Transcript AY145 Topics in Astrophysics

Cosmology & the Big Bang
AY16 Lecture 19, April 10, 2008
Introduction to Cosmology
Basic Principles
Fundamental Observations
The FRW Metric
“My husband adheres to the Big
Bang theory of creation.”
COSMOLOGY
The BIG Picture!
The cosmological model dominates much of
extragalactic astronomy, in fact much of
astronomy & astrophysics, even physics.
What is Cosmology? “The study of the large
scale structure & evolution of the Universe”
What is Cosmogony? “The study of the origin
of observable structures”
Cosmology is perhaps the oldest real science.
Its tied to our “World View.”
Changing Worldviews
Age
Universe
---------------------------------------------------------------------------------10,000 years BC --- The Valley you lived in
1,000 years BC --- Your Kingdom
300 years BC --- The Mediterranean
(for Egypto-Greco-Romans, at least!)
100 years AD --- The Earth + Celestial Sphere
400 years ago --- The Solar System
100 years ago --- The Milky Way
75 years ago --- The “Modern” Universe
(2 Billion Light -Years in *radius*)
Today
--- An Infinite Universe
(the visible part has a Radius of ~15 Gly)
A Brief History of Extragalactic
Astronomy:
~ 1750 Early Rumblings of “Island Universes”
from I. Kant, T. Wright, P. Laplace.
This seems to have been forgotten soon after.
1800’s Catalogs of Things but no understanding.
3
de la Caille, Messier, Herschel , Dreyer
William, Caroline & John
~ 1875 The Discovery of the Galaxy --- Kapteyn’s
Universe
1910 Removal of the Solar-Centric view
1900-1920 Shapley and the Great Debate
1907 Bohlin --- M31 Parallax
1918 Van Maanen --- M31 Parallax
1885 S Andromeda = SN1885a
large reflectors + photographic plates
1920 The Shapley-Curtis debate
Shapley + Globular Clusters + Cepheids
1924 Hubble & the Hooker --- NGC 6822
Cepheids, eventually M31 Cepheids
1910-1930 Theory! Einstein, Friedmann,
deSitter, Lemaitre, Tolman, Robertson …
1922 Opik’s M31 Mass-to-Light ratio
L = 4πr2 
GMm/r = ½ mv2
M = ½ v2r/G,
v2 1
1
so M/L = ½ G  4πD
and Opik estimated D of M31 at 450 kpc.
1929 Hubble (+Slipher)  Velocity-Distance
Law
1930’s Hubble’s Classification Scheme for
Galaxies (Tuning Fork Diagram)
N.B. Absolutely necessary but wrong
interpretation, set galaxy evolution back 20 years!
Hubble’s Galaxy Counts (Humason)
1937 Zwicky & Smith DARK MATTER
1940’s Galactic Dust, Stellar Populations,
Hubble Diagram
1948 Gamow & the Hot Big Bang
1950’s deVaucouleurs’ Local “Supergalaxy”
Rubin: Flows
Dicke: CM
HMS Velocities + Hubble Diagram
Baade & Sandage: H0 revisions
Minkowski: Radio galaxies
1960’s The Hubble Constant Debate
Tinsley: Stellar Evolution  Galaxy
Evolution
Greenstein & Schmidt: Quasars
Arp: Peculiar Galaxies
Spinrad &Taylor : Population Synthesis
Page: Galaxy Masses
1970’s Stability & Halos
Starbursts
H0!!! q0!!!
First Feeble Redshift Surveys
CMB Dipole
Galaxy Clusters & X-Rays
Gravitational Lenses
Galaxy Formation
1980’s Large-Scale Structure
Large Scale Flows & Cold Dark Matter
Galaxy Counts
H0!!!!
IRAS & Dusty Starbursts
1990’s COBE: 2.723 K + fluctuations
Biased galaxy Formation
Unified AGN Models
Λ!!!!!
Concordance Cosmology
HST and galaxy evolution
2000+ The Cosmic Web
Reionization
First Light
COSMOLOGY is a modern subject:
Today based on Principles & Observables
The basic framework for our current
view of the Universe rests on ideas and discoveries
(mostly) from the early 20th century.
