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The age of the Universe can be estimated from taking the inverse of the Hubble constant: $t_\text{universe} = 1/H_0 =d/v.$

enter image description here

It seems to me this method assumes that any given galaxy has been receding at a constant apparent velocity for the lifetime of the Universe. However, by including data from larger & larger distances, it can be seen that Hubble's law doesn't hold. The data suggest that the rate of expansion was actually slower in the past.

enter image description here

Why is it appropriate to use only nearby galaxies to estimate the age of the Universe using the Hubble time when the rate of expansion isn't constant?

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    $\begingroup$ It isn't, really. As you say, it's just an estimation, an order-of-magnitude age. To get the correct age, you have to integrate the Friedmann equation. $\endgroup$
    – pela
    Jul 26, 2016 at 20:51
  • $\begingroup$ I don't think that second diagram is from a reputable source. I tracked it to this blog post, which says it's based on the Union 2 SNeIa dataset, but the dataset doesn't contain velocities. My best guess is they plugged the redshifts into the SR redshift-speed formula, which is incorrect. The so-called velocity in Hubble's original chart is really $cz$. $\endgroup$
    – benrg
    Dec 6, 2023 at 21:05
  • $\begingroup$ NB: Hubble's law is between rate of change of proper distance and proper distance and always holds. What is plotted on the x and y-axes here are not proper distance or rate of change of proper distance, though they approximate to it at low redshift and hence the straight line. $\endgroup$
    – ProfRob
    Dec 7, 2023 at 16:06

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$H_0^{-1}$ is only a rough estimate for the age of the universe and you have correctly identified the reasons why it is likely to be just an approximate estimate.

A correct age estimation relies on knowing $H_0$ and the densities of matter and dark energy, so that the past expansion history of the universe can be correctly modelled. Even this relies on an assumption about how dark energy behaves.

A more interesting question is why a value of $H_0 \simeq 70$ km s$^{-1}$/ Mpc gives an $H_{0}^{-1}$ of 14 billion years, which is within a few per cent of the current best estimate for the age of the universe of 13.8 billion years. In the $\Lambda$CDM cosmology model, the reason for this cosmic coincidence is that the universal expansion underwent a period of decelaraton up until about 4 billion years ago, when it started to accelerate again (the red curve on the plot below).

As a result, a tangent to the red curve, showing the size of the universe versus time, at the present epoch almost goes to ${\rm size}=0$ about 14 billion years ago. If we were to go backwards or forwards in time by 5 billion years, the agreement between $H_0^{-1}$ and the age of the universe would not be so good. At earlier times $H_0^{-1}$ would have overestimated the age of the universe, whereas at later times $H_0^{-1}$ will underestimate the age of the universe.

Expansion history of the universe

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    $\begingroup$ I think you're taking what BMS has said a bit too far. He specifically mentioned that $H_o^{-1}$ is an estimate, which it is. $H_o^{-1}=14\:Gyr$ which is a pretty good estimate of the age of the universe, though of course not exact. I think his question can be better interpreted to be, why does using $H_o^{-1}$ produce such a good estimate, relatively speaking. $\endgroup$
    – zephyr
    Jul 27, 2016 at 14:05
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    $\begingroup$ @zephyr Then that is a different question and it is because we live in a universe that is flat and where the dynamics are not presently dominated by either matter or dark energy. $\endgroup$
    – ProfRob
    Jul 27, 2016 at 16:49

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