Redshift and distance measurements

Redshift definition

Redshift measures how much a wavelength has been stretched. It is defined by

1+z=rac{lambda_{obs}}{lambda_{emit}}

or

z=rac{lambda_{obs}-lambda_{emit}}{lambda_{emit}}

If $z>0$, light is redshifted. In cosmology, redshift usually indicates expansion of space during the light's journey.

Redshift and scale factor

Cosmological redshift relates directly to the scale factor:

1+z=rac{a(t_0)}{a(t_{emit})}

Often we set today's scale factor to

$a(t_0)=1$

so

a(t_{emit})=rac{1}{1+z}

High-redshift light was emitted when the universe was smaller.

Distance is not simple

In everyday life, distance is straightforward. In cosmology, expansion means several distance definitions are useful: comoving distance, proper distance, luminosity distance, angular diameter distance, and light-travel distance.

These are not always equal. The distinction matters because the universe expands while light travels.

Standard candles

A standard candle is an object whose intrinsic luminosity is known or can be calibrated. If luminosity $L$ is known and observed flux $F$ is measured, distance follows from

F=rac{L}{4pi d^2}

In cosmology, this becomes luminosity distance:

F=rac{L}{4pi d_L^2}

Cepheid variable stars and Type Ia supernovae are important standard candles.

Distance ladder

The cosmic distance ladder builds distances step by step. Parallax calibrates nearby stars. Cepheids calibrate distances to nearby galaxies. Type Ia supernovae extend measurements to far larger distances.

Each rung depends on calibration from the previous rung, so systematic errors can propagate.

Standard rulers

A standard ruler has known physical size. If its angular size is observed, distance can be inferred. Baryon acoustic oscillations provide a cosmological standard ruler. The angular diameter distance is defined through

heta=rac{ell}{d_A}

where $ell$ is physical size and $heta$ is angular size in radians.

Lookback time

Looking far away also means looking into the past. The lookback time is the difference between the current age of the universe and the age when the light was emitted. A galaxy at high redshift is seen as it was long ago.

Redshift is not simply distance; converting redshift to distance or time requires a cosmological model.

The big idea

Redshift tells how much the universe expanded while light traveled. Distances in cosmology require careful definitions and calibrated tools such as standard candles and standard rulers. Measuring the expansion history depends on linking redshift, brightness, angular size, and cosmological models.