On the Dark Energy EoS: Reconstructions and Parameterizations

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On the Dark Energy EoS Reconstructions and
Parameterizations
National Cosmology Workshop Dark Energy Week
_at_IHEP

• Dao-Jun Liu
• (Shanghai Normal University)
• 2008-12-9

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Outline

• Introduction
• Model-Independent Method reconstruction
• Parameterize the EoS
• functional form approach
• binned approach
• How to select a parameterization
• Discussions

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Introduction

• The quantities that describe DE
• EOS contain clues crucial to understanding the
  nature of dark energy.
• Deciphering the properties of EOS from data
  involves a combination of robust analysis and
  clear interpretation.

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Meeting point of observation and theory

• Comoving distance
• Luminosity distance
• Angular diameter distance

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Direct reconstruction

• Really model-independent, but
• Contains 1st and 2nd derivatives of comoving
  distance
• direct taking derivatives of data ----
  noisy
• fitting with a smooth function ---- bias
  introduced

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Another approach to non-parametric reconstruction
Shafieloo 2007
the Gaussian filter
A quantity needed to be given beforehand
Another choice the top-hat filter
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Two classes of parameterization

• Binned
• Functional form

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Non-binned Parameterizations (models)

• How to Parameterize the EOS functionally?
• Fit the data well
• the motivation from a physical point of view
  should be at the top priority
• Regular asymptotic behaviors both at late and
  early times
• Simplicity

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Single parameter models
Network of cosmic strings
Domain wall
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Two-parameter parameterizations

• The linear-redshift parameterization (Linear)
• The Upadhye-Ishak-Steinhardt parameterization
  (UIS) can avoid above problem,

not viable as it diverges for z gtgt 1 and
therefore incompatible with the constraints from
CMB and BBN.
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Two-parameter parameterizations
Sahni et al. 2003
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CPL Parameterization
Chevallier Polarski, 2001 Linder, 2003
Reduction to linear redshift behavior at low
reshift Well-behaved, bounded behavior for high
redshift high- accuracy in reconstructing many
scalar field EOS
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Two two-parameter parameterization families
Both have the reasonable asymptotical behavior at
high z. n 1 in both families corresponding
CPL. n 2 one in Family II is the
Jassal-Bagla-Padmanabhan parameterization (JBP),
which has the same EOS at the present epoch and
at high z, with rapid variation at low z.
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Multi-parameter parameterizations
Fast phase transition parameterization
Bassett et al 2002
Oscillating EOS
Feng et al 2002
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Multi-parameter parameterizations

• More parameters mean more degrees of freedom for
  adaptability to observations, at the same time
  more degeneracy in the determination of
  parameters.
• For models with more than two parameters, they
  lack predictability and even the next generation
  of experiments will not be able to constrain
  stringently.

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Summary of functional approach
Advantage Localization is guaranteed,
straightforward physical interpretation of
parameters is allowed
Drawback Fitting data to an assumed functional
form leads to possible biases in the
determination of properties of the dark energy
and its evolution, especially if the true
behavior of the dark energy EOS differs
significantly from the assumed form
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Binned parameterizations

• 1) dividing the redshift interval
• into N bins not
  necessarily equal widths

N?, bias ?
changing the binning variable from z to a or
lna is equivalent to changing the bins to
non-uniform widths in z.
2)
Baseline EOS, e.g. w_b -1
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Information localization problem
de Putter Linder 2007
The curves of information are far from sharp
spikes at z z, indicating the cosmological
information is difficult to localize and
decorrelate.
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The measure of uncertainty
Information within a localized region is also not
invariant when considering changes in the number
of bins or binning variable.
It is hard to define a measure of uncertainty in
the EOS estimation that does not depend on the
specific binning chosen.
de Putter Linder 2007
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Direct Binning

• simply considering the values in a small number
  of redshift bins.
• Localization is guaranteed, straightforward
  physical interpretation is allowed
• correlations in their uncertainties are retained

This is only just one kind of functional form of
parameterization !
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Principle Component Analysis (PCA)
Huterer Starkman, 2003

• effectively making the number of bins very large,
  diagonalizing the Fisher matrix and using its
  eigenvectors as a basis
• Selecting a small set of the best determined
  modes, i.e. the principle component and throwing
  away the others

Advantage the parameter uncertainties is
decorrelated

• Problems
• 1. Calculate eigenmodes in which coordiante?
• in principle, an infinite number of
  choice
• 2. Best determined is not well defined

de Putter Linder 2007
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uncorrelated bin approach

• using a small number of bins, diagonalizing and
  scaling the Fisher matrix in an attempt to
  localize the decorrelated EOS parameters

Using the square root of Fisher matrix as weight
matrix
The information is not fully localized !
4 bins
Huterer Cooray, 2005.
4 bins
Huterer Cooray, 2005.
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Summary of binned parameterizations

• Result depends on the scheme of binning, so they
  are not actually model independent
• EOS is discontinuous
• Decorrelated parameters that are not readily
  interpretable physically or phenomenally are of
  limited use. After all, our goal is understanding
  the physics, not obtaining particular statistical
  properties.

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Smoothing the bins

• Spline

bias
Zhao, Huterter, Zhang 2008
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Fitting data to the proposed models
Starobinsky et.al 2004
Non-parametric reconstruction
Daly Djorgovski 2004
Riess et.al ,2007
Polynomial parameterization
Zhao et.al. 2007.
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Fisher matrix method to fit data to the models
Goodness of fit
The distribution of errors in the measured
parameters
Fisher matrix
The error on the EOS
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How to compare these models

• Bayes factor

Under this circumstance, this method is invalid !
The above Bayes approach only works in the
condition that fittings of models are distinctly
different.
how do we compare them? Or, what parametrization
approach should be used to probe the nature of
dark energy in the future experiments? Needs
another figure of merit!
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• In this situation, a model that can be more
  easily disproved should be selected out !
• 1st candidate cosmological constant (no
  parameter model)
• 2nd candidate (1 parameter)
• So, today, distinguishing dark energy from a
  cosmological constant is a major quest of
  observational cosmology.

3rd candidate (2 parameter model) What?
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Figures of Merit
It does not work! Because the area of the error
ellipse has only relative meaning.
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The area of the band
LDJ et al, 2008
The justification of this measure lies in that
our ultimate goal is to constrain the shape of
w(z) as much as we can from the data.
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LDJ et al, 2008
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Conclusions

• Binned parameterizations are not strictly form
  independent.
• Although, the modes, and their uncertainties,
  depend on binning variable, PCA is useful in
  obtaining what qualities of the data are best
  constrained.
• In doing data fitting, physical motivated
  functional form parameterization and a binned EOS
  should be in compement with each other.
• To test a dynamical DE model, CPL
  parameterization may not be a preferred
  approach.

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Thank you!