Part I: The Expanding Universe | Chapter 3

Cosmological Redshift

How the expansion of the universe stretches photon wavelengths, its rigorous derivation from the FLRW metric, and the observational consequences that reveal cosmic history

1. Introduction: Redshift and the Expanding Universe

One of the most profound observational facts in all of physics is that the light from distant galaxies arrives at our telescopes with longer wavelengths than when it was emitted. This phenomenon — cosmological redshift — is not a Doppler effect due to galaxies moving through space, but rather a consequence of space itself expanding while the photon is in transit.

The redshift $z$ of a distant source is defined by the fractional change in wavelength:

\[ z \;\equiv\; \frac{\lambda_{\rm obs} - \lambda_{\rm emit}}{\lambda_{\rm emit}} \;=\; \frac{\lambda_{\rm obs}}{\lambda_{\rm emit}} - 1 \]

A redshift of $z = 1$ means the observed wavelength is twice the emitted wavelength;$z = 1100$ corresponds to the cosmic microwave background (CMB), whose photons have been stretched by a factor of $\sim 1100$ since the epoch of recombination. The most distant galaxies observed by JWST have redshifts approaching $z \sim 14$, meaning we see them as they were when the universe was only $\sim 300$ million years old.

Three Types of Redshift

Doppler Redshift

Caused by relative motion of source and observer through space. For $v \ll c$:$z \approx v/c$. Governed by special relativity.

Gravitational Redshift

Photons lose energy climbing out of gravitational wells. For a Schwarzschild source:$1 + z = (1 - 2GM/rc^2)^{-1/2}$.

Cosmological Redshift

Wavelength stretches with the scale factor $a(t)$ of the expanding universe:$1 + z = a_0/a(t_{\rm emit})$. This is our focus.

It is crucial to understand that cosmological redshift is neither a Doppler effect nor a gravitational effect — it is a fundamentally distinct phenomenon arising from the time-dependence of the metric. A photon propagating through an expanding FLRW spacetime has its wavelength stretched in direct proportion to the scale factor, regardless of the local motion or gravitational field of the source. We now derive this rigorously.

2. Derivation 1: Cosmological Redshift from the FLRW Metric

The FLRW line element for a spatially homogeneous and isotropic universe is:

\[ ds^2 = -c^2\,dt^2 + a^2(t)\left[\frac{dr^2}{1 - kr^2} + r^2\,d\Omega^2\right] \]

where $a(t)$ is the scale factor, $k$ is the spatial curvature parameter ($k = -1, 0, +1$), and $d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2$. Photons travel on null geodesics with $ds^2 = 0$.

Step 1: Radial Null Geodesics

Consider a photon emitted radially from a comoving source at coordinate $r = r_1$ at time $t_{\rm emit}$ and received by a comoving observer at $r = 0$ at time $t_{\rm obs}$. For a radial null geodesic ($d\theta = d\phi = 0$, $ds^2 = 0$):

\[ 0 = -c^2\,dt^2 + a^2(t)\,\frac{dr^2}{1 - kr^2} \]

\[ \Rightarrow \quad \frac{c\,dt}{a(t)} = -\frac{dr}{\sqrt{1 - kr^2}} \]

The minus sign arises because the photon travels toward decreasing $r$ (from $r_1$ to 0). Integrating both sides:

\[ \int_{t_{\rm emit}}^{t_{\rm obs}} \frac{c\,dt}{a(t)} = \int_0^{r_1} \frac{dr}{\sqrt{1 - kr^2}} \;\equiv\; f_k(r_1) \]

The right-hand side is a function only of the comoving coordinate $r_1$, which is time-independent for a comoving source.

