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.

5) The redshift and the time (or distance) relationship.

5) The redshift and the time (or distance) relationship.

Since in the Shrinking Matter Theory there is not receding speed, there is no reason to determine the distance based in the standard model (expanding universe), which is necessary determine the apparent receding speed to infer the distance based in the Hubble constant.

In the Shrinking Matter Theory, the size of the atom decrease along the time, so the time should be defined by the rule of lost in volume per unit of time (LVL). The LVL can be mathematically defined as d_VL /d_t.. The LVL should vary along the time, according to the size of the atoms, and this variance could be proportional to the surface or to the volume of the atoms along the time.

This would let us to two hypotheses, A, and B.

The hypothesis A proposes the LVL variance could be proportional to the surface of the atoms.

The hypothesis B proposes the LVL variance could be proportional to the volume of the atoms.

Now we can develop the two hypotheses to analyze the possibility of choose the one which best fit to the observations. 

 

5.1)  Hypothesis A:

This hypothesis proposes the LVL (d_VL/dt) varies proportionally to the surface S_f of the atom, so we can write:

LVL/S_f = constsnt = K_s =>

(d_VL / d_t) / S_f  = K_{s    =>}

d_t  = d_VL / (K_s  S_f)          (I)

Solving this system we have:

D = t = K_z  (x^2/3-1)   (IV)

x = (1+Z)   

The full solving for the above function can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

                                   

D = Distance  (Gly)

t = Time        (Gyr)

Kz = Stretching factor of the function so that fitting it to the measured observations at low redshifts.

The redshift Z can now be expressed in function of the time t as follow:

Z = ((t+K_z)/K_z)^3/2-1      (For hypothesis A)

The farthest bodies newly observed present redshift of about Z =8, so we should limit our researches in in the range of Z from 0 to 8

The distance D in the standard model for Z= 8 is 13.436113 Gly, 

The Bohr radius in the shrinking model r_o and  r_f  are:

r_o = 5,291773 x 10^-11 m (for Z=0)

r_f  = 2,899617 x 10^-10 m (for Z=8)

r_o  = Bohr radius of the hydrogen in the ground state at the present, in our local frame (m).

r_f   = Bohr radius of the hydrogen in the ground state  in the past frame (m).

For the hypothesis A, the above equation (IV), is the basic to define all the relationships in the Shrinking Matter Theory.

Now we can determine the best value for the constant Kz, so that calibrating the equation to the observed distances. This calibration must be done at low redshift, where we can determine the distances by parallax. This mean the above function should give us the same value when the redshift is null (zero), and at low redshifts give us neglected differences. This mean the tangent of the above function (IV), at Z= 0, should have the same as the tangent in function of the standard model (expansion universe).

In the standard model, (expanding universe), the equation to define the distance is:

                     

D = t =  ((x²-1) c 3.26) / ((x² + 1) H_o 10⁶)     mpc   (VI)

c = light speed = 299792458 m/s

H₀ = Hubble constant = 71 km/s/mpc

When deriving and matching the distance equations of the shrinking model and the standard model, we have:

K_z = ( 3 c 3.26) / (2 H_o 10⁶)   =>

c = light speed = 299792458 m/s

H₀ = Hubble constant = 71 km/s/mpc

The exact value of the conversion factor “3.26” is (3.261 633 430 119 53), and it should be used for taking of accurate results.

The full solving for the K_z constant can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

Kz = 20.65802330515155      (best value for Kz, for hypothesis A, for H_o = 71)

5.1.1) Shrinking speed SHV along the time in function of the redshift

The shrinking speed SHV of the Bohr radius can be defined as dr/dt_.

The Bohr radius in the past is defined by the function: 

r_f  =  r_o x^2/3      =>              

t = K_z  (x^2/3-1)   (IV) =>

SHV  = r_o / ( K_z (31 556 926.08)  (10)⁹  )                       

SHV  =  8.117414  (10)^-29  m/s   =  constant (for hypothesis A)

The Shrinking speed is constant along the time, for hypothesis A. 

The full solving for the SHV can be found here:

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

                                                                                                             

                        

5.1.2) Specific shrinking speed (SPV)

The specific shrinking speed is defined as  V_f  / r_f .

Vf : Shrinking speed in a reference frame SHV (IX).

r_f : Bohr radius  in a reference frame. r_f  (VIII).

So,     SPV = x ^-2/3/(K_z (31 556 926.08) 10⁹)       m/s /m  (X)

SPV = x ^-2/3 (3.085 677 581 492 37)(10)⁹ / (K_z  31 556 926.08)    Km/s /mpc   

The full solving for the SPV can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

For Z = 0  =>  x = 1  =>  SPV = 1.534 (10)^-18   m/s /m

The equatorial radius of the Earth is 6 378 136.3 m.

