# Friedmann Equation

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## 1 The Friedmann Equation

The Friedmann Equation is an equation of motion for the scale factor ${\displaystyle a(t)}$ 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 (${\displaystyle \Delta S=0}$) expanding, isotropic, homogeneous medium. Isotropy allows us to use ${\displaystyle r}$ as a scalar. Consider a thin, expanding spherical shell of radius ${\displaystyle a}$. Birkhoff’s Theorum states that even in GR, the motion of such a shell depends only on the enclosed mass ${\displaystyle M={4\pi \over 3}a^{3}\rho }$. Thus, the energy per unit mass per unit length is:

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

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

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

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

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

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 ${\displaystyle \epsilon }$, with ${\displaystyle \epsilon /c^{2}=\rho }$, giving us:

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

This is the Friedmann Equation.

### 1.1 Critical Density

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

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

we find that ${\displaystyle \rho _{0,cr}\approx 9{\rm {e}}-27}$ kg m${\displaystyle ^{-3}\approx 1{\rm {e}}11M_{\odot }}$ Mpc${\displaystyle ^{-3}}$. 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 ${\displaystyle t}$.

### 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

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

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

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