Galaxies and Cosmology

State Olber’s Paradox and how it is resolved

• If the universe is infinite, why do we have a dark night sky?

• resolved by finite age of the universe

State the Cosmological Principle

The Universe is both isotropic and homogenous

What are the meanings of isotropic and homogenous in the cosmological principle?

• Isotropic - no preferred direction

• Homogenous - no preferred location

At what scale does the cosmological principle hold?

scales > 100Mpc

Give the equation for redshift

z=\frac{\lambda_{obs}-\lambda_{em}}{\lambda_{obs}}=\frac{v}{c}

What is Hubble’s Law?

v=H_0r

Give the equations for how the distance and velocities between two galaxies change with time

\underline{r}_{12}\left(t\right)=a\left(t\right)^{}\underline{r}_{12}\left(t_0\right)

\underline{v}_{12}=\frac{1}{a}\frac{da}{dt}\underline{r}_{12}\left(t\right)

Define and give meaning to the Hubble Distance and Hubble time

Hubble time t_{h}=H_0^{-1} is an estimate for the age of the universe

Hubble distance d_{h}=\frac{c}{H_0}=ct_{h}is an estimate of the size of the observable universe

Give the equation of redshift in terms of scale factor

1+z=\frac{a_{obs}}{a_{em}}

What is the Cosmic Microwave Background and what is its temperature?

• Background radiation in the universe with near perfect blackbody spectrum

• Temperature of ~2.73K

What is the Cosmic Microwave Background evidence of?

Universe was initially hot and dense and has been expanding and cooling ever since due to scale factor expansion

Give the relation between the temperature of the CMB and the scale factor a(t)

T\left(t\right)\alpha \frac{1}{a\left(t\right)}

What speed do galaxies have beyond the Hubble distance and why isn’t this an issue?

• v > c

• Doesn’t violate GR or SR due to expansion of space

Sum up the equivalence principle and what it means for light and spacetime

• Standing on Earth’s surface is identical to accelerating in space at g

• same thing must apply to light

• mass curves spacetime 

Give details of Euclidean geometry of the universe

• angles of triangle add to pi

• must be infinite in extent to satisfy CP

Give details of spherical geometry of the universe

angles of triangle add to \pi+\frac{A}{R^2}

A is area of triangle, R is radius of sphere

Give details of hyperbolic geometry of the universe

angles of triangle add to \pi-\frac{A}{R^2} 

A is area of triangle, R is radius of saddle point

Give 3 key points on curvature

• R is radius of curvature, R approaches infinity as curvature does

• sign and curvature of R are indirectly observable

• curvature changes apparent size of objects

Define luminosity distance and when it is used

d_{L}=\left(\frac{L}{4\pi f}\right)^{\frac12} 

used for small distances

Give the equation for proper distance at time = t₀ and t_e

d_{p}\left(t_0\right)=c\int_{t_{e}}^{t_0}\!\frac{1}{a\left(t\right)}\,dt

d_{p}\left(t_{e}\right)=\frac{c}{1+z}\int_{t_{e}}^{t_0}\!\frac{1}{a\left(t\right)}\,dt 

Give the equation for horizon distance at t₀

d_{hor}\left(t_0\right)c\int_0^{t_0}\!\frac{1}{a\left(t\right)}\,dt