Cosmological Redshift

Equivalent forms

1 + z = \frac{a_0}{a_e}

z = \frac{\lambda_{obs}-\lambda_{emit}}{\lambda_{emit}}

Redshift is a direct clock: z tells you the exact ratio by which the universe has expanded since the light was emitted.

Symbol	Name	SI unit	Dimension	Range
$z$	Redshiftoutput Fractional wavelength stretch	dimensionless	1	0 – 1100
$ae$	Scale Factor at Emission a(t_emit), with a(today)=1	dimensionless	1	0.001 – 1

Dimensional analysis

1+z -> dimensionless

a_obs/a_emit -> dimensionless ratio

Both sides pure numbers.

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Discovery

Georges Lemaître · 1927

Lemaître showed redshift traces the expansion of space itself rather than ordinary Doppler motion, reinterpreting Slipher's spectra and Hubble's soon-to-come data.

Research status: stable

Real-world applications

Redshift surveys mapping large-scale structure
Dating the epoch a galaxy's light was emitted
CMB temperature-redshift tests of standard cosmology

Common misconceptions

Cosmological redshift is NOT a Doppler shift through space; it is the stretching of space
z can exceed 1 without any 'faster than light' contradiction
Photon energy is not conserved in an expanding universe — there is no global energy conservation

Experimental verification

The CMB blackbody at

z\sim 1100

(COBE/Planck), quasars out to z>7, and the tight relation between measured z and independent distance indicators all confirm the scale-factor interpretation. The CMB temperature at high z,

T = 2.725(1+z) K

, has been measured in distant gas clouds.

Derivation

A wave crest emitted at t_e and the next at t_e + dt_e travel null geodesics in an FLRW metric.

The comoving distance integral integral dt/a is the same for both crests, forcing dt_obs/dt_emit

= a_obs

/a_emit.

Since wavelength is proportional to the period, lambda_obs/lambda_emit

= a_obs

/a_emit

= 1+z

.

Limiting cases

a_

e = 1

⟶

z = 0

Light emitted here and now: no expansion, no shift.

a_

e = 0.5

⟶

z = 1

The universe was half its present size; wavelengths have doubled.

a_e -> 1/1101⟶

z \approx 1100

The cosmic microwave background, emitted at recombination when the universe

was \sim 1100x

smaller.

What if…

What if we observe

z = 1089

(the CMB)?

The light was emitted 380,000 years after the Big Bang, when the universe $was \sim 1090x$ smaller and 3000 K; it now peaks in microwaves at 2.7 K.

What if space contracted instead?

a_e > a_0 would give z < 0 — a blueshift, as would happen in a collapsing 'Big Crunch'.

1

A galaxy at z = 2

Given ·

z:: 2

Find · scale factor at emission

Steps

$1 + z = 1$ /a_e
a_ $e = 1/(1+z) = 1/3$
wavelengths stretched by factor 3

Result ·

1 + z = a_0

/a_e -> a_

e = 1/(1+z) = 1/3

. The universe was one third its current size; its light's wavelength has tripled.

Hubble-Lemaître Law→scale factor evolutioncmb temperature todayfriedmann equations