Questions about the accelerating Hubble expansion

[PeterDonis]

If H will approaches a constant value given by the cosmological constant, there will still be a continuous acceleration ?(exponential).

If $H$ is asymptotically approaching a constant value, the expansion of the universe is asymptotically approaching exponential. At what point during that process the expansion becomes accelerating depends on the details, but it will become accelerating at some point, and once it does, it will stay accelerating.

 

[DAH]

In a spatially flat universe the Hubble parameter is given by $$H^2=\frac{8\pi G(\rho_m+\rho_r)}{3}+\frac{\Lambda c^2}{3}$$
$\rho_m$ and $\rho_r$ decrease with time and $\Lambda$ stays constant, so the Hubble parameter decreases with time. In the remote future $\rho_m$ and $\rho_r$ will tend to zero and the Hubble parameter will become a constant given by $$H=\sqrt{\frac{\Lambda c^2}{3}}$$

 

[Kairos]

In a spatially flat universe the Hubble parameter is given by $$H^2=\frac{8\pi G(\rho_m+\rho_r)}{3}+\frac{\Lambda c^2}{3}$$
$\rho_m$ and $\rho_r$ decrease with time and $\Lambda$ stays constant, so the Hubble parameter decreases with time. In the remote future $\rho_m$ and $\rho_r$ will tend to zero and the Hubble parameter will become a constant given by $$H=\sqrt{\frac{\Lambda c^2}{3}}$$

Thank you. I understand that the densities decrease and lambda is left alone at the end. What I don't understand is why rho and lambda affect H in the same direction. I would have thought - and + signs since lambda increases expansion while rho slows it down.

 

[PeterDonis]

What I don't understand is why rho and lambda affect H in the same direction. I would have thought - and + signs since lambda increases expansion while rho slows it down.

First, what you mean by "expansion" here is not $H$, its $\dot{a}$. They're not the same.

Second, when you say "lambda increases expansion while rho slows it down", what you mean is that dark energy, by itself, causes expansion to accelerate, while rho, by itself, causes expansion to decelerate. But the equation for $H^2$, by itself, tells you nothing about acceleration or deceleration. What you need to look at is the equation for $\ddot{a}$.

 

[Kairos]

First, what you mean by "expansion" here is not $H$, its $\dot{a}$. They're not the same.

Second, when you say "lambda increases expansion while rho slows it down", what you mean is that dark energy, by itself, causes expansion to accelerate, while rho, by itself, causes expansion to decelerate. But the equation for $H^2$, by itself, tells you nothing about acceleration or deceleration. What you need to look at is the equation for $\ddot{a}$.

yes, that's why I spoke of expansion in this post, and not acceleration. dark energy favors expansion contrary to gravity

 

[PeterDonis]

that's why I spoke of expansion in this post, and not acceleration

And did so wrongly if you didn't mean what I said in my previous post. Expansion and acceleration are not the same thing.

dark energy favors expansion contrary to gravity

No, dark energy favors accelerated expansion.

 

[PeterDonis]

@Kairos you are clearly not engaging with what the rest of us are trying to tell you and we are just wasting time at this point. So this thread is closed. Please re-read the responses you have gotten more carefully.