luminosity distance, 0 . The best fit in my plot is > =
0.9995. On the right hand side in Figure 2c, the radial
106
J. Sci. Res. Stud.
velocity profile versus redshift was given. The blue circles
denote the redshift values for < 1.02 and red triangles
denote the redshift values for ≥ 1.02. The fit is suitable
for smaller redshifts but for higher redshifts (e.g. ≥
1.02), the stray points can easily be seen. However, the
fit value reached a good fit result as > = 0.9938.
The relationship between distance modulus and
redshift can be considered as a purely kinematic record
of the universe's expansion history. In the radial velocity
profile versus redshift in Figure 2c, the fit is suitable for
smaller redshifts but for higher redshifts (e.g. z ≥ 1.02),
the stray points can be seen.
This shows that the radial velocity values are not as
suitable as proper velocities for higher redshifts.
However, the fit value reached a good result as (> =
0.9938). The detailed explanation about the determination
of radial velocities will be given later on.
The Shell Model
Newton has postulated the existence of an absolute
space in which he thought the center of mass of the solar
system was at rest, with his laws being equally valid in all
other reference (inertial) frames which were moving
uniformly (with absolute velocity) relative to absolute
space (Rindler, 2006). In spite of this, it is possible to
reformulate the Newtonian mechanics without recourse to
absolute velocities, by formulating it in spacetime setting
and replace resulting Newtonian space with NeoNewtonian spacetime (Wuthrich, 2007). The Newtonian
cosmology is based essentially on Newtonian
hydrodynamics, but without uniquely defined boundary
condition at infinity the equations should arise from the
Newtonian approximation of general relativity (Rindler,
2006). It becomes from the enduring points of Newton's
infinite and stationary three dimensional (3D) Euclidean
continuum in space to Neo-Newtonian spacetime. The
Neo-Newtonain spacetime continuum is composed of
individual points at different times, and four dimensional
(4D) volume can be regarded as a collection of 3D
volumes of hyperplanes (Wuthrich, 2007). The frames in
the shell model are accelerating with respect to each
other. On the other hand, there is no energy exchange
between these inertial frames. Therefore, the
assumptions of Neo-Newtonian cosmology can be
applied to the shell model rather than classical Newtonian
theory. So, I used the different reference (inertial) frames
separately as in the Neo-Newtonian cosmology and
connected with the Minkowskian spacetime for
simultaneity. Simultaneity is the main difference between
Neo-Newtonian spacetime and Minkowski spacetime. All
these assumptions are explained in detail later on.
The distance from the Sun to the center of the MilkyWay and the heliocentric radial velocity towards to the
Galactic center are given as ⊙ = 8.5kpc and | D | =
−9 E
, respectively (Malkin, 2013). The velocity of the
Sun equals to ⊙ = 16.5 kms (Mendez et al., 1999). The
distortion of the spherical surface of the Local Group of
Galaxies (LGG) region was ignored because the
perturbing force due to the gravitational effect of the LGG
members decays quickly as 1⁄ F (Sawa and Fujimoto,
2005) and reduced the heliocentric coordinates to galaxycentered coordinates. The coordinate system is centered
on the Milky-Way, this is not only applied to the local
group of galaxies but also to all galaxies. In this problem,
an initial set of data that specifies the positions, masses
and velocities of two bodies for some particular point in
time was taken and then the motions of the two bodies
were determined.
In the beginning, the Euclidean metric was applied with
Eulerian angles to our Galactic coordinate system and
identified two different notation angles. The evolution of
the Universe sufficiently far from the classically presumed
Big-Bang. dH indicates the radial distance of supernovae
(today) from us. In Figures 3 and 4, r indicates the
position of Milky-Way's center (now) with respect to the
initial time moment as assessed by the observer sitting at
the edge of the whole object of concern, rI is the distance
of a supernova from the initial time moment, r is the
radius of the Universe soon after the initial time moment
and R is the maximum limit of the Universe, the size of
the Universe to be determined 13.5 billion light years.
The Universe, i.e., the initial relativistic energy of it, made
of tiny rest mass kernels, each bearing some sort of huge
energy and confined in a given space of size , all over
at rest, gets expanded with a zero initial velocity along
with a characteristic acceleration (of the order of 4.5 ×
10
). The initial radius and the initial size of the
Universe is 2 billion light years. Throughout the evolution
of the Universe, for shells initially around 0.862
ultimately tends to speed of light. Most of the layers
below the surface layers of the Universe, move faster
than the surface layer, thus catching up with it. The
galactic coordinates of radial distance of a supernovae
can be written from the Figure 4 as follows:
-1
dH = 'H ( 4 3 JKL, JK3 JKL, 4 L)
(4)
Here, the galactic latitude and the galactic longitude
equal to θ = b and φ = l, respectively. The study found the
distances of each 257 Type Ia supernovae with respect
to the center of Milky Way for the current time from the
observational radial velocities (see. SIMBAD Database).
r = (N , 0,0)
(5)
and N = N8O = (5 × 10 ) (9.46 × 10 " ) m = 4.73 × 10 " m
= 1532.88 Mpc (here, the age of the Milky Way Galaxy is
5 × 10 light years, 1 light years equal to 9.46 × 10 " m
and 3 × 10"
) and the radial velocity of the Sun with
respect to the center of Milky-Way is given as 8O =
(5 × 10 )( 3 × 10" )
= 1.49 × 10 E
. The velocity
coordinates of Milky-Way is VD = VD ( 4 3 JKL, JK3 JKL, 4 L).
