Do we know anything about the age of the universe? I am looking to understand how the age of the universe is calculated according to modern physics.
My understanding is very vague as the resources I have found do not seem to state consistently whether inflation is part of the standard model.
For example, starting at the Age of the Universe wikipedia page, the age is calculated precisely within +/- 21 million years according to the Lambda-CDM model.
And:

It is frequently referred to as the standard model...

and

The ΛCDM model can be extended by adding cosmological inflation, quintessence and other elements that are current areas of speculation and research in cosmology.

Then I read:

The fraction of the total energy density of our (flat or almost flat) universe that is dark energy, $ \Omega _{\Lambda }$, is estimated to be 0.669 ± 0.038 based on the 2018 Dark Energy Survey results using Type Ia Supernovae7 or 0.6847 ± 0.0073 based on the 2018 release of Planck satellite data, or more than 68.3% (2018 estimate) of the mass-energy density of the universe.8

So this is where the numbers come from. The Dark Energy Survey page on wikipedia states:

The standard model of cosmology assumes that quantum fluctuations of the density field of the various components that were present when our universe was very young were enhanced through a very rapid expansion called inflation.

which appears to contradict what was said about the standard model on the Age of the Universe page.
From there I read about supernovae and standard candles.
All these pages list so many theories and problems, it seems hard to me to say what we know for certain. i.e. something that no physicist would disagree with.
I am looking to understand what I have misunderstood here or whether this is a fair characterization:
It seems a very simple calculation from the Hubble constant gave us a
number for the age of the universe. But since the 1960's it's been
known that the universe is "flat" as accurately as we can measure i.e.
$ \Omega = 1 $, and though this falsifies the hypothesis (of Hubble's law), we've kept
the age to hang physical theories off, but in a way that can no longer
be justified from first principles and observations.
Surely we have made observations, and there are things we can infer from them. And my question is:
Is the age of the universe something we can infer from our observations without appealing to an empirically inconsistent model? And if so, how? And how do we get the numbers out of the equations?
 A: To compute the age of the universe, one must solve the equation:
$$\frac{1}{a}\frac{da}{dt} = H_0 \sqrt{\frac{\Omega_{\gamma,0}}{a^4}+\frac{\Omega_{m,0}}{a^3}+\frac{\Omega_{k,0}}{a} +\Omega_{\Lambda,0}}$$
where $\Omega_\gamma$, $\Omega_m$, $\Omega_k$, $\Omega_\Lambda$ are the densities of radiation, matter, curvature, and vacuum energy, and the subscript '0' denotes present-day quantities.  This expression comes directly from the Friedmann equation relating the Hubble parameter, $H=\dot{a}/a$, to the density, $\rho$,
$$H^2 = \frac{8\pi}{3m_{\rm Pl}^2}\rho.$$ The density parameter $\Omega$ is simply $\Omega = \rho/\rho_c = \frac{8\pi}{3m_{\rm Pl}^2H^2}$, where $\rho_c$ is the critical density.
Now, to solve this equation we simply need values for these density parameters of the individual components.  If we're going for an approximation, we can set $\Omega_{\gamma,0} \approx \Omega_{k,0} \approx 0$ and solve the resulting integral for $t$, $$ t = \frac{1}{H_0}\int_0^a \frac{da'}{a'\sqrt{\Omega_{m,0}/a'^3 + \Omega_{\Lambda,0}}}=\frac{1}{H_0}\int_0^a\frac{\sqrt{a'}da'}{\sqrt{\Omega_{m,0}+\Omega_{\Lambda,0}a'^3}}.$$
This can be solved analytically by taking $x=a^{3/2}$, giving $$t = \frac{2}{3H_0\sqrt{1-\Omega_{m,0}}}\arcsin\left(\sqrt{\frac{1-\Omega_{m,0}}{\Omega_{m,0}}}a^{3/2}\right).$$
To get the age of the universe, insert $a=1$ and the most up-to-date value of $
\Omega_{m,0}$.
One comment: inflation is not relevant here because we are starting the integration after inflation ends.  Inflation could have lasted for an arbitrarily long time, and the standard hot big bang is effectively taken to correspond to the end of inflation in our causal patch.
