Course on Dark Energy Cosmology at the Beach
Course on Dark Energy Cosmology at the Beach 2009 Eric Linder University of California, Berkeley Lawrence Berkeley National Lab JDEM constraints 11
Outline 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 22
Solving the Equation of Motion Klein-Gordon equation Transform to new variables Autonomous system where Copeland, Liddle, Wands 1998 Phys. Rev. D 57, 4686 Transform solution to Can add equation for EOS dynamics Caldwell & Linder 2005 Phys. Rev. Lett 95, 1413013 3
Equation of State Dynamics For robust solutions, pay attention to initial conditions, shoot forward in time, use 4 th order Runge-Kutta. For monotonic , can switch to as time variable, defining present as, e. g. =0. 72. 44
Asymptotic Behaviors Asymptotic behaviors can be physically interesting. Solve for critical points x (xc, yc)=0, y (xc, yc)=0. Check p={x, y} stability by sign of eigenvalues p =Mp. Copeland, Liddle, Wands 1998 Phys. Rev. D 57, 4686 Relevant to fate of universe. Crossing w=-1: 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. 55
From Data to Theory (and back) Fisher matrix gives lower limit for Gaussian See: Tegmark et al. astro-ph/9805117 likelihoods, quick and easy. Dodelson, “Modern Cosmology” Fij = d 2(- ln L) / dpi dpj = O(d. O/dpi) COV-1 (d. O/dpj) (pi) 1/(Fii)1/2 Example: O=dlum(z=0. 1, 0. 2, … 1), p=( m, w), COV=( d/d)d ij F w= k(d. Ok/d )(d. Ok/dw) k-2 F= ( F F w Fw Fww ) C= F-1 = ( 2( ) COV( , w) 2(w) ) Also called information matrix. Add independent data sets, or priors, by adding matrices. e. g. Gaussian prior on m=0. 28 0. 03 via 2 = ( m-0. 28)2/0. 032 66
Survival 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. 77
Bias 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)+ p 2]1/2 88
Design an Experiment Precision in measurement is not enough - one must beware degeneracies and systematics. . p 2 * p 1 Degeneracy: e. g. Aw 0+Bwa=const Degeneracy: hypersurface, e. g. covariance with m or Systematic: floor to precision, e. g. calibration Systematic: offset error in data or model, e. g. evolution 9 9
Orthogonal 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 10 10
Decorrelated 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) 11 11
Principal 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 w 0 -wa. 12 12
Common 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 3 10 -6 level 13 13
Controlling Systematics Controlling systematics is the name of the game. Finding more objects is not. Forthcoming experiments may deliver 100, 000 s of objects. But uncertainties do not reduce by 1/ N. Must choose cleanest probe/data, mature method, with multiple crosschecks. 14 14
Battle 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. ” 15 15
What makes SN measurement special? Control of systematic uncertainties Each supernova is “sending” us a rich stream of information about itself. Images Redshift & SN Properties Nature of Dark Energy Spectra data analysis physics 16 16
Astrophysical Uncertainties For accurate and precision cosmology, need to identify and control systematic uncertainties. Systematic Control Host-galaxy dust extinction Wavelength-dependent absorption identified with high S/N multiband photometry. Supernova evolution Supernova subclassified with high S/N light curves and peakbrightness 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. 17 17
Controlling Systematics Same SN, Different z Cosmology Same z, Different SN Systematics Control 18 18
Fitting Subsets perfect 19 19
Depth + Width + Resolution Subaru - best ground HST - space Weak lensing signal Bacon, Ellis, Refregier 2000 Kasliwal, Massey, Ellis, Miyazaki, Rhodes 2007 Weak lensing noise 20 20
Cluster Abundances Clusters -- largest bound objects. DE + astrophysics. Uncertainty in mass of 0. 1 dex gives wconst~0. 1 [M. White], w ~? Optical: light mass Xray: hot gas gravitational potential mass Traditional Difficult for z>1 Detects light, not mass Mass of what? Clean detections Difficult for z>1 Need optical survey for redshift Detects flux, not mass Only cluster center Assumes simple: ~ne 2 Sunyaev-Zel’dovich: hot escatter CMB mass Clean detections Indepedent of redshift Need optical survey for redshift Detects flux, not mass Assumes ~simple: ~ne. Te Weak Lensing: gravity distorts images of background galaxies Detect mass directly Can go to z>1 Line of sight contamination Efficiency reduced 21 21
Heterogeneous 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. 22 22
Design 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 { , w 0, wa, , g*}. 23 23
Discovery Space Dark energy may be a decades long mystery. Space wide-field surveys maximize the discovery space. 40 trillion pixels on sky! 20 x ground. Fundamental physics of inflation: • Weak lensing - ns primordial perturbation spectrum • Cluster abundances - non-Gaussianity Dark Matter maps “the skeleton of the universe” Imagine COSMOS x 2000! 24 24
Dark Energy – The Next Generation wide 104 the Hubble Deep Field area (and deeper) plus 107 HDF (almost as deep) deep Mapping 10 billion years / 70% age of universe colorful Optical + IR to see thru dust, to high redshift Euclid (ESA) Launch ~2015 25 25
The Next Physics What is dark energy? Current data do not tell us is the answer (or What is the fate of the universe? anything about dark energy at z>1). Odds How many dimensions are there? How are quantum gravity unified? it. against : Einstein+us failedphysics for 90 and years to explain 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. 26 26
Dark Energy Pessimism [2008 STSc. I Symposium: “We shall never be able to know the composition of dark energy” -- pessimistic physicist] 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 [2022? Cosmology on the Beach: Fiji has talks revealing the true nature of dark energy 27 27
“Acceleration” to the tune of The Beatles’ “Revolution” 28 28
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