11

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

22

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

33

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.

55

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}

66

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

99

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.

1313

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.

1515

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.

1818

Controlling SystematicsControlling Systematics

Same SN, Different z Cosmology Same z, Different SN Systematics Control

1919

Fitting SubsetsFitting Subsets

perfect

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~?

2222

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

2828

““Acceleration”Acceleration”

to the tune of The Beatles’ “Revolution”