Adventures in Parameter Estimation

Jason Dick

University of California, Davis

Motivation

• Goal: to gain information about basic physics through parameter estimation.

• Two major areas that have significant open questions today are dark energy and inflation.

• Upcoming experiments have the capability to provide significant information about the physics of these phenomena.

Inflation

• Best experimental test currently available for inflation is the Cosmic Microwave Background.

• Recent tentative detection of a departure from scale invariance in the three-year WMAP data release begs for further investigation.– Interesting because we expect some small departure

from scale invariance for most inflationary models.

• One way of measuring this better is to measure higher multipoles more accurately, as we will be doing with the SPT project.

Dark Energy

• Is dark energy a cosmological constant?• Theory gives little insight as to how dark

energy varies.• Theory-independent analysis.

– Want to look at those types of variation best-constrained by the data.

• Our solution: use eigenmodes.

What we found

• Our method appears to be an optimal method for detecting dark energy variation.

• But the eigenmodes we found cannot be physical.– Variation is too fast at the one-sigma error level to be

explained by dark energy.

• So for now, just a systematic error test.• When future data place better constraints on the

eigenmodes, there will be a possibility of detecting real variation.

Data

• Supernova data: Riess et. al. (astro-ph/0402512) and Astier et. al. (astro-ph/0510447)

• WMAP constraints: Obtained from chain available at the LAMBDA archive (http://lambda.gsfc.nasa.gov)

• BAO constraints: Eisenstein et. al. (astro-ph/0501171)

Parameterization

• Define: ρx(z) = ρc(0)aiei(z).

• Choose basis: e0 is constant, others vary

•Constant basis vector

•One varying vector

•Another varying vector

Diagonalization

• To describe our cosmology, we now have the parameters: ωm, Ωk, a0, a1-an, and the supernova parameters: M, α, β.

• Take Gaussian approximation to marginalize over all but a1-an.

• Diagonalize to get eigenvectors (a new basis):

Some example eigenmodes

•First varying mode

•Second varying mode

•Third varying mode

MCMC Analysis

• Don’t want to be limited by the Gaussian approximation.

• Using MCMC, estimate values and errors of best-measured modes only.

• The errors in each varying mode should be uncorrelated with all other varying modes.

SNLS + WMAP Results

SNLS + WMAP Results

SNLS + WMAP Results

SNLS + WMAP Results

When Gaussians Go Bad

• Adding more modes: degeneracies appear.

• Here it happens when the MCMC chain includes the 7th dark energy parameter.

• Four leftmost of each group of parameters are very poorly-constrained: some are off the graph!

SNLS + BAO + WMAP

But is the variation too fast?1-σ Variation of Eigenmodes

Why is this method useful?

• Estimation of energy density directly, instead of through integration of w(z), should result in tighter constraints on the density.

• Any real variation of dark energy should show up in the first eigenmode, as higher eigenmodes vary more quickly and are less likely to describe real physics.

• Use of eigenmode analysis should ensure that if the data can detect variation in dark energy, it should be detected by this method.– This bears investigation, however.

Possible issues

• The eigenmodes in dark energy density do not have an obvious connection with physics: this test only addresses the question as to whether or not dark energy varies, but the connection to specific dark energy models is not clear at this point.

• Have not tested method against many simulated data sets with different sorts of varying dark energy.– Main problem: how to allow dark energy to vary in

many different ways without biasing models?

Results of Dark Energy Analysis

• Good method for detecting deviation from constant without being tied to a particular theory.

• MCMC analysis is self-checking.• No detected variation: systematic error

test passed.• We expect this technique to be excellent

at discovering whether or not we have a cosmological constant for future data.

Moving on to Inflation

Any questions before we continue?

South Pole Telescope:Measuring the CMB at high resolution

• Instrument:– 960 bolometer array– 4000 deg2 survey area– Arcminute resolution

• Benefits for CMB science:– Large sky coverage will allow highly accurate

calibration with WMAP (and later Planck) results.– High resolution allows measurement of primary CMB

to high multipoles (up to about l=3000-4000).

What does this mean for constraining Inflation?

Lloyd Knox, 2006

Obtaining Cosmological Parameters

• Method is straightforward: libraries such as CMBEASY and CMBFAST are available and easy to use.

• But we need to develop a likelihood estimator for SPT data.

• Requires estimation of power spectrum and errors on the power spectrum.

Estimating the Power Spectrum

• Use MASTER-like algorithm, as described in Hivon et. al. (astro-ph/0105302).

• Algorithm parameterizes the power spectrum as follows:

• Pseudo Cl method first estimates the power spectrum of the map through a direct spherical harmonic transform, then compares it against a theory power spectrum that has been modified in the above way.– Method pioneered by Gorski in astro-ph/9403066

' ' ' ''

ll ll l l l l

l

C M F B C N

Calculating Mll’

• Calculation assumes statistical isotropy– Can we relax this?– May not need to.

1 2 3

3

2

1 2 323

2 1(2 1)

0 0 04l l ll

l l llM l W

Testing Mll’ on 1000deg2 map

( 1)

2 l

l lC

l

Estimating Fl

• Want to use Monte Carlo techniques to find Fl.

– Will be computationally difficult to simulate for every l value.

– Define Fl at discrete l values.

– Interpolate in between using cubic interpolation.

Still to come

• Estimating the beam profile contribution Bl

– Described in Wu et. al. 2001 (astro-ph/0007212).

• Estimating the noise contribution Nl

– Described in Hivon et. al. 2002 (astro-ph/0105302).

Next step:Simulating the detectors

• Computationally intensive.– 960 detectors!– Use small maps to start

• Atmosphere model– Modeled as a smooth gradient in temperature that slowly moves across

the field with time.– First approximation: remove with high-pass filter.– May implement more careful sky removal later.

• Detector noise– Modeled as white + 1/f

• Point sources– Can mask these out, so can simulate their effect by arbitrarily masking a

few small regions on the map.• Diffuse galactic sources

– Purposely measuring in area with low emission in our frequency bands.

WMAP dust map (W-band)

Courtesy: WMAP Science TeamLinear scale, -0.5 – 2.3 mK

Conclusions

• SPT will allow for excellent measurement of deviation from scale invariance.

• Highly complementary with current and future CMB experiments, such as WMAP, ACT, and Planck.

• Going online this summer!