# Why does comoving Hubble radius increase with time?

by ck99
Tags: comoving, hubble, increase, radius, time
 P: 61 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?
 Sci Advisor P: 1,686 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...
 P: 61 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?
P: 1,686
Why does comoving Hubble radius increase with time?

 Quote by 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?
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.
 P: 59 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.
P: 1,686
 Quote by 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 Proper Distance divided by the Proper Distance.
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.
P: 59
 Quote by 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.
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.
P: 1,686
 Quote by Deuterium2H I specifically said that comoving distances are constant.
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!
 And any equation involving the Hubble paramater (which involves the scale factor) is time-dependent, because it is based upon proper 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?
P: 59
 Quote by 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?
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.
Yes, it is the quantity $(aH)^{-1}$, which is a function of coordinate time, $t$.
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.