large scale structure matter perturbations beyond cold matter probes scott dodelson pasi 2006
TRANSCRIPT
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Large Scale Structure
Matter Perturbations Beyond Cold Matter Probes
Scott DodelsonPASI 2006
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Comoving Distance
Coordinate difference (comoving distance) is not the same as physical distance
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Can rewrite conformal horizon as integral over
Hubble radius (aH)-1
Perturbations outside the
horizon
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During inflation, fluctuates quantum mechanically around a smooth background
The mean value of is zero, but its variance is
Get contributions from all scales equally if
with n=1 (scale-invariant spectrum)
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Relation of potential to overdensity
32
2 ~~
aa
k
mG42 m
mx
)(
Then Poisson’s equation
in Fourier space becomes
Two results:
)()()( 24 kTkkPkkP n
~~a
Define
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How do perturbations evolve?
Matter only:
042 mGH
where δ=(ρ- ρm)/ ρm
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Non-Expanding Universe
04 mGH=0, so
If the universe is not expanding, the matter density is constant, so
mGte 4exponential growth, with time scale of order H0
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Expanding, Matter Dominated Universe
042 mGH
)3/(22/34 22 tHG m
In this universe, H=(2/3t) and
So,
03
2
3
42
tt
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Can solve this analytically
03
2
3
42
tt
03
2
3
4)1( ppp
Assume a solution of the form: δ=tp ; get an algebraic equation for p
with solutions:
3
8
9
1
2
1
6
1p
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Two modes: growing and decaying
Growing mode scales as a~t2/3
Gravitational accretion fights against dilution due to the expansion of the universe: Exponential growth changed to power law growth
Recall: ~~a
Gravitational wells remain constant in a matter dominated universe
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Beyond Cold Dark Matter
Dark Energy (important at late times z~1)
Radiation (important at early times z>1000)
Baryon Acoustic Oscillations (Remnant of Pre-recombination era)
Neutrino Mass (operates at all times)
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Dark EnergySolve the same differential equation
042 mGH
accounting for the new H(t) relation
Suppression in growth due to smooth component of the universe
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Radiation Dominated Era
03
4 2
k
Newton’s equations - with radiation as the source - reduce to
with analytic solution
33/
)3/cos()3/()3/sin()0(3)(
k
kkk
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Expect less power on small scales
For scales that enter the horizon well before equality,
So, we expect the transfer function to fall off as
23/
3/cos)0()(
EQ
EQEQ
k
k
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ConsequencesFor a scale invariant spectrum
On large scales,On small scales,
Log since structure grows slightly during radiation era when potential decays
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Power Spectrum sensitive to matter density
The turnover scale is the one that enters the horizon atthe epoch of matter-radiation equality:
Therefore, measuring the shape of the power spectrumwill give a precise estimate of mh
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Baryon Acoustic Oscillations
Eisenstein et al. 2006Baryons
Dark Matter
Apparent position of bump related to actual size (which is known!) and distance to galaxies at intermediate redshifts
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Neutrinos affect large scale structure
Since we know the neutrino abundance, we cancompute the energy density of a massive neutrino
This fraction of the total density does not participatein collapse on scales smaller than the freestreaming scale
At the relevant time, this scale is 0.02 Mpc-1 for a 1eV
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Qualitatively …
Colombi, Dodelson, & Widrow 1995
Structure is smoothed out in model with light neutrinos
CDM WarmDM C+HDM
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Even for a small neutrino mass, get large impact on structure: power spectrum is excellent probe of neutrino mass!
Quantitatively …
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Gravitational Lensing
Galaxy distribution
Lyman alpha forest
Probes of The Power Spectrum
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Deflection of Light first proposed by Einstein in
1912!
Einstein writes to George Hale (Director of Mount Wilson Observatory) in 1913. He mentions the 0.84’’ (2GM/Rc2) deflection expected from the Sun.
Wambsganss 1998
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The next total solar eclipse was August 21, 1914. An expedition was sent to observe in the region of greatest eclipse …
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Russian Crimean Peninsula
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1914 was not a good time to start a scientific expedition
in Europe
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The astronomers were captured by Russian soldiers and released a month
later … with no data
… which in retrospect is a good thing. Einstein improved his theory over the next several years. He eventually concluded that the deflection should be twice as large as the Newtonian result … And this was confirmed by the famous expeditions in 1919.
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Geodesic Equation
Affine parameter can be replaced by comoving distance
Since the transverse components are i , the geodesic equation becomes
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Evaluate the Christoffel Symbol
Derivative wrt a on the left cancels the second term on the right
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Consider the geometry
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With this boundary condition
with kernel
Define the distortion tensor
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Distortion Tensor
is the projected density, a measure of the convergence of light rays. I are the two components of shear.
