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?

We don't need knowledge of current universe size to determine it's age. Basically you just need an expansion rate Hubble constant. If you know the rate,- inverse of that would be expansion time. (Well it would be just approximate value, because Hubble constant changes with time) — Agnius Vasiliauskas, Aug 19 '22 at 15:17

seVenVo1d · Answer 1 · 2020-05-05T23:09:15.007

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

edited May 05 '20 at 23:09

answered May 05 '20 at 17:39

seVenVo1d

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Okay, thanks, I assume you get this equation from some derivation with the Friedman equations. But I still have a few questions: is the process I described in my question, a valid way to calculate the age of the universe? If it is does this calculation needs to know the current size of the universe? And how would such calculation would look like? If it isn't, then why does it seem to me, like every YouTube channel describe this way, as the way we measure the age of the universe. – Ofek Tevet May 05 '20 at 21:01
You said "From my research, the answer is something like:" Could you share the link = – seVenVo1d May 05 '20 at 22:31
I edited my answer – seVenVo1d May 05 '20 at 23:04
It is mostly the videos of the YouTube channels of scisowspace and PBS space time. https://youtu.be/tCn96DbBnB4 scishow space. https://youtu.be/Y6Vhh70Lw9w PBS space time. – Ofek Tevet May 06 '20 at 07:49
@OfekTevet I just looked briefly but as far as I can see they are only talking about the measurement of the $H_0$ which as I showed its a parameter that we put in to calculate the age of the universe. If you can give a timestamp maybe I can help you better. – seVenVo1d May 06 '20 at 08:18
@OfekTevet There are couple of $H_0$ measurements which do not use the distance ladder. As I said earlier before in late universe measurements we can use couple of methods, which some of them do not use the distance ladder at all. Such as H0LICOW (https://arxiv.org/abs/1907.04869). Another thing is there are also early universe methods that measures the $H_0$ such as CMB , BAO+BBN+DES without using the distance ladder technique. – seVenVo1d May 06 '20 at 08:35
@OfekTevet So the point is age of the universe depends on the $H_0$. And the measurements of the $H_0$ depends partially on the distance ladder, but there are many indepedent measurement techniques. – seVenVo1d May 06 '20 at 08:35
@OfekTevet In order to calculate the size of the observable universe you need to know the density parameters. But the same paramteres also gives you the age of the universe. Then, why would you go back to calculate the age of the universe from its distance. – seVenVo1d May 06 '20 at 08:41
@OfekTevet Also PBS is a good site but it does not give you the whole picture. Because its pop science. – seVenVo1d May 06 '20 at 08:42
Thanks for all the effort you put into the answer. Just two more things. 1. So just to be sure, the way we calculate the age of the universe, is by measuring $H_0$ and other parameters and plugging them into the Friedman equations right? – Ofek Tevet May 16 '20 at 00:32
Suppose that we somehow knew the current size of the universe and it's expansion rate at this moment in time, and in every moment before this one. Could we then calculate the age of the universe by some kind of backward numerical integration and figuring out for which t, it's size would be equal to zero? And could we do that without knowing the size of the universe by working with ratios of universe's size at t divided by the universe's current size?

Ofek Tevet

May 16 '20 at 00:38

Yes but there also other methods to estimate them. Such as looking for the age of the galaxies or some stars.. – seVenVo1d May 18 '20 at 12:38

score 1 · Answer 2 · answered Aug 19 '22 at 15:08

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.

Determining the size of the universe to calculate its age

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