Basics:
Einstein’s General Relativity
The Copernican Principle
Fundamental Principles:
• Cosmological Principle: (a.k.a. the
Copernican principle). There is no preferred
place in space --- the Universe should look
the same from anywhere
The Universe is HOMOGENEOUS and
ISOTROPIC.
we believe this is true to zeroth order
(i.e. on large scales, yes, on small scales, no)
A variant of the CP is
• The Perfect Cosmological Principle:
The Universe is also the same in time.
The STEADY STATE Model (XXX
it’s demonstrably wrong)
• The Anthropic Cosmological Principle:
We see the Universe in a preferrred state (time
etc.) --- when Humans can exist.
the ACP is almost the opposite of the PCP.
it leads to the Goldilocks Universe:
Not too hot, Not too cold
Not too dense, Not too empty
Not too young, Not too old….
• Relativistic Cosmological Principle:
The Laws of Physics are the same
everywhere and everywhen.
(!!!) absolutely necessary, often
assumed and forgotten. (!!!)
Fundamental Observations:
• The Sky is Dark at Night (Olber’s P.)
this implies
there must be some
limit to the observable Universe.
• The Universe is generally Expanding
It’s not static.
galaxies appear to be moving away
from us --- and each other.
Olber’s Paradox
Hubble’s Discovery of Expansion
• The Universe is Homogeneous on large
scales --- there exists an almost isotropic
microwave background (the CMB) of T~3K
a.k.a. relic radiation
• The Universe is not Empty. It has stuff in it,
stuff consistent with a hot origin (the
Universe has a temperature), i.e. contents
consistent with nuclear physics operating in
an initially hot, dense medium
COBE Fluctuations
dt/t
< 10-5,
i.e. much smoother than a
baby’s bottom!
Observational Cosmology
consists of taking these bases to build a more
detailed picture of the structure and evolution
of the Universe.
Sometimes to (1) Feed Theorists
(2) Kill Theories
(3) Explore
 generally support Gamow’s hot big bang
model
COSMOLOGICAL FRAMEWORK:
The Friedmann-Robertson-Walker
Metric
+
The Cosmic Microwave Background
= THE HOT BIG BANG
The Big Bang
WRONG!
WRONG!!!
WRONG ??
T
THE TRUTH BEHIND THE
BIG BANG THEORY
How my wife describes my job!
RIGHT!
Mathematical Cosmology
The simplest questions are Geometric.
How is Space measured?
Standard 3-Space Metric:
2
2
2
2
ds = dx + dy + dz
2
2 2
2
2
2
= dr + r dθ + r sin θ df
In Cartesian or Spherical coordinates in
Euclidean Space.
Now make our space Non-Static, but
“homogeneous” & “isotropic” 
2
2
2
2
2
ds = R (t)(dx + dy + dz )
And then allow transformation to a more
general geometry (i.e. allow nonEuclidean geometry) but keep isotropic
and homogeneous:
2
2 -2
ds = (1+1/4kr )
2
2
2
2
2
2
(dx +dy +dz )R (t)
2
2
where r = x + y + z , and k is a
measure of space curvature.
Note the Special Relativistic
Minkowski Metric
2
2
2
2
2
2
ds = c dt – (dx +dy + dz )
So, if we take our general metric and add the 4th
(time) dimension, we have:
2
2
2
2
ds = c dt – R (t)(dx +dy + dz )/(1+kr /4)
2
2
2
2
or in spherical coordinates and simplifying,
2
2
2
2
2
2
ds = c dt – R (t)[dr /(1-kr ) +
2
2
2
2
r (d +sin  df )]
which is the (Friedmann)-Robertson-Walker
Metric, a.k.a. FRW
• The FRW metric is the most general,
non-static, homogeneous and isotropic
metric. It was derived ~1930 by Robertson
and Walker.
