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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?

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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.

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  • $\begingroup$ Thanks! This is the kind of explanation I was looking for for the negative acceleration. Regarding the other part of the question, do you know if this deceleration could be what's mainly responsible for bringing $\dot{a}$ down to reasonable levels or is my math correct in that it can only decrease it by a factor of about 120? $\endgroup$ – Kris Walker Jul 29 '20 at 5:45
  • $\begingroup$ First, I've edited the answer: please have a look to make sure it still makes sense (I originally tried to keep things analytical by ignoring kinetic energy, but that's incorrect). As for your second question, I don't quite understand. The change in $\dot{a}$ is entirely due to $\ddot{a}$ (by definition). The proper way to study the behavior of $\dot{a}$ is to solve the coupled Friedmann and KG equations for a given potential. It's a good exercise if you haven't done it, though requires a numerical solution. $\endgroup$ – bapowell Jul 29 '20 at 13:15
  • $\begingroup$ Thanks, so I guess my initial hypothesis was more or less on the right track, great! Basically what I mean with the other question is this: between the end of inflation and the present day there are two deceleration mechnaisms, 1) the one we're discussing here and 2) the much more gradual but longer deceleration in the radiation-dominated (and later matter-dominated) universe. What I want to know is which of these two contributed most to the vast decrease in $\dot{a}$ between inflation and today. Though now I'm thinking that perhaps this would be better as a separate question. $\endgroup$ – Kris Walker Jul 29 '20 at 14:05

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