CMB Data Analysis

Stephen Feeney (UCL)illustrated in part by extensions to arXiv:1302.0014

with Hiranya Peiris (UCL) and Licia Verde (Barcelona & CERN)

First things first: where are the data?

• Planck Legacy Archive (Planck data only)

– www.sciops.esa.int/index.php?page=Planck_Legacy_Archive&project=planck

– popular products (webpage)

– full archive (Java application)

– precise list of products in Explanatory Supplement

But wait: there’s more!

• LAMBDA: “One-stop shopping for CMB researchers”

– lambda.gsfc.nasa.gov

– contains Planck Legacy Archive and WMAP, ACT, SPT & COBE, foreground and other datasets

– maps, likelihood functions, posterior samples, masks, noise properties, power spectra, parameters, source catalogues, publications...

– also links to all the tools you might need

1) So what can you do with CMB maps?

• Source detection: clusters, strings, bubble collisions, textures...

• Test topology and geometry of Universe

• Test for non-Gaussianity

• Work in pixel space, harmonic (alm) space or a mixture (wavelets)

• Apply filters, Minkowski Functionals, correlate with other signals...

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And how do you go about doing it?

• First, you will need HEALPix!

– Heirarchical Equal-Area isoLatitude Pixelization

– standard CMB map format

– FITS based (need CFITSIO lib)

– healpix.jpl.nasa.gov

– software available in F90, C, C++, IDL and Java, compiles with gfortran, ifort, icc, gcc

– also ported to Python (github.com/healpy/healpy)

A brief guide to HEALPix

• Both a specific pixelization of the sphere...

– Equal-area pixels, distributed on iso-latitude rings

– Different resolutions ( ) with nested pixels

• ... and a set of associated software routines

– I/O, plotting (IDL / Python), fast Fourier transforms, smoothing, segmentation...

– well-documented!

Npix

= 12N2

side

What data do you actually need?

• Simplest analyses (e.g. stacking): a map (and probably a mask)

• For likelihood/posterior analysis need noise & CMB properties

– noise map (WMAP); noise Cls and pixel “hits” (Planck)

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have to “build” atm!

2) Using likelihood funcs to constrain cosmology

• What if you want to

– constrain the underlying

– or add new parameters?

• Need a sampler and (at least one) likelihood function

• Where do you get them? What do they do?

cosmological parameters?

Likelihood functions

• For “simple” experiments, likelihood functions built into sampler: BAO, H0, etc

• WMAP and Planck (and ACT and SPT) have own, complex likelihood functions

– available for download (with or without data) from LAMBDA

• WMAP: Fortran90 code (needs CFITSIO & LAPACK)

• Planck: C, Fortran90 and Python (dependencies?)

What do the likelihood functions do?• Externally, black boxes that give you

– Internally: Cl likelihoods, estimate true Cl given theory & data

• WMAP: 2 ≤ l ≤ 1200, cleaned data [Bennett et al. 1212.5225]

– l < 32 temp: pixel-space (Gaussian), Gibbs sampling (partially pre-computed) [Dunkley et al. 0803.0586 and refs]

– high-l temperature: harmonic-space (Gaussian + log-normal) based on optimal estimator of Cls

– pol: pixel-space (low-l), MASTER estimate (high-l)

• Planck: 2 ≤ l ≤ 2500, dirty data [Planck XV 1303.5075]

– l < 50 temp: pixel-space (Gaussian), Gibbs sampling– high-l temp: harmonic ~Gaussian based on pseudo-Cls– low-l pol: same as WMAP

Pr(dCMB

|Ctheory

` )

How do you get your theory Cls?

• Using sampler: CosmoMC is king [Lewis & Bridle astro-ph/0205436]

– samples from posterior using MCMC ⇒ “chains”

– theory Cls generated using CAMB (Boltzmann code)– download from cosmologist.info/cosmomc

• Newest version uses Fortran2003 & requires ifort13: argh. Old versions do not.

• To run, compile, fiddle with params.ini, then go!

– params.ini defines data used, parameters sampled, prior ranges, accuracy settings, etc

– documentation exists...

How do you interpret your results?

• GetDist: function included in CosmoMC

– processes chains

– produces 1D & 2D marginalized

– control smoothing, plots, limits with distparams.ini

• NB: also MultiNest version of CosmoMC which calculates Bayesian Evidence (hooray!)

– nested sampler: samples intelligently from prior

posterior distributions, mean posterior, max posterior & max likelihood estimates, uncertainties...

3) Cheap & cheerful things to do with existing chains

• Planck and WMAP chains (samples from posterior) are public

• If no new fundamental parameters (i.e. new physics) needed, can post-process to

– add derived parameters...

