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When we say "age of the universe," I assume this means the time interval for the Euclidean FLRW metric $$ ds^{2} = -dt^{2} + a^{2}(t)\Big(dx^{2}+dy^{2}+dz^{2}\Big) $$ from when the scale factor was $a(0) = 0$ to today $a(t_{0})=1$.

First, before we go any further, is this true?

Now different histories of the rate of expansion presumably lead to different time intervals from $a=0$ to $a=1$ (and yes it is true the expansion rate changed a few times during the interval based on what dominated the universe).

Now inflation postulates that, in addition to the standard big bang model, there was an extremely rapid expansion early in the universe. Wouldn't this have an effect on our calculated age of the universe? If so, why is there always only one answer given to the age of the universe if different models lead to different ages? If not, then what exactly do we mean by "age of the universe" and why doesn't the question of whether or not unimaginably rapid expansion occurred not affect the age?

A comment helped me clarify some things about my (lack of) understanding. User Triatticus said,

Seeing as the time period of inflation itself was extremely short I don't see it adding substantially to the age of the universe [...]

This is true, and the point here is that without inflation it would take more time for the universe to expand to whatever size it needs to be. This is why I think the age should have to be affected.

Let me make this more specific. Wikipedia says,

The inflationary epoch lasted from $10^{−36}$ seconds after the conjectured Big Bang singularity to some time between $10^{−33}$ and $10^{−32}$ seconds after the singularity.

Within those $\Delta t = 10^{-33} - 10^{-36} \approx 10^{−33}$ seconds, the universe expanded by a factor of $x = a(t_\text{after})/a(t_\text{before})$. In other words, in an inflation model, this expansion by $x$ took $10^{−33}$ seconds. In a non-inflation model, wouldn't this expansion by $x$ take more time than that? If so, what would be the order of this time interval?

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  • $\begingroup$ From what I remember the idea of inflation was to settle the horizon problem. Seeing as the time period of inflation itself was extremely short I don't see it adding substantially to the age of the universe. Hopefully someone more versed in cosmology can enlighten you though (and me too since this was just from one semester of a cosmology class.) $\endgroup$
    – Triatticus
    Mar 31, 2022 at 20:48
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    $\begingroup$ "Seeing as the time period of inflation itself was extremely short I don't see it adding substantially to the age of the universe" Yes, this is true, but without inflation it would take more time for the universe to expand to whatever size it needs to be. $\endgroup$
    – Maximal Ideal
    Mar 31, 2022 at 21:04
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    $\begingroup$ @user110391 That's a really good point, I didn't think about that. $\endgroup$
    – Maximal Ideal
    Mar 31, 2022 at 21:19
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    $\begingroup$ You may find this question/answer useful: physics.stackexchange.com/questions/456907/… $\endgroup$
    – Andrew
    Mar 31, 2022 at 22:11
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    $\begingroup$ Related physics.stackexchange.com/questions/559620/… $\endgroup$
    – ProfRob
    Mar 31, 2022 at 22:30

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The age of the universe that is commonly quoted is the age since the end of any inflationary epoch and is based on the cosmological parameters we measure now.

There are inflationary models (e.g. the Ekpyrotic universe), or even alternatives to inflation, that hypothesise that the universe had a long and possibly infinite existence prior to inflation.

Inflation merely provides some sort of explanation for various features of our universe - its homogeneity and flat geometry - and sets initial conditions for the current expansionary epoch.

What happened prior to an inflationary episode is not currently amenable to observational tests.

Edit: Note that the addition of "dark energy" or inflationary terms into the cosmic expansion terms into the Friedmann equations makes the universe older than it otherwise would be. In the case of inflation, it makes it older by at least of order the length of the inflationary epoch. See Is cosmic inflation slow rather than fast? for more details.

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  • $\begingroup$ Ok, so now I understand that the "age of the universe" represents the time interval between now and inflation in inflationary models. But what is the meaning of the "age of the universe" in non-inflation big bang models? $\endgroup$
    – Maximal Ideal
    Mar 31, 2022 at 22:38
  • $\begingroup$ So wikipedia says, "The inflationary epoch lasted from $10^{−36}$ seconds after the conjectured Big Bang singularity to some time between $10^{−33}$ and $10^{−32}$ seconds after the singularity." Within those $\Delta t \approx 10^{-33}$ seconds, the universe expanded by a factor of $x = a(t_\text{after})/a(t_\text{before})$. In other words, in an inflation model, this expansion by $x$ took $10^{-33}$ seconds. In a non-inflation model, wouldn't this expansion by $x$ take more time than that? If so, what would be the order of this time interval? $\endgroup$
    – Maximal Ideal
    Mar 31, 2022 at 23:22
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    $\begingroup$ @MaximalIdeal inflation is slower. You need to read the answers to physics.stackexchange.com/questions/456907/… in terms of the time taken to go back to "zero". $\endgroup$
    – ProfRob
    Apr 1, 2022 at 5:54

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