Acceleration Equation

The second Friedmann equation describes the universe's rate of expansion, how quickly the expansion is speeding up or slowing down. In short the equation tells us that if there is matter present in the universe then either its expansion rate is decreasing or its contraction rate is increasing. Friedmann originally solved Einstein's field equations with the Robertson-Walker metric and found two solutions, one was the Friedmann equation and the second was an equation which when subtracted by the first Friedmann equation gave the second Friedmann equation. However I want to explain how I first learned about the acceleration which I believe is he most easy and accessible way to derive it. We still need to use the first Friedmann equation that I spoke about previously.

$\left ( \frac{\dot{a}}{a} \right )^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2}$

Now multiply each term of the equation by the scale factor squared and you get

$\dot{a}^2 = \frac{8 \pi G}{3 c^2} \rho a^2 - k c^2$

If you differentiate this equation with respect to time

$2 \ddot{a}\dot{a} = \frac{8 \pi G}{3 c^2} ( \dot{\rho} a^2 + 2\rho a \dot{a})$

Then divide by

$2 \dot{a} a$

Then the equation becomes

$\frac{\ddot{a}}{a} = \frac{4 \pi G}{3 c^2} \left ( \dot{\rho} \frac{a}{\dot{a}} +2 \rho\right )$

Finally substitute in the following term from the fluid equation

$\dot{\rho} \frac{a}{ \dot{a}} =-3 (\rho +P)$

This gives us the equation in its final form

$\frac{\ddot{a}}{a} =- \frac{4 \pi G}{3} \left ( \rho + \frac{3p}{c^2}\right)$

Although there is no force associated with pressure in a universe described by the Roberston-Walker metric, i.e., one that's isotropic, as you can see from the equation if we were to increase pressure we would further decelerate the expansion. On occasion you will see cosmologists who have mass density replaced with energy density. If you use natural units where c = 1 these become interchangeable. Though this need not concern us here.

The interesting thing about this equation is that if pressure p is positive then I can make the right hand side of the equation positive (if I also make the left hand side negative), then the equation tells me that if there is any matter in the universe the left hand side cannot be zero. So in standard cosmology with the cosmological constant omitted the expansion of the universe must be slowing down (or we equivalently say the contraction is spreading up). Perhaps, though we might speculate that I n the early universe the material content is such that it made the pressure negative in which the expansion would rapidly accelerate. This is known as "inflation" and is accepted by most cosmologists, with still a few reasonable sceptics but we'll discuss that in another post. For now I'll leave it at that.

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Acceleration Equation

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