Step 2: Comparing Successive Wave Crests

Now consider the next wave crest, emitted at $t_{\rm emit} + \delta t_{\rm emit}$ and received at $t_{\rm obs} + \delta t_{\rm obs}$. Since $r_1$ is unchanged:

\[ \int_{t_{\rm emit} + \delta t_{\rm emit}}^{t_{\rm obs} + \delta t_{\rm obs}} \frac{c\,dt}{a(t)} = f_k(r_1) = \int_{t_{\rm emit}}^{t_{\rm obs}} \frac{c\,dt}{a(t)} \]

Subtracting the first integral from the second:

\[ \int_{t_{\rm obs}}^{t_{\rm obs} + \delta t_{\rm obs}} \frac{c\,dt}{a(t)} = \int_{t_{\rm emit}}^{t_{\rm emit} + \delta t_{\rm emit}} \frac{c\,dt}{a(t)} \]

Since $\delta t_{\rm emit}$ and $\delta t_{\rm obs}$ are infinitesimally small (one oscillation period), $a(t)$ is effectively constant over each interval:

\[ \frac{\delta t_{\rm obs}}{a(t_{\rm obs})} = \frac{\delta t_{\rm emit}}{a(t_{\rm emit})} \]

Step 3: The Redshift-Scale Factor Relation

Since the period of the wave is related to its wavelength by $\delta t = \lambda/c$:

\[ \frac{\lambda_{\rm obs}}{a(t_{\rm obs})} = \frac{\lambda_{\rm emit}}{a(t_{\rm emit})} \quad\Rightarrow\quad \frac{\lambda_{\rm obs}}{\lambda_{\rm emit}} = \frac{a(t_{\rm obs})}{a(t_{\rm emit})} \]

Setting $a(t_{\rm obs}) = a_0 = 1$ (the convention that the present-day scale factor is unity):

The Cosmological Redshift Formula

\[ \boxed{1 + z = \frac{a_0}{a(t_{\rm emit})} = \frac{1}{a(t_{\rm emit})}} \]

The redshift directly encodes the ratio of the scale factor at observation to that at emission. A photon emitted when the universe was half its present size arrives with $z = 1$.

This is an exact result — it holds for all redshifts, all curvatures, and all matter content. It depends only on the FLRW symmetry and the null condition $ds^2 = 0$. Notice that:

The derivation is purely geometric — no reference to energy, momentum, or frequency was needed until the final identification of$\delta t$ with $\lambda/c$.
The result is independent of the curvature $k$— the function $f_k(r_1)$ cancelled completely.
Cosmological redshift differs fundamentally from Doppler shift: the Doppler formula requires a relative velocity in a common inertial frame, but in an expanding universe no such global frame exists. The source and observer are both at rest in comoving coordinates.

3. Derivation 2: Hubble's Law as the Low-z Limit

For nearby sources ($z \ll 1$), cosmological redshift reduces to the familiar Hubble's law. To see this, we Taylor-expand the scale factor around the present time $t_0$:

\[ a(t) = a(t_0) + \dot{a}(t_0)(t - t_0) + \frac{1}{2}\ddot{a}(t_0)(t - t_0)^2 + \cdots \]

With $a(t_0) = 1$ and defining the Hubble parameter $H_0 \equiv \dot{a}(t_0)/a(t_0) = \dot{a}(t_0)$:

\[ a(t_{\rm emit}) \approx 1 - H_0(t_0 - t_{\rm emit}) + \cdots \]

Therefore:

\[ 1 + z = \frac{1}{a(t_{\rm emit})} \approx \frac{1}{1 - H_0(t_0 - t_{\rm emit})} \approx 1 + H_0(t_0 - t_{\rm emit}) \]

For small lookback times, the proper distance to the source is$d \approx c(t_0 - t_{\rm emit})$, so:

Hubble's Law (Low-z Limit)

\[ \boxed{z \approx \frac{H_0 d}{c} \quad\Leftrightarrow\quad v \approx H_0 d} \]

Valid only for $z \ll 1$. At higher redshifts, the relationship between distance and redshift becomes nonlinear and depends on the cosmological model.

Recession Velocity: Can It Exceed c?

