•parallax •cepheid variables •type ia super novae · note that the first point is actually...

Physical Cosmology 2011/2012

Lecture 9

• H0 from the Hubble diagram–Basics

–Measuring distances•Parallax

•Cepheid variables

•Type Ia Super Novae

• H0 from other methods–Gravitational lensing

–Sunyaev-Zeldovich effect


H0 from the Hubble diagram

proper distance rRkr

drRs

020

1

pec0 vv sHs

rRrRR

R

rRrRs

00

0

0

00

v

s

large H0

small H0

“Hubble flow” Idea is very simple: Measure the velocities anddistances of a set of objects, plot graph andmeasure the slope → H0 (units kms-1 Mpc-1).

[kms-1]

[Mpc]


Typically vpec is of the order of a few hundred km/s (let’s say it’s500 kms-1). It’s bigger inside galaxy clusters than in between theclusters. Using this value and H0=72 kms-1 Mpc we can write

1

110

Hubble kmsMpc7Mpckms72

500v

sH

have to go beyond 7 Mpc for vHubble> vpec

At large distances v z s DL

z

z

L

zz

dz

H

cz

zH

dzczzD

02

1

,02

,0mat,03

mat,00

0

)1()1()1()1(

)()1()(


will depend on othercosmological parametersfor 5.0z

large H0

small H0

z

DL

Measuring velocities is easy and accurate

I

z,v


Arrows show theshift of theCalcium H and Klines


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 graph465±50 km/s/Mpc or513±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 areredshifted

• 1925: Hubble identifies Cepheid variables in the nearbygalaxies M31 and M33

• 1927: Arthur Holmes – “age of Earth’s crust is 1.6 – 3.0billion years”.

• 1929: Hubble’s constant first measured: value of 500km/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 togalaxies derived and published by Hubble. The second is from H. Robertson, also based primarilyon Hubble's data. Hubble himself finally weighed in in 1929 at 500 km/s/Mpc. Also, in 1930, theDutch astronomer, Jan Oort, thought something was wrong with Hubble's scale and published avalue of 290 km/s/Mpc, but this was largely forgotten.

Source: www.cfa.harvard.edu/~huchra

The first major revision to Hubble's value wasmade in the 1950's due to the discovery ofPopulation II stars by W. Baade. That wasfollowed by other corrections for confusion, etc.that pretty much dropped the accepted valuedown to around 100 km/s/Mpc by the early1960's.


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 aboutMV ~ -5

Type Ia super novae are the best for large distances. They havean absolute magnitude of MV ~ -19.3

Standard candles have to be accurately calibrated in order toremove systematic errors.

Measuring distances


The Distance Ladder diagram

The distance scale “ladder”

PhotometricGeometric


Hubble’s law: systematic errors

• Measured distances mostly depend onm – M = 5 log(d/10) + constant(where d is luminosity distance)

• getting M wrong changes d by a factor of

which does not affect linearity (it just changesthe slope)

• typical of the nature of systematic errors: verydifficult to spot– Oort(1931) expressed doubts very quickly

– Baade(1951) showed Hubble error

5est10 MM


Parallax measurement

Sun

Earth

LMC

For LMC the parallax angle is tiny

DEarth-Sun= 1 A.U. = 1 Astronomical Unit = 1.51011 m

DLMC = 52 kpc = 1.61021m

arcsec102.4102 510

moon degree0.5arcsec1800


Magnitudes and distance moduli

Observational astronomers use a logarithmic scale for fluxescalled apparent magnitudes

const.log5.2 Sm

S m The constant varies strongly forbetween different magnitude systems

Visible stars have apparent magnitude m = 1 → 6

The faintest galaxies observed have mV ~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 haveat a distance of 10pc. In SI units this gives

const.2.90log5.2 LM

The distance modulus is defined as

45.87log5 L DMm

21

4

S

LDL

with

This is as odd as it looks but is still used by observationalastronomers.


Type I and Type II Cepheids

Unfortunately, there are actually two types of Cepheid withdifferent P-L relations.

• Type I Cepheids have a similar chemical compositionto the Sun.

