1. Darkness at Night (Olbers’ Paradox)
Imagine you are a pre-20th century astronomer
How many stars would you expect to see?
Assumptions:
Speed of light is infinite
Universe is infinitely old
Universe is non-evolving
Universe is filled with a uniform density of stars, all of which are identical to the sun

2. How much flux does each thin spherical shell contribute to the total brightness of the sky?
# density of stars = n0
radius of shell = r
thickness of shell = dr

3. A “forest” of finite-sized stars
To figure out how much flux you should observe, consider what the flux would be if one star alone filled your entire field of view (i.e., you’re positioned at the photosphere of a star)

4. How Not to Solve Olbers’ Paradox
Embed the stars in gas (it will heat up and glow)
Embed the stars in dust (you simply move the problem out of the visible part of the spectrum to the IR instead)
Replace the stars with galaxies and call on redshift (again, you simply move the problem out of the visible part of the spectrum)

5. Resolution of Olbers’ Paradox in Static Universe
Can’t see the most distant stars because the light has not had time to reach us (universe is not infinitely old).
The edge of the observable universe is the “horizon”. The distance to the horizon changes constantly.

6. The Cosmological Principle
On a large enough scale, the universe is
both isotropic and homogeneous
ISOTROPY: There is no preferred direction in space. (All directions are alike.)
HOMOGENEITY: One randomly-chosen large volume of the universe will have the same physical properties (and identical physical laws) as another randomly-chosen large volume of the universe. (All places are alike.)

7. 2-Dimensional Examples of Isotropy and Homogeneity
Surface of plain white “cue” ball used for playing pool (billiards)
Infinite forest of identical trees

8. Isotropic Forest

9. Anisotropic Forest (trails = “preferred direction”)

10. Four 2-Dimensional Universes
Which are isotropic on a large scale (same in all directions)?
Which are anisotropic on a large scale (have preferred directions)?
Which are homogeneous on a large scale (all places are alike)?
Which are inhomogeneous on a large scale?
Which satisfy the Cosmological Principle?

12. Obviously isotropic and homogeneous
Satisfies the Cosmological Principle

14. Does not satisfy Cosmological Principle
Homogeneous on a large scale (the pattern repeats over and over), but is obviously anisotropic
Does not satisfy Cosmological Principle

16. Does not satisfy the Cosmological Principle
Homogeneous on a large scale, but is actually anisotropic (look at the diagonals!)
Does not satisfy the Cosmological Principle

18. On a large scale, this is both homogeneous and isotropic
How large is “Large Enough”? > 500 Mpc/h (give or take)
On a large scale, this is both homogeneous and isotropic
Satisfies the Cosmological Principle
This is actually a computer simulation of how galaxies may have formed in the universe after the Big Bang. It matches the real, observed universe remarkably well.
Yellow = high galaxy number density, black = low galaxy number density