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“ Science is facts; just as houses are made of stone, so is science made of facts;

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 (1854-1912) French mathematician.

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“ Science is facts; just as houses are made of stone, so is science made of facts;

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  1. 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 (1854-1912) French mathematician..

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

  4. 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. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Constant Proper time dt=0 Along a photon path dS=0 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) 5.1: Cosmological Distances Proper Distance (radial) ), DP Milne Universe (Wm=0, WL=0, k=-1, Rt)

  6. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Dp(to) 6 4 2 Dp(te) 6 Dp/DH 4 2 0.1 0.1 1 1 10 10 100 100 Redshift Redshift 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)

  7. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 3 0 1 2 3 r 2 1 t r = 0 r 0 1 2 3 D(t3) 3 r 2 1 r = 0 r 3 r D(t2) 2 1 r = 0 r 1 0 2 3 Hubbles Law D(t1) 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) D(t3) > D(t2) > D(t1)

  8. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 From the definition of the redshift Relate co-moving and proper separations via redshift • 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) 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  We define the co-moving co-ordinate system such that co-moving separations Dcm at the present epoch  proper separations Dp

  9. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Dc-m 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 dq DM Hogg 2000 Einstein De Sitter(Wm=1, WL=0) Concordance(Wm=0.3, WL=0.7) Open(Wm<<1, WL=0) 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) 5.1: Cosmological Distances Proper Motion Distance (Co-moving Transverse Motion Distance), 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

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

  11. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Milne Universe (Wm=0, WL=0) EdeS Universe (Wm=1, WL=0) Wo/2 Einstein De Sitter(Wm=1, WL=0) Concordance(Wm=0.3, WL=0.7) Open(Wm<<1, WL=0) Angular Size / redshift Gurvits et al. 1999 (angular size of quasars and radio galaxies) Hogg 2000 5.1: Cosmological Distances Angular Diameter Distance, DA 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!

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

  13. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 The Luminosity distance 2 factors of (1+z) from expanding Universe (1+z) (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 In Magnitudes: (Observable = Flux & Luminosity) Used to measure the distance to bright objects 5.1: Cosmological Distances Luminosity Distance, DL Integrating F(l) over all wavelengths gives Bolometric Flux {Wm-2}

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

  15. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Proper Motion Distance Angular Diameter Distance Luminosity Distance 5.1: Cosmological Distances Comparison of Distance Measures http://www.anzwers.org/free/universe/ • 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. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Proper Motion Distance Angular Diameter Distance Luminosity Distance 5.1: Cosmological Distances Derivation of Distance Measures

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

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

  19. 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 nLnor 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. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 M82, 3.3Mpc lo Observed 5.2: The K-Correction REDSHIFT z=0 The K-Correction 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. 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. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 (1+z) le = lobs Observed SED lobs le K-CORRECTION True Spectrum (rest frame) z Observed Spectrum(observed frame) True SED 5.2: 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) The K-Correction

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

  24. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 However: For simple case of power law SED f = Transmission of band Ln((1+z)no) = Ln((1+z)no) In general: In Magnitudes: K(z) : includes both the shape of the spectrum and the band pass transmission correction 5.2: The K-Correction In general: need to know the shape of the SED The K-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.109 - article with first known use of the term K-correction.

  25. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 Mannucci et al. 5.2: The K-Correction J (1.25mm) The K-Correction at optical wavelengths K-correction for galaxies in optical usually positive Optical K-correction  galaxies are fainter H (1.65mm) K (2.2mm)

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

  27. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 S d L 5.4: Cosmological Source Counts We would like to count the numbers of objects in the Universe Galaxy Number Counts • 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, Nearby Galaxies generally follow this distribution But have to consider cosmology at larger distances

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

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

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

  31. Chris Pearson : Observational Cosmology 5: Observational Tools - ISAS -2004 SCUBA 850mm Integral Counts 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 ISO 170mm Integral Counts 5.5: The Extragalactic Background • Can also determine the background light from number counts of galaxies if we can predict the counts to very faint fluxes The Background Intensity

  32. 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. 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：

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