• Type II Cepheids are lower metallicity since they areolder and are formed from more primordial material.

To make life really tricky, this means that Type I Cepheidsare found in the Milky Way and Type II in globular clusters.

This meant that first estimates of distances to Globularswere wrong since Type II Cepheids are about 1.5magnitudes (~4 times) fainter.


Period-Luminosity Relation for Cepheids and RR Lyrae

www.astro.livjm.ac.uk/courses/phys134/


The Large MagellanicCloud (LMC)

Irregular dwarf galaxyin the Local Group ofgalaxies


Uncertainties in thedistance to the Large

Magellanic Cloud(LMC)

From Mould et al, 2000, ApJ,529, 786.

Distribution ofpublished LMCdistance modulifrom the literature.


Measuring the Hubble constant involves several steps

• Measure distance to LMC to calibrate the Cepheid(P-L) relation (DLMC= 52kpc +/- 1 kpc)

• Parallax measurement of Cepheids by Hipparcossatellite. Only a few because Hipparcos parallaxmeasurements 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 distanceswith type Ia SN


Problems:

• peculiar velocities

• dust absorption – if objects become fainterbecause their light is absorbed you think they arefurther away than they are Hubble constantunderestimated

• Malmquist bias, pick systematically brighterobjects at high redshift Hubble constantoverestimated

• Calibration is tricky


Results:

2 Groups:

Tamman et al.: H0 = 60 +/-2.3 kms-1Mpc-1

Freedman et al.: H0 = 72 +/- 3 (statistical)+/- 7 (systematic)kms-1 Mpc-1

Discrepancy is mainly due to the different calibrationof 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 theChandrasekhar mass limit (~1.4 times the mass of the Sun). Whitedwarf is in a binary system and it gains mass by accreting materialfrom its companion. Just before it reaches the mass limit there’s arunaway nuclear explosion (fusion of Carbon). Because theChandrasekhar 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 eventpeople use the fact that the duration of the SN “flash” is correlatedwith the luminosity - empirical calibration of absolute luminosity.

Best standard candle known but rare, so not useful at smalldistances.


http://www-supernova.lbl.govhttp://www-supernova.lbl.gov


http://www-supernova.lbl.govhttp://www-supernova.lbl.gov

Hubble diagram fromthe SupernovaCosmology Project

mB is a measure for DL

Gravitational lensing: Determining H0

S

S

LO

OL

DOS small,large H0

DOS big,small H0

• Measuring the redshifts (i.e. radialvelocities) of the lens and the sourcecombined with an adopted cosmology(i.e. H0) defines exactly the geometry.

• This means we can determine theobserver-lens distance, the observer-source distance and the lens-sourcedistance.

• A model for the lens mass distributioncan be constructed that accuratelypredicts the observed lens images.

• We can also estimate the difference inpath length from the source to theobserver that corresponds to eachlensed image.

• However, all the distances aredependent on the assumed H0.

• For example, both cases shown on theright are consistent with the measuredredshifts and imaging data (measuredangles).

• We need to measure one of thedistances in the model independently sowe can set the scale and hence get H0.

angular separation onsky is the same in both

cases

Radial velocity = H0×distance


t

A BI

t

Using the detailed shape of how the light varies with time forthe various lensed images we can determine the time differenceΔt and therefore the path distance.

Gravitational Lensing: Measuring Ho

• For a variable source (and manylensed QSOs are variable) we canmeasure the time delay if we havetwo or more image paths.

• Δt = (distance)/c, and becauseHo 1/(distance) we are able tomeasure Ho directly.

• After many years of observation thetime delay (417 days) for the twoimages from the gravitationallylensed quasar 0957+561 wasdetermined with ~3% accuracy.

• The time delay directly gives thepath difference which is essentially astandard ruler that allows us todetermine H0.

• The largest uncertainty is the lensmodel.

Gravitational Lensing: Measuring Ho

• On the left of this image isthe lensing galaxy and fourimages of the lensedquasar.

• After subtraction of thefive bright images we cansee most of the Einsteinring.

• Such a system of fourimages plus a ringconstrains the model forthe galaxy massdistribution very well andso allows an improvedestimate of H0 using theobserved time delays.

