Novel Bayesian approaches to supernova type Ia cosmology

Transcription

1 Novel Bayesian approaches to supernova type Ia cosmology - MCMSki /01/14 -

2 The cosmological concordance model The ΛCDM 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!

3 Radiation era Dark matter era Dark energy era Big Bang TODAY End of the visible cosmos SN Type Ia time

4 The end of the visible Universe Data from the Planck satellite, 2013

5 10 12 bits 50x10 6 pixels 2500 harmonics 6 parameters model

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7 Low redshift cosmological probes Supernovae type Ia (z < 1.5) Baryonic acoustic oscillations (z~0.35) Kessler et al (SDDS collaboration) (2010) Δ distance modulus ΛCDM No DE Correlation function ΛCDM Padmanabhan et al (2012) 4 K. Mehta et al. Figure 13. The unreconstructed [left] and reconstructed [right] DR7 angle averaged correlation function. The error bars are the standard deviation of the 160 LasDamas simulations. These errors are however highly correlated from bin to bin and therefore no conclusions as to significance should be drawn from these figures. The solid line is the best fit model to these data. As in the simulations, the acoustic feature appears sharpened. z~0.35 z~1100 LasDamas simulations. The amplitude of the intermediatescale correlation function decreases due to the correction of redshift-space distortions, while the transition into the BAO feature at Mpc/h is sharpened. The correlated nature of the errors makes it di cult to quantitativelybao assess the impact of reconstruction on these data. Figure 14 plots the 2 surface for both before and after reconstruction. We note that the 2 minimum CMB after reconstruction is visibly narrower, indicating an improvement in the distance constraints. This improvement is also summarizedfigure in the 2. first 6dFGS, two reconstructed lines of Table SDSS DR7, 4 which and WiggleZ showsbao that data points. The black line represents the ΛCDM prediction using WMAP7 reconstruction reduces data only the(komatsu distanceet error al. 2011). fromthe 3.5% shaded to 1.9%. gray These distance constraints are also consistent 2 with the errors Acoustic scale distance the fiducial case. However, note that the physical observable is not, butd V /r s = (D V /r s ) fid. Comparing this across the three cosmologies (second column, Table 4), we find it insensitive to the choice of cosmology. The distance information from these BAO measurements may be summarized into a probability distribution p(d V /r s ), plotted in Figure 15 and summarized in the second column of Table 4. Unlike, these measurements no longer make reference to a fiducial cosmology. One may however freely convert between p( ) andp(d V /r s )bymultiplying the latter by (D V /r s ) fid. We use the results in Figure 15 to explore the cosmological consequences of these measurements in Paper III. If we assume a perfectly measured sound Figure 3. Plot of D V /r s normalized by the fiducial value. The open square is the Percival et al. (2010) BAO measurement. The black line is the WMAP7 ΛCDM model, red line shows the ef- Mehta et al (2012)

8 NASA/CXC/M. Weiss High z SN Team/ NASA/HST Type Ia supernovae Supernovae: core-collapse thermonuclear explosions of stars, emitting a large (~ erg, cf Lgalaxy ~ erg/s ) amount of energy (photons + neutrinos). Supernovae type Ia (SNIa): characterized by the 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). 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) SN1994D Single degenerate Double degenerate

9 SNIa cosmology Apparent rest-frame B-band magnitude From measurements in B, V, I, J,... band Distance modulus Absolute magnitude Unknown, but IF ~ constant unimportant ( standard candle ) Needs to be corrected via empirical correlations with other observables Luminosity distance dl (z,c) µ = m B M = 5 log 10 1Mpc + 25 Cosmological parameters Quantities of interest ΩM, ΩDE, w, w(z), H0 (degenerate with M), Redshift Measured via spectrum of the host galaxy Goal: From the measured multi-band light curves and redshift, infer constraints on the cosmological parameters. M M + linear corrections ( Phillips relations ) But: the devil is in the (statistical) detail! Our solution: March, RT et al, MNRAS 418(4): , 2011, e-print archive:

10 SNIa lightcurves CfA3 185 multi-band optical nearby SNIa SNLS SNLS-04D3gx Flux g r i z Guy et al (2007) MJD Hicken et al (2009)

11 Brightness-width relationship SNIa absolute magnitude Phillips (1993) residual scatter ~ 0.2 mag LC decline rate B band V band I band BRIGHTER FAINTER ~ factor of 3 Mandel et al (2011) Peak Magnitude Low-z calibration sample 16 M B B 0 µ Before dust correction Δ m 15 (B) Decline rate After dust correction Δ m Fig. 4. (left) Post-maximum optical decline sus posterior estimates of the inferred optical ab M B (black points) and the extinguished magn points). Each black point maps to a red poi Brighter SNIa are slow decliners dust extinction in the host galaxy. The intrinsic luminosity Phillips relation is reflected in the points, indicating that SN brighter in B have s

12 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 (AV) 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 (x1) and colour (c) correction extracted from LC alongside apparent B-band magnitude (mb) + covariance matrix. The distance modulus µ = m B M + width 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).

