Cosmology

This set of notes by Nick Strobel covers: cosmology. Some of the observational cosmology deals with finding distances to galaxies (necessary for determining the geometry of the universe) and has been dealt with in a previous set of notes. Most of these notes will be in outline form to aid in distinguishing various concepts. As a way to condense the text a bit, I'll often use phrases instead of complete sentences. Vocabulary terms are italicized.

Cosmology--The study of the nature, origin, and evolution of the universe.

Observations and Some Implications
Fate of Universe

Observations and Some Implications

Index

A. Universe has mass

Universe contains mass--universe is not empty. Newton knew that if universe is eternal (has existed forever) and static (no net pattern of motion), then there must be enough time for gravity to collapse the universe, but this has not happened! The ways to resolve this paradox: Universe is infinite (has infinite volume and mass) and/or expanding fast enough and/or noneternal (has a beginning and/or an end). Newton chose infinite universe.

B. Dark skies

Visible night sky is dark. If universe is infinite, eternal, and static, then sky should be as bright as the surface of the sun all the time!

Olbers' paradox

Heinrich Olbers (1826) popularizes the paradox (others before him: J.P.L. Cheseaux, Kepler (1610), Thomas Digges (1576)): If the universe is uniformly filled with stars, then no matter which direction you look, your line of sight will eventually intersect a star (or other bright thing). Now we know about galaxies, but the paradox remains.
Mathematics of Olbers' paradox
Closer look (involves a bit of math): Average star luminosity = , average star number density = , star at distance has brightness flux . Stars do get fainter (the flux decreases) with distance BUT there are more stars further out: thin spherical shell of thickness with a radius , volume of shell stars each with flux which, finally, means that the total flux of one shell = (no distance dependence!). Since there are lots of shells of stars lots of light. Intervening material absorbing the starlight will eventually heat up and radiate as much energy as it absorbed so the problem remains. Of course, stars are not points (ie., they do have finite size) and can block light from other stars so the total universe flux is not infinite but still LARGE-bright as the surface of a star!
Resolving the paradox
Resolving paradox: ??? Look at the assumptions used and find out whether or not they are valid.

C. Expanding universe Index

Universe is expanding. Edwin Hubble and Milton Humason discover this in 1920's. Long ago, galaxies much closer together. Expansion is one way to resolve Olbers' paradox. Doppler effect shorthand:

is the photon wavelength and $z=V_{recession}/c = delta l/l_{rest}$ so $l_{new} = l_{rest} * (1+z)$ due to expansion. Energy of a photon = hc/l (=h*f)

, so the energy is reduced by a factor of 1/(1+z)

. Remember that ``h'' is Planck's constant and ``f'' is the photon frequency. The time to receive the photon is also lengthened by a (1+z)

factor over the time it took to emit the photon. Since the luminosity = energy / time = flux times 4*pi*D^2

, the flux of any shell of stars (galaxies) we look at is reduced to L/[4*pi*D^2*(1+z)^2]

. Another benefit of their observation was that it implied a finite age for the universe, so even if the universe is infinite, the photons from very far away have not had enough time to reach us. All these factors explain our dark night sky.

D. Warped space

Mass (gravity) bends space. Some explanation:

Geodesic

Geodesic-shortest distance between two points OR where you go if you don't change directions.
General Relativity
Need to consider Einstein's General Relativity--gravity is curvature of spacetime around anything with mass. More mass more curvature. Curvature tells mass/energy how to move-along ``curved'' geodesic. Light travels along geodesics!
General Relativity proofs
Proofs of General Relativity:
a.
Solar eclipses (Eddington 1919): observed star positions close to Sun were shifted by the amount predicted by G.R. The starlight is bent around the Sun just as Einstein said it should so it appears to us that the stars have shifted their position compared to when the Sun is not close by.
b.
Gravitational lenses: see multiple images of far away quasar lensed by massive galaxy closer to us.
c.
Clocks high above ground are observed to run faster than those on the ground-time dilation closer to massive object (the Earth).
d.
Gravitational redshift observed in light from white dwarfs (have very strong surface gravity).