The recession velocity of a comoving object at proper distance $d_{\rm prop}$ is defined as:

\[ v_{\rm rec} = \dot{a}(t)\,r = H(t)\,d_{\rm prop}(t) \]

where $r$ is the comoving coordinate and $d_{\rm prop} = a(t)\,r$ is the proper distance. This is not a velocity in the special-relativistic sense — it is the rate of change of proper distance between comoving observers. There is no prohibition against$v_{\rm rec} > c$ because this does not represent motion through space but rather the expansion of space itself.

Numerical Example

Today, $H_0 \approx 67.4\;\text{km/s/Mpc}$. The Hubble radius $d_H = c/H_0$ is the distance at which the recession velocity equals $c$:

\[ d_H = \frac{c}{H_0} = \frac{3 \times 10^5\;\text{km/s}}{67.4\;\text{km/s/Mpc}} \approx 4{,}451\;\text{Mpc} \approx 14.5\;\text{Gly} \]

Galaxies beyond the Hubble radius are receding superluminally. The CMB ($z = 1100$) was emitted from matter that is now receding at$v_{\rm rec} \approx 3.2c$. Yet we still receive its photons, because the photons were emitted when the Hubble radius was much smaller and have been “surfing” the expansion ever since.

Peculiar Velocity vs. Recession Velocity

The total observed redshift of a galaxy combines two contributions:

\[ (1 + z_{\rm obs}) = (1 + z_{\rm cosmo})(1 + z_{\rm pec}) \]

where $z_{\rm cosmo}$ is the cosmological redshift due to expansion and$z_{\rm pec} \approx v_{\rm pec}/c$ is the Doppler shift from the galaxy's peculiar velocity $v_{\rm pec}$— its motion relative to the local comoving frame.

Peculiar velocities are typically$\sim 100{-}600\;\text{km/s}$ for galaxies in clusters, driven by local gravitational interactions. The Milky Way has $v_{\rm pec} \approx 620\;\text{km/s}$toward the Great Attractor.
Recession velocities grow with distance and are unbounded. For a galaxy at $d = 100\;\text{Mpc}$,$v_{\rm rec} \approx 6{,}740\;\text{km/s}$, far exceeding any peculiar velocity.
At low redshift ($z \lesssim 0.01$), peculiar velocities can significantly contaminate the Hubble flow, making precise $H_0$ measurements challenging.

4. Derivation 3: Redshift–Time Relation

The redshift serves as a convenient “clock” for the universe. We now derive the exact relationship between cosmic time and redshift, yielding expressions for the lookback time and the redshift-temperature relation.

Differential Time–Redshift Relation

From $1 + z = 1/a$, we differentiate:

\[ dz = -\frac{da}{a^2} = -\frac{\dot{a}\,dt}{a^2} = -(1+z)\,H(z)\,dt \]

where we used $H = \dot{a}/a$ and $a = 1/(1+z)$. Rearranging:

Time–Redshift Differential

\[ \boxed{dt = -\frac{dz}{(1+z)\,H(z)}} \]

The minus sign indicates that as we look to higher redshift, we look further back in time (decreasing $t$).

Lookback Time

The lookback time $t_L(z)$ is the time elapsed between when the light was emitted (at redshift $z$) and now ($z = 0$):

\[ t_L(z) = \int_0^z \frac{dz'}{(1+z')\,H(z')} \]

For the flat $\Lambda$CDM model, $H(z) = H_0\,E(z)$ where:

\[ E(z) = \sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda} \]

The age of the universe is obtained by taking $z \to \infty$:

\[ t_0 = \int_0^\infty \frac{dz}{(1+z)\,H_0\,E(z)} = \frac{1}{H_0}\int_0^\infty \frac{dz}{(1+z)\,E(z)} \]

For the Planck 2018 parameters ($\Omega_m = 0.315$, $\Omega_\Lambda = 0.685$,$H_0 = 67.4\;\text{km/s/Mpc}$), numerical integration gives$t_0 \approx 13.8\;\text{Gyr}$.

Cosmic Temperature–Redshift Relation

The CMB is a near-perfect blackbody. As the universe expands, the photon distribution maintains its Planckian form but with a temperature that scales inversely with the scale factor. Since$a = 1/(1+z)$:

Temperature–Redshift Relation

\[ \boxed{T(z) = T_0\,(1 + z)} \]

where $T_0 = 2.7255\;\text{K}$ is the present-day CMB temperature.

