Distance Measures in Cosmology

Distance Measures in Cosmology

The comoving distance $d_C$ is the fundamental coordinate distance, but astronomers use several derived distance measures depending on what is being observed. Each has a clear physical meaning and practical application.

Luminosity Distance

The luminosity distance $d_L$ relates the intrinsic luminosity $L$ of an object to its observed flux $F$:

$F = \frac{L}{4\pi d_L^2}$

In terms of the comoving distance:

$d_L = d_C \cdot (1+z)$

The extra factor of $(1+z)$ accounts for two effects: photon energies are redshifted by $(1+z)$, and photon arrival rate is also reduced by $(1+z)$. Luminosity distance is used with standard candles like Type Ia supernovae.

Angular Diameter Distance

The angular diameter distance $d_A$ relates the physical size $\ell$ of an object to its observed angular size $\theta$:

$\theta = \frac{\ell}{d_A}$

$d_A = \frac{d_C}{1+z}$

This is used with standard rulers like the Baryon Acoustic Oscillation (BAO) scale. Note that $d_A$ actually decreases beyond $z \approx 1.6$ in $\Lambda$CDM — very distant objects appear larger!

The Distance Modulus

For observational astronomy, the distance modulus $\mu$ converts luminosity distance to magnitudes:

$\mu = 5 \log_{10}\left(\frac{d_L}{10 \text{ pc}}\right) = 5 \log_{10}(d_L^{\text{Mpc}}) + 25$

since $d_L^{\text{Mpc}} \times 10^6 / 10 = d_L^{\text{Mpc}} \times 10^5$.

Your Task

Implement the following functions. The constant $\pi$ is available via import math. All other constants must be defined inside each function body.

• luminosity_distance_Mpc(d_comoving_Mpc, z) — returns $d_L = d_C(1+z)$ in Mpc
• angular_diameter_distance_Mpc(d_comoving_Mpc, z) — returns $d_A = d_C/(1+z)$ in Mpc
• distance_modulus(d_L_Mpc) — returns $\mu = 5\log_{10}(d_L^{\text{Mpc}} \times 10^6 / 10)$