Lecture 9 H 0 from the Hubble diagram Basics Measuring distances Parallax Cepheid variables Type Ia Super Novae H 0 from other methods Gravitational lensing Sunyaev-Zeldovich effect
H 0 from the Hubble diagram proper distance s dr R 0 R 2 0 1 kr r s v R R 0 R 0 0 s r R 0 H R r s 0 R r 0 r 0 v pec v [kms -1 ] large H 0 small H 0 Hubble flow Idea is very simple: Measure the velocities and distances of a set of objects, plot graph and measure the slope H 0 (units kms -1 Mpc -1 ). s [Mpc]
Typically v pec is of the order of a few hundred km/s (let s say it s 500 kms -1 ). It s bigger inside galaxy clusters than in between the clusters. Using this value and H 0 =72 kms -1 Mpc we can write v Hubble 500 72 H kms 0 1 Mpc 1 s 7 Mpc kms 1 have to go beyond 7 Mpc for v Hubble > v pec At large distances v z s D L D L ( z) (1 z) z 0 c dz H ( z) (1 z) c H 0 z 0 mat,0 (1 z) 3 (1 mat,0 dz ) (1 z) 2,0,0 1 2
z large H 0 small H 0 will depend on other cosmological parameters for z 0. 5 D L Measuring velocities is easy and accurate z,v I
Arrows show the shift of the Calcium H and K lines
The expanding universe At z << 1 all cosmological models predict a linear behaviour, z d first evidence: Edwin Hubble 1929 the possibility that the velocity-distance relation may represent the de Sitter effect slope of graph 465±50 km/s/mpc or 513±60 km/s/mpc (individual vs grouped) assumption of linearity no centre to expansion established by 1931 (Hubble & Humason) Hubble s original diagram
Hubble s Original Diagram (1929) Hubble underestimated the distances by about a factor of 7
Hubble s law (Early History) 1907: Bertram Boltwood dates rocks to 0.4 2.2 Gyr (using radio active decay, U-Pb) 1915: Vesto Slipher demonstrates that most galaxies are redshifted 1925: Hubble identifies Cepheid variables in the nearby galaxies M31 and M33 1927: Arthur Holmes age of Earth s crust is 1.6 3.0 billion years. 1929: Hubble s constant first measured: value of 500 km/s/mpc implies age of Universe ~2.0 Gyr Clearly something was wrong
Note that the first point is actually from a paper by G. Lemaitre in 1927 based on distances to galaxies derived and published by Hubble. The second is from H. Robertson, also based primarily on Hubble's data. Hubble himself finally weighed in in 1929 at 500 km/s/mpc. Also, in 1930, the Dutch astronomer, Jan Oort, thought something was wrong with Hubble's scale and published a value of 290 km/s/mpc, but this was largely forgotten. The first major revision to Hubble's value was made in the 1950's due to the discovery of Population II stars by W. Baade. That was followed by other corrections for confusion, etc. that pretty much dropped the accepted value down to around 100 km/s/mpc by the early 1960's. Source: www.cfa.harvard.edu/~huchra
Measuring distances Measuring distances is difficult Need a standard candle = object of known luminosity [or a standard ruler = object of known size] Cepheids are the best standard candle for small distances. The brightest ones have an absolute magnitude of about M V ~ -5 Type Ia super novae are the best for large distances. They have an absolute magnitude of M V ~ -19.3 Standard candles have to be accurately calibrated in order to remove systematic errors.
The distance scale ladder The Distance Ladder diagram Photometric Geometric
Hubble s law: systematic errors Measured distances mostly depend on m M = 5 log(d/10) + constant (where d is luminosity distance) getting M wrong changes d by a factor of 10 M M 5 which does not affect linearity (it just changes the slope) typical of the nature of systematic errors: very difficult to spot Oort(1931) expressed doubts very quickly Baade(1951) showed Hubble error est
Parallax measurement Earth Sun LMC For LMC the parallax angle is tiny D Earth-Sun = 1 A.U. = 1 Astronomical Unit = 1.510 11 m D LMC = 52 kpc = 1.610 21 m 210 4.210 10 5 arcsec moon 1800 arcsec 0.5 degree
Magnitudes and distance moduli Observational astronomers use a logarithmic scale for fluxes called apparent magnitudes m 2.5 log S const. S m The constant varies strongly for between different magnitude systems Visible stars have apparent magnitude m = 1 6 The faintest galaxies observed have m V ~30 This is a factor 10-29/2.5 ~ 3 10-12 fainter than a bright star.
