Cosmology

Olbers Paradox

Distances

Many different methods for measuring distances

Radar

How long for light to travel.

Example: Lunar Laser Ranging RetroReflector (LRRR) on the moon. Measures distance to solar system objects.

Parallax

On earth: about 100 parsecs max distance.

HST Max distance 3,000 parsecs (10,000 ly)

Gaia (10,000 parsecs with 20% error)

Milky way is at least 30,000 parsecs in diameter.

(Andromda is about 780,000 parsecs away)

Standard Candles

The flux (measured brightness) of an object depends on two things: how far away it is and its luminosity: $$\begin{equation} F = \frac{L}{4 \pi r^2} \end{equation}$$ where $L$ is the luminosity and $r$ is the distance between source and observer.

If we have a known value for $L$, then we can calculate $r$ based on the measured flux.

Some well understood physical processes can help lock in a value for $L$

Cepheids - Variable stars

Period of brightness fluctuations is related to apparent magnitude, $m$.

If we know the distance to some nearby ones (from parallax), we can calibrate the relationship.

Max: 30,000,000 parsecs (~100 million ly)

Useful for objects in this galaxy, and a few nearby ones.

Type 1A Supernova

A white dwarf in a binary system can gain mass from the companion until it reaches a limit, producing an explosion. This is a supernova. The luminosity of the resulting explosion can be predicted.

Max: 1 billion ly

Redshift

Earlier 20th century observations of galactic spectra indicated red-shifted spectral lines for most. Vesto Slipher did the first measurements of redshift in galactic spectra. This suggested they were moving away from us.

Hubble's Velocity vs. Distance curve

Hubble Parameter

The Hubble parameter is just the slope of the velocity-position data:

$$\begin{equation} v = H_0 d \end{equation}$$

The value of $H$ might depend on time, so we can call the Hubble constant the value it has today: $H_0 \equiv H(t_0)$

The current value for $H_0$ is around 70.

Light

Define redshift as change in wavelength w.r.t wavelength at time of emission, $e$.

$$\begin{equation} z \equiv \frac{\lambda_\textrm{0}- \lambda_\textrm{e}}{ \lambda_\textrm{e}} = \frac{\Delta \lambda}{ \lambda_\textrm{e}} \end{equation}$$

Expansion

If everything seems to moving away from everything else, then one possibility is that space is just getting bigger.

$$\begin{equation} r(t) = a(t) r_0 \end{equation}$$ where $r_0$ is the separation at the current time, and $a(t)$ is a dimensionless function known as the scale factor.

Wavelength - scale factor relation

$$\begin{equation} \frac{\lambda_e}{a(t_e)} = \frac{\lambda_0}{a(t_0)} \end{equation}$$

Thus, $$\begin{equation} 1 + z = \frac{\lambda_e}{\lambda_0} = \frac{a(t_0)}{a(t_e)} = \frac{1}{a(t_e)} \end{equation}$$

Thus, observing a quasar with $z = 6.4$ means we are looking at it when the universe was $$\begin{equation} a(t_e) = \frac{1}{7.4} = 0.135 \end{equation}$$ times smaller.

Hubble Flow

The motion of galaxies as part of the expansion is called the Hubble Flow

This can be distinguished from the peculiar velocity, which would be for example, the motion of the Milky Way towards Andromeda.

Earlier times

If we run the 'film' backwards, then earlier in time, everything would have been much closer together.

Two options: either things would be more densely packed in the beginning, or maybe there was just less stuff.

Big Bang Implications

The amount of He-3 present in the universe can be used to figure out the temperatures necessary in the very early universe. Then based on the expansions rates, we can figure out a temperature that the universe should be at now.

Rough estimates predicted 3-5 K.

If the universe were a blackbody, this should be convertible to a peak wavelength.

Cosmic Microwave Background

Isotropy/Anisotropy

Is it the same everywhere?

The CMB has a dipole anisotropy. This is likely due to the motion of the local group of galaxies compared to the rest of the visible universe, or the Hubble Flow.

How to make a CMB?

• Start with a really hot dense beginning: $T \gg 10^4 \; \textrm{K}$. At these temperatures, baryonic matter will be completely ionized.
• Any photons will scatter off electrons and the universe will be opaque.
• A dense, hot, and opaque medium will produce blackbody radiation.

• As the universe expands, the temperature cools. Around $T \sim 3000 \; \textrm{K}$, the protons and free electrons combine to form neutral atoms.
• Now, the blackbody photons are free to stream throughout the universe.

Cooling of the CMB

The energy density of a photon gas: $$\begin{equation} u = \left(\frac{4 \sigma}{c} \right)T^4 \end{equation}$$ and the pressure: $$\begin{equation} P = \frac{u}{3} \end{equation}$$

From the first law of thermodynamics. $$\begin{equation} dQ = dE + PdV \end{equation}$$

Since the universe is homogeneous and isotropic, we expect there to be no heat flow: $dQ = 0$.

Thus, we get: $$\begin{equation} \frac{dE}{dt} = - P(t)\frac{dV}{dt} \end{equation}$$

Substituting $u$ and $P$ from above, and taking derivatives, and simplifying: $$\begin{equation} \frac{1}{T}\frac{dT}{dt} = - \frac{1}{3V}\frac{dV}{dt} \end{equation}$$ since the volume of the universe is proportional to the scale factor cubed: $$\begin{equation} V(t) \propto a(t)^3 \end{equation}$$ we can write: $$\begin{equation} \frac{1}{T}\frac{dT}{dt} = - \frac{1}{a}\frac{da}{dt} \end{equation}$$ which can be reexpressed as: $$\begin{equation} \frac{d}{dt}(\ln T) = - \frac{d}{dt}(\ln a) \end{equation}$$ which implies: $$\begin{equation} T(t) \propto a(t)^{-1} \end{equation}$$

Olber's Paradox Resolution

The universe is not infinitely old.

Can't say how big it is, but only what we can see. (visible universe)