AY202a
Galaxies & Dynamics
Lecture 24:
Cosmological Distance
Ladder
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
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.)
Lemaitre 1927
Hubble 1929
Oort 1932
Baade 1952
Lemaitre 1927
Hubble 1929
Oort 1932
Baade 1952
deVaucouleurs ‘76
Cosmological
Distance
Ladder
DV++
102 +/- 5
S&T
52 +/- 2
!!!
Une construction solide et durable pour atteindre H0
Fukugita, Hogan & Peebles 1993
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
RR Lyraes
Cepheids
Pretty Good Distance Indicators --- Standard
Candles from the Leavitt Law (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.
Galactic/LMC Calibration
of Leavitt Law
H-band version
Welch et al
P-L Relation, LMC
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
(???)
•
Tully-Fisher
Basis for TF = L vs Vrot Law
The Back-of-the-Envelope (BOTE) approach:
½ mv2 = GMm/r
(A ha!)
Assume M/L ~ constant  M ~ L
 v2 ≈ 2GLC/r
(where C = M/L)
but we also have
L = <μ> π r2
mean surface brightness
For Spiral Galaxies, empirically
<μ>B ~ constant ~ 21.65 mag/sq-arcsec
= Freeman’s Law
thus r = (L/π<μ>) ½
v2
1/2
= 2GC(π<μ>)
L/L1/2
= 2GC (π<μ>)1/2 L1/2
so
L~
4
v
(4G2C2) π <μ>
A more complete and general derivation
of the L ~ v4 law involves assuming
self-similarity among most spiral
galaxies.
You can find the derivation in AHM (1979)
Surface
Brightness
Fluctuations
Tonry & Schneider
Image by J. Tonry
SBF in practice
Tonry & Schneider
’88
M32 vs NGC3379
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)
Dn-σ
PNLF
GCLF
MV ~ -7.3 σ ~ 1.4
magnitudes
From MW + M31
M31 IR
Nantais
TRGB = Tip of the Red Giant Branch
M31
TRGB in LMC
MI(TRGB) = -3.63 + 0.68[Fe/H] + 0.26[Fe/H]2
(Bellazzini et al ’04)
Sharp cut-off at the bright end of the
RGB Luminosity Function measured
using an “edge” detector
Jj
Jjj
Jjjj
Jjj
Jjj
Jjj
Jjj
jjj
HST H0 Key Project Team
•
Aaronson et al. 1985
Mould et al. 1989…..
HST Servicing
Mission
STS61
December
1993
Cepheid Light Curves N1326a
Matching P-L Relations
IC4182 (HST)
MW (Ground)
ITF
Calibration
SBF
Calibration
SN Ia
Calibration
Some
Convergence!
Warts!
Systematic Errors in
LMC distance (PL-calibration) !!! ~10%
Flow Field Corrections
Basic Photometry
Chemisty
Statistical Errors in
Fits to the Data
Calibrators
LMC Distance
Chemistry
[Fe/H] not
the same
everywher
e
as LMC
Global (High z) Methods
Sunyaev-Zeldovich Effect
CMB modified by hot intracluster gas
δT/T ~ ∫ ne k Te dl
Gravitational Lensing
Time delay between two (or multiple) images
depends on path length difference which can be
tied to a gravitational model for the lensing
galaxy/cluster .
SDSS
Lenses
H0 from CMB
requires lots of
priors! (i.e you
have to input the
cosmological
model…)
(Tegmark)
Riess et al 2009
NGC3982
Redo
cepheids
vs N4258
Maser
Galaxy
74
km/s/Mpc
Age of the Universe:
Ages of the Oldest things: stars,
galaxies, star clusters
Cosmological expansion age :
~ (1/H0) x geometric factors
Ages of the
World
Cooper’s Chronicle 1560
Bishop Ussher 1658
Globular Star Cluster M15
Just 15
years ago
GC ages
were
quoted as
16-18
Gyr!
!
“Well, this should give us some
valuable insight into the origin of the
Universe!”
“Because that’s the way it is.”
GUTS  Inflation  Ω = 1.0000000000
Globular Cluster Ages!
• Change in the He triple alpha reaction rate
He4 + He4 + He4 = C12
• 1997 Hipparcos recalibration of the
subdwarf distance scale - 10% more
distant
• Small change in the cosmic He abundance
Chaboyer et al. 1998
Ages of the Oldest Globular Clusters
Cosmological Age Calculation
For models w/o a Cosmological Constant,
for
q0 = 0
-1
t0 = H0
q0 = ½
-1
t0 = (2/3)H0
q0 > ½
-1
t0 = H0 (1/(1-2q0) + ..)
where
q0 = W/2 (if L = 0)
For the general case (with a CC),
the full form is:
t0 =
0
-1
-H0
∫∞
(1+z)[(1+z) (Wmz+1) –
2
(WLz(z+2))]
-1/2
dz
and a good approximation is
t0 =
-1
-1
1/2
(2/3) H0 sinn [(|1-Wa|/Wa) ]
1/2
/ |[1-Wa]|
Where
Wa
=
Wmatter -0.3*Wtotal
+ 0.3
and
sinn-1 = sinh-1
= sin-1
if
if
Wa </= 1
Wa > 1
(from Carroll, Press and Turner, 1992)
Also, for a flat model with L,
-1
t0 = (2/3)H0
-1/2
1/2
1/2
WL ln[(1+WL )/(1-WL) ]
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
JPH’s Favorite Guess Today:
H0 = 74 +/- 5 km/s/Mpc
The Universe is going to expand forever
Its current age is around
13.5 Billion Years, and
There is a good chance its FLAT with a
Cosmological constant =
Omega(Lambda) ~ 0.7
when all the systematics are included. Next
progress will probably be made locally with a better
Cepheid calibration.
•
Gold’s Law:
Complex problems often
have simple, easy to
understand, wrong
solutions.
Irwin’s Maxim:
•
Things are only simple
when you don’t know very
much.
Yogi’s Caution:
It ain’t what you don’t know,
Its what you know that ain’t so.
(originally S. Clemens = M. Twain)
Myths
Biases
“Traditional Wisdom” can lead
you down the garden path.
Tarzan’s Dilemma:
Don’t miss the forest for the trees.
Keep the Big Picture in mind.
Ask clear questions.
Know why you asked them.
Remember your assumptions!
(and communicate them)
“To My Data, Right or Wrong!”
Lets go
have
some
ice cream!