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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

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)

http://www3.plala.or.jp/MiTiempo/EnergyCirculationTheory.pdf

http://www3.plala.or.jp/MiTiempo/QuantumParticles.pdf


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)


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)


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


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.


(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.


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


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


(b) LD versus Relative Back in Time TBR

Magnitu

de o

f LD

to z

=0.0

5,

DM

0.0

5


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


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


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


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



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,


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)


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.

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