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Observational Cosmology: Tools and Distances

Understanding the distance measurements in cosmology is crucial for studying the expanding Universe. Different metrics and proper distances affect observations, highlighting the complexity of cosmological distances. Explore the intricacies of Hubble's law, cosmic redshift, and the relationship between co-moving and proper separations.

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Observational Cosmology: Tools and Distances

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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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