Absolute magnitudes are a logarithmic scale for luminosities. They are defined as the apparent magnitude a source would have at a distance of 10pc. In SI units this gives M 2.5 log L 90.2 const. The distance modulus is defined as m with M D L 5 log DL 87.45 L 4 S This is as odd as it looks but is still used by observational astronomers. 1 2
Type I and Type II Cepheids Unfortunately, there are actually two types of Cepheid with different P-L relations. Type I Cepheids have a similar chemical composition to the Sun. Type II Cepheids are lower metallicity since they are older and are formed from more primordial material. To make life really tricky, this means that Type I Cepheids are found in the Milky Way and Type II in globular clusters. This meant that first estimates of distances to Globulars were wrong since Type II Cepheids are about 1.5 magnitudes (~4 times) fainter.
Period-Luminosity Relation for Cepheids and RR Lyrae www.astro.livjm.ac.uk/courses/phys134 /
The Large Magellanic Cloud (LMC) Irregular dwarf galaxy in the Local Group of galaxies
Uncertainties in the distance to the Large Magellanic Cloud (LMC) From Mould et al, 2000, ApJ, 529, 786. Distribution of published LMC distance moduli from the literature.
Measuring the Hubble constant involves several steps Measure distance to LMC to calibrate the Cepheid (P-L) relation (D LMC = 52kpc +/- 1 kpc) Parallax measurement of Cepheids by Hipparcos satellite. Only a few because Hipparcos parallax measurements only go to a distance of 1kpc. Calibrate type Ia SN as standard candles. Measure distance-redshift relation at small distances (i.e. local galaxies) with Cepheids Measure distance-redshift relation at large distances with type Ia SN
Problems: peculiar velocities dust absorption if objects become fainter because their light is absorbed you think they are further away than they are Hubble constant underestimated Malmquist bias, pick systematically brighter objects at high redshift Hubble constant overestimated Calibration is tricky
Results: 2 Groups: Tamman et al.: H 0 = 60 +/-2.3 kms -1 Mpc -1 Freedman et al.: H 0 = 72 +/- 3 (statistical) +/- 7 (systematic) kms -1 Mpc -1 Discrepancy is mainly due to the different calibration of the period-luminosity relation of Cepheids.
HST Key Project, Cepheids From Freedman, W., et al, 2001
Type Ia SN Caused by the explosion of a white dwarf very near to the Chandrasekhar mass limit (~1.4 times the mass of the Sun). White dwarf is in a binary system and it gains mass by accreting material from its companion. Just before it reaches the mass limit there s a runaway nuclear explosion (fusion of Carbon). Because the Chandrasekhar limit is the same for all white dwarfs, to first order, the explosion always has approximately the same luminosity. To further establish the absolute luminosity of a given event people use the fact that the duration of the SN flash is correlated with the luminosity - empirical calibration of absolute luminosity. Best standard candle known but rare, so not useful at small distances.
http://www-supernova.lbl.gov http://www-supernova.lbl.gov
http://www-supernova.lbl.gov http://www-supernova.lbl.gov mb is a measure for DL Hubble diagram from the Supernova Cosmology Project
Gravitational lensing: Determining H0 Measuring the redshifts (i.e. radial velocities) of the lens and the source combined with an adopted cosmology (i.e. H0) defines exactly the geometry. This means we can determine the observer-lens distance, the observersource distance and the lens-source distance. A model for the lens mass distribution can be constructed that accurately predicts the observed lens images. We can also estimate the difference in path length from the source to the observer that corresponds to each lensed image. However, all the distances are dependent on the assumed H0. For example, both cases shown on the right are consistent with the measured redshifts and imaging data (measured angles). We need to measure one of the distances in the model independently so we can set the scale and hence get H0. Radial velocity = H0 distance S O L DOS big, small H0 angular separation on sky is the same in both cases S L DOS small, large H0 O
I t A B t Using the detailed shape of how the light varies with time for the various lensed images we can determine the time difference Δt and therefore the path distance.
Gravitational Lensing: Measuring Ho For a variable source (and many lensed QSOs are variable) we can measure the time delay if we have two or more image paths. Δt = (distance)/c, and because Ho 1/ (distance) we are able to measure Ho directly. After many years of observation the time delay (417 days) for the two images from the gravitationally lensed quasar 0957+561 was determined with ~3% accuracy. The time delay directly gives the path difference which is essentially a standard ruler that allows us to determine H0. The largest uncertainty is the lens model.
