1 1 eric linder university of california, berkeley lawrence berkeley national lab course on dark...
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Eric Linder University of California, BerkeleyLawrence Berkeley National Lab
Course on Dark EnergyCourse on Dark Energy Cosmology at the Beach 2009Cosmology at the Beach 2009
JDEM constraints
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OutlineOutline
Lecture 1: Dark Energy in Space
The panoply of observations
Lecture 2: Dark Energy in Theory
The garden of models
Lecture 3: Dark Energy in your Computer
The array of tools – Don’t try this at home!
In theory, there is no difference between theory and practice. In practice, there is. - Yogi Berra
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Solving the Equation of MotionSolving the Equation of Motion
Klein-Gordon equation
Transform to new variables
Autonomous system
where
Transform solution to
Copeland, Liddle, Wands 1998 Phys. Rev. D 57, 4686
Can add equation for EOS dynamics
Caldwell & Linder 2005 Phys. Rev. Lett 95, 141301
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Equation of State DynamicsEquation of State Dynamics
For robust solutions, pay attention to initial conditions, shoot forward in time, use 4th order Runge-Kutta.
For monotonic , can switch to as time variable, defining present as, e.g. =0.72.
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Asymptotic BehaviorsAsymptotic Behaviors
Asymptotic behaviors can be physically interesting. Solve for critical points x(xc,yc)=0, y(xc,yc)=0. Check stability by sign of eigenvalues p=Mp.
Copeland, Liddle, Wands 1998 Phys. Rev. D 57, 4686
Crossing w=-1:Relevant to fate of universe.
Phantom fields roll up potential so V>0, so wtot
∞<-1. Cannot cross w=-1 even with coupling. Quintessence can cross with coupling since w<wtot.
p={x,y}
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From Data to Theory (and back)From Data to Theory (and back)
Fisher matrix gives lower limit for Gaussian likelihoods, quick and easy.
Fij = d2(- ln L) / dpi dpj = O(dO/dpi) COV-1 (dO/dpj)
(pi) 1/(Fii)1/2
Example: O=dlum(z=0.1,0.2,…1), p=(m,w), COV=(d/d)d ij
Fw=k(dOk/d)(dOk/dw)k-2
2() COV(,w)
COV(,w) 2(w)C = F-1 =( )F Fw
Fw Fww
F = ( )Also called information matrix. Add independent data sets, or priors, by adding matrices.
e.g. Gaussian prior on m=0.280.03 via 2 = (m-0.28)2/0.032
See: Tegmark et al. astro-ph/9805117 Dodelson, “Modern Cosmology”
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Survival of the FittestSurvival of the Fittest
Fisher estimates give a N-dimension ellipsoid. Marginalize (integrate over the probability distribution) over parameters not of immediate interest by crossing out their row/column in F-1. Fix a parameter by crossing out row/column in F.
1 (68.3% probability enclosed) joint contours have 2=2.30 in 2-D (not 2=1). Read off 1 errors by projecting to axis and dividing by 1.52=2.30.
Orientation/ellipticity of ellipse shows degree of covariance (degeneracy).
Different types of observations can have different degeneracies (complementarity) and combine to give tight constraints.
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Bias from SystematicsBias from Systematics
Fisher estimation calculated around fiducial model, but can also compute bias due to offset (systematic).
Bias p in parameter p is related to offset O in observable, through U=O/p and covariance matrix C=O O. For diagonal covariance, simplifies to:
In statistics, often combine uncertainty and bias into Risk parameter:
R(p) = [2(p)+p2]1/2
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Design an ExperimentDesign an Experiment
Precision in measurement is not enough - one must beware degeneracies and systematics.
p2
p1
*
.
Degeneracy: e.g. Aw0+Bwa=const
Degeneracy: hypersurface, e.g. covariance with m
Systematic: offset error in data or model, e.g. evolution
or Systematic: floor to precision, e.g. calibration
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Orthogonal Basis AnalysisOrthogonal Basis Analysis
Eigenmodes: w(z) = i ei(z) For orthogonal basis, errors (i) are uncorrelated. “Principal components”.
Start with parameters {wi} in z bins. Diagonalize Fisher matrix F=ETDE: D is diagonal, rows of E give eigenvectors. NOTE: basis differs with model, experiment, and probe -- cannot directly compare.
Huterer & Starkman 2003
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Decorrelated BinsDecorrelated Bins
Bandpowers or decorrelated redshift bins diagonalize sqrt{F} to try to localize w(zi). Unlike for LSS, for dark energy they do not localize well, and confuse interpretation.
Also depends strongly on assumption of w(z>zmax)
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Principal Component AnalysisPrincipal Component Analysis
The uncertainties (i) have no physical meaning -- must interpret the signal-to-noise, not just the noise.
Even next generation experiments have only 2 components with S/N>3. Almost all models have 97-100% of the information in first 2 components. Eigenmode analysis does not improve over w0-wa.
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Common MistakesCommon Mistakes
• Neglecting M or S (SN or BAO absolute scale).
• Neglecting systematics.
• Claiming systematics, but still ’ing down errors.
• Thinking “self calibration” covers systematics; “self calibration” = “assuming a known form”.
• Using noise, not S/N, for PCA.
• Fixing w=-1 at high redshift.
