CMB photons that we receive today started their journey at t ∼ 0 on a sphere the last scattering surface They reach us today at t = t0 = 138 billion years We are also able to measure the proper distan

The prefactor 3 comes from the fact that the universe was expanding during the time it took for the CMB photons to travel from the last scattering surface to us. This means that the distance between us and the last scattering surface has increased by a factor of 3 since the photons were emitted.

To estimate this, we can use the formula for the scale factor of the universe:

a(t) = (t/t0)^(2/3)

where t0 is the present age of the universe.

At the time the CMB photons were emitted, the scale factor would have been a(t) = 10^(-3). Today, the scale factor is a(t0) = 1.

The proper distance to the last scattering surface is given by:

D = a(t0) * r

where r is the comoving distance to the last scattering surface.

Since the scale factor increased by a factor of 10^3, the comoving distance also increased by the same factor. Therefore,

r = D/10^3

Substituting this into the expression for the proper distance, we get:

D = a(t0) * r = a(t0) * D/10^3

Solving for D, we get:

D = 3 * a(t0) * ct0

where we have used the fact that a(t0) = 1 and c is the speed of light.

Therefore, the prefactor 3 comes from the expansion of the universe between the time the CMB photons were emitted and the present day.