13 PS1 data Most recent data set from PAN-STARRS1 survey 146 spectroscopically confirmed SNIa Cosmological fit: 112 PS1 at high-z (blue) low-z SNIa (red)

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.

15 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: parameters observed values (SALT2 fits) 2 log L = 2 = X i (µ(z i, C) [ˆm B,i M + ˆx 1,i ĉ i ]) 2 2 int + 2 fit 2 fit = 2 m B x c + correlations 2 int represents the intrinsic (residual) scatter determined by requiring Chi 2 /dof ~ 1

16 Problems of the standard analysis 2 log L = 2 = X i Form of the likelihood function is unjustified (µ(z i, C) [ˆm B,i M + ˆx 1,i ĉ i ]) 2 2 int + 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 Incorrectly normalized - missing term in front. Adding this in 2 log 2 int + fit 2 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! 2 fit

17 Bayesian hierarchical model For each SNIa, this relation holds exactly between latent (unobserved) variables: µ i (z i, C) =m B,i M i + x 1,i c i Derived variable Population-level hyperparameters to be estimated from the data Prior Prior M i N (M 0, 2 int) Parameters of interest Population hyper-parameters c i N (c?,r c ) x 1,i N (x?,r x ) [ˆm B,i, ĉ i, ˆx 1,i ] N ([m Bi,c i,x 1,i ], Ĉi) INTRINSIC VARIABILITY Latent variables NOISE, SELECTION EFFECTS Observed values

18 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

19 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): anagolous to µ i = m B,i M i + x 1,i c i y i = b + ax i x i p(x )=N xi (x?,r x ) POPULATION DISTRIBUTION y i x i N yi (b + ax i, 2 ) INTRINSIC VARIABILITY ˆx i, ŷ i x i,y i Nˆxi,ŷ i ([x i,y i ], 2 ) MEASUREMENT ERROR

20 INTRINSIC VARIABILITY + MEASUREMENT ERROR latent y observed y Kelly (2007) latent x observed x observed x Modeling the latent distribution of the independent variable accounts for Malmquist bias An observed x value far from the origin is more probable to arise from up-scattering (due to noise) of a lower latent x value than down-scattering of a higher (less probable) x value PDF latent distrib on observed x

21 The key parameter is noise/population variance σxσy/rx σxσy/rx small y i = b + ax i σxσy/rx large true Bayesian marginal posterior identical to profile likelihood true Bayesian marginal posterior broader but less biased than profile likelihood March, RT et al (2011)

22 Tests on simulated SNIa data Simulated N=288 SNIa with similar characteristics as SDSS +ESSENCE+SNLS+HST +Nearby sample Reconstruction of cosmological parameters over 100 realizations, comparing Bayesian hierarchical method with standard Chi 2. Simulated SNIa realization (colour coded according to survey ) March et al (2011)

23 Posterior sampling In the Bayesian hierarchical approach, we have 3 cosmological parameters: H0, ΩM, ΩK (w=1) or H0, ΩM, w (ΩK =0) 2 stretch/colour correction parameters: α, β 6 population-level parameters: M0, σ 2, x*, Rx, c*, Rc 3N (=864) latent variables Mi, x1i, ci Analytical marginalization over all latent variables and linear population-level parameters is possible in Gaussian case (no selection effects). Sampling of the remaining parameters via MultiNest. Alternatively, Gibbs sampling can be used to sample over all parameters (conditional distributions are Gaussian in the absence of selection effects. Including them introduces additional accept/reject step).

