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?
Best Answer
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.