The Lambda-CDM-model: when will the Age of the universe catch up with the Hubble time?

Question

Basic assumptions:

A) Within the Lambda-CDM-model we have two separate time measures:

• the Age of the universe - today roughly 13.8 GY, and
• the Hubble time - today roughly 14.4 GY

B) Very roughly, CDM came first, followed by Lambda

• CDM ruled 0-9 GY A.B.B.
• Lambda rules after 9 GY A.B.B.

C) According to theory:

• During CDM a(t)=t^(2/3) => Hubble Time = 1.5 * the Age of the universe.
• During Lambda the Hubble Time is constant, based on a(t) = exp(H*(t-t_now))

Questions, based on these very rough assumptions:

1. Currently the Hubble Time is still greater than the Age of the universe by some 600 MY, but Age is closing in. When will these two become equal?

2. Are there any reasonable estimates of the pure Lambda Hubble Time?

3. What will happen when Hubble Time equals Age of the universe?

Answer 1

In the approximation $Ω_m+Ω_Λ=1, Ω_r=Ω_k=0$, there is a closed formula for the Hubble parameter:

$$H(t) = H_\infty \coth \left( \tfrac32 H_\infty t \right) \qquad \text{where } ​H_\infty = \sqrt{Ω_{Λ_0}} H_0$$

From Planck data, $H_\infty \approx 56.3 \text{ km/s/Mpc} \approx 1/(17.4 \text{ Gyr})$ with an error of around 1%. I'm not sure this can be trusted, given the current disagreement between methods of measuring $H_0$, but I'll use it for this answer.

When will these two become equal?

$x\coth{\frac32 x}=1$ when $x\approx 0.85856$, so the Hubble time equals the age of the universe when $t \approx 0.85856/H_\infty \approx 14.9 \text{ Gyr}$.

Are there any reasonable estimates of the pure Lambda Hubble Time?

It's $1/H_\infty \approx 17.4 \text{ Gyr}$.

What will happen when Hubble Time equals Age of the universe?

Nothing. $c/H_\infty$ is the distance to the cosmological horizon in the de Sitter era, but $1/H$ at other times has no significance beyond being a crude estimate of the age of the universe.

Answer 2

The suggested value of the Lambda Hubble Time surprised me. It is by far much higher than I had expected.

As I see the Lambda-CDM, there are two processes: starting with 100 % CDM, ending with 100 % Lambda. And together they should all the time sum up to 100 %

With 100% CDM, H(now) would be 1/(20.7 Gyr), with 100% Lambda Hubble Time H(now) would thus be 1/(17.4 Gyr)

Yet I think we can agree on that the actual current value of H(now) = 1/(14.4 Gyr)

So what I cannot figure out is how [(100 - x) % 1/(20.7 Gyr) + x % 1/(17.4 Gyr)] could end up with a H(now) far outside the interval [17.4 Gyr - 20.7 Gyr]