# Formula for redshift of photon decoupling

• I
• Vick

#### Vick

TL;DR Summary
Formula for calculating the redshift at which photon decoupling (CMBR) occurs.
I have come across an old formula from my notes and I have no reference for it but it is using truncated digits in its formula to calculate the redshift for decoupling. The formula is nearly as accurate as the observed data from Planck 2018.

So I would like to figure out the derivation of those digits and any possible reference to it.

The formula is as follows:

redshift:
b2 = 0.56 * (1 + 21.1 * pbdp ^ 1.81) ^ -1
b1 = 0.0783 * pbdp ^  -0.238 * (1 + 39.5 * pbdp ^ 0.763) ^ -1
ms1 = m * (h / 100) ^ 2

zz1 = 1047.5 * (1 + 0.00124 * pbdp ^ -0.738) * (1 + b1 * ms1 ^ b2) * (0.0072973525664 / 0.0072973525664) ^ 2.08

where,
pbdp = physical baryon density parameter ##(\Omega_b h^2)## usually 0.02230
m = matter density ##(\Omega_m)## usually 0.3111
h = Hubble constant ##(H_0)## usually 67.66 km/sec/Mpc

At first glance, the 0.00729 appears to be the Fine-Structure constant, though it is dividing by itself...

Any ideas?