Göker
107
Figure 3. The Sketch of the Universe for the shell model.
Figure 4. The location and the distance of a SNe Ia from the initial time moment and the Milky-Way centered observation frame are
given, respectively.
108
J. Sci. Res. Stud.
The proper distance of supernovae rI is given by:
QI = Q − RH = (N − 'H 4 3 JKL, 'H JK3 JKL, 'H 4 L) (6)
The proper distance is related with the radial distance dH
of supernovae and the proper distance of the Milky Way
galaxy. The position of Milky-Way's center with respect to
the initial time moment is smaller than the maximum limit
of the expansion of the Universe (r ≪ R ). The cosmic
microwave background (CMB) radiation starts from
3 × 10" light years, but the evolution was started in the
shell model from 2 × 10T light years. The angle (U)
between the radial distance of supernovae from us and
the position of the center of Milky Way from the initial
time moment in Figure 4 can be written as:
4 U = |D V||6W |
Q d
V
(7)
X
'H = ('H 4 3 JK L + 'H JK 3 JK L + 'H 4 L)
4 Y=
[
DZ[ DV[ \6X
DZ DV
(8)
(9)
thus, a common α value for each supernovae was
obtained with respect to us in the beginning and took into
account this result as an initial radius soon after the initial
time moment.
NI = (N − 'H 4 3 JKL) + ('H JK 3 JK L) + 'H 4 L
HV
H
=
]
]V
=
^V
^
= (1 + )
(10)
(11)
Here, the size of the universe cannot be smaller than
some limited radius
(with respect to gravitational
interaction) and it is measured by the observer sitting at
the edge (final radius).
will be relatively shorter but
then leading to a quite big R (here R is the radius of the
final rest mass). The curvature parameter is given by:
E=
DV
D
(12)
Here, the radius r is assessed by the distant observer
while N is assessed by the observer sitting at the edge of
the whole object of concern and an assumption that this
radius corresponds to the center of the Milky-Way was
noted. If the Universe is inhomogeneous with an axial
symmetry, the global rotation in the study model is not
only time-dependent but also radial-dependent (Sun and
Chu, 2009).
N⊙ =
_V 8⊙
9[
=
`a×
bcc d` ×
× cf
eV d
= 1.5 × 10F
(13)
Here, g is the gravitational constant and 2⊙ is the solar
mass and equals 2 × 10F E5 and c is the speed of light.
The estimated mass equals 2 ∞ = 3 × 10F E5 for the
minimum size
as observed by an observer sitting at
the edge of the Universe and the Universe cannot be
squeezed beyond
. The radius r for the distant
observer is:
N = N hij )
_V 8
DV 9 [
-
(14)
here, M is the final rest mass. As observed by a distant
observer, the final radius of the Universe from the
theoretical calculations equals to:
=
_V 8V∞
9[
=
`a×
bcc d`F×
×
cf
k[ d
= 2 × 10
"
≈ 2 × 10 3J5ℎ$ mhnN
(15)
On the other hand, our Galaxy is equivalent to around
11
4 × 10 stars and there are about 10 billion galaxies.
So, the squeezed size of the Universe is up to:
= 1500 × 300 × 10 × 80 × 10
h$hN = 6 × 10
"
(16)
and this result corresponds to the same result obtained
by the observational calculations. According to this, the
difference between the theoretical and the observational
final radius of the Universe can be arisen from a missing
mass, maybe from the dark matter.
Throughout the evolution of the Universe, the shells
initially around 0.862
ultimately tend to the speed of
light. The thing is that most of the layers below the
surface layer of the Universe move faster than the
surface layer, thus catching up with it. Given a celestial
body of a residual mass 2 = 2( ) of density o and
final radius , the Universe has started from a beginning
where the entire relativistic energy of it was confined in a
given space of size,
Here, the density o for a
spherical shell equals to:
o =p
e
8V
+HVe
(17)
The velocity in the shell model can be written as follows
(Yarman and Kholmetskii, 2013):
< [ (D,DV )
9[
= 1 − )1 − qhij )
_V 8V∞ DVe
9 [ DV HVe
- − hij )
_V 8V∞ DVe
9[D
HVe
-r-
(18)
here, (N, N ) is the velocity of the shell which depends
not only on the distance r of the shell. The orbits are very
close to circular orbits. Here, R0 is the maximum limit of
the Universe, r0 indicates the position of Milky-Way's
center with respect to the initial time moment, r indicates
any layer after the initial time moment, r1 is the dimension
of the circle which is mentioned the radius of the
Universe soon after the initial time moment and R1 is the
maximum limit of the Universe soon after the initial time
moment.
