on the dark energy eos: reconstructions and parameterizations
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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 - PowerPoint PPT PresentationTRANSCRIPT
On the Dark Energy EoS: Reconstructions and Parameterizations
Dao-Jun Liu
(Shanghai Normal University)
2008-12-9
National Cosmology Workshop: Dark Energy Week @IHEP
Outline
Introduction Model-Independent Method: reconstruction Parameterize the EoS
functional form approach
binned approach How to select a parameterization Discussions
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.
Meeting point of observation and theory
Comoving distance:
Luminosity distance:
Angular diameter distance:
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 reconstructionShafieloo 2007
the Gaussian filter
Another choice:the ‘top-hat’ filter
A quantity needed to be given beforehand
Two classes of parameterization
Binned Functional form
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
Network of cosmic strings
Domain wall
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.
Two-parameter parameterizations
Sahni et al. 2003
CPL ParameterizationChevallier & 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
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
Fast phase transition parameterization:
Oscillating EOS:
Feng et al 2002
Bassett et al 2002
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
Drawback:Fitting data to an assumed functional form leads topossible 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
Advantage: Localization is guaranteed, straightforward physical interpretation of parameters is allowed
Binned parameterizations
1) dividing the redshift interval
into N bins not necessarily equal widths
2)
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 problemde 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
Information within a localized region is also not invariant when considering changes in the number of bins or binning variable.
de Putter & Linder 2007
It is hard to define a measure of uncertainty in the EOS estimation that does not depend on the specific binning chosen.
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)
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
Huterer & Starkman, 2003
de Putter & Linder 2007
Advantage: the parameter uncertainties is decorrelated
Problems:1. Calculate eigenmodes in which coordiante? in principle, an infinite number of choice2. “Best determined ” is not well defined
uncorrelated bin approach
using a small number of bins, diagonalizing and scaling the Fisher matrix in an attempt to localize the decorrelated EOS parameters
4 bins Huterer & Cooray, 2005. 4 bins Huterer & Cooray, 2005.
Using the square root of Fisher matrix as weight matrix
The information is not fully localized !
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
Spline
Zhao, Huterter, Zhang 2008
bias
Starobinsky et.al 2004
)1()( 10 awwaw
)1()( 10 awwaw
Polynomial parameterization
Riess et.al ,2007
))ln(sin()( 210 awwwaw
Zhao et.al. 2007.
Non-parametric reconstruction
Daly & Djorgovski 2004
Fitting data to the proposed models
Fisher matrix method to fit data to the models
Goodness of fit:
The distribution of errors in themeasured parameters:
Fisher matrix:
The error on the EOS:
How to compare these models
Bayes factor
Under this circumstance, this method is invalid !
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!
The above Bayes approach only works in the condition that fittings of models are distinctly different.
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
It does not work! Because the area of the error ellipse has only relative meaning.
The area of the band
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.
LDJ et al, 2008
LDJ et al, 2008
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.
Thank you!