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Bayesian Hierachical Models for SNIa Cosmology

Roberto Trotta19/09/2012 - www.robertotrotta.com

Thursday, 20 September 12

http://www.robertotrotta.com


astro.ic.ac.uk/icic


http://www.astro.ic.ac/icic

http://www.astro.ic.ac/icic

Roberto Trotta

The cosmological concordance model

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!

The ΛCDM cosmological concordance model is built on three pillars:


WMAP 7-years temperature power spectrum

Multipole moment ell10 100 500 1000

Best-fit 6-parameters

ΛCDM concordance

model

Jarosik et al (2010)

z ~ 1100


Roberto Trotta

WM

AP

team

12 Padmanabhan et al

Figure 13. The unreconstructed [left] and reconstructed [right] DR7 angle averaged correlation function. The error bars are the standarddeviation of the 160 LasDamas simulations. These errors are however highly correlated from bin to bin and therefore no conclusions asto significance should be drawn from these figures. The solid line is the best fit model to these data. As in the simulations, the acousticfeature appears sharpened.

LasDamas simulations. The amplitude of the intermediate-scale correlation function decreases due to the correction ofredshift-space distortions, while the transition into the BAOfeature at ⇠80� 100 Mpc/h is sharpened.

The correlated nature of the errors makes it di�cult toquantitatively assess the impact of reconstruction on thesedata. Figure 14 plots the �2 surface for ↵ both before andafter reconstruction. We note that the �2 minimum after re-construction is visibly narrower, indicating an improvementin the distance constraints. This improvement is also sum-marized in the first two lines of Table 4 which shows thatreconstruction reduces the distance error from 3.5% to 1.9%.These distance constraints are also consistent with the errorsestimated from the LasDamas simulations.

Figure 14 also plots the �2 surface for a template with-out a BAO feature, using the “no-wiggle” form of Eisenstein& Hu (1998). The lack of a well defined minimum eitherbefore or after reconstruction indicates that our distanceconstraints are indeed coming from the presence of a BAOfeature and not any broad band features in the correlationfunction. The di↵erence in �2 between the templates withand without a BAO feature also provides an estimate of thesignificance of the BAO detection in these data. Reconstruc-tion improves this detection significance from 3.3� (consis-tent with previous measurements) to 4.2�. This is not theonly measure of the detection significance possible; Paper IIdiscusses these in more detail.

As before, we would like to demonstrate the robustnessof the results to the various parameters of the reconstruc-tion algorithm. Table 4 lists the recovered distances varyingthe smoothing scale, input bias, growth rate (f), and priorpower spectrum; for each of these cases, we recover distancesconsistent with the fiducial choices of parameters.

Our final test is the impact of the assumed fiducial cos-mology. We consider two cases in Table 4: flat ⇤CDM cos-mologies with ⌦M = 0.2 and 0.35. In both of these cases,we adjust the Hubble constant and the baryon density ⌦b

to keep the physical densities ⌦bh2 and ⌦Mh2 equal to

their WMAP7 values. This prescription leaves the CMB un-changed, but alters the distance-redshift relation. We findthat the estimated values of ↵ are significantly di↵erent from

the fiducial case. However, note that the physical observableis not ↵, but DV /rs = ↵(DV /rs)

fid

. Comparing this acrossthe three cosmologies (second column, Table 4), we find itinsensitive to the choice of cosmology.

The distance information from these BAO measure-ments may be summarized into a probability distributionp(DV /rs), plotted in Figure 15 and summarized in the sec-ond column of Table 4. Unlike ↵, these measurements nolonger make reference to a fiducial cosmology. One may how-ever freely convert between p(↵) and p(DV /rs) by multiply-ing the latter by (DV /rs)

fid

. We use the results in Figure 15to explore the cosmological consequences of these measure-ments in Paper III. If we assume a perfectly measured soundhorizon, these measurements can be converted into a dis-tance measurement in Gpc. Using a sound horizon of 152.76Mpc, we get a distance to z = 0.35 of 1.356 ± 0.025 Gpc.Note that these numbers do not have h�1 factors in them.Of course, the sound horizon is not perfectly measured andits uncertainty must be taken into account when fitting forcosmologies. Paper III discusses the methodologies and re-sults in detail.