Example
Example of curved geodesics--plane flights to Europe go over Greenland and on a flat 2D map, the flight path appears to be curved.

E. Uniform Universe Index

Universe is uniform on large scales (billions of light years). More assumption than observation. Makes universe models simpler.

Cosmological Principle

Cosmological Principle-Universe is:
- homogeneous--no preferred observing position in the universe AND
- isotropic--see no difference in the structure of the universe as we look in different directions.
- The principle is a Copernican idea-we are not in a special place. Every observer at a given cosmological time will see the same thing, eg., same Hubble law.
- Universe can change (evolve) throughout time!
Perfect Cosmological Principle
Perfect Cosmological Principle-cosmological principle PLUS Universe does not change with time (no evolution). If the universe is expanding, then new matter must be being created (violates the conservation of mass law). However, the amount of new matter being created is quite small--one H atom/cm every years which is approximately equal to 1 H atom/Houston Astrodome every year--small amount!
Some proof of uniformity comes from cosmic microwave background radiation.

F. Universe without a center Index

No center to expansion in 3D space. In General Relativity 3D space is warped into the fourth dimension (which is perpendicular to the three dimensions we're familiar with). Analogy: let's say you're confined to the surface of a balloon and you only know about ``front'', ``back'', ``left'', and ``right'', but not ``up'' and ``down''. In your 2D universe you can't see the third dimension, yet you know that your 2D universe must be curved because if you walk in a straight line, you eventually arrive back at where you started! You also know that the angles of large triangles add up to a number larger than 180 degrees! In a truly flat universe, the angles would add up to exactly 180 degrees. So even though you cannot see the third dimension, you know your universe is curving into it. Ponder the example of curved geodesics above. In our 3D universe we see effects of curvature into the fourth dimension.

G. Cosmic Microwave Background Radiation

Predicted by George Gamov in 1948; observed by Arno Penzias and Robert Wilson in 1965.

Observation

We observe that the radiation has the same intensity and spectral character as a thermal continuous source at 3 K (2.735 0.06 K as measured by COBE in 1989) in every direction observed. Sky is uniformly (to high precision) bright in radio.
Interpretation
We interpret this background radiation to be the relic of the early universe. If so, then that means that the early universe was very uniform. Further out in space we look, the further back in time we look. Microwave radiation is coming from universe as it was 700,000 years after Big Bang when it was much hotter. Radiation has been redshifted by 1000 times!
More in-depth explanation
Closer look: Early universe (matter and radiation) was much more compact.
a.
Radiation density was so great that it dominated the expansion and conditions of the universe for first 10,000 years--remember E=mc so energy has a gravitational effect!
b.
Early universe was hot and opaque--freely moving electrons, protons, and neutrons scattering photons all about making the dense gas opaque with the result that it produced a continuous spectrum. Universe cools off as it expands. Eventually the early universe cooled to where electron and protons could combine to form Hydrogen atoms (recombination) and not be blown apart by photons. Photons could then travel long distances without running into some particle-universe became transparent. Recombination happened at a temperature of 3000 K. Okay, ``recombination'' is not really correct since this was the first time that the electrons combined with the protons.
c.
At this temperature (3000 K) universe was about 700,000 years old. Photons from this time are now reaching our radio telescopes.

H. Cosmic abundance of Helium and Hydrogen Index

Cosmic abundance of He/H.

Elements heavier than Helium

Elements heavier than Helium produced in cores of stars. (Extremely small amounts of Lithium-7 also produced in early universe.) Stars produce some of the Helium visible today, but not most of it. If all Helium was from stars, then galaxies should be much brighter than they are.
Nucleosynthesis
Early universe was hot and dense fusion possible. Universe was like core of star: 4 H He + energy. Happened from 1-3 minutes after Big Bang. Extremely small amounts of Lithium-7 also produced during early universe nucleosynthesis.
Proportions
Makes composition 10% He, 90% H (if use mass fraction, then the proportions are 25% He and 75% H).
Deuterium
Deuterium--Hydrogen isotope with 1 proton and 1 neutron in the nucleus. Amount of remaining Deuterium in the universe is very sensitive to temperature and density of the early universe. A denser universe would have had more Deuterium fused to Helium. A less dense universe would have had more Deuterium remaining. The amount of Helium is not as sensitive to the density of the early universe, so use deuterium as probe of early density.

