# The Cosmological Background Once upon a time, we were all kids, happily running around under the summer sun, without a care in the world, until social media entered our lives. While I might not be here to lecture you about the cons (and pros!) of social media, (olber's paradox) --> if there are so many stars, why is the night sky so dark? --> calculating starlight within an infinite universe: Where: $n$ = average density of stars $L$ = average stellar luminosity # Olbers' Paradox - An Initiation Ceremony ## Flux and an Infinite Universe Let's take a look at one of the thought experiments that had started it all. The **Olbers' Paradox**, named after amateur astronomer Heinrich Wilhelm Olbers, attempted to challenge the infinite-universe theory prevalent in the 1800s, asking: > "If the universe was truly infinite, should the night sky be filled with light, not be dark?" > \- Heinrich Olbers Now it's time for the fun bit - proving it by virtue of calculus! >[!Warning]- The Math behind a Paradox >We will take the volume of the universe to be infinite here. This means that we can consider everything as infinite. >Remember our flux equation from the Brightness article. If not, you can navigate back to it at the bottom of this page! >$F(r) = \frac{L}{4\pi r^2}$ >The inverse square law dictates that the luminosity depreciates as a function of its surface area, hence the relationship. We can imagine the Earth as being in a little shell surrounded by a bubble of stars. This is in line with the "eternal universe" paradox created by Olbers, where stars extend outwards from the Earth to infinitely. The diagram should help, hopefully! >![[Pasted image 20230710132507.png]] >We can therefore create a differential equation to show the change in flux over the area, allowing the total power on the Earth exerted by stars at an arbitrary distance $r$ away only be $J$. This creates the expression: >$dJ = \frac{Ln}{4\pi r^2}n \cdot r^2 dr$ >Where $n$ is the average stellar density across the universe. Simplify the expression: >$dJ = (\frac{Ln}{4\pi }) dr$ >We can therefore integrate by $dr$ to get: >$\int \, dJ = \int_{0}^\infty \frac{Ln}{4\pi} \, dr $ >$\frac{Ln}{4\pi}$ is a constant. This means we can take it as a constant. The limit is 0 - infinity as these stars subtend outwards infinitely in the paradox - ranging from our Earth to the radius to a star $r$. >$J = [\frac{Lnr}{4\pi} + c]^\infty_{0}$ >If we evaluate the integrals through substituting the values to the right we get an infinite value for the power $J$! This paradox shows that the power from the universe should be **infinite** --> this is the paradoxical nature of Olbers' Paradox. ## Understanding its Limitations However, the solution Olbers' paradox provides us has its own limitations we have virtually no way of proving. Perhaps we live in a universe small enough that we are fooled into believing that the number density of stars is equal everywhere, or one which is not isotropic - where only patches of universe are locally equal with equal stellar densities, meaning Olbers' Paradox would not completely stand. Other reasons for this includes: - Time, a variable which the paradox is dependent on to be infinite, may not be infinite on large scales - The observable universe may only take up a small portion of the true universe. This means that there may be hidden homogeneities within the universe we cannot see. $d = ct$ Where $c$ is the speed of light and $t$ is the age of the universe. # The Scale Factor and Expansion of the Universe A life of rigour is often one considered not worth living. Not to cosmologists, however, as they have created variables, differential equations and the like to generalise even the most trivial of things. The scale factor is one such thing. The scale factor of the Universe's expansion is given by the variable $a(t)$. We have to model the change in this as a function of time, allowing us to find the per unit change in the Universe's size. Since this is a scale factor, the scale factor doesn't actually have a unit - it's just given by itself, a scalar value. ![[Pasted image 20230710134952.png]] The above diagram illustrates the expansion of the universe from a time $t_1$ to a time $t_{2}$. We can actually generalise this using the relationship shown: $\frac{d a(t)}{dt} = \dot{a}$ Depending on the state of the universe, In an expanding universe, $\dot{a}$ > 0. In a static universe, $\dot{a}$ = 0. In a shrinking universe, $\dot{a}$ < 0. This is the rate of change of the expansion of the universe, so therefore these numbers are. # What's Next? For the flux calculations, go here: [["Brightness" - Luminosity and Magnitude]] To navigate back to anything Astro, go here: [[Everything Astronomy]]