Exemple of changing to a reference frame in the past

Suppose we search a galaxy and we detect the Lyα emission being three times greater than the Lyα in our world. The wavelenght is exactly 3647,07 Â.

The H Lyα in our frame is 1215,69 Â.

So, the apparent redshift Z(f) is Lyα(f)/Lyα(o)-1 = 2

The constant Kh is (1+Z(f))^(1/3) = (1+2)^(1/3) =1,44224957

So, Kh=1,44224957

Now we can determine the main constants of the reference frame in the past;

                        formula*                   local frame(o)             past frame(f)

h(f)                 h(o) x (Kh)                  6,62606957E-34 Js        9,5564460E-34 Js

Lyα(f)             Lyα(o) x (Kh)³              1215,69 Â                    3647,07 Â

r(f)                 r(o) x (Kh)²                 0,529 Â                       1,1007 Â

E(f)                 E(o) / (Kh)²                 10,204 eV                    4,906 eV

WDC(f)          WDC(o) x (Kh)               0,0028978 mK              0,0041796 mK

R∞(f)             R∞(o) / (Kh)³               10973730,68 m-¹          3657910,22 m-¹

T(f)                 T(o) / (Kh)²                 23836 ºK                     11459 ºK

(f) Observed frame in the past

(o) Local frame at the present

 * simplified formula

If we consider Ho=71 km/s/mpc and assume it is enough to determine the distance, we have:

distance= 3380,3 mpc   = 11,02 GLY

time (past) = 11,02 Gyr

Ho= Hublle constant