Cosmology & the Big Bang
AY16 Lecture 20, April 15, 2008
Mathematical Cosmology, con’t
Determination of Cosmological
Parameters
Inflation & the Big Bang
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
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.
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
The Hubble Constant:
• H0 = *current* expansion rate
•
•
= (velocity) / (distance)
= (km/s) / (Megaparsecs)
• named after Edwin Hubble who
discovered the relation in 1929.
The story of the Hubble Constant (never called
that by Hubble!) is the “Cosmological Distance
Ladder” or the “Extragalactic Distance Scale”
Basically, we need distances & velocities to
galaxies and other things.
Velocities are easy --- pick a galaxy, any galaxy,
get spectrum with moderate resolution, R ~
1000 (i.e λ/R ~ 5Å)
N.B. R = Linear Reciprocal Dispersion, get line
centroids to ~ 1/10 R ~ 0.5Å/5000Å ~ 1 part in
104 ~ 30 km/s
Spectral features in galaxies
Velocity Measurement
Radial Velocities (stars, galaxies) now usually
measured by cross-correlation techniques
pioneered by Simkin (1973), Schechter (1976)
& Tonry & Davis (1979). Accuracy depends on
Signal-to-Noise and resolution. Typically, for
S/N > ~ 20, errors are ~ 10% of Δλ, where
(remember)
R = λ/Δλ
Distances are Hard!
Hubble’s original estimates of galaxy distances
were based on brightest stars which were
based on Cepheid Variables
Distances to the LMC, SMC, NGC6822 &
eventually M31 from Cepheids.
 Find the brightest stars and assume they’re
the same (independent of galaxy type, etc.)
Cepheids
Pretty Good Distance Indicators --- Standard Candles from
the Period-Luminosity (PL)
relation: L ≈ P3/2  PLC relation
MV = -2.61 - 3.76 log P +2.60 (B-V)
but ya gotta find them!
H0 circa 1929 ~ 600 km/s/Mpc Wrong!
1. Hubble’s galactic calibrators not classical
Cepheids.
2. At large distances, brightest stars confused
with star clusters.
3. Hubble’s magnitude scale was off.
•
P-L Relation, LMC
deVaucouleurs ‘76
Cosmological
Distance
Ladder
Cosmological Distance Ladder
Find things that work as distance indicators
(standard candles, standard yardsticks) to
greater and greater distances.
Locally: Primary Indicators
Cepheids
MB ~ -2 to -6
RR Lyrae Stars MB ~ 0
Novae
MB ~ -6 to -9
Calibrate Cepheids via parallax, moving
cluster = convergent point method, expansion
parallax Baade-Wesselink, main sequence
(HR diagram) fitting.
Secondary Distance Indicators
Brightest Stars
(XX??)
Tully-Fisher (+ IRTF)
Planetary Nebulae LF
Globular Cluster LF
Supernovae of type Ia
Supernovae of type II (EPM)
Fundamental Plane (Dn-σ)
Faber-Jackson
Surface Brightness Fluctuations
Red Giant Branch Tip
Luminosity Classes
(XXX)
HII Region Diameters
(XXX)
HII Region Luminosities
(???)
Lemaitre 1927
Hubble 1929
Oort 1932
Baade 1952
•
Tully-Fisher
Surface
Brightness
Fluctuations
Tonry & Schneider
Baade-Wesselink --- EPM
EPM = Expanding Photospheres Method
Basically observe and expanding/contracting
object at two (multiple) times. Get redshift
and get SED. Then
L1 = 4πR12σT14 & L2 = 4πR22σT24
and R2 = R1 + v δt (or better ∫ vdt)
Fukugita, Hogan & Peebles 1993
HST H0 Key Project Team
•
WFPC2
footprint
Cepheid Light Curves N1326a
Matching P-L Relations
IC4182 (HST)
MW (Ground)
(matter):
0. Baryons from Nucleosynthesis
1. Sum up Starlight (count stars and/
or count galaxies)
2. Count and Weigh Galaxies
3. Use Global techniques:
Large Scale Structure
Large Scale Flows
Big Bang
Nucleosynthesis
For
H0=70km/s/Mpc
(baryons)
~ 0.04
Ωmatter:
Measure luminosity density = (sum of
all galaxies x their luminosity) per
unit volume (l/v) = L
Measure mean mass-to-light ratio for
galaxies (M/L)
Multiply: Mass density = (M /L) x (L)
How do we measure the Luminosity density?