R(t), the Scale Factor, is an unspecified
function of time (which is usually assumed
to be continuous)
and k = 1, 0, or -1 = the Curvature Constant
For k = -1 or 0, space is infinite
K = +1
Spherical
c < pr
K = -1
Hyperbolic
c > pr
K=0
Flat
c = pr
Consider Expansion in an Isotropic
Universe
a(t) = Universal Expansion factor
Expansion is self-similar and produces a
change in the frequency of received
radiation. If t0 = now, here, observed & t1 =
the time at which light is emitted from a
distant object in the scaled universe:
a(t)δt
t0+Δt0
t0
te+Δte
te = t1
1/Δte = υ1 frequency of emitted radiation

υ0
υ1
=
a(t1)
a(t0)
=
λ1
λ0
so υ1, λ1 = lab or rest frequency/wavelength and
υ0, λ0 = observed frequency/wavelength
and if we define z ≡ redshift =
R(t0)
a(t0)
Then 1 + z =
=
R(t1)
a(t1)
λ0 – λ1
λ0
For small z we can interpret this “redshift” in
terms of a Doppler shift, cz = v, or Doppler
velocity. For small Δt, if r0 = 0 (set origin to
us, the observer), we have:
r1 = c (t0 - t1)/R(t0)
d = r1 R(t0) = c(t0 -t1) the distance
and thus
1 dR(t1)
cz = c(t0 – t1)
R(t0) dt
so
v = cz = H0d
dR(t0) 1
dR(t1) 1
where H0 =
≈
dt R(t0)
dt
R(t1)
is the definition of Hubble’s Constant.
Note that this is true for small z only. This
formula for distance is NOT subject to
special relativity. The convention is to quote
apparent velocities as v = cz.
The apparent (radial) velocity of any object is made
up of three parts
v = vH + vP + vG
vH = the cosmological stretching of the metric,
a.k.a. the Hubble Flow
vP = the component of the “peculiar” velocity w.r.t.
the Hubble Flow that is an actual space velocity. vP
is Doppler, so computations of dynamical
properties like cluster velocity dispersions that
result from vP do require the (1+v2/c2) –1/2
correction.
vG = the gravitational redshift, usually tiny
Expansion Age & Hubble Distance
The Hubble Constant has units of inverse time;
its actually also a measure of the expansion age
of the Universe:
τH = H0-1 = 9.78x109 h-1 years = 3.09x1017 h-1 s
where H0 = 100 h km/s/Mpc
And the Hubble Distance is
DH = c/H0 = 3000 h-1 Mpc = 9.26x1025 h-1 m
What about the scale factor R(t)?
R(t) is specified by
Physics
we can use Newtonian Physics (the
Newtonian approximation) but now General
Relativity holds.
Start with Einstein’s (tensor) Field Equations
Gu = 8pTn + Lgu
Gu = Rn - 1/2 gu R
and
Where
Tn is the Stress Energy tensor
Rn
is the Ricci tensor
gu is the metric tensor
Gu is the Einstein tensor
and R

is the scalar curvature
Rn - 1/2 gu R = 8pTn + Lgu
is the Einstein Equation
The vector/scalar terms of the Tensor Field
Equations give the linear form
Einstein’s Equations:
(dR/dt) /R + kc /R = 8pGe/3c +Lc /3
2
2
2
2
energy density
2
2
2
2
2
2
CC
2
2(d R/dt )/R + (dR/dt) /R + kc /R =
-8pGP/c +Lc
3
pressure term
2
And Friedmann’s Equations:
(dR/dt) = 2GM/R + Lc R /3 – kc
2
2
 kc =
2 2
2
2
2
Ro [(8pG/3)ro – Ho]
if L = 0 (no Cosmological Constant)
or
(dR/dt) /R - 8pGro /3 =Lc /3 – kc
2
2
2
2
which is known as Friedmann’s Equation
/R
2
Note that if we assume Λ = 0, we have
(d2R/dt2)/R = - 4πG (ρ + 3P)
3
and in a matter dominated Universe, ρ >> P
So we can define a critical density by
combining the cosmological equations:
.2
2
3H
3
R
0
ρC =
=
8πG R2
8πG
And we define the ratio of the density to the
critical density as the parameter
Ω ≡ ρ/ρC
For a matter dominated, Λ=0 cosmology,
Ω > 1 = closed
Ω = 1 = flat, just bound
Ω < 1 = open
There are many possible forms of R(t), especially
when Λ and P are reintroduced. Its our job to find
the right one!
