Discontinuity at the transition between the inflationary and radiation-dominated universe

Question

Inflation is thought to be an extraordinarily brief period during the very early universe when the scale factor increased a remarkable $e^{60}$-fold, however this also comes with a corresponding increase in the expansion rate $\dot{a}=\mathrm{d}a/\mathrm{d}t$. When inflation ends, the exponential acceleration ceases and we transition to the radiation-dominated era where the expansion rate now evolves as $\dot{a}_\text{rad}(t)\propto t^{-1/2}$.

But what happened to the expansion rate, $\dot{a}_\text{end}$, at the end of inflation? Did it become an initial condition of the FLRW universe? Surely it's much too large to have decreased under $\ddot{a}_\text{rad}(t)$ in any reasonable amount of time, before the universe become too diluted? Did it undergo a large deceleration, $\ddot{a}(t)\ll 0$, a sort of "anti-inflation" right at the end of the inflation period? If so, what is the reason for this deceleration?

To make this more exact, consider inflation to begin and end at $t=0$ and $t=T$ respectively, and assume that inflation expands the universe such that $a(T)/a(0)=e^{60}$. We can then describe the inflation as

\begin{align} a(t<T)=a(0)\exp(ht)=a(T)\exp\left(60\left(\frac{t}{T}-1\right)\right) \end{align} where $h$ is the (constant) Hubble parameter during inflation. For $t>T$ we transition to the radiation-dominated universe and so to maintain continuity of the scale factor we must have \begin{align} a(t>T)=a(T)\sqrt{t/T} \end{align} So that's all well and good. Now let's consider $\dot{a}$. Taking the time derivatives of each of the above gives \begin{align} \dot{a}(t<T)=\frac{60}{T}a(T)\exp\left(60\left(\frac{t}{T}-1\right)\right)=\frac{60}{T}a(t<T) \end{align} and \begin{align} \dot{a}(t>T)=\frac{a(T)}{2T}\frac{1}{\sqrt{t/T}} \end{align} Now we find that the continuity is broken at $t=T$, with the former giving $\dot{a}(T)=60 a(T)/T$ and the latter giving $\dot{a}(T)=a(T)/2T$, which is not surprising since we are modelling this as an instantaneous transition between inflation and the radiation-doimated universe. So assuming this is valid we see that the expansion rate has decreased by a factor of 120, which is almost completely negligible when compared to the $e^{60}$-fold increase during inflation. So this can't be the decrease we desire.

What gives? Am I missing something crucial? How did we go from this ludicrously fast expansion to what we have today?

Additional question: assuming my math is even valid, is there a reason for the 120-fold decrease in the expansion rate? If we go by the standard "slow roll" explanation for a scalar field $\phi$ then inflation occurs when $V(\phi)\gg\dot{\phi}^2$ and stops when the potential decreases enough so that $V(\phi)\approx\dot\phi^2$, is this what causes the large negative acceleration required? If so is it able to be shown explicitly?

Answer 1

I think an example should help illustrate what's going on. First, there is no discontinuity when inflation ends: the rate of expansion goes smoothly from accelerating to decelerating, much like a ball rolling down a hill. To start, consider the Friedmann equation, $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3p),$$ where $p = w\rho$ relates the pressure and density of a scalar field. In general, $\rho = \dot{\phi}^2/2 + V(\phi)$, and $p = \dot{\phi}^2/2 - V(\phi)$, giving $$\frac{\ddot{a}}{a} = -\frac{8\pi G}{3}\left(\dot{\phi}^2-V(\phi)\right).$$

Even for simple potentials, there are in general no analytical expressions for $\dot{\phi}$ as a function of $\phi$, and so we can't do much better than this expression. But, this should make clear that as $\dot{\phi}$ grows (and $V(\phi)$ drops), $\ddot{a}$ also drops. Eventually, as $\dot{\phi}^2 > V$, the field has picked up enough speed that inflation ends; after this, the acceleration is negative.