In the lambda-CDM model describing an accelerating Universe, the Hubble parameter is currently decreasing with time. Will it continue to decrease forever?


The solution to the Friedmann equation in a flat universe is $$H^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3},$$ where $\rho$ is the matter density (including dark matter) and $\Lambda$ is the cosmological constant.

As the universe expands, $\rho$ of course decreases, but $\Lambda$ remains constant.

Thus the Hubble "constant" actually decreases from its current value $H_0$ and asymptotically tends towards $ H = \sqrt{\Lambda/3}$ as time tends towards infinity.

As $\Lambda = 3H_0^{2} \Omega_\Lambda$, and measurements suggest that $\Omega_{\Lambda} \simeq 2/3$, then $\Lambda \simeq 2H_0^2$, and the Hubble parameter will therefore decrease to approximately $\sqrt{2/3}$ of its present value if the cosmological constant stays constant.

Of course if $\Lambda = \Lambda(t)$, (ie not the basic $\Lambda$-CDM model) then the behaviour will be different.

EDIT: Another useful form of the solution (for the case of a constant vacuum energy density) is

$$H^2 = H_0^2 \left( \frac{\Omega_r}{a^4} + \frac{\Omega_M}{a^3} + \frac{\Omega_k}{a^2} + \Omega_{\Lambda}\right),$$ where $H_0$ is the Hubble parameter now, $a(t)$ is the scale factor of the universe, $\Omega_r$ is the current (i.e. $a=1$) ratio of the radiation density to the critical density and $\Omega_M$, $\Omega_k$ and $\Omega_{\Lambda}$ are the equivalent densities for the matter (baryonic and dark), curvature and (constant) vacuum energy densities.

As $a$ increases you can see that all three of the leading terms get smaller and the Hubble parameter decreases at all times. When $a$ is very large, $H$ approaches $\sqrt{\Omega_{\Lambda}} H_0$ as before.

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    $\begingroup$ Has the value of the Hubble parameter always been decreasing? From time=0 until now? $\endgroup$ – set5 May 5 '15 at 20:10
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    $\begingroup$ @mick: Yes it has, since from the Friedmann equation H decreases with density, which in turn decreases (for matter) or at least doesn't increase (for dark energy) with time (note though that things around the moment of the Big Bang after all still are quite uncertain and could have behaved differently from what our current knowledge suggests). $\endgroup$ – pela May 12 '15 at 11:29
  • $\begingroup$ @mick I have added a more general form which perhaps makes it easier to see that as long as the scale factor increases, H decreases (and providing that the dark energy doesn't get bigger in the early universe). In fact at very early (inflationary) epochs this cannot be assumed and the Hubble parameter is thought to be approximately constant, leading to exponential growth. $\endgroup$ – ProfRob May 12 '15 at 12:17

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