The distance of the Milky-Way from the initial time
moment equals to r0 = rMW = 1532.88Mpc and the radial
Göker
109
Figure 5. Relativistic radial velocities of 257 SNe Ia are given.
velocity of the Sun with respect to the center of the Milky12
-1
Way is given as VMW = 1.49x10 kms .
Of concern would have reached, but also on the initial
radius N of the portion of the cosmic egg. This
assumption seems reasonable for the Universe evolution
at times far enough from the chosen t = 0. However, at
the earlier stages of the Universe formation, the
interaction between layers may be present. The important
notation here is that the gravity in a flat space must not
be mixed with the spacetime curvature. For the flat
space, the curvature value k will be equal to zero and it
can be identified with two cases with the two different
ultimate velocities (Yarman and Kholmetskii, 2013):
(1) For k = 0 and r tends to infinity, (N, N ) tends to
_V 8V∞
= u, then Equation 18, becomes:
s]t (N ). If [
[ (D )
<vwx
V
9[
9 DV
= 1 − (1 − hijyu − 1z) = 1 − (2 − hiju)
The maximum plausible value of X for N =
Y (
)=
[ (H )
<vwx
V
9[
_V 8V∞
9 [ HV
= 1.089
(19)
is:
= 1 − (2 − hijy1.089z) = 0.0565 ≅ 0.057
(20)
(21)
and s]t = 0.057 gives maximum velocity for the radius
of the Universe as s]t ( ) = 0.24 .
(2) For k < 1, X becomes smaller than 1.089. Then
hij(u)= 2 or x =3K(2), X gives the value X = 0.693.
[
(DV )
<vwx
9[
and
= 1 − (2 − hijy0.693z) = 0.99
s]t
= 0.99
(22)
gives maximum velocity for the radius
of the Universe as
The
s]t (N ) = 0.99 ≅ .
corresponding coefficient N F ⁄ F
becomes X =
0.693/1.089 = 0.64. So, u = N F ⁄ F = 0.64 equals to
N ⁄ = 0.862. For the layer initially standing below the
layer of initial radius, 0.862
will ultimately acquire the
speed of light.
The large-scale view of the Universe from everywhere
is the same (cosmological principle) and let us suppose
that the Universe is flat and infinite, and uniformly filled
with galaxies. Thus, according to the Minkowskian
spacetime, the frame of each SNe has a different
expansion rate and distance measure, and the notion of
simultaneity in the direction of motion and the law for
velocities will be different. Three-vector notation was
used to find the real velocity of each SNe now (different
from the position of SNe today) and replace
with
⁄ . So, simultaneity in the direction of motion was used
to find the real velocities and transformed the velocities
according to the Lorentz transformations. In the
simultaneity, the events which occur simultaneously in
one reference frame will not occur simultaneously in any
other reference frame. A SNe has a velocity relative to
our Milky Way galaxy and this velocity is given as |H for
NI frame which is shown in Figure 4. On the other hand,
there is another velocity of SNe as measured in another
inertial frame moving with a velocity VS relative to Milky
Way. If a SNe has coordinates at time $ − ⁄ , then the
SNe will have coordinates at time t in NI . The real
velocity of SNe in the reference frame NI is given by:
′
|H
=}
<~X <~
•~X •~
€[
}
(23)
The relativistic radial velocity values of 257 SNe Ia from
the center of Milky Way galaxy is given in Table 1 and
plotted in Figure 5.
110
J. Sci. Res. Stud.
Figure 6. The correspondence between Minkowskian spacetime and Neo-Newtonian cosmology is
given.
COSMOLOGICAL IMPLICATIONS
The application of Minkowskian metric to relativistic
Newtonian cosmology
In the previous sections, the cosmological model was
clarified and the assumptions for this model in this
section was given. The origin in the shell model is the
center of Milky Way and the other lattice points are the
position of SNe where the light takes time to pass from
the position of the event to the observer at the origin and
the real position of SNe now. The expansion rate of the
Universe is called the Hubble parameter (H) and it is
written (# ) at the present epoch and depends on the
content of the Universe. In the classical Newtonian
spacetime, points in different hyperlines where the
different SNe are located in different times are spatially
related. Hence, event point in one inertial frame
represents the same event point over time (e.g. spacetime is absolute).
The motion and acceleration of the objects are uniform
(e.g. absolute velocity and absolute acceleration) in the
Newtonian spacetime. Newtonian mechanics is based on
the idea that all physical interactions like gravitation are
instantaneous (Prasad, 2011). Thus, the applications of
this spacetime are not suitable for the real Universe. On
the other hand, we can not totally ignore Newtonian
mechanics for the problems similar to the shell model,
and a relativistic version of Newtonian spacetime which is
called Neo-Newtonian spacetime will be helpful to this
problem. The correspondence between Minkowskian
space and Neo-Newtonian cosmology is given in Figure
6. The important points for each spacetime by following
Figure 6 was explained and the assumptions for the
study model was introduced by using the comparisons
between these two different spacetimes (the detailed
explanation for these models with the relativistic solutions
are given in the papers (Bain, 2004b; Wuthrich, 2007;
Flender and Schwarz, 2012):
(1) In the Neo-Newtonian spacetime, there are no
distance relations between different hyperplanes. So, the
motion of the objects are not uniform (e.g. relative
velocity). In the Minkowskian space, there is also no
absolute space but still there is a fixed spatio-temporal
geometric structure, similar to that of Neo-Newtonian
spacetime (Wuthrich, 2007). In the first assumption, the
inertial frames in the study model are located in different
hyperplanes because an assumption was made that the
Universe was formed inside a spherical shell with
expanding radius in a flat Universe but there is no energy
exchange between these inertial frames. This assumption
proves the first important point for the two spacetimes.