Proof: Photon energy density $u \propto T^4$(Stefan–Boltzmann) scales as $a^{-4}$ under adiabatic expansion (redshift $\times$ dilution), so $T \propto a^{-1} = 1 + z$.

Key Temperatures in Cosmic History

Epoch	Redshift $z$	Temperature	Cosmic Age
Today	0	$2.73\;\text{K}$	$13.8\;\text{Gyr}$
Recombination	$\sim 1100$	$\sim 3000\;\text{K}$	$\sim 380{,}000\;\text{yr}$
Matter–radiation equality	$\sim 3400$	$\sim 9300\;\text{K}$	$\sim 50{,}000\;\text{yr}$
BBN	$\sim 4 \times 10^8$	$\sim 10^9\;\text{K}$	$\sim 3\;\text{min}$
Electroweak transition	$\sim 10^{15}$	$\sim 10^{15}\;\text{K}$	$\sim 10^{-12}\;\text{s}$

5. Derivation 4: Tolman Surface Brightness Test

The Tolman surface brightness test is a classical cosmological test that distinguishes a genuinely expanding universe from alternative “tired light” hypotheses. Surface brightness $\text{SB}$ (flux per unit solid angle) scales differently depending on the cause of the redshift.

Derivation of the (1+z)^4 Dimming

Consider an extended source at redshift $z$ with intrinsic (rest-frame) surface brightness $\text{SB}_{\rm int}$. The observed surface brightness is affected by four factors:

Photon energy loss: Each photon's energy is reduced by a factor $(1+z)^{-1}$ due to the wavelength stretching$E = hc/\lambda$.
Time dilation: Photons arrive at a rate$(1+z)^{-1}$ times the emission rate, because the interval between successive photons is stretched by the same factor as the wavelength.
Angular size: The source subtends an angle$\Delta\theta = \ell/d_A$ where $d_A = d_L/(1+z)^2$ is the angular diameter distance. Thus the solid angle scales as $\Delta\Omega \propto d_A^{-2} \propto (1+z)^4/d_L^2$.

The bolometric flux is $F = L/(4\pi d_L^2)$ where $d_L = (1+z)d_M$ is the luminosity distance. The solid angle subtended by area $A$ is$\Delta\Omega = A/d_A^2 = A(1+z)^2/d_M^2$. The observed surface brightness is:

\[ \text{SB}_{\rm obs} = \frac{F}{\Delta\Omega} = \frac{L}{4\pi A} \cdot \frac{d_M^2}{d_L^2(1+z)^2} = \frac{\text{SB}_{\rm int}}{(1+z)^4} \]

using $d_L = (1+z)d_M \Rightarrow d_M^2/d_L^2 = (1+z)^{-2}$

Tolman Surface Brightness Relation

\[ \boxed{\text{SB}_{\rm obs} = \frac{\text{SB}_{\rm int}}{(1+z)^4}} \]

Surface brightness dims as the fourth power of $(1+z)$ in an expanding universe.

Distinguishing Expansion from “Tired Light”

In a “tired light” model, photons lose energy through some unknown mechanism as they travel, but space does not expand. In such a model:

Expanding Universe

Photon energy: $(1+z)^{-1}$ factor
Time dilation: $(1+z)^{-1}$ factor
Net SB dimming: $(1+z)^{-4}$
Time dilation of supernova light curves: observed

Tired Light (Static Universe)

Photon energy: $(1+z)^{-1}$ factor
Time dilation: none
Net SB dimming: $(1+z)^{-1}$ only
Time dilation of supernova light curves: not predicted

Observations strongly favor the $(1+z)^4$ dimming. The time dilation of Type Ia supernova light curves (first confirmed by Leibundgut et al. 1996 and Goldhaber et al. 2001) provides direct evidence: supernova durations stretch by exactly $(1+z)$, as predicted by expansion but not by tired light. Additionally, the Tolman test applied to galaxy surface brightness (Lubin & Sandage 2001) is consistent with the $(1+z)^4$ prediction after accounting for galaxy evolution.