24 The Nested Sampling algorithm Skilling (2006) introduced Nested Sampling as an algorithm originally aimed at the efficient computation of the model likelihood (Skilling, 2006). The idea is to map a multi-dimensional integral onto a 1D integral which is easy to compute numerical The method requires to sample uniformly from the fraction of the prior volume X(μ) above the iso-likelihood level μ Z θ 2 X(µ) = P ( )d L(x) Z L( )>µ Z 1 θ 1 P (d) = d L( )P ( ) = 0 X(µ)dµ 0 1 x Feroz et al (2008), arxiv: Trotta et al (2008), arxiv:

25 The MultiNest ellipsoidal sampling The MultiNest algorithm (Feroz & Hobson, 2007, 2008) uses a multi-dimensional ellipsoidal decomposition of the remaining set of live points to approximate the prior volume above the target iso-likelihood contour. Multimodal likelihood Highly degenerate likelihood target iso-likelihood contours ellipsoidal approximation multi-modal decomposition Decreasing prior fraction X

26 Marginal posterior (simulated data) w = 1 ΩK = 0 True value True value March et al (2011) Red/empty: Chi 2 (68%, 95% CL) Blue/filled: Bayesian (68%, 95% credible regions) Bayesian posterior is noticeably different from the Chi 2 CL: which one is best?

27 Coverage, bias and mean squared error Coverage of Bayesian 1D marginal posterior CR and of 1D Chi 2 profile likelihood CI computed from 100 realizations Bias and mean squared error (MSE) defined as ˆ is the posterior mean (Bayesian) or the maximum likelihood value (Chi 2 ). Red: Chi 2 Blue: Bayesian Results: Coverage March et al (2011) Coverage: generally improved (but still some undercoverage observed) Bias: reduced by a factor ~ 2-3 for most parameters MSE: reduced by a factor for all parameters

28 Cosmology results Combined sample w = 1 Kessler et al (SDDS collaboration) (2010) 288 SNIa Blue: Bayesian Red: Chi 2 March et al (2011) Marginal posteriors =0.12 ± 0.02 =2.7 ± 0.1 =0.13 ± 0.01 mag

29 Combined constraints Combined cosmological constraints on matter and dark energy content: w = 1 ΩK = 0 BAO Combined BAO CMB SNIa Combined CMB SNIa March, RT et al (2011)

30 The BayeSN approach Developed by K. Mandel (Mandel et al, 2009, 2011) and collaborators: fully Bayesian approach to LC fitting, including random errors, population structure, intrinsic variations/correlations, dust extinction and reddening, incomplete data Prior Distance modulus Dust population parameters Dust (Av, Rv) Apparent LC LC population parameters Absolute LC Prior Observed LC Redshift SN 1...N

31 Some results from BayeSN Mandel et al (2011) Dust absorption for each SNIa Population level analysis of correlations Hubble diagram: residual scatter reduced by ~2 using optical+nir LC Inclusion of NIR LC +NIR

32 The complete hierarchical model Redshift zi Redshift data Data Local calibration sample Population parameters Dust dust Latent variables Absorption dust,i Distance modulus µ Apparent light curves m ibt (nearby) Survey parameters E, C Standardization parameters t,, Optical spectra Near-infrared light curves SN environmental data i =1,...,M Light curves ˆm ibt Light curves SN Light curves M ibt Apparent light curves m ibt (distant) Data Cosmological sample Environment env Correlates c i i =1,...,M Distance modulus µ Survey parameters E, C Standardization parameters t,, Cosmological parameter C Light curve summary statistics Light curves Optical spectra ˆm ibt Near-infrared light curve Redshift zi Redshift data SN environmental data i =1,...,M Red arrows/boxes indicate elements/data that have never been explored before in such a multi-level setting

33 Summary Current and future SNIa surveys are becoming systematics limited: better modeling is required to use them as powerful and reliable probes of dark energy Bayesian multi-level models can capture the different layers and sources of uncertainties in SNIa Intrinsic population-level variability can be studied and constrained Full propagation of uncertainties to the level of cosmological parameters becomes possible with a consistent, principle Bayesian approach The Bayesian hierarchical model of March et al outperforms the standard Chi2 approach 2/3 of the time The BayeSN approach of Mandel et al offers a fully Bayesian modeling of SNIa LC

34 Conclusions and future work Powerful Bayesian methods can take SNIa cosmological inference to a next quantitative step: reduced systematics thanks to better modeling Required to deal with future large (~ 3000) samples Extension of our method to include survey selection effects ongoing Inclusion of multiple SNIa population, possible redshiftdependence of SNIa properties, correlation with other observables (galaxy mass, metallicity, spectral lines, etc) straightforward Bayesian model comparison (LCDM vs modified gravity) β(z) = β0 + β1z true true

35 Thanks! astro.ic.ac.uk/icic