Some irregular data points (z = 0.0141, 0.24, 0.271,
0.3, 0.352, 0.357, 0.358, 0.359, 0.363, 0.371 and 0.38)
were excluded. This unexpected deviation is probably
because of the observational errors or inconsistency
between the radial and the proper velocities of SNe. The
point 0.83c is the maximum relativistic velocity of a SNe
at the point NI , 0.25c is the turning point for higher
redshift velocities and 0.1c is the minimum relativistic
velocity limit of a SNe that can be decreased.
(2) In the Neo-Newtonian spacetime, the motion of the
objects in one reference frame is relative to the motion of
the objects in another reference frame. Similarly, the
reference frames are in relative uniform motion with
respect to each other (in which the laws of physics were
the same and the speed of light was the same) in the
Göker
Minkowskian theory (Wuthrich, 2007). In the study’s
second assumption, the velocities for different SNe Ia are
relative to each other because all these SNe are located
in different inertial frames and their motions will also be
different with respect to the center of Milky Way. This
assumption proves the second important point for the two
spacetimes.
(3) The acceleration is absolute for both of the different
spacetimes. This assumption is applicable to our model
because the frames in the shell model are accelerating
with respect to each other and it proves the third
important point for the two spacetimes. The actual
acceleration of the Universe, outward becomes the
residual effects of the initial acceleration, leading to a
velocity near c, for the farthest locations.
(4) Redshift space distortions are not only due to the
peculiar motion of galaxies but also to the spacetime
metric perturbations and local variation of the cosmic
expansion rate (Flender and Schwarz, 2012). It is clear
that perturbation theory is an important problem in the
Newtonian cosmology and it is not easy to eliminate the
distortion of the peculiar motion of galaxies. In spite of
this, the study made an assumption to decrease the local
perturbation of the galaxy and calculations were done
from the center of galaxy (galaxy-centered coordinates)
to make the model easier and to decrease the effect of
the perturbations originating from the rotation. The
distortion of the spherical surface of the LGG region was
also ignored because the perturbing force due to the
gravitational effect of the LGG members decays quickly
as 1⁄ F (Sawa and Fujimoto, 2005). This assumption is
related to the elimination of the local perturbation.
Therefore, the anisotropy and inhomogeneity will only be
due to the distribution of galaxies or other objects in the
inertial frames and Neo-Newtonian cosmology is a good
approximation for this. However, this will not affect the
metric and in the largest scales, the Universe will become
homogeneous and isotropic again. So, Minkowskian
metric is a good approximation for the whole Universe.
(5) In the Neo-Newtonian spacetime, between any two
events, there is an absolute temporal separation
(including zero separation or simultaneity) which is not
relative to any reference frame or state of motion. In twodimensional Neo-Newtonian spacetime, every straight
lines that does not point either to the left or to the right
points in a temporal direction (Wuthrich, 2007) as
indicated in Figure 6.
In spite of this, the axes of all light cones are vertical with
opening angles of 45° for the Minkowskian spacetime as
seen in Figure 6. The space consisted of three spatial
dimensions and one temporal dimension in Minkowskian
space. There is also a symmetry with respect to
coordinate transformations among inertial frames and it is
valid only in the absence of gravity, and time is also a
frame-dependent coordinate (Wuthrich, 2007). The
components of a four vector in the 4D Minkowski space
111
depend on the choice of reference frame, so it is
important to know how to transform these components
from one inertial reference frame to another (Prasad,
2011).
This is the main difference between these different
spacetimes. So, the study made an assumption for using
these two different spacetimes in the shell model.
Accordingly, it was better to use Neo-Newtonian
cosmology for the local motions and it gives the radial
velocity of SNe from us in today's coordinates. However,
the three-vector notation was used to find the real
velocity of each SNe now and replace the velocity in
the Neo-Newtonian cosmology by relativistic velocity ⁄ .
So, the simultaneity in the direction of motion was applied
to find the real velocities of SNe in different inertial
frames and transformed to velocities according to the
Lorentz transformations. In this way, different velocities
for different spacetimes in one metric could combined.
Neo-Newtonian theory assumes the ability to
distinguish between real and inertial forces. In the
Minkowskian theory, the reference frames are in relative
uniform motion with respect to each other (in which the
laws of physics were the same and the speed of light was
the same) while the simultaneity between the frames are
changing if the reference frames are accelerated with
respect to each other. The simultaneity is the main
difference of Minkowski spacetime from the NeoNewtonian cosmology (courtesy of J. Bain - Lecture
Notes).