Λ=0
Some of possible forms are:
Big Bang Models:
Einstein-deSitter
k=0
flat, open & infinite
expands
Friedmann-Lemaitre k=-1 hyperbolic “
“
“
k=+1 spherical, closed
finite, collapses
Leimaitre Λ ≠0 k=+1 spherical, closed
finite, expands
Non-Big Bang Models
Eddington-Lemaitre Λ≠0 k=+1 spherical,
closed, finite, static then expands
Steady State
k=0 flat, open,
infinite, stationary
deSitter
k=0 empty, no
singularity, open, infinite
H02 (Ω0 – 1) + 1/3 Λ0
k=
c2
≡ Radius of Curvature of the Universe
R(t)
A Child’s Garden
of Cosmological Models
E-L
F-L,0
EdS
SS,dS
L
F-L,C
t
Harrison’s
Classes
Geometric:
Closed
Open
Kinematic:
Bang
Whimper
Static
Oscillating
Cosmology is now the search for
three numbers:
• The Expansion Rate = Hubble’s Constant
= H0
• The Mean Matter Density = Ωmatter
• The Cosmological Constant = ΩΛ
Taken together, these three numbers describe
the geometry of space-time and its evolution.
They also give you the Age of the Universe.
The best routes to the first two are in the
Nearby Universe:
H0 is determined by measuring
distances and redshifts to galaxies. It
changes with time in real FRW models so
by definition it must be measured locally.
W(matter) is determined locally by
(1) a census, (2) topography, or (3) gravity
versus the velocity field (how things move
in the presence of lumps).
The cosmological constant is determined by
measuring geometry on large scales --- e.g. by
the supernovae Hubble Diagram (distance
versus redshift)
Or
by a difference technique, e.g. CMB shows us
that the Universe is Flat, but ΩM is only 0.3,
so ……
Lookback Time
For a Friedmann-Lemaitre Big-Bang Model,
the lookback time as a function of redshift is
τL = H0
-1
z
( 1+z
)
for q0=0; Λ=0
= 2/3 H0-1 [1 – (1 + z)-3/2] for q0=1/2, Λ=0
Cosmological Age Calculation
For models w/o a Cosmological Constant,
for
q0 = 0
t0 = H0
q0 = ½
t0 = (2/3)H0
q0 > ½
t0 =
-1
-1
-1
H0 (1/(1-2q0) +
where q0 = W/2 (if L = 0)
..)
For the general case (with a CC), the full
form is:
t0 =
-1
-H0
∫
0
(1+z)[(1+z) (Wmz+1) –
2
∞
-1/2
(WLz(z+2))]
dz
and a good approximation is
t0 = (2/3) H0 sinn [(|1-Wa|/Wa) ]
-1
-1
1/2
/ |[1-Wa]|
1/2
Where
Wa = Wmatter -0.3*Wtotal + 0.3
and
sinn-1 = sinh-1
= sin-1
Wa </= 1
if Wa > 1
if
(from Carroll, Press and Turner, 1992)
Also, for a flat model with L,
t0 = (2/3)H0
-1
-1/2
1/2
1/2
WL ln[(1+WL )/(1-WL) ]
Classical Cosmological Tests
Use observations to test/measure the
cosmological model:
Observables: Apparent magnitudes
Redshifts
Angular Diameters
Number Counts
Ages of Things (rocks, stars, star
clusters)
(1) Magnitude vs Redshift (The Hubble Diagram)
(assume or find a standard candle)
1
log cz
1/2
0
-1
V magnitude (corrected)
q0
(2) Counts vs Magnitude (Evolution?)
(3) Angular Diameter vs Redshift (Yardstick)
(4) Age Consistency (rocks vs expansion age)
(5) Density versus Redshift (galaxy evolution
again)