# Explanation on how inflation solves the horizon and flatness problem

• trv

#### trv

Hi, I'm totally lost on how inflation solves the horizon and flatness problem.

Flatness Problem

Explanation I have
d/dt(1/Ha)<0

and therefore

$|\Omega-1|$

is driven towards zero rather than away from it.

My Confusion
Doesn't inflation increase the volume of the universe, and hence wouldn't the density decrease rather than increase? Or am I misunderstanding the density parameter here?

Horizon Problem

Explanation I have

The quantity 1/Ha is the comoving Hubble length, and determines which two regions can communicate now. The particle horizon on the other hand, separates two regions that could never have communicated. The horizon problem is solved by the possibility of greatly reducing the comoving Hubble length. Hence, regions that cannot communicate today were in causal contact early.

My Understanding
The way I understand this is that, the comoving Hubble length gets smaller with time. Therefore region that were within the comoving Hubble length earlier and hence could communicate and affect each other, are now out of contact. So even though we look at space and see regions that can't communicate now, there was a time when they could do so.

Does what I have just said make sense?

My Confusion
Finally my question on this part is, how does the particle horizon bit come into the picture? Can someone try and explain?

Hi trv,

During inflation the energy density of the inflaton field is constant while the scale factor a increases very rapidly. The argument on Wikipedia writes the Friedmann equation in the form,

$$(\Omega^{-1} -1) \rho a^2 = \frac{-3kc^2}{8 \pi G}$$

All the factors on the RHS are constant so as $$\rho a^2$$ increases $$\Omega^{-1} -1$$ must decrease. This can only happen if $$\Omega$$ goes towards 1 which makes the Universe flat.

There is a very good lecture by Alan Guth (who came up with inflation) on YouTube

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Curvature goes as one over the scale factor squared. The stuff that drives inflation is nearly independent of scale. Therefore, as inflation progressed, the stuff that drove inflation came to dominate over the curvature (and anything else that happened to be around).