Fluid and Acceleration Equations

The Fluid and Acceleration Equations

The Friedmann Equation is an equation of motion balancing the kinetic and potential energy in the universe. It is an important equation governing the evolution of the scale factor ${\displaystyle a(t)}$ with energy density ${\displaystyle \epsilon (t)}$, but because ${\displaystyle a}$ and ${\displaystyle \epsilon }$ are both unknown, we need additional constraints in order to arrive at a solution.

The Fluid Equation

The first law of thermodynamics (${\displaystyle \Delta S=0}$) requires that any system with positive pressure must lose energy as the volume enclosing it expands. Thus, if ${\displaystyle U}$ is our internal energy and ${\displaystyle P}$ is our pressure:

${\displaystyle {dU \over dt}=-P{dV \over dt}.\,\!}$

In an expanding universe, ${\displaystyle U={E \over V}\cdot V=\epsilon a^{3}}$, where ${\displaystyle \epsilon }$ is the energy density of the universe. ${\displaystyle P}$ is the pressure of the photon gas, so:

${\displaystyle {\dot {\epsilon }}a^{3}+3\epsilon a^{2}{\dot {a}}=-P3a^{2}{\dot {a}},\,\!}$

which simplifies to:

${\displaystyle {{\dot {\epsilon }}=-3{{\dot {a}} \over a}(\epsilon +P)}\,\!}$

This is a a statement of the temperature loss of the universe due to adiabatic expansion.

The Acceleration Equation

The acceleration equation is not an independent equation; it is a combination of the Fluid equation and the Friedmann equation: ${\displaystyle {\frac {d}{dt}}}$(${\displaystyle 1^{st}}$ Friedmann Equation) gives us:

${\displaystyle 2{\dot {a}}{\ddot {a}}={8\pi \over 3}G{d \over dt}(\epsilon a^{2})={8\pi \over 3}Ga^{2}({\dot {\epsilon }}+2{{\dot {a}} \over a}\epsilon ).\,\!}$

Substituting the Fluid Equation for ${\displaystyle {\dot {\epsilon }}}$:

${\displaystyle 2{\dot {a}}{\ddot {a}}={8\pi \over 3}Ga^{2}(-{{\dot {a}} \over a}\epsilon -3{{\dot {a}} \over a}P)=-{8\pi \over 3}G{{\dot {a}} \over a}(\epsilon +3P).\,\!}$

Now we have our Acceleration Equation:

${\displaystyle {{{\ddot {a}} \over a}=-{4\pi G \over 3}(\epsilon +3P)}\,\!}$

The Acceleration Equation relates the acceleration of the expansion of the universe to the pressure of photon gas and the density of the universe. Note that if ${\displaystyle 3P\leq -\epsilon }$, we have an accelerating universe.

Compare this equation to the Newtonian equation for gravity, with ${\displaystyle M={4\pi \over 3}\epsilon x^{3}}$ and ${\displaystyle \epsilon _{eff}=\epsilon +3P}$:

${\displaystyle {\ddot {x}}={-G\cdot M \over x^{2}}={-4\pi \over 3}G\epsilon x\,\!}$