Novel Bayesian approaches to supernova type Ia cosmology Roberto Trotta - MCMSki 2014 07/01/14 - www.robertotrotta.com Wednesday, 8 January 14
The cosmological concordance model The Λ CDM cosmological concordance model is built on three pillars: 1. INFLATION: A burst of exponential expansion in the first ~10 -32 s after the Big Bang, probably powered by a yet unknown scalar field. 2. DARK MATTER: The growth of structure in the Universe and the observed gravitational effects require a massive, neutral, non-baryonic yet unknown particle making up ~25% of the energy density. 3. DARK ENERGY: The accelerated cosmic expansion (together with the flat Universe implied by the Cosmic Microwave Background) requires a smooth yet unknown field with negative equation of state, making up ~70% of the energy density. The next 5 to 10 years are poised to bring major observational breakthroughs in each of those topics! Roberto Trotta Wednesday, 8 January 14
Radiation Dark matter Dark energy era era era Big Bang TODAY time End of the visible SN Type Ia cosmos Wednesday, 8 January 14
The end of the visible Universe Data from the Planck satellite, 2013 Wednesday, 8 January 14
10 12 bits 50x10 6 pixels 2500 harmonics 6 parameters model Wednesday, 8 January 14
Wednesday, 8 January 14
Low redshift cosmological probes Baryonic acoustic oscillations (z~0.35) Supernovae type Ia (z < 1.5) Δ distance modulus Λ CDM Correlation function Λ CDM (SDDS collaboration) (2010) Padmanabhan et al (2012) Kessler et al No DE distance z~1100 z~0.35 Mehta et al (2012) Acoustic scale BAO CMB Roberto Trotta Wednesday, 8 January 14
Type Ia supernovae SN1994D • Supernovae: core-collapse thermonuclear explosions of stars, emitting a large (~ 10 51 erg, cf L galaxy ~ 10 44 erg/s ) amount of energy (photons + neutrinos). • Supernovae type Ia (SNIa): characterized by the High z SN Team/ NASA/HST lack of H in their spectrum, outcome of a CO white dwarf (WD) in a close binary system accreting mass above the Chandrasekhar limit (1.4 solar masses). Single Double degenerate degenerate • The nature of the donor star is still disputed: Single Degenerate (WD + Main sequence or Red giant or a He star companion) vs Double Degenerate (WD + WD merger) scenarios (or both) NASA/CXC/M. Weiss Roberto Trotta Wednesday, 8 January 14
SNIa cosmology Apparent rest-frame Luminosity Cosmological parameters B-band magnitude distance From measurements in B, Quantities of interest V, I, J, ... band Ω M , Ω DE , w, w(z), ✓ d L ( z, C ) ◆ H 0 (degenerate with M), µ = m B − M = 5 log 10 + 25 1 Mpc Redshift Absolute magnitude Distance modulus Measured via spectrum Unknown, but of the host galaxy IF ~ constant unimportant (“standard candle”) Goal: From the measured multi-band light curves and redshift, infer constraints on the Needs to be corrected via empirical correlations cosmological parameters. with other observables But: the devil is in the (statistical) detail! M → M + linear corrections Our solution: March, RT et al, MNRAS 418(4): (“Phillips relations”) 2308-2329, 2011 , e-print archive: 1102.3237 Roberto Trotta Wednesday, 8 January 14
CfA3 185 multi-band optical nearby SNIa SNIa lightcurves SNLS SNLS-04D3gx g r i z Flux Guy et al (2007) rors nges and 53100 53150 53200 r of MJD Hicken et al (2009) Roberto Trotta Wednesday, 8 January 14
Brightness-width relationship Low-z calibration sample SNIa absolute magnitude residual scatter After dust − 20 BRIGHTER ~ 0.2 mag correction − 19.5 ~ factor of 3 − 19 Peak Magnitude FAINTER − 18.5 − 18 LC decline rate Before dust − 17.5 B band correction − 17 Mandel et al (2011) M B − 16.5 B 0 − µ V band − 16 0.8 1 1.2 1.4 1.6 Δ m 15 (B) Phillips (1993) Decline rate I band Brighter SNIa are slow decliners Roberto Trotta Wednesday, 8 January 14
From lightcurves to distances • There are a few different lightcurve (LC) fitters on the market, with different philosophies/statistical approaches: • MLCS2k2 (Jha et al, 2007): color (A V ) and LC shape ( Δ ) parameters fitted simultaneously with cosmology. Color correction includes a dust extinction law correction. • SALT/SiFTO/SALT2 (Guy et al, 2007): LC shape (x 1 ) and colour (c) correction extracted from LC alongside apparent B-band magnitude (m B ) + covariance matrix. The distance modulus µ = m B − M + α × width − β × colour is subsequently estimated with cosmological parameters and remaining “intrinsic” scatter. • BayeSN (Mandel et al, 2009, 2011): Fully Bayesian hierarchical modeling of LC, including population-level distributions (see later). Roberto Trotta Wednesday, 8 January 14
