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Observational Cosmology Prof Simon Driver
1. An Expanding Universe 2. The Hot Big Bang
3. The Microwave Background 4. Building a model – geometry 5. Building a model – dynamics
6. The Einstein de Sitter & Milne Universes 7. Dark Energy
8. Inflation 9. Loose ends and future directions
Course Text: Introduction to Modern Cosmology by A.Liddle
Lecture 1: The Expanding Universe 1. The Copernican Principle 2. Olber’s Paradox 3. The idea of “Permanency” 4. The discovery of the expansion 5. Hubble’s law 6. The age of the Universe, Earth and Stars 7. The Big Bang and its three pillars:
- Big Bang Nucleosynthesis… - The Cosmic Microwave Background… - The age of the Universe
8. An Adiabatic Expansion 9. Equations of State 10. How density of matter and radiation scale with expansion
Course Text: Chapters 1 & 2 Wikipedia: Copernican Principle, Olber’s Paradox, Hubble’s Law
The Copernican Revolution • Cosmology is the study of the Universe • Pre-1593 world view was laid down by the Church • Earth at centre of an eternal, unchanging Universe
• Galileo, Copernicus, and Kepler challenged this authority by displacing the Earth from the centre of the Solar System
• 1543 Copernicus publishes... …so began the Scientific revolution
1473-1543
The Copernican Principle
Modern Cosmology begins with the following axiom:
This comments on space but not on time.
Universe still perceived as eternal and unchanging.
The sense of Permanency was an entrenched “known” from pre-1543 But everything starts and ends?
There is nothing special about the location of the Earth in the cosmos
Olber’s Paradox In 1826 Olber voiced a well known paradox:
This question pre-empts Einstein and Hubble by noting the impossibility of an infinitely old and infinitely large universe…
If the Universe is infinitely big with a uniform distribution of stars every line of sight will eventually intercept a star…
Why is the sky dark at night?
Olber’s Paradox • The fact that some stars are more distant is irrelevant:
• Flux from A: ~ L/d 2
• Flux per unit solid angle from A: ~ (L/d 2)/θ2
• As θ ∼ 1/d, this implies flux per unit solid angle constant
• If the Universe is infinite then the entire sky should be as bright as the surface of the sun!
A
B
Olber’s Paradox: Formally • Let n = the density of stars with
intrinsic luminosity L uniformly distributed to infinity
• No of sources within shell is:
• Flux of each source is:
• Total light from all shells is:
• But it is dark at night…
rdr
!
dn = n 4" r2dr
!
f =L
4" r2
!
I = dI" = f" dn =Ln4# r2
4# r2" dr = Lndr0
$
" = Ln r[ ]0$
=$
Solutions to Olber’s Paradox • Intervening dust - But dust will heat and reradiate • An edge to the stars - Violates Copernican Principle • Finite age to Stars/Universe - Violates permanency • Contractions/expansions - No noticeable effect unless extreme
Correct Solution: Universe has a finite age
Can see light from sources
Cannot see light from sources outside sphere
Problems with Permanency • Prior to the discovery of the Universal Expansion scientists were
already aware of problems: – Olber’s Paradox – Energy Conservation (for stars to shine indefinitely they would require
an infinite fuel reserve) – Ages of Earth, meteorites, and stars
• All of above point toward a Universe with a beginning (or at least to a problem with the notion of permanency!)
• Even Einstein missed his chance as he added a Cosmological Constant to GR to keep the Universe static.
“Everything has to have a start and an end” Kalagan, age 7, Feb, 2011
Nedlands Primary School
Hubble’s Discovery • Proved that M31 was external to our galaxy. • Hubble collected many galaxy images and spectra • Measured brightest stars and Cepheid variables to get distances • Measured offset of common spectral features to get velocity • Plotting distance v velocity he found: Hubble’s Law: A linear relation between a galaxy’s distance (d) and recession velocity (v) Today: Ho=72 km/s/Mpc ßUNITS!!
!
H0 =vd
Hubble’s Data
For these 5 bright ellipticals in nearby clusters we see that fainter galaxies have their Ca H & K lines
redshifted further
Simply by assuming that the brightest elliptical in a cluster is of comparable absolute magnitude we see Hubble’s law for ourselves
Shifting spectral features SAME GALAXY PLACED AT DIFFERENT DISTANCES, LIGHT IS STRETCH DURING TRAVEL
Hubble’s law appears to violate the Copernican Principle as it seems to place us at a special location:
Everything is moving away from us?
