# Inflation assumes $\ddot{a}<0$ after inflation?

## Main Question or Discussion Point

I'm reading introductions about inflation and one thing they assume is that presently $\ddot{a}<0$ where $a$ is the parameter in our FRW metric.

We know that this is observationally false, since in fact $\ddot{a}>0$.

But why do they assume this?

My guess is that in the big-bang model there is a period where radiation and matter dominates over the cosmological constant and it is in this regime that $\ddot{a}<0$.

Another guess could be that theorists of the time didn't know that $\ddot{a}>0$. But I don't think this is a good enough answer since people keep talking about inflation so there has to be some reasoning behind the assumption that $\ddot{a}<0$ after inflation.

So, what going on here?

Thanks.

Chalnoth
I'm reading introductions about inflation and one thing they assume is that presently $\ddot{a}<0$ where $a$ is the parameter in our FRW metric.
I think I'd need more context to understand what's going on here, but I don't think the current accelerated expansion is relevant to inflation at all. It is certainly the case that $\ddot{a} < 0$ for the period immediately following inflation, however, and for a few billion years afterwards.

It is certainly the case that $\ddot{a} < 0$ for the period immediately following inflation, however, and for a few billion years afterwards.
Thats what I needed to hear. Is your reasoning that matter and radiation dominated over the cosmological constant for those billion years?

bapowell
The universe began accelerating again (after inflation) when it was around 10 billion years old. Taking the end of inflation to correspond to the standard hot big bang, t=0, there was therefore a period of 10 billion years during which the expansion of the universe was decelerating in the sense of $\ddot{a} < 0$.

What caused the flip in the sign? Is it what I said about matter and radiation dominating over dark energy at first?

bapowell
Yes. The energy density associated with the CC, or whatever it is that's driving the accelerated expansion, is tiny. It didn't emerge as a dominant component of the energy density of the universe until after the radiation and matter red shifted sufficiently.

Ok thanks. I have a stupid question though that just occurred to me, which is somewhat unrelated to this thread:

what if dark energy is the inflaton field (of course we now sit at the bottom of the well of $V(\phi)$? Are there any hypothesis about that? Would that hypothesis be wrong?

Chalnoth