Non-parametric Reconstruction of the Hubble Expansion History

with a Monotonicity Prior

Hu Zhan1 & Youhua Xu1, 2

1National Astronomical Observatories of China2Nanjing University

COSMO-15 Sept 7-11, 2015Warsaw, Poland

MotivationH(z) is a direct probe of the cosmic expansion

history as well as the dark energy equation of state.

Need for model-independent constraints on intermediate quantities.Cosmological constant? w? w0 & wa? Quintessence?

Phantom? Modified gravity? Relatively model-independent constraints on P(k,z), d(z), g(z), H(z), etc. greatly facilitate DE/MG model tests.

Cross checks between methods.

Appropriate prior to regularize the reconstruction.

z

DEkm dzz

zwzz

H

H

a

aH

0

23

20

2

''1

'13exp11

BigBOSS/DESIZhan et al. 2009

Levi et al. 2013

Examples of Reconstruction & Forecasts

Zhao et al. 2012

Measuring H(z)

• Galaxy Ages

• Line-of-sight BAOs

• H0

Cepheids, SNe, maser, strong lensing, CMB etc. with a model.

sr

zczH

tz

zzH

1

Moresco et al. 2012

Reconstructing H(z) from SNeIa Data• Interpolate H(z) from a set of Hi in redshift

• Apparent magnitude of SNeIa:

Assumption: metric theory of gravity

(fairly model independent)

Straightforward to add curvature

Reconstructing H(z) from SNeIa Data• Bayesian inference

• MCMCAffine invariant MCMC ensemble sampling (Goodman & Weare 2010)

H(z)

The Monotonicity Prior

• It is reasonable to assume that H(z) increases with redshift, i.e., dH(z)/dz ≥ 0 or Hi ≤ Hi+1

• Observations are largely consistent with such an assumption.

z

DEkm dzz

zwzz

H

H0

23

20

2

''1

'13exp11

Moresco et al.(2012)

Test with Mock SNeIa Samples

• Cosmological parameters of mock samples:

• WFIRST SNeIa error models:

• Sample size: 2725

• z_max: 1.7

--- without MP

--- with MP

Bias Check & Improvement with Nbin

without mono. prior

with mono. prior

UNION2.1

SNLS3

JLA

Comparison & Forecast

* DESI

LSST: distances from Hi constraints on Hi

Summary• Reconstruction of time-varying cosmological quantities is

useful for model testing and cross checks.

• The monotonicity prior can significantly reduce the errors in reconstructed Hi when the data is poor. Similar priors

may be designed for other quantities of interest.

• The Hubble parameter can be constrained to a few percent level with multiple surveys under construction. SNeIa provide the best precision at z<0.5, and other methods will extend the measurements above z=2.