# Condition for inflation presented by Liddle

1. Jun 17, 2007

### Magister

I can't understand the condition for inflation that Liddle presents in his book,
Cosmological Inflation and Large-Scale Structure A.Liddle, Pg 51:

$$\frac{d}{dt} \frac{H^{-1}}{a} <0$$

Because $\frac{H^{-1}}{a}$ is the comoving Hubble length, the condition for inflation is that the comoving Hubble length, which is the most important characteristic scale of the expanding Universe, is decreasing with time. Viewed in coordinates fixed with the expansion, the observable Universe actually becomes smaller during inflation because the characteristic scale occupies a smaller and smaller coordinate size as inflation proceeds.

Shouldn't be the opposite? The the observable shouldn't became bigger instead of smaller? I know that this relation cames from the other one which states that during the inflation the scale factor is accelerating but the I am not getting the physical picture.

Thanks

Last edited: Jun 18, 2007
2. Jun 17, 2007

### smallphi

Inflation resolves the particle horizon problem with CMB. On page 27 of the book they calculate the particle horizon comoving radius at time of last scattering to be 90 Mpc/h. On the same page they calculate the comoving radius to the last scattering surface (a sphere centered on us from which the CMB we observe today was emitted) to be 5820 Mpc/h. The very high homogeneity of CMB suggests two diametrically opposite points on the last scattering sphere should have been in causal contact at time of last scattering, yet they are not within eachothers particle horizons at that time because their particle horizons were much smaller than the distance between them: 90 << 5820.

The explanation that inflation provides is that the comoving particle horizon radius (estimated as 1/Ha) before inflation was much bigger allowing for the two points to be in casual contact and homogenize with each other. Later during inflation 1/Ha shrinks. After end of inflation till the last scattering it expands again but not enough to be equal to the comoving radius of the last scattering surface. That creates the illusion that the two points have never been in causal contact but they were. That is shown on fig 3.2 in the book.

Now to answer your question. Shrinking of the particle horizon 1/Ha in comoving coordinates simply means that the particle horizon in 'physical' coordinates, 1/H, increases slower relative to the scale factor a. During inflation the scale factor increases exponentially while 1/H is approximately constant and the ratio 1/Ha shrinks exponentially. That means during inflation matter that was homogenized in the past is flowing out of the physical particle horizon radius 1/H, because the horizon is not expanding as fast as the universe. Matter flowing out of particle horizon in physical coordinates shows as shrinking of particle horizon in comoving coordinates because in comoving coordinates matter appears frozen so the radius must shrink for the matter to get outside of it. That creates at last scattering the apparent paradox of two points outside of eachothers physical horizon which are nevertheless at the same temperature. The explanation provided by inflation is that they were inside eachothers horizon before inflation.

Last edited: Jun 18, 2007
3. Jun 18, 2007

### Magister

Thanks a lot smallphi. I was missing the fact that matter is frozen in comoving coordinates and so in this coordinate system the horizon have to shrink.

4. Jun 18, 2007

### Magister

By the way, I have another question about inflation.
If the horizon was expanding at a much more slower rate than the scale factor, that means that the observable universe today is just a tiny part of the whole universe. When I say whole universe I mean the true vacuum bubble that causes the inflation, I not sure if that’s correct. It seems to me a bit strange to call a giant bubble of true vacuum the whole universe…
Anyway, what I would like to ask is if the multi universes theories came from this fact. Is our observable universe just a small part that stayed casually connected until now? Are there other parts of the true vacuum bubble that we still (or will never be), are not casually connected? Is this parts that are called multi universes?