In the Minkowskian spacetime, the Universe (the critical
density Universe) is spatially flat (e.g. the curvature
equals to zero k = 0 and the Hubble parameter is given
as Ω=1). The spatially flat Universe includes ∼70% dark
-3
energy, ∼25% cold dark matter, ∼5% barions and ∼10 %
radiation and neutrinos. Essentially, anisotropy from the
CMB radiation, measurements of the large-scale
structure, Hubble constant and Hubble parameters are all
consistent with a spatially flat model in the standard
theory. The Hubble time is given by $• = $ ≡ #
=
13.6±1.4 Gyr. The Hubble constant is equal to:
# =
<(HV )
HV
(24)
and for the Hubble parameter,
#=
<(H)
H
(25)
from Equation 11, = /(1 + ) and I include this term
in the equation above, to find:
•
•V
=
<(H)/H
<(HV )/HV
#=#
•(XV/(cƒ„))
XV /(cƒ„)
<(HV )/HV
(26)
=#
y<(HV /( \…))z( \…)
<(HV )
(27)
112
J. Sci. Res. Stud.
Velocity relates to redshift and so the ability to determine
distances out to high redshifts allows us to measure the
rate of expansion (Specian, 2005). From Equations 11,
24 and 25, the Hubble parameter equal to:
# = # (1 + ) †
<)
XV
(cƒ„)
<(HV )
‡
(28)
The Hubble law has been derived along with a
satisfactory calculation of the Hubble constant, and it
shows the linear dependence of the velocity on the
distance R. For the outermost shell, the acceleration
equals to (Yarman and Kholmetskii, 2013):
n (
)=
_V 8V∞
9 [ HV[
= 4.5 × 10
(29)
This acceleration value is not very big but it is still
capable to explain the dark energy quest throughout the
distant escape velocities reaching the velocity of light. In
order to model the Universe evolution at t > 0, the study
firstly considered an object of rest mass, 2 ∞ where the
inertial frames are originally infinitely far from each other.
The corresponding rest relativistic energy below the shell
size N corresponds to ∞ (N ) . So that the final total
energy remains the same but it will not be given the
energy calculations in this paper. The detailed
explanation about the energy can be found in the paper
Yarman and Kholmetskii (2013).
In the model, the velocity in the shell N will be much
smaller than the velocity of light, V0 (r0) << c. It is
important to note that due to the inequalities
ˆ1 − (N )/ < 1, ˆ1 − (N, N )/ < 1 and
(N ) <
(N, N ), I have hij )
_V 8V∞ DVe
9 [ DV HVe
- − hij )
_V 8V∞ DVe
9[D
HVe
- < 1. This
inequality supposes that there is no energy exchange
between layers of the initial cosmic egg (Yarman and
Kholmetskii, 2013). This causes the fact that every single
initial frame fuels its expansion energy by itself.
In order to calculate the Hubble constant, the initial
density of the Universe must be determined, o in which
requires the estimation of the initial Universe size . The
original density of the Universe equals to o = 3 ×
10 ‰ × 308 = 9 × 10 T E5 F which is 308 times
heavier than the average density of the Universe
(3 × 10 ‰ E5 F ).
The theoretical luminosity distance
The theoretical luminosity distance (?( ) is defined as the
apparent brightness of an object as a function of its
redshift, z and the Hubble parameter, H. It is given by
Riess et al. (1998):
?( = (1 + ) Š
6… ′
•(… ′ )
(30)
here c is the speed of light, z is the redshift and H is the
Hubble parameter which was indicated in earlier on. The
luminosity distance depends on the rate of
acceleration/deceleration or on the amount and types of
matter that form the Universe. If the Universe is
decelerating, this Universe has smaller ?( as a function
of z than their accelerating counterparts. If I assume that
_ 8
u = V[ V∞, the velocity equation (Equation 18) becomes:
9 DV
(N, N ) = )1 − q1 − hij(u) + hij ) V u-rD
D
/
(31)
and if the velocity Equation 31 is applied to the luminosity
distance in Equation 30, the result is given by:
?( = (1 + ) Š
?( =
9( \…)
Š
…
?( =
9( \…)
Š
…
?( =
9( \…)
Š
…
•V
•V
•V
… 6… ′
•(… ′ )
= (1 + ) Š
…
<vwx (HV )6… ′
6… ′
•(XV /(cƒ„′ ))
•( \… ′ )
•(XV )
(32)
( \… ′)<(HV /( \… ′ ))
‹
q
`
y
( \… ′)‹
( \… ′))
q
‘
(33)
[ c/[
Ž
Œt/(•)\Œt/) ŽV •-r •
6… ′
[
Ž
Œt/(•)\Œt/) V •( \… ′ )-r •
Ž
c/[
Œt/(•)\Œt/(••)z[ d
c/[
(34)
6… ′
[ c/[
(35)
Œt/(•)\Œt/`••( \… ′ )d’ -
From the assumptions for the ultimate velocities which
was given earlier, the first assumption indicates that
when k = 0, r tends to infinity and N = . This makes the
velocity as (N, N ) = s]t (N ) = s]t ( ) and (N, N ) =
(N (1 + )). For solving the integral, the necessary
substitutions are, a = 1-hij(u), b = hij(Eu) and Z =
1 + ′ and dz' = dZ.
