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What happened to the rate of expansion right after inflation ended?

Did the rapid inflationary expansion slow down a lot because the inflaton field had decayed, or because gravity from matter/radiation gradually decelerated the rapid expansion?

In other words, did inflationary expansion instantaneously turn into non-inflationary because inflaton field had decayed, or did inflationary expansion gradually decelerate to be non-inflationary because of gravity from matter/radiation?

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    $\begingroup$ We don't know whether inflation even happened. It's a theory that creates as many problems as it solves. The state of the art is not sufficient to say anything about the kind of thing being asked in this question. $\endgroup$ – user4552 Jan 18 '19 at 1:05
  • $\begingroup$ The observational evidence for inflation is summarized here: en.wikipedia.org/wiki/… $\endgroup$ – G. Smith Jan 18 '19 at 6:50
  • $\begingroup$ see my answer here and links therein physics.stackexchange.com/questions/454204/… $\endgroup$ – anna v Jan 18 '19 at 6:55
  • $\begingroup$ This is a substantial change from your original question. The proper way to do this is to leave your original question intact and add a dated update to the end of it. That way, readers can tell whether answers are answering your original question or your modified question. $\endgroup$ – G. Smith Jan 21 '19 at 6:02
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Actually my question is about the rate of expansion right after inflation ended when the field reach the bottom of the potential.

For the rate of expansion (the Hubble parameter) just take c over the Hubble radius:

enter image description here

So the Hubble parameter right after inflation in SI units was about

$$ \rm H = \frac{c}{r_H} = \frac{3\cdot 10^8 \ m/s}{10^{-28.5} \ m} = \frac{10^{37}}{s}$$

That number might vary depending on the specific model, but at least the order of magnitude is in that range.

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  • $\begingroup$ Right after inflation ended, did inflationary expansion instantly turn into non-inflationary, or did inflationary expansion gradually decelerate to non-inflationary due to gravity? $~$ I asked this because I don’t have any background in GR. $\endgroup$ – Forge Jan 18 '19 at 7:33
  • $\begingroup$ The function to transist from one to the other must at least be differentiable, although the transistion might have happened very fast but not instantaneous, but I don't know how long it took exactly. $\endgroup$ – Gendergaga Jan 18 '19 at 8:51
  • $\begingroup$ After inflation you have deceleration, during inflation the Hubble parameter (and therefore its reciprocal, the Hubble radius) was more or less constant and the expansion therefore accelerated (like it will be in the far future in the dark energy dominated era, so you can treat inflation like a very high cosmological constant which jumps or tunnels to its current lower level when inflation ends). $\endgroup$ – Gendergaga Jan 18 '19 at 19:38
  • $\begingroup$ “so you can treat inflation like a very high cosmological constant which jumps or tunnels to its current lower level when inflation ends). ”Not really. Both, the Inflaton field and the Cosmological Constant cause the universe to expand exponentially but their physical nature is different though. The Inflaton field must have coupled to matter and radiation whereas the CC doesn’t. Thus the latter isn’t a remnent of the former. $\endgroup$ – timm Jan 19 '19 at 4:09
  • $\begingroup$ @Forge “does inflationary expansion rate instantaneously change to non-inflationary inflation”. No, see my answer. $\endgroup$ – timm Jan 19 '19 at 4:14
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No, inflationary expansion did not “instantly turn into non-inflationary”. This epoch ended when the huge potential energy of the inflaton field started to decay into matter particles and radiation, a process which is called reheating. Before reheating the universe expanded exponentially and thereafter decelerated. Most physicists agree on this model. As far as I can tell the era where reheating happened was very short but certainly not zero.

