1. On the Dark Energy EoS: Reconstructions and Parameterizations Dao-Jun Liu (Shanghai Normal University) 2008-12-9 National Cosmology Workshop: Dark Energy Week @IHEP

2. Outline  Introduction  Model-Independent Method: reconstruction  Parameterize the EoS functional form approach binned approach  How to select a parameterization  Discussions

3. 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.

4. Meeting point of observation and theory  Comoving distance:  Luminosity distance:  Angular diameter distance:

5. 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

6. Another approach to non-parametric reconstruction Shafieloo 2007 the Gaussian filter Another choice: the ‘ top-hat ’ filter A quantity needed to be given beforehand

7. Two classes of parameterization  Binned  Functional form

8. 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

9. Single parameter models Network of cosmic strings Domain wall

10. 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.

11. Two-parameter parameterizations Sahni et al. 2003

12. 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

13. 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.

14. Multi-parameter parameterizations Fast phase transition parameterization: Oscillating EOS: Feng et al 2002 Bassett et al 2002

15. 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.

16. Summary of functional approach 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 Advantage: Localization is guaranteed, straightforward physical interpretation of parameters is allowed

17. 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

18. 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.

19. 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.

20. 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 !

21. 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 choice 2. “ Best determined ” is not well defined

22. 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 !

23. 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.

24. Smoothing the bins  Spline Zhao, Huterter, Zhang 2008 bias

25. Starobinsky et.al 2004 Polynomial parameterization Riess et.al,2007 Zhao et.al. 2007. Non-parametric reconstruction Daly & Djorgovski 2004 Fitting data to the proposed models

26. 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:

27. 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.

28.  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?

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

30. 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

32. 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.

33. Thank you!