The shrinking speed of the Earth would be:

SHV = (6378136,3) (1.534) (10)^-18   =>

SHV = 9.7839 (10)^-12  m/s

1 year  = 31 556 926.08 seconds, so,

SHV =  (31 556 926.08) (9.7839)(10)^-12  m/year    =>

SHV = 3.0875 (10) ^-4   m / yr

SHV = 0.30875        mm / yr

SHV = 308.75             m / Myr

              

5.2) Hypothesis B:

This hypothesis proposes the LVL (d_VL/dt) variance is proportional to the volume VL_f  of the atom, so we can write:

LVL/VL_f = constsnt = K_v =>

(d_VL / d_t) / VL_f  = K_v     

d_t = d_VL / (K_v  VL_f)       (XI) =>

t = K_z ln(x)          (XIII)

The full solving for the above function can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

x = (1+Z)  =>     (t/K_z) = ln(x) =>     x = e^(t/Kz)   =>  1+Z = e^(t/Kz) =>    

Z = e^(t/Kz) -1        (For hypothesis B)    

e = 2.718 281 828 459 05

K_Z = see below resolution

t : (Gyr)

Now we can determine the best value to the constant Kz, so that calibrating the equation to the observed distances. This calibration must be done at low redshift, where we can determine the distances by parallax. This mean the above function should give us the same value when the redshift is null (zero), and at low redshifts give us neglected differences.

This mean the tangent of the above function at Z= 0, should be the same as the tangent in the respective function of the standard model (expansion universe).

In the standard model, (expanding universe), the equation to define the distance is:

                     

D = t =  ((x²-1) c 3.26) / ((x² + 1) H_o 10⁶)     mpc   (VI)

c = light speed = 299792458 m/s

H0 = Hubble constant = 71 km/s/mpc

When deriving and matching the distance equations of the shrinking model and the standard model, we have:

K_z = ( c 3.26) / ( H_o 10⁶)   =>

K_Z = 13.772 015 536 767 7  (best value for Kz in the function (XIII), hypothesis B, for H_o = 71)

c = light speed = 299792458 m/s

H₀ = Hubble constant = 71 km/s/mpc

The exact value of the conversion factor “3.26” is (3.261 633 430 119 53), and it should be used for taking of accurate results.

The full solving for the K_z constant can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

5.2.1) The Shrinking speed SHV along the time in function of the redshift

The shrinking speed SHV of the Bohr radius can be defined as dr/dt_.

The Bohr radius in the past is defined by the function:

r_f  =  r_o x^2/3      =>                      dr = 2 r_o x ^-1/3 / 3          

 t = K_z ln(x)   (XIII )       =>      dt =  K_z / x (XII) =>

SHV = dr/dt = 2 r_o x^2/3/(3 K_z)            m / Gyr          =>

SHV =  2 r_o x^2/3/(3 K_z (31 556 926.08) 10⁹ )   m/s  (XIV)

   

x = (1+Z)

The full solving for the SHV can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

5.2.2) Specific shrinking speed (SPV)

The specific shrinking speed SPV is defined as V_f  / r_f .

V_f  : Shrinking speed in a reference frame SHV (XIV).

r_f : Bohr radius  in a reference frame.

r_f  = r_o x^2/3    =>

SPV = 2 / (3 K_z (31 556 926.08) 10⁹)    m/s /m    (XV)

SPV = 1,5339 (10)^-18   m/s /m

SPV = 2 (3.085 677 581 492 37)(10)¹⁰ / (3 K_z  31 556 926.08 )  km/s /mpc

SPV = 47.333  km/s /mpc

                     

5.2.3) The Shrinking acceleration (SHA) along the time in function of the redshift

The shrinking acceleration SHA is defined as the variation of the speed in function of the time, so, it can be defined mathematically as:

SHA  = dv / dt  =>

V = SHV  (IX)      and      d_t = (VII)

SHA = 4 r_o x^2/3 / (9 (K_z)² (31 556 926.08) (10)⁹ )  m/s   /Gyr  =>

SHA = 4 r_o x^2/3 / (9 (K_z)² (31 556 926.08) (10)¹² )  m/s   /Myr    (XVI)

The full solving for the SHV can be found here: 

https://app.box.com/s/lu5gmwwj5tfxzjrpu1y3kokdp7i5b0k6

5.2.4) Specific shrinking acceleration (SPA) along the time in function of the redshift

The specific acceleration SPA is defined as the shrinking acceleration SHA per unit of length.

This means as bigger the length of a body, as bigger the SHA.

The SPA can be defined as:      SPA = SHA / r_f

SPA = 4 / (9 (K_z)² (31 556 926.08) (10)¹² )  m/s /m  /Myr  =>  (XVII)

For hypothesis B,  SPA = Constant = 7.4245 (10)^-23   m/s /m /Myr

SPA = 4(3.085 677 581 492 37)(10)⁷ / (9 (K_z)² (31 556 926.08) )  km/s /mpc  /Myr  =>

For hypothesis B,  SPA = Constant = 2.2909 (10)^-3     km/s /mpc /Myr

The Graphic 01 presents the comparative evolution of the distance (Gly) or time (Gyr), for the shrinking Model hypothesis A “Time shrM A”, the shrinking Model hypothesis B “Time shrM B”, and the standard Model “Time stdM”.

Graphic  01