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The Friedmann Equations

Manual: General · Subject: Cosmology

Derive and interpret the equations governing cosmic expansion from general relativity.

Dynamics from General Relativity

Geometry plus matter

In homogeneous and isotropic cosmology, the spacetime metric is the Friedmann-Lemaître-Robertson-Walker metric. Plugging this geometry into Einstein's equations yields the Friedmann equations, which connect the expansion rate to the contents of the Universe. These equations are the backbone of modern cosmology.

Central equations

First Friedmann equation
H2=(a˙a)2=8πG3ρ−kc2a2+Λc23H^2 = \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}
Acceleration equation
a¨a=−4πG3(ρ+3pc2)+Λc23\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}
Continuity equation
ρ˙+3H(ρ+pc2)=0\dot\rho + 3H\left(\rho + \frac{p}{c^2}\right)=0
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Interpretation

Pressure gravitates in general relativity. A sufficiently negative pressure, as from a cosmological constant, can drive accelerated expansion.

In the acceleration equation, which term can cause a¨>0\ddot a > 0?

What does the parameter H(t)H(t) represent?

Matter sources in the Friedmann equations

Matter

  • p≈0p \approx 0
  • density scales as a−3a^{-3}

Radiation

  • p=ρc2/3p = \rho c^2/3
  • density scales as a−4a^{-4}