PS1 data • Most recent data set from PAN-STARRS1 survey • 146 spectroscopically confirmed SNIa • Cosmological fit: 112 PS1 at high-z (blue) + 201 low-z SNIa (red) Roberto Trotta Wednesday, 8 January 14
The importance of a principled approach • SNIa cosmology is now a mature field. Cosmological inferences (in particular, w(z)) are beginning to be dominated by systematic uncertainties: • Do SNIa properties evolve with z? • Are there multiple SNIa populations, with different characteristics? • Is dust extinction modeling adequate? • Can additional observables help in reducing “intrinsic” variability? • Can a data-based approach help in guiding theoretical understanding? • All of those questions are best addressed from a statistically principled standpoint: a complete (Bayesian) modeling including intrinsic variability, measurement errors, population-level distribution, observational effects can deliver superior insight. Roberto Trotta Wednesday, 8 January 14
Standard Chi 2 fits of SALT2 output • Standard analysis minimizes the likelihood (typically, C minimized with α , β fixed, then α , β minimized with C fixed), arbitrarily defined as: observed values (SALT2 fits) parameters c i ]) 2 ( µ ( z i , C ) − [ ˆ m B,i − M + α ˆ x 1 ,i − β ˆ − 2 log L = χ 2 = X σ 2 int + σ 2 fit i σ 2 fit = σ 2 m B + α 2 σ 2 x 1 + β 2 σ 2 c + correlations σ 2 int represents the “intrinsic” (residual) scatter determined by requiring Chi 2 /dof ~ 1 Roberto Trotta Wednesday, 8 January 14
Problems of the standard analysis c i ]) 2 ( µ ( z i , C ) − [ ˆ m B,i − M + α ˆ x 1 ,i − β ˆ − 2 log L = χ 2 = X σ 2 int + σ 2 fit i • Form of the likelihood function is unjustified • α , β appear in the variance, too - this is a problem of simultaneous estimation of the mean and of the variance. Chi 2 not the correct distribution. − 1 σ 2 int + σ 2 • Incorrectly normalized - missing term in front. Adding this in � � 2 log fit results in a (known) 6-sigma bias of β . • Chi 2 /dof ~ 1 prescription prevents by construction model checking and hypothesis testing • Marginalization (and use of fast Bayesian MCMC methods) impossible (profile likelihood “fudge” necessary) Principled Bayesian solution required! Roberto Trotta Wednesday, 8 January 14
Bayesian hierarchical model For each SNIa, this relation holds exactly between latent (unobserved) variables: µ i ( z i , C ) = m B,i − M i + α x 1 ,i − β c i Derived variable Prior Prior Population-level hyperparameters to be estimated from the data Parameters of Population M i ∼ N ( M 0 , σ 2 int ) interest hyper-parameters INTRINSIC VARIABILITY c i ∼ N ( c ? , R c ) Latent variables x 1 ,i ∼ N ( x ? , R x ) NOISE, SELECTION EFFECTS x 1 ,i ] ∼ N ([ m B i , c i , x 1 ,i ] , ˆ [ ˆ m B,i , ˆ c i , ˆ C i ) Observed values Roberto Trotta Wednesday, 8 January 14
Advantages of multi-layer model • The Bayesian hierarchical approach allows us to: • model explicitly the population-level intrinsic variability of SNIa • investigate the impact of multiple SNIa populations (e.g., different progenitor models) • determine/include correlations with other observables (galaxy mass, metallicity, age, spectral lines, etc) to reduce residual scatter in Hubble diagram • obtain a principled data likelihood that can be used with Bayesian MCMC/ MultiNest (marginal posteriors, Bayesian evidence for model selection) • derive a fully marginalized posterior on the residual (after colour and stretch correction) intrinsic scatter in the SNIa intrisic magnitude • investigate possible SNIa evolution (e.g., β (z)) and other systematics Roberto Trotta Wednesday, 8 January 14
At the heart of the method... • ... lies the fundamental problem of linear regression in the presence of measurement errors on both the dependent and independent variable and intrinsic scatter in the relationship (e.g., Gull 1989, Gelman et al 2004, Kelly 2007): µ i = m B,i − M i + α x 1 ,i − β c i anagolous to y i = b + ax i POPULATION x i ∼ p ( x | Ψ ) = N x i ( x ? , R x ) DISTRIBUTION y i | x i ∼ N y i ( b + ax i , σ 2 ) INTRINSIC VARIABILITY y i ([ x i , y i ] , Σ 2 ) x i , ˆ ˆ y i | x i , y i ∼ N ˆ MEASUREMENT ERROR x i , ˆ Roberto Trotta Wednesday, 8 January 14
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