Measuring clustering …
•! We want a way to quantify the amount of structure
that we see on various scales
•! The most common way of doing this is to measure the
two-point correlation function !(r)
•! We calculate the correlation function by estimating the
galaxy distances from their redshifts, correcting for
any distortions due to peculiar velocities, and counting
the number of galaxies within a given volume
•! Mathematically, the probability of finding a galaxy
within a volume "V1 and a volume "V2 is
Measuring clustering …
•! "P = n2[1+ !(r12)]"V1" V2
–! When !(r) > 0, then galaxies are clustered (which is what we see)
–! On scales of < 50h-1 Mpc, we can parameterize the correlation
function as a power-law: !(r) ~(r/r0)-# where #>0
–! Thus the probability of finding one galaxy within a distance r of
another is significantly increased (over random) when r< r0. r0 is
called the “correlation length”.
–! Note that the 2 point correlation function isn’t good for describing
one-dimensional filaments or two-dimensional walls. We need 3
and 4 point correlation functions for those. Much harder!
–! From the SDSS: r0=6.1 +/- 0.2 h-1 Mpc, #=1.75 over the scales 0.1 –
16 h-1 Mpc
–! "P = n2[1+ !(r12)]"V1" V2
–! Where n is the average spatial density of galaxies (number
per Mpc3) and r12 is the separation between the two regions
Measuring clustering …
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The Fourier transform of !(r) is the power spectrum
–! P(k), P(k) = 4$% !(r) [sin(kr)/kr] r2dr
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Las Campanas Redshift Survey
k is the wavenumber, small values of k correspond to large physical
scales
P(k) has the dimensions of volume. It will be at maximum close to the
radius r where !(r ) drops to zero.
Roughly speaking the power spectrum is a power-law at large k (small
physical scales) and turns over at small k (large physical scales)
We can combine information from different measurements (redshift
surveys, CMB, Ly& forest, weak lensing) to trace P(k) over a large range
of physical scales
The power spectrum provides strong constraints on the amount and
type of dark matter and dark energy in the universe
Measuring clustering …
•! We would also like to know how well the galaxies trace the mass
distribution, or in other words how biased are the galaxies
relative to the dark matter
•! We generally assume that the two densities are linearly related
such that:
–! Let 'x= '(x/(x be the density fluctuation of a given population
–! Linear biasing for galaxies implies 'galaxies=b'dark matter
–! Biasing may be a function of scale and of galaxy luminosity
•! We can measure relative biasing by measuring the power
spectrum of different populations
SDSS Power Spectrum, Tegmark et al (2004)
SDSS Power Spectrum, Tegmark et al (2004)
Biasing in the SDSS, Tegmark et al (2004)
Peculiar velocities & Bulk Flows
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Large scale structure causes peculiar velocities (different from the
Hubble Flow)
We can measure these if we have accurate distances to the galaxies by:
–! Vr = H0d + Vpec – so if we measure distance and radial velocity (and
assume the Hubble Constant) we can measure the peculiar velocity
of a galaxy
We are moving towards the Virgo cluster at ~270 km/s, this is called the
“Virgocentric infall”
We also measure a dipole anisotropy in the cosmic microwave
background which implies that the local group is moving at ~620 km/s
towards b=27, l=268.
This is due to a combination of our infall towards Virgo and the entire
Local Supercluster moving towards the general direction of the HydroCentaurus Supercluster (the Great Attractor)
–! Flows of superclusters are known as “bulk flows”
–! Measurements of the velocity field of galaxies can help put
constraints on the underlying mass field ==> measurement of dark
matter over large scales
Combined power spectrum, Tegmark et al (2004)
Dipole anisotropy in the Cosmic Microwave Background
(from COBE 1992). We are moving wrt. to the CMB at
~620 km/s !!
Corrections for Virgo infall, SBF distances, Tonry et al.
Bulk flows, Courteau & Dekel 2001
Bulk flows, Courteau & Dekel 2001
Bulk flows, Aaronson et al 1986
Great
Attractor
The GA lies in the “zone of avoidance”, it’s
hard to study …
Dark Matter
•! Inventory of the universe --2008 WMAP (+BAO+SNe) results:
–! )total=1.0052+/- 0.00064 (the universe is spatially flat!)
–! )*=0.721 +/- 0.0015 (but most of it is made of dark energy!)
–! )matter=0.279 +/- 0.015 (and even the matter is confusing …)
•! )baryon = 0.0462+/- 0.0015 but note that the baryon fraction observed in
stars and gas is only )* ~ 0.005 (so there must be some baryonic dark
matter)
•! )dark matter = 0.233 +/- 0.013 (and there is a LOT of non-baryonic matter!)
