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

Transcription

1 Frequency-based redshift for cosmological observation and Hubble diagram from the -D spherical model in comparison with observed supernovae Summary DIC1, Sep. 1-1, Castiglioncello (Italy) Shigeto Nagao, h.d. (Osaka, Japan) According to the formerly reported -D spherical model of the universe, factors on Hubble diagrams are discussed. he 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. he 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 1 z versus the frequency-based redshift with the reported Hubble diagrams from the Supernova Cosmology roject, which were time-dilated by 1 z and K-corrected, showed an excellent fit for the resent ime (radius of -D sphere) being.7 of its maximum.

2 -D Spherical Model of the Universe he space energy spreads with expansion in a 3-D surface of a -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 -D sphere is our Observed ime (ime or ), which we feel passing commonly for all of us. x: Radius vector of D sphere x ( x, θ) ( x, 1,, 3) (D spherical coordinate) hase ransition Space expansion x r r: 3D space vector corresponding to θ r ( r, 1, ) ( x, 1, ) x by D cylindrical coordinate: x x, r,, ) ( x, x,, ) < Definition of terms related to ime > x (Radius of universe): CU (Cosmic Unit): (ime of mission): (resent ime): (Relative ime of mission): Relative ratio of to. Radius of the D sphere of universe, equal to the Observed ime () Unit of x and, being one at its maximum when the space expansion stops. ime-related variables are expressed in the CU in this article. ime when the light was emitted. resent ime of universe, when the light reaches us. / B (Back in ime): Back in ime from present when the light was emitted. B BR (Relative Back in ime): Relative ratio of B to. BR B / C (ime Clear): ime when the space became transparent to light. CR (Relative ime Clear): Relative ratio of C to. CR C / DIC1, Castiglioncello (Italy) ( 1 1

3 Redshift for cosmological observation We compare the observed wavelength with that of the present atom Observed redshift z : emitted at reaches us at,, Wavelength-based redshift: z,. DIC1, Castiglioncello (Italy) 3 (present atom) 1 equal to frequency-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. z 1 C n C Frequency-based redshift: z 1 C C 1 C C 1 n n C C C C z z 1 z C C z.

4 Magnitude of LD to z=.5, Magnitude of LD to z=.5, Magnitude of light propagated distance LD to z=.5 Light of luminosity L emitted at reaches us now at. L Flux we observe now: F LD Its magnitude: m.5 lg F Relative magnitude to same luminosity at z=.5: DM m mz.5 m Light speed: Light propagated distance: LD Cx 1 5 lg log m 1 C C( x) K f D f M K 1 3 x 1 x x K dx dx K log log x 1 x lg log /1.5 /1.5 DM.5, a distance modulus (a) LD versus redshift z 1 (b) LD versus Relative Back in ime BR =. =.7 =. =. =.7 = : Constant light speed (for any ) : Constant light speed (for any ) Redshift z Relative Back in ime, BR DIC1, Castiglioncello (Italy)

5 Hubble diagram 3-D space expansion speed: For a given angle, variable x: dr d dr dx constant d x dx r x For a given radius x, variable : proportional to and to r Hubble s law Hubble diagram: o discuss the recessive velocity of the proper distance entatively provide that light speed has been constant. z Light propagated distance: LDC c c 1 c z 1 Multiply by 1 z (time dilation): 1 z LD c z k z proportional to z t' 1 zt Light-curve width of supernovae: dilated by 1 z Ratio of D to LD (LD-D conversion): A -B : C-B: n 1 CB A B proper distance at ( present distance, D ) light propagated distance (LD) (= 3D-space component of AB) n 1 n 1, D LD C n z 1 n 1 z 1 A B A B C DIC1, Castiglioncello (Italy) 5

6 Adjusted magnitude of present distance D to z=.5 For comparison with reported Hubble diagrams, take the following adjusted D: z 1 z D z LD z or D LD 1 ime dilation Reported Hubble diagrams from the SC: K-correction K xy : my M x DM Kxy LD-D conversion 1) cross-filter adjustment on absolute magnitude M y M x, plus ime-dilated and K-corrected DL (luminosity distance), m M DM z K M M DM z DM y ) difference in distance modulus DM z DM (= mag LD mag D) between observed / rest frames Frame-conversion part of K-correction corresponds to LD-D conversion. Adjusted magnitude of D to z =.5, DM adj.5 adj DM. 5 y : Add ime dilation and LD-D conversion to DM. 5 5 lg LD lg1 lg 1 lg LD1 1.5 lg1.5 lg lg LD lg lg1 lg LD1 1.5 lg1.5 lg.5 xy y x DM adj lg log lg 5 lg lg log 1 1lg1.5 5 lg /1.5 /1.5 DIC1, Castiglioncello (Italy)

7 Adjusted magnitude of D to z=.5, Adjusted magnitude of D to z=.5, Hubble diagram of adjusted D vs redshift he adjusted magnitude of D to z =.5, adj DM. 5 versus the redshift z =. =.7 = : Constant light speed (for any ) =. =.7 = : Constant light speed (for any ) Redshift z Redshift, z : reference adj DM. 5 based on LD c C 1 subject to constant light speed DIC1, Castiglioncello (Italy) 7

8 Adjusted magnitude of D to z=.5, Comparison with Hubble diagrams from the SC Superimposition on the Hubble diagram by erlmutter et al, z in a logarithmic scale 1 erlmutter et al. ApJ 1999 (SC) =. =.7 = Redshift z DIC1, Castiglioncello (Italy)

9 Adjusted magnitude of D to z=.5, Superimposition on the latest Hubble diagram from the SC, z in a uniform scale 1 Rubin et al. ApJ 13 (SC) =. =.7 =. - Redshift z DIC1, Castiglioncello (Italy) 9

10 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 D (proper distance at present) is given. he frame-conversion part of the K-correction corresponds to the LD-D conversion. Magnitude of 1 z D to z. 5 was compared with reported Hubble diagrams from the SC, which were time-dilated by 1 z and K-corrected. Superimposition on the reported Hubble diagrams from the SC showed an excellent fit. he graph for the resent ime. 7 very closely overlaps the line of flat m. 7 ΛCDM universe that Rubin et al concluded as the best fit. DIC1, Castiglioncello (Italy) 1