# Question #860db

Feb 1, 2016

This is going to be a long explanation. Buckle up.

#### Explanation:

In order to discuss this we have to declare first and foremost what we mean by the size of the universe. There's two possible options here. The spread of the matter in the universe, and the size of the empty space the matter can move into.

Cosmology gives us a couple of answers to these questions thanks to the Hubble constant, the Einstein Field Equations (EFEs), the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, the geometric curvature of the universe, and the Cosmological constant.

General relativity is a beast of a topic, so suffice it to say Einstein was a mega genius, and he gave us the Einstein Field Equations that tell us how matter and energy bend space time, and how the bends in space time shape the trajectories of matter and energy. It's very complex stuff, and absolutely worthwhile to study when you have all the necessary math and physics bases covered.

So this guy Friedmann makes an assumption about how the universe is spread out in a homogeneous and isotropic (even and symmetric) manner, uses that to construct a metric (a mathematical object for distances) and uses it to solve the EFEs. This gives us the Friedmann equations. We'll get back to that, but first some facts.

Fact 1. All elements will absorb specific wavelengths of light based on the specific atom in question. These are called absorption spectra. The elements in stars absorb the light they emit in these wavelengths creating dark bands in the spectrum of emitted light. In this way we can tell what the stars are made of, by comparing the dark bands to the known absorption spectra of the known elements.

Fact 2. When the source of a wave is moving, the length of the wave is compressed or stretched depending on it's relative motion to the emitter. This is the Doppler effect. It works for sound waves, as demonstrated by the sound of an approaching car, and also for light waves.

Fact 3. Stars of a certain type, as well as some celestial events occur with a characteristic brightness, or luminance. We call these standard candles. By looking for the emission spectra of these standard candles with sophisticated telescopes, we can find out how bright they appear to be, as opposed to how bright they should be. This tells us how far away they are.

So if you look at the emission spectra of the stars in galaxies outside our own, the absorption lines are either shifted left or right of where they should be. So the Doppler effect is at work, and that means the galaxies must be moving relative to each other. We can use the magnitude of the Doppler effect to tell how fast the galaxies are moving relative to each other. If you divide this velocity by the distance to that galaxy (from the standard candles) you get the Hubble constant. Every time. Every galaxy. The same constant. Think about that.

Since we have a constant that relates the distance of the galaxies to their speed, we can take a couple of them and fix them onto a grid. Now when they move, we say that the grid moves with them. These are called comoving coordinates. Since the galaxies always stay at a fixed place on the grid, we can consider the distance between them on the grid to stay the same up to some scale factor. This scale factor is time dependent and reflects how long the galaxies have been moving relative to each other. we call it $a$.

We can now express the Hubble constant as

$\frac{a '}{a}$ where $a '$ is the time rate of change of the scale factor.

So now we need to talk about the nature of the empty space in the universe. Long story short, there is nothing that says it can't be curved. I know what you're thinking, how can empty space be curved? Well it can. Thank Einstein for the wackiness that comes out of General Relativity. So we represent the three options for curvature with a constant called $k$.

$k = \left\{- 1 , 0 , 1\right\}$

So we have positive curvature (like a sphere), no curvature (flat space), and negative curvature (hard to conceptualize, but think saddle point). If space has a constant positive curvature than it must eventually loop around back onto itself and the universe can't be infinite. If it's flat it can be infinite, and negative curvature can give weird results. Infinite in some directions and not others.

So now we have the Friedmann equations.

${\left(\frac{a '}{a}\right)}^{2} = \left(\frac{8 \pi G}{3}\right) r - \left(\frac{k {c}^{2}}{{a}^{2}}\right) + \frac{L {c}^{2}}{a} ^ 2$

$\left(\frac{a ' '}{a}\right) = - \left(\frac{4 \pi G}{3}\right) r - \left(\frac{4 \pi G}{c} ^ 2\right) p + L {c}^{2} / 3$

Where $\frac{a '}{a}$ is the Hubble constant, $G$ is the universal gravitational constant, $r$ is the density of the universe, $p$ is the pressure, $k$ is universal curvature constant, $c$ is the speed of light, and $L$ is the cosmological constant

This gives a set of coupled ordinary differential equations, and if you know some dynamical systems you can use a phase plane diagram to really see what the various consequences for different values are.

Solving these equations breaks down into division into cases for this conversation though. The outcomes of the various values of $k$ and simlar cases for $L$ (positive, zero, or negative), gives us the possible outcomes of the universe as a whole, and whether or not it's entire vastness is even a finite thing to be measured. Some expand to infinity, some expand only to contract again and die in a big crunch. Some expand until they run out of energy and experience heat death.

Interestingly we don't have enough data yet as a species to tell what the outcome of our universe is. We don't even know if it's curved yet!

So, besides the question of whether the universe can even be measured, there's the question of how we would do it. You can use the Hubble constant to show the age of the universe since the big bang. I think it's 13.7 billion years or so. The problem with this is, there could be things in the universe that are more than 13.7 billion light years away, and their light hasn't even reached us yet.

So if the light leaving those objects hasn't even had time to get here since the dawn of the universe we have no way of measuring anything about them. Therefore we have no real way of ever telling how far away the farthest object in the universe is because of the same limitation. These are just some of the problems of living in a universe with a finite speed of light.

The real problem is that if we cannot travel faster than light, then we can never get far enough away in a reasonable amount of time to find the answers to these questions either.

So in conclusion, no it has not been measured, no it will not be measured, and no it probably cannot be measured.