# Why does comoving Hubble radius increase with time?

1. Feb 27, 2012

### ck99

I am looking at inflation at the moment, and it says in my textbook that (aH)^(-1) is constantly increasing in matter or radiation dominated epochs.

a is always positive and always increasing. This tells me that da/dt is positive. I think that setting the universe to MD/RD means that da/dt is decreasing with time (eg a decelerating universe as there is no cosmological constant driving expansion). So dt/da (which is another expression for comoving Hubble radius) is increasing with time.

Have I got this right?

2. Feb 27, 2012

### bapowell

da/dt decreasing in time is not a decelerating universe, it's a contracting one.

EDIT: This statement is obviously incorrect. See below for my efforts to redeem myself...

Last edited: Feb 27, 2012
3. Feb 27, 2012

### ck99

If todays value of da/dt is lower than yesterdays, but they are both greater than 0, doesn't that mean the universe is expanding, but that the rate of expansion is slowing down?

4. Feb 27, 2012

### bapowell

Errr...of course. My apologies. Yes, the comoving Hubble radius is indeed increasing in time during RD/MD, because as you say the universe is decelerating. You've undoubtedly noticed this is not the case during inflation.

5. Feb 27, 2012

### Deuterium2H

Ummm...no one has answered the question correctly so far.

Comoving Coordinates (and Comoving distances) do NOT increase with the expansion of the Universe, and do not increase in time. That is the whole point of the Comoving coordinate system.

Proper distances increase in time.

The Hubble Paramater is measuring the rate of change of Scale Factor (da/dt) divided by the Scale factor (a). The Scale factor is time-dependent, and is directly related to the Proper distance.

Last edited: Feb 27, 2012
6. Feb 27, 2012

### bapowell

I think you mean to say that comoving coordinate systems do expand with the universe, so that comoving distances have constant coordinates.

It does not follow that all quantities measured in comoving coordinates are constant -- what would be the point of using them then?? Any proper distance that is not increasing with the expansion will have non-constant comoving coordinates.

The Hubble radius, $H^{-1}$, measured with respect to comoving coordinates is the comoving Hubble radius, $(Ha)^{-1}$. It very much depends on time.

7. Feb 27, 2012

### Deuterium2H

I didn't say that "all quantities measured in comoving coordinates are constant". I specifically said that comoving distances are constant. And any equation involving the Hubble paramater (which involves the scale factor) is time-dependent, because it is based upon proper distance at a given (fixed) instant in time.

8. Feb 27, 2012

### bapowell

OK, well then what does this have to do with the OP? He's asking about the comoving Hubble parameter, which is the Hubble parameter in comoving coordinates. It is not a comoving distance!
I don't know what this has to do with the OP. Looks like you're making things more confused than they need to be. He's asking about the comoving Hubble radius. It is increasing in an RD/MD universe. So please tell me where we've gone wrong here?

9. Feb 27, 2012

### Deuterium2H

Perhaps I misunderstood the question of the OP. The term Comoving Hubble radius only makes sense when measured at a particular instant of cosmological time, and it is dependent upon the coordinate (proper) distance at the time of measuement.

And I agree, I have probably needlessly confused the question in the original post.

Last edited: Feb 27, 2012
10. Feb 27, 2012

### bapowell

Yes, it is the quantity $(aH)^{-1}$, which is a function of coordinate time, $t$.

11. Feb 27, 2012

### cepheid

Staff Emeritus
Yeah, your reasoning sounds alright to me. As you've already pointed out, just from the definition of the Hubble parameter $(aH)^{-1} = (\dot{a})^{-1}$. For the rad-dominated and matter-dominated cases, a~t1/2 and a~t2/3 respectively. (I believe that these only apply for models with no cosmological constant). Differentiating those, you get da/dt ~ t-1/2 or t-1/3 respectively. So a-dot decreases with time, which means that its reciprocal increases with time.

12. Feb 28, 2012

### ck99

Hooray! Nothing is simple in cosmology, is it? :D