Friedmann Equation

1 The Friedmann Equation

The Friedmann Equation is an equation of motion for the scale factor $a(t)$ in a homogeneous universe. A rigorous derivation requires General Relativity, but we can fake it with a quasi-Newtonian derivation. We will model the universe as an adiabatically ($\Delta S=0$) expanding, isotropic, homogeneous medium. Isotropy allows us to use $r$ as a scalar. Consider a thin, expanding spherical shell of radius $a$. Birkhoff’s Theorum states that even in GR, the motion of such a shell depends only on the enclosed mass $M={4\pi \over 3}a^{3}\rho$. Thus, the energy per unit mass per unit length is:

$E=\overbrace {{\frac {1}{2}}{\dot {a}}^{2}} ^{Kinetic}\overbrace {-{G\cdot M \over a}} ^{Potential}={\frac {1}{2}}{\dot {a}}^{2}-{4\pi \over 3}G\rho a^{2}.\,\!$

We define $k\equiv -{2E \over c^{2}}$, and we will show later that $k$ is a measure of the curvature of the universe:

$k{\begin{cases}>0&\,for\ E<0\ (bound)\\=0&\,for\ E=0\ (critical)\\<0&\,for\ E>0\ (unbound)\\\end{cases}}\,\!$

where $k$ has units of ${1 \over length^{2}}$ if $a$ is dimensionless. Substituting $k$ into the above energy equation, and solving for ${{\dot {a}} \over a}$, we get:

$H^{2}=\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\rho -{kc^{2} \over a^{2}}.\,\!$

In GR, we understand that it is not just mass that curves space, but also energy. Thus, we can generalize this equation to use an energy density $\epsilon$, with $\epsilon /c^{2}=\rho$, giving us:

$H^{2}=\left({{\dot {a}} \over a}\right)^{2}={\frac {8\pi }{3}}G{\frac {\epsilon }{c^{2}}}-{kc^{2} \over a^{2}}.\,\!$

This is the Friedmann Equation.

1.1 Critical Density

As discussed above, $k=0$ corresponds to a flat, critically bound universe. If we assume $k=0$, we can solve for the critical density of the universe: the value of $\epsilon /c^{2}=\rho$ that leads to a critically bound universe. To do this, we need a measure of what $H$ is. A lot of effort has gone into measuring $H_{0}$, the value of the Hubble parameter at the present day, with $H_{0}\approx 68$ km s$^{-1}$ Mpc$^{-1}$. Using that

$H_{0}^{2}={\frac {8\pi G}{3c^{2}}}\epsilon _{0,cr}={\frac {8\pi G}{3}}\rho _{0,cr},\,\!$

we find that $\rho _{0,cr}\approx 9{\rm {e}}-27$ kg m$^{-3}\approx 1{\rm {e}}11M_{\odot }$ Mpc$^{-3}$. This is called the critical density of our (present day) universe. Note that generally, the critical density is time dependent because the Hubble parameter changes with $t$.

1.2 In terms of the Dimensionless Density Parameter

The critical density is an important yardstick for determining how a universe behaves. It is common to measure energy densities relative to this yardstick. We define

$\Omega (t)\equiv {\frac {\epsilon (t)}{\epsilon _{cr}(t)}}={\frac {\rho (t)}{\rho _{cr}(t)}}\,\!$

to be the dimesionless density parameter. Using this definition and remembering that $\epsilon _{cr}={\frac {3c^{2}}{8\pi G}}H(t)^{2}$, we can rewrite the Friedmann equation as:

$1-\Omega (t)=-{\frac {kc^{2}}{a^{2}H^{2}}}\,\!$