6 DISCUSSION

We present the results of the density field reconstructionon the BAO feature on the SDSS DR7 LRG data. This isthe first application of reconstruction on a galaxy redshiftsurvey, resulting in a 1.8 factor reduction in the distanceerror to a z = 0.35, equivalent to a tripling of the surveyvolume. This is the first in a series of three papers; Paper IIdescribes the fitting of the correlation function, while PaperIII explores the cosmological implications of these results.

Our principal results and conclusions are :

(i) We modify the Eisenstein et al. (2007a) reconstructionalgorithm to account for the e↵ects of survey boundaries andredshift-space distortions and test it on the mock catalogsfrom the LasDamas suite of simulations. These mock cat-alogs have been designed to both match the SDSS surveygeometry as well as the redshift distribution and clusteringproperties of the SDSS LRG sample.

c� 0000 RAS, MNRAS 000, 000–000

Baryonic acoustic oscillations (z~0.35)

ΛCDM

Low redshift cosmological probesKe

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Supernovae type Ia (z < 1.5)

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4 K. Mehta et al.

Figure 2. 6dFGS, reconstructed SDSS DR7, and WiggleZ BAOdata points. The black line represents the !CDM prediction us-ing WMAP7 data only (Komatsu et al. 2011). The shaded grayregion is the e"ect of varying #mh2 within the 1! measurementerrors of WMAP7. We see that the BAO data is consistent withthe !CDM cosmological model.

on this plot with the width set by the uncertainty in !mh2.In this figure, we explore the e"ects of varying the equationof state parameter, w and the curvature of the Universe !K

respectively. The blue region corresponds to a flat Universewith w = !0.7, while the red region corresponds to a Uni-verse with a cosmological constant and !K = 0.01. !m isadjusted to keep the sound horizon constant. From this fig-ure, we see that changing w mostly changes the slope of theline on this plot while a non-zero !K mostly changes thevertical o"set. The relative distance measure from compar-ing the flux of SN are constrain only the slope of the lines,while the BAO data can measure an absolute distance andhence the vertical o"set. This explains why SN data is moree"ective at constraining w, while the BAO data is more ef-fective at constraining !K . The Riess et al. (2011) direct H0

measurement is also plotted in this figure assuming the fidu-cial sound horizon value. While the sound horizon varies byabout 1% within the WMAP7 results, this e"ect is subdom-inant to the quoted errors on H0. We explore the apparenttension between the BAO measurement and the direct mea-surement of H0 in Section 3.9.

Conventionally, the Hubble constant has been mea-sured by building a distance ladder from local measurementsout to measuring the cosmological Hubble flow. Conversely,the CMB and BAO data build an inverse distance ladderstarting from a distance measurement at the recombina-tion epoch. The CMB data provides an accurate measure-ment of the distance to the recombination redshift and ourBAO data provides a measurement of distance to z = 0.35,thereby building an inverse distance ladder. The combina-tion of these two datasets has the power to distinguish be-tween di"erent cosmological models. The supernovae dataextrapolate the distance measurements to lower redshiftand, therefore, precisely measure the expansion of the Uni-verse at z = 0, which is the Hubble constant, H0. In thefollowing sections we use a combination of these datasetsto explore a variety of cosmological models, and we use the

Figure 3. Plot of DV /rs normalized by the fiducial value. Theopen square is the Percival et al. (2010) BAO measurement. Theblack line is the WMAP7 !CDM model, red line shows the ef-fect of varying w and the blue line, the e"ect of varying #K .The shaded regions around these lines correspond to 1! uncer-tainty in #mh2 around the WMAP7 measurement. We see thatthe BAO data has the power to distinguish between various cos-mological models. The H0 point is the direct H0 measurementfrom Riess et al. (2011).

CMB+BAO+SN dataset to obtain robust measurements ofH0 and !m.

3.2 #CDM: The Vanilla Model

The WMAP7 measurements of the CMB give us very goodmeasurements of the various parameters in the “vanilla cos-mology” model, also known as the #CDM model. AddingBAO measurement to the WMAP7 results improves themeasurement of !m by about 40% and H0 by almost 30%.With reconstruction, we measure !m = 0.280 ± 0.014 andH0 = 69.8 ± 1.2 km/s/Mpc giving us a 1.7% measurementof the Hubble constant. Figure 4 shows the 68% and 95%confidence level contours for H0 vs !m and we can see theimprovement in these parameters by adding the BAO data.Table 1 shows the values for !mh2, !m, and H0 for variouscosmological models and the corresponding datasets used.

