Redshift and Universe Temperature: Connections with Distance

How Redshift Links Cosmic Temperature, Distance, and Lookback Time in an Expanding Universe

The James Webb Space Telescope has discovered the most distant galaxies ever observed. CEERS survey data now enables immersive 3D fly-throughs of cosmic history. At extreme distances, compact, star-forming galaxies dominate; closer in, galaxies appear more diffuse and quiescent. As space expands, it stretches light waves—creating a fundamental link between redshift (z), the temperature of the cosmic microwave background (CMB), physical distances, and lookback time.

Redshift as a Direct Measure of Cosmic Expansion

For nearby objects—our Sun or stars within the Milky Way—redshift z = 0: distance in light-years equals light-travel time. Sunlight (150 million km away) takes 500 seconds; light from a star 10 light-years away arrives after 10 years.

Beyond gravitationally bound structures—roughly beyond 5 million light-years—cosmic expansion dominates. Light from distant sources is stretched: the observed wavelength λₒ exceeds the emitted wavelength λₑ.

Redshift is defined as:

λₒ / λₑ = 1 + z

All spectral features—emission lines, absorption lines, and blackbody continuum peaks—shift by the same factor (1 + z). By matching laboratory spectra to observed ones, astronomers calculate z.

• Emission lines: electrons drop from higher to lower energy levels, emitting photons at precise, fixed wavelengths.
• Absorption lines: foreground atoms absorb background light at those same transition wavelengths.
• Thermal radiation: blackbody spectra peak according to Planck’s law.

When z > 0, space expanded during the photon’s journey—stretching all wavelengths proportionally by (1 + z).

Cosmic Temperature and Its Evolution Over Time

Today, the cosmic microwave background (CMB) has a temperature of T = 2.725 K. In the past, it was hotter: T(z) = T₀ × (1 + z).

Because all wavelengths stretch equally—including CMB photons—the temperature at emission for light observed at redshift z was 2.725 × (1 + z) K.

Observations confirm this scaling:

• Near us (z ≈ 0): T ≈ 2.7 K (blue data points).
• At high redshifts (z > 1): T rises linearly with (1 + z) (red data points).
• Perfect agreement with Big Bang cosmology.

Solar spectra show broad H/He lines and metal absorption from supernova ejecta—similar features appear in distant galaxies, but shifted predictably by redshift.

Distance and Lookback Time in the FLRW Metric

In an expanding universe, distances are not static. Light doesn’t travel a fixed path: the scale factor a(t) evolves from a = 1 today to a < 1 in the past.

Redshift relates directly to the scale factor at emission: z = 1/a − 1.

Lookback time t_L—the time elapsed since emission—is calculated as:

t_L = ∫₀^{t₀} dt = ∫₀^z dz' / [H₀ (1 + z') E(z')]

where H₀ is the Hubble constant and E(z) = √[Ω_m (1+z)^3 + Ω_Λ].

Luminosity distance is d_L = (1 + z) ∫ c dt / a(t).

• Comoving distance: χ = ∫ c dt / a(t).
• Angular diameter distance: d_A = χ / (1 + z).
• Luminosity distance: d_L = (1 + z) χ.

Practical calculations rely on numerical integration using standard cosmological parameters (Ω_m ≈ 0.3, Ω_Λ ≈ 0.7).

Real-World Measurements with JWST

CEERS data provides 3D galaxy coordinates—yielding precise z, T(z), and d(z). Compact galaxies at z > 10 represent the earliest epochs: hot, dense, rapidly evolving environments.

Example: a galaxy at z = 10.

• CMB temperature = 2.725 × 11 ≈ 30 K.
• Lookback time ≈ 13.2 billion years (assuming H₀ = 70 km/s/Mpc).
• Luminosity distance d_L ≈ 30 Gpc.

Spectroscopy identifies Lyα emission (1216 Å → 12768 Å at z = 10), confirming redshift and enabling age/distance calibration.

Key Takeaways

• Redshift z directly scales both wavelengths and temperatures: T(z) = 2.725 (1 + z) K.
• Distances and times require integration over the FLRW metric using E(z) and measured cosmological parameters.
• JWST observations robustly confirm cosmic expansion: compact, young galaxies dominate at high z.
• Spectroscopy remains the gold standard for measuring z—via H, He, and metal line identification.
• For z ≫ 1, cosmological expansion accounts for virtually 100% of the observed redshift.

— Editorial Team