Redshift Surveys + Φ(M)
Measure the Galaxy Luminosity Function
For a typical flux (magnitude = mL) limited
survey, we can see a galaxy of absolute
magnitude M to a distance
r = 10 ( mL -M - 25)/5 Mpc
V(M) = 4/3 π r3 (Survey Solid Angle)
then
Φ(M) = dN(M)/dM = N(M,M±dM/2)/V(M)
Φ(M) or Φ(L) is the number density of galaxies
of a given magnitude or luminosity in a
sample.
Early forms:
Zwicky
Hubble
N
Holmberg
M
Abell Form (circa 1960) = two power laws
Now use the Schechter LF Form:
Φ(L) dL = φ* (L/L*)α exp(-L/L*) d(L/L*)
or
Φ(M)dM = 0.4 φ* log[dex 0.4(M*-M)]α+1
exp[-dex 0.4(M*-M)] dM
where φ* = normalization (# / Mpc3)
α = faint end slope
M*, L* = characteristic mag or luminosity
Schechter Form
The Schechter form for the LF is derived from
Press-Schechter formalism for self-similar
galaxy formation (more later).
Is integrable(!) solution:
L = φ* L* Γ(α+2)
a Gamma function
Galaxy Luminosity Function:
Luminosity Density
The Luminosity Density is then just the
integral of the luminosity function:
∞
L=
or
L=
∫0
∫
L Φ(L) dL
∞
L(M) Φ(M) dM
0
(either way works)
Luminosity Density
Typical numbers:
B band
R band
K band
log L = 26.65
log L = 26.90
log L = 27.20
In units of ergs s-1 Hz-1 Mpc-3 for H0=70,
in Solar Units LB = 1.2 x 108 L/ Mpc3
Galaxy Masses and M/Ls
Galaxies are weighed via a large number of
techniques:
(a) Disk Dispersion (more later)
(b) Rotation Curves
(c) Velocity Dispersions
(d) Binary Galaxies
(e) Galaxy Groups
(f) Galaxy Clusters
Virial Theorem /Projected Mass
Hydrostatic Equilibrium
Gravitational Lensing
(g) Large Scale Flows
(e) Cosmic Virial Theorem
Galaxy Field Velocity Dispersion
In all cases, L = Σ LGal in the system.
(b) Rotation Curves
½ m1v(r)2 (sin i)2 GM(r)m1
r
 M(r) =
=
v(r)2 r
G 2
r2
(sin i)2
With m1 = test particle mass, i = inclination,
r = radius, v(r) = rotation speed at r
(d) Binary Galaxies
Must Model Projection Effects!
M ~
1
cos3i cos2φ
i = inclination angle
φ = orbital velocity angle
Abell 2142
Hot Gas in X-rays
Strong Gravitational Lensing
Galaxy Flows:
Observed galaxy “velocity” is composed of
several parts 
VO = VHubble + Vpeculiar + Vgrav + LSR
and
VP/VH = (1/3) (dr/r) 
0.66
•
Blue
1000 < V < 2000 km/s
VIRGO
•
The Local Supercluster
The Local Supercluster
We have an infall measure for the LSC and
from redshift surveys we have a pretty good
measure of δρ/ρ:
VP ~ 250 km/s
VH = 1100 + 250 km/s = 1350
δρ/ρ ~ 2.5

Ω ~ 0.25
In terms of M/LB Ratios
M/L populations
M/L rotation/dispersion
M/L galaxy satellites
M/L binaries
M/L galaxy groups
M/L Clusters
M/L CVT
M/L Flows
~ 1-5
~ 10
~ 25
~ 50
~ 100
~ 400
~ 3-500
~ 500
What’s This Saying?
