Determining the size of the universe to calculate its age

Question

Recently, I started to ponder again with the question of how do we know the age of the universe.

From my research, the answer is something like: "We measure the expansion rate directly with type ia supernovea, and extrapolate it from CMB data, and with that, we calculate when the universe would have size zero according to the expansion history".

But as far as I understand this explantion, we also need to know the current size of the universe independently of the current age of the universe (because we are trying to calculate its age, and if we need the age for the size, we have a circular dependency).

So my question is how do we know the size of the universe? Is it because we know the universe is infinite so we know it's size (although as far as I understand, this would lead to the question of how do we know the distance to the CMB, because we use that distance to prove that the universe is flat, and therfore infinite)? Is it somehow with the friedmann equations and additional parameters we measure with the CMB? Does it has something to do with the size of the visible universe (although as far as I understand, we know it thanks to the combanition of the age of the universe and its expansion rate)? Or do I misunderstand this explanation?

Answer 1

You don't need to know the size of the universe to calculate its age.

The age of the universe or cosmic time is a function of the current density parameters and the Hubble constant,

$$t_{uni}(H_0, \Omega_{m,0},\Omega_{\Lambda,0},\Omega_{r,0})=\frac{1}{H_0}\int_{0}^{\infty}\frac{dz}{(1+z)\sqrt{\Omega_{r,0}(1+z)^4+\Omega_{m,0}(1+z)^3+\Omega_{\Lambda,0}+\Omega_{\kappa}(1+z)^2}}$$

We measure the expansion rate directly with type in supernovae, and extrapolate it from CMB data, and with that, we calculate when the universe would have size zero according to the expansion history.

It's partly true. From the CMB measurements, we can calculate many cosmological parameters. As you can see from the above equation the most important parameters are $H_0$ and the density parameters.

For instance, $H_0$ measured by using two methods, the early universe (by using CMB and BAO) and the late universe (by using type Ia Supernova, TRGB etc,). Measurements showed that there is approximately 6 sigma difference between these two methods, which is called the Hubble Tension

But as far as I understand this explanation, we also need to know the current size of the universe independently of the current age of the universe. So my question is how do we know the size of the universe?

We cannot know the size of the universe, however, we can calculate the size of the observable universe (see particle horizon)

$$\eta(H_0, \Omega_{m,0}, \Omega_{\Lambda,0}, \Omega_{r,0}) = \frac{1}{H_0}\int_{0}^{\infty}\frac{dz}{\sqrt{\Omega_{r,0}(1+z)^4 + \Omega_{m,0}(1+z)^3 + \Omega_{\Lambda,0} + \Omega_{\kappa}(1+z)^2}}$$

As you can see they have the same parameters as the cosmic time.

Is it somehow with the Friedmann equations and additional parameters we measure with the CMB?

Yes by obtaining the cosmological parameters from various measurements we can calculate the age of the universe or the particle horizon.

Can we obtain the cosmic time from the particle horizon?

Well, I am not sure but I do not think it's possible. Expansion of the universe is governed by the Friedmann Equations and Friedmann Equation depends on the density parameters. Without knowing how the universe evolves and by just looking at a distance we cannot calculate its age. Because we do not know "how fast it evolved"

For more information about the measurements I found this wiki page

If you want to know where these equations come from I made the derivation in another question

Answer 2

If you were to measure the relative postions and velocities of two bodies that were part of an explosion in the past then there would be no need to know the extent of the fireball in order to work out when it happened. All you do is calculate how long ago the two bodies had zero separation.

You might initially assume the bodies move inertially - i.e. with no accelerating forces and that would give you an answer. But if the dynamics of the bodies were affected by e.g. friction or gravity, then you would need to use your knowledge of the physics to solve the equations of motion using the current positions and velocities of the bodies as boundary conditions.

This latter problem is more akin to getting an estimate for the age of the universe. We calculate at what time in the past the separation of two bodies would be zero. The separation of all pairs of bodies will be zero at this epoch - there is no need to know the "size of the universe" (which could be infinite as you say). For a precise estimate then one needs to take account of the non-uniform dynamics of the universe, which is done through measurements of the cosmological parameters and the Friedmann equations.

Note that this analogy only gets you so far. There is no central explosion point where the Big Bang happened - it happened everywhere. There is also no "distance to the CMB" - the CMB is measured here and now. There is a distance to the surface of last scattering, which is where the CMB we currently detect originated. However, this location (but not the distance) is observer-dependent. If we were to go to a different part of universe (at the same cosmic epoch), the distance to this last scattering surface would be the same but of course its location would then have to be different. The CMB constrains the flatness of the universe via the angular spectrum of its temperature inhomogeneities, not through an estimate of the distance to the last scttering surface.