# Presentation: 638272

**Eric Linder **
**University of California, Berkeley**
**Lawrence Berkeley National Lab**

**Course on Dark Energy**** **
***Cosmology at the Beach 2009***

**JDEM constraints**

## 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**

## Solving 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**

## Equation 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****. **

## Asymptotic Behaviors

**Asymptotic behaviors can be physically interesting. Solve for ****critical points**** ****x********(x****c****,y****c****)=0****,**** y********(x****c****,y****c****)=0****.  Check stability by sign of eigenvalues ********p********=*****M*****p****.  **

**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 w****tot****∞****<-1. Cannot cross w=-1 even with coupling.  Quintessence can cross with coupling since w<w****tot****.**

**p={x,y}**

## From Data to Theory (and back)

**Fisher matrix**** gives lower limit for Gaussian likelihoods, quick and easy. **
**F****ij**** = d****2****(- ln L) / dp****i**** dp****j ****= ********O****(dO/dp****i****) COV****-1**** (dO/dp****j****)**
******(p****i****) ********1/(F****ii****)****1/2**
**Example: O=d****lum****(****z=0.1,0.2,...1)****, p=(********m****,w), COV=(********d/d)d ********ij ****	****          ****F********w****=********k****(dO****k****/d********)(dO****k****/dw)********k****-2**

**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****/0.03****2**

**See: Tegmark et al. astro-ph/9805117 Dodelson, “Modern Cosmology”**

## 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. **

## 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**

## Design an Experiment

**Precision in measurement is not enough - one must beware degeneracies and systematics. **

**p****2**

**p****1**

*****

**Degeneracy: ****	****e.g. Aw****0****+Bw****a****=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**

## Orthogonal Basis Analysis

**Eigenmodes****: ****w(z) = ******** ********i**** e****i****(z)**** For orthogonal basis, errors ********(********i****)**** are uncorrelated.  ****“*****Principal components*****”.**
**Start with parameters {w****i****} in z bins.  ****	****	****	****Diagonalize Fisher matrix F=E****T****DE: D is diagonal, rows of E give eigenvectors.**** ****	****	****	****	****	****	****NOTE:**** ****basis differs with**** ****model, experiment, and probe -- *****cannot***** directly compare.**

**Huterer & Starkman 2003**

## Decorrelated Bins

**Bandpowers**** or decorrelated redshift bins diagonalize sqrt{F} to try to localize w(z****i****).  Unlike for LSS, for dark energy they do not localize well, and confuse interpretation.**

**Also depends strongly on assumption of w(z>z****max****)**

## 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****-w****a****.**

## 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 **

## Controlling Systematics

**Controlling 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.**

## 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.” **

## What makes SN measurement special? Control of systematic uncertainties

*Images*

*Spectra*

*Redshift & SN Properties*

**data**

**analysis**

**physics**

**Nature of**
**Dark Energy**

**Each supernova is “sending” us a rich stream of information about itself.**

- What makes SN measurement special?** ****Control of systematic uncertainties**

## Astrophysical 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.**

## Controlling Systematics

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

## Fitting Subsets

## Depth + Width + Resolution

**Subaru - best ground**

**HST - space**

**Weak lensing noise**

**Weak lensing signal**

**Kasliwal, Massey, Ellis, Miyazaki, Rhodes 2007**

## Cluster Abundances

**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**

**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: ~n****e****2**

**Clean detections**
**Indepedent of redshift**
**Need optical survey for redshift**
**Detects flux, not mass**
**Assumes ~simple: ~n****e****T****e**

**Detect mass directly**
**Can go to z>1**
**Line of sight contamination**
**Efficiency reduced**

**Clusters**** ****-- largest bound objects. ****DE + astrophysics.**** ****Uncertainty in mass of 0.1 dex gives ********w****const****~0.1**** **[M. White]**,**** ********w********~?**

## 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.**

## Design an Experiment

**How to design an experiment to explore dark energy?**
** ****Choose clear, robust, mature technique****s**
** ****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****,w****0****,w****a****,********,g*********}.**

## Discovery Space

***	******Space wide-field surveys***** *****maximize the ******	******	******discovery space*****. **
**Fundamental physics of inflation:**** **
** ****Weak lensing**** - n****s**** primordial perturbation spectrum **
** ****Cluster abundances**** - non-Gaussianity**
**Dark Matter maps -**

**40 trillion pixels on sky! 20x ground.**

**Discovery Space**

**“****the skeleton of the universe”**

**Imagine COSMOS x 2000!**

## Dark Energy – The Next Generation

**c****o****l****o****r****f****u****l**

- w i d e

**10****4 ******** the Hubble Deep Field area (and deeper) *****plus***** 10****7 ******** HDF (almost as deep)**

**deep**

**Mapping 10 billion years / 70% age of universe ****	**

**Optical + IR to see thru dust, to high redshift**

## The 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?**** **

## Dark 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**

## “Acceleration”

***to the tune of The Beatles’ “Revolution”***