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Example: Magnification
But
So
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Move beyond point images (QSOs) to extended objects
(galaxies)
HST CL0024
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Lensing producing elliptical images
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Move from Strong Lensing (multiple images) to Weak Lensing (small changes in shapes of extended
objects)
Jain, Seljak, & White (2000)
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Cosmic Shear field depends on cosmology: one of these has more
matter than the other
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Apply Limber formula for the Power Spectrum
One of these combinations - the B mode - vanishes. The other - the E mode
Has a power spectrum
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We can compute this power spectrum with knowledge of the nonlinear 3D
power spectrum
Dodelson, Shapiro, & White 2005
Points from ray tracing through a numerical simulation
Curve from integrating nonlinear power spectrum
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Need to measure Amplitude of Fluctuations
in Shear
Van Waerbeke & Mellier 2003
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Constraints on parameters
Contaldi, Hoekstra, Lewis: astro-ph/0302435
Matter Density
Amplitude of Matter fluctuations
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Several Upcoming Surveys
Panstarrs
LSSTSNAP
Dark Energy Survey
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What can we expect?
Hu & Tegmark 1999
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Tomography
Hu 2002
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Interesting Degeneracies
Abazajian & Dodelson 2003
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Sloan Digital Sky Survey
2.5 meter telescope in Apache Point, New Mexico
Collaboration of: Fermilab, Princeton, U. Chicago, U.Washington, Johns Hopkins, New Mexico State, Max Planck, Japan, Pittsburgh, …
Scheduled to end in 2005; has been extended until 2008; will cover ¼ of the sky
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Two surveys in one
Photometric survey: hundreds of millions of objects in 5 bands
Spectroscopic survey: ~1 million objects with spectra
Spectroscopic survey targets objects found in photometric survey. Reduces systematic effects (typically objects targeted for redshifts are found in different survey, leads to complicated selection function).
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5 Filters very efficient
Ultimately will get redshifts for ~750,000 galaxies; 100,000 QSOs
i’ and z’ bands especially important for high redshift QSOs. Lyman alpha line (1215Ang) redshifted to 1215*(1+z) Ang. Can get z>6 QSOs.
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SDSS Galaxy Power Spectrum
Tegmark et al. 2004
Corrects for luminosity bias
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In these probes [and all others], the observables arecomplicated functionals of the easy-to-predict lineardensity field, L.
N-Body interactions in Newtonian gravity
Galaxy formation including hydro, feedback from SN, star formation, …
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Simple biasing scheme valid on large scales
Assumed to hold on scales k≤0.2 h Mpc-1
Bias unknown so must be fit for: give up hope of determining amplitude of the power spectrum Cosmological constraints come from power spectrum shape
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Constraints on Neutrino Mass
Use as variables:
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Cmbgg OmOlCMB
Tegmark et al. 2004
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Cmbgg OmOlCMB
+
LSS
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Lyman alpha forestPhotons with energy > (n=1 to n=2 transition energy) get absorbed along the line of sight as they lose energy due to cosmic redshift.
Every absorption line corresponds to cloud of neutral hydrogen.
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Fluctuations in forest trace fluctuations in density
Gnedin & Hui, 1997
Flux
Baryon Density
Position along line of Sight
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Lyman alpha observes universe at early times
Sloan Digital Sky Survey (SDSS)
At high redshift, even small scales were linear!
Redshifts of Absorbers
Num
ber
of S
pect
ra
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SDSS Spectra of 3300 Quasars
McDonald et al. (2004)
11 redshift bins1D Power Spectrum of the Flux
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This is only half the battle!
Want to test cosmology Need to run simulations
which generate 1D flux spectra for every parameter set
Do likelihood analysis to see which simulations are closest to observations
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Constraints on running
WMAP+ACBAR+CBI+ SDSS Lyman alpha 7 cosmological parameters Consistent with no running
Abazajian et al (March 19, 11:10 CST)
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Hoping to resolve the issue, researchers are once again turning to quasars. So far, the results have been inconclusive: Two groups analyzing the same quasar data
have come up with starkly different answers. One, represented by Fermilab's Hui, sees no deviation from scale invariance. The other team, which included
Princeton University's Uros Seljak, claims to have spotted not only a significant deviation from scale invariance but also a change in the spectral index over
different scales, a quantity known as the "running" of the spectral index, far larger than most inflation models predict. If Seljak's team is correct, almost all inflationary
theories can be ruled out right away. Most physicists, however, are skeptical. Hui suggests that the differences between the two groups' conclusions arise from
differing assumptions about the properties of the telescopes as well as assumptions that went into the computer models that contribute to the analysis. "We're trying to
get to the bottom of it," he says.
Science, Vol 300, Issue 5620, 730-731 , 2 May 2003
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Conclusions
Coherent/Beautiful picture of formation and evolution of lumpy universe from smooth origins Requires Dark Matter Strong Constraints on Neutrino Mass Comparing observations with theory is very complex; Weak lensing is promising
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Notice the difference between these 2 pictures
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How can we extract information from the non-Gaussianity?
• Compute N-point functions: e.g. Bispectrum is
• Several groups have shown that there is much cosmological information stored in the bispectrum (e.g., Hui 1999; Takada & Jain 2004)
• Bispectrum vanishes at zeroth order• Need to be careful when computing
perturbatively (Dodelson, Kolb, Mataresse, Riotto, & Zhang 2005)
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We theorists have work to do!
Dodelson, Huterer, & Zhang 2005
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Super-horizon modes remain constant
Small decay through the transition era: radiation domination to matter domination
The time parameter y=a/aEQ