Gravitational Lensing: Measuring Ho On the left of this image is the lensing galaxy and four images of the lensed quasar. After subtraction of the five bright images we can see most of the Einstein ring. Such a system of four images plus a ring constrains the model for the galaxy mass distribution very well and so allows an improved estimate of H0 using the observed time delays. This should be contrasted with 0957+561 which had a galaxy plus cluster and only two quasar images
Sunyaev-Zeldovich effect X-ray emission Very rich galaxy clusters contain hot, tenuous gas (107-108 K) which emits Bremsstrahlung (free-free radiation) n V S 2 4 DL 2 cooling function n particle density V volume The S-Z effect The gas is ionized and the electrons scatter CMB photons. scattering changes energy of CMB photons
Sunyaev-Zeldovich effect (contours) and X-ray emission (colour) of the galaxy cluster Abell 2218 So how does the SZ effect help us determine Ho? http://astro.uchicago.edu/~kerry/sze_parameters.html http://astro.uchicago.edu/~kerry/sze_parameters.html
Probability for scattering, optical depth ne T dl is the scattering cross section ne is the electron density 3 Measure X-ray flux 2 ne c H0 n 2 c e 2 H0 c H 0 c ne Measure for the CMB (SZ effect) H0 So with these two measurements can get ne and H0! Results from gravitational lensing and SZ effect are in agreement with results from Hubble diagram but they have larger error bars.
Rich Clusters of Galaxies as Standard Rulers The x-ray radiation is free-free emission from a hot gas. This is due to one electron being slowed down by another one so the emission is a two electron event and obeys. 2 E ne dl However, if we look at the CMB through the cluster, a photon experiences an increase in energy due to inverse-compton scattering by an electron. This is a single electron event and obeys E E A ne dl We can measure the x-ray emission and the CMB energy change and deduce the density weighted path length through the cluster, A2/E = L. We can also measure the angular size of the x-ray emitting gas in the cluster, Θ which we can associate with distance L by assuming the cluster is spherical. Applying this to distant clusters gives the angular diameter distance from Θ and L. Given a sample of clusters with a range of distances it is possible to make a Hubble diagram.
The Sunyaev-Zel dovich (S-Z) Effect The gas density is very low and so multiple electron-photon collisions can be neglected. The optical depth is given by τe = ne σt L where ne is electron density, σt is the Thompson cross-section, and L is the path length. Electrons dominate the cross-section because the electron-photon crosssection is >> the nuclei-photon cross-section. A cluster contains many fast moving electrons and their interaction is with lower energy photons. This leads to inverse-compton scattering in which the photons gain energy. The frequency shift for a CMB photon scattered by an electron is given by: kt E E e me c 2 The photon energy increases and therefore the apparent temperature of the CMB also increases.
Rich Clusters of Galaxies: The Sunyaev-Zel dovich (S-Z) Effect This figure shows the distortions that one gets as a consequence of the S-Z effect both in terms of the change in flux that one sees and also in terms of the brightness temperature observed. The consequence is that the (remarkably uniform) cosmic microwave background radiation is distorted by the presence of a cluster of galaxies and this can be detected at radio wavelengths. At high frequencies the CMB intensity and temperature are increased by the cluster whereas at low frequencies they are decreased. =30Ghz is =1cm, observe here Kinetic SZE is due to bulk motion of the whole cluster wrt the CMB rest frame. The thermal SZE is due to the particle motion of cluster gas wrt the CMB rest frame.
The Sunyaev-Zel dovich (S-Z) Effect Radio telescopes are used therefore to look for 'dips' in the background in order to identify clusters independently of any concerns of galaxy overdensity. By combining these data with x-ray measurements of clusters we can measure the Hubble constant, Ho. However, quantifying the decrement is not easy since the effect is only of the order of ~ 10-4 even for the richest, most massive clusters.
The Sunyaev-Zel dovich (S-Z) Effect There is a good correlation between the S-Z effect and the distribution of x-ray emission over a cluster of galaxies. An example is shown in this figure. Here the S-Z effect data are shown as contours which overlay the image of xray emission in false colours for the Galaxy cluster CL0016+16. Generally however, it is very difficult to detect. The lower image shows the background fluctuations in another cluster, Abell 401. The full width half maximum resolution is just over six minutes of arc and the peak temperature difference that is detected is only 300 μk. The noise level is approximately 20μK.