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National Cosmology Workshop: Dark Energy Week @IHEP. On the Dark Energy EoS: Reconstructions and Parameterizations. Dao-Jun Liu (Shanghai Normal University) 2008-12-9. Outline. Introduction Model-Independent Method: reconstruction Parameterize the EoS functional form approach

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on the dark energy eos reconstructions and parameterizations

National Cosmology Workshop: Dark Energy Week @IHEP

On the Dark Energy EoS: Reconstructions and Parameterizations

Dao-Jun Liu

(Shanghai Normal University)


  • Introduction
  • Model-Independent Method: reconstruction
  • Parameterize the EoS

functional form approach

binned approach

  • How to select a parameterization
  • Discussions

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.
meeting point of observation and theory
Meeting point of observation and theory
  • Comoving distance:
  • Luminosity distance:
  • Angular diameter distance:
direct reconstruction
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

another approach to non parametric reconstruction
Another approach to non-parametric reconstruction

Shafieloo 2007

the Gaussian filter

A quantity needed to

be given beforehand

Another choice:

the ‘top-hat’ filter

two classes of parameterization
Two classes of parameterization
  • Binned
  • Functional form
non binned parameterizations models
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
single parameter models
Single parameter models

Network of cosmic strings

Domain wall

two parameter parameterizations
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 >> 1 and therefore incompatible with the constraints from CMB and BBN.

cpl parameterization
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

two two parameter parameterization families
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.

multi parameter parameterizations
Multi-parameter parameterizations

Fast phase transition parameterization:

Bassett et al 2002

Oscillating EOS:

Feng et al 2002

multi parameter parameterizations1
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.
summary of functional approach
Summary of functional approach


Localization is guaranteed,

straightforward physical interpretation of parameters is allowed


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

binned parameterizations
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.


Baseline EOS,

e.g. w_b = −1

information localization problem
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.

the measure of uncertainty
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

direct binning
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 !

principle component analysis pca
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


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

uncorrelated bin approach
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.

summary of binned parameterizations
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.
smoothing the bins
Smoothing the bins
  • Spline


Zhao, Huterter, Zhang 2008


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.

fisher matrix method to fit data to the models
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:

how to compare these models
How to compare these models
  • Bayes factor

Under this circumstance, this method isinvalid !

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!

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?

figures of merit
Figures of Merit

It does not work! Because the area of the error ellipse has only relative meaning.

the area of the band
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.

  • 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.