# B Relationship Between Density and the Hubble parameter

1. Jun 10, 2017

### mindarson

I'm just looking for conceptual clarification re: the relationship between matter density and the Hubble parameter in the Friedmann equation. Just for quick reference, the equation I'm looking at is

H2 = 8πGρ/3 - ka-2

(I'm working through Liddle's Intro text, and for now we're ignoring the cosmological constant.)

My question is pretty broad and basic. Looking at this equation, it seems that the Universe will expand at a faster rate if the density ρ of matter within it increases. Why would that be? Isn't it true that more matter density would generate more gravitation? And doesn't that slow the expansion down, rather than speeding it up?

In Liddle's book, he writes in a couple places that this model essentially works just like a gas-and-piston system. Is that to say that the Hubble parameter increasing with density is analogous to how a volume of gas might expand more quickly if we could somehow inject more gas particles into it, because more particles would be colliding with the piston and pushing it out?

But for the gas-piston system, there's no gravity force between the particles that might bring them closer together and contract them.

I'm just not clear as to how it can be that more matter density would lead to faster expansion. But that seems to be what the equation says.

Last edited: Jun 10, 2017
2. Jun 10, 2017

### Bandersnatch

The relationship does indeed mean that higher mass density leads to higher deceleration of the Hubble parameter. But one needs to look at it from the other way around, so to speak.
You start with the present expansion rate being constrained by observations. For the present Hubble parameter to be like it is, the past rate had to be higher, so that after all the deceleration it had underwent, it has reached the present value.
Furthermore, the matter density is $\rho \propto 1/a^3$, so in strictly expanding universes it increases into the past and decreases into the future.
So, as you look into the past (smaller scale factor), and towards higher mass density, then you see the Hubble parameter as higher. In the future (higher $a$ and lower mass density), the expansion rate goes down - all due to the gravitational pull of all that matter.

It should be $a^{-2}$, by the way.

3. Jun 10, 2017

### mindarson

Argh, none of this makes sense to me at all. It seems like you're saying that in the past, the Universe had both a) higher mass-energy density and b) faster expansion. But that makes no sense, because the gravitation from higher mass-energy density would slow down the expansion, right?

So if you start the Universe at a certain volume and with a certain mass-energy density, and then it expands a little bit, now it has a smaller mass-energy density, therefore smaller gravitational 'contraction', therefore faster expansion. So its expansion rate should INCREASE over time, not DECREASE.

(I'm not saying I'm correct about this, just trying to display my thinking so errors can be pointed out.)

4. Jun 10, 2017

### Bandersnatch

Ok. Let's first simplify and assume denisty is constant. In this case, you start with the universe at some time t, and expansion rate $H_t$. As time goes on, matter acts gravitationally upon itself, slowing the expansion down. The rate of slowing down is constant (since matter density is constant). When we reach the present epoch $t_0$, where $t_0>t$, the hubble parameter will have been decelerated to its presently observed value $H_0$.

If we then add the quality of matter density going down with expansion (so, with time), then you have essentially the same situation, with the expansion rate $H_t$ going down with time due to matter interacting with itself gravitationally, but now as the time goes on, the strength of gravitational self-interaction goes down due to gradually lowering density. You still get deceleration, but it's slower the more dilute the matter. In order to reach the present value, it had to be even higher in the past than in the previous case.

The mistake you're making is in trying to apply cause-effect relationship the wrong way. The time goes forward, so higher density in the past can not cause the universe to expand slower than now.
It's somewhat analogous to projectile motion - if you throw a ball really high up, then it'll be decelerated in gradually slower fashion (gravity goes down the higher it flies). If you then observe the ball at e.g. 10 000 km above Earth to have some velocity, then you can say that in the past the ball had higher velocity, both due to being decelerated on its way, and due to the rate of deceleration going up towards the past.

5. Jun 10, 2017

### mindarson

I'll press on with my studies with this analogy in mind. Thank you for the patient explanation! Your latest response will help me untie my mental knots about this subject.

6. Jun 11, 2017

### timmdeeg

It appears to me that you are confusing the rate of expansion, which is given by the Hubble parameter $H$ with the acceleration, resp. deceleration of the expansion which is given by the sign of $\ddot a$. If you look at the second Friedmann equation, you will see that $\rho$ yields a negative value for $\ddot a$, which means that the universe is expanding decelerated. Vice versa the universe expands accelerated in case $\ddot a$ is positive (which requires sufficient negative pressure).
Note that the value of $H$ is decreasing as long as the universe expands, with the exception of exponential expansion. In the latter case $H=const.$

Last edited: Jun 11, 2017