Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf ·...
Transcript of Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf ·...
![Page 1: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/1.jpg)
© All copy rights reserved by the author.
Hubble Diagram from the 4D Spherical Universe
(Nature of the time and 4DS model - Part 3)
Version 2020.06
Shigeto Nagao < Part 1 >
1. Nature of the time 2. 4-D Spherical Model of the universe
3. Michelson-Morley experiment 4. Acceleration factor 5. Expansion of the universe
< Part 2 >
6. Light speed 7. Redshift 8. Light propagated distance
< Part 3 >
9. Redshift for cosmological observation (revised)
10. Factors affecting the brightness
11. Hubble diagram
12. Comparison with Hubble diagrams from the Supernova Cosmology Project
![Page 2: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/2.jpg)
Hubble Diagram from the 4D Spherical Universe 2
9. Redshift for cosmological observation
Revision after newly reported energy circulation theory and quantum particles
Spectrum comparison with the present atom of the same element:
emitted at ET reaches us at PT
( ) ( )EE TT00 , → ( ) ( )PP TT ,
( ) ( )PP TT 00 , (present atom)
𝑧 + 1 =𝜆(𝑇𝑃)
𝜆0(𝑇𝑃)=
𝜈0(𝑇𝑃)
𝜈(𝑇𝑃) not compare 𝜆(𝑇𝑃) with 𝜆0(𝑇𝐸)
Light speed:
𝐶(𝑇𝐸) = 𝜈0(𝑇𝐸) ∗ 𝜆0(𝑇𝐸) ⟶ 𝐶(𝑇𝑃) = 𝜈(𝑇𝑃) ∗ 𝜆(𝑇𝑃)
Present atom: 𝐶(𝑇𝑃) = 𝜈0(𝑇𝑃) ∗ 𝜆0(𝑇𝑃)
![Page 3: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/3.jpg)
Hubble Diagram from the 4D Spherical Universe 3
Redshift from values at the emission
Wavelength-based redshift:
By space expansion: Wavelength is stretched by n.
𝑧𝜆𝑒 + 1 =
𝜆(𝑇𝑃)
𝜆0(𝑇𝐸)= 𝑛 =
𝑇𝑃
𝑇𝐸=
1
𝑇𝐸𝑅
Frequency-based redshift:
𝑧𝜈𝑒 + 1 =
𝜈0(𝑇𝐸)
𝜈(𝑇𝑃)=
𝐶(𝑇𝐸)
𝜆0(𝑇𝐸)
𝜆(𝑇𝑃)
𝐶(𝑇𝑃)= 𝑛
𝐶(𝑇𝐸)
𝐶(𝑇𝑃)=
1
𝑇𝐸𝑅
𝐶(𝑇𝐸)
𝐶(𝑇𝑃)
![Page 4: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/4.jpg)
Hubble Diagram from the 4D Spherical Universe 4
Redshift of actual measurement
From the energy circulation theory:
Atomic energy varies by the space expansion.
Energy of the single circulation 𝑖𝑆 in hidden-space dimensions:
𝐸(𝑖𝑆) = 𝑚0𝑐2 = 𝑚0𝑣𝑐2 = 𝑚0𝜇0
2𝜔02
Light energy / Photon energy:
𝐸𝛾 = ℎ𝜈2 = ℏ𝜔2 , 𝐸𝑝 = 𝐸𝛾 𝜈⁄ = ℎ𝜈 = ℏ𝜔
Space expansion by 𝑛 ⟹ 𝐸(𝑖𝑆), 𝑐2 and 𝜔02 decrease by 1 𝑛3⁄ . (𝑚0, 𝜇0: invariant)
Present atom of the same element: Light energy decreases by 1 𝑛3⁄ at emission.