?( =
?( =
9( \…)
•V
9( \…)ˆ
•V
Š
…
ˆ
ˆ” [ (
(]\“)[ –
(] “)[
(]\“”)[ )
'•
”†˜™š( “”) ˜™š‹ˆ(
—
(
(36)
][ )(
(]\“…)[ )bc/[ √
(]\“”)[ ) `][ \]“”
][ ˆ” [ (
(]\“”)[ )
d•‡
ϥ
…
(37)
The luminosity distance, ?( is expressed as distance
modulus ( − 2) for the cosmological models. The
predicted distance modulus (μpredicted) is given by:
0 = 0/DΠ=
− 2 = 5345 )
ž7
8/9
- + 25
(38)
The main importance in finding the luminosity distance is
that I would have the chance to compare the differences
with distance modulus between observational and
theoretical results. From the result of 0/DΠ0 which is
Göker
113
Figure 7. The correspondence between observed (shown by red asteriks) and
predicted luminosity distances (shown by blue dots) is given.
given in Figure 7, I could easily see the future structure
and expansion rate of the Universe. In the standard
theory, the Hubble parameter for ΩM = 0.29 and ΩΛ = 0.71
is written (Riess et al., 2007):
#( ) = # yΩ8 (1 + )F + Ω z
(39)
here ΩΛ is the density parameter for dark energy (or the
fraction of the vacuum energy), ΩM = ΩVM + ΩDM is the
density parameter for matter including dark matter and
# is the Hubble constant. ΩVM is the fraction of visible
matter and ΩDM is the fraction of dark matter. On the
other hand, it is necessary to find new parameter values
for given velocities in the shell model. For this, the new
Hubble parameter which was previously announced by
Arik et al. (2011) was used.
#( ) = # yΩ (1 + ) − Ω (1 + )F z
(40)
here Ω1 and Ω2 corresponds to ΩM and ΩΛ parameters in
the standard theory, respectively. The study found the
new possible values for the density parameter for matter
including dark matter and for dark energy as ΩM (=Ω1) =
0.14±0.06 and ΩΛ (=Ω2) = 0.86±0.02, respectively.
A New Model for the Hubble Parameter
Measurements of the CMB combined with the
observations of SNe Ia provide strong evidence of a
spatially flat Universe. The total energy density in a
spatially flat Universe is necessarily equal to the critical
density. In addition to explaining the accelerated
expansion, dark energy can account for the missing mass
which is necessary to reach the critical density. The
critical density is given by Riess et al. (2004).
o9D¡¢ ≡
F
‰+_
) - ⇒) - =
]£
]
]£
]
‰+_
¥V
F ]e(¦ƒc)
−
•
][
(41)
Here n is the scale factor and denotes today's values and
n£ is the scale size of the Universe. The effective equation
of state parameter (EoS) is equal to Riess et al. (2004).
§≡
/
¥
(42)
where, ρ is the density and p is the pressure, and
§ < − corresponds to our model. In this case, the
F
expansion will proceed for all time independent of the
curvature (Rindler, 2006). However, the problem with
EoS still needs improving because it mainly depends on
the redshift range and it is an unsteady parameter (e.g. it
should have different values for the redshifts
< 1
and ≳ 1).
The predicted curve show a different result than the
observational curve for the redshifts less than < 0.12 as
expected because observational results were improved
with some instrumental corrections while predicted
results are developed from the model. As seen from the
figure, the predicted values satisfy the observational
values very good for the redshift ≥ 0.28, especially for
higher redhsifts.
The EoS parameter needs special attention and so it
was not focused in this paper. Decreasing of the density
of radiation is faster than that of dust is that it is possible
for each photon also be redshifted during the expansion
and the scale factor of the Universe increases (Riess et
al., 2011). So, the Hubble parameter is given by:
#( ) =
]£
]
≡ ©)
‰+_ ¥V
F ][
-
(43)
114
J. Sci. Res. Stud.
Figure 8. The estimated values for the expansion history of the Universe are given.
The second time derivative of the scale factor nª is given
by (Riess et al., 2004):
nª = −
*_]+
F
(o + 3j)
(44)
During an accelerated expansion, nª will be positive. At z
< 1, the behavior is attributed to dark energy with
negative pressure (Riess et al., 2004). The density of the
Universe is related to the rate at which the Hubble
expansion is changing with time. The deceleration
parameter is given by Riess et al. (2004).
≡−
( )=
]ª (¢V )
]£ (¢V )[
+
(45)
6«
6…
= −) -) ]ª
]£
]£
]
(46)
The variation of Hubble parameter and deceleration
parameter versus redshift is given in Figure 8. An
empirical description of the time variation of the scale
factor a(t) can answer the most important question
whether the Universe is accelerating or decelerating
(Riess et al., 2004).
DISCUSSIONS AND CONCLUSIONS
In this work, the history of cosmic expansion over the last
13.5 Gyr was examined and the study used 257 SNe Ia
in the redshift range 0.1 < ≤ 1.55. 11 high-redshifts
which are all at z > 1.25 were also included, these highredshifts are important to determine the behavior of the
Universe. In the analysis, these steps were followed:
(1) Plotted the luminosity distances which have been
determined from the relations between light-curve shape
and luminosity.