A: This is not a full answer, but I think it will help if you separate out inflation from the rest of the picture. The age of the universe can be estimated in the first instance as the time elapsed since some very early epoch where the temperature was low enough that the Standard Model of particle physics applies to reasonable approximation. This means you can leave out the very early processes which remain very unknown in any case.
With this approach one then applies general relativity and standard physics to construct a model of the main components of the universe, and one can estimate the evolution with reasonable confidence; see a textbook for details. This is how the age is estimated.
A: What's called the "age of the universe" would more accurately be called the age of the most recent epoch in the universe's history. That epoch began with the end of inflation, or with the end of whatever noninflationary process created the incredibly uniform expanding quark-gluon plasma that eventually clumped into stars, planets and us. We don't know the age of everything that exists, and probably never will, but we know the age of the expanding cosmos that we live in.
The best current model of that cosmos (the simplest model that fits all the data) is called Lambda-CDM (ΛCDM). ΛCDM has a singularity of infinite density called the "big bang singularity", and times measured from that singularity are called times "after the big bang" (ABB). Our current location in spacetime is about 13.8 billion years ABB, and that's called the "age of the universe".
But no one believes that the singularity in ΛCDM has any physical significance. To get a correct model of the universe, you need to remove the singularity and some short time interval after it from the model, and graft some other model onto it.
The most popular candidates for models of the previous epoch are based on cosmic inflation. They fit all of the available data, but the amount of still-visible information about the universe of 13.8 billion years ago is small enough that we can't draw any definite conclusions. That's where things stand today.
(There's a disturbing possibility that that's where things will stand forever, because according to ΛCDM and semiclassical quantum mechanics, the total amount of information that we will ever be able to collect about the early universe is finite, and it may not be large enough to pin down the right model. Even the information that enabled us to pin down the parameters of ΛCDM will be inaccessible to far-future civilizations, according to ΛCDM.)
By this terminology, inflation ends a tiny fraction of a second ABB, and this has given rise to a common misconception that inflation only lasts a tiny fraction of a second. Actually, depending on the model, the inflationary epoch can last for essentially any amount of time, and whatever preceded it could have lasted for any amount of time, if time even has meaning at that point. None of this is counted in ABB times.
ABB times do include a fraction of a second that is literally meaningless, since it's from the early part of ΛCDM that we remove as unrealistic, but we can't calculate any ABB time to nearly that accuracy, so it doesn't really matter.
A: The rough idea is that under the assumptions contained in the cosmological principle, the application of Einstein's equations leads us to the equation
$$d(t) = a(t) \chi$$
where $d(t)$ is called the proper distance and $\chi$ is called the comoving distance between two points in space.  $a(t)$ is the time-dependent scale factor, which is by convention set to $1$ at the present cosmological time.
The rate at which this proper distance increases (assuming no change in the comoving distance $\chi$) is then
$$d'(t) = a'(t) \chi$$
The observation that distant galaxies are receding, and that the recession velocity is proportional to the observed proper distance with proportionality constant $H_0$ (Hubble's constant) tells us that $a'(0) = H_0$.  If we assume that $a'(t)$ is constant, then
$$d(t) = (1+H_0 t) \chi$$
and that when $t=-\frac{1}{H_0}$, the proper distance between any two points in space would be zero, i.e. the scale factor would vanish.  This leads us to a naive estimate of the age of the universe, $T = \frac{1}{H_0} \approx 14$ billion years.

