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← Back to Part IV: Classic Solutions

Chapter 21: FLRW Cosmology

The Friedmann-Lemaître-Robertson-Walker metric describes a homogeneous, isotropic universe. It's the foundation of modern cosmology, describing the expanding universe from Big Bang to present acceleration.

FLRW Metric

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

a(t) = scale factor, k = curvature (-1, 0, +1)

k = +1

Closed (spherical) universe. Finite volume, re-collapses.

k = 0

Flat universe. Infinite, expands forever (our universe!).

k = -1

Open (hyperbolic) universe. Infinite, expands forever.

Friedmann Equations

\( H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \)

First Friedmann equation (Hubble parameter H)

\( \frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3} \)

Second Friedmann equation (acceleration)

Density Parameters

The Friedmann equation can be rewritten in terms of dimensionless density parameters:

\( \Omega_m + \Omega_\Lambda + \Omega_k = 1 \)

where \( \Omega_i = \rho_i / \rho_{crit} \)

Critical Density

\( \rho_{crit} = \frac{3H^2}{8\pi G} \approx 10^{-26} \text{ kg/m}^3 \)

Our Universe (Planck 2018)

\( \Omega_m \approx 0.31, \Omega_\Lambda \approx 0.69, \Omega_k \approx 0 \)

Interactive Simulation: Universe Evolution

Run this Python code to visualize how the scale factor evolves in different cosmological models. The simulation compares ΛCDM with other theoretical universes.

FLRW Universe Evolution

Python

Solve Friedmann equations and compare cosmological models

flrw_cosmology.py107 lines

#!/usr/bin/env python3
"""
flrw_cosmology.py - FLRW universe evolution
Simulates scale factor evolution for different cosmological models
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# Cosmological parameters (Planck 2018 values)
H0 = 70  # km/s/Mpc (Hubble constant)
Omega_m = 0.3    # Matter density parameter
Omega_Lambda = 0.7  # Dark energy density parameter
Omega_r = 9e-5   # Radiation density parameter

def H(a, Omega_m, Omega_Lambda, Omega_r):
    """
    Hubble parameter H(a)/H0 as function of scale factor
    H^2 = H0^2 * (Omega_r/a^4 + Omega_m/a^3 + Omega_k/a^2 + Omega_Lambda)
    """
    return np.sqrt(Omega_r/a**4 + Omega_m/a**3 + Omega_Lambda)

def da_dt(a, t, Omega_m, Omega_Lambda, Omega_r):
    """Scale factor evolution: da/dt = a * H(a)"""
    return a * H(a, Omega_m, Omega_Lambda, Omega_r)

# Solve from early universe to far future
a0 = 1e-4  # Initial scale factor (early universe)
t_span = np.linspace(0, 3, 1000)  # Time in units of 1/H0

# Solve ODE for ΛCDM model
sol = odeint(da_dt, a0, t_span, args=(Omega_m, Omega_Lambda, Omega_r))
a_vals = sol.flatten()

# Find where a = 1 (today)
today_idx = np.argmin(np.abs(a_vals - 1))
t_today = t_span[today_idx]

# Convert to Gyr (1/H0 ≈ 14 Gyr for H0=70)
t_Gyr = (t_span - t_today) * 9.78 / (H0/100)

# Create figure with two panels
fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Left panel: Scale factor evolution
ax1 = axes[0]
ax1.plot(t_Gyr, a_vals, 'b-', linewidth=2, label='ΛCDM Universe')
ax1.axhline(y=1, color='red', linestyle='--', alpha=0.7, label='Today (a=1)')
ax1.axvline(x=0, color='red', linestyle='--', alpha=0.3)
ax1.fill_between(t_Gyr, 0, a_vals, alpha=0.2)
ax1.set_xlabel('Time from today (Gyr)', fontsize=12)
ax1.set_ylabel('Scale factor a(t)', fontsize=12)
ax1.set_title('Scale Factor Evolution in ΛCDM', fontsize=14)
ax1.legend(loc='upper left')
ax1.grid(True, alpha=0.3)
ax1.set_xlim(-14, 50)
ax1.set_ylim(0, 5)

# Mark key epochs
ax1.annotate('Big Bang', xy=(-13.8, 0.1), fontsize=10, color='orange')
ax1.annotate('Now', xy=(1, 1.1), fontsize=10, color='red')

# Right panel: Different cosmological models comparison
ax2 = axes[1]
cosmologies = [
    (1.0, 0.0, 0, 'Einstein-de Sitter (Ωm=1)', 'green'),
    (0.3, 0.7, 0, 'ΛCDM (Ωm=0.3, ΩΛ=0.7)', 'blue'),
    (0.3, 0.0, 0, 'Open Universe (Ωm=0.3)', 'orange'),
    (0.05, 0.95, 0, 'Dark Energy Dominated', 'purple'),
]

for Om, OL, Or, label, color in cosmologies:
    sol = odeint(da_dt, a0, t_span, args=(Om, OL, Or))
    # Recalculate time offset for each model
    a_model = sol.flatten()
    today_idx_model = np.argmin(np.abs(a_model - 1))
    t_today_model = t_span[today_idx_model]
    t_Gyr_model = (t_span - t_today_model) * 9.78 / (H0/100)
    ax2.plot(t_Gyr_model, a_model, linewidth=2, label=label, color=color)

ax2.axhline(y=1, color='gray', linestyle='--', alpha=0.5)
ax2.axvline(x=0, color='gray', linestyle='--', alpha=0.3)
ax2.set_xlabel('Time from today (Gyr)', fontsize=12)
ax2.set_ylabel('Scale factor a(t)', fontsize=12)
ax2.set_title('Comparing Cosmological Models', fontsize=14)
ax2.legend(loc='upper left', fontsize=9)
ax2.grid(True, alpha=0.3)
ax2.set_xlim(-14, 50)
ax2.set_ylim(0, 5)

plt.tight_layout()
plt.savefig('flrw_evolution.png', dpi=150, bbox_inches='tight')

# Print cosmological parameters
print("=" * 50)
print("ΛCDM Cosmology Simulation Results")
print("=" * 50)
print(f"\nCosmological Parameters:")
print(f"  Hubble constant H₀ = {H0} km/s/Mpc")
print(f"  Matter density Ωm = {Omega_m}")
print(f"  Dark energy density ΩΛ = {Omega_Lambda}")
print(f"  Radiation density Ωr = {Omega_r}")
print(f"\nDerived Values:")
print(f"  Age of universe: {abs(t_Gyr[0]):.2f} Gyr")
print(f"  Current Hubble time: {9.78/(H0/100):.2f} Gyr")
print("\n✓ Plot saved as 'flrw_evolution.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key Physical Results

Age of the Universe

For ΛCDM: t₀ ≈ 13.8 billion years. The exact value depends on H₀ and the density parameters.

Hubble Time

1/H₀ ≈ 14 Gyr sets the natural timescale. The actual age differs due to the varying expansion rate.

Future Evolution

With Λ > 0, expansion accelerates forever. The scale factor grows exponentially: a(t) ∝ e^Ht.

Matter-Λ Equality

Dark energy began dominating at z ≈ 0.4, about 5 billion years ago, when Ω_m = Ω_Λ.

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