𝐸𝛾(𝑇𝐸) = ℎ𝜈0(𝑇𝐸)2 , 𝐸𝛾(𝑇𝑃) = ℎ𝜈0(𝑇𝑃)
2
𝜈0(𝑇𝑃)
𝜈0(𝑇𝐸)=
𝐶(𝑇𝑃)
𝐶(𝑇𝐸)
![Page 5: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/5.jpg)
Hubble Diagram from the 4D Spherical Universe 5
Redshift from the present atom
𝑧𝜆 + 1 =𝜆(𝑇𝑃)
𝜆0(𝑇𝑃)=
𝜆(𝑇𝑃)
𝜆0(𝑇𝐸)
𝜆0(𝑇𝐸)
𝜆0(𝑇𝑃)= 𝑛 ×
𝐶(𝑇𝐸)
𝜈0(𝑇𝐸)
𝜈0(𝑇𝑃)
𝐶(𝑇𝑃)= 𝑛 =
1
𝑇𝐸𝑅
𝑧𝜈 + 1 =𝜈0(𝑇𝑃)
𝜈(𝑇𝑃)=
𝜈0(𝑇𝐸)
𝜈(𝑇𝑃)
𝜈0(𝑇𝑃)
𝜈0(𝑇𝐸)=
𝜈0(𝑇𝐸)
𝜈(𝑇𝑃)
𝐶(𝑇𝑃)
𝐶(𝑇𝐸)= 𝑛
𝐶(𝑇𝐸)
𝐶(𝑇𝑃)
𝐶(𝑇𝑃)
𝐶(𝑇𝐸)= 𝑛 =
1
𝑇𝐸𝑅
𝑧𝜆 = 𝑧𝜈 = 𝑧 =1
𝑇𝐸𝑅− 1
0.02 0.05 0.1 0.2 0.5 0.7
0.1
1
10
100
Relative back in Time, TBR = 1 - TER
Redshift from present atom, 𝒛
![Page 6: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/6.jpg)
Hubble Diagram from the 4D Spherical Universe 6
10. Factors affecting the brightness
(i) Factor by wavelength prolongation
Along with space expansion:
• Wavelength: stretched by EREP TTTn 1=
• Energy of a single photon h : decreases to n1
• Number of photons: increase to n-fold (preservation of the total energy)
• h times the number of photon: no change
• No effect by wavelength prolongation on the luminosity (total energy per unit time) (ii) Factors by scattering
Light speed:
−
−==
3
3
11
1)(
x
T
xxKffKxC C
EMD
• Electromagnetic interaction factor EMf is effect of scattering.
• Plural Time Clear CT values:
For Cosmic Microwave Background radiation: 380,000 years after Big Bang For “Reionization” of interstellar hydrogen by stars: 150 - 1,000 million years after Big Bang For light propagation in substances
• If the density is constant, the transmittance is ( ) ( )( ) )1(1 0
3
0 kxIdxTIxI C −=−= .
• Light from stars emitted during the reionization period: Not reach us or be darken by scattering
• Stellar light emitted later ( 072.0ERT , 8.12z ): Factor by scattering is negligible.
![Page 7: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/7.jpg)
Hubble Diagram from the 4D Spherical Universe 7
(iii) Factor by variation of the space energy density
• Luminosity: energy per unit-time
Flux: energy per unit-time per unit-area 24 r
LF
=
• Luminosity varies depending on the light speed at the time of detection because it is a per unit-time
value. L: function of the time of detection xT =
( )( )( ) ( )
22 44 r
xL
r
xCLxF
==
• For our observation, the variable x is fixed to the present time PT . Luminosity and Flux are
invariant by the time of emission ET .
• No effect by variation of the space energy density on brightness for our observation.
![Page 8: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/8.jpg)
Hubble Diagram from the 4D Spherical Universe 8
11. Hubble diagram – Magnitude of flux and distance
Light of luminosity L was emitted at ET and reaches us now at PT .