The acoustic standard ruler is calibrated by the WMAPmeasurement of !mh2. Komatsu et al. (2011) shows thatallowing for a running spectral index, dns/d ln k increasesthe errors on !mh2. Thus, we explore the e"ects of varyingthe running spectral index, dns/d ln k with the CMB andCMB+BAO datasets. We note that the nuisance param-eters used in our BAO fitting techniques (PaperII) makeour measurement of DV /rs insensitive to the running spec-tral index. Table 2 shows the e"ect of varying the runningspectral index on cosmological parameters. We see that therunning spectral index is consistent with 0: dns/d ln k =!0.024± 0.020 using the CMB+BAO dataset. We find thatincluding this parameter in the case of CMB data only, the!mh2 measurements are degraded by a factor of 1.4 from!mh2 = 0.1341 ± 0.0056 to 0.1393 ± 0.0080. This corre-sponds to an increased uncertainty in the measurements of!m, H0, and the spectral index ns. Adding the BAO dataimproves the measurement of !mh2, !m, H0, and ns and

Meh

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z~0.35

CMBBAO


Roberto Trotta

Type Ia supernovae

• Supernovae: core-collapse thermonuclear explosions of stars, emitting a large (~ 1051 erg, cf Lgalaxy ~ 1044 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

High

z S

N Te

am/

NASA

/HST

NASA

/CXC

/M. W

eiss

Progenitors of Type 1a Supernovae?

Images:NASA/CXC/M Weiss.

Accretion Merger

bimodal population?Imperial College

London arXiv:1102.3237

[email protected]

Single degenerate

Double degenerate

Progenitors of Type 1a Supernovae?

Images:NASA/CXC/M Weiss.

Accretion Merger

bimodal population?Imperial College

London arXiv:1102.3237

[email protected]


Roberto Trotta

SNIa cosmology

Absolute magnitude

Unknown, but IF ~ constant unimportant

(“standard candle”)

Needs to be corrected via empirical correlations with other observables

M → M + linear corrections

(“Phillips relations”)

Apparent rest-frame B-band magnitude

From measurements in B, V, I, J, ... band

Distance modulus

Luminosity distance

Cosmological parameters

Quantities of interestΩM, 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.

But: the devil is in the (statistical) detail!

Our solution: March, RT et al, MNRAS 418(4):2308-2329, 2011 , e-print archive: 1102.3237

µ = mB �M = 5 log10

✓dL(z, C)1 Mpc

◆+ 25


http://arxiv.org/abs/1102.3237

http://arxiv.org/abs/1102.3237

Roberto Trotta

SNIa lightcurves

J. Guy et al, SNLS Collaboration: SALT2 11

(U-band)Å<3900 λphase, for 3200<-20 0 20 40

m)

Δ(σ

0

0.05

0.1

0.15

0.2

(B-band)Å<5000 λphase, for 3900<-20 0 20 40

m)

Δ(σ

0

0.05

0.1

0.15

0.2

(V-band)Å<5700 λphase, for 5000<-20 0 20 40

m)

Δ(σ

0

0.05

0.1

0.15

0.2

(R-band)Å<7200 λphase, for 5700<-20 0 20 40

m)

Δ(σ

0

0.05

0.1

0.15

0.2

(I-band)Å<8800 λphase, for 7300<-20 0 20 40

m)

Δ(σ

0

0.05

0.1

0.15

0.2

Fig. 6 Estimated standard deviation of model photometric errorsas a function of phase, for several rest-frame wavelength rangesroughly corresponding from top to bottom to U, B, V , R andI!bands. Those model errors were evaluated from the scatter ofresiduals to the single light-curve fit.

)ÅWavelength (4000 6000 8000

mΔ

-0.5

0

0.5

Fig. 7 Di!erence between observed peakmagnitude in each bandof each SN from table 2 and the model prediction as a functionof the rest-frame e!ective wavelength of the filter used (gray tri-angles : SNLS SNe, gray squares : nearby SNe). The large blacksymbols represent the estimated dispersion in each wavelengthbin (triangles for SNLS, and squares for nearby SNe). The largecircles show the average di!erence in each wavelength bin for allSNe and the solid curve is a polynomial fit to the dispersion usedas an estimate of the K-correction scatter. Since uncertainties onB and V magnitudes at maximum enter in the normalization andcolor evaluation of the model, K-correction uncertainties are setto zero for B and V! band wavelengths.

