6) The SN1a distance ladder and the shrinking matter theory.

The Shrinking Matter Theory is characterized by the possibility of vary the Planck constant  along the time as the factor of the redshift of the emissions in the past.

This justifies the bigger size of the atoms and bodies in the past, as well the longer wavelength emissions and smaller energy and temperature.

The main relationship relative to the proprieties of the matter and the redshift is listed below.

h_f  =  h_o (1+Z)^1/3                        Plank constant

λ_f = λ_o (1+Z)                             wavelength emissions

_rf = r_o (1+Z)^2/3                          Bohr radius, energetic n level radius, and bodies size.

E_f = E_o (1+Z) ^-2/3                      Energy of the line emissions.

WDC_f = WDC_o (1+Z)^1/3          Wien Displacement Constant

R_∞f = R_∞o(1+Z) ^-1                     Rydberg constant

T_f = T_o (1+Z)^-2/3                       Temperature of the line emissions

The SN1a distance ladder or "cosmic distance ladder" is a system used to calculate the distances based in the hypothesis which their luminosity peak is constant, so, as fainter the flux received in our telescopies, as longer the distance from the Earth. The relationship between the distances and the flux is:

 (XVIII)

F₁ and D₁ are flux and distance of a near and known SN1a, which distance can be determined by parallax, used as standard reference.

F₂ is the measured flux of a more distant SN1a, and D₂ is the unknown distance to be calculated.

The absolute magnitude μ is a logarithm scale where:

  (XIX)

but, 

       

so,

     

                                                                                       

The adopted standard distance D₁ is 10 pc, so that simplify the equation, since log10 = 1. The equation so becomes:

             μ = 5 log D₂ - 5      (XX)                   

                                         D₂ : (pc)

This equation works well for low redshifts, but in the Shrinking Matter Theory the flux F₂ is affected by the redshift. In the past the energy of the emissions was smaller, and such energies were spread onto a bigger surface.  

The energy of the emissions in the past is defined by the function  E_f = E_o (1+Z) ^-2/3, and the surface by the function  S_f = 4 π  r_f ².

To nullify the effects of the redshift in flux F_2, we should replace it by the corrected flux F_2c .

The F_2c is defined by the function:

 

 

    F_2c = F₂ (1+Z)²     (XXI)        

Then, the relationship between the fluxes and the distances becomes:

  

 

   (XXII)

The absolute magnitude function for the Shrinking Matter Theory becomes:

           μ = 5 log D₂  + 5 log (1+Z) - 5      (XXIII)

                                     D₂ : (parsecs)

The graphic 02 presents the comparative evolution of the absolute magnitude μ.

The evolution of the expected μ to the Shrinking Matter Theory, hypothesis A is in red color.

The evolution of the expected μ to the Shrinking Matter Theory, hypothesis B is in green color.

The evolution of the expected μ to the Standard Model (expanding universe) is in black color.

The observed evolution of the absolute magnitude μ is represented by square blue points, which were extracted from Betoule et al 2014, Table F1, page 30,  “http://arxiv.org/pdf/1401.4064v2.pdf “.    

Graphic 02

The curve which best fit to the observational data is the hypothesis A of the Shrinking Matter Theory “shrM A abs mag”. No need for dark energy.