Of course, there is no particular reason to think that $a'(t)$ should be constant.  The dynamics of the scale factor are determined by the distribution of matter and radiation in the universe, and on its overall spatial curvature.  For example, if we assume that the universe is spatially flat and consists of dust and nothing else, then we find that
$$a(t) = (1+\frac{3}{2}H_0 t)^{2/3}$$
where $H_0$ is the current-day Hubble constant and $t$ is again measured from the present.  In such a universe, the scale factor would vanish when $t = -\frac{2}{3}\frac{1}{H_0}$, so the age of the universe would be 2/3 the naive estimate. More generally, if we model the contents of the universe as a fluid having a density/pressure equation of state $p = wc^2\rho$ for some number $w$, then we would find
$$a(t) = \left(1 + \frac{3(w+1)}{2}H_0 t\right)^\frac{2}{3(w+1)}$$
leading to respective ages
$$T = \frac{2}{3(w+1)}\frac{1}{H_0}$$

The $\Lambda_{CDM}$ model assumes that the universe can be appropriately modeled as a non-interacting combination of dust and cold dark matter $(w=0)$, electromagnetic radiation $(w=1/3)$, and dark energy, and having an overall spatial curvature $k$.  The Friedmann equation can be put in the form
$$\frac{\dot a}{a} = \sqrt{(\Omega_{c}+\Omega_b)a^{-3} + \Omega_{EM}a^{-4} + \Omega_ka^{-2} + \Omega_\Lambda a^{-3(1+w)}}$$
where $w$ is the equation of state parameter for the dark energy/cosmological constant and the $\Omega$'s are parameters which encapsulate the relative contributions of cold dark matter, baryonic (normal) matter, electromagnetic radiation, spatial curvature, and dark matter, respectively.  By definition, $\sum_i \Omega_i = 1$. Note that if we set all the $\Omega$'s to zero except for $\Omega_b=1$, we recover the solution for dust from before.
The electromagnetic contribution is small in the present day, so neglecting it is reasonable as long as $\Omega_{EM}a^{-4}\ll \Omega_ma^{-3} \implies a\gg \Omega_{EM}/\Omega_m$.  If additionally the universe is spatially flat so $\Omega_k=0$ (as per the Planck measurements) and $w=-1$ (consistent with dark energy being attributable to a cosmological constant), then this is reduced to
$$\frac{\dot a}{a} = \sqrt{(\Omega_{c}+\Omega_{b})a^{-3}+\Omega_\Lambda}$$
This can be solved analytically to yield
$$a(t) = \left(\frac{\Omega_c+\Omega_b}{\Omega_\Lambda}\right)^{1/3} \sinh^{2/3}\left(\frac{t}{T}\right)$$
where $T \equiv \frac{2}{3H_0\sqrt{\Omega_\Lambda}}$ and now $t$ is measured from the beginning of the universe.  Setting this equal to 1 allows us to solve for the time to the present day.
The Planck satellite measured $\Omega_b=0.0486,\Omega_c=0.2589,$ and $\Omega_\Lambda=0.6911$ (they don't add up to 1 because we've neglected $\Omega_{EM}$ and $\Omega_k$).  The result is an age of the universe
$$t =T\sinh^{-1}\left(\left[\frac{\Omega_\Lambda}{\Omega_c+\Omega_b}\right]^{1/2}\right) = \frac{2}{3H_0\sqrt{\Omega_\Lambda}}(1.194) \approx 13.84\text{ billion years}$$
The actual calculation is more careful, but this is the general idea.