Reductio ad absurdum:
1 SN/sec, 10 y survey gives d(z) to 0.003%
Every acoustic mode gives d(z) to 0.1%
Full sky space WL takes 1% shears to 310-6 level
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Controlling SystematicsControlling Systematics
Controlling systematics is the name of the game. Finding more objects is not.
Forthcoming experiments may deliver 100,000s of objects. But uncertainties do not reduce by 1/N.
Must choose cleanest probe/data, mature method, with multiple crosschecks.
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Battle RoyaleBattle Royale
Astronomer Royal (Airy): “I should not have believed it if I had not seen it!”
Astronomer Royal (Hamilton): “How different we are! My eyes have too often deceived me. I believe it because I have proved it.”
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Images
Spectra
Redshift & SN Properties
data analysis physics
Nature ofDark Energy
Each supernova is “sending” us a rich stream of information about itself.
What makes SN measurement special?What makes SN measurement special? Control of systematic uncertaintiesControl of systematic uncertainties
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
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Astrophysical UncertaintiesAstrophysical Uncertainties
Systematic Control
Host-galaxy dust extinction
Wavelength-dependent absorption identified with high S/N multi-band photometry.
Supernova evolution Supernova subclassified with high S/N light curves and peak-brightness spectrum.
Flux calibration error Program to construct a set of 1% error flux standard stars.
Malmquist bias Supernova discovered early with high S/N multi-band photometry.
K-correction Construction of a library of supernova spectra.
Gravitational lensing Measure the average flux for a large number of supernovae in each redshift bin.
Non-Type Ia contamination
Classification of each event with a peak-brightness spectrum.
For accurate and precision cosmology, need to identify and control systematic uncertainties.
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Controlling SystematicsControlling Systematics
Same SN, Different z Cosmology Same z, Different SN Systematics Control
2020
Depth + Width + ResolutionDepth + Width + Resolution
Bac
on
, E
llis
, R
efre
gie
r 20
00
Subaru - best ground
HST - space
Weak lensing noiseWeak lensing signal
Kas
liw
al, M
asse
y, E
llis
, Miy
azak
i, R
hod
es 2
007
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Cluster Cluster AbundancesAbundances
Optical: light mass Xray: hot gas gravitational potential mass
Sunyaev-Zel’dovich: hot e- scatter CMB mass
Weak Lensing: gravity distorts images of background galaxies
TraditionalDifficult for z>1Detects light, not massMass of what?
Clean detectionsDifficult for z>1Need optical survey for redshiftDetects flux, not massOnly cluster centerAssumes simple: ~ne
2
Clean detectionsIndepedent of redshiftNeed optical survey for redshiftDetects flux, not massAssumes ~simple: ~neTe
Detect mass directlyCan go to z>1Line of sight contaminationEfficiency reduced
Clusters -- largest bound objects. DE + astrophysics. Uncertainty in mass of 0.1 dex gives wconst~0.1 [M. White], w~?
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Heterogeneous DataHeterogeneous Data
Offsets due to different instruments, filters, sources can be a serious source of bias. “Stitching together” surveys, even with modest overlap, may give precision cosmology, but inaccurate results.
No need to stitch in z>2 – no leverage.
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Design an ExperimentDesign an Experiment
How to design an experiment to explore dark energy?
• Choose clear, robust, mature techniques
• Rotate the contours thru choice of redshift span
• Narrow the contours thru systematics control
• Break degeneracies thru multiple probes
• Use homogeneous data set
With a strong experiment, we can even test the framework of physics. Recall {m,w0,wa,,g*}.
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Dark energy may be a decades long mystery.
Space wide-field surveys maximize the discovery space.
Fundamental physics of inflation:
• Weak lensing - ns primordial perturbation spectrum
• Cluster abundances - non-Gaussianity
Dark Matter maps -
40 trillion pixels on sky! 20x ground.
Discovery SpaceDiscovery Space
“the skeleton of the universe”
Imagine COSMOS x 2000!
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Dark Energy – The Next GenerationDark Energy – The Next Generation
colorful
w i d e104 the Hubble Deep Field area (and deeper) plus 107 HDF (almost as deep)
deepdeep Mapping 10 billion years / 70% age of universe
Optical + IR to see thru dust, to high redshift
Launch ~2015Euclid (ESA)
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The Next PhysicsThe Next Physics
Current data do not tell us is the answer (or anything about dark energy at z>1).
Odds against : Einstein+us failed for 90 years to explain it.
Experiments to reveal dynamics (w-w) are essential to reveal physics. Space is the low risk option for dependable answers.
Expansion plus growth (e.g. SN+WL) is critical combination. We can test GR and can test geometry.
Space imaging mission gives optical-NIR and low-high z measurements, high resolution and low systematics; multiple probes and rich astronomical resources.
What is dark energy? What is the fate of the universe?
How many dimensions are there? How are quantum physics and gravity unified?
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Dark Energy PessimismDark Energy Pessimism
1835: “We shall never be able to know the composition of stars” -- Comte
1849: Kirchhoff discovers that the spectrum of electromagnetic radiation encodes the composition
[2008 STScI Symposium: “We shall never be able to know the composition of dark energy”
-- pessimistic physicist]
[2022? Cosmology on the Beach: Fiji has talks revealing the true nature of dark energy