Flux we observe now: ( )( )24 E
ETLD
LTF
=
Define its magnitude: ( ) ( )EE TFTm lg5.2 −
Define the relative magnitude to the same luminosity with 05.0=z :
( ) ( ) ( ) ( ) ( )05.1105.005.0 mTmzmTmTDM ERERER −==−
• This is a distance modulus : referred to as the “magnitude of LD to z=0.05”
• Light propagated distance LD: EMf can be ignored if 8.12z
( )
−+
−−−
−+
−−=
−
E
E
P
PT
TE
T
T
T
TKdx
xx
KTLD
P
E 11
11log
11
11log
1
( )
−−
−+
−+
−−−
−−
−+
−+
−−=
05.1/11
05.1/11
11
11loglg5
11
11
11
11loglg505.0
P
P
P
P
ERP
ERP
P
PER
T
T
T
T
TT
TT
T
TTDM
![Page 9: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/9.jpg)
Hubble Diagram from the 4D Spherical Universe 9
Magnitude of light propagated distance LD to z = 0.05
( )
−−
−+
−+
−−−
−−
−+
−+
−−=
05.1/11
05.1/11
11
11loglg5
11
11
11
11loglg505.0
P
P
P
P
ERP
ERP
P
PER
T
T
T
T
TT
TT
T
TTDM
Magnitu
de o
f LD
to z
=0.0
5,
DM
0.0
5
Redshift, z
TP = 0.6 TP = 0.7 TP = 0.8
- - - : Constant light speed (for any TP)
(a) LD versus redshift z
0.05 0.1 0.5 1 5 10
-2
0
2
4
6
8
10
Relative Back in Time, TBR
0.02 0.05 0.1 0.2 0.5 1
-2
0
2
4
6
8
10
TP = 0.6 TP = 0.7 TP = 0.8
- - - : Constant light speed (for any TP)
(b) LD versus Relative Back in Time TBR
Magnitu
de o
f LD
to z
=0.0
5,
DM
0.0
5
![Page 10: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/10.jpg)
Hubble Diagram from the 4D Spherical Universe 10
Hubble diagram
3-D space expansion speed: ( )
===dx
xd
dx
dr
dT
dr xr =
For a given angle , variable x: constant
For a given radius x, variable : proportional to and to r → Hubble’s law
Hubble diagram: To discuss the recessive velocity of the proper distance
Tentatively provide that light speed has been constant.
Light propagated distance: ( ) ( )1
1+
=−=−=z
zTcTTcTTcLD PERPEPC
Multiply by ( )z+1 (time dilation): ( ) zkzTcLDz PC =+1 → proportional to z
( ) tzt += 1' Light-curve width of supernovae: dilated by z+1
Conversion of LD to PD:
AP-BP: proper distance at TP (“present distance, PD”)
C-BP: light propagated distance (LD)
EREP TTTn 1==
( ) ( )12
11
2
11 +=−+= nn
BA
CB
EE
P,
( )
ERTz
z
n
n
LD
PD
+=
+
+=
+=
1
2
2
12
1
2
TE TP
AE
AP
BE
BP
C
![Page 11: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/11.jpg)
Hubble Diagram from the 4D Spherical Universe 11
Adjusted magnitude of PD for Hubble diagram
In order to compare the present distance with reported Hubble diagrams, take the following value:
( ) ( )( )
LDz
zzPDz
+
++=+
2
1211 or LD
TTPD
T ERERER
+
=1
211
Time dilation LD-PD conversion
In actual cosmological observation, in addition to the time dilation, K-correction is made to flux.
K-corrections: convert a measurement at redshift z to that in the rest frame at z = 0. Cross-filter adjustment in abs-Mag + difference in distance modulus between the two frames.
Theoretical part of K-corrections correspond to the LD-PD conversion.
