1. Observational Cosmology: 5. Observational Tools
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
Observational Cosmology: 5. Observational Tools
“Science is facts;
just as houses are made of stone, so is science made of facts;
but a pile of stones is not a house, and a collection of facts is not necessarily science.”
  —  Jules Henri Poincaré ( ) French mathematician..

2. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Cosmological Distances
Measurement of distance is very important in cosmology
The Universe is expanding as we measure distances
We must specify at what time (what epoch) the distance corresponds to!
The Distance we measure depends on
What we are measuring
How we are measuring
When we are measuring
These Cosmological Distances will not always agree !!!

3. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Hubble Distance
Hubble
Constant
Hubble
Parameter
Linear Relation v = cz = Hod HUBBLE’s LAW
Hubble Time
Hubble Distance
** We shall see that the vd linearity only holds true for low redshifts
True Cosmological Redshift given by
where R(to)=Ro, denotes the present epoch

4. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Proper Distance (radial)
The Robertson-Walker Metric defines the geometry of the Universe
The Proper distance, Dp = The radial distance between 2 events that happen at same (proper) time dt = 0
注意　Proper distances - depend on frame of measurement
Cannot measure radial distances at constant proper time dt = 0
Universe expands between measurements
We can only effectively measure distances along past light cone dS = 0
Proper distance at time of emission and observation will be different!

5. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Proper Distance (radial) ), DP
Constant Proper time dt=0
Along a photon path dS=0
Milne Universe (Wm=0, WL=0, k=-1, Rt)
Current Proper distance (in observers frame)
 Can see sources at distance >> c/Ho even though age is 1/Ho
Proper distance in emission frame
Smaller by a factor (1+z)
Einstein De Sitter Universe (Wm=1, WL=0, k=0, Rt2/3)
L Dominated Universe (Wm=0, WL>0, k=0, Rexp(Ho t) defining Ho= (L/3)1/2)

6. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Proper Distance (radial) ), DP
z<1 : See linear relation between distance and redshift (Hubbles Law)
Einstein De Sitter Universe (Wm=1, WL=0, k=0, Rt2/3)
Milne Universe (Wm=0, WL=0, k=-1, Rt)
L Dominated Universe (Wm=0, WL>0, k=0, Rexp(Ho t) defining Ho= (L/3)1/2)
Concordance Model (Wm=0.3, WL=0.7, k=0, (R - numerical solution)
0.1 1 10
100
Redshift
Dp(to) 2 4 6
Dp(te)
Dp/DH

7. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Co-moving Distance
Co-moving Co-ordinates (r,q,f) stay fixed  Not directly related to measurable quantities
At later times,
Co-moving coords scaled by scale factor R(t) 1 2 3
r = 0 r t 1 2 3
r = 0 r
D(t3)
D(t3) > D(t2) > D(t1)
Hubbles Law 1 2 3
r = 0 r
D(t2)
D(t1)

8. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Co-moving Seperations), Dcm
Proper distance not accessible since Universe expands during our measurements
However can measure the proper separations of objects (separated by small redshift)
Proper separations (proper sizes) are related to co-moving sizes by the scale factor R(t)
Normalizing the current value of the Scale Factor, Ro, to unity 
From the definition of the redshift
Relate co-moving and proper separations via redshift
We define the co-moving co-ordinate system such that
co-moving separations Dcm at the present epoch  proper separations Dp
Co-moving separation is distance we would measure TODAY if the measured objects are in the Hubble Flow
Used for measurements of Large Scale Structure (distance between walls , voids etc)

9. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Proper Motion Distance (Co-moving Transverse Motion Distance), DM
Proper Motion Distance
Not related to Proper Distance
Proper Motion = angular motion across the sky
Nothing to do with Relativity or RW metric ! 注意
Einstein De Sitter (Wm=1, WL=0)
Concordance (Wm=0.3, WL=0.7)
Open (Wm<<1, WL=0)
Hogg 2000 dq
Dc-m DM
Wm=0.3, WL=0.7, k=0
Dc-m : Comoving separation observed today between 2 points at same redshift separated by angle dq on sky
Where
and
The Proper Motion Distance
The Proper Motion Distance = ratio of transverse proper velocity to observed angular velocity
For R-W Metric, Proper Motion Distance  Co-moving Distance  Proper Distance DP(to)
(Obsrvable = motion) Used for measurements of angular motion (knots in Radio Jets)

10. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Angular Diameter Distance, DA dq l
Angular Diameter Distance = ratio of objects physical size to angular size
R(te) since need size at source
In frame of object
For flat Universe: Angular Diameter Distance  Proper Distance when light was emitted
(Observable = size) Used to convert angular sizes into real seperations at the source

11. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Angular Diameter Distance, DA
EdeS Universe (Wm=1, WL=0)
Milne Universe (Wm=0, WL=0)
Apparent angular size of object with fixed physical diameter l decreases to a minimum at a finite redshift (zmin = 1.25, EdeS)
Apparent angular size will then appear to grow larger to higher redshifts
Since the light rays emitted by the ends of the diameter l propagating through the slowing expansion of the Universe.
 Angular Diameter Distance and the corresponding angular size may be degenerate!
Angular Size / redshift
Gurvits et al. 1999
(angular size of quasars and radio galaxies)
Wo/2
Einstein De Sitter (Wm=1, WL=0)
Concordance (Wm=0.3, WL=0.7)
Open (Wm<<1, WL=0)
Hogg 2000

12. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Luminosity Distance, DL
Objects intrinsic Luminosity is power emitted by the source = L {Lo} (1Lo=3.85x1026Js-1)
Measure Flux, S, = Luminosity spread over sphere of 4pDL2
DL = The Luminosity distance
For a given normalized intensity distribution I(le), The energy emitted per unit time over bandwidth le+dle = dL=L I(le) dle
Observe energy spread over sphere of line element
Surface Area
Received Flux density {W/m2/m} is given by energy received per unit area per unit wavelength (dlo) over a time (dto)
corresponding to the photons emitted over (dte) where lo=(1+z) le
Photons losing energy  (1+z)
Number of emitted photons in time dte :
Photon arrival interval
increases  (1+z)
Number of received photons / unit time / unit area:
Energy Flux Density F(l)

13. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Luminosity Distance, DL
Integrating F(l) over all wavelengths gives Bolometric Flux {Wm-2}
The Luminosity distance
2 factors of (1+z) from expanding Universe
(1+z)
Photons lose energy as they travel from source to observer
Photons arrive less frequently at observer than when they were emitted from the source
(1+z)
In Magnitudes:
(Observable = Flux & Luminosity) Used to measure the distance to bright objects

14. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Luminosity Distance
General Luminosity Distance (L=0)
Mattig Formula
Einstein De Sitter (Wm=1, WL=0)
Concordance (Wm=0.3, WL=0.7)
Open (Wm<<1, WL=0)
Hogg 2000
EdeS Universe (Wm=1, WL=0)
Milne Universe (Wm=0, WL=0)
z<<1 
The Hubble Law
Also special case for W=2  dL=czHo-1
since in this case the scale factor Ro=c/Ho

15. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Comparison of Distance Measures
Proper Motion
Distance
Angular Diameter
Distance
Luminosity
Distance
The Luminosity Distance (DL) shows why distant galaxies are so hard to see - a very young and distant galaxy at redshift 15 would appear to be about 560 billion light years from us
Even though the Angular Diameter Distance (DA) suggests that it was actually about 2.2 billion light years from us when it emitted the light that we now see.
The Hubble Distance (DLT) tells us that the light from this galaxy has travelled for 13.6 billion years between the time that the light was emitted and today.
The Comoving Distance (DcM) tells us this same galaxy if seen today, would be about 35 billion light years from us.

16. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Derivation of Distance Measures
Proper Motion
Distance
Angular Diameter
Distance
Luminosity
Distance

17. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Co-moving Volume
Einstein De Sitter (Wm=1, WL=0)
Concordance (Wm=0.3, WL=0.7)
Open (Wm<<1, WL=0)
Hogg 2000
Co-moving Volume : volume containing constant number density of (non-evolving) objects in the Hubble Flow
EdeS Universe (Wm=1, WL=0)
Milne Universe (Wm=0, WL=0)

18. 5.1: Cosmological Distances
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.1: Cosmological Distances
Co-moving Volume
Co-moving Volume : volume containing constant number density of objects in the Hubble Flow
FOR ANY COSMOLOGY
Wk < 0
Wk = 0
Wk > 0

19. 05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
Definitions of Luminosity (Flux)
Measurement of Luminosity (or Flux) depends on out definition
Bolometric Luminosity
Total luminosity of a galaxy, LBOl { W or L or absolute mag}
Line Luminosity
Total luminosity of an emission line, {W or L, e.g. LHa,}
In band Luminosity
Luminosity emitted in a given wavelength interval {W or L or absolute mag}
Luminosity Density (Differential Luminosity)
Luminosity / unit frequency/wavelength, { WHz-1 or LHz-1 or Wmm-1 or Lmm-1 }
often represented as nLn or lLl (nLn= lLl) {W, or L or absolute mag}
Telescope instruments: Finite band width or specific observation frequency
 Use of bolometric Flux or Luminosity - rare
More common to measure Flux density as a function of wavelength (frequency) Fn, Ln
Must take care !!! Redshifted object is emitting flux at different wavelength than observed

20. BLUE---------------RED
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction
Observed lo
REDSHIFT
M82, 3.3Mpc
z=0
z=1
z=5
z=10
Object at redshift z emits light at le
Observed at (1+z) le = lo
Observe light from Bluer part of spectrum
Observe light from shorter wavelengths as z increases
e.g. Observation at 60mm at telecope corresponds to
30mm at z=1 in galaxy galaxy frame
10mm at z=5 in galaxy galaxy frame
5mm at z=10 in galaxy galaxy frame
BLUE RED
60mm

21. 5.2: The K-Correction The K-Correction 60mm at z=0 30mm at z=1
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction
60mm at z=0
30mm at z=1
10mm at z=5
5mm at z=10
Need to collect information about emission from sources single wavelength