MJD53100 53150 53200

Flux

griz

SNLS-04D3gx

Fig. 8 Observed light-curves points of the SN Ia SNLS-04D3gxat z=0.91 along with the light-curves derived from the model(solid line, trained without this SN). The dashed lines representthe 1 ! uncertainties of the model (both uncorrelated and K-correction errors).

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SNLS

CfA3185 multi-band optical nearby SNIa

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Roberto Trotta

Brightness-width relationship

LC decline rate

SNIa

abs

olut

e m

agnit

ude

BRIGHTER

FAINTER

~ factor of 3

residual scatter ~ 0.2 mag

Philli

ps (1

993)

13

Even SN with low extinction benefit from observations inthe H-band by reducing the uncertainty in the dust es-timate. Table 4 lists summary statistics of the marginalposterior distribution of each host galaxy dust parameterfor each SN, obtained from the MCMC samples.

5.2. Intrinsic Correlation Structure of SN Ia Lightcurves in the Optical-NIR

We use the hierarchical model to infer the intrinsiccorrelation structure of the absolute SN Ia light curves.This correlation structure captures the statistical rela-tionships between peak absolute magnitudes and declinerates of light curves in multiple filters at di!erent wave-lengths and phases. We summarize inferences about lightcurve shape and luminosity across the optical and nearinfrared filters; a more detailed analysis of the intrin-sic correlation structure of colors, luminosities and lightcurve shapes will be presented elsewhere.

5.2.1. Intrinsic Scatter Plots

The hierarchical model fits the individual light curveswith the di!erential decline rates model and infers theabsolute magnitudes in multiple passbands, corrected forhost galaxy dust extinction. For each individual SN lightcurve, we can use the inferred local decline rates dF tocompute the "m15(F ) of the light curve in each filter. Inthe left panel of Figure 4, we plot the posterior estimateof the peak absolute magnitude MB versus its canoni-cal "m15(B) decline rate with black points. The errorbars reflect measurement errors and the marginal uncer-tainties from the distance and inferred dust extinction.This set of points describes the well-known intrinsic lightcurve decline rate versus luminosity relationship (Phillips1993). We also show the mean linear relation betweenMB and "m15(B) found by Phillips et al. (1999), whoanalyzed a smaller sample of SN Ia. The statistical trendfound by our model is consistent with that analysis. Thered points are simply the peak apparent magnitudes mi-nus the distance moduli, B0 ! µ, which are the extin-guished peak absolute magnitudes MB + AB. Whereasthe range of extinguished magnitudes spans " 3 magni-tudes, the intrinsic absolute magnitudes lie along a nar-row, roughly linear trend with "m15(B).In the right panel, we plot the intrinsic and ex-

tinguished absolute magnitudes of SN Ia in the H-band. In contrast to the left panel, the di!erencesbetween the intrinsic absolute magnitudes and the ex-tinguished magnitudes are nearly negligible. Notably,there is no correlation between the intrinsic MH inthe NIR and optical "m15(B). This was noted previ-ously by Krisciunas et al. (2004a) and Wood-Vasey et al.(2008). The standard deviation of absolute magnitudesis much smaller in H than in B, demonstrating thatthe NIR SN Ia light curves are good standard can-dles (Krisciunas et al. 2004a,c; Wood-Vasey et al. 2008;Mandel et al. 2009). Theoretical models of Kasen (2006)indicate that NIR peak absolute magnitudes have rela-tively weak sensitivity to the input progenitor 56Ni mass,with a dispersion of " 0.2 mag in J and K, and " 0.1mag in H over models ranging from 0.4 to 0.9 solarmasses of 56Ni. The physical explanation may be tracedto the ionization evolution of the iron group elements inthe SN atmosphere.

