The Shape and Fate of the Universe Assumptions in cosmology Olber’s paradox Expansion of the Universe Curvature of the Universe Fate of the Universe

Assumptions in Cosmology Copernican principle: –We do not occupy a special place. –There are no special places. –The universe is homogeneous if viewed at sufficently large scales. –The laws of physics are the same everywhere.

Implications of the Copernican Principle The average density of matter and energy is the same throughout the Universe. The same Hubble expansion law is seen for all observers anywhere in the Universe. The curvature of the Universe is the same everywhere.

How can we test the Copernican principle? Does the Universe look the same in all directions? (Isotropy) Are the spectral lines from atoms the same in distant galaxies? Do the same laws of gravity apply in other galaxies?

Why is the night sky dark? (Olber’s Paradox 1826) Or what is the temperature of the sky? Assume universe is static, infinite, and full of stars like the Sun. Then every ray extending out from the Earth will eventually intersect a star. So, the brightness of the sky at that point will be determined by the surface brightness of that star. But surface brightness is independent of distance, so the whole sky should be as bright as the Sun.

Why is the night sky dark? (Olber’s Paradox 1826) One of the assumptions (static, infinite, and full of stars like the Sun) must be incorrect. Thus, to have a dark night sky, the Universe must be some combination of dynamic finite in time finite in extent

Hubble expansion v = H 0 d

Expansion of the Universe

Hubble expansion v = H 0 d Time = distance/velocity = d/H 0 d = 1/H 0 = 1/(71 km/s/Mpc) = 13.8 Gyr We can only see out to distances ~ c/H 0

Two dimensional geometry Only two directions: up/down and left/right : north/south and east/west All motion of particles, light confined to two dimensions Examples: black board, piece of paper, surface of sphere, surface of donut, surface of saddle

Geometry How are the diameter and circumference of a circle related? What is the sum of all of the angles in a triangle?

Geometry in flat space Circumference = 2   radius  = … 2  = 6.28… The sum of the angles in a triangle is 180º

Geometry in positively curved space Circumference < 2   radius The sum of the angles in a triangle > 180º

Geometry in negatively curved space Circumference > 2   radius The sum of the angles in a triangle < 180º

Curvature of the Universe The curvature of the Universe is determined by: the density of matter and energy –higher density produces positive curvature the expansion of the Universe –more rapid expansion produces negative curvature

Fate of the Universe

Critical Density At what density is the Universe balanced between expanding forever versus contracting? Consider a small mass at the surface of a sphere:

Curvature of the Universe The curvature of the Universe is determined by the density parameter  0  0 < 1  negative curvature  0 > 1  positive curvature

Fate of the Universe

Age of the Universe

Imagine that the circumference of a circle with a radius of 1000 Mpc was measured to be 5500 Mpc. This would imply that 1.The Universe is flat. 2.The Universe is positively curved and will eventually contract in a big crunch. 3.The Universe is negatively curved and will expand forever.