Two useful methods for the supernova cosmologist: (1) Including CMB constraints by using the CMB shift parameters (2) A model-independent photometric redshift estimator for SNe Ia Yun Wang June 28, 2007, Aspen Workshop on “Supernovae as Cosmological Distance Indicators”

Yun Wang, 6/28/07 (1)Including CMB constraints by using the CMB shift parameters R and l a ( Y. Wang & P. Mukherjee, astro-ph/ ) R=[  m H 0 2 ] 1/2 r(z CMB ) dimensionless distance to z CMB l a =  r(z CMB ) / r s (z CMB ) angular scale of the sound horizon at z CMB (R, l a ) have nearly uncorrelated measured values. (R, l a,  b h 2 ) provide an efficient summary of CMB data, independent of the dark energy model.

Yun Wang, 6/28/07

Gaussian fits: Normalized correlation matrices:  k =0 Rlala bh2bh2 R1.0  .0197 lala  .6283 bh2bh2 .0197  k0k0  k =0 R 1.71   0.03 lala   1.2 bh2bh   k0k0 Rlala bh2bh2 R1.0  lala  .6722 bh2bh 

Yun Wang, 6/28/07 All you have to do is to add this to your  2 tot :  2 CMB = [p i   p i data  ] Cov -1 (p i,p j ) [p j   p j data  ] with Cov(p i,p j ) = σ(p i ) σ(p j ) Cov(p i,p j ) norm data {p i }={R, l a,  b h 2 }

Yun Wang, 6/28/07 w(z)=w 0 +w a (1-a) WMAP SNe Ia (Riess et al. 2007, inc SNLS and nearby SNe) +SDSS BAO (Wang & Mukherjee 2007)

Yun Wang, 6/28/07 Model- independent constraints on dark energy (as proposed by Wang & Garnavich 2001) Wang & Mukherjee (2007)

Yun Wang, 6/28/07 Wang & Mukherjee (2007) [See Wang & Tegmark (2005) for the method to derive uncorrelated estimate of H(z) using SNe.]

Yun Wang, 6/28/07 (2) A model-independent photometric redshift estimator for SNe Ia (Y. Wang, astro-ph/ , ApJ, 654 (2007) L123 ) Accurate photo-z’s boost the cosmological impact of large photometric surveys of SNe. Derive a simple photo-z estimator for SNe Ia using imaging observables that reflect the properties of SNe Ia as calibrated standard candles. If SNe Ia were perfect standard candles, the most important observable is peak brightness. Use the maximum flux in the best-sampled band (say, i-band) to represent this. Use the fluxes in other bands at the same epoch to make an effective K-correction.

Yun Wang, 6/28/07 A model-independent photo-z estimator for SNe Ia (1) z 0 phot =c 1 +c 2 g f +c 3 r f +c 4 i f +c 5 z f +c 6 i f 2 g f =2.5 log(f g ), r f =2.5 log(f r ), i f =2.5 log(f i ), z f =2.5 log(f z ), f g, f r, f i, f z : fluxes in griz at the epoch of i max flux (2)  i 15 =2.5 log(f i 15d /f i ) f i 15d is the i-band flux at 15 days after the i flux max in the estimated restframe,  t 15d =15(1+ z 0 phot ) (3) z phot =z 0 phot +c 7  i 15

Yun Wang, 6/28/07 A model-independent photo-z estimator for SNe Ia The coefficients c i (i=1,2,…,7) are found by using a training set of SNe Ia with both griz light curves and measured spectro z’s Use jackknife technique to estimate bias- corrected mean and covariance matrix of c i

Yun Wang, 6/28/07 jackknife Consider a consistent statistic t t n : value of t calculated from a sample of size n For a single sample of n points, we extract n subsamples of size n-1 by omitting one element t n-1,i : omitting i-th element;  t n-1  =  i=1,n t n-1,i Bias-corrected estimate: t n J =t n +(n-1)(t n -  t n-1  ) Variance: V J (t n )=[(n-1)/n]  i (t n-1,i -  t n-1  ) 2

Yun Wang, 6/28/07 Demonstration using SNLS data Y. Wang (2007)

Yun Wang, 6/28/07 Using simulated data with zero A V Y. Wang, M. Wood-Vasey, G. Narayan, in prep.

Yun Wang, 6/28/07 Blind test on simulated data with A V  0 (z phot -z spec ) versus z spec (Y. Wang, M. Wood-Vasey, & G. Narayan, in prep.)

Yun Wang, 6/28/07 Blind test on simulated data with A V  0 (z phot -z spec ) versus A v (Y. Wang, M. Wood-Vasey, & G. Narayan, in prep.)

Yun Wang, 6/28/07 8.4m (6.5m clear aperture) telescope; FOV: 3.5 deg diameter;  m 10 6 SNe Ia y , z < 0.8, 6 bands,  t = 7d 20,000 sq deg WL & BAO with photo-z

Yun Wang, 6/28/07 ALPACA 8m liquid mirror telescope FOV: 2.5 deg diameter Imaging  =0.3-1  m 50,000 SNe Ia per yr to z=0.8, 5 bands,  t = 1d 800 sq deg WL & BAO with photo-z

Yun Wang, 6/28/07 Conclusions The CMB shift parameters (R and l a ) provide an efficient summary of the full CMB temperature power spectrum as far as dark energy constraints are concerned. Including R and l a is very easy and tightens SN cosmology constraints considerably. (Wang & Mukherjee 2007) The simple model-independent photo-z estimator derived in Wang (2007) works well for current SN Ia data, with  [(z phot - z spec )/(1+z spec )]=0.05 for SNe Ia not used in the training set. Preliminary studies (Wang, Wood-Vasey, & Narayan 2007) indicate that  [(z phot -z spec )/(1+z spec )]= can be achieved for SN Ia data with much higher S/N. This can boost the cosmological utility of large photometric surveys of SNe.