A: $\Lambda CDM$'s claim of the Universe being 13.8B years old should be taken with a grain of salt.
The Universe (as depicted by $\Lambda CDM$) has hypothetically underwent inflation only for a fraction of a second shortly after the Bang, negligible when compared with its current age. Therefore, you shouldn't be hung up on inflation when it comes to guessing its age, albeit inflation has allegedly left some permanent mark on $\Lambda CDM$, such as being nearly flat ($\Omega_k=0$) as you noticed.
That being said, you should be alarmed by $\Lambda CDM$'s inconsistent stories about the Universe's late-time history at low redshift (long after inflation was gone), evidenced by the contradicting numbers of Hubble constant ($H_0$) measurements (the "Hubble tension" is all over the place), which could have real implications on the uncertainty of dark energy density ($\Omega_\Lambda$) and the true age of the Universe.
The standard cosmology model $\Lambda CDM$ has been known as the "concordance model". Given the "Hubble tension" and other inconsistencies (check out the controversy surrounding $\sigma_8$), "discordance model" might be a more suitable name for $\Lambda CDM$.
Hence $\Lambda CDM$'s calculation of the Universe being 13.8B years young should not be taken too seriously, at least one should put a much higher error margin on the number.
A: I'm not too interested in providing an answer from the cosmological point of view. It is clear that the age of the universe derived in that way is model-dependent. The age thus obtained depends on certain assumptions (e.g. that the dark energy density remains constant).
I will just add a couple of additional age determination methods that rely on alternative "non-cosmological" methods, that provide at least some verification that the answers from cosmology are in the right ballpark.

*

*Stellar evolution calculations rely on very solid, non-controversial physics. These predict that stars spend most of their lives burning hydrogen in their cores before evolving away from the main sequence. By comparing the predictions of these models with the luminosity, temperature, surface gravity and chemical composition of stars, we can estimate their age; particularly those that have begun their evolution away from the main sequence. If we look around the solar neighborhood, we see a variety of stars with different ages. The oldest stars appear to be the ones with the most metal-poor composition and they have ages of around 12-13 billion years. The universe must be at least this old.


*When stars "die" the lowest mass objects will end their lives as white dwarfs. These cinders of carbon and oxygen are supported by electron degeneracy, release no internal energy and cool radiatively. The coolest, lowest luminosity white dwarfs we can see will be the ones that have been cooling longest. The cooling physics is relatively simple - if the lowest luminosity white dwarfs have temperatures of around 3000K and are a millionth of a solar luminosity, then one works out a cooling age of around 11-12 billion years. The progenitors of these objects will have had their own short lifetimes, so estimate #2 is consistent with estimate #1 and provides a minimum age for the universe.
At the moment, our observations of high redshift galaxies suggest that galaxy formation and the formation of the first stars occured relatively quickly after the universe was very small and hot. The first galaxies and stars were assembled at redshifts of at least 6. This in turn suggests that the pre-stellar "dark ages" were comparatively short. The age of the universe at a redshift of 6 is much less dependent on cosmological assumptions and parameters, but in any case is a small fraction ($<10$%) of the age of the universe now (e.g. in the concordance LCDM model, $z=6$ is only 0.94 billion years post big-bang, but this only changes to 0.86 billion if there is no dark energy) . Thus we can be reasonably sure that the age of the universe (or at least the time since the universe was very small and very hot) is perhaps only a billion or less years older than the oldest stars we can see.
You can mess about with cosmological parameters (and their time dependence) a bit to alter these results. But you can't make the universe much younger without conflicting with the evidence from old stars and white dwarfs. You also can't make it much older whilst simultaneously accounting for the lack of older stars in our own and other galaxies, the cosmic microwave background (and its temperature), the abundance of helium and deuterium in the universe or the rate of evolution of cosmic structure. I think most scientists would agree that the $\pm 21$ million year error bar implicitly assumes the LCDM model is correct (age calculated as per some of the other answers). The true error bar could be a factor of 10 higher, given the current debate about differences in $H_0$ derived from the CMB as opposed to the local universe, but probably not a factor of 100. Even a naive extrapolation back in time of the currently observed expansion rate gives an age of around 14 billion years.
It is also possible to avoid a singular big bang in the past altogether, by having the current phase of our universe's expansion beginning at the end of a previous contraction phase (a.k.a. the big bounce). In which case, the "real" age of the universe can be anything you like, with 13.8 billion years just being the time since the latest bounce.