Fate of Universe

Index

A. Mass dependence

Depends on mass (curvature of space). More mass means

more gravity slowing down expansion. Is there enough gravity to halt expansion and recollapse the universe or not? If there is enough matter (gravity) to recollapse the universe, we say the universe is ``closed''. If there is not enough matter, so the universe keeps expanding forever, we say the universe is ``open''. The mass density of the universe is usually specified by the symbol Omega

. If

, the universe is open; if Omega is greater than 1

, the universe is closed.

B. Critical density

Critical density--boundary density between case where universe has enough mass/volume to close universe and too little mass/volume to stop the expansion (expands forever). We derive critical density = 3H^2/(8*pi*G)

, where

is Hubble ``constant'' for our cosmological time. Current critical density is approximately 8*10^-30

g/cm

. A critical density universe has ``flat'' curvature. The symbol Omega

is the ratio of the current density and the critical density.

C. Changing Hubble constant

Hubble ``constant'' is different at different cosmological times. Gravity slows expansion, so the early universe was expanding faster than it is now. That means that the critical density was greater at earlier times.

D. Always open or always closed Index

If universe starts out with a density > critical density, then its density will always be > critical density. If universe starts out with a density < critical density, then its density will always be < critical density.

E. Cosmic inventory

Is universe open or closed? Do cosmic inventory of all mass from ordinary matter. Get 10-20 times too little mass to close universe.

F. Deuterium abundance

Use current Deuterium abundance (assuming it is primordial) to find ordinary matter mass density of early universe. Again, get 5-15 times too little mass to close universe.

G. Dark Matter Index

What about Dark Matter? There may be about 90 times more mass than visible matter. May be enough so universe mass density = critical density. Observations indicating presence of Dark Matter: (Lynx users: this section contains two complicated equations that look a lot nicer with a graphical browser. I hope you'll be able to decipher the plain text-based equations I've included here.)

Flat rotation curves

Flat rotation curves of spirals even though amount light producing matter falls off as go outward. (Remember $M_{encl}=V^2 *R/G$ .)
Faint gas shells around ellipticals
Ellipticals have faint gas shells that need massive ``dark'' haloes to contain them. Need dark haloes out to 300,000 light years around each galaxy. Find . If the haloes are larger, then . For the mathematically gifted: the amount of halo mass inside a given distance is
where the is the Boltzman constant, is the temperature of the gas, is the mean molecular weight of the gas, is the mass of a proton, and is the density of the gas.
Motion of galaxies in a cluster

Index
Galaxy cluster members are moving too fast to be gravitationally bound unless there is unseen mass. We assume that we do not live at a special time, so the galaxies in the cluster must have always been close to each other. For the mathematically inclined:
where is the distance of the galaxy from the center of the cluster, is the velocity of galaxy and is the mass of galaxy . The angle brackets denote an averaging of the quantities over time (square brackets for lynx users).
Hot gas in clusters
The existence of HOT ( fast moving) gas in galaxy clusters. To keep gas bound, need unseen mass.
Quasar spectra
Absorption lines from Hydrogen in Quasar spectra tells us that there is a lot of stuff between us and the quasars.
Inflation
Theory: We think that the very early universe experienced a time of ultra-fast expansion (called inflation). According to this theory, the universe is exactly between the open and closed cases, . But the amount of visible matter says so there must be a lot of dark matter out there. Analogy: Imagine taking a globe and expanding it to the size of the Earth. The globe is still curved but the local piece we'd be on would appear to be fairly flat. The small universe inflated a large amount and the part of the universe we can observe appears to be nearly flat.
A good book on the history of dark matter is The Dark Matter: Contemporary Science's Quest for the Mass Hidden in Our Universe by Wallace and Karen Tucker (New York: Morrow, 1988).