Redshift and the Scale Factor

Redshift and the Scale Factor

As the universe expands, photons travelling through space are stretched to longer wavelengths — a phenomenon called cosmological redshift. This is distinct from Doppler redshift; it is the fabric of space itself that is expanding.

The Scale Factor

The scale factor $a(t)$ describes the relative size of the universe at time $t$, normalised so that $a_0 = 1$ today. When a photon was emitted at time $t_e$, the scale factor was $a_e < 1$.

Redshift Definition

The cosmological redshift $z$ is defined by:

$1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}} = \frac{a_0}{a_e} = \frac{1}{a_e}$

So: $z = \frac{1}{a} - 1 \qquad \text{and} \qquad a = \frac{1}{1+z}$

Key Redshifts in Cosmology

CMB Temperature

The CMB photons were emitted at recombination ($z \approx 1089$) when the universe was opaque and hot. As space expanded by a factor of $1+z$, the photon wavelengths stretched by the same factor, cooling the radiation:

$T(z) = T_0 \cdot (1+z)$

where $T_0 = 2.725$ K is the CMB temperature today. At recombination, the temperature was $T \approx 2970$ K.

Your Task

Implement the following functions. The constant $T_0$ must be defined inside cmb_temperature_at_z.

• redshift_from_scale(a) — returns $z = 1/a - 1$
• scale_from_redshift(z) — returns $a = 1/(1+z)$
• cmb_temperature_at_z(z) — returns $T(z) = T_0(1+z)$ in Kelvin, with $T_0 = 2.725$ K