Redshift and Cosmological Distances

Relation between redshift, expansion, and observable cosmological distances.

Conceptual context

When we observe a distant galaxy, we are not only asking where it is. We are reading light that has traveled while the universe changed size. $z$ appears because a spectral line emitted with a certain $\lambda _em$ arrives with a larger $\lambda _obs$. That stretching is not itself a ruler, but it is a record of expansion. To convert it into $D_C$, $D_L$, $D_A$, or $\mu$, we need a cosmological model that tells us how expansion evolved during the light journey.

🟢 Essential level

$z$ measures how much light has been stretched between emission by a distant galaxy and detection by a telescope.

If $\lambda _obs$ is larger than $\lambda _em$, the light is shifted toward the red, which indicates that the universe was smaller when the light was emitted. At low redshift, a larger $z$ usually means a larger distance, so an approximate $D_Hubble$ can be estimated using the scale set by $H_0$ and $c$. At larger redshift, the relation is no longer a simple straight line. The whole expansion history matters, summarized by $E_z$, $\Omega _m$, $\Omega _k$, and $\Omega _\Lambda$. Redshift is therefore not distance itself, but a spectral key for calculating several observable cosmological distances.

Quasar 3C 273, identified in 1963 by Maarten Schmidt, was the first object in which such a large $z$, z ≈ 0.158, was recognized, revealing it to be an extraordinarily distant and luminous object rather than a peculiar nearby star, as had first been assumed before its spectrum was correctly interpreted.

🔵 Formal level

Basic formulas

The formal starting point compares the detected spectral line with its rest-frame reference. $z$ is obtained from $\lambda _obs$ and $\lambda _em$ as a dimensionless relation: matching wavelengths give zero z, while a larger observed wavelength gives positive z.

Redshift from wavelength

The same information can be read as the relative size of the universe at emission. $a_em$ connects to $z$ because cosmic expansion stretches all cosmological wavelengths.

Scale factor and redshift

For small $z$, the local approximation is used: $D_Hubble$ is nearly proportional to z, with a scale controlled by $c$ and $H_0$.

Low-z Hubble law

Real student questions

Is $z$ a velocity? Not exactly. At small z it can be approximated as local recession, but at large z it measures accumulated expansion.

Why are there several distances? Because brightness, angular size, and comoving coordinates ask different questions about the same light.

Is a galaxy with larger z always farther away? In the global cosmological model, usually yes, but local observations, peculiar velocities, and spectral line errors can break a naive reading.

Is $H_0$ enough for everything? Only near zero redshift. Cosmological distances require $E_z$.

Final synthesis

$z$ is the spectral signal of accumulated expansion. Turning it into distance requires choosing a model, parameters, and a distance type. The physics lives in that translation: measuring z is not enough; one must interpret the cosmological map it builds.