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We have Hubble's law:

$$v=Hd$$

where H is the Hubble parameter, which is decreasing in value, but it will be constant in the distant future. So, assuming the Standard Model of Cosmology holds true, will Hubble's law be accurate when the Hubble parameter becomes a constant?

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    $\begingroup$ Not clear what you are asking. The Hubble parameter is defined to be $\dot{a}(t)/a(t)$, so by definition Hubble's law will be still applicable. Do you mean will it be accurate to a greater range of distance? $\endgroup$
    – ProfRob
    Commented Jun 6, 2015 at 15:50
  • $\begingroup$ I thought Hubble's law was currently accurate for all distances. Is that not true? $\endgroup$
    – set5
    Commented Jun 6, 2015 at 16:20
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    $\begingroup$ No, that isn't true. The universal expansion was decelerating, it is now accelerating; that is why the Hubble parameter is changing. If the parameter changes with time, then when we look at distant objects in the past, it is clear that today's Hubble parameter will not accurately predict their recession velocities. $\endgroup$
    – ProfRob
    Commented Jun 6, 2015 at 16:36
  • $\begingroup$ Peculiar Velocity in cosmology: en.wikipedia.org/wiki/Peculiar_velocity#Cosmology $\endgroup$
    – Wayfaring Stranger
    Commented Jun 7, 2015 at 23:57
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    $\begingroup$ You got me thinking quite hard about this. You are correct, but only if the distance is the proper distance - the distance away now. In practice we cannot measure this for distant galaxies. Have a look at astro.ucla.edu/~wright/cosmo_02.htm $\endgroup$
    – ProfRob
    Commented Jun 8, 2015 at 8:47

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As per the above comments, there will still be v=Hd, if v is the rate of change of current distance, and d is the current distance-- as long as the universe continues to obey the cosmological principle. The nearby version of that will still look like correlating Doppler-shift velocity to distance indicators, and the faraway version will require a dynamical model, albeit a very simple one due to dark energy. It will lead to a constant H that will not be anything like the inverse of the age of the universe. And if you go too far into the future, you won't be able to see the CMB, or quasars, so alien cosmology will really struggle.

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