–! The evidence for dark matter has been with us significantly longer
than that for dark energy, so the field is more mature.
•! There are lots of dark matter candidates, lots of fun for particle
physicists!
–! But it’s always possible that we don’t understand gravity, so we
need to modify gravity (MOND – Modified Newtonian Dynamics)
ACO 3627,
Heart of the GA?
Evidence for Dark Matter
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X-ray halos of Elliptical Galaxies
(Possibly) dynamics of globular clusters and planetary nebulae around
elliptical galaxies (conflicting answers here!)
Flat rotation curves of spiral galaxies (best evidence!)
Kinematics of dwarf galaxies
Measurements of galaxy masses using “binary galaxies”
Measurements of the mass of galaxy clusters via:
–! X-ray gas
–! Motions
–! Gravitational lensing
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Measurements on the largest scales from peculiar velocities
Measuring the mass from peculiar velocities
•! Assume that the measured peculiar velocities are generated
locally
•! Then a galaxy only feels the gravitational pull of the nearest
large mass concentration
•! The velocities are related to the local gravitational potential,
which in turn is due to the mass distribution
•! Compare the observed velocity field to a density field (derived
from a galaxy redshift survey) and derive the matter density
distribution
•! Most results favor )dark matter < 0.3. The universe cannot be flat
(unless we have dark energy!)
Bullet Cluster: hot x-ray gas (red) vs dark matter (blue)
HC
Density contours from POTENT (Dekel 1994) and IRAS
redshift survey (Strauss & Willick 1995)
Density contours from POTENT and IRAS
redshift survey (Sigad et al. 1997)
Dark Matter (Baryonic)
•! Baryon inventory: from Big Bang nucleosynthesis calculations
and the observations of light element abundances (deuterium,
helium, and lithium) we can constrain )baryon = 0.04 +/- 0.01
(more soon).This is the constraint from the very early universe,
z~109.
Dark Matter (Baryonic)
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What are the missing baryons?
Some candidates:
–! Some of it may be in low density warm/hot intergalactic medium (WHIM) that
is difficult to measure (perhaps in the x-ray?)
–! Galaxies we don’t see (Low Surface Brightness Galaxies)
–! Cold dense clumps of hydrogen, of Jupiter-like masses ~10-3 M! located in
the halo. Microlensing surveys put strong constraints on the amount of mass
that can be present in these clumps. Also unless they are reheated (by
cosmic rays?) they would collapse.
–! Massive Compact Halo Objects (MACHOs)
–! These calculations are confirmed by the fluctuations in the Cosmic
Microwave Background (again, more soon!)This measures )baryon at
z~1000.
–! And observations of the Ly& forest – measures )baryon at z~3.
•! Baryonic matter is made of stuff we understand – neutrons &
protons (nucleons).
•! Everything we can observe directly is baryonic – stars, galaxies,
hot x-ray gas, etc.
•! White dwarfs
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–! But today (z~0) we only observe ) = 0.0024 +/- 0.001 in cold stars
and cold interstellar matter, and ) = 0.0026 in hot intracluster gas in
clusters.
–! So we only directly observe ~12% of )baryon at low redshift
Brown dwarfs
Planets
Black holes (stellar remnants)
The list goes on, but we can place some observational constraints…
Dark Matter (Baryonic)
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First unambiguous detection of a brown dwarf, 1995!
We can attempt to detect MACHOs via microlensing experiments
As a MACHO passes between us and a halo star, it will briefly amplify
the light due to gravitational lensing
Need to continously monitor a large number of stars to search for such
events, they are rare!
MACHO project started monitoring the LMC and fields in the Milky Way
bulge from 1992 to 1999 using a 50 inch telescope at Mt. Stromlo
observatory.
Other projects along these same lines include OGLE and AGAPE
Results indicate that machos may make up to 20% of the halo mass, the
masses are similar to white dwarf masses
Dark Matter (Baryonic)
•! Some more “exotic” baryonic dark matter candidates:
–! Remnants of Pop III stars, very massive objects (VMOs) with
M=103 – 106 M!. These remnants might coalesce into one
supermassive black hole (central engine of QSOs?)
•! But we would probably notice these if they were in the halo of
the Milky Way as they would move in and out of the plane of the
galaxy (~1 every 100 years or so)
–! Quantum black holes (primordial, created by quantum
fluctuations in the very early universe)
•! These are tiny with M~1012 kg, r=10-13 cm
•! Probably not though …
Microlensing events in the LMC, Sutherland et al 1996.