• This should be contrastedwith 0957+561 which hada galaxy plus cluster andonly two quasar images


Sunyaev-Zeldovich effectX-ray emissionVery rich galaxy clusters contain hot, tenuous gas(107-108 K) which emits Bremsstrahlung (free-freeradiation)

2

2

4 LD

VnS

cooling functionn particle densityV volume

The S-Z effectThe gas is ionized and the electrons scatter CMB photons.

scattering changes energy of CMB photons


Sunyaev-Zeldovicheffect (contours) andX-ray emission(colour) of the galaxycluster Abell 2218

So how does the SZeffect help usdetermine Ho?

http://astro.uchicago.edu/~kerry/sze_parameters.htmlhttp://astro.uchicago.edu/~kerry/sze_parameters.html


Results from gravitational lensing and SZ effect are inagreement with results from Hubble diagram but they havelarger error bars.

Measure for the CMB (SZ effect)

So with these two measurements can get ne and H0!

0H

cne

Probability for scattering, optical depth

dln Te is the scattering cross sectionne is the electron density

Measure X-ray flux0

2

2

0

3

0

2

H

cn

Hc

Hcn

e

e

Rich Clusters of Galaxies as Standard Rulers

• The x-ray radiation is free-free emission from a hot gas. This is due to oneelectron being slowed down by another one so the emission is a twoelectron event and obeys.

• However, if we look at the CMB through the cluster, a photon experiencesan increase in energy due to inverse-Compton scattering by an electron.This is a single electron event and obeys

• We can measure the x-ray emission and the CMB energy change anddeduce 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 thecluster, Θ which we can associate with distance L by assuming the clusteris spherical.

• Applying this to distant clusters gives the angular diameter distance fromΘ and L.

• Given a sample of clusters with a range of distances it is possible to makea Hubble diagram.

dlnE e 2

dlnAE

Ee

The Sunyaev-Zel’dovich (S-Z) Effect

• The gas density is very low and so multiple electron-photon collisions canbe 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 cross-section is >> the nuclei-photon cross-section.

• A cluster contains many fast moving electrons and their interaction is withlower energy photons. This leads to inverse-Compton scattering in whichthe photons gain energy.

• The frequency shift for a CMB photon scattered by an electron is givenby:

• The photon energy increases and therefore the apparent temperature of theCMB also increases.

2cm

kT

E

E

e

e

Rich Clusters of Galaxies: The Sunyaev-Zel’dovich (S-Z) Effect

• This figure shows thedistortions that one gets as aconsequence of the S-Zeffect both in terms of thechange in flux that one seesand also in terms of thebrightness temperatureobserved.

• The consequence is that the(remarkably uniform)cosmic microwavebackground radiation isdistorted by the presence ofa cluster of galaxies and thiscan be detected at radiowavelengths.

• At high frequencies theCMB intensity andtemperature are increased bythe cluster whereas at lowfrequencies they aredecreased.

=30Ghz is =1cm, observe here

•Kinetic SZE is due to bulk motion of the whole cluster wrtthe CMB rest frame.•The thermal SZE is due to the particle motion of cluster gaswrt the CMB rest frame.


• Radio telescopes are usedtherefore to look for 'dips' in thebackground in order to identifyclusters independently of anyconcerns of galaxy over-density.

• By combining these data withx-ray measurements of clusterswe can measure the Hubbleconstant, Ho.

• However, quantifying thedecrement is not easy since theeffect is only of the order of ~10-4 even for the richest, mostmassive clusters.


• There is a good correlation between theS-Z effect and the distribution of x-rayemission over a cluster of galaxies. Anexample is shown in this figure.

• Here the S-Z effect data are shown ascontours which overlay the image of x-ray emission in false colours for theGalaxy cluster CL0016+16.

• Generally however, it is very difficultto detect.

• The lower image shows the backgroundfluctuations in another cluster, Abell401.

• The full width half maximum resolutionis just over six minutes of arc and thepeak temperature difference that isdetected is only 300 μK. The noiselevel is approximately 20μK.

•parallax •cepheid variables •type ia super novae · note that the first point is actually...

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