(1) M/L maxes out ~ 450, 
ΩG = ΩM = 0.25 ± 0.05
(2) M/L grows with scale?!
Gravitating matter seems to be distributed
on a scale somewhat larger than galaxies.
and there’s more of it than Baryons 
Non-Baryonic Dark Matter exists
Cosmological Constant:
Cosmological Constant = Lambda
is measured by observing the
geometry of the Universe at large
redshift (distance)
Supernovae as standard candles
CMB Fluctuations vs Models
Essence Project, 2004
Levels of Certainty in Science
You bet:
A Dime
=
$0.1
Your Dog
=
$100
Your House =
$100,000
Your Firstborn = $100,000,000 ….
each x 1000
(except in New York and Boston where everything is x 10!!!)
WMAP Microwave Sky
Best Fit
b=0.04
CDM=0.27
L=0.71
T=1.02
+/- 0.02
Large scale geometry:
CMB Fluctuations as measured by
WMAP indicate that ΩT is
very nearly unity (1.02 +/- 0.02)
the Universe is FLAT
ΩΛ = ΩT
- ΩM
Contents:
 = (density of the Universe)/
(closure density)
= 1.02 +/- 0.02
+
(total) = (baryons) +
(neutrinos) + (Cold dark matter)
(Dark Energy)
Contents:
Omega (stars) =0.005 +/- 0.002
Omega(baryons) = 0.044 +/- 0.004
Omega(neutrinos) < 0.008
Omega(CDM) = 0.23 +/- 0.04
Omega(Dark Energy) = 0.73 +/- 0.04
Omega(Total) = 1
Contents of the Universe
0.005
0.045
0.24
0.71
Age of the Universe:
Ages of the Oldest things: stars,
galaxies, star clusters
Cosmological expansion age :
~ (1/H0) x geometric factors
Cosmological Age Calculation
In FRW Cosmologies, the age of the
Universe is calculated from
τ0 = -H0-1
∫
0
∞
dz
(1+z)[(1+z)2(ΩMz+1) – ΩΛz(z+2)]1/2
Where the terms are fairly self explanatory. We
need to know H0, ΩM and ΩΛ
The empty model has t0 =
-1
H0
The SCDM Flat model has t0 =
-1
(2/3) H0
For the general case (with a CC), the full
form is:
and a good approximation is
t0 = (2/3) H0 sinn [(|1-a|/a) ]
-1
-1
1/2
/ |[1-a]|
1/2
Where
a = matter -0.3*total + 0.3
and
sinn-1 = sinh-1
= sin -1
a </= 1
if a > 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
L ln[(1+L )/(1-L) ]
The Age of Flat Universes
H0/ΩΛ
0.0
0.6
0.7
0.8
55
65
70
75
11.9
10.0
9.4
8.7
15.1
12.7
11.9
11.1
17.1
14.5
13.6
12.6
18.5
16.2
15.1
14.0
Where Ωtotal = 1.00000, and the ΩΛ = 0 models
are the Standard CDM models in Gyr
Alternatives
ΩM = 0.3, ΩΛ = 0
gives τ0 = 0.79 H0-1 = 11.8 Gyr for H=65
(no Lambda)
ΩM = 0.25, ΩΛ = 0.6
gives τ0 = 0.97 H0-1 = 14.6 Gyr for H=65
(minimal Lambda)
JPH’s Favorite Guess Today:
H0 = 70 +/- 5 km/s/Mpc
The Universe is going to expand forever
Its current age is around
14 Billion Years, and
There is a good chance its FLAT with a
Cosmological constant =
(Lambda) ~ 0.7
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