1. Non-parametric Reconstruction of the Hubble Expansion History with a Monotonicity Prior Hu Zhan 1 & Youhua Xu 1, 2 1 National Astronomical Observatories of China 2 Nanjing University COSMO-15 Sept 7-11, 2015 Warsaw, Poland

2. H(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? w 0 & w a ? 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.

3. BigBOSS/DESI Zhan et al. 2009 Levi et al. 2013 Examples of Reconstruction & Forecasts Zhao et al. 2012

4. Galaxy Ages Line-of-sight BAOs H 0 Cepheids, SNe, maser, strong lensing, CMB etc. with a model. Moresco et al. 2012

5. Interpolate H(z) from a set of H i in redshift Apparent magnitude of SNeIa: Assumption: metric theory of gravity (fairly model independent) Straightforward to add curvature

6. Bayesian inference MCMC Affine invariant MCMC ensemble sampling (Goodman & Weare 2010) H(z)H(z)

7. It is reasonable to assume that H(z) increases with redshift, i.e., dH(z)/dz ≥ 0 or H i ≤ H i+1 Observations are largely consistent with such an assumption. Moresco et al.(2012)

8. Cosmological parameters of mock samples: WFIRST SNeIa error models: Sample size: 2725 z_max: 1.7

9. --- without MP --- with MP

10. without mono. prior with mono. prior

15. * DESI LSST: distances from H i  constraints on H i

16. Reconstruction of time-varying cosmological quantities is useful for model testing and cross checks. The monotonicity prior can significantly reduce the errors in reconstructed H i 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.