Chapter 4

Friedmann Equation

The scale factor $a(t)$ is clearly important, but we don't know what its functional form is yet. Using Newtonian gravity, we can derive an equation that relates $a$ with physical quantities, namely the average density $\rho(t)$ of the universe. We'll derive this equation in class, which is very close to the more correct, GR version of the equation.

Newtonian Friedmann equation:

\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho(t) + \frac{2 U}{r_s^2}\frac{1}{a(t)^2}

GR Friedmann equation:

\boxed{\boxed{\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3 c^2} \varepsilon(t) - \frac{\kappa c^2}{R_0^2}\frac{1}{a(t)^2}}}

In the Newtonian version, $U$ corresponds to the potential energy of expanding material. If $U < 0$ , the gravitational binding energy is greater than the kinetic energy of the material, meaning that the expansion will eventually stop, the mutual gravity of all the stuff overcoming its kinetic energy. This term corresponds to $\kappa$ in the GR version, with $U<0$ implying $\kappa = +1$ , or a universe with a positive curvature. In general, you can use this equation to relate these quantities at the present time (when they can be most easily measured):

H_0^2 = \frac{8\pi G}{3c^2}\varepsilon_0 - \frac{\kappa c^2}{R_0^2}

where $\varepsilon_0$ is the energy density of the universe today. If $H_0$ and $\varepsilon_0$ can be accurately measured, the curvature of the universe can be determined ( $\varepsilon_0$ cannot easily be measured it turns out, especially given the mysterious nature of most of the contents of the universe).

In practice, we can define a critical energy-density, $\varepsilon_{\rm crit}$ , which is required for $\kappa = 0$ , and if $\varepsilon > \varepsilon_{\rm crit}$ that means $\kappa = +1$ and if $\varepsilon < \varepsilon_{\rm crit}$ then $\kappa = -1$ .

\boxed{\varepsilon_{\rm crit}(t) \equiv \frac{3 c^2}{8\pi G}H(t)^2}

Instead of memorizing the critical density in some units (e.g., ${\rm GeV~m}^{-3}$ ), it's easier to parameterize the energy density by the critical density (i.e., put it into units of the critical density). We call this variable the density paramter:

\boxed{\Omega(t) \equiv \frac{\varepsilon(t)}{\varepsilon_{\rm crit}(t)}}

Rewriting the Friedmann equation with this definition, we get

1 - \Omega(t) = -\frac{\kappa c^2}{R_0^2 a(t)^2 H(t)^2}\, .

Fluid, Acceleration, & EoS Equations

With the Friedmann equation, we have two unknowns but only one equation, so we need at least one more equation (more if we introduce any more variables). Physics has been successful at finding solutions through deriving equations based on conserving properties of a system, like mass, energy, and momentum. Using energy conservation in thermodynamics, we can write an equation that relates energy density and pressure in an expanding volume.

Fluid Equation:

\dot{\varepsilon} + 3 \frac{\dot{a}}{a}(\varepsilon+P) = 0

By itself, this equation doesn't help us as it introduces another variable, the pressure $P$ . We'll fix this in a second. First, we can combine the Fluid and Friedmann equations to derive a third, but not independent equation.

Acceleration Equation:

\frac{\ddot{a}}{a} = -\frac{4\pi G}{3 c^2}(\varepsilon + 3P)

This equation can quickly tell us whether the universe's expansion is accelerating or decelerating — for normal matter, energy density and pressure are always positive, so classically the universe should always be decelerating. For the universe's expansion to accelerate (as we believe it is now), a more exotic substance is needed.

Although we've introduced $P$ , if we can come up with a third equation that specifies pressure in terms of $\varepsilon$ and/or $a$ , we can finally solve for $a(t)$ . Usually, the pressure can be expressed in terms of energy density, $P(\varepsilon)$ , and while its functional form could be complicated, $P \propto \varepsilon$ in practice, which is nice and simple. $P(\varepsilon)$ is called a substance's "equation of state."

Equation of State:

\boxed{P(\varepsilon) = w \varepsilon}

where $w$ is usually a constant (although it can be a function of $a$ ). The value of the constant depends on the substance. For example, normal matter in gaseous form typically follows the ideal gas law:

P = \frac{\rho}{\mu} k T = \frac{\varepsilon}{\mu c^2} k T = \frac{kT}{\mu c^2}\varepsilon

Here we used $E=mc^2$ to convert density to energy density, and $\mu$ is the mean molecular mass (average mass of a particle) of the gas.

Thus, the value of $w$ for something that follows the ideal gas law is

w = \frac{kT}{\mu c^2}\, .

A Maxwell-Boltzmann distribution tells us the distribution of velocities particles have in a gas if in equilibrium at a certain temperature $T$ (i.e., the fraction of particles with a given velocity). One property of the distribution is that the average square velocity $\left<v^2\right>$ is related to the temperature by $3kT = \mu \left< v^2 \right>$ . Plugging this into our equation for $w$ above:

w = \frac{1}{3}\frac{\left<v^2\right>}{c^2}

This form of the equation is much more useful; we can now easily determine $w$ for 2 types of substances: non-relativistic ( $v << c$ ) and relativistic ( $v \sim c$ ) substances. Most protons, electrons, and dark matter particles (whatever they are) are non-relativistic, while photons and neutrinos are relativistic. From the equation:

\begin{array}{lrr} {\rm Non-rel.} & v<<c: & w_{\rm nonrel} \rightarrow 0 \\ {\rm Rel.} & v \sim c: & w_{\rm rel} \rightarrow \frac{1}{3} \end{array}

Each substance will therefore evolve differently as the universe expands, but since their contributions sum together, we now have everything we need to solve for $a(t)$ .

Cosmological Constant

But, we're not quite done. It turns out GR also allows a constant term to be added to its equations, in the same way a constant term can be added to an indefinite integral. An analogy in Newtonian gravity is defining the gravitational potential energy to equal a non-zero constant value (instead of 0) as $r\rightarrow\infin$ . This constant is called the cosmological constant (usually denoted by $\Lambda$ ) and modifies the Friedmann equation:

\boxed{\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3 c^2} \varepsilon(t) - \frac{\kappa c^2}{R_0^2}\frac{1}{a(t)^2} + \frac{\Lambda}{3}}

This is the complete, most general form of the equation. Combining this equation with the fluid equation reveals that

P_\Lambda = -\varepsilon_\Lambda = -\frac{c^2}{8\pi G}\Lambda\\ {\rm so,}~w_\Lambda = -1\, .

While technically $\Lambda$ is a separate beast from energy density (it is a feature of the gravitational force itself, not an actual physical substance with energy), using this definition of $\varepsilon_\Lambda$ allows us to use the original Friedmann equation without this extra term. The cosmological constant is the "null hypothesis" or simplest guess for the nature of dark energy. Dark energy is anything with a sufficiently negative value for $w$ that would cause the universe's expansion to accelerate. In general, it could vary with time, in which case $w_{\rm DE}(a)$ depends on time or equivalently the scale factor.