6. Applications of Cosmological Redshift

Spectroscopic Redshift

The gold standard: identify known emission/absorption lines (Lyman-$\alpha$ at$1216\;\text{\AA}$, [O III] at $4959/5007\;\text{\AA}$, Ca II H&K) and measure $z = (\lambda_{\rm obs} - \lambda_0)/\lambda_0$. Precision: $\sigma_z \sim 0.0001$. Major surveys: SDSS ($>4$M spectra), DESI (30M+ spectra), Euclid.

Photometric Redshift

For faint/numerous galaxies: estimate $z$ from broad-band colors using the 4000 Å break or Lyman break as they shift through filters. Methods: template fitting (LePhare, EAZY) or machine learning. Accuracy: $\sigma_z/(1+z) \sim 0.02{-}0.05$, sufficient for statistical studies but not precise line-of-sight distances.

Redshift-Space Distortions & Large-Scale Structure

Peculiar velocities cause redshift-space distortions (RSD): on large scales, coherent infall toward overdensities produces the Kaiser “squashing” effect, while on small scales, random virial motions create “Fingers of God” elongations. RSD constrain the growth rate $f\sigma_8$, testing GR on cosmological scales. Redshift surveys (SDSS, DESI) map the cosmic web and measure the BAO scale ($\sim 150\;\text{Mpc}$ standard ruler), the Lyman-$\alpha$ forest (matter distribution at $z \sim 2{-}5$), and void statistics sensitive to dark energy and modified gravity.

7. Historical Context

The Discovery of Cosmic Expansion

Vesto Slipher (1912–1917): At Lowell Observatory, Slipher measured radial velocities of 25 “spiral nebulae” via Doppler shifts, finding most receding at up to $\sim 1{,}100\;\text{km/s}$. His data provided the velocities Hubble later used.

Georges Lemaître (1927): Independently derived the expansion law from GR and estimated $H_0 \approx 575\;\text{km/s/Mpc}$ using Slipher's velocities and Hubble's distances — the first to connect expansion theory with observation.

Edwin Hubble (1929): Using Cepheid distances to 24 galaxies, established $v = H_0 d$ with $H_0 \approx 500\;\text{km/s/Mpc}$(too large by $\sim 7\times$ due to Cepheid calibration errors resolved by Baade in 1952).

Milton Humason (1930s): Extended the Hubble diagram to$v \sim 40{,}000\;\text{km/s}$, confirming linearity well beyond Hubble's original sample.

The Modern Era: The “Hubble wars” (Sandage vs. de Vaucouleurs) were partially resolved by the HST Key Project ($H_0 = 72 \pm 8$). Today, Planck gives $67.4 \pm 0.5$ while SH0ES gives $73.0 \pm 1.0$— the $5\sigma$ “Hubble tension” may signal new physics.

8. Python Simulation: Hubble Diagram, Lookback Time & Cosmic Temperature

The following Python code computes and plots three key relationships involving cosmological redshift: (1) a Hubble diagram showing the deviation from the linear Hubble law at high $z$, (2) the lookback time as a function of redshift, and (3) the cosmic temperature evolution. We use numpy only (no scipy) and perform numerical integration via the trapezoidal rule.

Cosmological Redshift: Hubble Diagram, Lookback Time & Temperature

Python

Computes and plots the Hubble diagram (velocity vs distance), lookback time, and cosmic temperature as functions of redshift for LCDM cosmology using numpy only

cosmological_redshift.py201 lines

#!/usr/bin/env python3
"""
cosmological_redshift.py
Hubble diagram, lookback time, and cosmic temperature vs redshift.
Uses numpy only (no scipy). Numerical integration via trapezoidal rule.
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================
# Cosmological Parameters (Planck 2018)
# ============================
H0_km = 67.4            # Hubble constant [km/s/Mpc]
H0 = H0_km * 3.2408e-20 # Convert to [1/s]
c_km = 2.998e5           # Speed of light [km/s]
c_m = 2.998e8            # Speed of light [m/s]
Mpc = 3.0857e22          # 1 Mpc in meters
Gyr = 3.1557e16          # 1 Gyr in seconds
T0 = 2.7255              # CMB temperature today [K]

