Frequency-based redshift for cosmological observation and Hubble - - PowerPoint PPT Presentation

Frequency-based redshift for cosmological observation and Hubble diagram from the 4-D spherical model in comparison with observed supernovae

DICE2016, Sep. 12 - 16, Castiglioncello (Italy)

Shigeto Nagao, Ph.D.

(Osaka, Japan)

Summary

• According to the formerly reported 4-D spherical model of the universe, factors on Hubble diagrams

are discussed.

• The observed redshift is not the prolongation of wavelength from that of the source at the emission

but from the wavelength of spectrum of the present atom, equal to the redshift based on the shift of frequency from the time of emission.

• The K-correction corresponds to conversion of the light propagated distance (luminosity distance) to

the proper distance at present (present distance).

• Comparison of the graph of the present distance times

z  1

versus the frequency-based redshift with the reported Hubble diagrams from the Supernova Cosmology Project, which were time-dilated by z  1 and K-corrected, showed an excellent fit for the Present Time (radius of 4-D sphere) being 0.7 of its maximum.

DICE2016, Castiglioncello (Italy)

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4-D Spherical Model of the Universe

• The space energy spreads with expansion in a 3-D surface of a 4-D sphere. A vibration of the

intrinsic space energy in the 3-D space vests additional energy, which is light or a quantum particle. Radius x of the 4-D sphere is our “Observed Time” (Time or T ), which we feel passing commonly for all of us.

< Definition of terms related to Time > x (Radius of universe): Radius of the 4D sphere of universe, equal to the Observed Time (T) CU (Cosmic Unit): Unit of x and T, being one at its maximum when the space expansion stops. Time-related variables are expressed in the CU in this article. TE (Time of Emission): Time when the light was emitted. TP (Present Time): Present Time of universe, when the light reaches us. TER (Relative Time of Emission): Relative ratio of TE to TP.

P E ER

T T T /  TB (Back in Time): Back in Time from present when the light was emitted.

E P B

T T T  

TBR (Relative Back in Time): Relative ratio of TB to TP.

P B BR

T T T /  TC (Time Clear): Time when the space became transparent to light. TCR (Relative Time Clear): Relative ratio of TC to TP.

P C CR

T T T / 

Phase Transition Space expansion

r x

x: Radius vector of 4D sphere

) , , , ( ) , (

3 2 1

   x x   θ x

(4D spherical coordinate)

r: 3D space vector corresponding to θ ) , , ( ) , , (

2 1 2 1

     x r   r x by 4D cylindrical coordinate:

) , , , ( ) , , , (

2 1 2 1

     x x r x   x

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Redshift for cosmological observation

We compare the observed wavelength

 

P

T 

with that of the present atom

 

P

T 

. emitted at

E

T

reaches us at

P

T

   

E E

T T ,   →

   

P P

T T   ,

   

P E

T T   

   

P P

T T ,  

(present atom) Observed redshift z :

           

P E P P P P

T T T T T T z       1    

equal to frequency-based redshift



z .

Wavelength-based redshift:

• From space expansion:

Light speed does not vary. Wavelength is stretched by n.

• From light speed variation:

Wavelength prolongs in proportion to the speed.

               

E P ER E P E P E P E P

T C T C T T C T C T T T C T C n T T z         1 1  



Frequency-based redshift:                        

ER P E E P P E E P P P P E

T n T C T C T C T C n T T T T T T T T z 1 1                 



   

1 1   

E P ER

T C T C T z

1 1   

ER

T z z

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Magnitude of light propagated distance LD to z=0.05

Light of luminosity L emitted at TE reaches us now at TP. Flux we observe now:    2

4

E E

T LD L T F   

Its magnitude:

   

E E

T F T m lg 5 . 2    Relative magnitude to same luminosity at z=0.05:

         

05 . 1 1 05 .

