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

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. x : Radius vector of 4D sphere      θ x ( x , ) ( x , , , ) (4D spherical coordinate) 1 2 3 r : 3D space vector corresponding to θ x Phase Transition Space expansion r        r ( r , , ) ( x , , ) 1 2 1 2 x by 4D cylindrical coordinate:        x ( x , r , , ) ( x , x , , ) 1 2 1 2 < 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. T E (Time of Emission): Time when the light was emitted. T P (Present Time): Present Time of universe, when the light reaches us.  T T / T T ER (Relative Time of Emission): Relative ratio of T E to T P . ER E P   T B (Back in Time): Back in Time from present when the light was emitted. T T T B P E  T BR (Relative Back in Time): Relative ratio of T B to T P . T T / T BR B P T C (Time Clear): Time when the space became transparent to light.  T CR (Relative Time Clear): Relative ratio of T C to T P . T T / T CR C P 2 DICE2016, Castiglioncello (Italy)

Redshift for cosmological observation       T T We compare the observed wavelength with that of the present atom . P 0 P T T emitted at reaches us at E P             → T , T T , T 0 E E 0 P P              T , T T T (present atom) 0 E 0 P 0 P 0 P          T T T     P 0 P 0 E z 1 Observed redshift z : z .       equal to frequency-based redshift     T T T 0 P P P 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.            T C T T C T C T 1         P P P P P 1 C T 1 z n             P  z  1   T C T T C T T C T 0 E E E E ER E T C T ER E Frequency-based redshift: 1                    T T T T C T C T 1 z  z 1          0 E P P 0 E P E z 1 n n                  T T T T T C T C T T ER P 0 P 0 E 0 P E P ER 3 DICE2016, Castiglioncello (Italy)

Magnitude of light propagated distance LD to z=0.05 Light of luminosity L emitted at T E reaches us now at T P .   L         Flux we observe now: F T Its magnitude: m T 2 . 5 lg F T   2 E   E E 4 LD T E                DM T m T m z 0 . 05 m T m 1 1 . 05 Relative magnitude to same luminosity at z =0.05 : 0 . 05 ER ER ER    3 1 T         Light speed: C C ( x ) K f f K 1   D EM  3 x 1 x  x            1 1 T 1 1 T     K    T  T      P  P   Light propagated distance: P E LD T C x dx dx K log log       E      1 1 1 1 1 T T x x  T   T    E E P E                 1 1 T 1 1 T T 1 1 T 1 1 T / 1 . 05                 P P ER P P DM T 5 lg log 5 lg log , a distance modulus         0 . 05 ER          1 1 T 1 1 T T   1 1 T 1 1 T / 1 . 05      P P ER P P (b) LD versus Relative Back in Time T BR 10 (a) LD versus redshift z 10 8 8 T P = 0.6 T P = 0.7 Magnitude of LD to z=0.05, 6 Magnitude of LD to z=0.05, 6 T P = 0.8 - - - : Constant light speed 4 4 T P = 0.6 (for any T P ) T P = 0.7 2 2 T P = 0.8 - - - : Constant light speed 0 0 (for any T P ) -2 -2 0.05 0.1 0.5 1 5 10 0.02 0.05 0.1 0.2 0.5 1 Redshift z Relative Back in Time, T BR 4 DICE2016, Castiglioncello (Italy)

Hubble diagram     dr dr d x       3-D space expansion speed: r x dT dx dx For a given angle  , variable x : constant  For a given radius x , variable  : proportional to  Hubble’s law → and to r Hubble diagram: To discuss the recessive velocity of the proper distance Tentatively provide that light speed has been constant.     z          Light propagated distance: LD c T T c T 1 T c T  C P E P ER P z 1 Multiply by             → 1 z 1 z LD c T z k z ( time dilation ): proportional to z C P   t      t ' 1 z Light-curve width of supernovae: dilated by 1 z Ratio of PD to LD ( LD-PD conversion ): A P proper distance at T P (“ present distance, PD ”) A P -B P : A E C C-B P : light propagated distance (LD) (= 3D-space component of A E B P ) T E T P   n T T 1 T P E ER B E    CB 1   1   PD 2 n 2 z 1 2         P 1 n 1 n 1 , B P    A B 2 2 LD n 1 z 2 1 T E E ER 5 DICE2016, Castiglioncello (Italy)

Adjusted magnitude of present distance PD to z=0.05 For comparison with reported Hubble diagrams, take the following adjusted PD:   LD      2 z 1 1 1 2           1 z PD 1 z PD LD or   z 2 T T 1 T ER ER ER Time dilation LD-PD conversion Reported Hubble diagrams from the SCP: Time-dilated and K-corrected D L (luminosity distance)          →     K : 0 0 K-correction m M DM z m M DM K K M M DM z DM , xy y x xy y y xy y x M  1) cross-filter adjustment on absolute magnitude , plus M y x    2) difference in distance modulus 0 (= mag LD – mag PD ) between observed / rest frames DM z DM Frame-conversion part of K-correction corresponds to LD-PD conversion . adj DM DM Adjusted magnitude of PD to z = 0.05 , : Add Time dilation and LD-PD conversion to 0 . 05 0 . 05                          adj DM T 5 lg LD T lg 1 T lg 2 1 T lg LD 1 1 . 05 lg 1 . 05 lg 2 1 . 05 2 . 05 0 . 05 ER ER ER ER                    5 lg LD T lg T lg 1 T lg LD 1 1 . 05 2 lg 1 . 05 lg 2 . 05 ER ER ER                 1 1 T 1 1 T T 1 1 T 1 1 T / 1 . 05                 P P ER P P adj DM T 5 lg log 5 lg log         0 . 05 ER         1 1 T 1 1 T T 1 1 T 1 1 T / 1 . 05         P P ER P P            5 lg T 5 lg 1 T 10 lg 1 . 05 5 lg 2 . 05 ER ER 6 DICE2016, Castiglioncello (Italy)