Omega_r = 9.1e-5         # Radiation density parameter
Omega_m = 0.315          # Matter density parameter
Omega_L = 0.685          # Dark energy (cosmological constant)
Omega_k = 1.0 - Omega_r - Omega_m - Omega_L  # Curvature (~ 0)

# ============================
# Dimensionless Hubble parameter E(z)
# ============================
def E(z):
    """E(z) = H(z)/H0"""
    return np.sqrt(Omega_r * (1+z)**4
                 + Omega_m * (1+z)**3
                 + Omega_k * (1+z)**2
                 + Omega_L)

# ============================
# Trapezoidal integration (no scipy)
# ============================
def trapz_integrate(f, a, b, n=2000):
    """Integrate f from a to b using trapezoidal rule with n steps."""
    x = np.linspace(a, b, n)
    y = f(x)
    return np.trapezoid(y, x)

# ============================
# 1. Hubble Diagram: velocity vs distance
# ============================
z_arr = np.linspace(0.001, 5.0, 500)

# Comoving distance: d_C(z) = (c/H0) * integral_0^z dz'/E(z')
def comoving_distance(z_val):
    """Compute comoving distance in Mpc for a single z value."""
    integrand = lambda zp: 1.0 / E(zp)
    result = trapz_integrate(integrand, 0, z_val, n=2000)
    return (c_m / (H0 * Mpc)) * result  # in Mpc

d_C = np.array([comoving_distance(z) for z in z_arr])

# Proper distance today = comoving distance (for flat universe)
d_proper = d_C  # in Mpc

# Recession velocity: v_rec = H0 * d_proper
v_rec = H0_km * d_proper  # in km/s

# Linear Hubble law: v = c*z (valid for z << 1)
v_linear = c_km * z_arr

# Exact special-relativistic Doppler (for comparison):
# v_SR = c * [(1+z)^2 - 1] / [(1+z)^2 + 1]
v_SR = c_km * ((1+z_arr)**2 - 1) / ((1+z_arr)**2 + 1)

# ============================
# 2. Lookback time
# ============================
def lookback_time(z_val):
    """Compute lookback time in Gyr for a single z value."""
    integrand = lambda zp: 1.0 / ((1+zp) * E(zp))
    result = trapz_integrate(integrand, 0, z_val, n=2000)
    return result / (H0 * Gyr)  # in Gyr

t_lookback = np.array([lookback_time(z) for z in z_arr])

# Age of universe (integrate to large z ~ 10000 as proxy for infinity)
age_integrand = lambda zp: 1.0 / ((1+zp) * E(zp))
t_age = trapz_integrate(age_integrand, 0, 10000, n=5000) / (H0 * Gyr)

# Cosmic age at redshift z
t_cosmic = t_age - t_lookback

# ============================
# 3. Cosmic temperature
# ============================
T_z = T0 * (1 + z_arr)

# ============================
# Plotting
# ============================
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.suptitle('Cosmological Redshift: Key Relations ($\\Omega_m=0.315$, $\\Omega_\\Lambda=0.685$, $H_0=67.4$)',
             fontsize=14, fontweight='bold', color='white', y=0.98)
fig.patch.set_facecolor('#0a0a1a')

for ax in axes.flat:
    ax.set_facecolor('#0d0d2b')
    ax.tick_params(colors='white', which='both')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#444466')