05 .

m T m z m T m T DM

ER ER ER

     Light speed:

               

3 3

1 1 1 ) ( x T x x K f f K x C

C EM D

Light propagated distance:

   

                                    

 

E E P P T T T T E

T T T T K dx x x K dx x C T LD

P E P E

1 1 1 1 log 1 1 1 1 log 1

 

                                                      05 . 1 / 1 1 05 . 1 / 1 1 1 1 1 1 log lg 5 1 1 1 1 1 1 1 1 log lg 5

05 . P P P P ER P ER P P P ER

T T T T T T T T T T T DM

, a distance modulus

Magnitude of LD to z=0.05, 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

2 4 6 8 10 Relative Back in Time, TBR 0.02 0.05 0.1 0.2 0.5 1

• 2

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

Magnitude of LD to z=0.05,

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Hubble diagram

3-D space expansion speed:  

     dx x d dx dr dT dr

x r  

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 z T c T T c T T c LD

P ER P E P C

Multiply by 



z  1 (time dilation):

 

z k z T c LD z

P C

       1 → proportional to z

  t

z t     1 ' Light-curve width of supernovae: dilated by z  1 Ratio of PD to LD (LD-PD conversion): AP-BP: proper distance at TP (“present distance, PD”) C-BP: light propagated distance (LD) (= 3D-space component of AEBP)

ER E P

T T T n 1      

1 2 1 1 2 1 1      n n B A CB

E E P

,  

ER

T z z n n LD PD        1 2 2 1 2 1 2

TE TP AE AP BE BP C

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Adjusted magnitude of present distance PD to z=0.05

For comparison with reported Hubble diagrams, take the following adjusted PD:

      LD

z z z PD z         2 1 2 1 1

• r

LD T T PD T

ER ER ER

     1 2 1 1

Time dilation LD-PD conversion Reported Hubble diagrams from the SCP: Time-dilated and K-corrected DL (luminosity distance) K-correction

xy

K :

xy x y

K DM M m   

,

 

z DM M m

y y

 

→

 

DM z DM M M K

x y xy

   

1) cross-filter adjustment on absolute magnitude

x y

M M 

, plus

2) difference in distance modulus

 

DM z DM 

(= mag LD – mag PD) between observed / rest frames Frame-conversion part of K-correction corresponds to LD-PD conversion. Adjusted magnitude of PD to z = 0.05,

adj

DM

05 .

: Add Time dilation and LD-PD conversion to

05 .

DM

                         

05 . 2 lg 05 . 1 lg 2 05 . 1 1 lg 1 lg lg lg 5 05 . 2 05 . 1 2 lg 05 . 1 lg 05 . 1 1 lg 1 2 lg 1 lg lg 5

05 .

                  LD T T T LD LD T T T LD T DM

ER ER ER ER ER ER ER adj

   

05 . 2 lg 5 05 . 1 lg 10 1 lg 5 lg 5 05 . 1 / 1 1 05 . 1 / 1 1 1 1 1 1 log lg 5 1 1 1 1 1 1 1 1 log lg 5

05 .

                                                              

ER ER P P P P ER P ER P P P ER adj

T T T T T T T T T T T T T DM

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Hubble diagram of adjusted PD vs redshift

The adjusted magnitude of PD to z = 0.05,

adj

DM

05 .

versus the redshift

1 1  

ER

T z

• - - - : reference

adj

DM

05 .

based on

 

ER P C

T T c LD    1

subject to constant light speed

Adjusted magnitude of PD to z=0.05, 0.02 0.05 0.1 0.2 0.5 1

• 2

2 4 6 8 10 Redshift z TP = 0.6 TP = 0.7 TP = 0.8

• - - : Constant light speed

(for any TP) Adjusted magnitude of PD to z=0.05, Redshift, z 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

• 2

2 4 6 8 10 TP = 0.6 TP = 0.7 TP = 0.8

• - - : Constant light speed

(for any TP)

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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

2 4 6 8 10 TP = 0.6 TP = 0.7 TP = 0.8

Perlmutter et al. ApJ 1999 (SCP) Adjusted magnitude of PD to z=0.05,

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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

2 4 6 8 10 TP = 0.6 TP = 0.7 TP = 0.8

Adjusted magnitude of PD to z=0.05,

Redshift z

Rubin et al. ApJ 2013 (SCP)

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Conclusion

• Observed redshift is the frequency-based redshift from the time of emission.
• 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 

 PD

z   1

to

05 .  z

was compared with reported Hubble diagrams from the SCP, which were time-dilated by

z  1

and K-corrected.

• Superimposition on the reported Hubble diagrams from the SCP showed an excellent fit. The graph

for the Present Time 7 . 

P

T very closely overlaps the line of flat 27 .  m ΛCDM universe that Rubin et al concluded as the best fit.