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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.