Cosmic Calibration - Statistical Modeling for Dark Energy

UCSC Tracy Holsclaw, Bruno Sanso, and Herbie Lee

LANLUjjaini Alam, Katrin Heitmann, David Higdon and Salman Habib

LA-UR-09-05891

Overview

-The Universe is expanding (1920s)

-Observations have been made that the Universe is expanding at an accelerating pace (1998)

-Ordinary matter would mean the Universe is decelerating

-Dark energy is the unknown driver of this acceleration

-Dark energy has an equation of state relating its pressure to density; this equation is our focus to understand more about the nature of dark energy

-Our current method is based on Bayesian analysis of a differential equation by modeling w(z) directly, where w(z) is the equation of state parameter

Data of Interest

SNe – Supernovae Stars 397 measurements

We want to know whether w(z), the equation of state for Dark Energy, is constant or not

BAO – Baryon Acoustic CMB- Cosmic Microwave Oscillation Background large scale matter clustering probes the oldest light in the Universe

0 2 1000

Redshift (z)

Photos provided by NASA

-The main parameter of interest is w(z); we need to specify a form for w(z) -Three other unknown parameters also have to be estimated H0, Ωm, M, and σ2

-This leads to the following likelihood equation:

Likelihood Equation

sz

ss duuH

H

zcMzT

0010 )(

)1(log525)(

n

s C

CC

B

BB

s

ss zRzAzTn

ssCBm eHwL 1

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1

)(3

334 0)1)(1()1()1()(

s

duu

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Bz

BBmB duuH

z

czHzA

Cz

mC duuHzR0

)()(

SNe BAO CMB

-Prior sensitivity was examined and thus far it seems that the prior does not

change the outcome of the estimations

Overview of the Models

Ωr =0.247/ Ho2

π(Ho) ~ N(71.9, 2.72)

π(σ2) ~ IG(10, 9)

π(Ωm| Ho) ~ N(0.1326(100/ Ho)2, (0.0063(100/ Ho)2) 2)

π(α) ~ U(-25,1)

π(β) ~ U(-25,25)

Model 1: w(u) = α

Model 2: w(u) = α-βu/(1+u)

Gaussian Process: w(ui) ~ GP(-1, k2ρ|u-u’|2)

Model 1: w(u) = α

Dataset α M Ho Ωm σ2

SNe (-1.12,-0.80) (0.03,0.37) (66.4,77.2) (0.22,0.31) (1.02,1.34)

SNe+BAO (-1.16,-0.79) (-0.04,0.35) (64.3,76.4) (0.23,0.32) (1.03,1.37)

SNe+CMB (-1.05,-0.85) (0.07,0.32) (67.3,75.4) (0.23,0.30) (1.02,1.34)

SNe+BAO+CMB (-1.10,-0.84) (0.04,0.31) (66.3,75.5) (0.24,0.30) (1.03,1.35)

SNe SNe+BAO SNe+CMB SNe+BAO+CMB

Dataset α β M Ho Ωm σ2

SNe (-1.27,-0.47) (-2.00,3.49) (0.03,0.36) (66.0,76.6) (0.22,0.32) (1.03,1.34)

SNe+BAO (-1.54,-0.23) (-3.51,5.52) (-0.05,0.32) (63.4,75.4) (0.24,0.33) (1.03,1.36)

SNe+CMB (-1.17,-0.49) (-0.98,2.58) (0.05,0.34) (66.9,76.0) (0.23,0.30) (1.02,1.35)

SNe+BAO+CMB (-1.22,-0.32) (-1.09,4.26) (-0.01,0.34) (64.6,75.8) (0.24,0.32) (1.03,1.37)

SNe SNe+BAO SNe+CMB SNe+BAO+CMB

Model 2: w(u) = α – βu/(1+u)

Dataset M Ho Ωm σ2

SNe (0.02,0.37) (66.1,76.8) (0.22,0.31) (1.02,1.34)

SNe+BAO (0.05,0.29) (66.8,74.6) (0.24,0.30) (1.02,1.34)

SNe+CMB (0.07,0.33) (67.4,75.6) (0.23,0.30) (1.02,1.34)

SNe+BAO+CMB (0.07,0.30) (67.1,74.9) (0.24,0.30) (1.02,1.34)

SNe SNe+BAO SNe+CMB SNe+BAO+CMB

Gaussian Process Model

Model 1 Model 2 Gaussian Process Model

Extended Model

-The parametric Model 2 does not extend well to a larger domain, whereasthe Gaussian process model does

Conclusions

• The Gaussian process method provides a non-parametric fit to the equation of state

• The Gaussian process analysis has tighter estimated bands than one of the commonly used parametric forms

• Unlike the parametric models the fit presented here is coherent even when extended to z=1000

• All methods presented here have been tested with simulated datasets and correct results have been obtained

Future Work

• We would like to explore a new set of real data with nearly 600 Supernovae observations

• We want to show that this work will extend to future data sets with 2000 SNe points and 20 BAO points. And how this new data could shrink the uncertainty of the parameters

• The cosmologists would like to know where more observations (on the z axis) are needed to shrink uncertainty

• We want to understanding how measurement error in the Supernovae data affects the analysis

Cosmic Calibration - Statistical Modeling for Dark Energy

UCSC Tracy Holsclaw, Bruno Sanso, and Herbie Lee

LANLUjjaini Alam, Katrin Heitmann, David Higdon and Salman Habib