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!