0.8 1 1.2 1.4 1.6

−20

−19.5

−19

−18.5

−18

−17.5

−17

−16.5

−16Δ m15(B)

Peak

Mag

nitu

de

0.8 1 1.2 1.4 1.6Δ m15(B)

MBB0−µ

MHH0−µ

Fig. 4.— (left) Post-maximum optical decline rate !m15(B) ver-sus posterior estimates of the inferred optical absolute magnitudesMB (black points) and the extinguished magnitudes B0 ! µ (redpoints). Each black point maps to a red point through opticaldust extinction in the host galaxy. The intrinsic light curve width-luminosity Phillips relation is reflected in the trend of the blackpoints, indicating that SN brighter in B have slower decline rates.The blue line is the linear trend of Phillips et al. (1999). (right)Inferred absolute magnitudes and extinguished magnitudes in thenear infrared H-band. The extinction correction, depicted by thedi"erence between red and black points, is much smaller in H thanin B. The absolute magnitudes MH have no correlation with the!m15(B). The standard deviation of peak absolute magnitudes isalso much smaller for MH compared to MB .

These scatter plots convey some aspects of the popu-lation correlation structure of optical and near infraredlight curves that is captured by the hierarchical model.In the next section, we further discuss the multi-bandluminosity and light curve shape correlation structure interms of the estimated correlation matrices.Figure 5 shows scatter plots of optical-near infrared

colors (B!H,V !H,R!H, J!H) versus absolute mag-nitude (MB,MV ,MR,MH) at peak. The blue points arethe posterior estimates of the inferred peak intrinsic col-ors and absolute magnitudes of the SN, along with theirmarginal uncertainties. Red points are the peak apparentcolors and extinguished absolute magnitudes, includinghost galaxy dust extinction and reddening. These plotsshow correlations between the peak optical-near infraredcolors and peak optical luminosity, in the direction of in-trinsically brighter SN having bluer peak colors. In con-trast, the intrinsic J !H colors have a relatively narrowdistribution, and the near infrared absolute magnitudeMH is uncorrelated with intrinsic J !H color.

5.2.2. Intrinsic Correlation Matrices

Using the hierarchical model, we compute posterior in-ferences of the population correlations between the dif-ferent components of the absolute light curves of SN Ia.This includes population correlations between peak ab-solute magnitudes in di!erent filters, !(MF ,MF !), cor-relations between the peak absolute magnitudes andlight curve shape parameters (di!erential decline rates)in di!erent filters, !(MF ,dF !

), and the correlations be-tween light curve shape parameters in di!erent filters,!(dF ,dF !

). They also imply correlations between thesequantities and intrinsic colors. This information and itsuncertainty is captured in the posterior inference of thepopulation covariance matrix !! of the absolute light

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Decline rate

Before dust correction

After dust correction

Low-z calibration sample

Brighter SNIa are slow decliners

B band

V band

I band


Roberto Trotta

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

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).

µ = mB �M + ↵⇥ width� � ⇥ colour


Roberto Trotta

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

Standard Chi2 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:

�2fit = �2

mB+ ↵2�2

x1+ �2�2

c

+ correlations

�2int represents the “intrinsic” (residual) scatter

determined by requiring Chi2/dof ~ 1

observed values (SALT2 fits)parameters

�2 logL = �

2=

X

i

(µ(zi, C)� [mB,i �M + ↵x1,i � �ci])2

�

2int + �

2fit


Roberto Trotta

Problems of the standard analysis

• 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. Chi2 not the correct distribution.• Incorrectly normalized - missing term in front. Adding this in

results in a (known) 6-sigma bias of β. • Chi2/dof ~ 1 prescription prevents by construction model checking and hypothesis

testing• Marginalization (and use of fast Bayesian MCMC methods) impossible (profile

likelihood “fudge” necessary)

�1

2

log

��2int + �2

fit

�

Principled Bayesian solution required!

�2 logL = �

2=

X

i

(µ(zi, C)� [mB,i �M + ↵x1,i � �ci])2

�

2int + �

2fit


Roberto Trotta

Bayesian hierarchical model

For each SNIa, this relation holds exactly between latent (unobserved) variables:

Latent variables

µi(zi, C) = mB,i �Mi + ↵x1,i � �ci

Mi ⇠ N (M0,�2int)

ci ⇠ N (c?, Rc)

x1,i ⇠ N (x?

, R

x

)

Population-level hyperparametersto be estimated from the data

Populationhyper-parameters

Prior

Parameters of interest

PriorDerived variable

Observed values[mB,i, ci, x1,i] ⇠ N ([mBi , ci, x1,i], Ci)

INTRINSIC VARIABILITY

NOISE, SELECTION EFFECTS


Roberto Trotta

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

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

µi = mB,i �Mi + ↵x1,i � �ci

yi = b+ axi

x

i

⇠ p(x| ) = Nxi(x?

