Part IV, Chapter 21 | Page 1 of 3

The FLRW Metric and Friedmann Equations

The foundation of modern cosmology: describing the large-scale evolution of a homogeneous, isotropic universe

The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is the exact solution to Einstein's field equations describing a homogeneous, isotropic universe. It is the mathematical backbone of modern cosmology, encoding the expansion of the universe from the Big Bang through the present accelerated expansion driven by dark energy.

The Cosmological Principle

The starting point of modern cosmology is the cosmological principle: on sufficiently large scales (greater than ~100 Mpc), the universe is spatially homogeneous and isotropic.

Homogeneity

The universe looks the same at every point. There is no preferred location -- no center of the universe. Mathematically, this means the spatial geometry admits a transitive group of isometries: for any two points, there exists a symmetry transformation mapping one to the other.

Isotropy

The universe looks the same in every direction from any given point. The cosmic microwave background confirms this to about 1 part in 100,000. Mathematically, at each point the spatial geometry is invariant under rotations: the rotation group SO(3) acts as an isometry group.

Together, homogeneity and isotropy severely constrain the possible spacetime geometries. The most general metric compatible with these symmetries is the FLRW metric, determined up to a single free function a(t) and a discrete parameter k.

The FLRW Metric

The metric for a homogeneous, isotropic universe can be written in the form:

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

The FLRW line element in comoving coordinates with cosmic time t

The key elements of this metric are:

Scale factor a(t): A dimensionless function of cosmic time that encodes the expansion history. By convention, $a(t_0) = 1$ today.
Curvature parameter k: Takes values $k = -1, 0, +1$, determining the spatial geometry.
Comoving coordinates: $(r, \theta, \phi)$ are constant for freely falling observers carried along with the expansion (the "comoving observers").
Cosmic time t: The proper time measured by comoving observers. All comoving observers agree on the time since the Big Bang.

Spatial Curvature: Three Geometries

k = +1: Closed

$$d\Sigma^2 = d\chi^2 + \sin^2\chi\,d\Omega^2$$

Spherical spatial geometry (3-sphere). Finite volume, no boundary. Parallel lines converge. The sum of triangle angles exceeds $\pi$. If dominated by matter, this universe eventually re-collapses.

k = 0: Flat

$$d\Sigma^2 = d\chi^2 + \chi^2\,d\Omega^2$$

Euclidean spatial geometry. Infinite volume. Parallel lines remain parallel. Standard angle sum in triangles. Observations indicate our universe is flat to within $|\Omega_k| < 0.002$.

k = -1: Open

$$d\Sigma^2 = d\chi^2 + \sinh^2\chi\,d\Omega^2$$

Hyperbolic spatial geometry. Infinite volume. Parallel lines diverge. Triangle angle sum is less than $\pi$. Expands forever even without dark energy.

The coordinate $\chi$ used above is related to r by the substitution$r = S_k(\chi)$ where $S_k(\chi) = \sin\chi, \chi, \sinh\chi$ for$k = +1, 0, -1$ respectively.

Deriving the Friedmann Equations

The Friedmann equations follow from inserting the FLRW metric into Einstein's field equations$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}/c^4$. For the FLRW metric, the non-vanishing Christoffel symbols are:

$$\Gamma^0{}_{ij} = \frac{a\dot{a}}{c^2}\,\gamma_{ij}, \quad \Gamma^i{}_{0j} = \frac{\dot{a}}{ca}\,\delta^i{}_j$$

$$\Gamma^r{}_{rr} = \frac{kr}{1-kr^2}, \quad \Gamma^r{}_{\theta\theta} = -r(1-kr^2), \quad \Gamma^\theta{}_{r\theta} = \frac{1}{r}, \quad \text{etc.}$$

The Ricci tensor components for the FLRW metric are:

$$R_{00} = -\frac{3\ddot{a}}{ac^2}$$

$$R_{ij} = \frac{1}{c^2}\left(a\ddot{a} + 2\dot{a}^2 + 2kc^2\right)\gamma_{ij}$$

where $\gamma_{ij}$ is the spatial metric and dots denote $d/dt$

The stress-energy tensor for a perfect fluid (the appropriate matter model for cosmology) is:

$$T_{\mu\nu} = \left(\rho + \frac{p}{c^2}\right)u_\mu u_\nu + p\,g_{\mu\nu}$$

where $\rho$ is the energy density, p is the pressure, and $u^\mu$ is the 4-velocity of the comoving fluid

Inserting these into Einstein's equations, the (0,0) component gives the first Friedmann equation:

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

The expansion rate is determined by the energy density, curvature, and cosmological constant

Here $H = \dot{a}/a$ is the Hubble parameter, which measures the instantaneous rate of expansion. Its present value is the Hubble constant$H_0 \approx 67.4$ km/s/Mpc (Planck 2018).

The spatial (i, j) components yield the second Friedmann equation (also called the acceleration or Raychaudhuri equation):

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

Ordinary matter ($\rho + 3p/c^2 > 0$) decelerates the expansion; a cosmological constant accelerates it

The second Friedmann equation reveals a remarkable feature: the expansion decelerates when the strong energy condition $\rho + 3p/c^2 > 0$ is satisfied (which is true for all known forms of ordinary matter and radiation). Acceleration requires either a cosmological constant $\Lambda > 0$ or exotic matter with sufficiently negative pressure.

The Continuity Equation

The two Friedmann equations are not independent: they are related through the conservation of stress-energy, $\nabla_\mu T^{\mu\nu} = 0$. For the FLRW metric, this gives the continuity equation (also called the fluid equation):

$$\dot{\rho} + 3H\left(\rho + \frac{p}{c^2}\right) = 0$$

Energy conservation in an expanding universe: the $3H$ term accounts for the dilution due to expansion

This equation has a simple physical interpretation. Consider a comoving volume element$V \propto a^3$. The total energy in this volume is $E = \rho c^2 V$. The rate of change of energy equals minus the work done by pressure:

$$\frac{d(\rho c^2 a^3)}{dt} = -p\frac{d(a^3)}{dt}$$

First law of thermodynamics for an expanding universe (with dS = 0 for adiabatic expansion)

Any two of the three equations (first Friedmann, second Friedmann, continuity) can be derived from the other one. In practice, we typically work with the first Friedmann equation and the continuity equation, as they form a closed system once an equation of state $p = p(\rho)$ is specified.

Interactive Simulation: Universe Evolution

Run this Python code to visualize how the scale factor evolves in different cosmological models. The simulation compares the standard ΛCDM model with other theoretical universes, solving the Friedmann equations numerically.

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

Physical Significance of the Friedmann Equations

The Friedmann equations are the master equations of cosmology. They connect the geometry of the universe (the scale factor and curvature) to its matter content (energy density and pressure). Combined with an equation of state for the cosmic fluid, they completely determine the expansion history a(t). All of observational cosmology -- from the cosmic microwave background to the accelerating expansion -- is encoded in these deceptively simple equations.

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General Relativity