(2) Determined the radial velocities and proper velocities
of 257 SNe Ia.
(3) Identified the shell model.
(4) Compared the differences between Neo-Newtonian
cosmology and Minkowskian space, and applied these
theories to the shell model with some assumptions.
(5) Found the new theoretical distance modulus and
connected the distance modulus with the new Hubble
parameter.
(6) From this new Hubble parameter values, the critical
density and deceleration parameter were calcultaed. The
detailed explanation of these steps is given as follows:
(1) The main idea of this paper is finding the past and
future behaviors of the Universe. In the shell model, the
study assumed that anisotropy and inhomogeneity will
only be due to the distribution of galaxies or other objects
but not to the metric. In this model, the author accepted
that the Universe had started to expand from a spherical
distribution at the beginning and it had an initial radius
before it started its expansion. In the largest scales, the
Universe will become homogeneous and isotropic again.
The maximum radius of the Universe will be
=6×
10 "
before it started to collapse. The Minkowskian
Göker
metric was used in general but for the local analysis, the
Neo-Newtonian cosmology was applied to inertial frames.
It is important to note that in this paper, the study of
relativistic calculations of Minkowskian metric or NeoNewtonian cosmology was not done but took into account
astrophysical applications of these theories.
In the shell model, the Universe was accepted as
overall electrically neutral, the gravitational interaction
dominates at larger scales and the radiation was
neglected. the earliest stages of the Universe formation
were not considered and possible effects discussed for
these stages as indicated first by Yarman and
Kholmetshii (2013). The time moments where the
Universe has a radius of cosmological scale are
important for the shell model. In that time, the gravitation
is sufficiently weak and so that the Newtonian laws
become applicable. This clearly demonstrates that the
assumptions of Neo-Newtonian cosmology was a good
choice to understand the evolution of the Universe in the
flat spacetime.
(2) The observational distance modulus μ0 was plotted
versus redshift for 257 SNe Ia in Figure 1 and found the
fit value as > = 0.9906. So, the distance modulus is in a
good correlation even for higher redshifts as seen in this
figure but what about the radial velocities? Figure 8a
indicates the measurement of #( ) from plotted data.
Figure 8b shows the derived quantity n£ versus redshift
and in this figure, a negative sign of the slope of the data
indicates the sign of the acceleration of the expansion. In
Figure 8c, the future of the expanded Universe is clearly
seen and the positive value of
( ) shows pure
deceleration as we assumed in our calculations.
(3) Heliocentric radial velocity indicates the radial velocity
of an object from us by using the position of the Sun and
rotation around the center of Milky Way in today's
coordinates. The radial velocities strayed from the plot for
redshift z > 0.8 as seen in Figure 2 and this deviation
from the plot line is more distinctive for higher redshifts
≥ 1. It is because the radial velocity is related with the
light of SNe and for higher distances there is more dust in
the extragalactic medium and the light of furthest SNe is
absorbed more than the nearby ones. In spite of this, the
fit value gives the good result as > = 0.9938. As seen
from radial velocities, the Universe follows the steady
path rather than acceleration soon after the redshift ≥
1. The radial velocities of higher redshifts remain steady
with respect to my reference system. In contrast to this,
the relativistic radial velocities of SNe which corresponds
to the real position of SNe now will be different.
(4) If the real velocity of Sne is to be found, the peculiar
velocity and the radial velocity of each SNe Ia will have to
be calcuated together by using Lorentz transformation as
given in Equation 23. The effects of peculiar velocities of
cosmological parameters include our own peculiar
motion, SNe's motion and coherent bulk motion. These
velocities of 257 SNe Ia which was found by Equation 23
is given in Table 1 and Figure 5. In Figure 5, the point
115
0.83c is the maximum relativistic velocity of SNe at the
point NI , 0.25c is the turning point for higher redshift
velocities and 0.1c is the minimum velocity limit of SNe
that can be decreased. This decrease was started at
≅ 0.4 which corresponds to the maximum velocity 0.83c
and after ≅ 0.6, the SNe (or the galaxies) expand in the
very steady (or regular) way with respect to each other.
Relativistic velocities are very important to see the predict
future of the Universe. However, some irregular data
points in the redshift ranges were found, z = 0.0141 and
0.24 ≤ z ≤ 0.38. This unexpected deviation is probably
because of the observational errors or inconsistency
between the radial and proper velocities of SNe. This is
because the proper velocities are related directly with the
expansion rate and the redshift of SNe while the radial
velocities are related with the distance of supernova with
respect to us. Figure 5 corresponds to the estimated
velocity results in the shell model, for example, (i) for k =
0, r → ∞ and
(N, N ) = s]t ( ) = 0.24 , this value
satisfies the value DI ( ) = 0.25 ; (ii) for k < 1,
s]t (N ) ≅ 0.99 and this value is also very close to the
value DI (N ) = 0.83 in Figure 5. This shows that for the
layer initially standing below the layer of initial radius,
N ⁄ = 0.862 that will ultimately acquire the velocity of
light.