, R

x

) POPULATION DISTRIBUTION

yi|xi ⇠ Nyi(b+ axi,�2) INTRINSIC VARIABILITY

x

i

, y

i

|xi

, y

i

⇠ Nxi,yi([xi

, y

i

],⌃2) MEASUREMENT ERROR


latent x

laten

t yINTRINSIC VARIABILITY

observed x

obse

rved

y

+ MEASUREMENT ERROR

observed x

Kelly

(200

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observed x

latent distrib’on

PDF• 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


The key parameter is noise/population variance σxσy/Rx

σxσy/Rx small

Bayesian marginal posterior identical to profile likelihood

true

σxσy/Rx large

Bayesian marginal posterior broader but less biased than

profile likelihood Mar

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yi = b+ axi


Roberto Trotta

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 Chi2.

Simulated SNIa realization(colour coded according to “survey”)

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Roberto Trotta

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, σ2, 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 (see Mike Hobson’s talk)

• 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).


Roberto Trotta

Marginal posterior (simulated data)

w = 1 ΩK = 0

Red/empty: Chi2 (68%, 95% CL)Blue/filled: Bayesian (68%, 95% credible regions)

True valueTrue value

Bayesian posterior is noticeably different from the Chi2 CL: which one is “best”?

Mar

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Roberto Trotta

Coverage, bias and mean squared error

• Coverage of Bayesian 1D marginal posterior CR and of 1D Chi2 profile likelihood CI computed from 100 realizations

• Bias and mean squared error (MSE) defined as

is the posterior mean (Bayesian) or the maximum likelihood value (Chi2).✓

Cove

rage

Red: Chi2 Blue: Bayesian Results:

Coverage: generally improved (but still some undercoverage observed)

Bias: reduced by a factor ~ 2-3 for most parameters

MSE: reduced by a factor 1.5-3.0 for all parameters

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Roberto Trotta

Cosmology results

288 SNIa

Kess

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n) (2

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Combined sample w = 1

Red: Chi2Blue: Bayesian

Marginal posteriors

↵ = 0.12± 0.02 � = 2.7± 0.1 � = 0.13± 0.01 mag

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(201

1)


Roberto Trotta

Combined constraints

• Combined cosmological constraints on matter and dark energy content:

CMB

CMBBAO

BAO

SNIa

SNIa

Combined

Combined

w = 1 ΩK = 0

Mar

ch, R

T et

al (2

011)


Roberto Trotta

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

Dust populationparameters

LC population parameters

Prior

Prior

Dust (Av, Rv)

Distance modulus

Observed LC

Absolute LC

Apparent LC

Redshift

SN 1...N


Dust absorption for each SNIa Population level analysis of correlations

Inclusion of NIR LC

Hubble diagram: residual scatter reduced by ~2 using optical+NIR LC

+NIR

Mandel et al (2011)Some results from BayeSN


The complete hierarchical model

Latent variablesPopulation parameters

DataCosmological sample

Dust

Light curves

Absorption

Light curves

Environment CorrelatesLight curve summary statistics

Optical spectra

Near-infrared light curves

SN environmental data

Redshiftzi

Apparent light curves

(nearby)

Apparent light curves

(distant)

Redshift data

Optical spectra

Near-infrared light curve

SN environmental data

DataLocal calibration sample

Survey parameters

E, C

dust

env

SN

i = 1, . . . ,M

Distance modulus

Redshiftzi

Survey parameters

E, C

Standardization parameters

Cosmological parameter

C

Light curves

i = 1, . . . ,M

Light curves

Redshift datai = 1, . . . ,M

Mibt

mibt

mibt

mibt

mibt

µ

dust,i

ci

⌫t,↵,⌥

Distance modulusµ

Standardization parameters⌫t,↵,⌥

Red arrows/boxes indicate elements/data that have never been explored before in such a multi-level setting


Roberto Trotta

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


Roberto Trotta

Conclusions and future work

• Extension of our method to include survey selection effects ongoing

• Inclusion of multiple SNIa population, possible redshift-dependence of SNIa properties, correlation with other observables (galaxy mass, metallicity, spectral lines, etc) straightforward

• Bayesian model comparison (LCDM vs modified gravity)

true

true

β(z) = β0 + β1z

• 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


Thanks!

www.robertotrotta.com

astro.ic.ac.uk/icic




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