Universal Expansion
Milky Way
Universal Expansion Q) What is so special about our location ? A) Nothing !
Consider:
According to Hubble’s Law:
I see: But if we jump to your location, you see:
Me You
v v 2v 3v
3v 2v v v
The Universal Expansion • A “vector jump” to another galaxy will result in that galaxy
seeing all others moving away from it. • Only an expansion or contraction can produce a centre-less
but dynamic Universe.
The Age of the Universe • If we extrapolate back at constant velocity every galaxy was
coincident at a time of d/v=1/Ho
• So from 1/H0 we can calculate an approximate age for the Universe:
!
tAge =1Ho
=175s.Mpc /km
tAge =175
"106 " 3"1016
103#
$ %
&
' ( = 4 "1017 s
tAge = 4 "1017 " 1365.25 " 24 " 60 " 60#
$ %
&
' ( yrs
tAge =1.267 "1010 yrstAge )13Gyrs
Big Bang v Steady-State • GR without the Cosmological Constant provided a basis for the
expansion
• But a model has to make predictions to gain credibility
• Big Bang provided one explanation and one prediction:
– Big Bang Nucleosynthesis --- explained the 4He and other light element abundances (1948)
– The Cosmic Microwave Background --- predicted the ubiquitous background radiation (1948)
• Unlike the expansion the CMB was predicted before its discovery
• Big Bang model adopted over Steady State following CMB
• Both follow from the idea that as the Universe expands it cools
Abundances in the Solar System
Very high Helium abundance not expected via stellar nucleosynthesis
Yield from SN
Data v prediction (400σ errorbars)
• If U self-contained it must expand without losing energy: (1st law of thermodynamics)
• Can use E=mc2 and rewrite with m= ρ(4πr3/3) where r is some physical radius for expanding region of density ρ.
[Uses: Chain rule + d(x3)=3x2d
• Use dot notation: i.e.,
• Rearrange to get the Fluid Equation:
Adiabatic Expansion
!
dE = "pdV
!
dEdt
=d(4"r3#c 2 3)
dt=43"r3c 2 d#
dt+ #4"r2c 2 dr
dt= $4"pr2 dr
dt= $pdV
!
dxdt
= x•
!
43"r3c 2 #
•
+ 4"r2c 2# r•
= $4"r2pr•
!
"•
+ 3 r•
r(" +
pc 2) = 0
Equations of State • We have an expression for how the density of U depends on the
density and pressure of its contents.
• We know about two kinds of stuff:
– Matter - uniform diluted stationary matter exerts no pressure, p=0
– Radiation - photons exert radiation pressure given by, p=ρc2/3 [From Thermodynamics, see also Problem 4.2]
This can be generalised into an equation of state: w=0 for normal matter, 1/3 for photons (and -1 for dark energy).
!
p = w"c 2
How radiation and matter scale • Matter:
– Subbing w=0 into EoS and then Fluid Eqn gives:
• Radiation: – Subbing w=1/3 into EoS and then Fluid Eqn gives:
• In an adiabatically expanding Universe matter dilutes with length cubed and radiation with length to the fourth.
• This means radiation dominates over matter in the very early Universe with serious implications…
!
"•
+ 3" r•
r= 0, 1
r3ddt
("r3) = 0, i.e., "M # r$3
!
"•
+ 4" r•
r= 0, 1
r4ddt
("r4 ) = 0, i.e., "R # r$4
Early Universe radiation dominated
Figure Credit: Pearson Education Inc. Pearson Addison-Wesley
Lecture 1: The Expanding Universe 1. The Copernican Principle 2. Olber’s Paradox 3. The idea of “Permanency” 4. The discovery of the expansion 5. Hubble’s law 6. The age of the Universe, Earth and Stars 7. The Big Bang and its three pillars:
- Big Bang Nucleosynthesis… - The Cosmic Microwave Background… - The age of the Universe
8. An Adiabatic Expansion 9. Equations of State 10. How density of matter and radiation scales with expansion
Course Text: Chapters 1 & 2 Wikipedia: Copernican Principle, Olber’s Paradox, Hubble’s Law
