cosmology & the big bang ay16 lecture 20, april 15, 2008 mathematical cosmology, con’t...
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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)2/R
2 + kc
2/R
2 = 8Gc2+c
2/3
energy density CC
2(d2R/dt
2)/R + (dR/dt)
2/R + kc
2/R
2 =
-8GPc3+c2
pressure term
And Friedmann’s Equations:
(dR/dt)2
= 2GM/R + c2R
2/3 – kc
2
kc2 = Ro
2[(8G/3)o – Ho]2
if = 0 (no Cosmological Constant)
or
(dR/dt)2/R2 - 8Go /3 =c
2/3 – kc
2/R2
which is known as Friedmann’s Equation
Note that if we assume Λ = 0, we have
(d2R/dt2)/R = (ρ + 3P)
and in a matter dominated Universe, ρ >> P
So we can define a critical density by combining the cosmological equations:
ρC = =
4πG3
3 R2.
8πG R2
3H02
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 = openThere 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
k =
≡ Radius of Curvature of the Universe
H02 (Ω0 – 1) + 1/3 Λ0
c2
R(t)
t
F-L,0
E-L
F-L,C
L
SS,dS
EdSA Child’s Garden
of Cosmological Models
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 ( ) for q0=0; Λ=0
= 2/3 H0-1 [1 – (1 + z)-3/2] for q0=1/2, Λ=0
z1+z
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.)
CepheidsPretty Good Distance Indicators --- Standard Candles from
the Period-Luminosity (PL)
relation: L ≈ P3/2 PLC relationMV = -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 ‘76Cosmological 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 1932Baade 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 = 4πR12σT1
4 & L2 = 4πR22σT2
4
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:
N
M
Holmberg
Hubble
Zwicky
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 = ∫ L Φ(L) dL
or
L = ∫ L(M) Φ(M) dM
(either way works)
∞
0
∞
0
Luminosity DensityTypical numbers:
B band log L = 26.65
R band log L = 26.90
K band 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
M(r) = (sin i)2
With m1 = test particle mass, i = inclination,
r = radius, v(r) = rotation speed at r
r r2
=
v(r)2 r
G 2
(d) Binary Galaxies
Must Model Projection Effects!
M ~
i = inclination angle
φ = orbital velocity angle
1
cos3i cos2φ
Abell 2142Hot 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) () 0.66
• Blue 1000 < V < 2000 km/sBlue 1000 < V < 2000 km/s
•
The Local Supercluster
VIRGO
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 ~ 1-5 M/L rotation/dispersion ~ 10 M/L galaxy satellites ~ 25 M/L binaries ~ 50 M/L galaxy groups ~ 100 M/L Clusters ~ 400 M/L CVT ~ 3-500 M/L Flows ~ 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
=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
0.71
0.24
0.005 0.045
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 ∫
Where the terms are fairly self explanatory. We need to know H0, ΩM and ΩΛ
(1+z)[(1+z)2(ΩMz+1) – ΩΛz(z+2)]1/2
dz
∞
0
The empty model has 0 = H0-1
The SCDM Flat model has 0 = (2/3) H0-1
For the general case (with a CC), the full form is:
and a good approximation is
0 = (2/3) H0-1 sinn-1
[(|1-a|/a)1/2
]
/ |[1-a]|1/2
Where
a = matter -0.3*total + 0.3
and
sinn-1 = sinh-1 if a </= 1
= sin -1 if a > 1
(from Carroll, Press and Turner, 1992)
Also, for a flat model with L,
0 = (2/3)H0-1
-1/2
ln[(1+1/2
)/(1-)1/2
]
The Age of Flat Universes
H0/ΩΛ 0.0 0.6 0.7 0.8
55 11.9 15.1 17.1 18.5 65 10.0 12.7 14.5 16.2 70 9.4 11.9 13.6 15.1 75 8.7 11.1 12.6 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