# --- Panel 1: Hubble Diagram ---
ax1 = axes[0, 0]
ax1.plot(d_proper / 1000, v_rec / 1000, color='#a855f7', linewidth=2.0, label='$v_{rec} = H_0 d_{proper}$')
ax1.plot(d_proper / 1000, v_linear / 1000, color='#f472b6', linewidth=1.5, linestyle='--', label='Linear: $v = cz$')
ax1.plot(d_proper / 1000, v_SR / 1000, color='#38bdf8', linewidth=1.5, linestyle=':', label='SR Doppler')
ax1.axhline(y=c_km/1000, color='#fbbf24', linewidth=1.0, linestyle='-.', alpha=0.7, label='$v = c$')
ax1.set_xlabel('Proper Distance [Gpc]', fontsize=11)
ax1.set_ylabel('Velocity [$10^3$ km/s]', fontsize=11)
ax1.set_title('Hubble Diagram: Velocity vs Distance', fontsize=12)
ax1.legend(fontsize=9, facecolor='#1a1a3a', edgecolor='#444466', labelcolor='white')
ax1.set_xlim(0, max(d_proper/1000))
ax1.set_ylim(0, max(v_rec/1000)*1.05)
ax1.grid(True, alpha=0.15, color='#666688')

# --- Panel 2: Redshift vs Distance (Hubble Diagram) ---
ax2 = axes[0, 1]
d_linear_hubble = c_km * z_arr / H0_km  # d = cz/H0 (linear approx)
ax2.plot(z_arr, d_proper, color='#a855f7', linewidth=2.0, label='Exact $d_{proper}(z)$')
ax2.plot(z_arr, d_linear_hubble, color='#f472b6', linewidth=1.5, linestyle='--', label='Linear: $d = cz/H_0$')
ax2.set_xlabel('Redshift $z$', fontsize=11)
ax2.set_ylabel('Proper Distance [Mpc]', fontsize=11)
ax2.set_title('Distance vs Redshift', fontsize=12)
ax2.legend(fontsize=9, facecolor='#1a1a3a', edgecolor='#444466', labelcolor='white')
ax2.grid(True, alpha=0.15, color='#666688')

# --- Panel 3: Lookback Time ---
ax3 = axes[1, 0]
ax3.plot(z_arr, t_lookback, color='#a855f7', linewidth=2.0, label='Lookback time $t_L(z)$')
ax3.plot(z_arr, t_cosmic, color='#38bdf8', linewidth=1.5, linestyle='--', label='Cosmic age $t(z)$')
ax3.axhline(y=t_age, color='#fbbf24', linewidth=1.0, linestyle='-.', alpha=0.7,
            label=f'Age of universe: {t_age:.2f} Gyr')
ax3.set_xlabel('Redshift $z$', fontsize=11)
ax3.set_ylabel('Time [Gyr]', fontsize=11)
ax3.set_title('Lookback Time & Cosmic Age vs Redshift', fontsize=12)
ax3.legend(fontsize=9, facecolor='#1a1a3a', edgecolor='#444466', labelcolor='white')
ax3.grid(True, alpha=0.15, color='#666688')

# --- Panel 4: Cosmic Temperature ---
ax4 = axes[1, 1]
z_temp = np.linspace(0, 20, 500)
T_temp = T0 * (1 + z_temp)
ax4.semilogy(z_temp, T_temp, color='#f97316', linewidth=2.0, label='$T(z) = T_0(1+z)$')
ax4.axhline(y=T0, color='#fbbf24', linewidth=1.0, linestyle='-.', alpha=0.7,
            label=f'$T_0 = {T0}$ K')
ax4.axhline(y=3000, color='#ef4444', linewidth=1.0, linestyle=':', alpha=0.7,
            label='Recombination ($\\sim 3000$ K)')
ax4.set_xlabel('Redshift $z$', fontsize=11)
ax4.set_ylabel('CMB Temperature [K]', fontsize=11)
ax4.set_title('Cosmic Temperature vs Redshift', fontsize=12)
ax4.legend(fontsize=9, facecolor='#1a1a3a', edgecolor='#444466', labelcolor='white')
ax4.grid(True, alpha=0.15, color='#666688')

plt.tight_layout(rect=[0, 0, 1, 0.96])
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
plt.close()