– or add new data via importance sampling...

– or calculate new statistics such as the Bayesian evidence or profile likelihood, to extend existing results

Importance sampling

• If have new independent data, can re-weight existing posterior

• wnew = Lnew * wold (where weight ∝ posterior)

• Built-in to CosmoMC (which file?)

• Rapidly assess new data (or test prior dependence)

• Need new data to not change things too much...

Extensions beyond parameter estimation

• Lots of cosmological analyses rely on parameter estimation

– posterior mean and 68% / 95% limits

• Not always ideal: see Neff

– open to biasing by degeneracies

– can’t select between models 2 3 4 5 6 7Neff

W7+SPT+BAO+H0+Union21 Neff+1k+fi+w+nsrunW7+CMB+LRG+SN+H02W7+CMB+BAO+SN+H03 Neff+1k+fi+wW7+CMB+LRG+H04W7+CMB+BAO+H05W7+H0+WL+BAO+H(z)+Union26W7+SPT+WiggleZ+H(z)+BAO+SNLS7W9+SPT+WiggleZ+H(z)+BAO+SNLS8W7+SPT+BAO+H09W7+SPT+BAO+H0+Union210 Neff+fi+wW7+ACT+SPT+BAO+H011 Neff+1k+fiW7+ACT+SPT+BAO+H012W7+BAO+H013 Neff+1kW7+SPT+WiggleZ+H(z)+BAO+SNLS14W7+CMB+LRG+H015W7+CMB+BAO+H016W7+ACT+SPT+LRG+H017W7+SPTSZ+BAO+H018W7+SDSS+H019W7+SDSS+H0+Union220W7+SDSS+H0+Union2+4He+D/H21W7+H0+WL+BAO+H(z)+Union222W7+SPT+BAO+H023W7+SNLS+BAO+BOSS24W7+SPT+BAO+H025 Neff+fi4He26D/H27D/H+4He28W7+D/H29W7+SPT(agnostic)30W7+SPT31W7+ACT+SPT+BAO+H032W7+ACT+SPT+LRG+H033W7+SPT+BAO+H034W7+SPT35W7+ACT+BAO+H036W7+ACT37W7+LRG+H038W7+BAO+H039W5+LRG+maxBGC+H040W5+CMB+BAO+fgas+H041W5+LRG+H042W5+BAO+SN+H043W7+H0+SDSS+SN+CHFTLS44W7+SPT+H(z)+H045W7+H0+WL+BAO+H(z)+Union246W7+ACBAR+BAO+H0+ACT47W7+ACBAR+ACT+SPT+SDSS+H048W7+ACBAR+ACT+SPT+SDSS+MSH049W7+SPT+BAO+H050W7+SPT51W7+H052W7+SPT+BAO+H053W7+SPT54W7+SPT+BAO+H055W9+ACT+SPT+BAO+H056 Neff

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Generating more informative statistics

• Fundamental question: is Universe ΛCDM or ΛCDM+Neff?

• Parameter constraints insufficient, need posterior ratio

• Can therefore perform model selection!

Pr(⇤CDM+Ne↵ |d)Pr(⇤CDM|d) =

Pr(d|⇤CDM+Ne↵)

Pr(d|⇤CDM)=

Pr(Ne↵ |d,⇤CDM+Ne↵)

Pr(Ne↵ |⇤CDM+Ne↵)

����Neff=3.046

using Savage-Dickey Density Ratio(Dickey [1971]): models nested

Bayes’ Theorem, modelsequally likely a priori

easily obtained by binning chains!

Planck evidence ratios

• No evidence for additional neutrinos!

– odds ~6:1 in favour of ΛCDM

What if we don’t trust our priors?

• Check: are hints present in likelihood?

• Use profile likelihood ratio

– ratio of conditional to unconditional maximum likelihoods

– PLR

– prior-“independent”

– not rigorous model selection, but informative

• PLR(Neff ≠3.046) > n2/2 indicates n-sigma “evidence”

(N⇤e↵) =

max[Pr(d|✓⇤CDM, Ne↵ = N⇤e↵)]

max[Pr(d|✓⇤CDM, Ne↵)]

Planck profile likelihood ratios

• Even with discrepant HST data, not even 2 sigma

The end

What could end the Neff debate?

• Planck polarisation

– polarisation peaks sharper

– pin down phase shift: must be neutrinos (ΔNeff ~ 0.18)

• Precise local measurements of H0 and age of the Universe

– see Verde, Jimenez & Feeney (arXiv:1303.5341)

• CMB lensing helps break degeneracies (and measure mass!)

Neutrino mass

Neutrino mass and number of species

Number of species assuming one sterile