22. 5.2: The K-Correction The K-Correction True SED
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction
Due to redshift effect s, the true galaxy SED and that seen from Earth are different
Need to know about emission from sources at one single wavelength but we have ensemble le = lo/ (1+z)
Need a CORRECTION  This correction is called the K-CORRECTION
The significance of the correction depends on the shape of the galaxy SED
Generally the K-correction is constructed from model galaxy spectral energy distributions (SED)
True Spectrum (rest frame)
Observed Spectrum(observed frame)
(1+z) le = lobs
lobs le
K-CORRECTION z
True SED
Observed SED

23. 5.2: The K-Correction The K-Correction
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction
Observed flux at observed frequency, no, (c=nl)
Corresponding to the luminosity at ne
True Spectrum (rest frame)
Observed Spectrum(observed frame)
(1+z) le = lobs
lobs le
The Flux density
{W/m2/Hz}
Flux at no related to Luminosity at no by
K-CORRECTION -
The K-Correction depends on the assumed Spectral Energy Distribution (SED)

24. 5.2: The K-Correction The K-Correction
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction
In general: need to know the shape of the SED
However: For simple case of power law SED
In general:
f = Transmission of band
Ln((1+z)no) = Ln((1+z)no)
In Magnitudes:
K(z) : includes both the shape of the spectrum
and the band pass transmission correction
German Carl Wilhelm Wirtz observed a systematic redshift of nebulae,. He used the equivalent in German of K-correction.
Carl Wilhelm, 1918, Astronomische Nachrichten, volume 206, p article with first known use of the term K-correction.

25. 5.2: The K-Correction Optical K-correction  galaxies are fainter
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction at optical wavelengths
Mannucci et al.
J (1.25mm)
K-correction for galaxies in optical usually positive
Optical K-correction
 galaxies are fainter
H (1.65mm)
Band
l (mm) U
0.36 B
0.44 V
0.55 R
0.7 I
0.9 J
1.25 H
1.65 K
2.2 L
3.4 M
5.0 N
10.2 Q
20.0
K (2.2mm)

26. 05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.2: The K-Correction
The K-Correction at infrared - millimetre wavelengths
At longer wavelengths
K-Correction becomes NEGATIVE
SED is climbing the dust emission hump
1mm 10  100mm emission
Galaxies brighter
850mm
Sub-mm Galaxies
Constant brightness out to z~10
Sub-mm GOOD for High-z Universe!

27. 5.4: Cosmological Source Counts
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.4: Cosmological Source Counts
Galaxy Number Counts
We would like to count the numbers of objects in the Universe
For a flat, non expanding Universe (Euclidean Universe)；
Uniformly distributed galaxies with number density = n
Galaxy luminosity = L
Number of galaxies to distance ( d ) = N galaxies
Measure a Flux, S, d L S
Nearby Galaxies generally follow this distribution
But have to consider cosmology at larger distances

28. 5.4: Cosmological Source Counts
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.4: Cosmological Source Counts
Galaxy Number Counts
In general:
Number of sources brighter than Flux, S
Number of sources is found by integrating over all volumes and fluxes
Assume co-moving density is constant
Proper
Volume
Co-moving
Volume
The Luminosity is given by

29. 5.4: Cosmological Source Counts
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.4: Cosmological Source Counts
Galaxy Number Counts
In general:
For a population of sources of luminosity, L
Number of sources brighter than Flux, S
Number of sources is found by integrating over all volumes and fluxes
For any Cosmology

30. 5.5: The Extragalactic Background
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.5: The Extragalactic Background
The Background Intensity
The total observed flux summed over all galaxies produces an average background intensity
dN is the number of galaxies per unit solid angle

31. 5.5: The Extragalactic Background
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.5: The Extragalactic Background
The Background Intensity
Can also determine the background light from number counts of galaxies if we can predict the counts to very faint fluxes
But Very difficult to predict galaxy counts to faint flux levels
On the other hand: Measured integrated background light  integral contraints on faint counts of galaxies
 can be directly related to total amount of metal production ( SF) over history of the Universe
R
SCUBA 850mm Integral Counts
ISO 170mm Integral Counts

32. 5.3: Summary Summary Distance Measurements in Cosmology depend on
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.3: Summary
Summary
Distance Measurements in Cosmology depend on
Epoch of measurement
Measuring tool (a candle or a yardstick)
Hubble Distance - Measures Light Travel Time
Co-moving distance - What we would measure if the measurement were at the current epoch
Proper Motion Distance - Measured in our frame from angular motion of object (=DCM for RW)
Angular Diameter Distance - distance to a source of angular size q in source frame
Luminosity Distance - Distance assuming source luminosity spread over spherical surface today
For Luminosity Distance require K-Correction to relate observed and emission frames
Using the above observational tools - can construct models
Number Counts of Galaxies
The Distribution of sources as function of redshift
The Background Radiation due to these sources

33. 終 次： 5.3: Summary Observational Cosmology 5. Observational Tools
05/12/2018
Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004
5.3: Summary
Summary 終
Observational Cosmology
5. Observational Tools
Observational Cosmology
6. Galaxy Number Counts 次：