# ============================
# Print numerical results
# ============================
print("=" * 65)
print("  COSMOLOGICAL REDSHIFT — NUMERICAL RESULTS")
print("=" * 65)
print(f"\n  Cosmological Parameters (Planck 2018):")
print(f"    H0       = {H0_km} km/s/Mpc")
print(f"    Omega_m  = {Omega_m}")
print(f"    Omega_L  = {Omega_L}")
print(f"    Omega_r  = {Omega_r}")
print(f"    T0 (CMB) = {T0} K")
print(f"\n  Derived quantities:")
print(f"    Hubble radius d_H = c/H0 = {c_km/H0_km:.1f} Mpc = {c_km/H0_km/1000:.2f} Gpc")
print(f"    Hubble time 1/H0 = {1/(H0*Gyr):.2f} Gyr")
print(f"    Age of universe   = {t_age:.2f} Gyr")

print(f"\n  {'z':>6} {'d_proper [Mpc]':>16} {'v_rec [km/s]':>14} {'v_rec/c':>10} {'t_lookback [Gyr]':>18} {'T(z) [K]':>10}")
print("  " + "-" * 78)
for z_sample in [0.01, 0.1, 0.5, 1.0, 2.0, 3.0, 5.0]:
    idx = np.argmin(np.abs(z_arr - z_sample))
    print(f"  {z_arr[idx]:6.2f} {d_proper[idx]:16.1f} {v_rec[idx]:14.0f} {v_rec[idx]/c_km:10.3f} {t_lookback[idx]:18.2f} {T0*(1+z_arr[idx]):10.1f}")

print(f"\n  Key redshifts:")
print(f"    z=1100 (CMB):  T = {T0*1101:.0f} K,  a = {1/1101:.6f}")
print(f"    z=10 (early galaxies):  T = {T0*11:.1f} K")
print(f"    z=1 (half the age):  T = {T0*2:.2f} K")
print(f"\n  Superluminality: Objects beyond d_H = {c_km/H0_km:.0f} Mpc")
print(f"  recede faster than light. This is NOT a violation of SR")
print(f"  because recession velocity is not a relative velocity")
print(f"  in a local inertial frame.")
print("\n" + "=" * 65)

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Code will be executed with Python 3 on the server

References

Slipher, V.M. (1917). Proc. Amer. Phil. Soc. 56, 403. — First systematic radial velocity measurements of nebulae.
Lemaître, G. (1927). Ann. Soc. Sci. Bruxelles A47, 49. — First derivation of the expansion law from GR.
Hubble, E. (1929). PNAS 15, 168. — Observational discovery of the velocity–distance relation.
Tolman, R.C. (1930). PNAS 16, 511. — Original surface brightness test derivation.
Humason, Mayall & Sandage (1956). AJ 61, 97. — Extension of the Hubble diagram.
Goldhaber, G. et al. (2001). ApJ 558, 359. — Confirmation of cosmological time dilation in SNe Ia.
Lubin & Sandage (2001). AJ 121, 2289. — Tolman test verification of $(1+z)^4$ dimming.
Planck Collaboration (2020). A&A 641, A6. — Definitive cosmological parameters.
Weinberg, S. (2008). Cosmology. Oxford. — Graduate-level treatment.
Ryden, B. (2017). Introduction to Cosmology, 2nd ed. Cambridge. — Pedagogical derivation of redshift relations.

← Friedmann Equations Distance Measures →

Epoch	Redshift \(z\)	Temperature	Cosmic Age
Today	0	\(2.73\;\text{K}\)	\(13.8\;\text{Gyr}\)
Recombination	\(\sim 1100\)	\(\sim 3000\;\text{K}\)	\(\sim 380{,}000\;\text{yr}\)
Matter–radiation equality	\(\sim 3400\)	\(\sim 9300\;\text{K}\)	\(\sim 50{,}000\;\text{yr}\)
BBN	\(\sim 4 \times 10^8\)	\(\sim 10^9\;\text{K}\)	\(\sim 3\;\text{min}\)
Electroweak transition	\(\sim 10^{15}\)	\(\sim 10^{15}\;\text{K}\)	\(\sim 10^{-12}\;\text{s}\)