(5) The inertial frames in my model are located in
different hyperplanes because I assumed that the
Universe was formed inside a spherical shell with
expanding radius in a flat Universe and there is no
energy exchange between these inertial frames as
indicated previously by Yarman and Kholmetshii (2013).
The velocities for different SNe Ia are relative to each
other because all these SNe are located in different
inertial frames and their motions will also be different with
respect to the center of the Milky Way. The frames are
also accelerating with respect to each other, so we can
say that the acceleration is absolute. It was better to use
Neo-Newtonian cosmology for the local motions and it
gives the radial velocity of SNe from us in today's
coordinates. However, the three-vector notation was
used to find the real velocity of each SNe now and
replace the velocity in the Neo-Newtonian cosmology to
relativistic velocity / . So, the simultaneity in the
direction of motion was applied to find the real velocity of
SNe in different inertial frames and transformed to
velocities according to the Lorentz transformations.
(6) The luminosity distance (?( ), the theoretical one, is
defined as the apparent brightness of an object as a
function of its redshift. In Figure 7, the predicted curve
shows a different result than the observational curve for
the redshifts less than z < 0.12 as expected because
observational results were improved with some
instrumental corrections while predicted results are
developed from the model. As seen from the figure, the
predicted values satisfy the observational values very
well for higher redhsifts. For the redshifts z < 0.3,
irregularity (or observational error) is unavoidable in the
116
J. Sci. Res. Stud.
observational calculations because of the extinction from
the Milky Way. The deceleration in the past, acceleration
in the present and possible deceleration in the future can
easily be seen from Figure 7. In this figure, the transition
from acceleration to the steady path starts from the
redshift z = 0.28±0.01 and the transition between the two
epochs from the relativistic radial velocities of SNe Ia is
constrained to be at z = 0.39±0.007. This shows that the
redshift value z = 0.3 assigns the transition between two
different epochs and exact observational results will
improve our calculations.
(7) From the study’s calculations, very small positive
outward acceleration was found out for the expansion of
the Universe and this is also a signature of dark energy.
Another evidence of the dark matter is that the distance
from theoretical calculations was found out to be as
= 2 × 10 " but the squeezed size of the Universe up
to
= 6 × 10 " which corresponds to the same result
obtained by the observational calculations. The difference
between the theoretical and observational final radius of
the Universe can also arise from missing mass and this
mass may be indicated by the dark matter. In addition to
explaining the accelerated expansion, dark energy can
account for the missing mass which is necessary to reach
the critical density. It also showed that at the earlier
stages of the Universe evolution, the acceleration was
negative.
The matter density decreases with time in an
expanding Universe while the vacuum energy density
remains constant or increases a little. The new Hubble
parameters which corresponds to dark matter were found
and dark energy parameters in the standard theory are
found as Ω1 = 0.14±0.06 and Ω2 = 0.86±0.02, respectively
from the new Hubble equation, Equation 40. The dark
energy with high density Ω2 and decreasing matter
density Ω1 corresponds to a low temperature Universe
and tends to increase the rate of expansion of the
Universe. This result is clearly seen in Figure 7. The
expansion of the Universe at the present time appears to
be accelerating at redshift z < 1 but at redshift ≥ 1, the
expansion of the Universe will be decelerating as seen in
Figure 7. The same result has been found in different
papers (Riess et al., 2004, 2011; Saadat, 2012).
(8) The ratio of Hubble parameter to Hubble's constant
(H/H0) for today's values are shown for 257 Type Ia SNe
in Figure 8a and the expansion history is also indicated in
this figure. Hubble parameter is related with the density of
the Universe and from the density value, the author found
the negative acceleration of expansion
< 0 for today's
values. The study also found the scale size of the
Universe n£ from Equation 44 and deceleration parameter
( ) from Equation 47. Decreasing of the density of
radiation is faster than that of dust and this makes
possible for each photon also be redshifted during the
expansion and the scale factor of the Universe increases
as seen in Figure 8b. The study found the similar
acceleration for the scale factor, even not taking into
account the radiation. The deceleration parameter
( ) corresponds to the positive value, this indicates that
the Universe will decrease after expansion as seen in
Figure 8c.
In this paper, the evolution of the shell model in NeoNewtonian cosmology for the local motions was
explained and in Minkowskian spacetime for the whole
universe with astrophysical applications. It needs more
accurate observational data received from the space and
ground-base telescopes. In addition to this the model can
improve with extra limitations, maybe relativistic
calculations will help to this. This calculations will be for
future work.
ACKNOWLEDGEMENTS
The author would like to thank to Prof. Graham S. Hall
from the University of Aberdeen for his valuable
comments and Prof. Mikhail Sheftel from the Bogazici
University for his help in the first draft of the paper and
TUBITAK (The Scientific and Technological Research
Council of Turkey) for their financial support. The author
wishes to thank Tolga Yarman and Metin Arik who led
this work with their deep knowledge in cosmology during
his studies in their Cosmology group. He also would like
to thank anonymous referee(s) for valuable comments
and guidance.
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