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  • $\begingroup$ Thanks. I’m not sure if I understand your answer correctly. $~$ After reheating, does inflationary expansion rate instantaneously change to non-inflationary expansion rate, or does inflationary expansion rate gradually decelerate to non-inflationary rate due to gravity? $\endgroup$ – Forge Jan 20 '19 at 4:12
  • $\begingroup$ It happens gradually. In the beginning of reheating the negative pressure due to the Inflaton field still dominates the minor amount of matter already there which means exponential expansion turns gradually into accelerated expansion. With more and more matter created (along with decaying Inflaton field) matter starts to dominate decreasing negative pressure and at a certain point acceleration turns to deceleration. You can consider this mathematically as the turning point of a curve. $\endgroup$ – timm Jan 20 '19 at 4:41
  • $\begingroup$ What is the reason that the rapid inflationary expansion slowed down a lot after inflation ended?$~$Is it because inflaton field decayed, or because gravity from matter/radiation gradually decelerated the rapid expansion? $\endgroup$ – Forge Jan 20 '19 at 17:26
  • $\begingroup$ Both is correct, the inflaton field decayed and by this matter was produced. In other words repelling gravity was gradually replaced by attractive gravity. $\endgroup$ – timm Jan 21 '19 at 20:15
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    $\begingroup$ It is really enlightening to digest the message of the 2. Friedmann equation in this context. Omitting all natural constants it says that the evolution of the universe depends simply on $(\rho_m+3p)$ with $\rho_m$ matter density and $p$ pressure. The sign of the brace expression decides if the universe expands accelerated (-) or decelerated (+). As mentioned the pressure due to the CC and the Inflaton field is negative. If $\rho_m=0$ then the universe expands exponentially. $\endgroup$ – timm Jan 22 '19 at 15:29
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The inflationary expansion gradually decelerated. The change was not instantaneous, but “gradually” means over something like a billionth of a trillionth of a trillionth of a second!

In most inflationary models, the value of a scalar field filling all of space transitions from a nonzero value to a zero value as the field “rolls down its potential energy hill” to the bottom. The potential energy here is the energy of the field interacting with itself.

For example, a simple and classic model (but one no longer used in inflationary models) is the “Mexican hat” potential, $$V(\phi)=a(|\phi|^2-b^2)^2.$$ At high temperature, the field has the value 0 and the nonzero potential energy drives inflation. As the field evolves toward a value with $|\phi|=b$, the potential energy gradually drops to zero and inflation gradually stops.

Update to new questions posed by OP on 1/29:

Did the rapid inflationary expansion slow down a lot because the inflaton field had decayed, or because gravity from matter/radiation gradually decelerated the rapid expansion?

Inflation slows down and stops because the inflaton field decays.

In other words, did inflationary expansion instantaneously turn into non-inflationary because inflaton field had decayed, or did inflationary expansion gradually decelerate to be non-inflationary because of gravity from matter/radiation?

Neither. Inflation gradually turns into non-inflationary expansion because the inflaton field decays. Nothing happens instantaneously in this process.

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  • $\begingroup$ Thank you. Actually my question is about the rate of expansion right after inflation ended when the field reach the bottom of the potential. $~$Right after inflation ended, did inflationary expansion instantly turn into non-inflationary, or did inflationary expansion gradually decelerate to non-inflationary due to gravity? $\endgroup$ – Forge Jan 18 '19 at 3:44
  • $\begingroup$ By the time the field is at the bottom of the hill, inflation has stopped, and done so in a gradual way. Nothing happened instantly. $\endgroup$ – G. Smith Jan 18 '19 at 3:56
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    $\begingroup$ What happened before the "top of the hill"? When inflation started, did it instantaneously begin at a higher rate of expansion (compared to the previous non-inflationary expansion) and this rate then decreased (deceleration), or did the rate increase from the previous non-inflationary rate to a maximum before reducing? $\endgroup$ – Chappo Hasn't Forgotten Monica Jan 18 '19 at 4:19
  • $\begingroup$ I think the standard answer to what heppened before inflation is that universe was so hot that it was radiation-dominated, rather than dominated by the potential energy of the scalar field “at the top of the hill”. A radiation-dominated universe expands like $t^{1/2}$,. When the temperature dropped to the point that the energy in the scalar field dominated, the expansion became exponential with time (i.e. “inflationary”) because scalar fields have large negative pressure. When the energy in the scalar field decreased to zero, radiation again dominated, and then later matter dominated. $\endgroup$ – G. Smith Jan 18 '19 at 6:43
  • $\begingroup$ In the later universe, we are converting back to exponential expansion, because the energy density of matter has decreased and the constant energy density of dark energy has been dominating. $\endgroup$ – G. Smith Jan 18 '19 at 6:48

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