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The expansion history of the Universe can found by measuring the differential age evolution of cosmic chronometers. This yields a measurement of the Hubble parameter H(z) as a function of redshift.

see: Jimenez & Loeb (2002); Moresco et al. (2018).

The basic idea of the Cosmic Chronometer method is that the Hubble parameter is related to the differential redshift-time relation as $H(z) = \frac{-1}{1+z}\times\frac{dz}{dt}\tag1$

(1) is derived from

$H(z)= \frac{\dot a}{a}$

$a=\frac{1}{1+z}$ and

$\frac{da}{dt}=\frac{da}{dz} \times \frac{dz}{dt}$

Then the quantities $dz$ and $dt$ need to be measured for a passively evolving system such as a group of stars, $dz$ is found from spectroscopic measurements.

For $dt$ the ‘cosmic chronometer’ was a direct spectroscopic observable (the 4000 ˚A break) known to be linearly related with the age of the stellar population

enter image description here

and a plot of $H(z)$ against $z$ can be obtained like the one above.

“The solid line and the dashed regions...show the fiducial flat LCDM cosmology... H0 = 67.8 km/s/Mpc, m = 0.308. For comparison an Einstein-de Sitter model is shown...”

• Moresco et al DOI:10.1088/1475-7516/2016/05/014

The method doesn’t need absolute ages of stars, but the difference in ages of the stars.

The question is this:

Since both the actual age of a star and the difference in ages of a pair of stars are both subject to time dilation, has this properly been accounted for in the method?

For example, with easy numbers.

If time dilation caused a 10.2 million year old star at a redshift of $z$ to appear to be 5.1 million years old, and a 10 million year old star further away at redshift $z+dz$ to appear to be 5 million years old, the $dt$ would be measured as 0.1 million years, but in reality it should be 0.2 million years.

It doesn’t seem to be taken into account, or is it already there in equation (1)?

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The method in summary is to observe two galaxies that were assumed to have formed at the same cosmic epoch but are observed at different redshifts. The difference in their redshift is $\Delta z$ and then by measuring some spectral feature that depends on age (in their own frame of reference), to estimate the difference in age between them $\Delta t$. The idea is to choose a pair of galaxies that formed at a considerably higher redshift than where they are now observed; where "considerably higher" I think means that the time since they formed is a lot longer than the observed age difference between them.

The key thing here is that the calibration of whatever age indicator you are using is done (or calculated) in the rest frame. When we look at a galaxy at a redshift of 1, we see their stellar populations as they were about 7.9 billion years ago. A galaxy at redshift 1.1 is seen as it was 8.3 billion years ago (these numbers based on a standard cosmological model). We therefore expect to see a difference in the age of their populations of about 0.4 billion years, providing they were both formed at the same time at a much higher redshift than they have now. There is no further time dilation to insert into these ages or age differences.

You might be confused by the time dilation that must be taken into account if we were to see something change with time. For example if we could observe a very distant star pulsating and the pulsation period gave the age, then yes, that would need to be corrected for time dilation to estimate the rest frame pulsation period. But that correction is already done when observing spectral features - you just blueshift it back to where it was in the rest frame of the galaxy.

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  • $\begingroup$ Thanks for the answer. In your example of a star at redshift 1 whose light left it 7.9byr ago and star at redshift 1.1 whose light left it 8.3 byr (difference 0.4). The universe is 13.8byr old, the first would be 5.9byr old when the light left it (if had formed shortly after the Big Bang). During these 5.9byrs wouldn’t it have appeared to us to have evolved only for about half this time due to the time dilation at redshift of about 1 and appear to be 2.95byrs old, similarly for the other (5.5 becoming 2.75), giving a measured age difference to us of 0.2byrs instead of the true 0.4byrs? $\endgroup$ Mar 28 at 13:39
  • $\begingroup$ @JohnHunter The light left the first star when it was 5.9 Gyr old (in its own frame of reference - that is what this value of time means). Therefore when that light gets to us, it looks like it was emitted by a star that was 5.9 Gyr old, but redshifted by a factor of $1+z$. There isn't any other correction to apply. $\endgroup$
    – ProfRob
    Mar 28 at 14:50
  • $\begingroup$ Thanks, will have a think about it... $\endgroup$ Mar 28 at 18:03

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