Adjusted magnitude of PD to z = 0.05, adjDM 05.0 : Add Time dilation and LD-PD conversion to 05.0DM
( ) ( ) ( ) ( )( ) ( ) ( )( )
( ) ( ) ( ) ( )( )05.2lg05.1lg205.11lg1lglglg5
05.205.12lg05.1lg05.11lg12lg1lglg505.0
+−−+−−=
−−−+++=
LDTTTLD
LDTTTLDTDM
ERERER
ERERERERadj
( )
( ) 05.2lg505.1lg101lg5lg5
05.1/11
05.1/11
11
11loglg5
11
11
11
11loglg505.0
+−+−−
−−
−+
−+
−−−
−−
−+
−+
−−=
ERER
P
P
P
P
ERP
ERP
P
PER
adj
TT
T
T
T
T
TT
TT
T
TTDM
![Page 12: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/12.jpg)
Hubble Diagram from the 4D Spherical Universe 12
Hubble diagram of adjusted PD vs redshift
The adjusted magnitude of PD to z = 0.05, adjDM 05.0 versus the redshift 11 −= ERTz
- - - - : reference adjDM 05.0 based on ( )ERPC TTcLD −= 1 subject to constant light speed
Ad
juste
d m
ag
nitu
de
of P
D t
o z
=0.0
5,
0.02 0.05 0.1 0.2 0.5 1
-2
0
2
4
6
8
10
Redshift z
TP = 0.6
TP = 0.7
TP = 0.8
- - - : Constant light speed (for any TP)
Ad
juste
d m
ag
nitu
de
of P
D t
o z
=0.0
5,
Redshift, z 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-2
0
2
4
6
8
10
TP = 0.6 TP = 0.7 TP = 0.8
- - - : Constant light speed (for any TP)
![Page 13: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/13.jpg)
Hubble Diagram from the 4D Spherical Universe 13
12. Comparison with Hubble diagrams from the SCP
Superimposition on the Hubble diagram by Perlmutter et al, z in a logarithmic scale
Redshift z
0.02 0.05 0.1 0.2 0.5 1
-2
0
2
4
6
8
10
TP = 0.6 TP = 0.7 TP = 0.8
Perlmutter et al. ApJ 1999 (SCP)
Adju
ste
d m
agn
itude o
f P
D t
o z
=0.0
5,
![Page 14: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/14.jpg)
Hubble Diagram from the 4D Spherical Universe 14
Superimposition on the latest Hubble diagram from the SCP, z in a uniform scale
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-2
0
2
4
6
8
10
TP = 0.6
TP = 0.7
TP = 0.8
Adju
ste
d m
agn
itu
de
of P
D to
z=
0.0
5,
Redshift z
Rubin et al. ApJ 2013 (SCP)
![Page 15: Hubble Diagram from the 4D Spherical Universe - plala.or.jp › MiTiempo › HubbleDiagram.pdf · Hubble Diagram from the 4D Spherical Universe 11 12. Comparison with Hubble diagrams](https://reader030.fdocuments.in/reader030/viewer/2022041108/5f0c11c97e708231d43397ec/html5/thumbnails/15.jpg)
Hubble Diagram from the 4D Spherical Universe 15
Conclusion of comparison with the SCP
➢ Observed redshift is from values of the present atom.
➢ Light propagated distance LD is equal to the luminosity distance.
➢ Ratio of converting LD to the present distance PD (proper distance at present) is given.
➢ The frame-conversion part of the K-correction corresponds to the LD-PD conversion.
➢ Magnitude of (1 + 𝑧) ⋅ 𝑃𝐷 to 𝑧 = 0.05 was compared with reported Hubble diagrams
from the SCP, which were time-dilated by 1 + 𝑧 and K-corrected.
➢ Superimposition on the reported Hubble diagrams from the SCP showed an excellent
fit. The graph for the Present Time 𝑇𝑃 = 0.7 very closely overlaps the line of flat Ωm =
0.27 ΛCDM universe that Rubin et al concluded as the best fit.
The expansion of universe is not accelerating.
Decelerating by the Original time
Constant by the Observed Time.